SAMPTA’09, International Conference on SAMPling
Theory and Applications
Laurent Fesquet, Bruno Torrésani
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Laurent Fesquet, Bruno Torrésani. SAMPTA’09, International Conference on SAMPling Theory and Applications. Laurent Fesquet and Bruno Torrésani. pp.384, 2010. <hal-00495456>
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SAMPTA'09
SAMPling Theory and Applications
Centre International de Rencontres
Mathématiques
Marseille Luminy
MAY 18-22, 2009
Editors: Laurent Fesquet and Bruno Torrésani
Organized by TIMA, INP Grenoble and LATP, Université de Provence
http://www.latp.univ-mrs.fr/SAMPTA09
SAMPTA'09
2
SAMPTA'09 Participants
SAMPTA'09, the 8th international conference on Sampling Theory and
Applications, was organized in Marseille-Luminy, on May 18-22, 2009. The
previous conferences were held in Riga (Latvia) in 1995, Aveiro (Portugal) in 1997,
Loen (Norway) in 1999, Orlando (USA) in 2001, Salzburg (Austria) in 2003,
Samsun
(Turkey)
in
2005
and
Thessaloniki
(Greece)
in
2007.
The purpose of SAMPTA's is to bring together mathematicians and engineers
interested in sampling theory and its applications to related fields (such as signal
and image processing, coding theory, control theory, complex analysis, harmonic
analysis, differential equations) to exchange recent advances and to discuss open
problems.
SAMPTA09 gathered around 160 participants from various countries and
scientific areas, The conference benefited from the infrastructure of CIRM, the
Centre International de Rencontres Mathématiques, an institute mainly sponsored
by the french Centre National de la Recherche Scientifique (CNRS) and the French
Mathematical Society (SMF).
SAMPTA'09
3
Organizing committee:
General
chairs
Local
committee
• Laurent Fesquet (TIMA, Grenoble Institute of Technology)
• Bruno Torrésani (LATP, Université de Provence, Marseille)
•
•
•
•
•
Sandrine Anthoine (I3S, CNRS, Sophia-Antipolis)
Karim Kellay (LATP, Université de Provence, Marseille)
Matthieu Kowalski (LATP, Université de Provence, Marseille)
Clothilde Melot (LATP, Université de Provence, Marseille)
El Hassan Youssfi (LATP, Université de Provence, Marseille
Program committee:
Program
chairs
Special
sessions
organizers
SAMPTA'09
• Yonina Eldar (Electrical Engineering, Technion, Israel Institute
of Technology)
• Karlheinz Gröchenig (Fakultät für Mathematik, University of
Vienna)
• Sinan Gunturk (Mathematics Department, Courant Institute,
New York)
• Michael Unser (Biomedical Imaging Group, EPFL Lausanne)
• Bernhard Bodmann (Dept of Mathematics, University of
Houston, USA)
• Pierluigi Dragotti (Dept of Electronic and Alectrical
Engineering, Imperial College London, UK)
• Yonina Eldar (Electrical Engineering, Technion, Israel Institute
of Technology, Israel)
• Laurent Fesquet (TIMA, INP Grenoble, France)
• Massimo Fornasier (RICAM, Linz University, Austria)
• Hakan Johansson (Dept of Electrical Engineering, Linkoping
University, Sweden)
• Gitta Kutyniok (Universität Osnabrück, Germany)
• Pina Marziliano (Nanyang Technological University, Singapore)
• Götz Pfander (International University Bremen, Germany)
• Hölger Rauhut (Hausdorff Center for Mathematics, University
of Bonn, Germany)
• Jared Tanner (School of Mathematics, University of
Edimburgh, Scotland)
• Christian Vogel (ISI, ETH Zurich, Switzerland)
• Ozgur Yilmaz (Dept of Mathematics, University of British
Columbia, Vancouver, Canada)
4
Acknowledgements
The conference was extremely successful, thanks mainly to the
participants, whose scientific contributions were remarkably good. Thanks are
also due to the members of the organizing committee and the program committee,
as well as all the reviewers who participated in the selection of contributions.
We would also like to thank the CIRM staff for the practical organization
(accomodation, conference facilities,...) and their constant availability, and the
Faculté des Sciences de Luminy for lending a conference room for the plenary
sessions.
The secretary staff at LATP was instrumental in all aspects of the
organization, from the scientific part to the social events.
Finally, we would like to thank the sponsors of the conference: CIRM (CNRS
and French Mathematical Society), Université de Provence, the European
Excellence Center for Time-Frequency Analysis (EUCETIFA), the City of Marseille
and the Conseil Général des Bouches du Rhône for their financial support.
SAMPTA'09
5
SAMPTA'09
6
SampTA Technical Program
Monday 18 May 2009
09:10 - 09:30 Opening Session – Amphi 8
09:30 - 10:30 Plenary talk - Amphi 8 – Chair: K. Gröchenig
Gabor frames in Complex Analysis , Yura Lyubarskii
10:30 - 11:00 Coffee break
11:00 - 12:00 Plenary talk - Amphi 8 – Chair: K. Gröchenig
A Prior-Free Approach to Signal and Image Denoising: the SURE-LET Methodology, Thierry
Blu
12:00 - 14:00 Lunch
14:00 - 16:00 Special session – Auditorium
Sparse approximation and high-dimensional geometry - Chair: J. Tanner
#192. Dense Error Correction via L1-Minimization, John Wright, Yi Ma
#204. Recovery of Clustered Sparse Signals from Compressive Measurements, Volkan Cevher, Piotr
Indyk, Chinmay Hegde, Richard G. Baraniuk
#206. Sparse Recovery via lq-minimization for 0 < q ≤ 1, Simon Foucart
#210. The Balancedness Properties of Linear Subpaces and Signal Recovery
Robustness in Compressive Sensing, Weiyu Xu
#207. Phase Transitions Phenomena in Compressed Sensing, Jared Tanner
#167 Optimal Non-Linear Models, Akram Aldroubi, Carlos Cabrelli,Ursula Molter
14:00 - 16:00 General session – room 1
General sampling - Chair: Y. Lyubarskii
#78. Linear Signal Reconstruction from Jittered Sampling, Alessandro Nordio, Carla-Fabiana
Chiasserini, Emanuele Viterbo
#81. Zero-two derivative sampling, Gerhard Schmeisser
#115. On average sampling restoration of Piranashvili-type harmonizable processes, Andriy Ya.
Olenko, Tibor K. Pogany
#86. Uniform Sampling and Reconstruction of Trivariate Functions, Alireza Entezari
#184. On Subordination Principles for Generalized Shannon Sampling Series, Andi Kivinukk and Gert
Tamberg
#104. The Class of Bandlimited Functions with Unstable Reconstruction under Thresholding, Holger
Boche, Ullrich J. Mönich
16:00 - 16:30 Coffee break
16:30 – 18:30 Special Session – Auditorium
Compressed sensing - Chair: Y. Eldar
#150. Sampling Shift-Invariant Signals with Finite Rate of Innovation, Kfir Gedalyahu, Yonina C.
Eldar
#105. Compressed sensing signal models - to infinity and beyond?, Thomas Blumensath, Mike Davies
#110. Compressed sampling Via Huffman Codes, Akram Aldroubi, Haichao Wang, Kourosh
Zaringhalam
#126. On Lp minimisation, instance optimality, and restricted isometry constants for sparse
approximation, Michael Davies, Rémi Gribonval
#73. Signal recovery from incomplete and inaccurate measurements via ROMP, Deanna Needell,
Roman Vershynin
#169 Sparse approximation and the MAP, Akram Aldroubi, Romain Tessera
16:30 – 18:30 Special Session – room 1
Frame theory and oversampling - Chair: B. Bodmann
#118. Invariance of Shift Invariance Spaces, Akram Aldroubi, Carlos Cabrelli, Christopher Heil, Keri
Kornelson, Ursula Molter
#188. Gabor frames with reduced redundancy, Ole Christensen, Hong Oh Kim, Rae Young Kim
#141. Gradient descent of the frame potential, Peter G. Casazza, Matthew Fickus
#201. Error Correction for Erasures of Quantized Frame Coefficients, Bernhard G. Bodmann, Peter G.
Casazza ,Gitta Kutyniok, Steven Senger
#199. Linear independence and coherence of Gabor systems in finite dimensional spaces, Götz E.
Pfander
SAMPTA'09
7
Tuesday 19 May 2009
09:10 - 10:30 Special Session – Auditorium
Efficient design and implementation of sampling rate conversion, resampling and signal
reconstruction methods - Chair: H. Johansson and C. Vogel
#194. Efficient design and implementation of sampling rate conversion, resampling, and signal
reconstruction methods, Håkan Johansson, Christian Vogel
#171. Structures for Interpolation, Decimation, and Nonuniform Sampling Based on Newton's
Interpolation Formula, Vesa Lehtinen, Markku Renfors
#79. Chromatic Derivatives, Chromatic Expansions and Associated Function Spaces, Aleksandar
Ignjatovic
#84. Estimation of the Length and the Polynomial Order of Polynomial-based Filters, Djordje Babic,
Heinz G. Göckler
09:10 - 10:30 General Session – room 1
Time frequency and frames - Chair: J.P. Antoine
#121. An Efficient Algorithm for the Discrete Gabor Transform using full length Windows, Peter L.
Søndergaard
#82. Matrix Representation of Bounded Linear Operators By Bessel Sequences, Frames and Riesz
Sequence, Peter Balazs
#124. Nonstationary Gabor Frames, Florent Jaillet, Peter Balazs, Monika Dörfler
#140. A Nonlinear Reconstruction Algorithm from Absolute Value of Frame Coefficients for Low
Redundancy Frames, Radu Balan
10:30 - 11:00 Coffee break
11:00 - 12:00 Plenary talk – Amphi 8 – Chair: A. Aldroubi
Harmonic and multiscale analysis of and on data sets in high-dimensions, Mauro Maggioni
12:00 - 14:00 Lunch
14:00 - 16:00 Special session – Auditorium
Geometric multiscale analysis I - Chair: G. Kutyniok
#74. Analysis of Singularities and Edge Detection using the Shearlet Transform, Glenn Easley,
Kanghui Guo, Demetrio Labate
#98. Discrete Shearlet Transform: New Multiscale Directional Image Representation, Wang-Q Lim
#125. The Continuous Shearlet Transform in Arbitrary Space Dimensions, Frame Construction, and
Analysis of singularities, Stephan Dahlke, Gabriele Steidl, Gerd Teschke
#193. Computable Fourier Conditions for Alias-Free Sampling and Critical Sampling, Yue M. Lu,
Minh N. Do, Richard S. Laugesen
#149. Compressive-wavefield simulations, Felix J. Herrmann, Yogi Erlangga, Tim T. Y. Lin
#164. Analysis of Singularity Lines by Transforms with Parabolic Scaling, Panuvuth Lakhonchai,
Jouni Sampo, Songkiat Sumetkijakan
14:00 - 16:00 Special session – room 1
Sampling and communication - Chair: G. Pfander
#146. Erasure-proof coding with fusion frames, Bernhard G. Bodmann, Gitta Kutyniok, Ali Pezeshki
#175. Operator Identification and Sampling, Götz Pfander, David Walnut
#116. A Kashin Approach to the Capacity of the Discrete Amplitude Constrained Gaussian Channel,
Brendan Farrell, Peter Jung
#147. Irregular and Multi-channel sampling in Operator Paley-Wiener spaces, Yoon Mi Hong, G.
Pfander
#136. Low-rate Wideband Receiver, Moshe Mishali, Yonina Eldar
#151. Representation of operators by sampling in the time-frequency domain, Monika Dörfler, Bruno
Torrésani
16:00 - 16:30 Coffee break
16:30 - 17:30 Special session – Auditorium
Geometric multiscale analysis II - Chair: G. Kutyniok
#120. Geometric Wavelets for Image Processing: Metric Curvature of Wavelets, Emil Saucan, Chen
Sagiv, Eli Appleboim
#102. Image Approximation by Adaptive Tetrolet Transform, Jens Krommweh
#202. Geometric Separation using a Wavelet-Shearlet Dictionary, David L. Donoho, Gitta Kutyniok
SAMPTA'09
8
16:30 - 17:30 General session – room 1
Sparsity and compressed sensing - Chair: R. Gribonval
#127. Sparse Coding in Mass Spectrometry, Stefan Schiffler, Dirk Lorenz,Theodore Alexandrov
#161.Quasi-Random Sequences for Signal Sampling and Recovery, Miroslaw Pawlak, Ewaryst
Rafajlowicz
17:30 - 18:30 Poster session
#75. Sparse representation with harmonic wavelets, Carlo Cattani
#85. Reconstruction of signals in a shift-invariant space from nonuniform samples, Junxi Zhao
#92. Spline Interpolation in Piecewise Constant Tension, Masaru Kamada, Rentsen Enkhbat#95. The
Effect of Sampling Frequency on a FFT Based Spectral Estimator, Saeed Ayat
#99. Nonlinear Locally Adaptive Wavelet Filter Banks, Gerlind Plonka and Stefanie Tenorth
#111. Continuous Fast Fourier Sampling, Praveen K. Yenduri, Anna C. Gilbert
#134. Double Dirichlet averages and complex B-splines, Peter Massopust
#135. Sampling in cylindrical 2D PET, Yannick Grondin, Laurent Desbat, Michel Desvignes
#148. Significant Reduction of Gibbs' Overshoot with Generalized Sampling Method, Yufang Hao,
Achim Kempf
#156. Optimized Sampling Patterns for Practical Compressed MRI, Muhammad Usman, Philip G.
Batchelor
#160. A study on sparse signal reconstruction from interlaced samples by l1-norm minimization, Akira
Hirabayashi
#162. Multiresolution analysis on multidimensional dyadic grids, Douglas A. Castro, Sônia M.
Gomes, Anamaria Gomide, Andrielber S. Oliveira, Jorge Stolfi
#165. Adaptive and Ultra-Wideband Sampling via Signal Segmentation and Projection, Stephen D.
Casey, Brian M. Sadler
#174. Non-Uniform Sampling Methods for MRI, Steven Troxler
#187. On approximation properties of sampling operators defined by dilated kernels, Andi Kivinukk,
Gert Tamberg
Wednesday 20 May 2009
09:10 - 10:30 Special Session – Auditorium
Sampling and industrial applications - Chair: L. Fesquet
#182. A coherent sampling-based method for estimating the jitter used as entropy source for True
Random Number Generators, Boyan Valtchanov, Viktor Fischer, Alain Aubert
#91. Orthogonal exponential spline pulses with application to impulse radio, Masaru Kamada, Semih
Özlem, Hiromasa Habuchi
#117. Effective Resolution of an Adaptive Rate ADC, Saeed Mian Qaisar, Laurent Fesquet, Marc
Renaudin
#157. An Event-Based PID Controller With Low Computational Cost, Sylvain Durand, Nicolas
Marchand
09:10 -10:30 General Session – room 1
Wavelets, multiresolution and multirate sampling – Chair: D. Walnut
#189. Asymmetric Multi-channel Sampling in Shift Invariant Spaces, Sinuk Kang,Kil Hyun Kwon
#89. Sparse Data Representation on the Sphere using the Easy Path Wavelet Transform,
Gerlind Plonka, Daniela Rosca
#114. On the incoherence of noiselet and Haar bases, Tomas Tuma, Paul Hurley
#138. Adaptive compressed image sensing based on wavelet modeling and direct sampling, Shay
Deutsch, Amir Averbuch, Shai Dekel10:30 - 11:00
Coffee break
11:00 - 12:00 Plenary talk – Amphi 6 – Chair: G. Teschke
Recent Developments in Iterative Shrinkage/Thresholding Algorithms, Mario Figueiredo
12:00 - 14:00 Lunch
14:00 - 23:00 Social event
Thursday 21 May 2009
09:10 - 10:30 General Session – Auditorium
Adaptive techniques – Chair: N. Marchand
#68. Adaptive transmission for lossless image reconstruction, Elisabeth Lahalle, Gilles Fleury, Rawad
SAMPTA'09
9
Zgheib
#129. A fully non-uniform approach to FIR filtering, Brigitte Bidegaray-Fesquet, Laurent Fesquet
#72. Sampling of bandlimited functions on combinatorial graphs, Isaac Pesenson, Meyer Pesenson
#01. Pseudospectral Fourier reconstruction with the inverse polynomial reconstruction method,
Karlheinz Groechenig, Tomasz Hrycak
09:10 - 10:30 General Session – room 1
General sampling – Chair: A. Jerri
#112. Geometric Sampling of Images, Vector Quantization and Zador's Theorem, Emil Saucan, Eli
Appleboim, Yehoshua Y. Zeevi
#168. On sampling lattices with similarity scaling relationships, Steven Bergner, Dimitri Van De Ville,
Thierry Blu, Torsten Möller
#83. Scattering Theory and Sampling of Bandlimited Hardy Space Functions, Ahmed I. Zayed,
Marianna Shubov
#119. Sampling of Homogeneous Polynomials, Somantika Datta, Stephen D. Howard, Douglas
Cochran
10:30 - 11:00 Coffee break
11:00 - 12:00 Plenary talk – Amphi 6 – Chair: S. Güntürk
A Taste of Compressed Sensing, Ron DeVore
12:00 - 14:00 Lunch
14:00 - 16:00 Special Session – Auditorium
Sampling using finite rate of innovation principles I - Chair: P. Dragotti and P. Marziliano
#113. The Generalized Annihilation Property --- A Tool For Solving Finite Rate of Innovation
Problems, Thierry Blu
#100. Sampling of Sparse Signals in Fractional Fourier Domain, Ayush Bhandari, Pina Marziliano
#153. A method for generalized sampling and reconstruction of finite-rate-of-innovation signals,
Chandra Sekhar Seelamantula, Michael Unser
#80. An ``algebraic'' reconstruction of piecewise-smooth functions from integral measurements,
Dima Batenkov, Niv Sarig, Yosef Yomdin
#108. Estimating Signals With Finite Rate of Innovation From Noisy Samples: A Stochastic
Algorithm, Vincent Y. F. Tan, Vivek K. Goyal
14:00 - 16:00 Special Session – room 1
Mathematical aspects of compressed sensing - Chair: H. Rauhut
#195. Orthogonal Matching Pursuit with random dictionaries, Paweł Bechler, Przemysław
Wojtaszczyk
#178. A short note on nonconvex compressed sensing, Rayan Saab, Ozgur Yilmaz
#190. Domain decomposition methods for compressed sensing, Massimo Fornasier, Andreas
Langer, Carola-Bibiane Schönlieb
#197. Free discontinuity problems meet iterative thresholding, Rachel Ward, Massimo Fornasier
#198. Concise Models for Multi-Signal Compressive Sensing, Mike Wakin
#196. Average case analysis of multichannel Basis Pursuit, Yonina Eldar, Holger Rauhut
16:00 - 16:30 Coffee break
16:30 - 17:30 Special session – Auditorium
Sampling using finite rate of innovation principles II - Chair: P. Dragotti and P. Marziliano
#96. Distributed Sensing of Signals Under a Sparse Filtering Model, Ali Hormati, Olivier Roy, Yue M.
Lu, Martin Vetterli
#154. Multichannel Sampling of Translated, Rotated and Scaled Bilevel Polygons Using Exponential
Splines, Hojjat Akhondi Asl, Pier Luigi Dragotti
16:30 - 17:30 General session – room 1
Signal Analysis and compressed sensing – Chair: A. Cohen
#176. General Perturbations of Sparse Signals in Compressed Sensing, Matthew Herman, Thomas
Strohmer
#203. Limits of Deterministic Compressed Sensing Considering Arbitrary Orthonormal Basis for
Sparsity, Arash Amini, Farokh Marvasti
#185. Analysis of High-Dimensional Signal Data by Manifold Learning and Convolutions, Mijail
Guillemard, Armin Iske
17:30 - 18:30 Poster session
SAMPTA'09
10
See Tuesday poster session.
Friday 22 May 2009
09:10 - 10:30 General Session – Auditorium
Kernels and unusual Paley-Wiener spaces – Chair: G. Schmeisser
#131. Geometric Reproducing Kernels for Signal Reconstruction, Eli Appleboim, Emil Saucan,
Yehoshua Y. Zeevi
#137. Concrete and discrete operator reproducing formulae for abstract Paley-Wiener space, John R.
Higgins
#132. Multivariate Complex B-Splines, Dirichlet Averages and Difference Operators, Brigitte Forster,
Peter Massopust
#143. Explicit localization estimates for spline-type spaces, José Luis Romero
09:10 - 10:30 General Session – room 1
Reconstruction, time and frequency analysis - Chair: R. Balan
#70. Daubechies Localization Operator in Bargmann-Fock Space and Generating Function of
Eigenvalues of Localization Operator, Kunio Yoshino
#97 Optimal Characteristic of Optical Filter for White Light Interferometry based on Sampling
Theory, Hidemitsu Ogawa and Akira Hirabayashi
#90. Signal-dependent sampling and reconstruction method of signals with time-varying bandwidth,
Modris Greitans , Rolands Shavelis
#177. A Fast Fourier Transform with Rectangular Output on the BCC and FCC Lattices, Usman Raza
Alim, Torsten Moeller
10:30 - 11:00 Coffee break
11:00 - 12:00 Plenary talk – Amphi 6
Compressed Sensing in Astronomy, Jean-Luc Starck
12:00 - 14:00 Lunch
14:00 - 16:00 Special Session – Auditorium
Sampling and quantization - Chair: O. Yilmaz
#180. Finite Range Scalar Quantization for Compressive Sensing, Jason N. Laska, Petros Boufounos,
Richard G. Baraniuk
#107. Quantization for Compressed Sensing Reconstruction, John Z. Sun, Vivek K Goyal
#106. Determination of Idle Tones in Sigma-Delta Modulation by Ergodic Theory, Nguyen T. Thao
#172. Noncanonical reconstruction for quantized frame coefficients, Alexander M. Powell
#166. Stability Analysis of Sigma-Delta Quantization Schemes with Linear Quantizers, Percy Deift,
Sinan Güntürk, Felix Krahmer
14:00 - 16:00 Special Session – room 1
Sampling and inpainting - Chair: M. Fornasier
#191. Image Inpainting Using a Fourth-Order Total Variation Flow, Carola-Bibiane
Schönlieb, Andrea Bertozzi, Martin Burger, Lin He
#139. Reproducing kernels and colorization, Minh Q. Ha, Sung Ha Kang, Triet M. Le
#123. Edge Orientation Using Contour Stencils, Pascal Getreuer
#71. Image Segmentation Through Efficient Boundary Sampling, Alex Chen, Todd Wittman,
Alexander Tartakovsky, Andrea Bertozzi
#103. Report on Digital Image Processing for Art Historians, Bruno Cornelis, Ann Dooms, Ingrid
Daubechies, Peter Schelkens
#158. Smoothing techniques for convex problems. Applications in image processing, Pierre Weiss,
Mikael Carlavan, Laure Blanc-Féraud, Josiane Zerubia
SAMPTA'09
11
SAMPTA'09
12
SAMPTA'09
Special Sessions
SAMPTA'09
13
SAMPTA'09
14
Special session on
Sparse approximation
and
high-dimensional geometry
Chair: Jared TANNER
SAMPTA'09
15
SAMPTA'09
16
Recovery of Clustered Sparse Signals
from Compressive Measurements
Volkan Cevher(1) , Piotr Indyk(1,2) , Chinmay Hegde(1) , and Richard G. Baraniuk(1)
(1) Electrical and Computer Engineering, Rice University, Houston, TX
(2) Computer Science and Artificial Intelligence Lab, MIT, Cambridge, MA
Abstract:
ing algorithms for dimensionality reduction.
We introduce a new signal model, called (K, C)-sparse, to
capture K-sparse signals in N dimensions whose nonzero
coefficients are contained within at most C clusters, with
C < K ≪ N . In contrast to the existing work in
the sparse approximation and compressive sensing literature on block sparsity, no prior knowledge of the locations and sizes of the clusters is assumed. We prove that
O (K + C log(N/C))) random projections are sufficient
for (K, C)-model sparse signal recovery based on subspace enumeration. We also provide a robust polynomialtime recovery algorithm for (K, C)-model sparse signals
with provable estimation guarantees.
Fortunately, it is possible to design CS recovery algorithms that exploit the knowledge of structured sparsity models with provable performance guarantees [3, 5,
6]. In particular, the model-based CS recovery framework
in [3] generalizes to any structured-sparsity model that has
a tractable model-based approximation algorithm. This
framework has been applied productively to two structured signal models: block sparsity and wavelet trees
with robust recovery guarantees from O (K) measurements [3]. To recover signals that have structured sparsity, problem-specific convex relaxation approaches are
also used in the literature with recovery guarantees similar to those in [3]; e.g., for block sparse signals, see [5, 6].
1.
In this paper, we introduce a new structured sparsity
model, called the (K, C)-model, that constrains the Ksparse signal coefficients to be contained within at most
C-clusters. In contrast to the block sparsity model in [5,
6], our proposed model does not assume prior knowledge
of the locations and sizes of the coefficient clusters. We
show that O (K + C log(N/C))) random projections are
sufficient for (K, C)-model signal recovery using a subspace counting argument. We also provide a polynomialtime model-based approximation algorithm based on dynamic programming and a CS recovery algorithm based
on the model-based recovery framework of [3]. In contrast to the clustered sparse recovery algorithm based on
the probabilistic Ising model in [7], the (K, C)-model has
provable performance guarantees.
Introduction
Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for the acquisition of sparse or
compressible signals in an appropriate basis [1, 2]. By
sparse, we mean that only K of the N basis coefficients
are nonzero, where K ≪ N . By compressible, we mean
the basis coefficients, when sorted, decay rapidly enough
to zero so that they can be well-approximated as K-sparse.
Instead of taking periodic samples of a signal, CS measures inner products with random vectors and then recovers the signal via a sparsity-seeking convex optimization
or greedy algorithm. The number of compressive measurements M necessary to recover a sparse signal under
this framework grows as M = O (K log(N/K))
In many applications, including imaging systems and
high-speed analog-to-digital converters, such a saving can
be dramatic; however, the dimensionality reduction from
N to M is still not on par with state-of-the-art transform
coding systems. While many natural and manmade signals
can be described to a first-order as sparse or compressible, their sparse supports often have an underlying domain specific structure [3–6]. Exploiting this structure in
CS recovery has two immediate benefits. First, the number
of compressive measurements required for stable recovery
decreases due to the reduction in the degrees of freedom of
a sparse or compressible signal. Second, true signal information can be better differentiated from recovery artifacts
during signal recovery, which increases recovery robustness. Only by exploiting a priori information on coefficient structure in addition to signal sparsity, can CS hope
to be competitive with the state-of-the-art transform cod-
SAMPTA'09
The paper is organized as follows. Section 2 provides
the necessary theoretical and algorithmic background on
model-based CS. Section 3 introduces the (K, C)-model,
derives its sampling bound for CS recovery, and describes
a dynamic programming solution for optimal (K, C)model approximation. Section 4 discusses the aspect of
compressibility and highlights some connections to the
block sparsity model. Simulation results are given in
Section 5 to demonstrate the effectiveness of the (K, C)model. Section 6 provides our conclusions.
2. Model-based CS Background
N
A K-sparse signal
vector x lives in ΣK ⊂ R , which
N
is a union of K subspaces of dimension K. Other than
its K-sparsity, there are no further constraints on the support or values of its coefficients. A union-of-subspaces
17
signal model (a signal model in the sequel for brevity) endows the K-sparse signal x with additional structure that
allows certain K-dimensional subspaces in ΣK and disallows others [4, 8].
More formally, let x|Ω represent the entries of x corresponding to the set of indices Ω ⊆ {1, . . . , N }, and let
ΩC denote the complement of the set Ω. A signal model
MK is then defined as the union of mK canonical Kdimensional subspaces
MK =
m
K
[
m=1
Xm , Xm := {x : x|Ωm ∈ RK , x|ΩCm = 0}.
Each subspace Xm contains all signals x with supp(x) ∈
Ωm . Thus, the signal model MK is defined by the set of
possible supports {Ω1 , . . . , ΩmK }. Signals from MK are
called K-model sparse. Likewise, we may define McK
to be the set of c-wise differences of signals belonging
to MK . Clearly, MK ⊆ ΣK and M4K ⊆ Σ4K . In
the sequel, we will use an algorithm M(x; K) that returns
the best K-term approximation of the signal x under the
model MK .
If we know that the signal x being acquired is Kmodel sparse, then we can relax the standard restricted
isometry property (RIP) [1] of the CS measurement matrix
Φ and still achieve stable recovery from the compressive
measurements y = Φx. The model-based RIP MK -RIP
requires that
(1 − δMK )kxk22 ≤ kΦxk22 ≤ (1 + δMK )kxk22
(1)
hold for signals x ∈ MK [4, 8], where δMK is the modelbased RIP constant.
Blumensath and Davies [4] have quantified the number of measurements M necessary for a subgaussian CS
matrix to have the MK -RIP with constant δMK and with
probability 1 − e−t to be
12
2
ln(2mK ) + K ln
+ t . (2)
M≥ 2
cδMK
δMK
This bound can be used to recover
the conventional CS
N
result by substituting mK = K
≈ (N e/K)K .
To take practical advantage of signal models in CS,
we can integrate them into a standard CS recovery algorithm based on iterative greedy approximation. The key
modification is surprisingly simple [3]: we merely replace
the best K-term approximation step with the best K-term
model-based approximation M(x; K). For example, in the
CoSaMP algorithm [9], the best LK-term approximation
(with L a small integer) is modified to incorporate a best
LK-term model-based approximation. The resulting algorithm (see [3]) then inherits the following model-based
CS recovery guarantee at each iteration i, when the measurement matrix Φ has the M4K -RIP with δM4K ≤ 0.1:
kx − x
bi k2 ≤ 2−i kxk2 + 20 kx − xMK k2
!
1
+ √ kx − xMK k1 + knk2 ,
K
SAMPTA'09
where xMK = M(x; K) is the best model-based approximation of x within MK .
3. The (K, C)-Model
Motivation: The block sparsity model is used in applications where the significant coefficients of a sparse signal
appear in designated blocks on the ambient signal dimension, e.g., group sparse regression problems, DNA microarrays, MIMO channel equalization, source localization in sensor networks, and magnetoencephalography [3,
5, 6, 10–14]. It has been shown that recovery algorithms
provably improve standard CS recovery by exploiting this
block-sparse structure [3, 5].
The (K, C)-model generalizes the block sparsity
model by allowing the significant coefficients of a sparse
signal to appear in at most C clusters of unknown size and
location (Figure 1(a)). This way, the (K, C)-model further accommodates additional applications in, e.g., neuroscience problems that are involved with decoding of natural images in the primary visual cortex (V1) or understanding the statistical behavior of groups of neurons in
the retina [15]. In this section, we formulate the (K, C)model as a union of subspaces and pose an approximation
algorithm on this union of subspaces.
To define the set of (K, C)-sparse signals, without
loss of generality, we focus on canonically sparse signals
in N + 2 dimensions whose first and last coefficients are
zero. Consider expressing the support of such signals via
run-length coding with a vector β = (β1 , . . . , β2C+1 )
(βj 6= 0), where βodd counts the number of continuous
zero-signal values and βeven counts the number of continuous nonzero-signal values (i.e., clusters).
The (K, C)-sparse signal model M(K,C) is
Definition:
defined as
M(K,C) =
(
x∈R
N+2
2C+1
X
βi = N + 2,
i=1
C
X
β2i = K
i=1
)
.
(3)
Sampling Bound: The number of subspaces m(K,C) in
M(K,C) can be obtained by counting the number of positive solutions to the following integer equations:
β1 + β2 + . . . + β2C+1
=
N + 2,
β2 + β4 + . . . + β2C
=
K,
which can be rewritten as
β1 + β3 + . . . + β2C+1 = N + 2 − K,
β2 + β4 + . . . + β2C = K.
(4)
Note that the number of positive integer solutions to the
following problem:
β1 + β2 + β3 + . . . + βn = N,
−1
is given by N
n−1 . Then, we can count the solutions to the
two of decoupled problems in (4) and multiply the number
of solutions to obtain m(K,C) :
N +1−K
K −1
m(K,C) =
.
(5)
C
C −1
18
4. Additional Remarks
Plugging (5) into (2), we obtain the sampling bound
for M(K,C) :
N
M = O K + C log
.
C
Compressibility: Just as compressible signals are
nearly K-sparse and live close to the union of subspaces ΣK in RN , (K, C)-compressible signals are nearly
(K, C)-model sparse and live close to the restricted union
of subspaces M(K,C) . Here, we rigorously introduce a
(K, C)-compressible signal model in terms of the decay
of their (K, C)-model approximation error.
(6)
Note that the (K, C)-sampling bound (6) becomes the
standard CS bound of M = O K log N
K when C ≈ K.
Model Approximation Algorithm: In this section we
focus on designing an algorithm M(x; K, C) for finding
the best (K, C)-model approximation to a given signal
x. The algorithm uses the principle of dynamic programming [16]. For simplicity, we focus on the problem of
finding the cost of the best (K, C)-clustered signal approximation in ℓ2 . This solution generalizes to the best
(K, C)-clustered signal approximation in ℓp for p ≥ 1.
The actual sparsity pattern can be then recovered using
standard back-tracing techniques; see [16] for the details.
We first define the ℓ2 error incurred by approximating
x ∈ RN by the best approximation in M(K,C) :
σM(K,C) (x) ,
Ms =
min
k⋆ =0...k
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(7)
.
We use the restricted amplification property (RAmP)
and the nested approximation property (NAP) in [3] to
ensure that the (K, C)-model based CoSaMP recovery
possesses the following guarantee for (K, C)-model scompressible signals at each iteration i:
SM
kx − x
bi k2 ≤ 2 kxk2 + 35 knk2 + s (1 + ln⌈N/K⌉) ,
K
(8)
−i
)
cost[i, j ⋆ , k ⋆ , c⋆ ] × cost[j ⋆ + 1, j, k − k ⋆ , c − c⋆ ] .
The correctness of the algorithm follows from the following observation. Let v be the best (k, c)-clustered approximation of xi:j . Unless all entries of xi:j can be included
in the approximation v (in which case j − i + 1 ≥ k and
the entry has been already computed during initialization),
then there must exist an index l ∈ [i, . . . , j] such that xl is
not included in v. Let l⋆ = l if l < j, and l⋆ = j − 1 otherwise. Let k ⋆ be the number of non-zero entries present
in the left segment of v i:l⋆ , and let c⋆ be the number of
clusters present in that left segment. Then, it must be the
case that v i:l⋆ is the best (k ⋆ , c⋆ )-approximation to xi:l ,
and v l+1:j is the best (k − k ⋆ , c − c⋆ )-approximation to
x(l⋆ +1):j . Otherwise, those better approximations could
have been concatenated together to yield an even better
(k, c)-approximation of xi:j . Thus, the recursive formula
will identify the optimal split and compute the optimal approximation cost.
The cost table contains O N 2 KC entries. Each entry can be computed in O (N KC) time.
Thus, the running
time of the algorithm is O N 3 K 2 C 2 .
)
Define SM as the smallest value of S for which this condition holds for x and s.
min
j ⋆ =i...j−1
x ∈ RN : σMj(K,C) (x) ≤ S(jK)−1/s ,
N
1 ≤ K ≤ N, S < ∞, j = 1, . . . ,
K
(Main loop) All other cost entries can then be computed using the following recursion:
min
(
(Initialization) When either c = 0 or k = 0, the signal approximation costs can be computed directly, since
cost[i, j, 0, c] = kxi:j k22 and cost[i, j, k, 0] = kxi:j k22 ,
for all valid indices i, j, k, c. Moreover, for all entries
i, j, k, c such that c > 0 and j − i + 1 ≤ k, we have
cost[i, j, k, c] = 0 since we can include all j − i + 1 coordinates of the vector xi:j in the approximation.
c⋆ =0...c
kx− x̄k2 = kx−M(x; K, C)k2 .
The decay of the (K, C)-model approximation error in (7)
defines the (K, C)-compressibility of a signal. Then, a set
of (K, C)-model s-compressible signals is given by
The algorithm M(x; K, C) computes an array
cost[i, j, k, c], where 1 ≤ i ≤ j ≤ N , 0 ≤ k ≤ K,
and 0 ≤ c ≤ C. At the end of the algorithm, each entry
cost[i, j, k, c] contains the smallest cost of approximating
xi:j , the signal vector restricted to the index set [i, . . . , j],
using at most k non-zero entries that span at most c clusters. M(x; K, C) performs the following operations.
cost[i, j, k, c] =
(
inf
x̄∈M(K,C)
when Φ has the M4(K,C) -RIP with δM4(K,C) ≤ 0.1 and the
(ǫK , r)-RAmP with ǫK ≤ 0.1 and r = s − 1.
Simulation via Block Sparsity: It is possible to recover
(K, C)-sparse signals by using the block sparsity model if
we are willing to pay an added penalty in terms of the
number of measurements. To demonstrate this, we define
uniform blocks of size K/C (e.g., average cluster length)
on the signal space. Then, it is straightforward to see that
the number of active blocks B in the block sparse model
is upper-bounded by
B ≤ 2(C − 1) +
K − 2(C − 1)
≤ 3C.
K/C
(9)
To reach this upper bound, we first construct a (K, C)sparse signal that has (C − 1)-clusters with 2 coefficients
and a single cluster with the remaining sparse coefficients.
We then place the clusters with two coefficients at the
boundary of the block sparse model so that each cluster activate two blocks in the block sparse model to arrive at (9).
Then, the (K, C)-equivalentblock sparse model requires
M = O BK/C + B log N
B samples, where B = 3C.
19
(K,C)−based recovery
CoSaMP
0.8
0.8
0.6
0.4
0.2
0
2
(a) A (10, 2)-model signal
1
Average distortion
Probability of perfect signal recovery
1
0.6
0.4
0.2
(K,C)−based recovery
CoSaMP
2.5
3
3.5
M/K
4
(b) Reconstruction probability
4.5
5
0
2
2.5
3
3.5
M/K
4
4.5
5
(c) Recovery error
Figure 1: Monte Carlo simulation results for (K, C)-model based recovery with K = 10, C = 2..
5.
Experiments
In this section we demonstrate the performance of (K, C)model based recovery. Our test signals are the class of
length-100 clustered-sparse signals with K = 10, C = 2.
We run both the CoSaMP algorithm as well as (K, C)model based CoSaMP algorithm [3] until convergence for
1000 independent trials. In Fig. 1(a), a sample realization
of the signal is displayed. It is evident from Figs. 1(b) and
(c) that enforcing the structured sparsity model in the recovery process significantly improves CS reconstruction
performance. In particular, Fig. 1(b) demonstrates that approximately 85% of the signals are almost perfectly recovered at M = 2.5K, whereas CoSaMP fails to recover
any signals at this level of measurements. Instead, traditional sparsity-based recovery requires M ≥ 4.5K to
attain comparable performance. Similarly, Figure. 1(c)
displays the rapid decrease in average recovery distortion
of our proposed method, as compared to the conventional
approach. The (K, C)-sparse approximation algorithm
codes are available at dsp.rice.edu/software/KC.
6.
Conclusions
In this paper, we have introduced a new sparse signal
model that generalizes the block-sparsity model used in
the CS literature. To exploit the provable model-based CS
recovery framework of [3], we developed a dynamic programming algorithm that computes, for any given signal,
its optimal ℓ2 -approximation within our clustered sparsity model. We then demonstrated that significant performance gains can be made by exploiting the clustered
signal model beyond the simplistic sparse model that are
prevalent the CS literature.
Acknowledgments
The authors would like to thank Marco F. Duarte for useful discussions and Andrew E. Waters for converting the (K, C)-model
MATLAB code into C++. VC, CH and RGB were supported by
the grants NSF CCF-0431150 and CCF-0728867, DARPA/ONR
N66001-08-1-2065, ONR N00014-07-1-0936 and N00014-081-1112, AFOSR FA9550-07-1-0301, ARO MURI W311NF-071-0185, and the Texas Instruments Leadership University Program. PI is supported in part by David and Lucille Packard Fellowship and by MADALGO (Center for Massive Data Algorithmics, funded by the Danish National Research Association) and
by NSF grant CCF-0728645.
SAMPTA'09
References:
[1] E. J. Candès, “Compressive sampling,” in Proc. International Congress of Mathematicians, vol. 3, (Madrid,
Spain), pp. 1433–1452, 2006.
[2] D. L. Donoho, “Compressed sensing,” IEEE Trans. Info.
Theory, vol. 52, pp. 1289–1306, Sept. 2006.
[3] R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hegde,
“Model-based compressive sensing,” 2008. Preprint. Available at http://dsp.rice.edu/cs.
[4] T. Blumensath and M. E. Davies, “Sampling theorems for
signals from the union of finite-dimensional linear subspaces,” IEEE Trans. Info. Theory, Dec. 2008.
[5] Y. Eldar and M. Mishali, “Robust recovery of signals from
a union of subspaces,” 2008. Preprint.
[6] M. Stojnic, F. Parvaresh, and B. Hassibi, “On the reconstruction of block-sparse signals with an optimal number
of measurements,” Mar. 2008. Preprint.
[7] V. Cevher, M. F. Duarte, C. Hegde, and R. G. Baraniuk,
“Sparse signal recovery using Markov Random Fields,” in
Proc. Workshop on Neural Info. Proc. Sys. (NIPS), (Vancouver, Canada), Dec. 2008.
[8] Y. M. Lu and M. N. Do, “Sampling signals from a union of
subspaces,” IEEE Signal Processing Mag., vol. 25, pp. 41–
47, Mar. 2008.
[9] D. Needell and J. Tropp, “CoSaMP: Iterative signal recovery from incomplete and inaccurate samples,” Applied and
Computational Harmonic Analysis, June 2008.
[10] J. Tropp, A. C. Gilbert, and M. J. Strauss, “Algorithms
for simultaneous sparse approximation. Part I: Greedy pursuit,” Signal Processing, vol. 86, pp. 572–588, Apr. 2006.
[11] Y. Kim, J. Kim, and Y. Kim, “Blockwise sparse regression,” Statistica Sinica, vol. 16, no. 2, p. 375, 2006.
[12] L. Meier, S. van de Geer, and P. Buhlmann, “The group
lasso for logistic regression,” Journal of Royal Stat. Society: Series B (Statistical Methodology), vol. 70, no. 1,
pp. 53–71, 2008.
[13] F. Parvaresh, H. Vikalo, S. Misra, and B. Hassibi, “Recovering Sparse Signals Using Sparse Measurement Matrices
in Compressed DNA Microarrays,” IEEE Journal of Selected Topics in Sig. Proc., vol. 2, no. 3, pp. 275–285, 2008.
[14] I. F. Gorodnitsky, J. S. George, and B. D. Rao, “Neuromagnetic source imaging with FOCUSS: a recursive weighted
minimum norm algorithm,” Electroenceph. and Clin. Neurophys., vol. 95, no. 4, pp. 231–251, 1995.
[15] P. J. Garrigues and B. A. Olshausen, “Learning Horizontal
Connections in a Sparse Coding Model of Natural Images,”
in Advances in Neural Info. Proc. Sys. (NIPS), 2008.
[16] T. H. Corman, C. E. Leiserson, R. L. Rivest, and C. Stein,
Introduction to Algorithms. MIT Press and McGraw-Hill,
New York, USA, 2001.
20
Special session on
Compressed Sensing
Chair: Yonina ELDAR
SAMPTA'09
21
SAMPTA'09
22
Compressed sensing signal models - to infinity
and beyond?
Thomas Blumensath and Michael Davies
IDCOM & Joint Research Institute for Signal and Image Processing
Edinburgh University, King’s Buildings, Mayfield Road, Edinburgh, UK
thomas.blumensath@ed.ac.uk, mike.davies@ed.ac.uk
Abstract:
Compressed sensing is an emerging signal acquisition
technique that enables signals to be sampled well below
the Nyquist rate, given a finite dimensional signal with a
sparse representation in some orthonormal basis. In fact,
sparsity in an orthonormal basis is only one possible signal model that allows for sampling strategies below the
Nyquist rate. We discuss some recent results for more
general signal models based on unions of subspaces that
allow us to consider more general structured representations. These include classical sparse signal models and
finite rate of innovation systems as special cases.
We consider the dimensionality conditions for two aspects of the compressed sensing inverse problem: the existence of one-to-one maps to lower dimensional observation spaces and the smoothness of the inverse map.
On the surface Lipschitz smoothness of the inverse map
appears to limit the applicability of compressed sensing to
infinite dimensional signal models. We therefore discuss
conditions where smooth inverse maps are possible even
in infinite dimensions. Finally we conclude by mentioning some recent work [14] which develops the these ideas
further allowing the theory to be extended beyond exact
representations to structured approximations.
1. Introduction
Since Nyquist and Shannon we are used to sampling continuous signals at a rate that is twice the bandwidth of the
signal. However recently, under the umbrella title of compressed sensing, researchers have begun to explore how
and when signals can be recovered using much fewer samples, but relying on known signal structure. Importantly
the papers by Candes, Romberg and Tao [4], [5], [6] and
by Donoho [8] have shown that under certain conditions
on the signal sparsity and the sampling operator (which
are often satisfied by certain random matrices), finite dimensional signals can be stably reconstructed when the
number of observations is of the order of the signal sparsity and only logarithmically dependent on the ambient
space dimension. Furthermore the reconstruction can be
performed using practical polynomial time algorithms.
Here we discuss a generalization of the sparse signal
model that enables us to consider more structured signal
types. We are interested in when the signals can be stably reconstructed (or in some cases approximated). We
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finish the paper by considering the implications of these
results for ∞-dimensional signal models and extending
from structured representations to structured approximation.
2.
Signal models and problem statement
The problem can be formulated as follows. A continuous
or discrete signal f from some separable Hilbert space is
to be sampled. This is done by using M linear measurements {hf, φn i}n , where h·, ·i is the inner product and
where {φn } is a set of vectors from the Hilbert space under consideration. Through the choice of an appropriate
orthonormal basis,
PN ψ we can replace f by the vector x
such that f = i=1 ψi xi . Let Φ ∈ RM ×N be the sensing
matrix with entries hψi , φj i. The observation can then be
written as
y = Φx.
(1)
In compressed sensing it is paramount to consider signals
x that are highly structured and in the original papers, x
was assumed to be an exact k-sparse vector, i.e. a vector
with not more than k non-zero entries (we discuss a relaxation of this in section 6.). This naturally defines the signal model as a union of N -choose-k k-dimensional subspaces, K.
A nice generalization of this model, introduced in [12], is
to consider the signal x to be an element from a union of
arbitrary subspaces A, defined formally as
A=
L
[
j
Sj , Sj = {y = Ωj a, Ωj ∈ RN ×kj , a ∈ Rkj },
(2)
where the Ωj are bases for linear subspaces. This general signal model incorporates many previously considered compressed sensing settings, including:
• The exact k-sparse signal model, K
• Finite Rate of Innovation (FRI) [15] signal models, if
we allow an uncountable number of subspaces (e.g.
filtered streams of Dirac functions)
• signals that are k-sparse in a general, possibly redundant dictionary
• exact k-sparse signals whose non-zero elements form
a tree
23
• multi-dimensional signals that are k-sparse with
common support
Importantly this model allows us to incorporate additional
structure which can in turn be advantageous by for example reducing signal complexity (as in the tree-constrained
sparse model).
The aim of compressed sensing is to select a linear sampling operator, Φ, such that there exists a unique inverse
−1
map Φ|K : Φ(K) 7→ K. Moreover, for stability, we generally desire Φ(K) to be a bi-Lipschitz embedding of K.
In standard compressed sensing this stability is captured
by the restricted isometry property [1].
When considering the union of subspaces model we can
similarly look for a Φ with a unique stable (Lipschitz) in−1
verse map Φ|A
: Φ(A) 7→ A. Below we will discuss
both necessary and sufficient conditions for this.
3. Existence of a unique inverse map
In [12] it was shown that a necessary condition for a
unique inverse map to exist is that M ≥ Mmin :=
maxi6=j ki + kj . If this is not the case we can find a vector
x ∈ Si ⊕ Sj , x 6= 0 such that Φx = 0. The authors further
go on to show that when there are a countable number of
finite dimensional subspaces then the set of such sampling
operators, Φ giving a unique inverse is dense.
In [3] we presented a slight refinement of this result for
the case where the number of subspaces is finite. In this
case almost every sampling operator, Φ, M ≥ Mmin has a
unique inverse on A. Furthermore even when maxi ki <
M < Mmin for almost every Φ the set of points in A
without a unique inverse has zero measure (with respect
to the largest subspace).
All this suggests that we might be able to perform compressed sensing from only slightly more observations than
the dimension of the signal model, i.e. M > dim(A).
Unfortunately we have so far ignored the issue of stability
which we will see presents additional complications.
4. Stability of the inverse map
We now consider when the inverse mapping for the union
of subspaces model is stable. Here we are particularly interested in the Lipschitz property of this inverse map and
we derive conditions for the existence of a bi-Lipschitz
embedding from A into a subset of RM .
The Lipschitz property is an important aspect of the map
which ensures stability of any reconstruction to perturbations of the observation and in effect specifies the robustness of compressed sensing against noise and quantization errors. Furthermore, in the k-sparse model, the
bi-Lipschitz property has also played an important role
in demonstrating the existence of efficient and robust reconstruction algorithms through the k-restricted isometry
property (RIP) [4, 5, 6, 8].
A natural extension of the k-restricted isometry for the
union of subspaces model is [12, 3]:
Definition: (A-restricted isometry) For any matrix Φ
and any subset A ⊂ RN we define the A-restricted isom-
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etry constant δA (Φ) to be the smallest quantity such that
(1 − δA (Φ)) ≤
kΦxk22
≤ (1 + δA (Φ)),
kxk22
(3)
holds for all x ∈ A.
If we define the set Ā = {x = x1 + x2 : x1 , x2 ∈ A}
then δĀ < 1 controls the Lipschitz constants of Φ and
−1
Φ|A (in the standard compressed sensing this is directly
equivalent to δ2m ). Specifically let us define:
kΦ(y1 ) − Φ(y2 )k2
kΦ|−1
A (x1 )
−
≤ KF ky1 − y2 k2
Φ|−1
A (x2 )k2
≤ KI kx1 − x2 k2
(4)
(5)
then a straight forward consequence of the Ā-RIP definition is that:
p
KF ≤ 1 + δĀ
(6)
1
(7)
KI
≤√
1−δĀ
Note, as always with RIP, it is prudent to consider appropriate scaling of Φ to balance the upper and lower inequalities in (3).
The following results, proved in [3], give neccessary and
sufficient conditions for Φ to be an A-restricted isometry.
4.1
Sufficient conditions
Theorem 1 For any t > 0, let
µ
µ ¶
¶
2
12
M≥
ln(2L) + k ln
+t ,
cδA
δA
(8)
then there exist a matrix Φ ∈ RM ×N and a constant c > 0
such
(1 − δA (Φ))kxk22 ≤ kΦxk22 ≤ (1 + δA (Φ))kxk22
(9)
holds for all x from the union of L arbitrary k dimensional
subspaces A. What is more, if Φ is generated by randomly
drawing i.i.d. entries from an appropriately scaled subgaussian distribution then this matrix satisfies equation
(9) with probability at least
1 − e−t .
(10)
The proof follows the same lines as the construction of
random matrices with k-RIP [1].
In contrast to the previous results on the existence of a
unique inverse map this sufficient condition is logarithmic
in the number of subspaces considered.
4.2
Necessary conditions
We next show that the logarithmic dependence on L is in
fact necessary. This can be done by considering the distance between the optimally packed unit norm vectors in
A as a function of the number of observations. To this
end it is useful to define a measure of separation between
vectors in the different subspaces:
24
Definition:
S (∆(A) subspace separation) Let
A =
i Si be the union of subspaces Si and let
A/Si be the union of subspaces with the ith subspace
excluded. The subspace separation of A is defined as
inf
∆(A) = inf sup
i
xi ∈Si
kxi k2 =1
xj ∈A/Si
kxj k2 =1
kxi − xj k2
(11)
Theorem 2 Let A be the union of L subspaces of dimension no more than k. In order for a linear map
Φ : A 7→ RN to exist such that it has a Lipschitz constant
KF and such that its inverse map ΦA −1 : Φ(A) 7→ A has
a Lipschitz constant KI , it is necessary that
ln
ln(L)
³
´.
4KF KI
∆(A)
(12)
Therefore, for a fixed subspace separation, the necessary
number of samples grows logarithmically with the number
of subspaces.
This last fact suggests that extending the compressed sensing framwork to infinite dimensional signals may be problematic. For example, it implies that the log(N ) dependence in the standard k-sparse signal model
p is necessary
(from the easily derived bound ∆(A) ≥ 2/k) and therefore such a framework does not directly map to infinite
dimensional signal models.
5. 2 routes to infinity
Most of the results in compressed sensing assume that the
ambient signal space, N , is finite dimensional. This also
implies in the case of the k-sparse signal model (k < ∞)
that the number of subspaces, L, in the signal model is also
finite. In fact we would ideally like to understand when we
can perform compressed sensing when either or both the
quantities, N and L, are infinite. Specifically when might
a stable unique inverse for Φ|A exist based upon a finite
number of observations.
For example the Finite Rate of Innovation (FRI) sampling
framework introduced by Vetterli et al. [15] provides sampling strategies for signals composed of the weighted sum
of a finite stream of diracs. In this case both N and L are
uncountably infinite while M > 2k is sufficient to reconstruct the signal.
Below we consider two possible routes to infinity and
comment on their stability. Note other routes to infinity also exist, such as when we let k, M and N → ∞
while keeping k/M and M/N finite [9], or in the blind
multi-band signal model [10, 13], where the sampling rate,
M/N , is finite but where M, N → ∞.
5.1 k, L finite and N infinite
We begin with the easy case that the reader might consider
to be a bit of a cheat.
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U :=
L
M
Si
(13)
i=1
We can now state the following necessary condition for
the existence of an A-restricted isometry in terms of ∆(A)
and the observation dimension.
M≥
Consider a signal model A ⊂ H, where H is an infinite
dimensional separable Hilbert space (i.e. N = ∞). Assume that both k and L are finite. In this case the union
of subspace model A automatically lives within a finite
dimensional subspace, U ⊂ H defined as:
Note that dim(U ) ≤ kL < ∞. We can therefore first
project onto the finite dimensional subspace U and then
apply the above theory to guarantee both the existence and
stability of inverse mappings in this setting.
Two signal models that naturally fit into this framework
are: the block-based sparsity model [11], which is related to the multiple measurement vectors problem and has
been used recently in a blind multi-band signal acquisition
scheme [13]; and the tree-based sparsity model where the
usual k-sparse model is constrained to form a rooted subk
tree where L ≤ (2e)
k+1 independent of N [3] and naturally
occurs in multi-resolution modelling. This model has also
been recently extended to include tree-compressible signals [14]: see section 6..
5.2 k finite, L and N infinite
From Theorem 2 the only way in which the number of
subspaces can be infinite (or even un-countable) while permitting a stable inverse mapping, Φ|−1
A , with M finite is
if the subspace separation, ∆(A) = 0. In such a case
the union of subspace model may often form a nonlinear
signal manifold. Note also that when we have an uncountable union of k-dimensional subspaces the dimension of
the signal model may well be greater than k.
As an example let us consider the case of a simple Finite
Rate of Innovation process [15]. Such models can be described as an uncountable union of subspaces and the key
existence results from [12] immediately apply. However
this tells us nothing about stability. For simplicity we will
limit ourselves to a basic form of periodic FRI signal on
T = R/Z which can be written as:
x(t) = G(τ, a)(t) :=
k−1
X
i=0
ai ψ(t − τi )
(14)
where ψ are also periodic on T, τ = {τ1 , . . . , τk } and
a = {a1 , . . . , ak } ∈ Rk .
In [15] the possibility of a periodic Dirac stream is considered, i.e. ψ(t) = δ(t), t ∈ [0, 1]. Here we avoid the Dirac
stream by restricting to the case where ψ(t) ∈ L2 (T) and
directly consider the signal model defined by the parametric mapping:
G : U × Rk 7→ L2 (T)
(15)
where U = {τ ∈ Rk : τi < τj , ∀i < j}. Individual
subspaces can be identified with a given τ . Furthermore
the continuity of the shift operator implies that for any
ψ(t) ∈ L2 (T), the associated union of subspace model,
A has ∆(A) = 0. Equivalently we can only find a finite
SL ′
number of subspaces, Sj′ , whose union, A′ := j Sj′ ⊂
25
A has ∆(A′ ) ≥ ǫ > 0). Theorem 2 can then be used to
lower bound the Lipschitz constants of any embedding in
terms of the number of subspaces, L′ of any such A′ .
We have seen that Theorem 2 does not preclude a stable
embedding for such systems. However there is clearly
more work needed to determine when such models can
have finite dimensional stable embeddings. One possible
avenue of research would be to examine the recently derived sufficient conditions for stable embedding of general
smooth manifolds [7, 2].
6. ...and beyond?
In reality all the union of subspace models we have considered are an idealization. In practise we can expect to, at
most, be able to approximate a signal by one from a union
of subspaces model. In traditional compressed sensing
this is the difference between finding a sparse representaion of an exact k-sparse signal and finding a good sparse
approximation of a compressible signal (i.e. one that is
well approximated by a k-sparse signal).
Recent work at Rice university [14] has shown that for
the special case of restricted k-sparse models (such as the
tree-restricted sparsity) the exact union of subspace model
can be extended to approximate union of subspace models
that are subsets of compressible signal models.
In order to go beyond exact representations further conditions are introduced. Notably:
1. Nested Approximation Property (NAP) - this specifies
sets of models, MK , that are naturally nested.
2. Restricted Amplification Property (RAmP) - this imposes additional regularity on the sensing matrix Φ
when acting on the difference between the MK subspaces and the MK−1 subspaces (in the k-sparse
case it is interesting to note that the RAmP condition
is automatically satisfied by the k-RIP condition).
There are therefore a number of interesting open questions. For example, are such additional conditions typically necessary to go beyond exact subspace representations? Furthermore can these additional tools be applied
successfully to arbitrary union of subspace models (i.e.
ones that are not subsets of the standard k-sparse model)?
7. Acknowledgements
This research was supported by EPSRC grants D000246/1
and D002184/1. MED acknowledges support of his position from the Scottish Funding Council and their support
of the Joint Research Institute with the Heriot-Watt University as a component part of the Edinburgh Research
Partnership.
[2] R Baraniuk and M Wakin. Random projections of
smooth manifolds. Foundations of Computational
Mathematics, 2007.
[3] T. Blumensath and M. E. Davies. Sampling theorems for signals from the union of linear subspaces.
Awaiting Publication, IEEE Transactions on Information Theory, 2008.
[4] E. Candès, J. Romberg, and T. Tao. Robust uncertainty principles: Exact signal reconstruction from
highly incomplete frequency information. IEEE
Transactions on information theory, 52(2):489–509,
Feb 2006.
[5] Emmanuel Candès and Justin Romberg. Quantitative
robust uncertainty principles and optimally sparse
decompositions. Foundations of Comput. Math,
6(2):227 – 254, 2006.
[6] Emmanuel Candès and Terence Tao. Near optimal
signal recovery from random projections: Universal
encoding strategies? IEEE Trans. on Information
Theory, 52(12):5406 – 5425, 2006.
[7] K. L. Clarkson. Tighter bounds for random projections of manifolds. In Proceedings of the twentyfourth annual symposium on Computational geometry, pages 39–48, 2008.
[8] D. Donoho. Compressed sensing. IEEE Trans. on
Information Theory, 52(4):1289–1306, 2006.
[9] D. Donoho and J. Tanner. Counting faces of
randomly-projected polytopes when the projection
radically lowers dimension. Journal of the AMS,
2009.
[10] Y. C. Eldar. Compressed sensing of analog signals. submitted to IEEE Trans. on Signal Processing,
2008.
[11] Y. C. Eldar and M. Mishali. Robust recovery of signals from a union of subspaces. Submitted to IEEE
Trans. Inf Theory, arXiv.org 0807.4581, 2008.
[12] Y. Lu and M. Do. A theory for sampling signals from
a union of subspaces. IEEE transactions on signal
processing, 56(6):2334–2345, 2008.
[13] M. Mishali and Y. C. Eldar. Blind multiband signal reconstruction: Compressed sensing for analog
signals. IEEE Trans. Signal Proc., 57(3):993–1009,
2009.
[14] M.F. Duarte R. G. Baraniuk, V. Cevher and
C. Hegde. Model based compressed sensing. Submitted to IEEE Transactions on Information Theory,
2008.
[15] M. Vetterli, P. Marziliano, and T. Blu. Sampling signals with finite rate of innovation. IEEE Transactions on Signal Processing, 50(6):1417–1428, 2002.
References:
[1] R. Baraniuk, M. Davenport, R. De Vore, and
M. Wakin. A simple proof of the restricted isometry
property for random matrices. Constructive Approximation, 28(3):253–263, 2008.
SAMPTA'09
26
Special session on
Frame Theory
and
Oversampling
Chair: Bernhard BODMANN
SAMPTA'09
27
SAMPTA'09
28
Gradient descent of the frame potential
Peter G. Casazza (1) and Matthew Fickus (2)
(1) Department of Mathematics, University of Missouri, Columbia, MO 65211 USA.
(2) Department of Mathematics & Statistics, Air Force Institute of Technology, WPAFB, OH 45433 USA.
pete@math.missouri.edu, matthew.fickus@afit.edu
Abstract:
Unit norm tight frames provide Parseval-like decompositions of vectors in terms of possibly nonorthogonal collections of unit norm vectors. One way to prove the existence
of unit norm tight frames is to characterize them as the
minimizers of a particular energy functional, dubbed the
frame potential. We consider this minimization problem
from a numerical perspective. In particular, we discuss
how by descending the gradient of the frame potential,
one, under certain conditions, is guaranteed to produce a
sequence of unit norm frames which converge to a unit
norm tight frame at a geometric rate. This makes the gradient descent of the frame potential a viable method for
numerically constructing unit norm tight frames.
1.
Introduction
The analysis operator of some finite sequence of vectors
{fm }M
m=1 in an N -dimensional Hilbert space HN is the
operator F : HN → CM , (F f )(m) := hf, fm i. The
corresponding frame operator is F ∗ F : HN → HN ,
F ∗F f =
M
X
hf, fm ifm .
m=1
Generally speaking, frame theory is the study of how
∗
{fm }M
m=1 may be chosen in order to guarantee that F F
M
is well-conditioned. In particular, {fm }m=1 is a frame for
HN if there exists frame bounds 0 < A ≤ B < ∞ such
that AI ≤ F ∗ F ≤ BI, and is a tight frame if A = B, that
is, if F ∗ F = AI.
Typically, one’s choice of fm ’s is restricted according to
some nonlinear, application-specific constraints. Of particular interest is the case of unit norm tight frames, that is,
tight frames for which kfm k = 1 for all m = 1, . . . , M ;
such frames, known to exist for any M ≥ N , provide
Parseval-like decompositions in terms of vectors of unit
length, even though these vectors are possibly nonorthogonal. Despite an ever-growing list of specific constructions of such frames, little is known about the manifold
structure of the set of all unit norm tight frames.
In the hunt for unit norm tight frames, the frame potential,
specifically defined as:
FP({fm }M
m=1 ) :=
M
X
m,m′ =1
SAMPTA'09
|hfm , fm′ i|2
M
for any sequence {fm }M
m=1 ∈ HN , is a useful tool.
Specifically, the frame potential quantifies the total orthogonality of a system of vectors by measuring the total potential energy stored within that system under a certain force which encourages orthogonality. Regarded as a
functional over
M
M
SM
N = {fm }m=1 ∈ HN : kfm k = 1, m = 1, . . . , M ,
one may show that when M ≥ N , the local minimizers of
the frame potential are precisely the unit norm tight frames
of M elements for HN . In particular, as the frame potential is continuous and SM
N is compact, one may conclude
that such frames indeed exist for any M ≥ N .
In this paper, we consider the minimization of the frame
potential from a numerical perspective. In particular, in
the next section, we compute the gradient of the frame
M
potential, namely a specific direction {gm }M
m=1 ∈ HN
M
in which to push {fm }m=1 so as to achieve the greatest
instantaneous decrease of FP. Then, in an improvement
over typical uses of gradient descent, we compute an exact
step size in which to travel in this direction so as to produce a certain decrease in potential. In the third section,
we estimate the size of this decrease in relation to how far
the frame potential is from its minimum; under sufficient
conditions, this estimate may be used to show that by descending the gradient of the frame potential, one may produce a sequence of unit norm frames which converge to a
unit norm tight frame at a geometric rate.
The frame potential was introduced in [1], with its domain
of optimization being later generalized in [4]. It has been
used to characterize tight filter bank frames [5, 6]. The
frame potential may also be used to prove the existence
of tight fusion frames [3], and the local minimizers of the
fusion frame potential are themselves a subject of interest [7, 9]. Further generalizations of the frame potential
are considered in [2, 8].
2.
The gradient of the frame potential
Our goal is to numerically minimize the frame potential
over SM
N . As our domain of optimization is a product of
spheres as opposed to the entire space HM
N , our approach
departs from the classical theory of gradients. In particuM
M
M
lar, given {fm }M
m=1 ∈ SN and any {gm }m=1 ∈ HN such
that hfm , gm i = 0 for all m = 1, . . . , M , we shall compute the rate of change of the frame potential as each fm
29
is pushed along a great circle with tangent velocity gm .
We then define the gradient of FP to be that particular
{gm }M
m=1 which makes this directional derivative as large
as possible. We begin with the following result, which
gives the first two derivatives of the frame potential of a
single parameter family of frames:
Lemma 1 (Lemma 2 of [3]). For any set of twicedifferentiable parameterized curves {fm (·)}M
m=1 in HN ,
the first two derivatives of ϕ(t) := FP({fm (t)}M
m=1 ) are:
ϕ̇(t) = 4ReTr Ḟ (t)F ∗ (t)F (t)F ∗ (t)) ,
ϕ̈(t) = 4ReTr F̈ (t)F ∗ (t)F (t)F ∗ (t)
+ 4kḞ (t)F ∗ (t)k2HS
∗
+ 2kḞ (t)F (t) + F
∗
To compute the terms in (4), note that
f˙m (t) = −kgm k sin(kgm kt)fm + cos(kgm kt)gm
(5)
for any m such that gm 6= 0. As (5) also immediately
holds when gm = 0, we have f˙m (0) = gm for all m.
Thus, by Lemma 1,
ϕ̇(0) = 4ReTr Ḟ (0)F ∗ (0)F (0)F ∗ (0)
= 4ReTr Ḟ (0)F ∗ F F ∗
= 4Re
(t)Ḟ (t)k2HS ,
M
X
hḞ (0)F ∗ F F ∗ em , em i
M
X
hF ∗ F fm , f˙m (0)i
M
X
hF ∗ F fm , gm i.
m=1
where Ḟ (t) and F̈ (t) are the analysis operators of
M
¨
{f˙m (t)}M
m=1 and {fm (t)}m=1 , respectively.
We now use Lemma 1 along with Taylor’s theorem to
asymptotically estimate the change in frame potential one
M
obtains by perturbing a given {fm }M
m=1 ∈ SN along any
choice of great circles. To be precise, letting:
⊥
M
:= {gm }M
⊕fm
m=1 ∈ HN : hfm , gm i = 0, ∀m ,
= 4Re
m=1
= 4Re
(6)
m=1
Next, as taking the derivative of (5) yields f¨m (t) =
−kgm k2 fm (t) for any m, we have:
we have the following:
M
M
⊥
Theorem 2. For any {fm }M
m=1 ∈ SN , {gm }m=1 ∈ ⊕fm ,
let:
=
fm (t) := cos(kgm kt)fm + (sin(kgm kt)/kgm k) gm
kfm (t) − fm k2 ≤ t2
m=1
M
X
kgm k2 ,
M
X
hF̈ (t)F ∗ (t)F (t)F ∗ (t)em , em i
M
X
hF ∗ (t)F (t)fm (t), f¨m (t)i
M
X
hF ∗ (t)F (t)fm (t), −kgm k2 fm (t)i
m=1
whenever gm 6= 0 and let fm (t) := fm otherwise. Then,
M
{fm (t)}M
m=1 ∈ SN for any t ∈ R, and satsifies:
M
X
Tr(F̈ (t)F ∗ (t)F (t)F ∗ (t))
(1)
=
m=1
=
m=1
m=1
as well as:
=−
FP({fm (t)}M
m=1 )
≤
FP({fm }M
m=1 )
+ 4tRe
M
X
M
X
kgm k2 kF (t)fm (t)k2 .
(7)
m=1
hF ∗ F fm , gm i
In particular, combining (7) with Lemma 1 gives:
m=1
+ 8M t
2
M
X
kgm k2 .
(2)
m=1
ϕ̈(t) = −4
M
X
kgm k2 kF (t)fm (t)k2
m=1
Proof. It is straightforward to show that kfm (t)k = 1 for
all m = 1, . . . , M and all t ∈ R. To show (1), note that
for any m such that gm 6= 0, we have:
2
kfm (t) − fm k2 = cos(kgm kt) − 1 + sin2 (kgm kt)
= 4 sin2 (kgm kt/2)
≤ kgm k2 t2 .
+ 2kḞ ∗ (t)F (t) + F ∗ (t)Ḟ (t)k2HS .
ϕ(t) ≤ ϕ(0) + tϕ̇(0) + 12 t2 max |ϕ̈(s)|.
s∈R
(8)
To bound (8), note that by (5),
(3)
As (3) also immediately holds for any m such that gm = 0,
we may sum (3) over all m to conclude (1). To show (2),
we apply Taylor’s theorem to ϕ(t) = FP({fm (t)}M
m=1 ) at
t = 0:
SAMPTA'09
+ 4kḞ (t)F ∗ (t)k2HS
kF (t)k2HS =
M
X
kfm (t)k2 = M,
M
X
kf˙m (t)k2 =
m=1
kḞ (t)k2HS =
m=1
M
X
kgm k2 ,
m=1
(4)
30
M
over all {gm }M
m=1 ∈ SN and all t ∈ R. We note immediately from (12) that the optimal {gm }M
m=1 and t are not
unique, though we now show that their product is. Indeed,
for any fixed m, letting Pm denote the orthogonal projection of HN onto the orthogonal complement of fm , we
have:
and thus, taking absolute values of (8), we have:
|ϕ̈(t)|
M
X
≤4
kgm k2 kF (t)fm (t)k2 + 4kḞ (t)F ∗ (t)k2HS
m=1
+ 2kḞ ∗ (t)F (t) + F ∗ (t)Ḟ (t)k2HS
M
X
≤4
2
kgm k
kF (t)k22 kfm (t)k2
+ 4kḞ (t)F
∗
RehF ∗ F fm + 2M tgm , 2M tgm i
= RehF ∗ F fm + 2M tgm , 2M tPm gm i
(t)k2HS
m=1
∗
∗
+ 2 kḞ (t)F (t)kHS + kF (t)Ḟ (t)kHS
≤4
M
X
= RehPm F ∗ F fm + 2M tgm , 2M tgm i
2
=
m=1
= 16M
kgm k2 .
(9)
m=1
In light of the Taylor expansion (2), one, in light of
Cauchy’s inequality, might expect the gradient of FP,
M
namely the {gm }M
m=1 ∈ HN which maximizes the linear
term
M
X
hF ∗ F fm , gm i,
Re
gm = Pm F ∗ F fm
= F ∗ F fm − hF ∗ F fm , fm ifm
= F ∗ F fm − kF fm k2 fm ,
M
Theorem 3. For any {fm }M
m=1 ∈ SN , the minimizer of
⊥
the bound in (2) over all t ∈ R and {gm }M
m=1 ∈ ⊕fm is
given by t = −1/(4M ) and
In particular, there exists
M
X
{f˜m }M
m=1
m = 1, . . . , M.
∈ SM
N such that:
kf˜m − fm k2
kgm k2 = hF ∗ F fm , gm i
≤
M
1 X
kF ∗ F fm k2 − kF fm k4 , (10)
16M 2 m=1
and such that:
M
FP({f˜m })M
m=1 ) − FP({fm }m=1 )
≤−
M
1 X
kF ∗ F fm k2 − kF fm k4 . (11)
2M m=1
Proof. We seek to minimize:
4tRe
M
X
hF ∗ F fm , gm i + 8M t2
m=1
M
X
kgm k2
m=1
M
2 X
=
RehF ∗ F fm + 2M tgm , 2M tgm i (12)
M m=1
SAMPTA'09
= F ∗ F fm , F ∗ F fm − kF fm k2 fm
= kF ∗ F fm k2 − kF fm k4 ,
which, when substituted into (1) and (2) yields (10)
and (11), respectively, where f˜m := fm (−1/4M ).
Note that as kF fm k4 = |hF ∗ F fm , fm i|2 ≤ kF ∗ F fm k2
for all m = 1, . . . , M , Theorem 3 provides a direction and
M
step size in which to travel from a given {fm }M
m=1 ∈ SN
so as to produce a concrete decrease in frame potential.
In the next section, we estimate the size of this decrease
in terms of how far the current potential is from its minimum, and in so doing, provide an upper bound on the rate
at which repeated applications of Theorem 3 will asymptotically produce a unit norm tight frame.
3.
m=1
(13)
as claimed. Moreover, in light of (13), we have:
m=1
∗
to be given by gm = F F fm for all m = 1, . . . , M . Indeed, one may show that this would be the correct gradient if the frame potential was being regarded as a functional over the entire space HM
N . However, as we are
M
⊥
optimizing over SM
N , we require that {gm }m=1 ∈ ⊕fm ,
M
and as such, instead take {gm }m=1 to be the projection of
⊥
{F ∗ F fm }M
m=1 onto ⊕fm . In the next result, we formally
verify that such a choice is indeed optimal.
with equality if and only if Pm F ∗ F fm + 4M tgm = 0.
Thus, to minimize (12), and consequently to minimize the
upper bound in (2), we may take t = −1/(4M ) and
Substituting (7) and (9) into (4) yields (2).
gm = F ∗ F fm − kF fm k2 fm ,
kPm F ∗ F fm + 4M tgm k2 − kPm F ∗ F fm k2
≥ − 14 kPm F ∗ F fm k2 ,
kgm k2 kF (t)k2HS + 12kḞ (t)k2HS kF (t)k2HS
M
X
1
4
Gradient descent of the frame potential
We now consider the gradient descent of the frame potential: by repeatedly applying Theorem 3, we hope to
produce a sequence of unit norm frames which are converging to a unit norm tight frame. Here, the main idea
is to estimate the right hand side of (11) as a proportion
of the difference between the current value of the frame
potential and its minimum.
To be clear, in [1], the minimum value of FP over SM
N
is found to be M 2 /N ; we now show how the quantity
2
FP({fm }M
m=1 ) − M /N is a good metric on the tightness
M
of {fm }m=1 . Indeed, letting {λn }N
n=1 be the eigenvalues
of the corresponding frame operator F ∗ F , we have:
N
X
n=1
λn = Tr(F ∗ F ) = Tr(F F ∗ ) =
M
X
kfm k2 = M.
m=1
(14)
31
M
In particular, (14) implies that {fm }M
m=1 ∈ SN is tight if
M
and only if λn = N for all n = 1, . . . , N . Moreover, as
N
∗ 2 X
∗ 2
)
=
kF
F
k
=
Tr
(F
F
)
=
λ2n ,
FP({fm }M
m=1
HS
n=1
another consequence of (14) is that:
FP({fm }M
m=1 )
=
N
X
(λn −
M
N
N
X
(λn −
M 2
N)
+ 2(0) +
FP({f˜m })M
m=1 ) −
and thus:
≤ 1−
M2
N
FP({fm }M
m=1 ) −
=
N
X
(λn −
M 2
N) .
(15)
n=1
That is, the difference between the frame potential and its
minimum is the square of the distance of the eigenvalues
of F ∗ F from their optimal values. Using this fact, one
may show:
M
Theorem 4. For any {fm }M
m=1 ∈ SN ,
kF ∗ F fm k2 − kF fm k4
m=1
M2
N
, (16)
where δ is defined as:
δ := inf max min |hfm , en i|,
m
kF ∗ F fm k2 − kF fm k4
hfm , en ien
2
n=1
−
D
F ∗F
N
X
hfm , en ien , fm
E
2
n=1
=
N
X
λn hfm , en ien
2
−
n=1
=
λ2n |hfm , en i|2 −
N
X
λn −
n=1
=
N
X
hfm , en ihλn en , fm i
2
n=1
N
X
N
X
λn |hfm , en i|2
2
(18)
n=1
n=1
N
X
2
λp |hfm , ep i|2 |hfm , en i|2 ,
(19)
p=1
where the equality of (18) and (19) arises from the fact that
they both represent the variance of the random variable
{λn }N
n=1 with respect to the probability density function
{|hfm , en i|2 }N
n=1 .
SAMPTA'09
, (20)
M2
N
δ2
2M
FP({fm }M
m=1 ) −
M2
N
, (21)
By repeatedly applying Theorem 5, one produces a sequence of unit norm frames whose tightness, measured in
terms of (15), improves at a geometric rate, provided all
δ’s remain above some positive lower bound; finding such
a bound is a subject of current research.
Acknowledgments
Casazza and Fickus were supported by NSF DMS
0704216 and AFOSR F1ATA07337J001, respectively.
The views expressed in this article are those of the authors and do not reflect the official policy or position of
the United States Air Force, Department of Defense, or
the U.S. Government.
References:
For sake of space, we omit the complete proof of Theorem 4; the main idea is to let {en }N
n=1 be an orthonormal
eigenbasis of F ∗ F , and note that for any m = 1, . . . , M ,
N
X
M2
N
(17)
n
where the infimum is taken over all orthonormal bases
{en }N
n=1 of HN .
= F ∗F
FP({fm }M
m=1 ) −
where δ is given in (17).
4.
≥ δ 2 FP({fm }M
m=1 ) −
N +1
16M
and such that:
M2
N ,
n=1
M
X
kf˜m − fm k2 ≤
m=1
n=1
=
M
Theorem 5. For any {fm }M
m=1 ∈ SN , there exists
M
M
{f˜m }m=1 ∈ SN such that:
M
X
M 2
N)
+
The significance of Theorem 4 is that it bounds the decrease in frame potential given in Theorem 3 in terms
M
of (15), that is, how far {fm }M
m=1 ∈ SN is from being
tight. Indeed, using Theorem 4, one may show:
[1] J.J. Benedetto and M. Fickus. Finite normalized tight
frames. Adv. Comput. Math., 18:357–385, 2003.
[2] I. Bengtsson and H. Granström. The frame potential,
on average. Preprint.
[3] P.G. Casazza and M. Fickus. Minimizing fusion
frame potential. To appear in Acta Appl. Math.
[4] P.G. Casazza, M. Fickus, J. Kovačević, M. Leon and
J. Tremain. A physical interpretation of tight frames.
In C. Heil, editor, Harmonic analysis and applications, pp. 51–76, 2006.
[5] M. Fickus, B.D. Johnson, K. Kornelson, and K. Okoudjou. Convolutional frames and the frame potential. Appl. Comput. Harmon. Anal., 19:77–91, 2005.
[6] B.D. Johnson and K. Okoudjou. Frame potential
and finite abelian groups. Contemp. Math., 464:137–
148, 2008.
[7] P. Massey. Optimal reconstruction systems for erasures and for the q-potential. Preprint.
[8] P. Massey and M. Ruiz. Minimization of convex
functionals over frame operators. To appear in
Adv. Comput. Math.
[9] P. Massey, M. Ruiz and D. Stojanoff. The structure
of minimizers of the frame potential of fusion frames.
Submitted.
32
Gabor frames with reduced redundancy
Ole Christensen (1) , Hong Oh Kim (2) and Rae Young Kim (3)
(1) Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark.
(2) Department of Mathematical Sciences, KAIST, Daejeon, Korea.
(3) Department of Mathematics, Yeungnam University, Gyeongsan-si,Korea.
Ole.Christensen@mat.dtu.dk, kimhong@kaist.edu, rykim@ynu.ac.kr
This work was supported by the Korea Science and Engineering Foundation (KOSEF) Grant funded by the Korea
Government(MOST)(R01-2006-000-10424-0) and by the Korea Research Foundation Grant funded by the Korean
Government (MOEHRD) (KRF-2006-331-C00014).
Abstract:
2. The range 2N1−1 < b < N1
Considering previous constructions of pairs of dual Gabor
We first cite a result from [2]. It yields an explicit conframes, we discuss ways to reduce the redundancy. The
struction of dual Gabor frames:
focus is on B-spline type windows.
1.
Introduction
We will consider Gabor systems in L2 (R), i.e., families of
functions {Emb Tn g}m,n∈Z, where
Emb Tn g(x) := e
2πimbx
g(x − na).
If there exists a constant B > 0 such that
X
|hf, Emb Tn gi|2 ≤ B ||f ||2 , ∀f ∈ L2 (R),
m,n∈Z
then {Emb Tn g}m,n∈Z is called a Bessel sequence. If there
exist two constants A, B > 0 such that
X
A ||f ||2 ≤
|hf, Emb Tn gi|2 ≤ B ||f ||2 , ∀f ∈ L2 (R),
m,n∈Z
then {Emb Tn g}m,n∈Z is called a frame.
If
{Emb Tn g}m,n∈Z is a frame with dual frame
{Emb Tn h}m,n∈Z , then
X
f=
hf, Emb Tn hiEmb Tn g, f ∈ L2 (R),
m,n∈Z
where the series expansion converges unconditionally in
L2 (R).
Our starting point is the duality condition for Gabor
frames, originally due to Ron and Shen [4]. We use the
version due to Janssen [3]:
Lemma 1..1 Two Bessel sequences {Emb Tn g}m,n∈Z and
{Emb Tn h}m,n∈Z form dual Gabor frames for L2 (R) if
and only if
X
g(x − n/b + k)h(x + k) = bδn,0 (1..1)
k∈Z
for a.e. x ∈ [0, 1].
The Bessel condition in Lemma 1..1 is always satisfied
for bounded windows with compact support, see [1]. Note
that if g and h have compact support, we only need to
check a finite number of conditions in (1..1). In this paper
we will usually choose b so small that only the condition
for n = 0 has to be verified.
SAMPTA'09
Theorem 2..1 Let N ∈ N. Let g ∈ L2 (R) be a realvalued bounded function with supp g ⊂ [0, N ], for which
X
g(x − n) = 1.
(2..1)
n∈Z
Let b ∈]0, 2N1−1 ]. Consider any scalar sequence
−1
{an }N
n=−N +1 for which
a0 = b and an + a−n = 2b, n = 1, 2, · · · N − 1, (2..2)
and define h ∈ L2 (R) by
h(x) =
N
−1
X
an g(x + n).
(2..3)
n=−N +1
Then g and h generate dual frames {Emb Tn g}m,n∈Z and
{Emb Tn h}m,n∈Z for L2 (R).
The above result can be extended:
Corollary 2..2 Consider any b ≤ 1/N. With g and an as
in Theorem 2..1, the function
!
N
−1
X
h(x) =
an g(x + n) χ[0,N ] (x)
(2..4)
n=−N +1
is a dual frame generator of g.
Proof. Consider the condition (1..1) for n = 0; only the
values of h(x) for x ∈ [0, N ] play a role, so since the
condition holds for the function in (2..3), it also holds for
the function in (2..4).
The cut-off in (2..4) yields a non-smooth function. However, for any b < 1/N, we might modify h slightly and
obtain a smooth dual generator:
In particular, we obtain the following:
Corollary 2..3 Consider any b < 1/N, and take ǫ <
1/b − N. With g as in Theorem 2..1, the function h(x) =
b, x ∈ [0, N ] has an extension to a function of desired
smoothness, supported on [−ǫ, N + ǫ], which is a dual
frame generator of g.
33
Proof. The choice an = b, n = −N + 1, . . . , N − 1,
leads to
0.7
0.6
N
−1
X
0.5
an g(x + n) = b, x ∈ [0, N ].
0.4
n=−N +1
0.3
0.2
Given ǫ < 1/b − N and any functions φ1 : [−ǫ, 0[→ R
and φ2 :]N, N + ǫ] → R, the function
φ1 (x),
x ∈ [−ǫ, 0[,
PN −1
a
g(x
+
n)
=
b,
x
∈ [0, N ],
n=−N +1 n
h(x) =
φ2 ,
x ∈]N, N + ǫ],
0,
x∈
/ [−ǫ, N + ǫ],
0.1
0
-2
2
4
6
x
Figure 1: B3 and the dual generator h3 in (2..5).
0.25
0.20
will satisfy (1..1); in fact, for n 6= 0, the support of the
functions g(· ± n/b) and h are disjoint, and for n = 0
we are (for all relevant values of x) back at the function in
(2..4). The functions φ1 and φ2 can be chosen such that
the function h has the desired smoothness.
The assumptions in Theorem 2..1 are tailored to B-splines,
defined inductively by
0
0.15
0.10
0.05
K
1
0
1
2
3
4
Figure 2: The function h in (3..13)..
B1 := χ[0,1] , BN +1 := BN ∗ B1 .
h3 in (2..5) from [−2, 0] to [−1/2, 0] and from [3, 5] to
[3, 31/2] and obtain the dual
Direct calculations shows that
B2 (x)
=
x
2−x
0
if x ∈ [0, 1],
if x ∈ [1, 2],
otherwise,
and
B3 (x)
=
1 2
2x
2
−x + 3x − 32
1 2
x − 3x + 29
2
0
if x ∈ [0, 1],
if x ∈ [1, 2],
if x ∈ [2, 3],
otherwise.
In general, the functions BN are (N − 2)−times differentiable piecewise polynomials (explicit expressions are
known). Furthermore, supp BN = [0, N ], and the partition of unity condition (2..1) is satisfied.
In case g = BN , the dual generators in Theorem 2..1 are
splines, of the same smoothness
as BN itself. By comP −1
a
pressing the function N
n=−N +1 n g(x + n) from the interval [−N + 1, 0] to [−ǫ, 0] and from [N, 2N − 1] to
[N, N + ǫ] we obtain a dual in (2..3) with the same features:
Example 2..4 For the B-spline B3 (x) and b = 1/5, Theorem 2..1 yields the symmetric dual
1/2 x2 + 2 x + 2,
x ∈ [−2, −1[,
2
−1/2
x
+
1,
x ∈ [−1, 0[,
1 1,
x ∈ [0, 3[,
h3 (x) =
−1/2 x2 + 3 x − 7/2, x ∈ [3, 4[,
5
1/2 x2 − 5x + 25/2,
x ∈ [4, 5[,
0,
x∈
/ [0, 5[.
(2..5)
See Figure 1.
Now, for b = 1/4, we can use Corollary 2..3 for ǫ <
4 − 3 = 1. Taking ǫ = 1/2, we compress the function
SAMPTA'09
h(x) =
1/2 (4x)2 + 2 (4x) + 2,
x ∈ [−1/2, −1/4[,
2
−1/2
(4x)
+
1,
x ∈ [−1/4, 0[,
1,
x
∈ [0, 3[,
1 −1/2 (4(x − 3) + 3)2 + 3 (4(x − 3) + 3) − 7/2,
x ∈ [3, 3 + 1/4[,
4
2
1/2
(4(x
−
3)
+
3)
−
5(4(x
− 3) + 3) + 25/2,
x ∈ [3 + 1/4, 3 + 1/2[,
0,
x∈
/ [−1/2, 3 + 1/2[.
8 x2 + 8 x + 2,
−8 x2 + 1,
1
1,
=
−8 x2 + 48 x − 71,
4
8 x2 − 56 x + 98,
0,
See Figure 2.
x ∈ [−1/2, −1/4[,
x ∈ [−1/4, 0[,
x ∈ [0, 3[,
x ∈ [3, 3 + 1/4[,
x ∈ [3 + 1/4, 3 + 1/2[,
x∈
/ [−1/2, 3 + 1/2[.
3. B2 and 1/2 < b < 1
In the following discussion, we consider dual windows associated with a Gabor frame {Emb Tn B2 }m,n∈Z generated
by the B-spline B2 . The arguments can be extended to
general functions supported on [0, 2]. Take any function
h with values specified only on [0, 2] and such that
X
B2 (x + k)h(x + k) = 1, x ∈ [0, 1].
(3..1)
k∈Z
In fact, due to the support of B2 , only the values for h(x)
for x ∈ [0, 2] play a role for that condition. We know that
34
for any b ≤ 1/2 the function generates – up to a certain
scalar multiple – a dual of g.
Now consider any 1/2 < b < 1; that is, we have 1 <
1/b < 2.
Similarly, considering (3..3) for
x ∈ [0, 1] = [0, 2 − 1/b] ∪ [2 − 1/b, 1]
leads to (3..5) and
Lemma 3..1 Assume that h(x), x ∈ [0, 2] is chosen such
that (3..1) is satisfied. The the following hold:
B2 (x + 1/b − 2)h(x − 2) + B2 (x + 1/b − 1)h(x − 1)
(i) If
X
= 0, x ∈ [2 − 1/b, 1];
B2 (x − 1/b + k)h(x + k) = 0, x ∈ R, (3..2)
k∈Z
and
X
(3..7)
the equation (3..7) only involves h(x) for
x ∈ [−1/b, −1] ∪ [1 − 1/b, 0],
B2 (x + 1/b + k)h(x + k) = 0, x ∈ R, (3..3)
and (3..5) implies that
k∈Z
then
h(x − 1) =
B2 (x − 1/b)h(x) + B2 (x − 1/b + 1)h(x + 1) = 0,
i.e.,
x ∈ [1/b, 2],
(3..4)
B2 (x + 1/b − 1)h(x − 1) + B2 (x + 1/b)h(x) = 0
h(x) =
−B2 (x + 1/b)h(x)
, x ∈ [0, 2 − 1/b],
B2 (x + 1/b − 1)
−B2 (x + 1/b + 1)h(x + 1)
, x ∈ [−1, 1 − 1/b].
B2 (x + 1/b)
For the proof of (ii), the condition
h(x) = 0, x ∈
/ [0, 2] ∪ [−1, 1 − 1/b] ∪ [1 + 1/b, 3],
x ∈ [0, 2 − 1/b].
(3..5)
implies that (3..6) and (3..7) are satisfied. By construction,
(3..2) and (3..3) are satisfied.
These equations determine h(x) for
x ∈ [−1, 1 − 1/b] ∪ [1 + 1/b, 3].
(ii) If h(x) for x ∈ [−1, 1 − 1/b] ∪ [1 + 1/b, 3] is chosen
such that (3..4) and (3..5) are satisfied, and
h(x) = 0, x ∈
/ [0, 2] ∪ [−1, 1 − 1/b] ∪ [1 + 1/b, 3],
Lemma 3..1 shows that if we want that (3..1), (3..2), and
(3..3) hold for some b ∈]1/2, 1], then h in general will take
values outside [0, 2]. However, the proof shows that we
under certain circumstances can find a solution h having
support in [0, 2]. In that case, the support will actually be
a subset of [0, 2]:
then (3..2) and (3..3) hold.
Proof. We consider (3..2) for x ∈ [1, 2], and split into two
cases:
For x ∈ [1, 1/b], (3..2) yields that
0
= B2 (x − 1/b + 1)h(x + 1)
+B2 (x − 1/b + 2)h(x + 2);
(3..6)
the equation only involve h(x) for
x ∈ [2, 1 + 1/b] ∪ [3, 2 + 1/b].
For x ∈ [1/b, 2], (3..2) yields that
0 = B2 (x − 1/b)h(x) + B2 (x − 1/b + 1)h(x + 1);
since h(x) is known, this implies that
h(x + 1) =
−B2 (x − 1/b)h(x)
, x ∈ [1/b, 2],
B2 (x − 1/b + 1)
that is,
h(x) =
−B2 (x − 1/b − 1)h(x − 1)
, x ∈ [1/b + 1, 3].
B2 (x − 1/b)
SAMPTA'09
Corollary 3..2 Let b ∈]1/2, 1]. Assume that supp h ⊆
[0, 2] and that (3..1) and (3..2) holds. Then
h(x) = 0, x ∈ [0, 2 − 1/b] ∪ [1/b, 2].
(3..8)
Proof. According to the proof of Lemma 3..1, we obtain
that h(x) = 0 on [1/b+1, 3] by requiring that h(x) = 0 for
x ∈ [1/b, 2]; and we obtain that h(x) = 0 on [−1, 1 − 1/b]
by requiring that h(x) = 0 for x ∈ [0, 2 − 1/b].
If supp h ⊆ [0, 2], the condition (3..8) implies that h at
most can be nonzero on the interval [2 − 1/b, 1/b] having
length 2/b − 2. In order for (3..1) to hold, this interval
must have length at least 1; thus, we need to consider b
such that 2/b − 2 ≥ 1, i.e., b ≤ 2/3. Note that if b ≤ 2/3,
then 2/b ≥ 3 : that is, because B2 and h are supported on
[0, 2], Janssen’s duality conditions in (1..1) are automatically satisfied for n = ±2, ±3, . . . .
Corollary 3..3 Consider b ∈]1/2, 2/3]. Then there exists
a function h with supp h ⊆ [0, 2] such that (3..1) and (3..2)
hold; and bh(x) is a dual generator of B2 for these values
of b.
35
Proof. For x ∈ [0, 2 − 1/b] ∪ [1/b, 2], let h(x) = 0. For
x ∈ [0, 1], the equation (3..1) means that
1.5
1.25
1.0
xh(x) + (1 − x)h(x + 1) = 1.
0.75
0.5
This implies that
0.25
xh(x)
=
1, x ∈ [1/b − 1, 1],
(1 − x)h(x + 1) =
1, x ∈ [0, 2 − 1/b];
0.0
0.5
0.0
1.0
1.5
2.0
x
that is,
Figure 3: The function h in (3..13)..
h(x) =
1
, x ∈ [1/b − 1, 1],
x
(3..9)
Put
h(x) = 6x − 2, x ∈ [1/3, 1/2].
and
h(x) =
1
, x ∈ [1, 3 − 1/b].
2−x
(3..10)
Finally, for x ∈ [2 − 1/b, 1/b − 1] and x ∈ [3 − 1/b, 1/b],
choose h(x) such that
xh(x) + (1 − x)h(x + 1) = 1.
By construction, bh(x) is a dual generator.
For b = 3/5 we will now explicitly construct a continuous dual generator h of B2 with support in [0, 2]. Putting
Corollary 3..2, (3..9), and (3..10) together, we can state a
result about how a dual window supported on [0, 2] must
look like on parts of [0, 2]:
Lemma 3..4 For b = 3/5, every dual generator of B2
with support in [0, 2] has the form
h(x) =
0
1
x
1
2−x
0
if x ≤ 1/3;
if x ∈ [2/3, 1];
if x ∈ [1, 4/3];
if x ≥ 5/3.
That is, we only have freedom on the definition of h on
]1/3, 2/3[∪]4/3, 5/3[.
Note that on [2/3, 4/3], the function h is symmetric
around x = 1. We will now show that it is possible to
define h on ]1/3, 2/3[∪]4/3, 5/3[ in such a way that h becomes symmetric around x = 1.
First, we note that this form of symmetry means that
h(1 − x) = h(1 + x), x ∈]1/3, 2/3[.
(3..11)
Put together with the duality condition, we thus require
that
xh(x) = 1 − (1 − x)h(1 − x), x ∈]1/3, 2/3[. (3..12)
The condition (3..12) shows that must define h(1/2) =
1. Now, taking any continuous function h defined on
[1/3, 1/2] with the properties that h(1/3) = 0 and
h(1/2) = 1, the condition (3..12) shows how to define h(x) on ]1/2, 2/3[; and, finally, the condition (3..11)
shows how to define h on ]4/3, 5/3[ such that the resulting
function is a symmetric dual generator.
SAMPTA'09
Then, for x ∈ [1/2, 2/3],
1 − (1 − x)h(1 − x)
x
−6x2 + 10x − 3
.
=
x
The condition h(1 + x) = h(1 − x), x ∈]1/3, 2/3[ can
also be expressed as h(x) = h(2 − x), x ∈]4/3, 5/3[.
Thus, for x ∈ [4/3, 3/2] we arrive at
h(x)
=
h(x) = h(2 − x) =
−6x2 + 14x − 7
, x ∈ [4/3, 3/2];
2−x
while, for x ∈ [3/2, 5/3],
h(x) = h(2 − x) = 6(2 − x) − 2 = 10 − 6x.
We have arrived at the following conclusion:
Lemma 3..5 For b = 3/5, the function
0
if x ≤ 1/3;
if x ∈ [1/3, 1/2];
6x −2 2
−6x +10x−3
if
x ∈ [1/2, 2/3];
x
1
if x ∈ [2/3, 1];
h(x) = x 1
if x ∈ [1, 4/3];
2−x
−6x2 +14x−7
if
x ∈ [4/3, 3/2];
2−x
10 − 6x
if x ∈ [3/2, 5/3];
0
if x ≥ 5/3
(3..13)
is a continuous symmetric dual generator of B2 .
References:
[1] Christensen, O.: Frames and bases. An introductory
course. Birkhäuser 2007.
[2] Christensen, O. and Kim, R. Y.: On dual Gabor
frame pairs generated by polynomials. J. Fourier
Anal. Appl., accepted for publication.
[3] Janssen, A.J.E.M.: The duality condition for WeylHeisenberg frames. In ”Gabor analysis: theory
and applications” (eds. H.G. Feichtinger and T.
Strohmer). Birkhäuser, Boston, 1998.
[4] Ron, A. and Shen, Z.: Frames and stable bases for
shift-invariant subspaces of L2 (Rd ). Canad. J. Math.
47 no. 5 (1995), 1051–1094.
36
Linear independence and coherence of Gabor
systems in finite dimensional spaces
Götz E. Pfander (1) ,
(1) Jacobs University, 28759 Bremen, Germany.
g.pfander@jacobs-university.de
Abstract:
This paper reviews recent results on the geometry of Gabor systems in finite dimensions. For example, we discuss
the coherence of Gabor systems, the linear independence
of subsets of Gabor systems, and the condition number
of matrices formed by a small number of vectors from a
Gabor system. We state a result on the recovery of signals that have a sparse representation in certain Gabor systems. The results listed here are obtained by the author in
collaborations with Jim Lawrence, Felix Krahmer, Peter
Rashkov, Jared Tanner, Holger Rauhut, and David Walnut
linear independence
1.
Introduction and Notation
The theory of Gabor systems in the Hilbert space of square
integrable functions on the real line has received significant attention during the last ten to twenty years (see, for
example, [4, 6, 8, 7] and references within). Much of the
research concentrates on showing that certain Gabor systems are frames or Riesz bases for their closed linear span.
The seemingly simpler concept of linear independence of
vectors in a Gabor system was addressed in [10]. There,
it was conjectured that any finite set of time–frequency
shifted copies of a single square integrable function is linear independent. This conjecture still remains to be resolved.
In the last years, in part due to the emergence of the theory of compressed sensing and sparse signal recovery, the
structure of Gabor systems in finite dimensional spaces
has received increased attention. Such finite Gabor systems on finite Abelian groups are described below.
We let G denote a finite Abelian group. Its dual group
b consists of the group homomorphisms ξ : G 7→ S 1 .
G
b ⊆ CG = {f : G −→ C}, the latter being
We have G
the space of complex valued functions on G. The support size of f ∈ CG is kf k0 := |{x : f (x) 6= 0}|.
G
The Fourier
P transform of f ∈b C is normalized to be
b
f (ξ) = x∈G f (x) ξ(x), ξ ∈ G.
Translation operators Tx , x ∈ G, and modulation operb on CG are unitary operators given by
ators Mξ , ξ ∈ G,
(Tx f )(t) = f (t − x) and (Mξ f )(t) = f (t) · ξ(t). Timeb
frequency shift operators π(λ), λ = (x, ξ) ∈ G × G,
are the unitary operator on CG represented by π(λ)f =
b
Tx ◦ Mξ f , λ = (x, ξ) ∈ G × G.
SAMPTA'09
b ⊆ CG is called (full)
The system {π(λ)g : λ ∈ G × G}
Gabor system with window g ∈ CG , it consists of |G|2
vectors in a |G| dimensional space.
The short-time Fourier transform with respect to g is given
by
Vg f (λ) = hf, π(λ)gi =
X
y∈G
f (y)g(y − x)ξ(y),
b
f ∈ CG , λ = (x, ξ) ∈ G × G.
We shall not make a distinction between the linear mapb
ping Vg : CG −→ CG×G and its matrix representation
with respect to the Euclidean basis.
Full Gabor systems in finite dimensions share an important and very useful property: for any g 6= 0, the collection
{π(λ)g}λ∈G×Gb forms a uniform tight finite frame for CG
with frame bound n2 kgk2 , that is,
X
b
λ∈G×G
|hf, π(λ)gi|2 = n2 kgk2 kf k2 .
This is a simple consequence of the representation theory
of the Weyl–Heisenberg group [9, 12].
In this paper we are concerned with properties of subsets of full Gabor systems. In Section 2, we consider
the linear independence of subsets of |G| elements of
{π(λ)g}λ∈G×Gb . Recall that a finite set of vectors in
CG is in general linear position if any subset of at most
|G| of these vectors are linearly independent. While being a classical concept in mathematics, it is also relevant
for communications, namely, for information transmission
through a so-called erasure channel [2]. In fact, a frame
n
F = {xk }m
k=1 in C is called maximally robust to erasures if the removal of any l ≤ m − n vectors from F
leaves a frame.
Moreover, we consider the coherence of Gabor systems
in Section 3. We state probabilistic estimates of the coherence of a full Gabor system with respect to a randomly
generated window. In Section 4, we consider the condition
number of matrices formed by a small subset of a Gabor
system.
The results presented below were obtained over the last
few years in collaboration with Jim Lawrence and David
Walnut [12], Felix Krahmer and Peter Rashkov [11], and
Holger Rauhut and Jared Tanner [14, 13].
37
2.
Gabor systems in general linear position
The following simple observations illustrate the usefulness of Gabor systems which are in general linear position.
Proposition 1 [11, 12] For g ∈ CG \ {0}, the following
are equivalent:
1. {π(λ)g}λ∈G×Gb are in general linear position.
2. For all f ∈ CG \{0} we have kVg f k ≥ |G|2 −|G|+1.
3. For all f ∈ CG , Vg f is completely determined by its
values on any set Λ with |Λ| = n.
4. {π(λ)g}λ∈G×Gb is maximally robust to erasures.
5. The |G| × |G|2 matrix Vg has the property that every
minor of order n is nonzero.
Corollary 2 [12] If {π(λ)g}λ∈G×Gb are in general linear
g k0 = |G|.
position, then kgk0 = |G| and kb
Unfortunately, not each finite Abelian groups G permits
the existence of a vector g ∈ CG satisfying one and therefore all conditions listed in Proposition 1. For example,
for the group G = Z2 × Z2 , no such g exists [11]. The
situation is different for G = Zp . Recall that E is of full
measure if the Lebesgue measure of CG \ E is 0.
Theorem 3 [12] If |G| is prime, that is, G = Zp , p prime,
then there is a dense open set E of full measure in CG such
that for every g ∈ E, the elements of the full Gabor system
{π(λ)g}λ∈G×Gb are in general linear position. That is, for
almost all g we have kVg f k ≥ |G|2 −|G|+1 for all f 6= 0.
Rudimentary numerical experiments encourage us to ask
the following question.
Question 4 [12] For G cyclic, that is, G = Zn , n ∈ N,
exists g ∈ CG so that the conclusions of Proposition 1,
and, therefore, kVg f k ≥ |G|2 − |G| + 1, f ∈ CG , hold
In fact, for |G| prime, Theorem 3 can be strengthened.
Theorem 5 [11] Let G = Zp , p prime. For almost every
g ∈ CG , we have
kVg f k0 ≥ |G|2 − kf k0 + 1
(1)
for all f ∈ CG \ {0}. Moreover, for 1 ≤ k ≤ |G| and
1 ≤ l ≤ |G|2 with k + l ≥ |G|2 + 1 there exists f with
kf k0 = k and kVg f k0 = l.
Proposition 6 [11] If |G| is not prime, then Vg has zero
minors for all g ∈ CG . Hence, there is no g ∈ CG such
that (1) holds for all f ∈ CG .
Numerical experiments for Abelian groups of order less
than or equal to 8, as well as our result for all cyclic groups
of prime order, indicate that the following question might
have an affirmative answer.
SAMPTA'09
Question 7 [11] For every cyclic group G and almost every g ∈ CG , does
hold?
(kf k0 , kVg f k0 ), f ∈ CG \{0}
= ( kf k0 , kfbk0 +|G|2 −|G| ), f ∈ CG \{0}
The following result improves on Theorem 5. It allows for
the construction of Gabor based equal norm tight frames
of p2 elements in Cn , n ≤ p. To our knowledge, the only
previously known equal norm tight frames that are maximally robust to erasures are so-called harmonic frames
(see Conclusions in [2]).
Proposition 8 [11] There exists a unimodular g ∈ CZp , p
prime, that is, a g with |g(x)| = 1 for all x ∈ G satisfying
the conclusions of Theorem 5.
To construct an equal norm tight frame, we choose a
g ∈ (S 1 )p satisfying the conclusions of Proposition 8.
We remove p − n components of the equal norm tight
frame {π(λ)g}λ∈G×Gb . The resulting frame remains an
equal norm tight frame which is maximally robust to erasure. Note that this frame is not a Gabor frame proper.
Reducing the number of vectors in the frame to m ≤ p2
vectors leaves an equal norm frame which is maximally
robust to erasure but which might not be tight. With the
restriction to frames with p2 elements, p prime, we have
shown the existence of Gabor frames which share the usefulness of harmonic frames when it comes to transmission
of information through erasure channels.
Background and more details on frames and erasures can
be found in [2, 15] and the references cited therein.
Note that Theorem 5 has as direct consequence
Theorem 9 [11] Let g ∈ CZp , p prime, satisfy the conclusion of Theorem 5. Then any f ∈ CZp with kf k0 ≤ 12 |Λ|,
cp , is uniquely determined by Λ and rΛ Vg f .
Λ ⊂ Zp ×Z
Here, only the support size of f is known. No additional
information on the support of f is required to determine
f.
In terms of sparse representations,
X we consider the question whether any vector f =
cλ π(λ)g can be deterλ∈Λ
mined by a few entries of f in case that |Λ| is small.
Theorem 10 [11] Let g ∈ CZp , p prime, satisfy the conclusion of Theorem 5. Then any f ∈ CZp with f =
P
c
λ∈Λ cλ π(λ)g, Λ ⊂ Zp ×Zp is uniquely determined by
B and rB f whenever |B| ≥ 2|Λ|.
Note that similar to before, the efficient recovery of f from
2|Λ| samples of f in Theorem 10 does not require knowledge of Λ.
The question asking how to recover f from a small number
of entries of f efficiently will be briefly addressed with
Theorem 14
38
3.
Coherence of Gabor systems
In the following we restrict our attention to cyclic groups
G = Zn , n ∈ N. We consider the so-called Alltop window
hA [15] with entries
3
1
hA (x) = √ e2πix /n ,
n
x = 0, . . . , n−1,
(2)
and the randomly generated window hR with entries
1
hR (x) = √ ǫx ,
n
x = 0, . . . , n−1,
(3)
where the ǫx are independent and uniformly distributed on
the torus {z ∈ C, |z| = 1}.
For khk2 = 1, the coherence of a full Gabor systems is
µ =
max
(ℓ,p)6=(ℓ′ ,p′ )
|hMℓ Tp h, Mℓ′ Tp′ hi|.
(4)
In [16] it is shown that the coherence of {π(λ)hA : λ ∈
b n } ⊆ Cn given in (2) satisfies
Zn × Z
1
µ = √
n
(5)
Theorem 12 [13] Let ε, δ ∈ (0, 1) and |Λ| = S. Suppose
that
δ2 n
(7)
S≤
4e(log(S/ε) + c)
with c = log(e2 /(4(e−1))) ≈ 0.0724. Then kIΛ −
Ψ∗Λ ΨΛ k ≤ δ with probability at least 1 − ε; in other
words the minimal and maximal eigenvalues of Ψ∗Λ ΨΛ satisfy 1 − δ ≤ λmin ≤ λmax ≤ 1 + δ with probability at
least 1 − ε.
Remark 13 [13] Assuming equality in condition (7) and
solving for ε we deduce
2
e2
δ n
∗
S exp −
P kIΛ − ΨΛ ΨΛ k > δ) ≤
4(e−1)
4eS
2
δ n
= CS exp −
4eS
with C ≈ 1.075.
Theorem 12 allows us to guarantee theX
successful use of
efficient algorithms to determine f =
cλ π(λ)g from
λ∈Λ
for n prime. This is close to optimal since as the lower
bound for the coherence of frames with n2 elements in
1
Cn is µ ≥ √n+1
[16].
Unfortunately, the coherence (4) of hA applies only for n
prime. For arbitrary n we now consider the random window hR .
Theorem 11 [14] Let n ∈ N and choose a random window hR with entries
1
hR (x) = √ ǫx ,
n
x = 0, . . . , n−1,
where the ǫx are independent and uniformly distributed on
the torus {z ∈ C, |z| = 1}. Let µ be the coherence of the
associated Gabor dictionary (4), then for α > 0 and n
even,
2
α
P µ ≥ √ ≤ 4n(n−1)e−α /4 ,
n
while for n odd,
n−1 2
n+1 2
α
P µ ≥ √ ≤ 2n(n−1) e− n α /4 + e− n α /4 .
n
(6)
Up to the constant factor α, the coherence in Theorem 11
1
with high
comes close to the lower bound µ ≥ √n+1
probability. (The probability depends on α).
4.
Conditioning of submatrices of Vg
For applications such as sparse signal recovery, not only
linear independence of subsets of Gabor systems is required. It is rather needed, that small subsets of Gabor
systems form well-conditioned matrices.
2
Throughout this section, we let Ψ = Vg ∈ Cn×n with
g = hR being the randomly generated unimodular winb we denote by ΨΛ
dow described in (3). For Λ ⊆ G×G
the matrix consisting only of those columns indexed by
λ ∈ Λ.
SAMPTA'09
a few entries of f in case that |Λ| is small. Here, we will
concentrate on algorithms based on Basis Pursuit. Basis
Pursuit seeks the solution of the convex problem
min kxk1
x
subject to Ψg x = y,
(8)
P
where kxk1 = λ∈Z2 |xλ | is the ℓ1 -norm of x. Efficient
n
convex optimization techniques for Basis Pursuit can be
found in [1, 3, 5].
Theorem 14 [13] Assume x is an arbitrary S-sparse coefficient vector. Choose the random unimodular Gabor
window g = hR defined in (3), that is, with random entries independently and uniformly distributed on the torus
{z ∈ C, |z| = 1}. Assume that
S≤C
n
log(n/ε)
(9)
for some constant C. Then with probability at least 1 − ε
Basis Pursuit (8) recovers x from y = Ψx = Ψg x.
References:
[1] Stephen Boyd and Lieven Vandenberghe. Convex
Optimization. Cambridge Univ. Press, 2004.
[2] Peter G. Casazza and Jelena Kovačević. Equal-norm
tight frames with erasures. Adv. Comput. Math.,
18(2-4):387–430, 2003. Frames.
[3] S.S. Chen, D.L. Donoho, and M.A. Saunders.
Atomic decomposition by Basis Pursuit. SIAM J.
Sci. Comput., 20(1):33–61, 1999.
[4] O. Christensen. An introduction to frames and Riesz
bases. Applied and Numerical Harmonic Analysis.
Birkhäuser Boston Inc., Boston, MA, 2003.
[5] D.L. Donoho and Y. Tsaig. Fast solution of l1-norm
minimization problems when the solution may be
sparse. Preprint, 2006.
39
[6] H.G. Feichtinger and T. Strohmer, editors. Gabor
Analysis and Algorithms: Theory and Applications.
Birkhäuser, Boston, MA, 1998.
[7] H.G. Feichtinger and T. Strohmer, editors. Advances in Gabor Analysis. Applied and Numerical
Harmonic Analysis. Birkhäuser Boston Inc., Boston,
MA, 2003.
[8] K. Gröchenig. Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis.
Birkhäuser, Boston, MA, 2001.
[9] A. Grossmann, J. Morlet, and T. Paul. Transforms associated to square integrable group representations. I. General results. J. Math. Phys.,
26(10):2473–2479, 1985.
[10] C. Heil, J. Ramanathan, and P. Topiwala. Linear
independence of time–frequency translates. Proc.
Amer. Math. Soc., 124(9), September 1996.
[11] F. Krahmer, G.E. Pfander, and P. Rashkov. Uncertainty principles for time–frequency representations
on finite abelian groups. Appl. Comp. Harm. Anal.,
2008. doi:10.1016/j.acha.2007.09.008.
[12] J. Lawrence, G.E. Pfander, and D. Walnut. Linear
independence of Gabor systems in finite dimensional
vector spaces. J. Fourier Anal. Appl., 11(6):715–726,
2005.
[13] G.E. Pfander and H. Rauhut. Sparsity in time–
frequency representations. 2008. Preprint.
[14] G.E. Pfander, H. Rauhut, and J. Tanner. Identification of matrices having a sparse representation. IEEE
Trans. Signal Proc., 2008. to appear.
[15] T. Strohmer and R.W. Heath, Jr. Grassmannian
frames with applications to coding and communication. Appl. Comput. Harmon. Anal., 14(3):257–275,
2003.
[16] Thomas Strohmer and Robert W.jun. Heath. Grassmannian frames with applications to coding and
communication. Appl. Comput. Harmon. Anal.,
14(3):257–275, 2003.
SAMPTA'09
40
Error Correction for Erasures
of Quantized Frame Coefficients
Bernhard G. Bodmann(1) , Peter G. Casazza(2) , Gitta Kutyniok(3) and Steven Senger(2)
(1) Department of Mathematics, University of Houston, Houston, TX 77204, USA.
(2) Department of Mathematics, University of Missouri, Columbia, MO 65211, USA.
(3) Institute of Mathematics, University of Osnabrück, 49069 Osnabrück, Germany.
bgb@math.uh.edu, pete@math.missouri.edu, kutyniok@math.uni-osnabrueck.de,
senger@math.missouri.edu
Abstract:
In this paper we investigate an algorithm for the suppression of errors caused by quantization of frame coefficients
and by erasures in their subsequent transmission. The erasures are assumed to happen independently, modeled by
a Bernoulli experiment. The algorithm for error correction in this study embeds check bits in the quantization
of frame coefficients, causing a possible, but controlled
quantizer overload. If a single-bit quantizer is used in conjunction with codes which satisfy the Gilbert Varshamov
bound, then the contributions from erasures and quantization to the reconstruction error is shown to have bounds
with the same asymptotics in the limit of large numbers of
frame vectors.
1.
Introduction
The versatility of redundant systems, in particular frames,
has been demonstrated by their resilience to erasures and
by their usefulness to suppress quantization errors. In the
context of finite frames, the statistical error estimates by
Goyal, Kovačević, Vetterli and Kelner [7, 6] were to the
authors’ knowledge the first instance of a combined analysis of erasures and quantization.
In recent years, the robustness of finite frames against erasures has been more extensively studied, for instance, in
[5, 14, 9, 2, 10]. These studies typically provide estimates
for the (average and worst case) blind reconstruction error,
meaning all erased (unknown) coefficients are set to zero
and the reconstruction relies on a fixed synthesis operator. It is well-known that if the frame vectors related to the
non-erased coefficients still form a spanning set, then the
frame operator of those can be inverted, leading to perfect
reconstruction. However, the latency caused by the wait
until all coefficients have been transmitted and the computational cost of inverting the frame operator make perfect
reconstruction less practicable.
On the other hand, Benedetto, Powell and Yilmaz [1] investigated an easily implementable, active error correction
for the compensation of quantization errors with so-called
sigma-delta algorithms, which provide highly accurate reconstruction.
Recently, Boufounos, Oppenheim and Goyal [4] introducedSAMPTA'09
an erasure correction scheme with strong similarities to quantization-noise shaping, offering the possibility of a combined treatment of both types of errors.
The idea of pre-compensation and error-forward projection deserves to be explored further, but the algorithm by
Boufounos, Oppenheim and Goyal is computationally still
more costly than a simple application of sigma-delta quantization.
The need for results on low-complexity quantization-anderasure correcting algorithms motivated the present study,
which investigates a rather simple strategy for error compensation, a modified sigma-delta algorithm with embedded check bits. The error correction algorithm we present
allows precise bounds on quantization errors and also on
the effect of erasures from unreliable transmissions of
frame coefficients.
2. PCM quantization and blind reconstruction
We first revisit erasure-averaged error bounds for PCM
quantization of frame coefficients and blind reconstruction
after transmission.
Definition. Let H be a d-dimensional Hilbert space. A
frame F = {f1 , f2 , . . . fN } for H is a spanning set. If all
vectors in the frame
the same norm, we call F equalPhave
N
norm. If x = A1 j=1 hx, fj ifj for all x ∈ H, then we
say that F is A-tight.
Quantizing frame coefficients simply means mapping
them to a finite set of values.
Definition. A function Q on R is called a quantizer with
accuracy ǫ > 0 on the interval [−L, +L] if it has a finite range A and for any x ∈ [−L, +L], Q(x) satisfies
|x − Q(x)| ≤ ǫ. The range A of the quantizer Q is also
called the alphabet. If this alphabet consists of all integer multiples of a fixed step-size δ contained in the interval [−L − δ/2, +L + δ/2] and the quantizer assigns to
x ∈ [−L, +L] the unique value mδ, m ∈ Z, satisfying
(m − 21 )δ < x ≤ (m + 12 )δ then we call Q the uniform
mid-tread quantizer with step-size δ [3]. Alternatively, if
the alphabet is A = (Z + 21 )δ ∩ [−L − δ/2, +L + δ/2]
and if Q assigns to x ∈ [−L, +L] the value (m+ 21 )δ such
that mδ < x ≤ (m + 1)δ, then we speak of the so-called
uniform mid-riser quantizer with step-size δ. In the latter part of this study, we focus on the single-bit mid-riser
quantizer which rounds the input to A = {−δ/2, +δ/2}.
We want to apply this quantizer to frame coefficients.41
Definition. Given a quantizer Q, the PCM quantization
of a vector x in a real Hilbert space H of dimension
dim(H) = d, equipped with an A-tight frame F =
{fj }N
j=1 , is defined by
N
QF (x) =
1 X
Q(hx, fj i)fj .
A j=1
Remark. We recall that the PCM quantization error resulting from a uniform quantizer Q with accuracy ǫ > 0 on
[−L, +L], and a N/d-tight equal-norm frame F applied
to any input vector x ∈ H satisfying kxk ≤ L is in norm
bounded by
N
d X
|
uj hfj , vi|
kvk=1 uj ∈{±1} N
j=1
kQF (x) − xk ≤ max
≤
max
N
X
√
d √
|hfj , vi|2 )1/2 = dǫ .
( N ǫ)(
N
j=1
This is in contrast to erasures, where the bound on the reconstruction error depends on the norm of the input vector.
Definition. Given a probability measure P on the set of
erasures, and the analysis operator V belonging to an Atight frame, we define the erasure-averaged reconstruction
error to be
1
e(V, P) = E[k V ∗ E(ω)V − Ik] .
A
Hereby, E[·] is the expectation with respect to the probability measure P on Ω = {0, 1}N , and E : Ω → RN ×N is
a random diagonal matrix with entries Ej,j = ωj .
Theorem. Let H be a real Hilbert space of dimension d,
equipped with an A-tight equal-norm frame F. If all the
frame coefficients are erased with a probability 0 ≤ p ≤
1, independently of each other, then the erasure-averaged
reconstruction error is bounded by
1
p ≤ E[k V ∗ E(ω)V − Ik] ≤ pd .
A
Proof. The lower bound uses Jensen’s inequality and
the convexity of the norm on the real vector space of
Hermitian operators [12]. The upper bound relies on
the identity for the operator norms kV ∗ (I − E)V k =
k(I − E)V V ∗ (I − E)k and on the bound for entries
in the Grammian, |(V V ∗ )j,k | ≤ kfj kkfk k = 1, derived from the Cauchy-Schwarz inequality, which implies
E[k(I − E(ω))V V ∗ (I − E(ω))k] ≤ N p.
Thus,√for a vector x for which pkxk is bigger than
(δ/2) d, the bound on the worst case error due to erasures dominates that of PCM quantization.
A similar phenomenon happens when the quantization is
obtained with first and higher-order sigma delta quantization. For sufficiently large N , the bound for the worst-case
quantization error, see e.g. [3], is smaller than the worstcase erasure error. This motivates investigating active error correction for erasures.
3.
Sigma-delta quantization with embedded
check bits
Our main
goal is to make the two error bounds for erasures
SAMPTA'09
and quantization comparable. To this end, we use systematic binary error-correcting codes for packets of quantized
coefficients, and replace a portion of the output from the
sigma-delta quantizer by the check bits.
Definition. A binary (n, k)-code is an invertible map
C : Zk2 → Zn2 .
The minimum distance of this code is the minimal number of bits by which any two code words (elements in
the range of C) differ. A systematic (n, k)-code simply
appends check bits, meaning q = (q1 , q2 , . . . qk ) maps
to C(q) = (q1′ , q2′ , . . . qn′ ) such that qj′ = qj for all
j ∈ {1, 2, . . . k}.
The relevance of this definition is that among any block of
n transmitted bits, the minimum distance is the number of
bit erasures that cannot be corrected any more.
The reconstruction strategy we study is given by incorporating check bits in the output of the quantizer, which are
used by the receiver to correct a portion of the erased bits.
The remaining, incorrectible bits are then omitted from reconstruction.
As already mentioned, we will exploit a particular accompanying quantization strategy, which we briefly explain.
Definition. Let Q be the binary mid-riser quantizer with
stepsize δ > 0 and let F = {f1 , f2 , . . . fN } be an N/dtight frame for a d-dimensional real Hilbert space H.
Also, assume that C is a binary (n, k)-code. Given an
input vector x ∈ H, then the C-embedded
sigma-delta
PN
quantization of x is QF ,C (x) = Nd j=1 qj fj , where the
sequence {qj }∞
j=1 associated with the initialization value
u0 = 0 is defined by
(
Q(hx, fm+j i + um+j−1 ),
1 ≤ j ≤ k,
qm+j :=
C((qm+1 , qm+2 , . . . qm+k ))j , else ,
for any m ∈ {0, n, 2n, . . . }, and j ∈ {1, 2, . . . n}, and the
map for updating the internal variable is
um+j := hx, fm+j i − qm+j + um+j−1 .
Our first main theorem is the stability of this modified
sigma-delta algorithm.
Theorem. Let Q be a binary mid-riser quantizer with stepsize δ > 0, let F = {f1 , f2 , . . . fN } be an N/d-tight
equal-norm frame for a d-dimensional real Hilbert space
H, and let C be a systematic binary (n, k)-code, such that
n divides N . If kxk ≤ αδ/2, α < 1, and
k≥
n
(1 + α)
2
then in the course of the C-embedded first-order sigmadelta quantization, the internal variable is bounded by
|uj | ≤
δ
k2
1
((n − k + 1) + (n − k)α) ≤ δ(k −
+ )
2
n
2
for all j ∈ {1, 2, . . . N }.
Proof. We proceed by induction. At the end of the first
block of n bits, if all n − k check bits were chosen incorrectly and the input is taken to be the worst case, then uN
reaches the maximum magnitude stated in the theorem.
42In
the course of quantizing the next block, due to the bound
on the input, each bit allows the quantizer to recover at
least δ/2 − αδ/2. With the inequality k ≥ n2 (1 + α) we
deduce
1 α
1
k( − ) ≥ (n − k)(1 + α)
2
2
2
which means uj is contained in [−δ/2, δ/2] before the
next check bit is encountered.
Similarly as in [1] and [3], we deduce an error estimate
from the bound on the internal variable.
The relevant quantity in this estimate is derived from the
frame geometry, as in [3],
T (F) = k(f1 −f2 )±(f2 −f3 )±· · ·±(fN −1 −fN )±fN k .
We define the maximal error caused by quantization to be
eq(V, δ, α) =
max
kxk≤αδ/2
kQF ,C (x) − xk ,
where V is the analysis operator of the frame F.
Theorem. Under the same assumptions as in the preceding
theorem,
eq(V, δ, α) ≤
d δ
( ((n − k)(1 + α) + 1)T (F) .
N 2
Proof. This is an immediate consequence of the bound on
the internal variable and the proof in [3].
In comparison with the unmodified first order sigma-delta
quantization, we have a bound that is worse by at most a
factor of 2(n − k). However, the advantage of the embedded check bits is the ability to correct erasures in each
block.
Assume the initial probability measure applies an erasure
with a probability of p to each coefficient. Assume that
the code C has minimal distance np + t with t > 0. Let
P′ denote the probability measure governing the erasures
remaining after the error correction has been applied in
each block of length n.
Definition. The combination of quantization, erasures and
error correction gives the reconstruction error
ec(V, δ, α, P′ ) = E[ max
kxk≤αδ/2
k
1 X
ωj qj fj − xk] ,
A j
where ωj = 0 means that the j-th coefficient is erased.
The following lemma helps bound the probability of erasures remaining, if the weight of the code is larger than the
expected number of erasures before correction.
Lemma. (Hoeffding [8]). Let E[ωj ] = 1 − p and assume that the minimum distance of C is bounded below
by n(p + ǫ), ǫ > 0. The probability p′ of an individual coefficient being erased after the error correction is applied
is bounded by
p′ ≤ exp(−2nǫ2 ) .
Now we can combine the two error estimates for quantization and erasures.
Theorem. Let ǫ > 0, assume C has minimal distance
n(p + ǫ). Let P′ be the probability measure governing the
erasures after the error correction has been applied. Under
the additional assumptions of the preceding theorem,
SAMPTA'09
dδ
exp(−2nǫ2 ) .
ec(V, δ, α, P ) ≤ eq(V, C, δ, α) +
2
′
Proof. First we apply Minkowski’s inequality to separate
the error caused by quantization and by erasures. The
expected number of erasures is N p′ , with p′ bounded in
accordance with the preceding lemma. Each erased coefficient has magnitude δ/2, so the norm of the vectors
which are omitted in the reconstruction can at most be
δdp′ /2.
The remaining question is which asymptotics can be
achieved for the minimum distance with a suitable sequence of codes.
To this end, we quote a version of the Gilbert-Varshamov
bound.
Lemma. Let 0 ≤ q ≤ 1/2, then there exist infinitely many
systematic linear (n, k)-codes with minimum distance at
least nq and rate
k
≥ 1 − H2 (q) ,
n
where H2 (q) = −q log2 q − (1 − q) log2 (1 − q) is the
binary entropy.
Proof. The usual form of the Gilbert Varshamov bound for
linear codes [11, Ch. 17] can be re-stated as a bound for
the maximal number of erasures that can be corrected by
certain codes. In this form, it states the existence of linear
codes for which any n−d+1 rows of the generator matrix
have rank k if d ≥ nq, meaning up to d − 1 erasures can
be corrected. Permuting the rows so that the first k have
maximal rank and right-multiplying by the inverse of this
k × k block gives the generator matrix for a systematic
code that can correct the same number of erasures.
We are ready to state the final result.
Theorem. Let 0 ≤ p < q ≤ 1/2, H2 (q) ≤ (1 − α)/2,
0 < α < 1 and denote ǫ = q − p. Consider the sequence of systematic linear codes provided by the GilbertVarshamov bound for minimum distance bounded below
2
by nq and let N ≥ ne2nǫ , then
ec(V, δ, α, P′ ) ≤
dδ
((2 ln N H2 (q)/ǫ2 + 1)T (F)
2N
1
+ 2 ln N ) .
2ǫ
2
Proof. From the assumption, we have e2nǫ ≤ N and thus
n ≤ 2ǫ12 ln N . By the Gilbert-Varshamov bound,
n − k ≤ nH2 (q) ≤
1
ln N H2 (q) .
2ǫ2
Using the Hoeffding inequality on the error due to the remaining erasures gives
2
e−2nǫ ≤
n
1 ln N
≤ 2
.
N
2ǫ N
Thus, the two error terms have the same asymptotic behavior.
We note that this error bound is only worse by a term logarithmic in N compared to the quantization error without
erasures. We also remark that even in the lossy regime,
when the error correction fails with near certainty in any
packet, then we still have p′ ≤ p and thus
43
ec(V, δ, α, P′ ) ≤dδ((
ln N
p
H2 (q)/ǫ2 + 1)T (F) + ) .
N
2
Acknowledgment
This work was partially supported by NSF DMS 0704216, NSF DMS 08-07399 and by the Deutsche
Forschungsgemeinschaft (DFG) under Heisenberg Fellowship KU 1446/8-1.
References:
[1] J. J. Benedetto, A. M. Powell, and O. Yilmaz, SigmaDelta quantization and finite frames, IEEE Trans. Inform. Theory 52:1990–2005, 2006.
[2] B. G. Bodmann and V. I. Paulsen. Frames, graphs
and erasures. Linear Algebra Appl. 404:118–146,
2005.
[3] B. G. Bodmann and V. I. Paulsen. Frame Paths and
Error Bounds for Sigma-Delta Quantization. Appl.
Comput. Harmon. Anal. 22:176–197, 2007.
[4] P. Boufounos, A. V. Oppenheim, and V. K. Goyal.
Causal Compensation for Erasures in Frame Representations. IEEE Trans. Signal Proc. 3:1071–1082,
2008.
[5] P. Casazza and J. Kovačević, Equal-norm tight
frames with erasures. (English summary) Frames.
Adv. Comput. Math. 18:387–430, 2003.
[6] V. K. Goyal, J. Kovačević, and J. A. Kelner. Quantized frame expansions with erasures. Appl. Comp.
Harm. Anal. 10:203–233, 2001.
[7] V. K. Goyal, J. Kovačević, and M. Vetterli. Quantized frame expansions as source-channel codes for
erasure channels. In: Proc. Data Compr. Conf.,
Snowbird, UT, Mar. 1999.
[8] W. Hoeffding, Probability inequalities for sums of
bounded random variables, J. Amer. Stat. Assoc. 58
(301):13–30, 1963.
[9] R. B. Holmes and V. I. Paulsen, Optimal frames for
erasures, Linear Algebra Appl. 377:31–51, 2004.
[10] D. Kalra, Complex equiangular cyclic frames and
erasures, Linear Algebra Appl. 419:373–399, 2006.
[11] F. J. MacWilliams and N. J. A. Sloane, The theory of
error-correcting codes. North-Holland, Amsterdam,
1977
[12] D. Petz, Spectral scale of self-adjoint operators and
trace inequalities, J. Math. Anal. Appl. 109:74–82,
1985.
[13] M. Püschel and J. Kovačević, Real, Tight Frames
with Maximal Robustness to Erasures, Proc. Data
Compr. Conf., Snowbird, UT, March 2005, pp. 63–
72.
[14] T. Strohmer and R. Heath, Grassmannian frames
with applications to coding and communication,
Appl. Comput. Harmon. Anal. 14:257–275, 2003.
SAMPTA'09
44
Special session on
Efficient Design and Implementation
of Sampling Rate Conversion, Resampling
and Signal Reconstruction Methods
Chair: Hakan Johansson and Christian Vogel
SAMPTA'09
45
SAMPTA'09
46
Structures for Interpolation, Decimation, and
Nonuniform Sampling Based on Newton’s
Interpolation Formula
Vesa Lehtinen and Markku Renfors
Department of Communications Engineering, Tampere University of Technology
P.O.Box 553, FI-33101 Tampere, Finland
{vesa.lehtinen,markku.renfors}@tut.fi
where
Abstract:
m–1
The variable fractional-delay (FD) filter structure by Tassart
and Depalle performs Lagrange interpolation in an efficient
way. We point out that this structure directly corresponds to
Newton’s interpolation (backward difference) formula, hence
we prefer to refer to it as the Newton FD filter. This structure
does not function correctly when the fractional delay is made
time-variant, e.g., in sample rate conversion. We present a
simple modification that enables time-variant usage such as
fractional sample rate conversion and nonuniform resampling.
We refer to the new structure as the Newton (interpolator)
structure. Almost all advantages of the Newton FD structure
are preserved. Furthermore, we suggest that by transposing the
Newton interpolator we obtain the transposed Newton structure which can be used in decimation as well as reconstruction
of nonuniformly sampled signals, analogously to the transposed Farrow structure. The presented structures are a competitive alternative for the Farrow structure family when low
complexity and flexibility are required.
1. Introduction
In [1][2][3], Tassart and Depalle as well as Candan derive an
efficient implementation structure for FD filters, depicted in
Fig. 1, from Lagrange’s interpolation formula. It turns out that
the obtained filter structure directly corresponds to Newton’s
(backward difference) interpolation formula [4] (with some
subexpression sharing) which indeed is equivalent with Lagrange interpolation [5]. Newton’s backward difference formula is
f (t + τ) = ∑
∞
m=0
τ(m)∆m f ( t )
---------------------------- ,
m!
(1)
τ(m) = ∏
k=0
(τ + k )
(2)
is the rising factorial, and ∆ is the backward difference operator such that ∆ m f ( t ) = ∆ m – 1 f ( t ) – ∆ m – 1 f ( t – 1 ) and
∆ 0 f ( t ) = f ( t ) , resulting in
∆m f ( t ) = ∑
m
k=0
m
( – 1 ) k f ( t – k ).
k
(3)
Newton’s backward difference formula provides an efficient means to realise piecewise-polynomial interpolation for
DSP. Its complexity is only O(M) (where M is the interpolator
order)–cf. equivalent Lagrange implementations based on the
Farrow structure [6] having O(M2) complexity [3]. The subfilters are multiplier-free and extremely simple. The structure is
modular, as highlighted by the grey shading in Fig. 1, and the
interpolator order can be changed in real time [3].
Unfortunately, the structure presented in Fig. 1 does not
function correctly in sample rate conversion (SRC). Because
the multiplications are performed between the subfilters, making them time-variant will result in incorrect output. This is
because each output sample should only depend on the current
value of the delay parameter D; in Fig. 1, past values of D contribute to the output through the delayed paths through the subfilters. Therefore, the structure in Fig. 1 is only useful in
single-rate, time-invariant or slowly-varying fractional-delay
filtering.
We propose a slightly modified structure that allows arbitrary resampling, including increasing the sample rate by arbitrary, also fractional, factors (fractional interpolation). We
also point out that the structure can be transposed to obtain a
decimator structure that possesses all the advantages of the
Newton interpolation structure.
This work was supported by the Graduate School in Electronics,
Telecommunications and Automation (GETA).
SAMPTA'09
47
1 – z –1
1–z –1
–D+1
-------------2
–D
...
1–z –1
1–z –1
– D+M –1
-----------------------M
–D+2
-------------3
...
Figure 1. The fractional-delay filter structure proposed in [1][3], based on Newton’s interpolation formula.
1–z –1
H&S
1
1–z –1
H&S
–
1⁄2
1⁄3
H&S
–
D ( t )–1
1–z –1
1⁄M
H&S
–
–
D(t )
...
1–z –1
H&S
...
D ( t )–2
D ( t )–M +1
Figure 2. The Newton interpolator structure suitable for sample rate conversion. The hold & sample
(H&S) blocks perform the sampling at the output sample instants.
2. The Newton structure for interpolation
In order to allow fractional SRC and arbitrary resampling, the
Newton structure must work correctly with a time-variant
fractional delay. This is achieved through two simple steps:
(i) We invert the summation order at the output part of the
structure from that presented in [1][3] (this was already done
in [2]). (ii) The time-varying multiplications can now be implemented in the high-rate part between the adders. The improved structure is shown in Fig. 2. We refer to it as the
Newton interpolator structure or the Newton structure for
short. Also the improved structure is modular, permitting
changing the interpolator order in real time. In single-rate FD
filtering, the improved structure is equivalent to [1][2][3].
In Fig. 2, the H&S blocks stand for hold & sample, i.e.,
each output sample obtains the value of the previously arrived
input sample.
In fractional interpolation, i.e., increasing the sample rate
by a fractional factor, we use the common notation illustrated
in Fig. 3. The time interval between the previous input sample
and the next output sample to be generated is expressed using
the fractional interval variable µ which is normalised with respect to the input sample interval so that µ ∈ [0, 1) .
Interpolation of uniformly spaced input samples can be
modelled as convolution [5], leading to the generic model depicted in Fig. 4 [7]. The continuous-time (CT) linear time-invariant (LTI) model filter is piecewise polynomial, with
M + 1 pieces, each with duration equal to the input sample interval T in . Hence the impulse response length is ( M + 1 )T in .
SAMPTA'09
Input samples
Output samples
T in
µ l–1 T in
( k –1 )T in
µ l+1 T in
( k +1 )T in
k T in
( l–1 )T out
lT out
( l+1 )T out
Figure 3. Definition of the fractional interval µ for
interpolation.
x[n]
CT
@ F in
H CT ( f )
1
x CT ( t ) = -------F in
Figure 4.
factors.
y[n]
= y CT ( nT out )
x CT ( t )
DT
y CT ( t )
@ F out
-
∑n x [ n ]δ t – ------F in
n
The generic model for SRC by arbitrary
The composite transfer function of m cascaded subfilters
is
( 1 – z –1 ) m = ∑
m
n=0
m
( – 1 ) n z –n ,
n
(4)
cf. (3). The output of the interpolator is
48
D(t )
D ( t )–1
D ( t )–2
D ( t )–M +1
...
A&D
A&D
1
1–z –1
–
A&D
1⁄2
–
1–z –1
–
A&D
1⁄3
1–z –1
–
A&D
1⁄M
1–z –1
...
Figure 5. The transposed Newton structure for decimation and
reconstruction of signals from nonuniformly spaced samples.
y ( ( k + µ )T in ) = ∑
M
n=0
Input samples
h ( ( n + µ )T in )x [ k – n ]
m ( D0 – µ )m
= ∑
x [ k – n ] ( –1 ) n ∑
( – 1 ) m ----------------------- n
n=0
m=n
m!
M
M
T out
(2.1)
where
n<0∨n>m
( l–1 )T in
(5)
for m ≥ 0 , and
m–1
( x )m = ∏
k=0
(x – k)
(6)
is the falling factorial. The delay of the interpolator is D 0 T in .
The parameter D 0 can be chosen quite freely, but the best amplitude response and linear phase response are obtained with
D 0 = ( M + 1 ) ⁄ 2 [1].
The continuous-time model impulse response of the interpolator is then (cf. the expression of the filter input in Fig. 4)
m ( D0 – µ )m
M
1
( – 1 ) n + m ------------------------.
h ( ( n + µ )T in ) = ------- ∑
n
T in m = n
m!
(7)
The reversed summation order in the high-rate part comes
with a price: the structure is more costly to pipeline than those
in [1][3] because the signal paths cannot share pipeline registers.
3. The transposed Newton structure
There exists a duality1 between decimation and interpolation
that allows transforming a decimator into an interpolator and
vice versa through network transposition [7]. By transposing
the Newton interpolator, we obtain the structure depicted in
Fig. 5. We refer to this as the transposed Newton structure.
The transpose is obtained by inverting the flow direction of all
signals and replacing each block with its dual. For instance,
the H&S block is replaced with the accumulate & dump
1. There exist a number of definitions for duality,
including the adjoint. Here we use the generalised
duality/transpose as defined in [7].
SAMPTA'09
µ l+1 T out
µ l–1 T out
( k –1 )T out
m = 0,
n
Output samples
( k +1 )T out
k T out
lT in
( l+1 )T in
Figure 6. Definition of the fractional interval µ for the
transposed structure (dual of interpolation).
(A&D) block, which sums up all its input samples since the
previous output sample. This is also the most straightforward
way to obtain the transposed Farrow structure from the Farrow
structure2 [9].
The output samples of the transposed Newton structure are
uniformly spaced, but the input samples may arrive at arbitrary
time instants. The generic SRC model (Fig. 4) is valid also for
the transposed Newton structure. The model impulse response
is again piecewise-polynomial, now with the piece duration
equal to the output sample interval. The model impulse response is obtained by replacing T in with T out in (7) and redefining µ according to Fig. 6 (reflecting the duality between
decimation and interpolation). For an input sample arriving at
time instant t , the fractional interval is
t
t
µ ( t ) = --------– --------- ∈ [0, 1).
T out
T out
(8)
For fractional decimation, the fractional interval for the lth input sample is
lT in
lT in
– ---------.
µ l = --------T
T out
out
(9)
The impulse response in the generic model is now
m ( D0 – µ )m
M
1
h ( ( n + µ )T out ) = --------- ∑
( – 1 ) n + m ----------------------- n
m!
T out m = n
(10)
2. The structure in [8] (transposed structure I in [9])
is not the true transpose of the Farrow structure
even though the duality of responses holds.
49
with integer n. Again, D 0 = ( M + 1 ) ⁄ 2 for the best response.
In the frequency response, the model filter has M + 1 zeros at
each (nonzero) integer multiple of the output sample rate,
hence realising antialiasing regardless of the decimation factor.
The transposed Newton structure is able to receive input
samples at arbitrary time instants, which makes it a potential
building block for reconstruction of signals from nonuniformly spaced samples (e.g., in algorithms like [10][11]), as earlier
suggested for the transposed Farrow structure in [12].
The transposed Newton structure shares the advantages
and disadvantages of the Newton interpolator, such as modularity, O ( M ) complexity and the inefficient zero locations.
4. Computational complexity
In interpolation by factor R, the Newton structure will perform ( 1 + R )M additions and ( 1 + R )M multiplications per
input sample on average. In decimation by R, the transposed
Newton structure will perform ( R – 1 ) ( 1 + M ) + 2M additions and ( 1 + R )M multiplications per output sample. The
first term in the addition count comes from the A&D block.
Multiplication by a constant inverse of a small integer requires
only few additions/subtractions.
Unambiguous complexity comparison between the proposed structures and alternatives, mainly the Farrow family,
would require specifying the implementation technology and
the SRC factor. However, the following points can be made:
(i) The basis multipliers are more complex in the Newton
structures (integer part present in the time-variant coefficients)
than in Farrow structures (no integer part). Hence, large SRC
factors are unfavourable to the Newton family. (ii) If the Lagrange response suffices, the ultimate simplicity of the subfilters makes the Newton family superior to the Farrow structure
when the SRC factor is small. (iii) The response of the Newton
structures can be improved only by increasing the order (i.e.,
number of stages). In designs with a low oversampling factor
and/or strict performance requirements, this may lead to a very
high filter order. In such cases, an optimised Farrow design
with a non-Lagrange response will have a lower complexity
and smaller delay.
5. Conclusions
The proposed structures allow efficient piecewise Newton interpolation for SRC and arbitrary resampling as well as its dual
for decimation and reconstruction of nonuniformly sampled
signals. The advantages of the proposed structures include
SAMPTA'09
low, O(M) complexity (high orders are feasible at the cost of a
long delay), very simple subfilters and run-time adjustability
of the filter order. As a drawback, the basis multipliers running
at the high-rate end of the filter have longer wordlengths than
in the Farrow counterparts.
Due to their simplicity, the Newton structures may be useful as building blocks of more complicated algorithms for interpolation, decimation, and reconstruction of nonuniformly
sampled signals.
References:
[1]
S. Tassart and Ph. Depalle, “Fractional delays using Lagrange
interpolators,“ in Proc. Nordic Acoustic Meeting, Helsinki,
Finland, 12–14 June, 1996.
[2] S. Tassart, Ph. Depalle, “Analytical approximations of fractional delays: Lagrange interpolators and allpass filters,” in
Proc. IEEE Int. Conf. Acoust., Speech, and Signal Proc.
(ICASSP’97), 21–24 Apr 1997, pp. 455–458.
[3] Ç. Candan, “An efficient filtering structure for Lagrange interpolation,” IEEE Signal Processing Letters, Vol. 14, No. 1, Jan
2007, pp. 17–19.
[4] E.W. Weisstein, "Newton’s Backward Difference Formula."
Available: http://mathworld.wolfram.com/
NewtonsBackwardDifferenceFormula.html.
Visited: 22 Jan 2008.
[5] E. Meijering, “A chronology of interpolation: From ancient astronomy to modern signal and image processing,” in Proc. of
the IEEE, Vol. 90, No. 3, Mar 2002, pp. 319–342.
[6] C.W. Farrow, “A continuously variable digital delay element,”
in Proc. IEEE Int. Symp. Circ. Syst. (ISCAS’88), Espoo, Finland, June 1988, pp. 2641–2645.
[7] R.E. Crochiere, L.R. Rabiner, Multirate Digital Signal
Processing, Prentice-Hall, 1983.
[8] T. Hentschel, G. Fettweis, “Continuous-time digital filters for
sample-rate conversion in reconfigurable radio terminals,” in
Proc. European Wireless, Dresden, Germany, Sep 2000, pp.
55–59.
[9] D. Babic, J. Vesma, T. Saramäki, M. Renfors, “Implementation of the transposed Farrow structure,” in Proc. IEEE Int.
Symp. Circ. Syst., May 2002, pp. IV-5–IV-8.
[10] F. Marvasti, M. Analoui, M. Gamshadzahi, “Recovery of signals from nonuniform samples using iterative methods,” IEEE
Trans. Signal Proc., Vol. 39, No. 4, Apr 1991, pp. 872–878.
[11] F.A. Marvasti, P.M. Clarkson, M.V. Dokic, U. Goenchanart,
C. Liu, “Reconstruction of speech signals with lost samples,”
IEEE Trans. Signal Proc., Vol. 40, No. 12, Dec 1992,
pp. 2897–2903.
[12] D. Babic and M. Renfors, “Reconstruction of non-uniformly
sampled signal using transposed Farrow structure,” in Proc.
Int. Symp. Circ. Syst. (ISCAS), Vancouver, Canada, May 2004,
Vol. III, pp. 221–224.
50
Chromatic Derivatives, Chromatic Expansions
and Associated Function Spaces
Aleksandar Ignjatović
School of Computer Science and Engineering, University of New South Wales,
and National ICT Australia (NICTA), Sydney, Australia;
ignjat@cse.unsw.edu.au
Abstract:
We present the basic properties of the chromatic derivatives and the chromatic expansions as well as a motivation for introducing these notions. The chromatic derivatives are special, numerically robust linear differential operators; the chromatic expansions are the associated local
expansions, which possess the best features of both the
Taylor and the Nyquist expansions. This makes them potentially useful in fields involving sampled data, such as
signal and image processing.
1. Motivation
The Nyquist–(Whittaker–Kotelnikov–Shannon)
expanP∞
sion f (t) =
f
(n)
sin
π(t
−
n)/π(t
−
n)
of a
n=−∞
π-band limited signal of finite energy f (t) ∈ BL(π) is
of global nature, because it requires samples of the signal at integers of arbitrarily large absolute value. On the
other hand, since signals from BL(π) are analytic functions, they can
be represented by the Taylor expanPalso
∞
sion, f (t) = n=0 f (n) (0) tn /n!. Such expansion is of
local nature, because the values of the derivatives f (n) (0)
are determined by the values of the signal in an arbitrarily
small neighborhood of zero.
While the Nyquist expansion has a central role in digital
signal processing, the Taylor expansion is of very limited
use there, for several reasons.
(1) Numerical evaluation of higher order derivatives of a
signal from its samples is very noise sensitive; in general,
one is cautioned against numerical differentiation of signals given by empirical samples.
(2) The Taylor expansion of a signal f ∈ BL(π) converges non-uniformly; its truncations are unbounded and
have rapid error accumulation.
(3) The Nyquist expansion of a signal f ∈ BL(π) converges to f in BL(π) and thus the action of a filter A on
any f ∈ BL(π) can be expressed using the samples of f
and the impulse response A[sinc ] of A, i.e.,
A[f ](t) =
∞
X
n=−∞
2.
Chromatic Derivatives
To explain our notions, we first consider normalized and
rescaled Legendre polynomials PnL (ω) which satisfy
Z π
1
L
P L (ω) Pm
(ω)dω = δ(m − n),
2π −π n
and then define operator polynomials
µ
¶
d
1 L
n
.
Kt = n Pn i
dt
i
(2)
It is easy to verify that for f ∈ BL(π) and its Fourier
transform f[
(ω) we have
Z π
1
n
(ω) ei ωt dω.
K [f ](t) =
in PnL (ω)f[
2π −π
Figure 1 compares the plots of PnL (ω) and ω n /π n for
n = 15 to n = 18, which are the transfer functions (save
a factor of in ) of the operators Kn and of the (normalized) derivatives 1/π n dn /dtn , respectively. While the
transfer functions of the normalized “standard derivatives”
1/π n dn /dtn obliterate the spectrum of the signal, leaving only its edges which in practice contain mostly noise,
the transfer functions of operators Kn form a family of
well separated, interleaved and increasingly refined comb
filters. Due to their spectrum preserving property, we call
the operators Kn the chromatic derivatives associated with
the Legendre polynomials. Both analytic estimates and
empirical tests have shown that the chromatic derivatives
2
0.5
1
-3
-2
1
-1
2
3
-3
-2
-1
1
2
3
-1
f (n) A [sinc ] (t − n).
(1)
In contrast, the polynomials obtained by truncating the
Taylor series do not belong to BL(π) and nothing similar to (1) holds for the Taylor expansion.
SAMPTA'09
The chromatic derivatives and the chromatic expansions
and approximations were introduced to obtain local signal
representations which do not suffer from these problems.
-0.5
-2
Figure 1: Graphs of PnL (ω) (left) and ω n /π n (right), for
n = 15 − 18.
51
can be accurately and robustly evaluated from samples of
the signal taken at a small multiple (2 to 4) of the usual
Nyquist rate, thus solving problem (1) associated with numerical evaluation of the standard derivatives, mentioned
above. Chromatic expansions, on the other hand, were introduced to solve problems (2) and (3).
3. Chromatic Approximations
Proposition 1 Let Kn be the chromatic derivatives associated with the Legendre polynomials, and let f (t) be any
analytic function; then for all t,
f (t) =
P∞
n
n=0 (−1)
K n [f ](u) K n [sinc ](t − u). (3)
If f ∈ BL(π) the series converges uniformly and in L2 .
The series in (3) is denoted by CE[f, u](t) and is called
the chromatic expansion of f (t) associated with the Legendre polynomials; a truncation of this series up to first
n + 1 terms is denoted by CA[f, n, u](t) and is called a
chromatic approximation of f (t). Just like a Taylor approximation, a chromatic approximation is also a local approximation: its coefficients are the values of differential
operators Km [f ](u) at a single instant
¯ u, and for all k ≤ n,
f (k) (u) = dk /dtk CA[f, n, u](t)¯t=u .
Figure 2 compares the behavior of the chromatic approximation (black) of a signal f ∈ BL(π) (gray) with the behavior of the Taylor approximation of f (t) (dashed). Both
approximations are of order sixteen. The plot reveals that,
when approximating a signal f ∈ BL(π), a chromatic approximation has a much gentler error accumulation when
moving away from the point of expansion than the Taylor
approximation of the same order.
Functions Kn [sinc ](t) appearing in the chromatic expansion associated with the Legendre
polynomials are given
√
by Kn [sinc ](t) = (−1)n 2n + 1 jn (πt), where jn is
the spherical Bessel function of the first kind of order n.
Thus, unlike the monomials that appear in the Taylor formula, functions Kn [sinc ](t) belong to BL(π) and satisfy
|Kn [sinc ](t)| ≤ 1 for all t ∈ R. Consequently, the chromatic approximations are bounded on R and belong to
BL(π). Also, as Proposition 1 asserts, the chromatic approximation of a signal f ∈ BL(π) converges in BL(π).
Thus, if A is a filter, then A commutes with the differential operators Kn and for every f ∈ BL(π), we have the
2.0
1.5
1.0
0.5
-15
-10
5
-5
10
15
following analogue of (1):
A[f ](t) =
∞
X
n=0
(−1)n Kn [f ](0) Kn [A[ sinc ]](t).
Thus, while local, the chromatic expansion possesses the
features that make the Nyquist expansion useful in signal processing. This, together with numerical robustness
of the chromatic derivatives, makes chromatic approximations applicable in fields involving empirically sampled
data, such as digital signal and image processing.
The next proposition demonstrates another remarkable
feature of the chromatic derivatives which is relevant to
signal processing.
Proposition 2 Let Kn be the chromatic derivatives associated with the (re-scaled and normalized) Legendre polynomials, and f, g ∈ BL(π). Then
∞
X
K n [f ](t)2 =
K n [f ](t)K n [g](t) =
n=0
K
n
Z
∞
Z
∞
f (x)2 dx;
f (x)g(x)dx;
−∞
n=0
∞
X
∞
−∞
n=0
∞
X
Z
[f ](t)Ktn [g(u
− t)] =
−∞
f (x)g(u − x)dx.
Thus, the sums on the left hand side of the above equations
do not depend on the choice of the instant t.
Note that the above equations provide local representations of the usual norm, the scalar product and the convolution, respectively, which are defined in L2 globally,
as improper integrals.
Given the above properties of the Legendre polynomials,
it is natural to ask if other families of orthonormal polynomials have similar properties. This question was answered
in [1].
4.
General Chromatic Derivatives
Let M : Pω → R be a linear functional on the vector
space Pω of real polynomials in the variable ω. Such M
is called a moment functional and µn = M(ω n ) is the
moment of M of order n.
Definition 1 A moment functionals M is chromatic if it
satisfies the following conditions (condition (iii) is not essential, but simplifies the technicalities):
(i) M is positive definite;
1/n
(ii) lim supn→∞ µn /n < ∞;
(iii) M is symmetric, i.e., µ2n+1 = 0 for all n.
-0.5
-1.0
-1.5
Figure 2: Chromatic approximation (black) and Taylor’s
approximation (dashed) of a signal from BL(π) (gray).
For functionals M which satisfy conditions (i) and (iii)
there exists a family of real polynomials {PnM (ω)}n∈N ,
such that PnM (ω) contains only powers of ω of the same
parity as n and which are orthonormal with respect to M;
i.e., for all m, n,
M
M(Pm
(ω) PnM (ω)) = δ(m − n).
SAMPTA'09
52
The family {PnM (ω)}n∈N is a family of orthonormal polynomials which corresponds to a symmetric positive definite moment functional M just in case there exists a sequence of positive reals {γn }n∈N such that
M
Pn+1
(ω) =
1
γn
ω PnM (ω) −
γn−1
γn
M
Pn−1
(ω).
(4)
For every positive definite moment functional there exists
a non-decreasing bounded function a(ω), called an m–
distribution function, such that for the associated Stieltjes
integral we have
R∞ n
(5)
ω da(ω) = µn ,
−∞
R∞ M
M
P (ω) Pm (ω) da(ω) = δ(m − n).
(6)
−∞ n
If M is chromatic, then condition (3) implies that
{PnM (ω)}n∈N is a complete system in L2a(ω) .
Let ϕ ∈ L2a(ω) ; we can define a corresponding function
fϕ : R → C by
R∞
(7)
fϕ (t) = −∞ ϕ(ω)eiωt da(ω),
and one can show that (7) can be differentiated under the
integral sign any number of times. Setting
¡ ¢
Kn = i1n PnM (ω) i ddt
we get that for all t
Kn [fϕ ](t) =
R∞
−∞
in PnM (ω) ϕ(ω) eiωt da(ω),
(8)
i.e., hϕ(ω)eiωt , PnM (ω)ia(ω) = (− i)n Kn [fϕ ](t). Thus,
ϕ(ω)eiωt = (− i)n Kn [fϕ ](t)PnM (ω), and by Parseval’s
Theorem, for every t ∈ R,
P∞
2
n
2
iωt k2
a(ω) = kϕ(ω)ka(ω) .
n=0 |K [fϕ ](t)| = kϕ(ω)e
P∞
Thus, if ϕ ∈ L2a(ω) , then the sum n=0 |Kn [fϕ ](t)|2 converges to a constant function on R.
If we let
R∞
(9)
m(t) = −∞ eiωt da(ω),
then (5) implies m(k) (0) = ik µk . It can be shown that
condition (iii) of Definition 1 implies that m(t) is analytic
at every t ∈ R (moreover, it is analytic on a strip in C; see
[2]). For the chromatic approximation associated with M,
Pn
CAM [f, n, u](t) = k=0 (−1)k Kk [f ](u)Kk [m](t − u),
one can show that
¯
¯2
P∞
|fϕ (t) − CAM [fϕ , n, u](t)| < k=n+1 ¯Kk [fϕ ](u)¯ .
P∞
Thus, fϕ (t) = k=0 (−1)k Kk [fϕ ](u) Kk [m](t − u), and
the convergence is uniform on R.
Definition 2 LM
2 denotes
P∞ the space of functions analytic
on R which satisfy k=0 Kk [f ](0)2 < ∞.
Let f (t) ∈ LM
2 ; then
P∞
ϕf (ω) = k=0 (−i)k Kk [f ](0)PkM (ω)
belongs to
L2a(ω)
and for all t,
R∞
f (t) = −∞ ϕf (ω) eiωt da(ω).
On the space LM
2 one can now introduce locally defined
norm, inner product and convolution using equations from
Proposition 2, and for every fixed u, the chromatic expansion of an f ∈ LM
2 is just the Fourier series of f in the
orthonormal and complete base {Kun [m(t − u)]}n∈N .
SAMPTA'09
5.
Examples
Example 1. (Legendre polynomials/Spherical Bessel
functions) Let√Ln (ω) be the Legendre polynomials; if we
set PnL (ω) = 2n + 1 Ln (ω/π), then
Rπ L
L
P (ω)Pm
(ω) d2πω = δ(m − n).
−π n
The corresponding recursion coefficients
pin equation (4)
are given by the formula γn = π(n+1)/ 4(n + 1)2 − 1.
In thisp case m(t) = sinc t, and Kn [m](t) =
(−1)n (2n + 1) jn (πt), where jn (x) is the spherical
Bessel function of the first kind of order n. The corresponding space LM
2 consists of all analytic functions
which belong to L2 and have a Fourier Transform supported in [−π, π].
Example 2. (Chebyshev polynomials of the first
kind/Bessel functions) Let PnT (ω) be the family of
orthonormal polynomials obtained by normalizing and
rescaling the Chebyshev polynomials of the
√ first kind,
Tn (ω), by setting P0T (ω) = 1 and PnT (ω) = 2 Tn (ω/π)
for n > 0. In this case
Rπ T
T
P (ω)Pm
(ω) q dω ω 2 = δ(n − m).
−π n
π 2 1−( π )
The corresponding function
(9) is m(t) = J0 (πt) and
√
Kn [m](t) = (−1)n 2 Jn (πt) for n > 0, where Jn (t)
is the Bessel function of the first kind of order n. In
the recurrence
√ relation (4) the coefficients are given by
γ0 = π/ 2 and γn = π/2 for n > 0. The corresponding space LM
2 consists of analytic functions whose
Fourier transform f[
(ω) is supported in (−π, π) and satisRπ p
fies −π 1 − (ω/π)2 |f[
(ω)|2 dω < ∞. The chromatic
expansion of a function f (t) is the Neumann series
√ P∞
n
f (t) = f (0)J0 (πt) + 2
n=1 K [f ](0)Jn (πt).
Thus, the chromatic expansions corresponding to various
families of orthogonal polynomials can be seen as generalizations of the Neumann series, while the families of
corresponding functions {Kn [m](t)}n∈N can be seen as
generalizations (and a uniform representation) of some familiar families of special functions.
Example 3. (Hermite polynomials/Gaussian monomial
functions) Let Hn (ω) be the Hermite polynomials; then
the polynomials given by PnH (ω) = (2n n!)−1/2 Hn (ω)
satisfy
R∞
−∞
H
PnH (ω)Pm
(ω)
2
−ω
e√
π
dω = δ(n − m).
The corresponding function defined by (9)
√ is m(t) =
2
2
−t
/4
n
n
n
−t
/4
e
and K [m](t) = (−1) t e
/ 2n n!. The
corresponding
recursion coefficients are given by γn =
p
(n + 1)/2. The corresponding space LM
2 consists of analytic functions whose Fourier transform f[
(ω) satisfies
R∞
2
|f[
(ω)|2 eω dω < ∞. The chromatic expansion of
−∞
2
f (t) is just the Taylor expansion of f (t) et
2
by e−t /4 .
/4
, multiplied
53
6. Weakly Bounded Moment Functionals
7.
To study local (i.e., non-uniform) convergence of chromatic expansions, we somewhat restrict the class of moment functionals we consider.
If M is weakly bounded, the
functions do not
Pperiodic
∞
n
belong to LM
K
[sin
ωt]2 diverges.
2 ; for example,
n=0
We now consider some inner product spaces in which pure
harmonic oscillations have finite positive norms ([3, 2]).
Definition 3 Let M be a symmetric positive definite
moment functional and let γn > 0 be such that (4) holds.
(i) M is weakly bounded if there exist some M ≥ 1, some
0 ≤ p < 1 and some integer r, such that for all n ≥ 0,
1/M ≤ γn ≤ M (n + r)p and γn /γn+1 ≤ M 2 .
(ii) M is bounded if there exists some M ≥ 1 such that
1/M ≤ γn ≤ M for all n ≥ 0.
Thus, every bounded moment functional is also weakly
bounded with p = 0. Functionals in our Example 1 and
Example 2 are bounded. For bounded moment functionals
the corresponding m-distribution a(ω) has a finite support
and consequently m(t) is a band-limited signal. However, m(t) can be of infinite energy (i.e., not in L2 ) as
is the case in our Example 2. Moment functional in Example 3 is weakly bounded but not bounded (p = 1/2).
We note that all important examples of classical orthogonal polynomials which correspond to weakly bounded
moment functionals in fact satisfy a stronger condition
0 < limn→∞ γn /np < ∞ for some 0 ≤ p < 1.
Lemma 3 If M is a weakly bounded moment functional,
1/k
then limk→∞ (µk /k!)
P∞ n= 0.n Thus, M is chromatic;
moreover, m(z) = n=0 i µn z /n! is an entire function
on C.
Lemma 4 Let M be weakly bounded and p < 1 as in
Definition 3(i); then for every integer k ≥ 1/(1 − p) there
exists K > 0 and a polynomial P (x) such that for every
n ∈ N and every z ∈ C,
k
|Kn [m](z)| < |Kz|n P (|z|)e|Kz| /n!1−p .
This Lemma is used to prove the following Proposition.
Proposition 5 Let M be as in Lemma 4, f (z) an entire
function and u ∈ C. If limn→∞ |f (n) (u)/n!1−p |1/n = 0,
then the chromatic expansion of f (z) centered at u converges everywhere to f (z), and the convergence is uniform
on every disc of finite radius.
Thus, if M is bounded (p = 0) and f is an entire function, then the chromatic expansion CE[f, u](t) converges
to f (t) for all t.
Many well known equalities for the Bessel functions Jn (t)
are just the special cases of chromatic expansions. For
example, the chromatic expansions of f (t) = eiωt , f (t) =
1 and f (t) = m(t + u) yield
P
n M
n
eiωt = ∞
n=0 i Pn (ω) K [m](t);
³
´
Qn γ2k−2
P∞
2n
m(t) + n=1
k=1 γ2k−1 K [m](t) = 1,
P∞
m(t + u) = n=0 (−1)n Kn [m](u)Kn [m](t),
which generalize the following well known equalities:
P
ei ωt = J0 (t) + 2 ∞
in Tn (ω)Jn (t);
P∞ n=1
J0 (t) + 2 n=1 J2n (t) = 1;
P∞
J0 (t + u) = J0 (u)J0 (t) + 2 n=1 (−1)n Jn (u)Jn (t).
SAMPTA'09
Non-Separable Inner Product Spaces
Definition 4 Assume again that M is weakly bounded
and let p be as in Definition 3. We denote by C M the vector
space of analytic functions such that the sequence
Pn
νnf (t) = 1/(n + 1)1−p k=0 Kk [f ](t)2
converges uniformly on every finite interval.
Proposition 6 Let f, g ∈ C M and
Pn
σnf g (t) = 1/(n + 1)1−p k=0 Kk [f ](t)Kk [g](t);
then the sequence {σnf g (t)}n∈N converges to a constant
function. In particular, νnf (t) is constant.
Corollary 7 Let C0M be the vector space consisting of
analytic functions f (t) such that limn→∞ νnf (t) = 0;
then in the quotient space C2M = C M /C0M the limit
limn→∞ σnf g (t) is independent of t and defines a scalar
product on C2M .
Proposition 8 Let M correspond to Chebyshev polynomials as in our Example
√ 2; then functions fω (t) =
√
2 sin ωt and gω (t) = 2 cos ωt for all 0 < ω < π form
an uncountable orthonormal system of vectors in C2M .
Proposition 9 Let M correspond to Hermite polynomials as in our Example 3; then for all ω > 0 functions
fω (t) = sin ωt and gω (t) = cos ωt form an uncountM
able orthogonal system of vectors in C2M , and kfω k =
√
2
M
4
ω
/2
kgω k = e
/ 2π.
Conjecture 1 Assume that for some 0 ≤ p < 1 the recursion coefficients γn in (4) are such that γn /np converges
to a finite positive limit. Then, for the corresponding family of orthogonal polynomials we have
Pn
0 < limn→∞ 1/(n + 1)1−p k=0 PkM (ω)2 < ∞
for all ω in the support sp(a) of the corresponding mdistribution function a(ω). Thus, in the corresponding
space C2M all pure harmonic oscillations with positive frequencies ω ∈ sp(a) have finite positive norm and are mutually orthogonal.
Detailed presentation of the theory of chromatic
derivatives can be found in our references; preprints
of some unpublished manuscripts are available at
http://www.cse.unsw.edu.au/˜ignjat/diff.
References:
[1] A. Ignjatovic. Local approximations based on orthogonal differential operators. Journal of Fourier Analysis and Applications, 13(3), 2007.
[2] A. Ignjatovic. Chromatic derivatives and associated
function spaces. manuscript, 2008.
[3] A. Ignjatovic. Chromatic derivatives and local approximations. to appear in: IEEE Transactions on
Signal Processing, 2009.
54
Estimation of the Length and the Polynomial
Order of Polynomial-based Filters
Djordje Babic(1), and Heinz G. Göckler(2)
(1) Faculty of Computer Science, University Union, Belgrade, Knez Mihailova 6/VI, 11000 Belgrade, Serbia.
(2) DISPO, Faculty of Electrical Engineering and Information Sciences, Ruhr-Universität, Bochum, Germany.
djbabic@raf.edu.rs, goeckler@nt.rub.de
Abstract:
In many signal processing applications it is beneficial to
use polynomial-based interpolation filters for sampling
rate conversion. Actual implementations of these filters
can be performed effectively by using the Farrow
structure or its modifications. In the literature, several
design methods have been proposed. However,
estimation formulae for the number of polynomialsegments defining the finite length of the underlying
continuous-time filter impulse response and the order of
polynomials have not been known. This contribution
presents estimation formulae for the length and the
polynomial order of polynomial-based filters for various
types of requirements. The formulae presented here can
save time in designing, since they provide good starting
values of length and order for a given set of
requirements.
1. Introduction
In many signal processing applications it is required to
determine signal samples at arbitrary positions between
existing samples of a discrete-time signal. In these cases,
it is beneficial to use polynomial-based interpolation
filters. For these filters, an efficient overall
implementation can be achieved by using a continuoustime impulse response ha(t) having the following
properties [1], [2]; First, ha(t) is nonzero only in a finite
interval 0≤t<NT with N being an integer. Second, in each
subinterval nT≤t<(n+1)T, for n=0, …, N−1, ha(t) is
expressible as a polynomial of t of a given (low) order
M. Third, ha(t) is symmetric with respect to t = NT/2 to
guarantee phase linearity of the resulting overall system.
The length of polynomial segments, T, can be selected to
be equal to the input Tin or output Tout sampling interval,
a fraction of the input or output sampling interval, or an
integer multiple of the input or output sampling interval.
The advantage of the above system lies in the fact that
the actual implementation can be efficiently performed
by using the Farrow structure [3] or its modifications [4],
[5].
In the literature, several design methods have been
proposed [1], [2], [4]. However, estimation formulae for
the number N of polynomial-segments and the order M
of polynomial have not been known. This contribution
presents the missing estimation formulae for the length N
SAMPTA'09
and polynomial order M for various types of
requirements. The formulae presented subsequently can
save time for the filter designers, because they get
suitable starting values for N and M that can be used for
the given set of requirements. The formulae can also be
used to estimate implementation costs of Farrow filter as
subsystem of general sampling rate converters, for
example, in optimal factorization of multistage
decimation (interpolation).
2. Polynomial-based filters
As it has been originally suggested in [1], [2] when
deriving the modified Farrow structure for interpolation,
it is beneficial to construct ha(t) as follows:
N −1 M
ha (t ) = ∑ ∑ cm (n) f m (n, T , t )
n =0 m = 0
(1)
where the number of polynomial segments N is an
integer. The basis functions fm(n, T, t), as defined in [1],
are given by
m
2(t − nT )
− 1 for nT ≤ t < (n + 1)T
f m (n, T , t ) =
T
otherwise,
0
(2)
where the common polynomial order of all segments is
M. The coefficients cm(n) are the adjustable parameters
being related to each other by
cm (n) for m even
cm ( N − 1 − n) =
− cm (n) for m odd
(3)
for n = 0, 1,…, N−1, as consequence of the symmetry
properties required above. The resulting ha(t) is
characterized by the following properties: (i) ha(t) is
nonzero for 0≤ t < NT and zero elsewhere; (ii) in each
subinterval nT ≤ t < (n +1)T for n = 0 , …, N−1, ha(t) is
expressed as a polynomial of degree M; (iii) ha(t) is
symmetric about t = NT/2, that is, ha(NT−t) = ha(t) .
Based on Property (iii), it is guaranteed that the resulting
overall system has a linear phase, a very attractive
property for many applications. Furthermore, the
generation of the above ha(t) guarantees that, in the
frequency domain, the zero-phase frequency response,
when omitting the linear-phase term, is expressible as
(see [1] for details)
N / 2 −1 M
H a ( j 2πf ) = ∑ ∑ cm (n)Gm (n, T , f ) ,
n =0 m =0
(4)
where Gm(n, T, f ) is the Fourier transform of
55
g m (n, T , t ) = (− 1) f m (n, T , t − NT / 2)
m
+ f m ( N − 1 − n, T , t − NT / 2) .
(5)
Since the above approximating function is linear with
respect to the unknown coefficients cm(n), it enables one
to optimize the overall filter to meet the given criteria in
a manner similar to that used for synthesizing various
types of linear-phase FIR filters [6]. In the above, T, the
length of the polynomial segments, can be used to define
different implementation structures as discussed in [4],
[5]. As seen in [4], [5], T can be chosen as T = βTin or T =
βTout, where β is unity, an integer, or one divided by an
integer. The selection depends on whether decimation or
interpolation is under consideration, and on the structural
needs for efficient implementation. The actual
implementation can be efficiently performed by using the
Farrow structure [3] or its modifications [4], [5].
For all these structure the number of fixed coefficients
depends on the number N of polynomial segments and
the order M of the polynomial in each segment. The total
number of multipliers, exploiting the symmetry
properties of (3), is given by
for N even
N ⋅ ( M + 1) / 2
S=
, (6)
( N − 1)(M + 1) / 2 + ( M + 1) / 2 for N odd.
For the purpose of illustration, the modified Farrow
structure [1] is used with T=Tin. It should be pointed out
that, in a practical realization, the coefficients’ symmetry
of the FIR branches will be exploited, and a single delay
line can be shared with all branches.
3. Review of minimax design method
This section reviews minimax design method of
polynomial-based filters of arbitrary length and order, as
presented in [1], [2], for which we estimate N and M.
To this end, we assume a lowpass signal x(n)↔X(ejΩin).
Its sampling rate Fin=1/Tin shall be converted by an
arbitrary ration according to Fout=RFin yielding
y(l)↔Y(ejΩout). In case of R>1 (R<1) the system realizes
interpolation (decimation). The ultimate aim is to
determine a continuous-time, finite-length impulse
response ha(t) of the sampling rate conversion system
such that the Fourier transform of ha(t) meets following
requirements [4] , [7]:
(1 − δ p ) ≤ H a ( f ) ≤ (1 + δ p ) for f ≤ f p = αF / 2
Ha ( f ) ≤ δs
for f ∈ Φ s ,
(7)
where
[F / 2, ∞ ]
∞
Φ s = kF − f p , kF + f p
k =1
F − f ,∞
p
[
[
]
for Case A
]
for Case B
(8)
for Case C.
In all three cases, the signal is preserved according to the
given tolerance in the passband region [0, fp].
Furthermore, the aliasing components are attenuated in
the defined manner. In Case A, all components aliasing
into the baseband [0, F/2] are attenuated. In Case B, all
SAMPTA'09
components aliasing into the passband [0, fp] are
attenuated, but aliasing is allowed in the transition band
[fp, F/2]. In Case C, aliasing into the transition band [fp,
F/2] is allowed only from the band [F/2, F+fp]. In the
above discussion and in (7) and (8) F stands for Fout in a
decimation case, and Fin in an interpolation case.
The minimax optimization method introduced in [1], [2]
is probably the most convenient and the most flexible
solution for designing polynomial-based interpolation
filters:
Minimax Optimization Problem: Given N, M, and a
compact subset Φ ⊂ [0,∞) as well as a desired function
D( f ) being continuous for f ∈ Φ and a weight function
W( f ) being positive for f ∈ Φ , find the (M +1)N/2
unknown coefficients cm(n) to minimize
δ ∞ = max W ( f )[H a ( f ) − D( f )]
f ∈Φ
(9)
subject to the given time-domain conditions of ha(t).
Here, Ha( f ) is the real-valued frequency response and
D(f ) is the desired function according to specifications.
(For details refer to [2]). The design procedure has been
generalized, and modified for optimization of prolonged
and transposed prolonged polynomial-based filters [4].
The minimax design method has several design
parameters. First of all, the design parameters include
passband and stopband regions Φp and Φs. The desired
filter may have several passbands and stopbands as
stated in [2]. Next, the minimum stopband attenuation δs,
and maximum allowable passband ripple δp are also
included. Other design parameters are the number of
polynomial segments N and the order M of the
polynomial, which determine the number of multipliers
in the overall structure, see (6). Finally, some weighting
function can be used to give different weights to
passband and stopband [2]. Hence we give estimation
formulae for the number N of polynomial segments and
the order M of polynomial for a minimax design.
4. Estimation of N and M
In the previous section, we have seen that the number of
polynomial segments N and the order M of the
polynomial, are the design parameters that highly
influence the performance of the filter in the frequency
domain. Furthermore, the cost of realization, i.e. the
number of multipliers, of a filter can be estimated by
introducing the required values for N and M into (6). It
would be very beneficial to estimate N and M by only
using the given specifications of the filter in the
frequency domain. Similar order estimation formulae
exist for FIR filters, for example Kaiser order estimation
[6], [8]. In the actual implementation, polynomial-based
filters can be modeled as FIR filters [4]. Thus, we can
start from the Kaiser formula and adapt it to polynomialbased filters. To this end, a lot of filters were designed,
by using different system specifications, in order to adapt
the Kaiser formula to polynomial-based case. The
obtained estimation formula for the number of
polynomial segments N, is rather similar to Kaiser
formula for the order estimation of FIR filters. The
56
A − 10 log10 (W ) − 8.4
N e = 2 s
30.4( f s − f p ) / F
65
60
Stopband attenuation As in dB
where As=-20log10(δs) is the required attenuation in
stopband, and W=δp/δs represents weighting between
required tolerances in passband and stopband.
The next problem is to find the minimum value of the
polynomial order M to meet the specifications. It has
been observed that the required value of M depends on
the type of requirements from (7) and (8). Never the less,
it is possible to consider the following estimate as good
starting point for all three types of requirements:
55
50
45
40
35
2
4
6
8
10
12
14
16
18
A − 20 ⋅ log10 (W )
+ log10 (W ) +1.
Me = s
2 .5
20
Number of polynomial segments N
(a)
65
55
s
Stopband attenuation A in dB
(12)
It has been observed that if transition band is relatively
large to the sampling frequency, that is when (fs-fp)/F
≥0.5, the required value of polynomial order M is
lowered by one. The estimation formula cannot be used
when the transition band is very small, i.e., in the case
when (fs-fp)/F<0.1. However, even in this border
situation required value of M is always smaller than Me
given by (12). Thus, the estimation formula (12) for the
polynomial order M can be used to estimate the upper
border for M for all types of requirements.
60
50
45
40
35
(11)
5. Design Examples
0
1
2
3
4
5
6
7
Polynomial order M
(b)
Fig. 1. Case A specifications: The passband and
stopband edges are at fp=0.4Fin and at fs=0.5Fin,
and stopband weighting W=100. (a) The curves
are shown for M equals 0 to 7. Dashed line is plot
obtained from the estimation formula for N. (b)
The curves are shown for N equals 2 to 20.
Dashed line is plot obtained from the estimation
formula for M.
estimation formula for N, which can be found in [9], is
not accurate enough. Hence, we propose the more
accurate formula:
− 20 log10 ( δ pδ s ) − 8.4
N e = 2
30.4( f s − f p ) / F
(10)
where δp and δs are the maximum deviations of the
amplitude response from unity for f∈[0,fp] and the
maximum deviation from zero for f∈Φs, respectively.
Here, x stands for the smallest integer which is larger
or equal to x. It has been observed that in most cases the
above estimation formula is rather accurate with only a
2% error. The formula above is valid for all three types
of requirements, i.e., A, B, and C, as given by (7) and
(8). However, if the transition band is narrow, i.e., in the
case when (fs-fp)/F≤0.1, the required value of N should
be increased by 2. Further, in the case of very narrow
transition band ((fs-fp)/F ≤0.05) the formula can not be
used.
The kernel of the estimation formula for the number N of
polynomial segments can be expressed in a different
form:
SAMPTA'09
This part gives several examples to illustrate the
performance of the presented formulae. To illustrate this,
the following specifications are considered:
Case A specifications: The passband and stopband edges
are at fp=0.4Fin and at fs=0.5Fin.
Case B specifications: The passband and stopband edges
are at fp=0.35Fin and at fs=0.65Fin.
Case C specifications: The passband and stopband edges
are at fp=0.35Fin and at fs=0.65Fin.
In each case, several filters have been designed in
minimax sense with the passband weighting equal to
unity and stopband weightings of W=100. The degree of
the polynomial in each subinterval M varies from 0 to 7.
The number of intervals N varies from 2 to 20. Recall
that N is an even integer. Figures 1 give the results for
Case A, the similar results for Case B are given in Fig. 2,
and for Case C in Fig. 3. It can be observed that the
estimation formulae are relatively good, as they estimate
the border performance for the given set of requirements
(dashed lines in Figs 1-3).
6. Conclusions
In this paper, the estimation formulae for the number N
of polynomial segments and the polynomial order M are
presented. It has been shown that these estimates give the
border performance of the filter for the given set of
specifications. Formulae for N and M can be used to
estimate the starting value of these two parameters in
minimax optimization. Furthermore, the formulae for N
and M can be used to estimate implementation costs of
57
110
110
100
100
Stopband attenuation A in dB
120
90
90
s
Stopband attenuation As in dB
120
80
70
60
80
70
60
50
50
40
40
30
2
4
6
8
10
12
14
16
18
30
20
2
4
6
Number of polynomial segments N
8
10
(a)
14
16
18
20
(a)
130
120
120
110
110
Stopband attenuation A in dB
130
Stopband attenuation As in dB
12
Number of polynomial segments N
100
s
100
90
80
70
90
80
70
60
60
50
50
40
0
1
2
3
4
5
6
7
Polynomial order M
40
0
1
2
3
4
5
6
7
Polynomial order M
(b)
(b)
Fig. 2. Case B specifications: The passband and
stopband edges are at fp=0.35Fin and at fs=0.65Fin,
and stopband weighting W=100. (a)The curves
are shown for M equals 0 to 7. Dashed line is plot
obtained from the estimation formula for N. (b)
The curves are shown for N equals 2 to 20.
Dashed line is plot obtained from the estimation
formula for M.
Fig. 3. Case C specifications: The passband and
stopband edges are at fp=0.35Fin and at fs=0.65Fin,
and stopband weighting W=100. (a) The curves
are shown for M equals 0 to 7. Dashed line is plot
obtained from the estimation formula for N. (b)
The curves are shown for N equals 2 to 20.
Dashed line is plot obtained from the estimation
formula for M.
the Farrow based filters for the given set of requirements.
Formulae can also be used to estimate implementation
costs of composed sampling rate converters containing
Farrow, for example, in optimal factorization for
multistage decimation (interpolation).
Processing SMMSP’02 , Toulouse, France,
September 2002, pp. 57−64.
[5] D. Babic, Techniques for sampling rate conversion
by arbitrary factors with applications in flexible
communications receivers, Doctoral Thesis, Tampere
University of Technology, 2004.
[6] T. Saramäki, “Finite impulse response filter design,”
Chapter 4 in Handbook for Digital Signal Processing,
edited by S. K. Mitra and J. F. Kaiser, John Wiley &
Sons, New York, 1993.
[7] D. Babic, J. Vesma, T. Saramäki, M. Renfors,
“Implementation of the transposed Farrow structure,”
in Proc. 2002 IEEE Int. Symp. Circuits and Systems,
Scotsdale, Arizona, USA, 2002, vol. 4, pp. 4−8.
[8] J.F. Kaiser, "Nonrecursive Digital Filter Design
Using the - sinh Window Function," Proc. 1974
IEEE Symp. Circuits and Systems, (April 1974),
pp.20-23.
[9]T. Saramäki, "Multirate Signal Processing," Lecture
Notes, http://www.cs.tut.fi/~ts/
References:
[1] J. Vesma and T. Saramäki, “Interpolation filters with
arbitrary frequency response for all-digital receivers,”
in Proc. 1996 IEEE Int. Symp. Circuits and Systems,
Atlanta, Georgia, May 1996, pp. 568−571.
[2] J. Vesma and T. Saramäki, “Polynomial-based
interpolation Filters - Part I: Filter synthesis,"
Circuits, Systems, and Signal Processing, vol. 26, no.
2, pp. 115-146, March/April 2007.
[3] C. W. Farrow, “A continuously variable digital delay
element,”in Proc. 1988 IEEE Int. Symp. Circuits and
Systems, Espoo, Finland, June 1988, pp. 2641−2645.
[4] D. Babic, T. Saramäki, M. Renfors, “Conversion
between arbitrary sampling rates using polynomialbased interpolation filters,” in Proc. 2nd Int. TICSP
Workshop on Spectral Methods and Multirate Signal
SAMPTA'09
58
Special session on
Geometric Multiscale Analysis
Chair: Gitta Kutyniok
SAMPTA'09
59
SAMPTA'09
60
The Continuous Shearlet Transform in
Arbitrary Space Dimensions, Frame
Construction, and Analysis of Singularities
S. Dahlke (1) , G. Steidl (2) and G. Teschke (3)
(1) Philipps-Universität Marburg, FB12 Mathematik und Informatik, Hans-Meerwein Straße,
Lahnberge, 35032 Marburg, Germany.
(2) Universität Mannheim, Fakultät für Mathematik und Informatik, Institut für Mathematik,
68131 Mannheim, Germany.
(3) University of Applied Sciences Neubrandenburg, Institute for Computational Mathematics
in Science and Technology, Brodaer Str. 2, 17033 Neubrandenburg, Germany.
dahlke@mathematik.uni-marburg.de, steidl@math.uni-mannheim.de, teschke@hs-nb.de
Abstract:
This note is concerned with the generalization of the continuous shearlet transform to higher dimensions. Similar
to the two-dimensional case, our approach is based on
translations, anisotropic dilations and specific shear matrices. We show that the associated integral transform
again originates from a square-integrable representation of
a specific group, the full n-variate shearlet group. Moreover, we verify that by applying the coorbit theory, canonical scales of smoothness spaces and associated Banach
frames can be derived. We also indicate how our transform can be used to characterize singularities in signals.
So far, the shearlet transform is well developed for problems in R2 . However, for analyzing higher-dimensional
data sets, there is clearly an urgent need for further generalizations and applications. This is exactly the concern of
this paper. One particular field of application is the geometrical structure analysis of multi-dimensional data, e.g.
multimodal spectral measurements in meteorology.
To our best knowledge, it seems that there exist only few
results in this direction: some important progress has been
achieved for the curvelet case in [1] and for surfacelets in
[16]. However, for the shearlet approach the question has
been completely open.
2.
1.
Introduction
Modern technology allows for easy creation, transmission
and storage of huge amounts of data. Confronted with a
flood of data, such as internet traffic, or audio and video
applications, nowadays the key problem is to extract the
relevant information from these sets. To this end, usually the first step is to decompose the signal with respect
to suitable building blocks which are well–suited for the
specific application and allow a fast and efficient extraction. In this context, one particular problem which is
currently in the center of interest is the analysis of directional information. Due to the bias to the coordinate
axes, classical approaches such as, e.g., wavelet or Gabor transforms are clearly not the best choices, and hence
new building blocks have to be developed. In recent
studies, several approaches have been suggested such as
ridgelets [2], curvelets [3], contourlets [7], shearlets [14]
and many others. For a general approach see also [13].
Among all these approaches, the shearlet transform stands
out because it is related to group theory, i.e., this transform can be derived from a square-integrable representation π : S → U(L2 (R2 )) of a certain group S, the socalled shearlet group, see [5]. Therefore, in the context
of the shearlet transform, all the powerful tools of group
representation theory can be exploited.
SAMPTA'09
Multivariate Continuous Shearlet Transform
In this section, we introduce the shearlet transform on
L2 (Rn ). This requires the generalization of the twodimensional parabolic dilation matrix and of the shear matrix, respectively. Let In denote the (n, n)-identity matrix
and 0n , resp. 1n the vectors with n entries 0, resp. 1. For
a ∈ R∗ := R \ {0} and s ∈ Rn−1 , we set
!
a
0Tn−1
Aa :=
1
0n−1 sgn(a)|a| n In−1
and
Ss :=
1
sT
0n−1
In−1
.
Lemma 1 The set R∗ × Rn−1 × Rn endowed with the
operation
(a, s, t) ◦ (a′ , s′ , t′ ) = (aa′ , s + |a|1−1/n s′ , t + Ss Aa t′ )
is a locally compact group S which we call full shearlet
group. The left and right Haar measures on S are given
by
1
dµl (a, s, t) = n+1 da ds dt
|a|
61
and
ω3 ✻
1
da ds dt.
dµr (a, s, t) =
|a|
In the following, we use only the left Haar measure and
use the abbreviation dµ = dµl . For f ∈ L2 (Rn ) we
define
1
−1
π(a, s, t)f (x) = fa,s,t (x) := |a| 2n −1 f (A−1
a Ss (x − t)).
(1)
It is easy to check that π : S → U(L2 (Rn )) is a mapping
from S into the group U(L2 (Rn )) of unitary operators on
L2 (Rn ). Recall that a unitary representation of a locally
compact group G with the left Haar measure µ on a Hilbert
space H is a homomorphism π from G into the group of
unitary operators U(H) on H which is continuous with
respect to the strong operator topology.
1
2
❇
A nontrivial function ψ ∈ L2 (Rn ) is called admissible, if
Z
|hψ, π(a, s, t)ψi|2 dµ(a, s, t) < ∞.
S
If π is irreducible and there exits at least one admissible
function ψ ∈ L2 (Rn ), then π is called square integrable.
The following result shows that the unitary representation
π defined in (1) is square integrable.
Theorem 3 A function ψ ∈ L2 (Rn ) is admissible if and
only if it fulfills the admissibility condition
Z
|ψ̂(ω)|2
Cψ :=
dω < ∞.
(2)
n
Rn |ω1 |
Then, for any f ∈ L2 (Rn ), the following equality holds
true:
Z
|hf, ψa,s,t i|2 dµ(a, s, t) = Cψ kf k2L2 (Rn ) . (3)
PP
PP
PP
PP
PP
P
P
❇
✠ ω2
Lemma 2 The mapping π defined by (1) is a unitary representation of S.
2
❇
❇
❇
✲
ω1
❇❇
Figure 1: Support of the shearlet ψ̂ for ω1 ≥ 0.
3.1
Shearlet Coorbit Spaces
We consider weight functions w(a, s, t) = w(a, s) that
are locally integrable with respect to a and s, i.e.,
n
′ ′ ′
w ∈ Lloc
1 (R ) and fulfill w ((a, s, t) ◦ (a , s , t )) ≤
′ ′ ′
w(a, s, t)w(a , s , t ) and w(a, s, t) ≥ 1 for all
(a, s, t), (a′ , s′ , t′ ) ∈ S. For 1 ≤ p < ∞, let
Lp,w (S) := {F measurable :
kF kLp,w (S) :=
Z
S
p1
|F (g)| w(a, s, t) dµ(a, s, t) < ∞},
p
p
and let L∞,w be defined with the usual modifications. In
order to construct the coorbit spaces related to the shearlet
group we have to ensure that there exists a function ψ ∈
L2 (Rn ) such that
SHψ (ψ) = hψ, π(a, s, t)ψi ∈ L1,w (S).
S
(4)
In particular, the unitary representation π is irreducible
and hence square integrable.
Fortunately, it turns out that (4) can be satisfied in our setting.
An example of a continuous shearlet can be constructed
as follows: Let ψ1 be a continuous wavelet with ψ̂1 ∈
C ∞ (R) and supp ψ̂1 ⊆ [−2, − 12 ] ∪ [ 12 , 2], and let ψ2 be
such that ψ̂2 ∈ C ∞ (Rn−1 ) and supp ψ̂2 ⊆ [−1, 1]n−1 .
Then the function ψ ∈ L2 (Rn ) defined by
1
ω̃
ψ̂(ω) = ψ̂(ω1 , ω̃) = ψ̂1 (ω1 ) ψ̂2
ω1
Theorem 4 Let ψ be a Schwartz function such that
supp ψ̂ ⊆ ([−a1 , −a0 ] ∪ [a0 , a1 ]) × Qb ,where Qb :=
[−b1 , b1 ] × · · · × [−bn−1 , bn−1 ]. Then we have that
SHψ (ψ) ∈ L1,w (S), i.e.,
is a continuous shearlet. The support of ψ̂ is depiced for
ω1 ≥ 0 in Fig. 1.
3.
Multivariate Shearlet Coorbit Theory
In this section we want to establish a coorbit theory based
on the square integrable representation (1) of the shearlet
group. We mainly follow the lines of [4]. For further information on coorbit space theory, the reader is referred to
[8, 9, 10, 11, 12].
SAMPTA'09
khψ, π(·)ψikL1,w (S) =
Z
|SHψ (ψ)(a, s, t)| w(a, s, t) dµ(a, s, t) < ∞.
S
For ψ satisfying (4) we can consider the space
H1,w := {f ∈ L2 (Rn ) : SHψ (f ) ∈ L1,w (S)},
(5)
with norm kf kH1,w := kSHψ f kL1,w (S) and its anti∼
dual H1,w
, the space of all continuous conjugate-linear
∼
functionals on H1,w . The spaces H1,w and H1,w
are
π-invariant Banach spaces with continuous embeddings
∼
H1,w ֒→ H ֒→ H1,w
, and their definition is independent
of the shearlet ψ. Then the inner product on L2 (Rn ) ×
62
∼
L2 (Rn ) extends to a sesquilinear form on H1,w
× H1,w ,
∼
therefore for ψ ∈ H1,w and f ∈ H1,w the extended representation coefficients
k(cλ (f ))λ∈Λ kℓp,w ≤ Ckf kSCp,w
SHψ (f )(a, s, t) := hf, π(a, s, t)ψiH∼
1,w ×H1,w
are well-defined. Now, for 1 ≤ p ≤ ∞, we define the
shearlet coorbit spaces
∼
: SHψ (f ) ∈ Lp,w (S)}
SCp,w := {f ∈ H1,w
Shearlet Banach Frames
ℓp,w := {c = (cλ )λ∈Λ : kckℓp,w := kcwkℓp < ∞},
where w
=
(w((a, s, t)λ ))λ∈Λ .
versely,
if
(c
(f
))
∈
ℓp,w ,
λ
λ∈Λ
P
f = λ∈Λ cλ π((a, s, t)λ )ψ is in SCp,w and
kf kSCp,w ≤ C ′ k(cλ (f ))λ∈Λ kℓp,w .
The Feichtinger-Gröchenig theory provides us with a machinery to construct atomic decompositions and Banach
frames for our shearlet coorbit spaces SCp,w . In a first
step, we have to determine, for a compact neighborhood
U of e ∈ S with non-void interior, so-called U –dense
sets. A (countable) family X = ((a, s, t)λ )λ∈Λ in S is
said to be U -dense if ∪λ∈Λ (a, s, t)λ U = S, and separated if for some compact neighborhood Q of e we have
(ai , si , ti )Q ∩ (aj , sj , tj )Q = ∅, i 6= j, and relatively separated if X is a finite union of separated sets.
1
1
(7)
Then the sequence
1
{(ǫαj , βαj(1− n ) k, S
1
βαj(1− n ) k
j ∈ Z, k ∈ Z n−1 , m ∈ Z n , ǫ ∈ {−1, 1}}
(8)
is U -dense and relatively separated.
Next we define the U –oscillation as
k oscU kL1,w (S) < 1/kSHψ (ψ)kL1,w (S) .
(9)
Then, the following decomposition theorem, which was
proved in a general setting in [8, 9, 10, 11, 12], says that
discretizing the representation by means of an U -dense set
produces an atomic decomposition for SCp,w .
Theorem 6 Assume that the irreducible, unitary representation π is w-integrable and let an appropriately normalized ψ ∈ L2 (Rn ) which fulfills
sup
|hψ, π(u)ψi| ∈ L1,w (S)
u∈(a,s,t)U
(10)
be given. Choose a neighborhood U of e so small that
k oscU kL1,w (S) < 1.
(11)
Then for any U -dense and relatively separated set X =
((a, s, t)λ )λ∈Λ the space SCp,w has the following atomic
decomposition: If f ∈ SCp,w , then
X
f=
cλ (f )π((a, s, t)λ )ψ
(12)
λ∈Λ
SAMPTA'09
if
ii) there exist two constants 0 < D ≤ D′ < ∞ such
that
D kf kSCp,w ≤
k(hf, π((a, s, t)λ )ψiH∼
)
k
≤ D′ kf kSCp,w ;
1,w ×H1,w λ∈Λ ℓp,w
(16)
u∈U
M hψ, π(a, s, t)i :=
(15)
Then, for every U -dense and relatively separated family
X = ((a, s, t)λ )λ∈Λ in G the set {π((a, s, t)λ )ψ : λ ∈ Λ}
is a Banach frame for SHp,w . This means that
oscU (a, s, t) :=
sup |SHψ (ψ)(u ◦ (a, s, t)) − SHψ (ψ)(a, s, t)|.
(14)
Theorem 7 Impose the same assumptions as in Theorem
6. Choose a neighborhood U of e such that
if
and
only
i) f
∈
SCp,w
(hf, π((a, s, t)λ )ψiH∼
∈
ℓ
;
)
×H
λ∈Λ
p,w
1,w
1,w
Aαj γm) :
Conthen
Given such an atomic decomposition, the problem arises
under which conditions a function f is completely determined by its moments hf, π((a, s, t)λ )ψi and how f can
be reconstructed from these moments. This is answered
by the following theorem which establishes the existence
of Banach frames.
Lemma 5 Let U be a neighborhood of the identity in S,
and let α > 1 and β, γ > 0 be defined such that
[α n −1 , α n ) × [− β2 , β2 )n−1 × [− γ2 , γ2 )n ⊆ U.
(13)
with a constant C depending only on ψ and with ℓp,w being defined by
(6)
with norms kf kSCp,w := kSHψ f kLp,w (S) . It holds that
SC1,w = H1,w and SC1,1 = L2 (Rn ).
3.2
where the sequence of coefficients depends linearly on f
and satisfies
iii) there exists a bounded, linear reconstruction operatorR from ℓp,w to SCp,w such that
)
R (hf, ψ((a, s, t)λ )ψiH∼
= f.
1,w ×H1,w λ∈Λ
It can be checked that the conditions (10), (11) and (15)
can be satisfied, see [6] for details.
4.
Analysis of Singularities
In this section, we deal with the decay of the shearlet transform at hyperplane singularities. The following analysis
generalizes techniques and results presented in [15] for
two dimensions. An (n − m)-dimensional hyperplane in
Rn , 1 ≤ m ≤ n − 1, not containing the x1 -axis can be
written w.l.o.g. as
xm+1
0
x1
..
..
..
. + P . = . ,
xm
| {z }
xA
|
xn
{z
xE
0
}
63
p1T
P := ... ∈ Rm,n−m .
pTm
Then we obtain for
νm := δ(xA + P xE )
with the Delta distribution δ that
Z
ν̂m (ω) =
δ(xA + P xE )e−2πi(hxA ,ωA i+hxE ,ωE i) dx
Rn
Z
=
e−2πi(−hP xE ,ωA i+hxE ,ωE i) dxE
Rn−m
= δ(ωE − P T ωA ).
(17)
The following theorem describes the decay of the shearlet
transform at hyperplane singularities. We use the notation
SHψ f (a, s, t) ∼ |a|r as a → 0, if there exist constants
0 < c ≤ C < ∞ such that
c|a|r ≤ SHψ f (a, s, t) ≤ C|a|r
as a → 0.
Theorem 8 Let ψ ∈ L2 (Rn ) be a shearlet satisfying ψ̂ ∈
C ∞ (Rn ). Assume further that ψ̂(ω) = ψ̂1 (ω1 )ψ̂2 (ω̃/ω1 ),
where supp ψ̂1 ∈ [−a1 , −a0 ] ∪ [a0 , a1 ] for some a1 >
a0 ≥ α > 0 and supp ψ̂2 ∈ Qb . If
(sm , . . . , sn−1 ) = (−1, s1 , . . . , sm−1 ) P
and
(t1 , . . . , tm ) = −(tm+1 , . . . , tn ) P T ,
then
SHψ νm (a, s, t) ∼ |a|
1−2m
2n
as a → 0.
(18)
Otherwise, the shearlet transform SHψ νm decays rapidly
as a → 0.
Similar results can be derived for point singularities, see
again [6] for details.
References:
[1] L. Borup and M. Nielsen, Frame decomposition of
decomposition spaces, J. Fourier Anal. Appl., to appear.
[2] E. J. Candès and D. L. Donoho, Ridgelets: a key to
higher-dimensional intermittency?, Phil. Trans. R.
Soc. Lond. A. 357 (1999), 2495–2509.
[3] E. J. Candès and D. L. Donoho, Curvelets - A
surprisingly effective nonadaptive representation for
objects with edges, in Curves and Surfaces, L. L.
Schumaker et al., eds., Vanderbilt University Press,
Nashville, TN (1999).
[4] S. Dahlke, G. Kutyniok, G. Steidl, and G. Teschke,
Shearlet Coorbit Spaces and Associated Banach
Frames, Preprint Nr. 2007-5, Philipps-Universitt
Marburg, 2007.
SAMPTA'09
[5] S. Dahlke, G. Kutyniok, P. Maass, C. Sagiv, H.-G.
Stark, and G. Teschke, The uncertainty principle associated with the continuous shearlet transform, Int.
J. Wavelets Multiresolut. Inf. Process. 6 (2008), 157181.
[6] S. Dahlke, G. Steidl, and G. Teschke, The continuous
shearlet transform in arbitrary space dimensions,
Preprint Nr. 2008–7, Philipps-Universität Marburg
2008.
[7] M. N. Do and M. Vetterli, The contourlet transform:
an efficient directional multiresolution image representation, IEEE Transactions on Image Processing
14(12) (2005), 2091–2106.
[8] H. G. Feichtinger and K. Gröchenig, A unified
approach to atomic decompositions via integrable
group representations, Proc. Conf. “Function Spaces
and Applications”, Lund 1986, Lecture Notes in
Math. 1302 (1988), 52–73.
[9] H. G. Feichtinger and K. Gröchenig, Banach spaces
related to integrable group representations and their
atomic decomposition I, J. Funct. Anal. 86 (1989),
307–340.
[10] H. G. Feichtinger and K. Gröchenig, Banach spaces
related to integrable group representations and
their atomic decomposition II, Monatsh. Math. 108
(1989), 129–148.
[11] H. G. Feichtinger and K. Gröchenig, Non–
orthogonal wavelet and Gabor expansions and
group representations, in: Wavelets and Their Applications, M.B. Ruskai et.al. (eds.), Jones and Bartlett,
Boston, 1992, 353–376.
[12] K. Gröchenig, Describing functions: Atomic decompositions versus frames, Monatsh. Math. 112 (1991),
1–42.
[13] K. Guo, W. Lim, D. Labate, G. Weiss, and E. Wilson,
Wavelets with composite dilations and their MRA
properties. Appl. Comput. Harmon. Anal. 20 (2006),
220–236.
[14] K. Guo, G. Kutyniok, and D. Labate, Sparse multidimensional representations using anisotropic dilation und shear operators, in Wavelets und Splines
(Athens, GA, 2005), G. Chen und M. J. Lai, eds.,
Nashboro Press, Nashville, TN (2006), 189–201.
[15] G. Kutyniok and D. Labate, Resolution of the wavefront set using continuous shearlets, Trans. Amer.
Math. Soc. 361 (2009), 2719-2754.
[16] Y. Lu and M.N. Do, Multidimensional directional filterbanks and surfacelets IEEE Trans. Image Process.
16 (2007) 918–931.
64
Compressive-wavefield simulations
Felix J. Herrmann, Yogi Erlangga, and Tim. T. Y. Lin
Department of Earth and Ocean Sciences, the University of British Columbia, Canada
fherrmann,yerlangga,tlin@eos.ubc.ca
Abstract:
Full-waveform inversion’s high demand on computational
resources forms, along with the non-uniqueness problem,
the major impediment withstanding its widespread use on
industrial-size datasets. Turning modeling and inversion
into a compressive sensing problem—where simulated data
are recovered from a relatively small number of independent simultaneous sources—can effectively mitigate this
high-cost impediment. The key is in showing that we can
design a sub-sampling operator that commutes with the
time-harmonic Helmholtz system. As in compressive sensing, this leads to a reduction in simulation cost. Moreover,
this reduction is commensurate with the transform-domain
sparsity of the solution, implying that computational costs
are no longer determined by the size of the discretization
but by transform-domain sparsity of the solution of the
CS problem which forms our data. The combination of
this sub-sampling strategy with our recent work on implicit solvers for the Helmholtz equation provides a viable
alternative to full-waveform inversion schemes based on
explicit finite-difference methods.
1.
Introduction
With the recent resurgence of full-waveform inversion—
i.e., adjoint-state methods applied to solve PDEconstrained optimization problems—the computational
cost of solving forward modeling has become one of the
major impediments withstanding successful application of
this technology to industry-size data volumes. To overcome
this impediment, we argue that further improvements will
depend on a problem formulation with a computational
complexity that is no longer strictly determined by the
size of the discretization but by transform-domain sparsity
of its solution. In this new paradigm, we bring computational costs in par with our ability to compress solutions to
certain PDEs. This premise is related to two recent developments. First, there is the new field of compressive sensing
[CS in short throughout the paper, 4, 5]—where the argument is made, and rigorously proven—that compressible
signals can be recovered from severely sub-Nyquist sampling by solving a sparsity promoting program. Second,
there is in the seismic community the recent resurgence
of simultaneous-source acquisition [1, 13, 2, 18, 12], and
continuing efforts to reduce the cost of seismic modeling,
imaging, and inversion through phase encoding of simultaneous sources [16, 21, 13, 12] and the removal of subsets
SAMPTA'09
of angular frequencies [22, 17, 15, 12] or plane waves [24].
All these approaches correspond to instances of CS. By
using CS principles, we have been able to remove the associated sub-sampling interferences through a combination
of exploiting transform-domain sparsity, properties of certain sub-sampling schemes, and the existence of sparsity
promoting solvers.
2.
Compressive full-waveform inversion
Full-waveform inversion entails solving PDE-constrained
optimization problems of the following type:
1
min kRM d − DU k22
U, m 2
s.t. H[m]U = B,
(1)
where d and U are the observed data volumes and the
solution of the multi-source (in its columns)-frequency
Helmholtz equation over the domain of interest, D represents the detection operator that extracts the simulated data
from time-harmonic solutions at the receiver locations, H
a matrix with the discretized multi-frequency Helmholtz
equation, and B a matrix with the frequency-transformed
source distributions in its columns. In the above optimization problem (from which—after casting Eq. 1 in its unconstrained form—most quasi-Newton type full-waveform
inversion schemes derive), solutions for the unknown velocity model, m, and for the wave equation, U, that minimize
the energy mismatch are pursued. Because Eq. 1 is nonlinear in the model variables collected in the vector m,
solutions of Eq. 1 require multiple solves of the (implicit)
Helmholtz equation. Even after preconditioning (yielding
a complexity for this solver of O(n4 ) in 2-D [7, 6]), this
may prove computationally prohibitive. We address this
problem by using CS [20, 12] to reduce the size of the
seismic data volume through y = RMd where
sub sampler
}|
{ random phase encoder
Ω
RΣ
z
1 ⊗ I ⊗ R1
{
}|
∗
..
îθ
RM =
⊗ I F3 ,
F2 diag e
.
Ω
RΣ
⊗
I
⊗
R
n s′
n s′
z
with F2,3 the 2,3-D Fourier transforms, and θ =
Uniform([0, 2π]) a random phase rotation. The matrices
RΩ and RΣ represent CS-subsampling matrices (see Figure 1) acting along the rows (frequency coordinate) and
columns (source coordinate) of the data volume, respectively. As shown by [12] application of this CS-sampling
65
matrix, RM, to the data is equivalent to applying it to the
source wavefields directly, which turns single-impulsive
sources into a smaller set (n′s ≪ ns with ns the number
of separated single-impulsive sources) of time-harmonic
simultaneous sources that are randomly phase encoded
and that have for each source-experiment a different set
of angular frequencies missing—i.e., there are n′f ≪ nf
(with nf the number of frequencies of fully sampled data)
frequencies non-zero (see Figure 1). This implies that the
sub-sampling operator commutes with the Helmholtz system and this allows us to recast Eq. 1 into the following
reduced form (consisting of fewer frequencies and fewer
right-hand sides):
1
min ky − DUk22
U, m 2
s.t.
H[m]U = B,
(2)
where the underlined quantities are related to the reduced
Helmholtz system.
3.
The time-harmonic Helmholtz system
Since their inception, iterative implicit matrix-free solutions to the time-harmonic Helmholtz equation have been
plagued by lack of numerical convergence for decreasing
mesh sizes and increasing angular frequencies [19]. The
inclusion of deflation—a way to handle small eigenvalues
that lead to slow convergence [7, 6]—can successfully remove this impediment, bringing 2- and 3-D solvers for the
time-harmonic Helmholtz into reach. For a given source
(right-hand side b) and angular frequency ω (:= 2πf , with
f the temporal frequency in Hz), the frequency-domain
wavefield u is computed with a Krylov method that involves the following system of equations:
H[ω]M−1 Qû = b,
u = M−1 Qû,
where H[ω], M, and Q represent the discretized
monochromatic Helmholtz equation, the preconditioner,
and the projection matrices, respectively. As shown by
[8, 9], convergence is guaranteed by defining the preconditioning matrix M in terms of the discretized shifted or
ω2
damped Helmholtz operator M := −∇ · ∇ − c(x)
2 (1 −
√
β î), î = −1, with β > 0. With this preconditioning,
the eigenvalues of HM−1 are clustered into a circle in
the complex plane. By the action of the projector matrix
Q, these eigenvalues move towards unity on the real axis.
These two operations lower the condition number, which
explains the superior performance of this solver.
4.
Source-solution CS-sampling equivalence
Aside from the required number of frequencies, the computational cost of full-wavefield simulation is determined by
the number of sources—i.e., the number of right-hand sides.
In the current simulation paradigm, the number of sources
coincides with the number of single-impulsive source simulations. As prescribed by CS, this number can be reduced by designing a survey that consists of a relatively
small number of simultaneous experiments with simultaneous sources that contain subsets of angular frequencies.
Mathematically, we can accomplish this by applying a CSsampling matrix, RM, to the individual-impulsive sources
collected in the vector s. If we can show that the solution
SAMPTA'09
from this set of “compressed” sources s = RMs, is identical to the compressively sampled solution yielded from
modeling the complete, we are in the position to speed
up our computations. This speed up is the result of a decreased number of experiments and angular frequencies
that are present in the simultaneous source vector. For this
to work, the solution y must be equivalent to the solution
y, obtained by compressively sampling the full solution.
More specifically, we need to demonstrate that the solutions
for the full and compressed systems are equivalent—i.e.,
y = y in
B = D∗
s
|{z}
impulsive sources
HU = B
y = RMDU := RMd
B = D∗ s = D∗
HU = B
y = DU.
RMs
| {z }
sim. sources
Here, H = diag(H[ωi ]) is the block-diagonal discretized
Helmholtz equation for each ωi := 2πi · ∆f, i = 1 · · · nf ,
with nf the number of frequencies and ∆f its sample interval. The adjoint (denoted by ∗ ) of the detection matrix
D injects the individual sources into the multiple righthand sides, B = [b1 b2 · · · bns ], with ns the number of
shots. This detection matrix extracts data at the receiver
positions. Its adjoint inserts data at the co-located source
positions. Each column of U contains the wavefields for
all frequencies induced by the shots located in the columns
of B. Consequently, the full simulation requires the inversion of the block-diagonal system (for all shots), followed by a detection—i.e., we have d = DH−1 B, with
H−1 = diag(H−1 [ωi ]), i = 1 · · · ns . After CS sampling,
this volume is reduced to y = RMd by applying the
flat rectangular CS-sampling matrix RM (defined explicitly in the next section) to the full simulation. Applying
RM directly to the sources s leads to a compressed system H, which after inversion gives y. To illustrate why
y is equivalent to y, consider a compressive sampling of
the solution over frequency by the subsampling matrix
RΩ (for clarity, we removed the orthonormal measurement
matrix). This restriction matrix removes arbitrary rows
from the right-hand side. By virtue of the block-diagonal
structure of our system, we have RΩ H−1 = H−1 RΩ
with H−1 = diag(H−1 [ωi ]), i ⊂ {1 · · · nf }, yielding
RΩ U = H−1 B = U, where B := RΩ B. This means
that frequency subsampling the right-hand side, followed
by solving the system for the corresponding frequencies, is
the same as solving the full system, followed by frequency
subsampling. A similar argument holds when subsampling
the shots (removing arbitrary columns of B). Now, we have
the reduced system RΩ U(RΣ )∗ = H−1 B = U, with B :=
RΩ B(RΣ )∗ . Using Kronecker products, these relations
can be written succinctly as (RΣ ⊗ RΩ )vec (U) = vec (U)
and (RΣ ⊗ RΩ )vec (B) = vec (B) with vec (·) being a
linear operator that maps a matrix into a lexicographicallysorted array. The inversion of HU = B is easier because it
involves only a subset of angular frequencies and simultaneous shots—i.e., {U, B} contain only n′s columns with n′f
frequency components each. Finally, the matrix D extracts
the compressed data from the solution.
66
5.
Recovery by sparsity promotion
Aside from CS sampling the recovery from simultaneous
simulations depends on a sparsifying transform that compresses seismic data, is fast, and reasonably incoherent with
the CS sampling matrix. We accomplish this by defining
the sparsity transform as the Kronecker product between
the 2-D discrete curvelet transform [3] along the sourcereceiver coordinates, and the discrete wavelet transform
along the time coordinate—i.e., S := C ⊗ W with C, W
the curvelet- and wavelet-transform matrices, respectively.
We reconstruct the seismic wavefield by solving the following nonlinear optimization problem
e = arg min kxk1
x
subject to Ax = y,
subsampling ratios. As expected, the SNR for the simple
model is better because of the reduced complexity, whereas
the numbers in Table 1 for the complex model confirm
increasing recovery errors for increasing subsampling ratios. Moreover, the bandwidth limitation of seismic data
explains improved recovery with decreasing frequencyto-shot ratio for a fixed subsampling ratio. Because the
speedup of the solution is roughly proportional to the subsampling ratio, we can conclude that speedups of four to
six times are possible with a minor drop in SNR.
(3)
x
e = S∗ x
e the reconstruction, A := RMS∗ the CS
with d
matrix, and y (= y) the compressively simulated data
(cf. Equation 2-right). Equation 3 is solved by SPGℓ1
[23], a projected-gradient algorithm with root finding.
6.
Computational complexity analysis
According to [19], the cost of the iterative Helmholtz
solver equals nf ns nit O(nd ), typically with nit = O(n)
the number of iterations. For d = 2 and assuming
ns = nf = O(n), this cost becomes O(n5 ). Under
the same assumption, the cost of a time-domain solver
is O(n4 ). The iterative Helmholtz solver can only become
competitive if nit = O(1), yielding an O(n4 ) computational complexity. [7, 6] achieve this by the method explained earlier. Despite this improvement, this figure is
still overly pessimistic for simulations that permit sparse
representations. As long as the simulation cost exceeds
the ℓ1 -recovery cost (cf. Equation 3), CS will improve
on this result. This reduction depends on the cost of A,
which is dominated by the CS-matrix. For naive choices,
such as Gaussian projections, these sampling matrices cost
O(n3 ) for each frequency, which offers no gain. However,
with our choice of fast O(n log n) projections with random convolutions [20], we are able to reduce this cost to
O(n2 log n). Note that these costs are of the same order
as those of calculating the sparsifying transforms. Now,
the leading order cost of the ℓ1 recovery is reduced to
O(n3 log n), which is significantly less than the cost of
solving the full Helmholtz system, especially for large
problems (n → ∞) and for extensions to d = 3.
7.
Example
To illustrate CS-recovery quality, we conduct a series of
experiments for two velocity models, namely the complex
model used in [10], and a simple single-layer model. These
models generate seismic lines that differ in complexity.
During these experiments, we vary the subsampling ratio
and the frequency-to-shot subsampling ratio. All simulations are carried out with a fully parallel Helmholtz solver
for a spread with 128 collocated shots and receivers sampled at a 30 m interval. The time sample interval is 0.004 s
and the source function is a Ricker wavelet with a central
frequency of 10 Hz. By solving Equation 3, we recover
the full simulation for the two datasets. Comparison between the full and compressive simulations in Figure 2
shows remarkable high-fidelity results even for increasing
SAMPTA'09
Subsample ratio
0.25
n′f /n′s
0.15
0.07
recovery error (dB)
2
1
0.5
14.3
18.2
22.2
12.1
14.5
16.5
8.6
10.2
10.7
Speed up (%)
400
670
1420
Table 1: Signal-to-noise ratios based on the complex model,
e 2
dk
SNR = −20 log10 ( kd−
kdk2 ) for reconstructions with the
curvelet-wavelet sparsity transform for different subsample
and frequency-to-shot ratios..
8.
Discussion, extensions, and conclusions
Compressive sampling (CS) can be considered a paradigm
shift because objects of interest that exhibit transformdomain sparsity can be recovered from degrees of subsampling commensurate their sparsity. This new paradigm
can be applied to reduce the computational complexity of
solving PDEs that lie at the heart of PDE-constrained optimization problems. In this paper, we demonstrate that
this principle leads to simultaneous source experiments
that reduce the cost of computer simulations. Similar cost
reductions are possible during actual acquisition in situations where we have control over the physical sources;
such as during acquisition on land [14]. These results are
exciting because CS decouples simulation- and acquisitionrelated costs from the discretization size. Instead, these
costs depend on sparsity. Because the image space is even
sparser after focusing seismic energy, we obtain further improvements when we extend CS principles to promote joint
sparsity through mixed (1, 2)-norm minimization [11].
References
[1] Craig J. Beasley. A new look at marine simultaneous sources. The
Leading Edge, 27(7):914–917, 2008.
[2] A. J. Berkhout. Changing the mindset in seismic data acquisition.
The Leading Edge, 27(7):924–938, 2008.
[3] E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying. Fast discrete
curvelet transforms. Multiscale Modeling and Simulation, 5:861–
899, 2006.
[4] E.J. Candès, J. Romberg, and T. Tao. Stable signal recovery from
incomplete and inaccurate measurements. Communications on Pure
and Applied Mathematics, 59(8):1207–1223, 2006.
[5] D. L. Donoho. Compressed sensing. IEEE Transactions on Information Theory, 52(4):1289–1306, 2006.
[6] Y. A. Erlangga and F. J. Herrmann. An iterative multilevel method
for computing wavefields in frequency-domain seismic inversion.
In SEG Technical Program Expanded Abstracts, volume 27, pages
1957–1960. SEG, November 2008.
[7] Y A Erlangga and R Nabben. On multilevel projection Krylov
method for the preconditioned Helmholtz system. 2007. Submitted
for publication.
67
Figure 1: Compressive sampling with simultaneous sources. (a) Amplitude spectrum for the source signatures emitted by each source as part
of the simultaneous-source experiments. These signatures appear noisy
in the shot-receiver coordinates because of the phase encoding (cf. Equation 1). Observe that the frequency restrictions are different for each
simultaneous source experiment. (b) CS-data after applying the inverse
Fourier transform. Notice the noisy character of the simultaneous-shot
interferences..
[8] Y A Erlangga, C Vuik, and C W Oosterlee. On a class of preconditioners for solving the Helmholtz equation. Applied Numerical
Mathematics, 50:409–425, 2004.
[9] Y A Erlangga, C Vuik, and C W Oosterlee. Comparison of multigrid
and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation. Applied Numerical Mathematics,
56:648–666, 2006.
[10] F. J. Herrmann, U. Boeniger, and D. J. Verschuur. Non-linear
primary-multiple separation with directional curvelet frames. Geophysical Journal International, 170:781–799, 2007.
[11] Felix J. Herrmann. Compressive imaging by wavefield inversion
with group sparsity. Technical Report TR-2009-01, UBC-SLIM,
2009.
[12] Felix J. Herrmann, Yogi A. Erlangga, and Tim T.Y. Lin. Compressive simultaneous full-waveform simulation. TR-2008-09. to appear
in geophysics. 2009.
[13] C.E. Krohn and R. Neelamani. Simultaneous sourcing without
compromise. In Rome 2008, 70th EAGE Conference & Exhibition,
page B008, 2008.
[14] Tim T Y Lin and Felix J Herrmann. Designing simultaneous acquisitions with compressive sensing. In Amsterdam 2009, 71th EAGE
Conference & Exhibition, 2009.
[15] T.T.Y. Lin, E. Lebed, Y. A. Erlangga, and F. J. Herrmann. Interpolating solutions of the helmholtz equation with compressed sensing.
In SEG Technical Program Expanded Abstracts, volume 27, pages
2122–2126. SEG, November 2008.
[16] S. A. Morton and C. C. Ober. Faster shot-record depth migrations using phase encoding. In SEG Technical Program Expanded Abstracts,
volume 17, pages 1131–1134. SEG, 1998.
[17] W. Mulder and R. Plessix. How to choose a subset of frequencies in
frequency-domain finite-difference migration. 158:801–812, 2004.
[18] N. Neelamani, C. Krohn, J. Krebs, M. Deffenbaugh, and J. Romberg.
Efficient seismic forward modeling using simultaneous random
sources and sparsity. In SEG International Exposition and 78th
Annual Meeting, pages 2107–2110, 2008.
[19] C. D. Riyanti, Y. A. Erlangga, R.-E. Plessix, W. A. Mulder, C. Vuik,
and C. Oosterlee. A new iterative solver for the time-harmonic wave
equation. Geophysics, 71(5):E57–E63, 2006.
[20] J. Romberg. Compressive sensing by random convolution. submitted, 2008.
[21] L. A. Romero, D. C. Ghiglia, C. C. Ober, and S. A. Morton.
Phase encoding of shot records in prestack migration. Geophysics,
65(2):426–436, 2000.
[22] Laurent Sirgue and R. Gerhard Pratt. Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies.
Geophysics, 69(1):231–248, 2004.
[23] E. van den Berg and M. P. Friedlander. Probing the pareto frontier
for basis pursuit solutions. SIAM Journal on Scientific Computing,
31(2):890–912, 2008.
[24] Denes Vigh and E. William Starr. 3d prestack plane-wave, fullwaveform inversion. Geophysics, 73(5):VE135–VE144, 2008.
SAMPTA'09
Figure 2: Comparison between conventional and compressive simulations in for simple and complex velocity models. (a) Crossing-planes
view of the seismic line for the simple model. (b) The same for the
complex model. (c). Recovered simulation (with a SNR of 28.1 dB) for
the simple model from 25 % of the samples with the ℓ1 -solver running to
convergence. (d) The same but for the complex model now with a SNR
of 18.2 dB..
68
Computable Fourier Conditions for Alias-Free
Sampling and Critical Sampling
Yue M. Lu (1)(2) , Minh N. Do (2) and Richard S. Laugesen (2)
(1) Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Lausanne, Switzerland
(2) University of Illinois at Urbana-Champaign, Urbana IL 61801, USA
yue.lu@epfl.ch, minhdo@illinois.edu, Laugesen@illinois.edu
Abstract:
We propose a Fourier analytical approach to the problems
of alias-free sampling and critical sampling. Central to this
approach are two Fourier conditions linking the above sampling criteria with the Fourier transform of the indicator
function defined on the underlying frequency support. We
present several examples to demonstrate the usefulness of
the proposed Fourier conditions in the design of critically
sampled multidimensional filter banks. In particular, we
show that it is impossible to implement any cone-shaped frequency partitioning by a nonredundant filter bank, except for
the 2-D case.
1
Introduction
The search for alias-free sampling lattices for a given frequency support, and in particular for those lattices achieving
minimum sampling densities, is a fundamental issue in various signal processing applications that involve the design
of efficient acquisition schemes for bandlimited signals. As
a special case of alias-free sampling, the concept of critical sampling also plays an important role in the theory and
design of critically sampled (a.k.a. maximally decimated)
multidimensional filter banks [9].
The study of alias-free (and critical) sampling lattices is a
classical problem [8, 4]. So far, most existing work in the
literature approaches the problem from a geometrical perspective: The primary tools employed include the theories
from Minkowski’s work [2], as well as various geometrical
intuitions and heuristics.
In this paper, we propose a Fourier analytical approach to the
problems of alias-free sampling and critical sampling. Central to this approach are two Fourier conditions linking the
above sampling criteria with the Fourier transform of the indicator function defined on the underlying frequency support
(see Theorem 1 and Proposition 2). An important feature of
the proposed conditions is that they open the door to purely
analytical and computational solutions to the sampling lattice selection problem.
The rest of the paper is organized as follows. In Section 2,
we briefly review some relevant concepts on sampling bandlimited signals. We present in Section 3 a novel condition
linking the alias-free sampling (as well as critical sampling)
with the
Fourier transform of the indicator function defined
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on the given frequency support. In Section 4, we present an
application of the proposed Fourier conditions in the design
of multidimensional nonredundant filter banks. We conclude
the paper in Section 5. The material in this paper was presented in part in [5] and [7]. As a novel aspect, we present in
this paper a different proof for Theorem 1, which provides
important new insights into this key result.
Notation: The Fourier transform of a function f (ω) defined
on RN is defined by
Z
b
f (x) =
f (ω) e−2πj x·ω dω.
(1)
RN
Calligraphic letters, such as D, represent bounded and
open frequency domains in RN , with m(D) denoting the
Lebesgue measure (i.e. volume) of D. Given a nonsingular
matrix M and a vector τ , we use M (D + τ ) to represent
the set of points of the form M (ω + τ ) for ω ∈ D. Finally,
we denote by 1D (ω) the indicator function of the domain
D, i.e., 1D (ω) = 1 if ω ∈ D and 1D (ω) = 0 otherwise.
2 Background
In multidimensional multirate signal processing, the sampling operations are usually defined on lattices, each of
which can be generated by an N × N nonsingular matrix
M as
def
ΛM = {M n : n ∈ ZN }.
(2)
We denote by Λ∗M the corresponding reciprocal lattice
(a.k.a. polar lattice), defined as
def
Λ∗M = {M −T ℓ : ℓ ∈ ZN }
(3)
In the rest of the paper, when it is clear from the context what
the generating matrix is, we will drop the subscripts in ΛM
and Λ∗M , and use Λ and Λ∗ for simplicity.
Let f (x) be a continuous-domain signal, whose Fourier
transform is bandlimited to a bounded open set D ⊂ RN .
def
The discrete-time Fourier transform of the samples s[n] =
f (M n) is supported in [9]
!
[
T
S =M
(D + k) .
(4)
k∈Λ∗
For appropriately chosen sampling lattices, the aliasing components in (4) do not overlap with the baseband frequency
69
support D. In this important case, we can fully recover
the original continuous-domain signal f (x) by applying an
ideal interpolation filter spectrally supported on D to the discrete samples s[n].
Definition 1 We say a frequency support D allows an aliasfree M -fold sampling, if different shifted copies of D in (4)
are disjoint, i.e.,
D ∩ (D + k) = ∅ for all k ∈ Λ∗ \ {0} .
(5)
Furthermore, we say D can be critically sampled by M , if
in addition to the alias-free condition in (5), the union of the
shifted copies also covers the entire spectrum, i.e.,
[
(D + k) = RN , up to a set of measure zero. (6)
Proof Consider the autocorrelation function
Z
RD (ω) = 1D (τ ) 1D (τ − ω) dτ .
Clearly, RD (ω) ≥ 0 for all ω. Meanwhile, we can verify
that supp RD (ω) = (D − D). Thus, we can apply Lemma 1
and obtain that, D allows an M -fold alias-free sampling if
and only if
Z
X
RD (k) = RD (0) = 1D (τ ) dτ = m(D).
k∈Λ∗
Applying the Poisson summation formula to the above
equality (see Appendix A of [7] for a justification of the
pointwise equality), we have
k∈Λ∗
m(D) =
The focus of this work is to present two Fourier analytical
conditions for alias-free sampling and critical sampling. Our
discussions will be based on the following geometrical argument [2], which can be easily verified from (5).
Proposition 1 The alias-free sampling condition in (5) is
equivalent to requiring
Λ∗ ∩ (D − D) = {0} ,
(7)
def
where D − D = {ω − τ : ω, τ ∈ D} is the Minkowski sum
of the open set D and its negative −D.
3
Fourier Analytical Conditions
In this section, we study the problems of alias-free sampling
and critical sampling with Fourier techniques. The key observation is a link between the alias-free sampling condition
and the Fourier transform of the indicator function 1D (ω)
defined on the frequency support D.
31
Alias-Free Sampling
Lemma 1 Let D be a frequency region, and f (ω) a positive
function supported on (D − D), i.e., f (ω) > 0 for ω ∈
(D − D) and f (ω) = 0 otherwise. Then D allows an M fold alias-free sampling if and only if
X
f (k) = f (0).
(8)
k∈Λ∗
Proof By construction, (8) holds if and only if Λ∗ ∩ (D −
D) = {0}. Applying Proposition 1, we are done.
Theorem 1 A frequency region D allows an M -fold aliasfree sampling if and only if
X
b D (n)|2 = m(D),
|M |
|1
(9)
n∈Λ
b D (x) is the Fourier transform of 1D (ω), and |M |
where 1
is the absolute value of the determinant of M .
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X
RD (k) = |M |
X
n∈Λ
k∈Λ∗
bD (n).
R
(10)
From the definition of RD (ω), its Fourier transform is
bD (x) = |1
b D (x)|2 . Substituting this formula into (10),
R
we are done.
32 Critical Sampling
Here we focus on the special case of critical sampling, and
begin by mentioning, without proof, a standard result:
Lemma 2 A frequency support D can be critically sampled by a sampling matrix M if and only if M is an aliasfree sampling matrix for D with sampling density 1/|M| =
m(D).
Proposition 2 A frequency support D can be critically sampled by a matrix M if and only if
b D (0) = m(D) =
1
1
|M |
b D (n) = 0
and 1
(11)
for all n ∈ Λ \ {0}.
Proof Suppose (11) holds. Then it follows that
X
b D (n)|2 = |1
b D (0)|2 = m(D) ,
|1
|M |
n∈Λ
and hence from Theorem 1, M is an alias-free sampling
1
matrix for D. Meanwhile, since m(D) = |M
| , we can apply
Lemma 2 to conclude that D is critically sampled by M . By
reversing the above line of reasoning, we can also show the
necessity of (11).
Remark: The result of Proposition 2 is previously known in
various disciplines. In approximation theory, the condition
(11) is often called the interpolation property (see, for example, [4]). The usefulness of this condition in the context of
lattice tiling was first pointed out by Kolountzakis and Lagarias [3] and applied to investigate the tiling of various high
dimensional shapes.
70
33
Computational Aspects
The Fourier conditions proposed in Theorem 1 and Proposition 2 can lead to practical computational algorithms for testing alias-free and critical sampling. Here, we briefly comment on two important computational aspects in applying
the proposed conditions.
First, as a prerequisite to using the proposed Fourier condibD (x). This evalutions, we must know the expression for 1
ation can be a cumbersome task if we need to do the derivation by hand for each given D. However, when the frequency
regions D are arbitrary polygonal and polyhedral domains,
b D (x) via the
we can obtain the closed-form expressions for 1
divergence theorem [1, 7].
Another potential issue in practical implementations is that
the Fourier conditions in (9) and (11) both involve an infinite
number of lattice points. We show in [7] that the infinite sum
in (9) can be well-approximated by a truncated finite sum.
Moreover, with high probability, we actually only need to
evaluate the Fourier transform on a very small number of
points in a lattice (e.g. 4 points in 2-D) in order to show
aliasing occurs, thus ruling out the lattice.
4
Application: Filter Bank Design
In this section we present an application of Proposition 2
in the design of multidimensional critically sampled filter
banks.
41
Frequency Partitioning of Critically Sampled
Filter Banks
Consider a general multidimensional filter bank, where each
channel contains a subband filter and a sampling operator.
As an important step in filter bank design, we need to specify the ideal passband support of each subband filter, all of
which form a partitioning of the frequency spectrum.
Not every possible frequency partitioning can be used for
filter bank implementation though. In particular, if we want
to have a nonredundant filter bank, then the ideal passband
support of each subband filter must be critically sampled by
the sampling matrix in that channel. Consequently, whenever given a possible frequency partitioning, we must first
perform a “reality check” of seeing whether the above condition is met, before proceeding to actual filter design.
The critical sampling condition is commonly verified geometrically (i.e. by drawing figures). Although intuitive and
straightforward, this geometrical approach becomes cumbersome when the shape of the passband support is complicated, or when we work in 3-D and higher dimensional
cases. Applying the result of Proposition 2, we propose in
the following a computational procedure, which can systematically check and determine the critical sampling matrices
of a given polytope region. Notice that the algorithm only
searches among integer matrices, since the filter banks considered here operate on discrete-time signals.
Procedure 1 Let D be a given polytope-shaped frequency
support
region.
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ω2
0
1
ω3
ω2
0,3
0,1
0,0
5
3
4
4
3
5
2
1
(a)
1,3
0,2
0
2
1,2
1,1
1,0
3,3
3,2
3,1
3,0
1
ω1
2,3
2,2
2,1
2,0
ω2
ω1
ω1
1
3,0
3,1
3,2
0
0
3,3
2,0
2,1
2,2
2,3
(b)
1,0
1,1
1,2
1,3
0,0
0,1
0,2
0,3
(c)
Figure 1: The ideal frequency partitioning of several filter
banks. (a) A directional filter bank which decomposes the
frequency cell (− 12 , 12 ]2 into 6 subbands. (b) A directional
multiresolution frequency partitioning. (c) A 3-D directional frequency decomposition with pyramid-shaped passband supports.
1. Calculate δ = 1/m(D). From (11), any matrix M that
can critically-sample D must satisfy |M | = δ. If δ
is not an integer, then stop the procedure, since in this
case it is impossible for D to be critically sampled by
any integer matrix.
b D (x).
2. Construct a closed-form formula [7] for 1
3. Based on the Hermite normal form, construct an exhaustive list of matrices of determinant δ, each corresponding to a distinct sampling lattice [7].
4. For every matrix M in the above list, test the following
condition
b D (M n) = 0 for all n ∈ ZN \{0} with knk∞ ≤ r,
1
(12)
where r is a large positive integer.
5. Present all the matrices in the list that satisfy (12). If
there is no such matrix, then D cannot be critically
sampled by any integer matrix.
To be clear, the expression (12) is a necessary condition
for D to be critically sampled by M . It is not sufficient
since we only check for integer points within a finite radius r, and so in principle, even if M satisfies (12) for all
b D (M n) 6= 0 for some n
knk∞ ≤ r, it might happen that 1
with knk∞ > r. However, by choosing r sufficiently large,
we can gain confidence in the validity of the original infinite condition (11) as required in Proposition 2. We leave
the quantitative analysis of this approximation to [7]. In the
following examples, we choose r = 10000.
Example 1 Figure 1(a) presents the frequency decomposition of a directional filter bank (DFB). Applying the algorithm in Procedure 1, we can easily verify that this frequency
decomposition can be critically sampled. The corresponding
sampling matrices, denoted by M k for the kth subband, are
6 3
M0 = M1 = M2 =
.
0 1
M 3 , M 4 and M 5 can be inferred by symmetry.
Example 2 We show in Figure 1(b) a directional and multiresolution decomposition of the 2-D frequency spectrum.
Applying Procedure 1 confirms that such a frequency partitioning can be critically sampled as well. The sampling
71
matrices for two representative subbands (marked as dark
regions in the figure) are
4 0
8 4
M0 =
and M 1 =
.
0 4
0 4
Example 3 Figure 1(c) shows an extension of the original
2-D DFB to the 3-D case [6]. Applying Procedure 1, we
find that the 3-D frequency partitioning shown in Figure 1(c)
cannot be critically sampled; in other words, redundancy is
unavoidable for a 3-D DFB.
42
Critical Sampling of General Cone-Shaped
Frequency Regions in Higher Dimensions
The result in Example 3 can be generalized to higher dimensions, and to cases where the subbands take different
directional shapes. As an application of the Fourier condition in Proposition 2, we show here a much more general
statement: it is impossible to implement any cone-shaped
frequency partitioning by a nonredundant filter bank, except
for the 2-D case.
We consider the following ideal subband supports in N -D:
D = {ω : a ≤ |ωN | ≤ b, (ω1 , . . . , ωN −1 ) ∈ ωN B}, (13)
N −1
where B is some bounded set in R
. Geometrically, D
takes the form of a two-sided cone in RN , truncated by hyperplanes |ωN | = a and |ωN | = b, where 0 ≤ a < b. The
“base” region B in (13) is the intersection between the cone
and the hyperplane ωN = 1.
The formulation in (13) is flexible enough to characterize,
up to a rotation, any directional subband shown in Figure 1. For example, the 3-D pyramid-shaped subband (1, 1)
in Figure 1(c) can be presented by a = 0, b = 12 , and
B = [− 21 , 0]2 . However, the class of frequency shapes that
can be described by (13) is far beyond those shown in Figure 1, since the formulation (13) allows for arbitrary configuration of the cross section heights a and b (not necessarily
the dyadic decomposition as in Figure 1(b)) and arbitrary
shape for the base B (not necessarily lines or squares).
Lemma 3 If a frequency support D can be critically sampled by an integer matrix M , then
bD (|M | n) = 0, for all n ∈ ZN \ {0}.
1
(14)
Proof It is easy to verify that, for any integer matrix M , the
vector |M | n belongs to the lattice Λ generated by M . The
condition (14) then follows from (11) in Proposition 2.
Theorem 2 For arbitrary choice of 0 ≤ a < b and the
base shape B, the frequency domain support D given in (13)
cannot be critically sampled by any integer matrix in N dimensions, N ≥ 3.
Remark: For 2-D, we established the positive result in Examples 1 and 2.
Proof We argue by contradiction. Suppose for N ≥ 3, and
for some particular choices of 0 ≤ a < b and B, the corresponding frequency region D in (13) can be critically sampled by
an integer matrix M . It then follows from (14) in
SAMPTA'09
Lemma 3 that
b D (0, . . . , 0, |M | n) = 0, for all n ∈ Z \ {0}.
1
(15)
From the definition of D, we have
bD (0, . . . , 0, x)
1
Z
Z
−2πj x ωN
=
dωN e
1 dω1 . . . dωN −1
a≤|ωN |≤b
ωN B
Z
=
e−2πj x ω m(ω B) dω
a≤|ω|≤b
Z
=
e−2πj x ω |ω|N −1 m(B) dω
a≤|ω|≤b
= 2 m(B)
Z
b
ω N −1 cos(2πx ω) dω.
a
After a change of variable, we can now rewrite (15) as
R 2π|M |b N −1
ω
cos(n ω) dω = 0, for all n ∈ Z \ {0}, which
2π|M |a
is impossible when N ≥ 3 by Appendix C of [7].
5 Conclusions
By linking the alias-free (and critical) sampling of a given
frequency support region with the Fourier transform of the
indicator function, we presented two simple yet powerful
conditions for checking alias-free sampling and critical sampling. We demonstrated the usefulness of the proposed conditions in the design of multidimensional critically sampled
filter banks. As an interesting result, we show that it is impossible to construct a nonredundant directional filter bank
with a general cone-shaped frequency decomposition, except for the 2-D case.
References:
[1] L. Brandolini, L. Colzani, and G. Travaglini. Average decay of
Fourier transforms and integer points in polyhedra. Ark. Mat.,
35:253–275, 1997.
[2] P. M. Gruber and C. G. Lekkerkerker. Geometry of Numbers. Elsevier Science Publishers, Amsterdam, second edition,
1987.
[3] M. N. Kolountzakis and J. C. Lagarias. Tilings of the line by
translates of a function. Duke Math. J., 82(3):653–678, 1996.
[4] H. R. Künsch, E. Agrell, and F. A. Hamprecht. Optimal lattices
for sampling. IEEE Trans. Inf. Theory, 51(2):634–47, Feb.
2005.
[5] Y. M. Lu and M. N. Do. Finding optimal integral sampling
lattices for a given frequency support in multidimensions. In
Proc. IEEE Int. Conf. on Image Proc., San Antonio, USA,
2007.
[6] Y. M. Lu and M. N. Do. Multidimensional directional filter
banks and surfacelets. IEEE Trans. Image Process., 16(4):918–
931, April 2007.
[7] Y. M. Lu, M. N. Do, and R. S. Laugesen. A computable Fourier
condition generating alias-free sampling lattices. IEEE Trans.
Signal Process., to appear, 2009.
[8] D. P. Peterson and D. Middleton. Sampling and reconstruction
of wavenumber-limited functions in N -dimensional Euclidean
spaces. Inform. Contr., 5:279–323, 1962.
[9] P. P. Vaidyanathan. Multirate Systems and Filter Banks.
Prentice-Hall, Englewood Cliffs, NJ, 1993.
72
Analysis of Singularities and Edge Detection
using the Shearlet Transform
Glenn Easley (1) , Kanghui Guo (2) , and Demetrio Labate (3)
(1) System Planning Corporation 1000 Wilson Boulevard, Arlington, VA 22209, USA.
(2) Missouri State University, Springfield, MO 65804, USA.
(3) University of Houston, 651 Phillip G Hoffman, Houston, TX 77204-3008, USA.
geasley@sysplan.com, KanghuiGuo@MissouriState.edu, dlabate@math.uh.edu
Abstract:
The continuous curvelet and shearlet transforms have recently been shown to be much more effective than the traditional wavelet transform in dealing with the set of discontinuities of functions and distributions. In particular,
the continuous shearlet transform has the ability to provide
a very precise geometrical characterization of general discontinuity curves occurring in images. In this paper, we
show that these properties are useful to design improved
algorithms for the analysis and detection of edges.
1. Introduction
One of the most useful properties of the wavelet transform
is its ability to deal very efficiently with the discontinuities of functions and distributions. Consider, for example,
a function f on R2 which is smooth except for a discontinuity at x0 ∈ R2 , and let Wψ f (a, t) be the continuous
wavelet transform of f . This is defined as the mapping
Z
¡
¢
Wψ f (a, t) = a−1
f (x) ψ a−1 (x − t) dx,
R2
where a > 0, t ∈ R2 and ψ ∈ L2 (R2 ) is an appropriate
well-localized function. Then Wψ f (a, t) decays rapidly
as a → 0 everywhere, unless t is near x0 [5]. Hence,
the wavelet transform is able to signal the location of the
singularity of f through its asymptotic decay at fine scales.
It was recently shown that certain “directional” extensions
of the wavelet transform have the ability to provide a
much finer description of the set of singularities of a function. Namely, the recently introduced curvelet and shearlet
transforms are able to identify not only the location of singularities of a function, but also the orientation of discontinuity curves. In particular, using the continuous shearlet
transform, one can precisely characterize the geometrical
information of general discontinuity curves, including discontinuity curves which contain irregularities such as corner and junction points.
In this paper, we show that one can take advantage of the
properties of the shearlet transform to design improved algorithms for the analysis and detection of edges in images.
Indeed, multiscale techniques based on wavelets have a
history of successful applications in the study of edges.
With respect to traditional wavelets, the shearlet framework has the ability to capture directly the information
about edge orientation and this is useful to improve the
SAMPTA'09
robustness of edge detection algorithms in the presence of
noise.
The paper is organized as follows. In Section 2. we recall the definition of the shearlet transform and its main
results concerning the analysis of edges. In Section 3.
we present some representative numerical experiments of
edge detection, comparing the shearlet approach against
wavelets and other standard edge detection techniques.
2.
The Shearlet Transform
For a > 0 s ∈ R and t ∈ R2 , let Mas be the matrices
µ
√ ¶
a − as
Mas =
√
0
a
and, corresponding to those, let ψast (x)
=
1
2
−1
(x − t)), where ψ ∈ L2 (R³
). It is
| det Mas |− 2 ψ(Mas
a 0 ´
useful to notice that Mas = Bs Aa , where Aa =
√
0 a
³ 1 −s ´
and Bs =
. Hence to each matrix Mas are
0 1
associated two distinct actions: an anisotropic dilation
produced by the matrix Aa and a shearing produced by
the non-expansive matrix Bs .
For f ∈ L2 (R2 ), the continuous shearlet transform is defined as the mapping
f → SHψ f (a, s, t) = hf, ψast i, a > 0, s ∈ R, t ∈ R2 .
The generating function ψ is chosen to be a well localized function satisfying appropriate admissibility conditions [7, 4], so that each f ∈ L2 (R2 ) satisfies the generalized Calderòn reproducing formula:
Z Z ∞Z ∞
da
hf, ψast i ψast 3 ds dt.
f=
a
R2 −∞ 0
The significance of the shearlet representation is that any
function f is broken up with respect to well-localized analyzing elements defined not only at various scales and locations, as in the traditional multiscale approach, but also
at various orientations associated with the shearing parameter s. Figure 1 shows the frequency support of the shearlet analyzing functions ψ̂ast for some values of s and a.
Thanks to this directional multiscale decomposition, the
continuous shearlet transform is able to precisely capture
the geometry of edges through its asymptotic decay at fine
73
ξ2
(a, s) =
otherwise, if s = s0 corresponds to one of the normal
directions of Γ at t then
1
(a, s) = ( 32
, 1)
❅
❅
❘
3
0 < lim a− 4 |SHψ B(a, s0 , t)| < ∞.
+
a→0
( 14 , 0)
❅
❅
❘
❅
ξ1
✻
1
, 0)
(a, s) = ( 32
Thus, the continuous shearlet transform has rapid asymptotic decay, as a → 0, everywhere except for locations t
on the edges and orientations s which are normal to the
edges. We refer to [7, 4, 3] for additional detail, including
a more precise description of the behavior at the corner
points. We also refer to [1] for some similar (even if more
restricted) results based on the curvelet transform.
2.1
Figure 1: Frequency support of same representative shearlet analyzing functions ψ̂ast .
scales (a → 0). To precisely describe these properties, let
us introduce the following model of images. S
L
Let Ω = [0, 1]2 and consider the partition Ω = n=1 Ωn ∪
Γ, where:
1. each “object” Ωn , for n = 1, . . . , L, is a connected
open set;
SL
2. the set of edges of Ω is given by Γ = n=1 ∂Ω Ωn ,
where each boundary ∂Ω Ωn is a piecewise smooth
curve of finite length.
Hence, we consider the space of images u ∈ I(Ω) of the
form
u(x) =
L
X
un (x) χΩn (x) for x ∈ Ω\Γ
n=1
where, for each n = 1, . . . , L, un ∈ C01 (Ω) has bounded
partial derivatives, and the sets Ωn are pairwise disjoint
in measure. We have the following result, which is a significant refinement with respect to the simple detection of
singularities obtained using traditional wavelets.
Theorem 2.1. Let f ∈ I(Ω).
(i) If t ∈
/ Γ, then, for each N ∈ N
lim a−N SHψ f (a, s, t) = 0.
a→0+
(ii) If t ∈ Γ is a regular point and s does not correspond
to the normal direction of Γ at t then
lim a−N SHψ B(a, s, t) = 0,
a→0+
for all N > 0;
otherwise, if s = s0 corresponds to the normal direction of Γ at t then
3
0 < lim a− 4 |SHψ B(a, s0 , t)| < ∞.
+
a→0
(iii) If t ∈ Γ is a corner point and s does not correspond
to any of the normal directions of Γ at t, then
9
lim+ a− 4 |SHψ B(a, s, t)| < ∞;
a→0
SAMPTA'09
Lipschitz regularity
The notion of Lipschitz regularity is a method to quantitatively describe the local regularity of functions and distributions.
Given α ≥ 0, a function f is Lipschitz α at x0 ∈ R2 if
there exists a positive constant K and a polynomial px0 of
degree m = ⌊α⌋ such that, for all x in a neighborhood of
x0 :
α
|f (x) − px0 (x)| ≤ K |x − x0 | .
(1)
A function f is uniformly Lipschitz α over an open set
Ω ⊂ R2 if there exists a constant K > 0, independent of
x0 , such that the above inequality holds for all x0 ∈ Ω.
If f is uniformly Lipschitz α > m in a neighborhood of
x0 , then f is necessarily m times differentiable at x0 . Also
notice that if 0 ≤ α < 1, then px0 = f (x0 ) and condition
(1) becomes
α
|f (x) − f (x0 )| ≤ K |x − x0 | .
If f is Lipschitz α with α < 1 at x0 , then f is not differentiable at x0 . The closer the Lipschitz exponent is to 0,
the more “singular” the function is. If f is bounded but
discontinuous at x0 , then it is Lipschitz 0 at x0 , indicating
the presence of an edge.
Also recall that if f (x) is Lipschitz α, then its primitive
g(x) is Lipschitz α + 1 (the converse however is not true;
that is, if a function is Lipschitz α at x0 , then its derivative
need not be Lipschitz α - 1 at the same point). This observation explains the following definition which extends the
concept of Lipschitz regularity to distributions.
Let α be a real number. A tempered distribution f is uniformly Lipschitz α on Ω ⊂ R2 if its primitive is uniformly
Lipschitz α + 1 on Ω ⊂ R2 .
It follows that a distribution may have a negative Lipschitz
exponent. For example, one can show that if f is a Dirac
delta distribution centered at x0 , then f is Lipschitz -1 at
x0 . We refer to [8] and to the references indicated there
for more details.
The function ψ satisfies the property that for each n ∈ N,
there exists a constant cn > 0 such that
|ψ(x)| ≤ cn (1 + |x|)−n
for all x ∈ R2 (for details, seeR [4], p. 26). As a conRsequence, weα obtain kψk1 = R2 |ψ(x)| dx < ∞, and
|ψ(x)||x| dx < ∞.
R2
The following result (whose proof is reported in the
appendix) is an adaptation of a similar theorem about the
74
continuous wavelet transform due to Jaffard
[6]. If we
R
assume ψ has n vanishing moments, i.e. tk ψ(t) dt = 0
for all k = 0, . . . , n − 1, we would need to add the
condition α ≤ n. However, the general construction
of ψ implies that ψ has an infinite number of vanishing
moments. Thus this assumption is unnecessary.
Theorem 2.2. If f ∈ L2 (R2 ) is Lipschitz α > 0 at t0 ,
then there exists a constant C > 0 such that, for all a < 1,
¯ 1
¯´
³
1
3
¯
¯
|SHψ f (a, s, t)| ≤ C a 2 (α+ 2 ) 1 + ¯a− 2 (t − t0 )¯ .
The theorem can be extended to the case where f is a
distribution. In addition, the estimation of the decay of
the shearlet transform of the Dirac delta and other distributions was computed in [7]. These results show that, for
locations t corresponding to delta-type singularities, the
shearlet transform has a very different behavior from edge
points. In fact, the amplitude of |SHψ f (a, s, t0 )| grows
1
like O(a− 4 ) as a → 0. Similarly, for spike singularities,
one can show that the amplitude of the shearlet transform
increases at fine scales. This shows that classification of
points by their Lipschitz regularity is important as it can
be used to distinguish true edge points from points corresponding to noise. This principle was already exploited,
for example, in [8].
3. Shearlet-based Edge Detection
Taking advantage of the theoretical observations reported
above, a discrete version of the shearlet transform was developed and applied to the purpose of locating and identifying edges in images. Because of space limitations,
we will limit ourselves to presenting a few numerical
demonstrations. A detailed account of the discrete shearlet transform and shearlet-based edge detection algorithms
is found in [2, 10].
Figures 2 and 3 compare a shearlet-based edge detection
routine against a wavelet-based routine using a consistent
set of predetermined default parameters. For a base-line
comparison against standard routines, we also used the
Sobel and Prewitt methods using their default parameters. The results highlight the superior performance of the
shearlet-based method. To assess the performance of the
edge detector, we have given the value of the Pratt’s Figure of Merit (FOM), which is a fidelity measure ranging
from 0 to 1, with 1 indicating a perfect edge detector [9].
Acknowledgments DL acknowledges partial support
from NSF DMS 0604561 and DMS (Career) 0746778.
4.
Appendix: Proof of Theorem 2.2.
Proof of Theorem 2.2. Since f is Lipschitz α at t0 , there
is a polynomial pt0 (x) and a constant K > 0 such that
Since SHψ pt0 (a, s, t) = 0, then
≤
≤
=
≤
≤
≤
|SHψ f (a, s, t)|
Z
−1
−3/4
|ψ(A−1
a
a Bs (x − t))| |f (x) − pt0 (x)| dx
R2
Z
−1
α
|ψ(A−1
K a−3/4
a Bs (x − t))| |x − t0 | dx
2
Z R
3/4
|ψ(y)| |t + Bs Aa y − t0 |α dy
Ka
R2
µ
Z
α 3/4
|ψ(y)| |y|α dy
kBs kα kAa kα
K2 a
R2
¶
Z
α
|ψ(y)| |t − t0 | dy
+
R2
µ
Z
α 3/4
K2 a
C(s)α aα/2
|ψ(y)| |y|α dy
R2
¶
Z
α
|ψ(y)| dy
+ |t − t0 |
R2
´
³
1
3
C a 2 (α+ 2 ) 1 + |a−1/2 (t − t0 )|α .
Here we have used the fact that kAa k = a1/2 , i.e. the
largest eigenvalue of the matrix Aa . Similarly kBs k is the
largest eigenvalue of the matrix Bs , which is 1.
References:
[1] E. J. Candès and D. L. Donoho, “Continuous
curvelet transform: I. Resolution of the wavefront
set”, Appl. Comput. Harmon. Anal., Vol. 19, pp.
162–197, 2005. 162–197.
[2] G. Easley, D. Labate, and W-Q. Lim “Sparse Directional Image Representations using the Discrete
Shearlet Transform”, Appl. Comput. Harmon. Anal.
Vol. 25, pp. 25–46, 2008.
[3] K. Guo and D. Labate, “Characterization and analysis of edges using the Continuous Shearlet Transform”, preprint, 2008
[4] K. Guo, D. Labate and W. Lim, “Edge analysis and
identification using the continuous shearlet transform”, to appear in Appl. Comput. Harmon. Anal..
[5] M. Holschneider, Wavelets. Analysis tool, Oxford
University Press, Oxford, 1995.
[6] S. Jaffard “Pointwise smoothness, two-localization
and wavelet coefficients”, Publicacions Mathematique, Vol. 35, pp. 155–168, 1991.
[7] G. Kutyniok and D. Labate, “Resolution of the
Wavefront Set using Continuous Shearlets”, Trans.
Am. Math. Soc., Vol. 361 pp. 2719-2754, 2009.
[8] S. Mallat and W. L. Hwang, Singularity detection
and processing with wavelets, IEEE Trans. Inf. Theory, vol. 38, no. 2, 617-643, Mar. 1992.
[9] W.K. Pratt, Digital Image Processing, Wiley Interscience Publications, 1978.
[10] S. Yi, D. Labate, G. R. Easley, and H. Krim, “A
Shearlet Approach to Edge Analysis and Detection”,
to appear in IEEE Trans. Image processing, 2008.
|f (x) − pt0 (x)| ≤ K |x − t0 |α .
SAMPTA'09
75
Figure 2: Results of edge detection methods. From top left, clockwise: Original image, noisy image (PSNR=28.10 dB),
Sobel result (FOM=0.24), shearlet result (FOM=0.44), wavelet result (FOM=0.29), and Prewitt result (FOM=0.23).
Figure 3: Results of edge detection methods. From top left, clockwise: Original image, noisy image (PSNR=24.58 dB),
Sobel result (FOM=0.15), shearlet result (FOM=0.45), wavelet result (FOM=0.27), and Prewitt result (FOM=0.15).
SAMPTA'09
76
Discrete Shearlet Transform : New Multiscale
Directional Image Representation
Wang-Q Lim
Department of Mathematics, University of Osnabrück, Osnabrück, Germany
wlim@mathematik.uni-osnabrueck.de
Abstract:
It is now widely acknowledged that analyzing the intrinsic
geometrical features of an underlying image is essentially
needed in image processing. In order to achieve this, several directional image representation schemes have been
proposed. In this report, we develop the discrete shearlet
transform (DST) which provides efficient multiscale directional representation. We also show that the implementation of the transform is built in the discrete framework
based on a multiresolution analysis. We further assess the
performance of the DST in image denoising and approximation applications. In image approximation, our adaptive approximation scheme using the DST significantly
outperforms the wavelet transform (up to 3.0dB) and other
competing transforms. Also, in image denoising, the DST
compares favorably with other existing methods in the literature.
1. Introduction
Sharp image transitions or singularities such as edges are
expensive to represent and intergrating the geometric regularity in the image representation is a key challenge to
improve state of the art applications to image compression and denoising. To exploit the anisotropic regularity
of a surface along edges, the basis must include elongated
functions that are nearly parallel to the edges.
Several image representations have been proposed to capture geometric image regularity. They include curvelets
[1], contourlets [2] and bandelets [3]. In particular, the
construction of curvelets is not built directly in the discrete
domain and they do not provide a multiresolution representation of the geometry. In consequence, the implementation and the mathematical analysis are more involved
and less efficient. Contourlets are bases constructed with
elongated basis functions using a combination of a multiscale and a directional filter bank. However, contourlets
have less clear directional features than curvelets, which
leads to artifacts in denoising and compression. Bandelets are bases adapted to the function that is represented.
Asymptotically, the resulting bandelets are regular functions with compact support, which is not the case for contourlets. However, in order to find bases adapted to an
image, the bandelet transform searches for the optimal geometry. For an image of N pixels, the complexity of this
SAMPTA'09
best
bandelet basis algorithm is O(N 3/2 ) which requires
extensive computation [3].
Recently, a new representation scheme has been introduced [4]. These so called shearlets are frame elements
which yield (nearly) optimally sparse representations [5].
This new representation system is based on a simple and
rigorous mathematical framework which not only provides a more flexible theoretical tool for the geometric
representation of multidimensional data, but is also more
natural for implementations. As a result, the shearlet approach can be associated to a multiresolution analysis [4].
However constructions proposed in [4] do not provide
compactly supported shearlets and this property is essentially needed especially in image processing applications.
In fact, in order to capture local singularities in images efficiently, basis functions need to be well localized in the
spatial domain.
In this report, we construct compactly supported shearlets
and show that there is a multiresolution analysis associated
with this construction. Based on this, we develop the fast
discrete shearlet transform (DST) which provides efficient
directional representations.
2.
Shearlets
A family of vectors {ϕn }n∈Γ constitutes a frame for a
Hilbert space H if there exist two positive constants A, B
such that for each f ∈ H we have
X
|hf, ϕn i|2 ≤ Bkf k2 .
Akf k2 ≤
n∈Γ
In the event that A = B, the frame is said to be tight.
Let us next introduce some notations that we will use
throughout this paper. For f ∈ L2 (Rd ), the Fourier transform of f is defined by
Z
ˆ
f (x)e−2πix·ω dx.
f (ω) =
Rd
Also, for t ∈ Rd and A ∈ GLd (R), we define the following unitary operators:
Tt (f )(x) = f (x − t)
and
1
DA (f )(x) = |A|− 2 f (A−1 x).
Finally, for q ∈ ( 21 , 1] and a > 1, we define
¶
µ
µ q
¶
a
0
1 1
and B0 =
A0 =
1
0 1
0 a2
(1)
77
and
A1 =
µ 1
a2
0
0
aq
¶
and B1 =
µ
1
1
¶
0
.
1
(2)
We are now ready to define a shearlet frame as follows.
For c ∈ R+ , ψ01 , . . . , ψ0L , ψ11 , . . . , ψ1L ∈ L2 (R2 ) and φ ∈
L2 (R2 ), we define
i,0
Ψ0c = {ψjkm
: j, k ∈ Z, m ∈ Z2 , i = 1, . . . , L},
i,1
Ψ1c = {ψjkm
: j, k ∈ Z, m ∈ Z2 , i = 1, . . . , L},
Figure 1: Examples of shearlets in the spatial domain. The
i,0
top row illustrates shearlet functions ψjk0
associated with
matrices A0 and B0 in (1). The bottom row shows shearlet
i,1
functions ψjk0
associated with matrices A1 and B1 in (2)..
and
Ψ2c = {Tcm φ : m ∈ Z2 }
i,0
∪{ψjkm
: j ≥ 0, −2j ≤ k ≤ 2j , m ∈ Z2 , i = 1, . . . , L}
i,1
∪{ψjkm
: j ≥ 0, −2j ≤ k ≤ 2j , m ∈ Z2 , i = 1, . . . , L}
Theorem 3..1. [7] For i = 1, . . . , L, we define
ψ0i (x1 , x2 ) = γ i (x1 )θ(x2 ) such that
where
′
i,ℓ
ψjkm
= DA−j B −k Tcm ψℓi
ℓ
ℓ
|ω1 |α
|γ̂ (ω1 )| ≤ K1
(1 + |ω1 |2 )γ ′ /2
i
(3)
for ℓ = 0, 1, m ∈ Z2 , i = 1, . . . , L and j, k ∈ Z. If Ψpc
i,ℓ
is a frame for L2 (R2 ), then we call the functions ψjkm
in
the system Ψpc shearlets.
i,ℓ
Observe that each element ψjkm
in Ψpc is obtained by applying an anisotropic scaling matrix Aℓ and a shear matrix Bℓ to fixed generating functions ψℓi . This implies that
the system Ψpc can provide window functions which can
be elongated along arbitrary directions. Therefore, the
geometrical structures of singularities in images can be
efficiently represented and analyzed using those window
functions. In fact, it was shown that 2-dimensional piecewise smooth functions with C 2 -singularities can be approximated with nearly optimal approximation rate using
shearlets. We refer to [5] for details. Furthermore, one
can show that shearlets can completely analyze the singular structures of piecewise smooth images [6]. In fact,
this property of shearlets is useful especially in signal and
image processing, since singularities and irregular structures carry essential information in a signal. For example,
discontinuities in the intensity of an image indicate the
presence of edges. Figure 1 displays examples of shearlets which can be elongated along arbitrary direction in
the spatial domain.
3. Construction of Shearlets
In this section, we will introduce some useful sufficient
conditions to construct compactly supported shearlets.
Using these conditions, we will show that the system Ψpc
can be generated by simple separable functions associated
with a multiresolution analysis. Furthermore, this leads to
the fast DST, and we will discuss this in the next section.
We first discuss sufficient conditions for the existence
of compactly supported shearlets.³ For this,´ let α >
1
max (1, (1 − p)γ) and γ > max α+1
be fixed
p , 1−p
positive numbers for 0 < p < 1. We choose α′ , γ ′ > 0
such that α′ ≥ α + γ and γ ′ ≥ α′ − α + γ. Then we obtain
SAMPTA'09
the
following results [7].
and
|θ̂(ω1 )| ≤ K2 (1 + |ω1 |2 )−γ
′
/2
.
If
ess
inf
|ω1 |≤1/2
|θ̂(ω1 )|2 ≥ K3 > 0
(4)
and
ess
inf
a−q ≤|ω1 |≤1
L
X
i=1
|γ̂ i (ω1 )|2 ≥ K4 > 0,
(5)
then there exists c0 > 0 such that Ψ0c is a frame for L2 (R2 )
for all c ≤ c0 .
Observe that the functions ψ01 , . . . , ψ0L are separable functions, and the one-dimensional scaling function θ and
wavelets γ i can be chosen with sufficient vanishing moments in this case.
We now show some concrete examples of compactly supported shearlets using Theorem 3.1. Assume that a = 4
and q = 1 in (1) and (2). Let us consider a box spline [1]
of order m defined as follows.
³ sin πω ´m+1
1
e−iǫω1 ,
θ̂m (ω1 ) =
πω1
where ǫ = 1 if m is even, and ǫ = 0 if m is odd. Obviously, we have the following two scaling equation:
θ̂m (2ω1 ) = m0 (ω1 )θ̂m (ω1 )
and
m0 (ω1 ) = (cos πω1 )m+1 e−iǫπω1 .
Let α′ and γ ′ be positive real numbers as in Theorem 3.1.
We now define
´ℓ
√ ³
ψ̂01 (ω) = (i)ℓ 2 sin πω1 θ̂m (ω1 )θ̂m (ω2 )
and
³
ω1
πω1 ´ℓ
ψ̂02 (ω) = (i)ℓ sin
θ̂m ( )θ̂m (ω2 ),
2
2
78
where ℓ ≥ α′ and m + 1 ≥ γ ′ . Then, by Theorem 3.1, ψ01
and ψ02 generate a frame Ψ0c for c ≤ c0 with some c0 > 0.
There are infinitely many possible choices for ℓ and m.
For example, one can choose ℓ = 9 and m = 11.
Define
φ(x1 , x2 ) = θm (x1 )θm (x2 ),
´ℓ
√ ³
ψ̂11 (ω) = (i)ℓ 2 sin πω2 θ̂m (ω2 )θ̂m (ω1 )
and
³
πω2 ´ℓ
ω2
ψ̂12 (ω) = (i)ℓ sin
θ̂m ( )θ̂m (ω1 ).
2
2
Figure 2: Examples of anisotropic discrete wavelet decomposition: (a) Anisotropic discrete wavelet decomposition by W, (b) Anisotropic discrete wavelet decomposif
tion by W.
Then similar arguments show that ψ11 and ψ12 generate a
frame Ψ1c for c ≤ c0 with some c0 > 0. Furthermore, the
functions φ, ψℓi for ℓ = 0, 1 and i = 1, 2 generate a frame
Ψ2c with c ≤ c0 for some c0 > 0.
where fJ (n) = h f, D2−J I2 Tn φ i. For h = 0, 1, let us
define maps Dhk,j : ℓ2 (Z2 ) → ℓ2 (Z2 ) by
X k,j
(Dhk,j x)(d) =
dh (d, m)x(m)
4. Discrete Shearlet Transform
where dk,j
h (d, m) = h D
In the previous section, we constructed compactly supported shearlets generated by separable functions associated with a multiresolution analysis. In this section, we
will show that this multiresolution analysis leads to the
i,ℓ
fast DST which computes hf, ψjkm
i. To be more specific,
we let a = 4 and q = 1 in (1) and (2). For notational
convenience, we let n = (n1 , n2 ), m = (m1 , m2 ), d =
(d1 , d2 ) ∈ Z2 and I2 be a 2 by 2 identity matrix.
Let θ ∈ L2 (R) be a compactly supported function such
that {θ(· − n1 ) : n1 ∈ Z} is an orthonormal sequence and
X
√
(6)
h(n1 ) 2θ(2x1 − n1 ).
θ(x1 ) =
n1 ∈Z
Define
γ(x1 ) =
X
n1 ∈Z
√
g(n1 ) 2θ(2x1 − n1 )
(7)
such that γ has sufficient vanishing moments and the pair
of the filters h and g is a pair of conjugate mirror filters.
We assume that γ and θ satisfy decay conditions (4) and
(5) in Theorem 3.1. We also define
φ(x1 , x2 ) = θ(x1 )θ(x2 ),
ψℓ1 (x1 , x2 ) = γ(xℓ+1 )θ(x2−ℓ )
and
(8)
xℓ+1
)θ(x2−ℓ )
(9)
2
for ℓ = 0, 1. Then Theorem 3.1 can be easily generalized
to show that the functions ψ01 , ψ02 , ψ11 , ψ12 and φ generate a
shearlet frame Ψ2c with c < c0 for some c0 > 0.
Let J be a positive odd integer. Based on a multiresolution
analysis associated with the two-scale equation (6), we can
now easily derive a fast algorithm for computing shearlet
i,ℓ
coefficients hf, ψjkm
i for ℓ = 0, 1,j = 1, . . . , J−1
2 , and
−2j ≤ k ≤ 2j as follows.
First, assume that
X
fJ (n)D2−J I2 Tn φ
(10)
SAMPTA'09 f =
1
ψℓ2 (x1 , x2 ) = 2− 2 γ(
n∈Z2
m∈Z2
Also we define
k/2j
Bh
X
H(ω1 ) =
Tm φ, Td φ i and x ∈ ℓ(Z2 ).
h(n1 )e−2iπω1
n1
and
G(ω1 ) =
X
g(n1 )e−2iπω1 .
n1
hj , gj0
Finally, we let
and gj1 be the Fourier coefficients of
QJ−j−1 ³ k ´
for J − j > 0,
H 2 ω2
H
(ω
)
=
j 2
k=0
QJ−2j−2
0
k
Gj (ω1 ) = k=0
H(2 ω1 )G(2J−2j−1 ω1 ),
Q
G1 (ω ) = J−2j−1 H(2k ω )G(2J−2j ω ),
1
1
1
j
k=0
(11)
respectively. Then we obtain
1,0
hf, ψjkm
i = (((D0k,j fJ ) ∗r hj )↓2J−j ∗c g 0j )↓2J−2j (m),
hf, ψ 2,0 i = (((Dk,j f ) ∗ h ) J−j ∗ g 1 ) J−2j+1 (m),
J
r j ↓2
c j ↓2
0
jkm
k,j
1,1
0
hf, ψjkm
J−j ∗r g )↓2J−2j (m),
h
)
i
=
(((D
f
)
∗
j
J
c
↓2
j
1
2,1
k,j
hf, ψjkm i = (((D1 fJ ) ∗c hj )↓2J−j ∗r g 1j )↓2J−2j+1 (m),
(12)
where ∗c and ∗r are convolutions along the vertical and
horizontal axes respectively, ↓ 2j is the downsampling by
2j and h(n) = h(−n) for given filter coefficients h(n).
From (12), we observe that the shearlet transform
i,ℓ
hf, ψjkm
i is the application of the shear transformation
D k/2j to f ∈ L2 (R2 ) followed by the wavelet transform
Bℓ
associated with anisotropic scaling matrix Aℓ . In this case,
applying Dℓk,j to fJ ∈ ℓ2 (Z2 ) corresponds to applying
the shear transform D k/2j in the discrete domain. Thus
Bℓ
we simply replace the operator Dℓk,j by the discrete shear
ℓ
transform Pk,j
for fJ ∈ ℓ2 (Z2 ), where we define the dis0
1
crete shear transforms Pk,j
and Pk,j
as follows:
(
¡
¢
0
(Pk,j
fJ )(n) = fJ n1 + ⌊(k/2j )n2 ⌋, n2 ,
¡
¢
(13)
1
(Pk,j
fJ )(n) = fJ n1 , n2 + ⌊(k/2j )n1 ⌋ .
1
0
Let M be a fixed positive integer. Since Pk,j
and Pk,j
79
2
are unitary operators on ℓ(Z ), we can extend the shearlet
transform defined in (12) to a linear transform S consisting
of finitely many orthogonal transforms SkM and S̃kM where
0
f 1 (fJ )
SkM (fJ ) = WPk,M
(fJ ) and S̃kM (fJ ) = WP
k,M
f are the wavelet transform associated with
and W and W
an anisotropic sampling matrices A0 and A1 , respectively.
f we refer to [7].
For the precise definitions of W and W,
In this case, the linear transform S, which we call DST, is
defined by
M
M
M
M
S = (S−2
M , . . . , S2M , S̃−2M , . . . , S̃2M )
for a given M ∈ Z+ . Notice that redundancy of the DST
is K = 2M +2 + 2 and the DST merely requires O(KN )
operations for an image of N pixels. It is obvious that the
inverse DST is simply the adjoint of S with normalization.
5. Image Approximation Using DST
In this section, we present some results of the DST in image compression applications. In this case, we use adaptive image representation using the DST. The main idea of
this is similar to the matching pursuit introduced by Mallat
and Zhong [8]. The matching pursuit selects vectors one
by one from a given basis dictionary at each iteration step.
On the other hand, our approximation scheme searches the
optimal directional index k0 at each iteration step so that
corresponding the orthogonal transform SkM0 or S̃kM0 provides an optimal nonlinear approximation with P nonzero
terms among all possible 2M +2 + 2 orthogonal transforms
in S. For a detailed description of this algorithm, we refer to [7]. For numerical tests, we compare the performance of the DST to other transforms such as the discrete
biorthogonal CDF 9/7 wavelet transform (DWT)[9] and
contourlet transform (CT)[2] in image compression (see
Figure 3). We used only 2 directions (horizontal and vertical) and 4 level decomposition for our DST. In this case,
our numerical tests indicate that only a few iterations (15) can give significant improvement over other transforms
and computing time is comparable to the wavelet transform. For more results, we refer to [8].
Figure 3: Compression results of ’Barbara’ image of
size 512 × 512: The image is reconstructed from 5024
most significant coefficients. Top left: Zoomed original
image, Top right: Zoomed image reconstructed by the
DWT (PSNR = 25.11), Bottom left: Zoomed image reconstructed by the CT (PSNR = 25.88), Bottom right:
Zoomed image reconstructed by the DST with only 1 iteration step (PSNR = 26.73).
References:
[1]
[2]
[3]
[4]
6. Conclusion
[5]
We have constructed compactly supported shearlet systems which can provide efficient directional image representations. We also have developed the fast discrete implementation of shearlets called the DST. This algorithm
consists of applying the shear transforms in the discrete
domain followed by the anisotropic wavelet transforms.
Applications of our proposed transform in image approximation and denoising were studied. In image approximation, the results obtained with our adaptive image representation using the DST are significantly superior to those
of other transforms such as the DWT and CT both visually
and with respect to PSNR.
In denoising, we studied the performance of the DST coupled with a (partially) translation invariant hard tresholding estimator. Our results indicate that the DST consistently outperforms other competing transforms. For deSAMPTA'09
tailed
numerical results, we refer to [7].
[6]
[7]
[8]
[9]
E. Candes and D. Donoho, ”New tight frames of
curvelets and optimal representations of objects
with piecewise C 2 singularities,” Commun. Pure
Appl. Math, vol. 57, no. 2, pp. 219-266, Feb. 2004.
M. Do and M. Vetterli, ”The contourlet transform:
An efficient directional multiresolution image representation,” IEEE Trans. Image Process., vol. 14,
no. 12, pp. 2091-2106, Dec. 2005.
G. Peyre and S. Mallat, ”Discrete Bandelets with
Geometric Orthogonal Filters,” Proceedings of
ICIP, Sept. 2005.
D. Labate, W. Lim, G. Kutyniok and G. Weiss
”Sparse Multidimensional Representation using
Shearlets”, Proc. of SPIE conference on Wavelet
Applications in Signal and Image Processing XI,
San Diego, USA, 2005.
K. Guo and D. Labate, ”Optimally Sparse Multidimensional Representation using Shearlets,”, SIAM
J Math. Anal., 39 pp. 298-318, 2007.
K. Guo, D. Labate and W. Lim, ”Edge Analysis and
identification using the Continuous Shearlet Transform”, to appear in Appl. Comput. Harmon. Anal.
W. Lim, ”Compactly Supported Shearlet Frames
and Their Applications”, submitted.
S. Mallat and S. Zhang, ”Matching Pursuits With
Time-Frequency Dictionaries,” IEEE Trans. Signal
Process., pp. 3397-3415, Dec. 1993.
A. Cohen, I. Daubechies and J. Feauveau,
”Biorthogonal bases of compactly supported
wavelets,” Commun. on Pure and Appl. Math.,
45:485-560, 1992.
80
Image Approximation by Adaptive Tetrolet
Transform
Jens Krommweh
Department of Mathematics, University of Duisburg-Essen, Campus Duisburg, 47048 Duisburg, Germany.
jens.krommweh@uni-due.de
Abstract:
In order to get an efficient image representation we introduce a new adaptive Haar wavelet transform, called Tetrolet Transform. Tetrolets are Haar-type wavelets whose
supports are tetrominoes which are shapes made by connecting four equal-sized squares. The corresponding filter
bank algorithm is simple but enormously effective. Numerical results show the strong efficiency of the tetrolet
transform for image compression.
1.
Introduction
The main task in every kind of image processing is finding an efficient image representation that characterizes the
significant image features in a compact form. In the last
years a lot of methods have been proposed to improve
the treatment with orientated geometric image structures.
Curvelets [1], contourlets [2], shearlets [5], and directionlets [10] are wavelet systems with more directional sensitivity than classical tensor product wavelets.
Instead of choosing a priori a basis or a frame one may
adapt the function system depending on the local image
structures. Wedgelets [3] and bandelets [7] stand for this
second class of image representation schemes which is a
wide field of further research. Very recent approaches are
the grouplets [8] or the EPWT [9] which are based on an
averaging in adaptive neighborhoods of data points.
In [6] we have introduced a new adaptive algorithm whose
underlying idea is similar to the idea of digital wedgelets
where Haar functions on wedge partitions are considered.
We divide the image into 4 × 4 blocks, then we determine in each block a tetromino partition which is adapted
to the image geometry in this block. Tetrominoes are
shapes made by connecting four equal-sized squares, each
joined together with at least one other square along an
edge. On these geometric shapes we define Haar-type
wavelets, called tetrolets, which form a local orthonormal
basis. The main advantage of Haar-type wavelets is the
lack of pseudo-Gibbs artifacts. The corresponding filter
bank algorithm decomposes an image into a compact representation.
The tetrolet transform is also very efficient for compression of real data arrays.
SAMPTA'09
2. The Adaptive Tetrolet Transform
2.1 Definitions and Notations
Let be I = {(i, j) : i, j = 0, . . . , N − 1} ⊂ Z2 the
index set of a digital image a = (a[i, j])(i,j)∈I with N =
2J , J ∈ N. We determine a 4-neighborhood of an index
(i, j) ∈ I by N4 (i, j) := {(i − 1, j), (i + 1, j), (i, j −
1), (i, j + 1)}. An index that lies at the boundary has three
neighbors, an index at the vertex of the image has two
neighbors.
A set E = {I0 , . . . , Ir }, r ∈ N, of subsets Iν ⊂ I is
a!disjoint partition of I if Iν ∩ Iµ = ∅ for ν '= µ and
r
ν=0 Iν = I.
In this paper we consider disjoint partitions of the index
set I that satisfy two conditions for all Iν :
1. each subset Iν contains four indices, i.e. #Iν = 4,
2. every index of Iν has a neighbor in Iν , i.e. ∀(i, j) ∈
Iν ∃(i" , j " ) ∈ Iν : (i" , j " ) ∈ N4 (i, j).
We call such subsets Iν tetromino, since the tiling prob2
lem of the square [0, N ) by shapes called tetrominoes is
a well-known problem being closely related to our partitions of the index set I = {0, 1, . . . , N − 1}2 . We shortly
introduce this tetromino tiling problem in the next subsection.
2.2 Tilings by Tetrominoes
Tetrominoes were introduced by Golomb in [4]. They are
shapes formed from a union of four unit squares, each connected by edges, not merely at their corners. The tiling
problem with tetrominoes became popular through the famous computer game classic ’Tetris’. Disregarding rotations and reflections there are five different shapes, the so
called free tetrominoes, see Figure 1.
2
It is clear that every square [0, N ) can be covered by
tetrominoes if and only if N is even. But the number of
different coverings explodes with increasing N . There are
117 solutions for disjoint covering of a 4 × 4 board with
four tetrominoes. As represented in Figure 2, we have 22
Figure 1: The five free tetrominoes.
81
In other words, we first divide the index set I of an im2
age a into N16 squares Qi,j and then we consider the admissible tetromino partitions there. Among the 117 solutions we compute an optimal partition in each image block
such that the wavelet coefficients defined on the tetrominoes have minimal l1 -norm.
3. Detailed Description of the Algorithm
Figure 2: The 22 fundamental forms tiling a 4 × 4 board.
Regarding additionally rotations and reflections there are
117 solutions.
fundamental configurations (disregarding rotations and reflections). One solution (first line) is unaltered by rotations
and reflections, four solutions (second line) give a second
version applying the isometries. Seven forms can occur in
four orientations (third line), and ten asymmetric cases in
eight directions (last line).
2.3
The Idea of Tetrolets
In the two-dimensional classical Haar case, the low-pass
filter and the high-pass filters are just given by the averaging sum and the averaging differences of each four pixel
values which are arranged in a 2 × 2 square, i.e., with
Ii,j = {(2i, 2j), (2i + 1, 2j), (2i, 2j + 1), (2i + 1, 2j + 1)}
for i, j = 0, 1, . . . , N2 − 1, we have a dyadic partition
E = {I0,0 , . . . , I N −1, N −1 } of the image index set I. Let
2
2
L be a bijective mapping which maps the four pixel pairs
(i, j) to the scalar set {0, 1, 2, 3}, that means it brings the
pixels into a unique order. Then we can determine the lowN
−1
2
pass part a1 = (a1 [i, j])i,j=0
as well as the three high-pass
N
2
−1
parts wl1 = (wl1 [i, j])i,j=0 for l = 1, 2, 3 with
a1 [i, j] =
"
"[0, L(i" , j " )] a[i" , j " ]
"
"[l, L(i" , j " )] a[i" , j " ],
(1)
(2)
where the coefficients "[l, m], l, m = 0, . . . , 3, are entries
from the Haar wavelet transform matrix
1
W :=
1
1
1
1 1 1 −1 −1
=
.
2 1 −1 1 −1
1
−1
−1
(3)
Going into detail our main attention shall be turned to step
2 of the algorithm where the adaptivity comes into play.
−1
We start with the input image a0 = (a[i, j])N
i,j=0 with
J
N = 2 , J ∈ N. In the rth-level, r = 1, . . . , J − 1,
we apply the following computations.
1. Divide the low-pass image ar−1 into blocks Qi,j of
size 4 × 4, i, j = 0, . . . , 4Nr − 1.
2. In each block Qi,j we compute analogously to (1)
and (2) the pixel averages for every admissible tetromino covering c = 1, . . . , 117 by
"
ar,(c) [s] =
"[0, L(m, n)] ar−1 [m, n],
as well as the three high-pass parts for l = 1, 2, 3
"
r,(c)
"[l, L(m, n)] ar−1 [m, n],
wl [s] =
(c)
(m,n)∈Is
s = 0, . . . , 3, where the coefficients are given in (3)
and L is the mapping mentioned above. Then we
choose the covering c∗ such that the l1 -norm of the
tetrolet coefficients becomes minimal
1
Obviously, the fixed blocking by the dyadic squares Ii,j
is very inefficient because the local structures of an image
are disregarded. Our idea is, to allow more general partitions such that the local image geometry is taken into account. Namely, we use tetromino partitions. As described
in the previous subsection we shall restrict us to 4 × 4
blocks. This leads to a third condition for the desired disjoint partition E of the index set I introduced in Section
2.1:
3. Each 4 × 4 square Qi,j := {4i, . . . , 4i + 3} ×
{4j, . . . , 4j + 3}, i, j = 0, 1, . . . , N4 − 1, is covered
by four subsets (tetrominoes) I0 , . . . , I3 .
SAMPTA'09
Table 1: Adaptive tetrolet decomposition algorithm.
(c)
(i! ,j ! )∈Ii,j
("[l, m])3l,m=0
Adaptive Tetrolet Decomposition Algorithm
−1
J
Input: Image a = (a[i, j])N
i,j=0 with N = 2 , J ∈ N.
1. Divide the image into 4 × 4 blocks.
2. Find in each block the sparsest tetrolet representation.
3. Rearrange the low- and high-pass coefficients of
each block into a 2 × 2 block.
4. Store the tetrolet coefficients (high-pass part).
5. Apply step 1 to 4 to the low-pass image.
Output: Decomposed image ã.
(m,n)∈Is
(i! ,j ! )∈Ii,j
wl1 [i, j] =
The rough structure of the tetrolet filter bank algorithm is
described in Table 1.
c∗ = arg min
c
3 "
3
"
r,(c)
|wl
[s]|.
(4)
l=1 s=0
Hence, for every block Qi,j we get an optimal tetro∗
r,(c∗ )
r,(c∗ )
r,(c∗ )
let decomposition [ar,(c ) , w1
, w2
, w3
].
By doing this, the local structure of the image block
is adapted. The best configuration c∗ is a covering whose tetrominoes do not intersect an important
structure like an edge in the image ar−1 . Because the
tetrolet coefficients become as minimal as possible
a sparse image representation will be obtained. We
have to store for each block Qi,j which covering c∗
has been chosen, since this information is necessary
for reconstruction.
82
3. In order to be able to apply further levels of the tetrolet decomposition algorithm, we rearrange the entries
∗
r,(c∗ )
of the vectors ar,(c ) and wl
into 2 × 2 matrices,
) r,(c∗ )
*
∗
a
[0] ar,(c ) [2]
r
∗
∗
,
a|Q =
i,j
ar,(c ) [1] ar,(c ) [3]
and in the same way wl|r Q
i,j
for l = 1, 2, 3.
4. After finding a sparse representation in every block
Qi,j for i, j = 0, . . . , 4Nr − 1, we store (as usually
done) the low-pass matrix ar and the high-pass matrices wlr , l = 1, 2, 3, replacing the low-pass image
ar−1 by the matrix
) r
*
a
w2r
.
w1r w3r
After a suitable number of decomposition steps, one can
apply a shrinkage to the tetrolet coefficients in order to get
a sparse image representation.
4.
An Orthonormal Basis of Tetrolets
We describe the discrete basis functions which correspond
to the above algorithm. Remember that the digital image
a = (a[i, j])(i,j)∈I is a subset of l2 (Z2 ). For any tetromino Iν of I we define the discrete functions
+
1/2, (m, n) ∈ Iν ,
φIν [m, n] :=
0, else,
+
"[l, L(m, n)], (m, n) ∈ Iν ,
ψIl ν [m, n] :=
0,
else.
Due to the underlying tetromino support, we call φIν and
ψIl ν tetrolets. As a straightforward consequence of the orthogonality of the standard 2D Haar basis functions and
the disjoint partition of the discrete space by the tetromino
supports, we have the following essential statement.
Theorem 1 For every admissible covering {I0 , I1 , I2 , I3 }
of a 4 × 4 square Q ⊂ Z2 the tetrolet system
{φIν : ν = 0, 1, 2, 3} ∪ {ψIl ν : ν = 0, 1, 2, 3; l = 1, 2, 3}
is an orthonormal basis of l2 (Q).
5.
Cost of Adaptivity: Modified Tetrolet
Transform
We will address the costs of storing additional adaptivity
information. Our observations will lead to some relaxed
versions of the tetrolet transform in order to reduce these
costs.
It is well known that a vector of length N and with entropy E can be stored with N · E bits. Hence, the entropy
describes the required bits per pixel (bpp) and is an appropriate measure for the quality of compression.
In the following, we propose three methods of entropy reduction in order to reduce the adaptivity costs. An application of these modified transforms as well as of combinations of them is given in the last section.
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The simplest approach of entropy reduction is reduction
of the symbol alphabet. The tetrolet transform uses the
alphabet {1, . . . , 117} for the chosen covering in each image block. If we restrict ourselves to 16 essential configurations that feature different directions we considerably
reduce the entropy as well as the computation time.
A second approach to reduce the entropy is to change the
distribution of the symbols. Relaxing the tetrolet transform we could ensure that only very few tilings are preferred. Hence, we allow the choice of an almost optimal
covering c∗ in (4) in order to get a tiling which is already
frequently chosen. More precisely, we replace (4) by the
two steps:
1. Find the set of almost optimal configurations that satisfy
3 "
3
"
l=1 s=0
r,(c)
|wl
[s]| ! min
c
3 "
3
"
r,(c)
|wl
[s]| + θ
l=1 s=0
with a predetermined tolerance parameter θ.
2. Among these tilings choose the covering c which is
chosen most frequently in the previous image blocks.
Using an appropriate relaxing parameter θ, we achieve a
satisfactory balance between low entropy (low adaptivity
costs) and minimal tetrolet coefficients.
The third method also reduces the entropy by optimization
of the tiling distribution. After an application of an edge
detector we use the classical Haar wavelet transform inside flat image regions. In the image blocks that contain
edges we make use of the strong adaptivity of the proposed
tetrolet transform.
More details of the modified versions can be found in [6].
6. Numerical Experiments
We apply a complete wavelet decomposition of an image
and use a shrinkage with global hard-thresholding.
The detail ’monarch’ image in Figure 3 shows the enormous efficiency in handling with several directional edges
due to the high adaptivity. It can be well noticed that the
tetrolet transformation gives excellent results for piecewise constant images. Though the tetrolets are not continuous the approximation of the ’cameraman’ image in
Figure 4 illustrates that even for natural images the tetrolet
filter bank outperforms the tensor product wavelets with
the biorthogonal 9-7 filter bank, since no pseudo-Gibbs
phenomena occur. This confirms the fact already noticed
with wedgelets [3] and bandelets [7]: While nonadaptive
methods need smooth wavelets for excellent results, well
constructed adaptive methods need not. See [6] for more
numerical examples.
Considering the adaptivity costs we compare the standard
tetrolet transform with its modified versions. Of course,
reduction of adaptivity cost produces a loss of approximation quality. Hence, a satisfactory balance is necessary.
For a rough estimation of the complete storage costs of the
compressed image with N 2 pixels we apply a simplified
scheme
costf ull = costW + costP + costA ,
83
10
10
20
20
30
30
40
40
50
50
60
50
50
100
100
150
150
200
200
60
250
10
20
30
40
50
60
10
10
20
30
40
50
250
50
60
100
150
200
250
50
100
150
200
250
50
100
150
200
250
10
20
20
30
30
40
40
50
50
60
60
50
50
100
100
150
150
200
200
250
250
50
10
20
30
40
50
60
10
20
30
40
50
100
150
200
250
60
Figure 3: Approximation with 256 coefficients. (a) Input, (b) classical Haar, PSNR 18.98, (c) Biorthogonal 9-7,
PSNR 21.78, (d) Tetrolets, PSNR 24.43.
Figure 4: Approximation with 2048 coefficients. (a) Input, (b) classical Haar, PSNR 25.47, (c) Biorthogonal 9-7,
PSNR 27.26, (d) Tetrolets, PSNR 29.17.
References:
where costW = 16 · M/N 2 are the costs in bpp of storing
M non-zero wavelet coefficients with 16 bits. The term
costP gives the cost for coding the position of these M co2
M
N 2 −M
M
log2 ( N N−M
). The
efficients by − N
2 log2 ( N 2 ) −
2
N2
third component appearing only with the tetrolet transform
contains the cost of adaptivity, costA = E · R/N 2 , for R
adaptivity values and the entropy E previously discussed.
Table 2 presents some results for the monarch detail image
(Fig. 3) where different versions of the tetrolet transform
are compared with the tensor product wavelet transformation regarding to quality and storage costs. We have tried
to balance the modified tetrolet transform such that the full
costs are in the same scale as with the 9-7 filter. For the
relaxed versions we have used the parameter θ = 25.
Tensor Haar
Tensor 9-7 filter
Tetrolet
Tetro 16
Tetro rel
Tetro edge
Tetro 16 edge rel
coeff
300
300
256
256
256
256
256
PSNR
19.58
22.62
24.43
23.56
24.51
24.24
23.48
entropy
0.53
0.30
0.32
0.43
0.21
costf ull
1.55
1.55
1.86
1.64
1.66
1.77
1.55
Table 2: Comparison between tensor wavelet transforms
and the different versions of the tetrolet transform regarding quality (PSNR) and storage cost (costf ull in bpp).
7.
Acknowledgments
The research is funded by the project PL 170/11-1 of the
Deutsche Forschungsgemeinschaft (DFG). This is gratefully acknowledged.
SAMPTA'09
[1] E.J. Candes and D.L. Donoho. New tight frames
of curvelets and optimal representations of objects
with piecewise C 2 singularities. Communications
on Pure and Applied Mathematics, 57(2):219–266,
2004.
[2] M.N. Do and M. Vetterli. The contourlet transform:
an efficient directional multiresolution image representation. IEEE Transactions on Image Processing,
14(12):2091–2106, 2005.
[3] D.L. Donoho. Wedgelets: Nearly-minimax estimation of edges. Annals of Statistics, 27(3):859–897,
1999.
[4] S.W. Golomb. Polyominoes. Princeton University
Press, 1994.
[5] K. Guo and D. Labate. Optimally sparse multidimensional representation using shearlets. SIAM Journal
on Mathematical Analysis, 39(1):298–318, 2007.
[6] Jens Krommweh. Tetrolet transform: A new adaptive Haar wavelet algorithm for sparse image representation. 2009.
[7] E. Le Pennec and S. Mallat. Sparse geometric image
representations with bandelets. IEEE Transactions
on Image Processing, 14(4):423–438, 2005.
[8] S. Mallat. Geometrical grouplets. Applied and
Compu-tational Harmonic Analysis, 26(2):161–180,
2009.
[9] G. Plonka. Easy path wavelet transform: A new
adaptive wavelet transform for sparse representation
of two-dimensional data. Multiscale Modeling and
Simulation, 7(3):1474–1496, 2009.
[10] V. Velisavljevic, B. Beferull-Lozano, M. Vetterli, and
P.L. Dragotti. Directionlets: Anisotropic multidirectional representation with separable filtering. IEEE
Transactions on Image Processing, 15(7):1916–
1933, 2006.
84
Geometric Wavelets for Image Processing:
Metric Curvature of Wavelets
Emil Saucan (1) , Chen Sagiv (2) and Eli Appleboim (3)
(1) Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel.
(2) SagivTech Ltd. Israel.
(3) Electrical Engineering Department, Technion - Israel Institute of Technology, Haifa 32000, Israel.
semil@tx.technion.ac.il, chensagivron@gmail.com, eliap@ee.technion.ac.il
Abstract:
We introduce a semi-discrete version of the FinslerHaantjes metric curvature to define curvature for wavelets
and show that scale and curvature play similar roles with
respect to image presentation and analysis. More precisely, we show that there is an inverse relationship between local scale and local curvature in images. This allows us to use curvature as a geometrically motivated automatic scale selection in signal and image processing, this
being an incipient bridging of the gap between the methods employed in Computer Graphics and Image Processing.
A natural extension to ridgelets and curvelets is also given.
Further directions of study, in particular the development
of a curvature transform and the study of its link with
wavelet and the scale transforms are also suggested.
1. Introduction
The versatility and adaptability of wavelets for a variety
of tasks in Image Processing and related fields is too well
established in the scientific community, and the bibliography pertaining to it is far too extensive, to even begin to
review it here.
We do, however, stress the fact that the multiresolution
property of wavelets has been already applied in determining the curvature of planar curves [1] and to the intelligence and reconstruction of meshed surfaces (see, e.g.
[18], [26], amongst many others). Moreover, the intimate
relation between scale and differentiability in natural images has also been stressed [10].
We have presented in [24] and other related works, an
extension of Shannon’s Sampling Theorem when images
are viewed as higher dimensional objects (i.e. manifolds),
rather than 2-dimensional signals. More precisely, our approach to Shannon’s Sampling Theorem is based on sampling the graph of the signal, considered as a manifold,
rather than sampling of the domain of the signal, as is customary in both theoretical and applied signal and image
processing, motivated by the framework of harmonic analysis. The main tool for proving our geometric sampling
theorem, resides in the confluence of Differential Topology and Differential Geometry. More precisely, we consider piecewise-linear (P L) approximations of the manifold, where the geometric feature (i.e. curvature) determines the proper size and shape-ration of the simplices of
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the constructed triangulation.
Naturally, the question is whether the implementation of
the geometric sampling scheme is feasible. We do not
address here the purely geometric aspects, that would
be highly relevant in Computer Graphics implementation
(besides, these were partly addressed in [24]). Instead,
we focus on the far more important and popular Image
Processing tool of wavelets. The versatility and adaptability of wavelets to a variety of tasks in Image Processing
and related fields is too well established in the scientific
community, and the bibliography pertaining to it is far to
extensive, to even begin to review it here.
Unfortunately, in contrast to Computer Graphics experts,
for many investigators concerned with wavelets applications, piecewise-linear approximations are not necessarily among their most familiar tools. It is, therefore, a
challenge to consider the integration of tools practiced
by both communities. Although it may appear to be a
surprising result to those primarily familiar with classical wavelets, the Strömberg wavelets [27], are based on
piecewise-linear functions. Another, more intriguing issue
is whether one can replace the intuitive trade-off between
scale and curvature, by a formal concept of wavelet curvature, in particular in cases such as those of the Strömberg
wavelets, or, in the more difficult case of Haar wavelets
that are not even piecewise linear.
Interestingly enough, this can be done by using metric curvatures [2] (and [21] for a short presentation). It turns out
that the best candidate, for the desired metric curvature is
the Finsler-Haantjes curvature, due to its adaptability to
both continuous and discrete settings.
A more suitable approach to surface reconstruction could,
for example, implement ridgelets [5], or the more generalized, curvelets [6].
2.
Mathematical Background
The central mathematical concept of the present paper
is the following metric notion of curvature suggested by
Finsler and developed by Haantjes [12]:
Definition 1 Let (M,d) be a metric space, let c : I =
∼
[0, 1] → M be a homeomorphism, and let p, q, r ∈
c(I), q, r 6= p. Denote by qr
b the arc of c(I) between
q and r, and by qr segment from q to r. We say that c has
85
✁
r
☎
C
✟
qr
✆✂
✝
qr
p
✡
✠
✞
✄
q
Figure 2: A piecewise-linear wavelet.
Figure 1: A metric arc and a metric segment.
3
Finsler-Haantjes curvature κF H (p) at the point p iff:
κ2F H (p) = 24 lim
q,r→p
l(qr)
b − d(q, r)
¡
¢3 ;
d(q, r))
(1)
where “l(qr)”
b denotes the length, in intrinsic metric induced by d, of qr
b – see Figure 1. (Here we assume that
the arc qr
b has finite length.)
3. Finsler-Haantjes Curvature of Wavelets
In [23] we have introduced, in the context of both vertex
and edge weighted graphs, a discretization of the FinslerHaantjes curvature, (for applications in DNA analysis).
Here we consider a semi-discrete (or semi-continuous)
version, as follows:
Let ϕ be the typical piecewise-linear wavelet depicted in
d be the arc of curve between the points A
Figure 2, let AE
and E, and let d(A, E) is the length of the line-segment
d = a + b + c + d and d(A, E) = e + f .
AE. Then l(AE)
2
Then κF H (ϕ) = 24[(a + b + c + d) − (e + f )]/(a +
b + c + d)3 . Note that, in addition to the “total” curvature
of ϕ, one can also compute the “local” curvatures at the
“peaks” B and D: κ2F H (B) = 24(a + c − e)/(a + b)3 and
κ2F H (D) = 24(c + d − f )/(a + b)3 , as well as the mean
curvature of these peaks: κ = [κF H (B) + κF H (B)]/2.
Even if these variations may prove to be useful in certain
applications, we believe that the correct approach, in the
sense that it best corresponds to the scale of the wavelet,
would be to compute the total curvature of ϕ.
Let us compare the relationship between curvature and
scale, for a concrete piecewise-linear wavelet – the Meyer
wavelet [19] – see Figure 3. The results indicating the relationship between scale and curvature, for this wavelet,
can be seen in the graph in Figure 4.
However, had the definition of Finsler-Haantjes curvature
been limited solely to piecewise-linear wavelets, its applicability would have also been diminished. We show,
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1/2
2
-1
Note that, while highly intuitive and definable for a very
large class of curves in general rather metric spaces, this
definition of curvature would remain some esoteric notion,
without the following theorem (see [2]):
Theorem 2 Let c ∈ C 3 (I) be a smooth curve in R3 , and
let p ∈ c be a regular point. Then κF H (p) exists and,
moreover, κF H (p) = k(p) – the classical (differential)
curvature of c at p.
1
0
-1
Figure 3: The Meyer wavelet.
however, that it is also definable for the “classical” Haar
wavelets, in a rather straightforward manner. For example,
consider the basic Haar wavelet and Haar scaling function,
illustrated in Figure 5. Then for the scaling function we
d = d(A, B) + d(B, C) + d(C, D) = 3, while
have: l(AE)
d(A, D) = 1. Analogously, for the Haar wavelet we get:
d = d(M, N, ) + d(N, P ) + d(P, R) + d(R, S) +
l(AE)
d(S, T ) = 5 and d(M, T ) = 1. The expression for κHF
follow easily in both cases and we present the results for
the first 10 scales in Figure 6 and Figure 7, respectively.
Moreover, while perhaps of lesser interest, it should be
mentioned that κHF (ϕ) can also be computed for smooth
wavelets, using the
p classical formula for the arc-length:
R
d =
1 + (ϕ′ )2 .
l(AE)
Suppϕ
4.
Ridgelets and beyond
The wavelet curvature definition introduced above is applicable, through standard methods, for image processing
goals, by using separable 2-dimensional wavelets. However, while practical in many cases, this presumption contravenes to real geometric structure of images, as emphasized, for instance, in [24]. In addition, as it has already
been pointed out by Candès [5], “that wavelets can efficiently represent only a small range of the full diversity of
interesting behavior”, since wavelets can cope well with
pointlike singularities, but they are not fitted for the analysis and reconstruction of singularities of dimension greater
that 0, that are distributed along lines (and more general
curves), planes (and other surfaces), etc. It is therefore
natural to ask whether the notion of curvature defined for
wavelets can be extended to ridgelets as well.
The perhaps somewhat surprising answer is that such an
extension is not only possible, it is in fact more straight-
86
Figure 4: Curvature as a function of scale: Meyer
wavelets.
C
N
1/2 P
1
1
1
1
A
D
M
Q
B
1
Figure 6: Curvature as a function of scale: The Haar scaling functions.
T
1
1
S
R
1/2
Figure 5: The Harr scaling function and wavelet.
forward and canonical. Indeed, 2-dimensional ridgelets
are, in fact, piecewise C 2 surfaces (with line singularities).
For these geometrical objects an almost standard notion
of curvature exists: the principal curvatures (i.e maximal
and minimal normal sectional curvatures – see [8]) at any
point of the surfaces. For ridgelets, we consider only the
maximal absolute curvature at points on the ridges (since,
along the ridge-line, curvature is 0 (cf. [8]) – see Figure
8. The sectional curvature of curves normal to the ridge
is then computed using the method described in the previous section. (See also [22] for the application of the this
method to piecewise-flat surfaces.)
Note that similar consideration apply with regard to
curvelets (and, evidently, to nonseparable 2-dimensional
wavelets as well). However, as far as curvature is concerned, there exists a basic difference between curvelets
and ridgelets, which is a direct consequence of the difference between the geometric models employed. Namely,
as already noted above, the principal curvature associated
with the feature of interest (i.e. the ridge) vanishes. In
consequence, Gaussian curvature, being the product of the
principal curvatures, will also equal 0 for any point on the
ridge (see Figure 8). In contrast, curvelets, being modeled
on more flexible types of surfaces, can – and will – exhibit Gaussian curvatures different from 0, both positive
and negative.
This geometric analysis can also be applied to shear-
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Figure 7: Curvature as a function of scale: The Haar
wavelets.
lets. As Figure 9 illustrates, shearlets display “peaks”
of high positive Gauss curvature. In consequence, they
are ideally suited for modeling phenomena which, in geometric terms, are characterized by positive curvature
concentrated at specific points. In view of this, shearlets may be viewed, in the context of our geometric approach, as a complementary tool to ridgelets. Indeed,
recall that ridgelets were developed as an extension of
wavelets, befitting the modeling of line-type singularities.
Point type singularities can still occur in conjunction to
1-dimensional singularities (not least as noise), hence a
combination of both type of tools, in a common, integrated
“dictionary” is, indeed, required. The geometric approach
presented above enables us to build such a “dictionary” in
natural manner.
5.
Future work – Theory and Applications
As we have seen, curvature can serve as a local scale estimator that is natural, i.e. intrinsic to the geometry of the
image. Moreover, it can be easily calculated and used for
image analysis and enhancement, especially in edge detection and texture discrimination (since in both cases curvature either large and/or exhibits a large variation). Results
87
Figure 8: Lines of curvatures on a ridgelet (after [9]).
should be validated using previous work of Brox & Weickert [3] and Lindenberg [17]. It’s extension to ridgelets
(and curvelets) should be compared with such benchmark
works as [6]. Moreover, in view of such works as [4], [15],
[16] (to cite only a few), further applications to image
compression also impose themselves as naturally stemming from our curvature analysis. In addition, feature extraction is also a natural application for our method, since
it allows for a better correlation between the internal scale
of he image (i.e. curvature) and wavelets’ scale. (In fact,
experiments in this direction are currently in progress.)
On the theoretical end of the spectrum, one would like
to develop a full multi-curvature analysis framework,
where images are constructed using basis functions that
are curve-related to one another. This is not an impossible task as it seems, since, as we have already mentioned, we have shown in [24] that image sampling and
reconstruction based on their curvature is possible. In
fact, in the said paper, we have proven that, in the geometric approach, the radius of curvature (see [8]) substitutes for the condition of the Nyquist rate, even in the
1-dimensional case. Since (sectional) curvature is defined
as 1/(curvature radius), the relationship between scale
and curvature becomes even clearer, in the light of the results presented herein. Therefore, we aim at presenting a
curvature transform, akin to the wavelet transform and to
the scale transform of [7]. Of course, in the context of
curvatures of ridgelets and curvelets one should consider
the appropriate types of transforms.
We conclude with a further natural application of metric
curvatures, lying at the confluence of theory and practice, namely to the fractals and their use, in conjunction
with wavelets or independent of them, to image processing
(see, e.g. [11], [13]). While a metric curvature – namely
Menger’s metric curvature (see [2], [21]) – was already
applied in a purely theoretical context to fractal analysis [20], our geometric method allows for a more flexible
and coherent approach, that provides a unified treatment
of wavelets (including their extensions mentioned above)
and fractals.
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Figure 9: Lines of curvatures on shearlets (after [14]).
Note the high positive curvature concentrated at the
“apex”.
6.
Acknowledgments
The authors would like to thank Professor Yehoshua Y.
Zeevi for possing the problem, and to Professor Peter
Maass, for his constructive critique and encouragement.
The first author would also like to thank Professor Shahar Mendelson – his warm support is gratefully acknowledged.
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[21] Emil Saucan. Curvature – Smooth, Piecewise-Linear
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Meshes. IEEE Transaction on Visualization and Computer Graphics, (10)2:113–122, 2004.
[27] Jan-Olov Strömberg. A modified Franklin system
and high order spline systems on Rn as unconditional
bases for Hardy spaces. In W. Beckner, editor, Conference on Harmonic Analysis in honor of A. Zygmund,
pages 475–494, Wadeworth International Group, Belmont, California, 1983.
89
SAMPTA'09
90
Analysis of Singularity Lines by
Transforms with Parabolic Scaling
Panuvuth Lakhonchai (1) , Jouni Sampo (2) and Songkiat Sumetkijakan (1)
(1) Department of Mathematics, Chulalongkorn University, Phyathai Road, Patumwan, Bangkok 10330, Thailand.
(2) Department of Applied Mathematics, Lappeenranta University of Technology, Lappeenranta, Finland.
panuvuth@hotmail.com, jouni.sampo@lut.fi, songkiat.s@chula.ac.th
Abstract:
Using Hart Smith’s, curvelet, and shearlet transforms, we
investigate L2 functions with sufficiently smooth background and present here sufficient and necessary conditions, which include the special case with 1-dimensional
singularity line. Specifically, we consider the situation
where regularity on a line in a non-parallel direction is
much lower than directional regularity along the line in a
neighborhood and how this is reflected in the behavior of
the three transforms.
1. Introduction
Wavelet transforms, both continuous and discrete, have
proved to be a very efficient tool in detecting point singularities. However, due to its isotropic scaling, wavelet
transforms are not ideal tools in detecting one-dimensional
singularities like singularity lines or curves. Recently,
wavelet-like transforms with parabolic scaling, such as
Hart Smith’s and curvelet transforms, were introduced and
applied successfully in edge detection. Our goal is then to
investigate how these transforms can be used in detecting
point, line, and curve singularities. New necessary and
new sufficient conditions for an L2 (R2 ) function to possess Hölder regularity, uniform and pointwise, with exponent α > 0 are given. Similar to the characterization
of Hölder regularity by the continuous wavelet transform,
the conditions here are in terms of bounds of the Smith
and curvelet transforms across fine scales. However, due
to the parabolic scaling, the sufficient and necessary conditions differ in both the uniform and pointwise cases, with
larger gap in pointwise regularities. Naturally, global conditions for pointwise singularities can be weakened. We
then investigate functions with sufficiently smooth background in one direction and potential singularity in the
perpendicular (non-parallel) direction. Specifically, sufficient and necessary conditions, which include the special
case with one-dimensional singularity line, are derived for
pointwise Hölder exponent. Inside their “cones” of influence, these conditions are practically the same, giving
near-characterization of direction of singularity.
2. Directional Regularity
We shall restrict our definition to a real-valued function f
of two variables. Generalization to a function of several
SAMPTA'09
variables is straightforward. For a given positive exponent
α not in N, its pointwise, uniform, and directional Hölder
(or Lipschitz) regularities are defined as follows. Fix a
point u ∈ R2 at which regularity is under investigation. f
is said to be pointwise Hölder regular with exponent α at
u, denoted by f ∈ C α (u), if there exists a polynomial Pu
of degree less than α and a constant C = Cu such that for
all x in a neighborhood of u
|f (x) − Pu (x − u)| ≤ Ckx − ukα .
(1)
If there exists a uniform constant C so that for all u in an
open subset Ω of R2 there is a polynomial Pu of degree
less than α such that (1) holds for all x ∈ Ω, then we say
that f is uniformly Hölder regular with exponent α on Ω
or f ∈ C α (Ω). The uniform Hölder exponent of f on Ω is
defined to be
αl (Ω) := sup{α : f ∈ C α (Ω)},
(2)
and the pointwise Hölder exponent is defined in an analogous manner. Following [9], the local Hölder exponent of
f at u is defines as
αl (u) = lim αl (In ).
n→∞
where {In }n∈N is a family of nested open sets in R2 , i.e.
In+1 ⊂ In , with intersection ∩n In = {u}.
In order to define directional regularity, let v ∈ Rd be a
fixed unit vector representing a direction and u be a point
in Rd . f is said to be pointwise Hölder regular with exponent α at u in the direction v, denoted by f ∈ C α (u; v),
if there exist a constant C = Cu,v and a polynomial Pu,v
of degree less than α such that
|f (u + λv) − Pu,v (λ)| ≤ C|λ|α
(3)
holds for all λ in a neighborhood of 0 ∈ R. We next
define directional regularity on a set Ω1 ⊆ R2 . Let Ω2
be an open neighborhood of Ω1 representing a set on
which the Hölder estimate holds. Then f is said to be
in C α (Ω1 , Ω2 ; v) if there exists a constant C = Cv so
that for all u ∈ Ω1 there is a polynomial Pu,v of degree less than α such that (3) holds for all λ ∈ R with
u + λv ∈ Ω2 . If Ω1 = Ω2 , then we denote C α (Ω1 , Ω2 ; v)
simply by C α (Ω1 ; v). Of course, the directional pointwise and uniform Hölder exponents could be defined in
the same way as (2). In the pointwise case, this directional
91
Hölder exponent measures one-dimensional regularity of
f at u on the line passing through u and parallel with v.
See [5]. For C α (Ω1 , Ω2 ; v), the set Ω1 in our context of
line singularity will usually be a line and v points in a direction that is nonparallel with the line. In this situation,
f ∈ C α (Ω1 , Ω2 ; v) has a ridge along the line provided
that hte regularity in the direction of the line is sufficiently
high. See Theorem 4.
3. Three Transforms with Parabolic Scaling
3.1 Hart Smith Transform
Originally defined in [10], the Hart Smith transform was
described in [1, 2] as follows. For a given ϕ ∈ L2 (R2 ),
we define
³
´
3
ϕabθ (x) = a− 4 ϕ D a1 R−θ (x − b) ,
for θ ∈ [0, 2π), b ∈ R2 , and 0 <
a0 , where a0 is a
³ a<´
1 √1
fixed coarsest scale, D a1 = diag a , a , and R−θ is the
matrix affecting planar rotation of θ radians in clockwise
direction. Hart Smith transform can then be defined as
Γf (a, b, θ) := hϕabθ , f i .
We define vector v θ := Rθ (0, 1)T so that v θ is parallel to
the major axis of the ellipse kvka,θ = 1.
Reconstruction Formula [10, 1, 2]
There exists a Fourier multiplier M of order 0 so that
whenever f ∈ L2 (R2 ) is a high-frequency function supported in frequency space kξk > a20 , then, in L2 (R2 )
Z
a0
0
=
Z
Z
2π
0
a0
0
Z
2π
0
Z
Z
R2
hϕabθ , M f i ϕabθ db dθ
da
a3
R2
hϕabθ , f i M ϕabθ db dθ
da
.
a3
(4)
3.2 Continuous Curvelet Transform
Following Candès and Donoho[1, 2], the continuous
curvelet transform (CCT) is defined in the polar coordinates (r, ω) of the Fourier domain. Let W ¡be a¢positive
real-valued C ∞ function supported inside 12 , 2 , called
a radial window, and let V be a real-valued C ∞ function
supported on [−1, 1], called an angular window, for which
the following admissibility conditions hold:
Z
∞
W (r)
0
2 dr
r
=1
and
Z
1
V (ω)2 dω = 1.
−1
At each scale a, 0 < a < a0 , γa00 is defined by
¡ √ ¢
3
γd
a00 (r cos(ω), r sin(ω)) = a 4 W (ar) V ω/ a
SAMPTA'09
γabθ (x) = γa00 (Rθ (x − b)) ,
(5)
for x ∈ R2 .
(6)
The continuous curvelet transform of f ∈ L( R2 ) is
Γf (a, b, θ) = hγabθ , f i
for 0 < a < a0 , b ∈ R2 , and θ ∈ [0, 2π).
The admissibility conditions (5) and the polar coordinate
design of curvelets yield the following:
Reconstruction formula [2]
There exists a bandlimited purely radial function Φ such
that for all f ∈ L2 (R2 ),
Z a0 Z 2π Z
da
f = f˜ +
hγabθ , f i γabθ db dθ 3 , (7)
a
0
R2
0
R
where f˜ = R2 hΦb , f i Φb db and Φb (x) = Φ(x − b).
For analysis of singularities of f , the low frequency part f˜
is not an issue as it is always C ∞ . Unlike Smith transform,
curvelet transform does not use a true affine parabolic scaling as a slightly different generating function γa00 is used
at each scale a > 0.
3.3
This gives a true affine transform that uses parabolic scaling. For each scale a and direction θ, let us define the
norm
°
°
°
°
kvka,θ := °D a1 R−θ v ° for v ∈ R2 .
f=
for r ≥ 0 and ω ∈ [0, 2π). For each 0 < a < a0 , b ∈ R2 ,
and θ ∈ [0, 2π), a curvelet γabθ is defined by
Continuous Shearlet Transform
We will follow mainly the definitions and notations in
G. Kutyniok and D. Labate[6]. Let ψ1 , ψ2 ∈ L2 (R) and
ψ ∈ L2 (R2 ) be given by
µ ¶
ξ2
, ξ1 6= 0, ξ2 ∈ R, (8)
ψ̂(ξ1 , ξ2 ) = ψ̂1 (ξ1 )ψ̂2
ξ1
where ψ1 satisfies the admissibility condition and ψ̂1 ∈
C0∞ (R) with supp ψ̂1 ⊂ [−2, − 21 ] ∪ [ 21 , 2] while ψ̂2 ∈
C0∞ (R) with supp ψ̂2 ⊂ [−1, 1], ψ̂2 > 0 on (−1, 1), and
kψk2 = 1. Given such a shearlet function ψ, a continuous
shearlet system is the family of functions ψast , a ∈ R+ ,
s ∈ R, t ∈ R2 , where
¡
¢
3
−1
ψast = a− 4 ψ D−1
a Bs (· − t)
µ
¶
1 −s
where Bs is the shear matrix
and Da is the di0 1
µ
¶
a √0
agonal matrix
. The continuous shearlet trans0
a
form of f is then defined for such (a, s, t) by
SHψ f (a, s, t) = hf, ψast i .
Many properties of the continuous shearlet are more evident in the frequency domain. So we note here that each
ψ̂ast is supported on the set
¯
¯
¾
½
¯ √
1
2 ¯ ξ2
(ξ1 , ξ2 ) :
≤ |ξ1 | ≤ , ¯¯ − s¯¯ ≤ a .
2a
a ξ1
Reconstruction Formula [6]
Let ψ ∈ L2 (R2 ) be a shearlet function. Then, for all f ∈
L2 (R2 ),
Z Z Z
da
(9)
f=
hψast , f i ψast 3 ds dt in L2 .
a
R2 R R+
92
If supp fˆ ⊂ C =
then
f=
Z
R2
Z
2
−2
Z
0
1
¯ ¯
n
o
¯ ¯
(ξ1 , ξ2 ) : |ξ1 | ≥ 2 and ¯ ξξ12 ¯ ≤ 1 ,
hψast , f i ψast
da
ds dt in L2 .
a3
(10)
Even though the second reconstruction formula (10) is
valid only for functions with frequency support in the
union C of two infinite horizontal trapezoids, it has the advantage that the integral involves only scales a and shear
parameters s in bounded sets. A complementary shearlet
(v)
system ψast can be similarly defined so that one has a reˆ
construction
¯ for
¯ f owith supp f ⊂
n formula which is valid
¯ ξ2 ¯
(v)
C
= (ξ1 , ξ2 ) : |ξ2 | ≥ 2 and ¯ ξ1 ¯ > 1 . Finally, ev-
ery f ∈ L2 (R2 ) can be decomposed into three functions
with frequency supports in C, C (v) , and W = [−2, 2]2 .
The former two functions can then be reconstructed from
(v)
ψast and ψast respectively, while the latter is C ∞ . Therefore, regularity analysis can be carried out by considering
the continuous shearlet transform with respect to these two
shearlet systems. For more details, see [6].
4. Common Properties of the Transforms
We shall suppose from this point onward that ϕ̂ ∈ C ∞
and that there exist C1′ > C1′ > 0 and C2 > 0 such that
supp(ϕ̂) ⊂ ([−C1′ , −C1 ]∪[C1 , C1′ ])×[−C2 , C2 ]. This assumption ensures that all our three kernel functions, Hart
Smith, curvelet, and shearlet functions, have Fourier supports away from the Y -axis, which in turns results in crucial properties needed to prove our main results.
4.1 Vanishing Directional Moments
A function f of two variables is said to have an L-order
vanishing directional moments along a direction v =
(v1 , v2 )T 6= 0 if
Z
bn f (bv+w)db = 0,
R
for all w ∈ R2 and 0 ≤ n < L.
Lemma 1: Let v = (v1 , v2 )T be a unit vector.
1. There exists C < ∞ (independent
√ of a, b and θ) such
that if |θ + arctan( vv21 )| ≥ C a then the curvelet
functions γabθ and the Smith functions ϕabθ and
M ϕabθ have vanishing directional moments of any
order L < ∞ along the direction v.
¯
¯
√
¯
¯
2. If ¯s + vv12 ¯ > a then the shearlet functions ψast
have vanishing directional moments of any order L <
∞ along the direction v. Here, if v2 is 0 then vv12
are treated as ∞ so that the assumed inequality holds
for all a ∈ (0, 1) and s ∈ [−2, 2], hence ψast has
vanishing directional moments of any order L < ∞
along the direction v = (v1 , 0).
SAMPTA'09
4.2
Smoothness and Decay Properties
Lemma 2: For each N = 1, 2, ... there is a constant CN
such that for all x ∈ R2 and ν ∈ N20
|∂ ν γabθ (x)| ≤
and
CN a−3/4−|ν|
°2N
°
°
°
1 + °D a1 R−θ (x − b)°
√
CN a−3/4−|ν| ( a + |s|)ν2
|∂ ψast (x)| ≤
°
°2N .
1 + °D1/a B−s (x − t)°
ν
(11)
(12)
Moreover, (11) also holds for functions ϕabθ and M ϕabθ .
5.
Singularity Lines
Let φabθ denote any of the γabθ , ϕabθ , or M ϕabθ . Let us
quote the following results.[8, 7]
Theorem 1: Let f ∈ L2 (R2 ), u ∈ R2 , and assume that
α > 0 is not an integer. If there exist α′ < 2α, θ0 ∈
[0, 2π], and A, C < ∞ such that |hφabθ , f i| is bounded by
Ã
° ′!
°
° b − u °α
√
5
α+
°
Ca 4 1 + °
,
if |θ − θ0 | ≥ A a
° a1/2 °
Ã
° ′!
°
° b − u °α
√
α+ 43
°
,
if |θ − θ0 | ≤ A a
1 + ° 1/2 °
Ca
°
a
for all a ∈ (0, a0 ), b ∈ R2 , and θ ∈ [0, 2π), then f ∈
C α (u).
Theorem 2: Let f ∈ L2 (R2 ), u ∈ R2 , and assume that
α > 0 is not an integer. If there exist α′ < 2α, −2 ≤
s0 ≤ 2, and C, C ′ < ∞ such that, for each 0 < a < 1,
−2 ≤ s ≤ 2, and t ∈ R2 , |hψast , PC1 f i| is bounded by
Ã
°
° α′ !
°
°
√
t
−
u
5
α+
°
°
,
if |s − s0 | > C ′ a,
Ca 4 1 + ° a1/2 °
Ã
°
° ′!
° t − u °α
√
3
α+
°
°
,
if |s − s0 | ≤ C ′ a,
Ca 4 1 + ° a1/2 °
(13)
and
Ã
°α′ !
°
¯D
E¯
°
°
5
t
−
u
¯
¯ (v)
°
, (14)
¯ ψast , PC2 f ¯ ≤ Caα+ 4 1 + °
° a1/2 °
then f ∈ C α (u).
holds if the inequality
E
D Similar statement
(v)
(13) holds for ψast , PC2 f and the inequality (14) holds
for hψast , PC1 f i.
Theorem 3 Let f be bounded with local Hölder exponent
α ∈ (0, 1] at point u and f ∈ C 2α+1+ε (R2 , v θ0 ) for some
θ0 ∈ [0, 2π) with any fixed ε > 0. Then there exist α′ ∈
[α − ε, α] and A, C < ∞ such that for a > 0 and b ∈ R2 ,
|hφabθ , f i| is bounded by
√
α+ 5
if |θ − θ0 | ≥ A a,
Ca 4 , Ã
°α′ !
°
°b − u°
√
α′ + 34
°
,
if |θ − θ0 | ≤ A a.
1+°
Ca
° a °
93
For s0 ∈ [−2, 2] and u = (u1 , u2 ) ∈ R2 , let Γu denote
the vertical line passing though u and Γu,s0 denote the
line passing through u with slope − s10 . Observe that we
may write Γu = Γu,0 so that (x1 , x2 ) ∈ Γu,s0 if and only
if x1 = −s0 (x2 − u2 ) + u1 . Recall that if Γ ⊆ R2 and
ρ > 0, then Γ(ρ) is the ρ-neighborhood of Γ, i.e. the set
of all points whose distance to Γ is less than ρ.
Theorem 4 Let f ∈ C α (Γu,s0 , Γu,s0 (ρ) ; (1, 0)) and
bounded for some α ∈ (0, 1], u ∈ R2 , s0 ∈
[−2, 2] and ρ > 1.
Suppose also that f is in
C 2α+1+ε (Γu,s0 (ρ) ; Bs0 (0, 1)) for some fixed ε > 0.
Then there exists C < ∞ such that if 0 < a < a0 < 1 and
t ∈ Γu (r) with r < ρ/2 and s ∈ [−2, 2], the continuous
shearlet transform hψast , f i is bounded in magnitude by
√
5
if |s − s0 | > a,
Caα+ 4 ,µ
¯
¯α ¶
¯ d (t, u) ¯
3
¯ , if |s − s0 | ≤ √a,
Caα+ 4 1 + ¯¯ s0
¯
a
[7] P. Lakhonchai, J. Sampo, and S. Sumetkijakan.
Shearlet transforms and hölder regularities. 2009.
Preprint.
[8] J. Sampo and S. Sumetkijakan. Estimations of
Hölder regularities and direction of singularity by
Hart Smith and curvelet transforms. Journal of
Fourier Analysis and Applications, 15(1):58–79,
2009.
[9] S. Seuret and J. Lévy Véhel. The local Hölder function of a continuous function. Appl. Comput. Harmon. Anal., 13(3):263–276, 2002.
[10] Hart F. Smith. A Hardy space for Fourier integral
operators. J. Geom. Anal., 8(4):629–653, 1998.
[11] S. Yi, D. Labate, G.R. Easley, and H. Krim. Edge
detection and processing using shearlets. 2008.
Preprint.
where ds0 (t, u) = |t1 + s0 t2 − u1 − s0 u2 | denotes the distance between the parallel lines with slope − s10 (vertical
line if s0 = 0) and passing through t and u respectively.
Edge analysis has been done successfully using the continuous shearlet transform ([11, 4, 3, 6]). They consider the
shearlet transform of the characteristic function of a set
with piecewise smooth boundary and found that, at a regular boundary point t, the shearlet transform decays like
a3/4 if s = s0 = ± vv12 and decays rapidly at other s 6= s0 ,
where v = (v1 , v2 ) is the normal vector of the boundary
curve at t. Since this characteristic function has Hölder
exponent 0 (bounded and discontinuous) at any boundary
point in the normal direction, this decay rate of a3/4 at
s = s0 = 0 agrees with that of Theorem 4. However,
when s0 6= 0 the two directions in Theorem 4 along which
regularity is assumed are not perpendicular. More comparisons of our results and the aforementioned work are
needed.
References:
[1] Emmanuel J. Candès and David L. Donoho. Continuous curvelet transform. I: Resolution of the wavefront set. Appl. Comput. Harmon. Anal., 19(2):162–
197, 2005.
[2] Emmanuel J. Candès and David L. Donoho. Continuous curvelet transform. II: Discretization and
frames. Appl. Comput. Harmon. Anal., 19(2):198–
222, 2005.
[3] K. Guo, Labate D., and W-Q. Lim. Edge analysis and identification using the continuous shearlet
transform. Appl. Comput. Harmon. Anal., 2008. In
Press.
[4] K. Guo and D. Labate. Characterization and analysis of edges using the continuous shearlet transform.
2008. Preprint.
[5] S. Jaffard. Multifractal functions: Recent advances
and open problems. Menuscript, 2004.
[6] G. Kutyniok and D. Labate. Resolution of the wavefront set using continuous shearlets. Trans. AMS.,
105(1):157–175, 2007.
SAMPTA'09
94
Geometric Separation using a
Wavelet-Shearlet Dictionary
David L. Donoho (1) and Gitta Kutyniok (2)
(1) Department of Statistics, Stanford University, Stanford, CA 94305, USA.
(2) Institute of Mathematics, University of Osnabrück, 49069 Osnabrück, Germany.
donoho@stanford.edu, kutyniok@uni-osnabrueck.de
Abstract:
Astronomical images of galaxies can be modeled as a
superposition of pointlike and curvelike structures. Astronomers typically face the problem of extracting those
components as accurate as possible. Although this problem seems unsolvable – as there are two unknowns for
every datum – suggestive empirical results have been
achieved by employing a dictionary consisting of wavelets
and curvelets combined with ℓ1 minimization techniques.
In this paper we present a theoretical analysis in a model
problem showing that accurate geometric separation can
be achieved by ℓ1 minimization. We introduce the notions
of cluster coherence and clustered sparse objects as a machinery to show that the underdetermined system of equations can be stably solved by ℓ1 minimization. We prove
that not only a radial wavelet-curvelet dictionary achieves
nearly-perfect separation at all sufficiently fine scales, but,
in particular, also an orthonormal wavelet-shearlet dictionary, thereby proposing this dictionary as an interesting alternative for geometric separation of pointlike and curvelike structures. To derive this final result we show that
curvelets and shearlets are sparsity equivalent in the sense
of a finite p-norm (0 < p ≤ 1) of the cross-Grammian
matrix.
1. Introduction
Cosmological data analysts face tasks of geometric separation. Gravitation, acting over time, drives an initially quasi-uniform distribution of matter in 3D to concentrate near lower-dimensional structures: points, filaments, and sheets. It would be desirable to process single ‘maps’ of matter density and somehow extract three
‘pure’ maps containing just the points, just the filaments,
and just the sheets around which matter is concentrating.
However, this problem contains three unknowns for every
datum which seems impossible to solve on mathematical
grounds.
Surprisingly, astronomer Jean-Luc Starck and collaborators have recently been empirically successful in numerical experiments with component separation. They used
two or more overcomplete frames, each one specially
adapted to particular geometric structures, and were able
to obtain separation despite the fact that the underlying
system of equations is highly underdetermined.
Here we analyze such approaches in a mathematical
SAMPTA'09
framework where we can show that success stems from
an interplay between geometric properties of the objects
to be separated, and the harmonic analysis for singularities of various geometric types.
1.1
Singularities and Sparsity
As a mathematical idealization of ’image’, consider a
Schwartz distribution f with domain R2 . The distribution f will be given singularities with specified geometry:
points and curves.
We plan to represent such an ’image’ using tools of harmonic analysis; in particular bases and frames. While
many such representations are conceivable, we are interested here just in those bases or frames which can sparsely
represent f .
The type of basis which best sparsifies f depends on the
geometry of its singularities. If the singularities occur at
a finite number of (variable) points, then wavelets give
what is, roughly speaking, an optimally sparse representation. If the singularities occur at a finite number of smooth
curves, then one of the recently studied directional multiscale representations (curvelets or shearlets) will do the
best job of sparsification.
Since we are concerned with f being a mixture of content
types, i.e., points and curves, presumably both systems are
needed to represent f sparsely.
1.2
Minimum ℓ1 Decomposition and Perfect Separation
In the early 1990’s, R. R. Coifman, Wickerhauser and coworkers became interested in the problem of representing signals using more than one basis and started a first
heuristic exploration motivated intuitively, see [5]. A few
years later, one of us worked with S .S. Chen to develop a
formal, optimization-based approach to the multiple-basis
representation problem [4]. Given bases Φi , i = 1, 2, one
solves the following problem
(BP)
min kα1 k1 +kα2 k1 subject to S = Φ1 α1 +Φ2 α2 ,
thereby exploiting that the ℓ1 norm has a tendency to find
sparse solutions when they exist. This can be regarded as
the starting point for ℓ1 decomposition techniques. For
theoretical work on this topic we refer to, e.g., [6, 10, 15,
16], and for empirical work see, for instance, [9,12,14,15].
95
For further references we would like to mention the survey
paper [1].
1.3 A Geometric Separation Problem
The work just cited, while suggestive and inspiring, concerns discretely indexed signal/image processing, and so
is either empirical or else rigorously analytical but not directly relevant to geometric separation tasks, which will
involve always continuum ideas.
In this paper we develop related methods in a mathematical setting where the notion of successful separation can
be made definitionally precise and can be established by
mathematical analysis. For this, we pose a simple but clear
model problem of geometric separation.
Consider a ‘pointlike’ object P made of point singularities:
P
X
|x − xi |−1 .
P=
Figure 1: Frequency tilings of radial wavelets and
curvelets as well as of orthonormal wavelets and shearlets
(from left to right).
Since the scaling subband of each pair are similar as illustrated in Figure 1, we can define two families of filters
(FjC )j and (FjS )j which allows to decompose a function
f into pieces fjC (resp. fjS ) with different scales j. The
piece fjC (resp. fjS ) at subband j arises from filtering f
using FjC (resp. FjS ):
i=1
Consider as well a curvelike object C, a singularity along
a closed curve τ : [0, 1] 7→ R2 :
Z
C = δτ (t) dt,
where δx is the usual Dirac Delta at x. By this choice, we
arrange that one of the two distributions does not become
dramatically larger than the other as we go to finer and
finer scales; rather the ratio of energies is more or less
independent of scale. This makes the separation problem
challenging at every scale.
Now assume that we observe the ‘Signal’
f = P + C,
(1)
however, the distributions P and C are unknown to us. The
Geometric Separation Problem now consists in recovering
P and C from knowledge of f .
fjC = FjC ⋆ f and fjS = FjS ⋆ f,
so that the Fourier transform fˆiC (resp. fˆjS ) is supported
in the scaling subband of scale j of the associated pair of
tight frames. The filters are defined in such as way, that
we can reconstruct the original function from these pieces
using the formula
f=
X
FjC ⋆ fjC =
j
X
FjS ⋆ fjS ,
f ∈ L2 (R2 ).
j
For the precise construction of those filters and further
properties, we refer to [7].
We can now use these tools to attack the Geometric Separation Problem scale-by-scale. For this, we filter the model
problem (1) to derive the sequences of filtered images
fjC = PjC + CjC and fjS = PjS + CjS for all scales j. (2)
1.4 Two Geometric Frames
1.5
We focus on two pairs of overcomplete systems for representing the object f :
In Section 2 we will develop and analyze the decomposition technique based on ℓ1 minimization we intend to employ, first in a very general Hilbert space setting. These results will then be applied to the scale-dependent Geometric Separation Problem (2) proving that the radial waveletcurvelet as well as the orthonormal wavelet-shearlet dictionary achieves nearly-perfect separation at all sufficient
fine scales (Theorems 1 and 3). The sparsity equivalence
between curvelets and shearlets we derive in Subsection
3.2 thereby allows transference of this result from the radial wavelet-curvelet to the orthonormal wavelet-shearlet
dictionary.
• Radial Wavelets – a tight frame with perfectly
isotropic generating elements.
• Curvelets – a highly directional tight frame with increasingly anisotropic elements at fine scales.
as well as the pair
• Orthonormal Separable Meyer Wavelets – an orthonormal basis of perfectly isotropic generating elements.
• Shearlets – a highly directional tight frame with increasingly anisotropic elements at fine scales and a
unified treatment of both the continuous and digital
setting.
We pick these because, as is well known, point singularities are coherent in wavelets and curvilinear singularities
are coherent in curvelets/shearlets. For the precise definitions we refer to [2, 3], [11, 13], as well as [7].
SAMPTA'09
2.
Outline
General Component Separation
We now first study the behavior of ℓ1 minimization in
the general two-frame case. Suppose we have two tight
frames Φ1 , Φ2 in a Hilbert space H, and a signal vector
S ∈ H. We know a priori that there exists a decomposition
S = S10 + S20 ,
96
where S10 is sparse in Φ1 and S20 is sparsely represented
by Φ2 . Our analysis will center on the use of cluster coherence to exploit the geometric structure of the sparse
expansions rather than merely the fact that the vector is
sparse.
Typically, separation results employ the notion of mutual
coherence between two tight frames Φ = (φi )i and Ψ =
(ψj )j ,
µ(Φ, Ψ) = max max |hφi , ψj i|,
j
• the nonzeros of sparse vectors often do not arise in arbitrary patterns, but are rather highly structured, and
that
• the interactions between the dictionary elements in
ill-posed problems are not arbitrary, but rather geometrically driven.
These key observations lead to the following new notion.
Definition 1. Given tight frames Φ = (φi )i and Ψ =
(ψj )j and an index subset S associated with expansions
in frame Φ, we define the cluster coherence
µc (S; Φ, Ψ) = max
j
X
|hφi , ψj i|.
2.2 Component Separation by ℓ1 Minimization
Now consider the following optimization problem:
argminS1 ,S2 kΦT1 S1 k1 + kΦT2 S2 k1
subject to S = S1 + S2 .
Notice that in this problem, the norm is placed on the analysis coefficients rather than on the synthesis coefficients
as in (BP) to avoid ‘self-terms’ in the frame expansions.
The introduction of cluster coherence now ensures that the
principle (S EP) gives a successful approximate separation.
2δ
,
1 − 2µc
where
µc = max(µc (S1 ; Φ1 , Φ2 ), µc (S2 ; Φ2 , Φ1 )).
3.
Geometric Separation of Pointlike and
Curvelike Structures
3.1
Radial Wavelet-Curvelet Dictionary
The concepts of the previous section will now be applied
to S = fjC = PjC + CjC , our signal of interest from (2).
The tight frames are Φ1 , the full radial wavelet frame, and
Φ2 , the full curvelet tight frame. The subsignals S1⋆ , S2⋆ we
derive by applying the optimization problem (S EP) will be
relabel to Wj , the wavelet component, and Cj , the curvelet
component.
The main difficulty in applying Proposition 1 consists in
choosing the sets of significant coefficients suitably. We
achieve this by using microlocal analysis to understand
heuristically the location of the significant coefficients in
phase space. Roughly speaking, we then employ the HartSmith phase space metric defined by
d((b, θ); (b′ , θ′ )) = |heθ , b − b′ i| + |heθ′ , b − b′ i|
+|b − b′ |2 + |θ − θ′ |2
i∈S
Thus cluster coherence bounds between a single member
of frame Ψ and a cluster of members of frame Φ, clustered at S, in contrast to mutual coherence, which can be
thought of as singleton coherence.
A related notion called ‘cumulative coherence’ was introduced in [16], but notice that here we fix a specific set of
significant coefficients and do not maximize over all such
subsets. The key idea for our analysis is that the index subsets we consider are not abstract, but have a specific geometric interpretation. Maximizing over all subsets with
a common combinatorial property would prohibit utilizing this interpretation, hence cumulative coherence is not
suitable for our purposes.
SAMPTA'09
kS1⋆ − S10 k2 + kS2⋆ − S20 k2 ≤
i
whose importance was shown by [6], as a means to impose
conditions on the interactions between the dictionary elements. However, this notion is too weak for our purposes.
Our novel contribution to sparse recovery and ℓ1 minimization consists in exploiting the facts that
(S1⋆ , S2⋆ ) =
k1S1c ΦT1 S10 k1 + k1S2c ΦT2 S20 k1 ≤ δ.
Let (S1⋆ , S2⋆ ) solve (S EP). Then
2.1 Cluster Coherence
(S EP)
Proposition 1 ( [7]). Suppose that S can be decomposed
as S = S10 + S20 so that each component Si0 is relatively
sparse in Φi , i = 1, 2, i.e.,
to define an ‘approximate’ set of significant wavelet coefficients
Λ1,j
=
{wavelet lattice}
∩{(b, θ) : d((b, θ); W F (P)) ≤ ηj aj }
and an ‘approximate’ set of significant curvelet coefficients
Λ2,j
=
{curvelet lattice}
∩{(b, θ) : d((b, θ); W F (C)) ≤ ηj aj }
for carefully chosen ηj ; W F denotes the wavefront set.
Tedious, highly technical estimates then lead to the following separation result:
Theorem 1 ( [7]). A SYMPTOTIC S EPARATION USING A
R ADIAL WAVELET-C URVELET D ICTIONARY.
kWj − PjC k2 + kCj − CjC k2
→ 0,
kPjC k2 + kCjC k2
j → ∞.
This result shows that components are recovered asymptotically: at fine scales, the energy in the curvelike component is all captured by the curvelet coefficients and the
energy in the pointlike component is all captured by the
wavelet coefficients.
97
4.
3.2 Sparsity Equivalence
We now aim to show that curvelets and shearlets are sparsity equivalent in the sense that, for 0 < p ≤ 1, the ℓp
norm of the curvelet coefficient sequence is finite if and
only if the same is true for the shearlet coefficient sequence.
First we observe that for two tight frames Φ = (φi )i and
Ψ = (ψj )j , their cross-Grammian matrix
M (i, j) = hφi , ψj i
contains all information on the relation between coefficient sequences ΦT S and ΨT S for some signal S. Sparsity equivalence can therefore be proven by analyzing the
p-norm, 0 < p ≤ 1 defined by
³
X
|M (i, j)|p )1/p ,
kM kp = max (sup
i
(sup
j
j
X
|M (i, j)|p )1/p
i
´
of a cross-Grammian matrix M .
Now setting (ση )η to be the shearlet tight frame and (γµ )µ
to be the curvelet tight frame, we derive the following result. We remark that the low frequency part has to be dealt
with particular care, but for these technicalities we refer
to [7].
Proposition 2 ( [8]). For all 0 < p ≤ 1,
k(hση , γµ i)η,µ kp < ∞.
Using basic estimates from frame theory and the previous proposition, we can show that shearlets and curvelets
are indeed sparsity equivalent, thereby allowing us to easily transfer results about sparsity from one system to the
other.
Theorem 2 ( [8]). Let f ∈ L2 (R2 ) and 0 < p ≤ 1. Then
k(hf, ση i)η kp < ∞ if and only if k(hf, γµ i)µ kp < ∞.
3.3 Orthonormal Wavelet-Shearlet Dictionary
Similar to Subsection 3.1, S = fjS = PjS + CjS (see (2)) is
now our signal of interest, and the tight frames are Φ1 , the
full orthonormal wavelet frame, and Φ2 the full shearlet
tight frame. The subsignals S1⋆ , S2⋆ , we derive by applying
the optimization problem (S EP) will be relabel to Wj , the
wavelet component, and Sj , the shearlet component.
The results from Subsection 3.2 as well as similar correspondences between radial wavelets and orthonormal
wavelets now form the backbone for the transfer of Theorem 1 to the orthonormal wavelet-shearlet dictionary.
Careful application of those to the key estimates in the
proof of Theorem 1 leads to a similar result for the orthonormal wavelet-shearlet dictionary.
Theorem 3 ([7]). A SYMPTOTIC S EPARATION USING AN
O RTHONORMAL WAVELET-S HEARLET D ICTIONARY.
kWj − PjS k2 + kSj − CjS k2
→ 0,
kPjS k2 + kCjS k2
SAMPTA'09
j → ∞.
Conclusion
We first considered signals, being a superposition of two
subsignals, each of which is relatively sparse with respect
to some tight frame. As a model procedure for separation we considered ℓ1 minimization of the analysis (rather
than synthesis) frame coefficients. By introducing cluster coherence as a new concept for analyzing the interaction of the two tight frames by taking the geometry of
the sparse component expansions into account, we derived an estimate for the ℓ2 norm of the separation error.
We then considered signals, which are a superposition of
pointlike and curvelike structures. Using the previously
derived estimate, we proved that for both pairs of tight
frames (radial wavelets/curvelets) as well as (orthonormal wavelets/shearlets) at sufficiently fine scale, nearlyperfect separation is achieved using the model procedure,
thereby proposing the orthonormal wavelet-shearlet dictionary as an interesting alternative for geometric separation of pointlike and curvelike structures. The sparsity
equivalence between curvelets and shearlets we further
proved thereby allows to derive this separation result only
for one dictionary and easily transfer it to the other one.
Acknowledgment
The authors would like to thank Emmanuel Candès,
Michael Elad, and Jean-Luc Starck, for numerous discussions on related topics. The second author would like to
thank the Department of Statistics at Stanford University
and the Department of Mathematics at Yale University for
their hospitality and support during her long-term visits.
The authors would also like to thank the Newton Institute
of Mathematics in Cambridge, UK for providing an inspiring research environment which led to the completion
of a significant part of this work during their stay. This
work was partially supported by NSF DMS 05-05303 and
DMS 01-40698 (FRG), and by Deutsche Forschungsgemeinschaft (DFG) Heisenberg Fellowship KU 1446/8-1.
We further thank the anonymous referee for useful comments and suggestions.
References:
[1] A.M. Bruckstein, D.L. Donoho, and M. Elad. From
Sparse Solutions of Systems of Equations to Sparse
Modeling of Signals and Images. SIAM Review
51:34–81, 2009.
[2] E. J. Candès and D. L. Donoho. Continuous curvelet
transform: I. Resolution of the wavefront set. Appl.
Comput. Harmon. Anal. 19:162–197, 2005.
[3] E. J. Candès and D. L. Donoho. Continuous curvelet
transform: II. Discretization of frames. Appl. Comput. Harmon. Anal. 19:198–222, 2005.
[4] S. S. Chen, D. L. Donoho, and M. A. Saunders.
Atomic decomposition by basis pursuit. SIAM Review 43:129–159, 2001.
[5] R. R. Coifman and M. V. Wickerhauser. Wavelets
and adapted waveform analysis. A toolkit for signal processing and numerical analysis, In Different
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[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
perspectives on wavelets (San Antonio, TX, 1993),
47:119–153, Proc. Sympos. Appl. Math., Amer.
Math. Soc., Providence, RI, 1993.
D. L. Donoho and X. Huo. Uncertainty principles
and ideal atomic decomposition. IEEE Trans. Inform. Theory 47:2845–2862, 2001.
D. L. Donoho and G. Kutyniok. Microlocal Analysis of the Geometric Separation Problem. Preprint,
2009.
D. L. Donoho and G. Kutyniok. Sparsity Equivalence of Anisotropic Decompositions. Preprint,
2009.
M. Elad, J.-L. Starck, P. Querre, and D. L. Donoho.
Simultaneous cartoon and texture image inpainting
using morphological component analysis (MCA).
Appl. Comput. Harmon. Anal. 19:340–358, 2005.
R. Gribonval and M. Nielsen. Sparse representations in unions of bases. IEEE Trans. Inform. Theory
49:3320–3325, 2003.
K. Guo, G. Kutyniok, and D. Labate. Sparse Multidimensional Representations using Anisotropic Dilation und Shear Operators. In Wavelets und Splines
(Athens, GA, 2005), G. Chen und M. J. Lai, eds.,
Nashboro Press, Nashville, TN (2006), 189–201.
M. Kowalski and B. Torrésani. Sparsity and Persistence: mixed norms provide simple signal models
with dependent coefficients. Signal, Image and Video
Processing, to appear.
G. Kutyniok and D. Labate. Resolution of the Wavefront Set using Continuous Shearlets. Trans. Amer.
Math. Soc. 361:2719–2754, 2009.
F. G. Meyer, A. Averbuch, and R. R. Coifman. Multilayered Image Representation: Application to Image
Compression. IEEE Trans. Image Proc. 11:1072–
1080, 2002.
J.-L. Starck, M. Elad, and D. L. Donoho. Image decomposition via the combination of sparse representations and a variational approach. IEEE Trans. Image Proc. 14:1570–1582, 2005.
J. A. Tropp. Greed is good: algorithmic results for
sparse approximation. IEEE Trans. Inform. Theory
50:2231–2242, 2004.
SAMPTA'09
99
SAMPTA'09
100
Special session on
Sampling and Communication
Chair: Götz PFANDER
SAMPTA'09
101
SAMPTA'09
102
A Kashin Approach to the Capacity of the
Discrete Amplitude Constrained Gaussian
Channel
Brendan Farrell (1) and Peter Jung (2)
(1) Heinrich-Hertz Lehrstuhl, Technische Universität Berlin, Einsteinufer 25, 10587 Berlin, Germany.
(2) Fraunhofer German-Sino Lab for Mobile Communications - MCI, Einsteinufer 37, 10587 Berlin, Germany.
brendan.farrell@mk.tu-berlin.de, peter.jung@hhi.fhg.de
Abstract:
We derive an explicit lower bound on the capacity of the
discrete amplitude–constrained Gaussian channel by proving the existence of tight frames that permit redundant
vector representations with small coefficients. Our method
encodes the information in subspaces that are optimal in
terms of the power to amplitude ratio. In a recent paper,
Lyubarskii and Vershynin discuss how the work of Kashin
(1977) implies the existence of such representations, and
they term them Kashin respresentations. We use this work
from frame theory to address the relationship between signal redundancy, peak–to–average power ratio and achievable data rates.
1.
Introduction
Communication at high data rates and with moderate cost
on hardware and complexity provide challenging topics
in engineering and applied mathematics. An important
problem in this direction is efficient signaling and coding under an amplitude constraint. In general, the cost
for high data rate is related to a power budget. However,
in practical communication systems, there sometimes exist disruptive or non-linear effects that only occur at high
signal amplitudes. The information–theoretic treatment
of amplitude–constrained channel is completely different
from the power–constrained channel. On the other hand,
coding for power–constrained Gaussian channels is well
understood. Clearly, if a loss in data rate is accepted, signals can be constructed with lower maximum amplitude.
The optimal scaling between power and amplitude and an
explicit relation to achievable rates will be given in this
paper. In this case, the data-rate loss is caused by considering redundant representations. Here, the original vectors
are expanded with respect to a particular frame and the coefficients are then transmitted.
We show that there exist frames which allow the standard
coding approach to be used for the amplitude-constrained
channel. Our result is Theorem 2, which comes at the end
of the paper. This theorem states that for the amplitude
constrained, Gaussian channel the rate
1
Signal Power
log 1 + λmin
(1)
2λmin
Noise Power
is achievable for a redundancy λmin that is an explicit
function of the peak-to-average power ratio. We note
SAMPTA'09
that by making the amplitude constraint compatible with
Gaussian codebooks, we make the developed tools and
understanding of Gaussian codebooks applicable to the
amplitude–constrained channel. Results from frame theory, thus, allow us to address a question in information
theory. While the results used from functional analysis
are well known there, we show a new application.
1.1
The Information–Theoretic Problem
The capacity of a communication channel is the maximum
amount of information per unit of time that can be sent
from a sender through the channel to the receiver. Shannon made this operational concept mathematically rigorous by formulating it in terms of entropy [7]. In [7] Shannon addressed the discrete–time model:
Y = X + Z,
(2)
for the noisy channel, where X and Y denote the (real)
channel input and output, and the additive noise Z is a
Gaussian random variable with variance σ 2 . Let X n be
a random vector in Rn according to a distribution to be
determined and Z n the random vector having n identical
independent distributed (iid) copies of Z. Shannon introduced two concepts of a capacity for this model. The information capacity C (i) is the supremum of the information rates:
1
(3)
sup I(X n ; Y n )
C (i) = lim
n→∞ n µn ∈F n
taken over all distributions µn of X n from a particular subset F n ⊂ P n of probability distributions P n .
I(X n ; Y n ) denotes the mutual information between the
random variables X n and Y n and is equal to the entropy
of Y n minus the entropy of Y n given X n , I(X n ; Y n ) =
h(Y n ) − h(Y n |X n ). From its concavity in µn it follows
that the optimum µnopt is at least achieved for a product distribution, i.e. single letter coding with a measure µ = µ1
is optimal in this sense. Shannon considered an averaged
power constraint P which corresponds to the set F = F 1
of single–letter distributions:
Z
(4)
F = {µ ∈ P | |x|2 dµ(x) ≤ P }
or equivalently
n
1X
E|xi |2 ≤ P.
n i=1
(5)
103
He found that the optimum µopt is attained for a Gaussian
distribution with variance P and that
C (i) =
1
P
log(1 + 2 ).
2
σ
(6)
Shannon further showed with a so called coding theorem
that it is even possible to get arbitrary close to that value
justifying the term channel capacity. That is, for each rate
nR
R < C (i) there exist 2nR codewords {X(ω)}ω=2
in Rn
ω=1
nR
n
(called a (2 , n) code) such that X(ω) + Z can be distinguished at the receiver with error probability going to
zero as n increases. (X will now denote codewords and
be indexed by ω.) Each admissible
Pn codeword satisfies the
average power constraint n1 i=1 |Xi (ω)|2 ≤ P ; however, to achieve the capacity it may be necessary to use
codewords
having maximum amplitudes which scale with
√
n.
We address an additive, white Gaussian noise (AWGN)
channel under the assumption that there is both a power
constraint,
n
1X
|Xi (ω)|2 ≤ P,
(7)
n i=1
and a strict amplitude constraint:
max |Xi (ω)| ≤ A,
i=1,...,n
(8)
for two positive, real numbers P and A and for all ω =
1, ..., 2nR .
The information capacity under a constraint A on the amplitudes of the signals was solved by Smith [8]. Similar to
the Gaussian channel with power constraint only, Smith
showed that the capacity of the amplitude–constrained
channel is attained when the entries xi are independent.
The set of (single–letter) input distributions is in this case:
F = {µ ∈ P | µ({|x| > A}) = 0}.
(9)
Smith found that the optimum measure µopt has discrete
and finite support. Similar results are known for other
noise densities (see for example [6]). A characterization
of the number of mass points in the Gaussian case is unknown. For a given assumption on this number the values
and the positions can be computed. From this Smith gave
an algorithm which numerically computes C (i) . Smith
establishes an algorithm to determine the optimal input
probability measure given the constraints A, P and σ 2 .
However, to date there is not a general strategy applicable
for a practical range of these parameters.
1.2
Frames and Banach Geometry
We will work strictly with real numbers.
Pn We have the
following norms for Rn : kxklpn = ( i=1 |xi |p )1/p and
n
n = maxi=1,...,n |xi |. B
kxkl∞
p will denote the unit ball in
n
R with respect to the ℓp -norm. We denote by UnN an ndimensional subspace of RN , N ≥ n. We will often speak
of a matrix U ∈ Rn×N whose rows are orthonormal and
span UnN or whose columns constitute a tight frame for
Rn .
SAMPTA'09
n
Definition 1. A set of vectors {ui }N
i=1 ⊂ R is a tight
n
frame for R if
kxk22
=
N
X
i=1
= |hx, ui i|2
(10)
for all x ∈ Rn .
It follows that the columns of an n×N matrix U constitute
a tight frame for Rn if and only if U U ∗ = In , where In
denotes the identity matrix of size n. In the proof of the
coding theorem (see, for example, [1]) for the Gaussian
channel with average power constraint P , the constructed
√
codewords X ∈ Rn satisfy the constraint kXkℓn2 ≤ nP .
Similarly, in the amplitude constrained channel codewords
must satisfy kXkℓn∞ ≤ A. In other words, admissible
signals X for the amplitude constraint channel lie in a
n
. And for a power conscaled cube, i.e. X ∈ A · B∞
strained channel
the
signals
are
contained
in an increasing
√
ball X ∈ nP · B2n .
Of course the difficult aspect of this channel is the amplitude constraint. We do not require that the random input
variables {xi }ni=1 be independent, which allows us to use
redundant representations.
The basic idea for our approach is the following: given N
n
n
vectors {ui }N
i=1 spanning R , N > n, a vector x ∈ R
may be expressed, in general, in multiple ways as a linear
combination of the vectors {ui }N
i=1 :
x=
N
X
bi u i .
(11)
i=1
In light of the amplitude constraint, the question is
whether one of the possible expressions (10) satisfies
N ≤ A. If this is possible, then we may transmit the
kbkl∞
vector b and suffer an efficiency loss of N − n symbols.
The representation (10) is called a Kashin representation
[5] of the vector x if kbk∞ ≤ Ckxk2 . We first address a
general frame setting and then focus on the Kashin representations in Section 3.
2.
General Frame Setting
As we have seen, the capacity of the discrete Gaussian
channel with average power constraint P and noise variance σ 2 is 12 log(1 + σP2 ). This means, If R < 12 log(1 +
P
nR
codewords, and all adσ 2 ) is the rate, then there are 2
missible codewords for
the power con√ this channel satisfy
straint kX(ω)k2 ≤ nP , ω = 1, ..., 2nR . If one has a
n
tight frame {ui }N
i=1 for R , N = [λn], then one can also
achieve the rate:
λP
1
log(1 + 2 )
2λ
σ
(12)
nR
by transmitting codewords {Y (ω)}2ω=1 ⊂ RN satisfying
U Y (ω) = X(ω) for ω = 1, ..., 2nR .
Since columns of U ∈ Cn×N form a tight frame for Rn ,
2 = kY (ω)kl2 , and thus:
kU X(ω)klN
n
N
n
1 X
1 X
|Yi (ω)|2 =
|Xi (ω)|2 ≤ P.
N i=1
λn i=1
(13)
104
The key point is that a vector Y (ω) that satisfies U Y (ω) =
X(ω) is, in general, not unique. For a given additional
constraint, one may ask if there exists a set Y ⊂ RN satisfying the additional constraint and a tight frame with matrix U such that:
U Y = {x|x ∈ Rn , kxk2 = 1}.
(14)
The existence of such a set and a corresponding tight
1
log(1 + λP
frame is sufficient to imply that 2λ
σ 2 ) is an
achievable rate for the discrete Gaussian channel with the
additional constraint.
The additional constraint of interest here is the amplitude
N ≤ A for all
constraint; that is, it is required that kY (ω)kl∞
nR
2
codewords Y (ω). Thus, for
√ a given codebook {X(ω)}ω=1
satisfying kX(ω)kl2n ≤ nP for all ω, we would like to
nR
determine a second codebook {Y (ω)}2ω=1 ⊂ RN satisfyN ≤ A and a tight frame so that U Y (ω) =
ing kY (ω)kl∞
X(ω) for all ω. For completeness and clarity, we include
the communication strategy. The next section will show
that Step 2 is possible for an appropriate λ.
Communication Strategy:
n
1. The set of vectors {ui }N
i=1 form a tight frame for R
and are known to both transmitter and receiver.
2. Each codeword X(ω) satisfies the power constraint ,
and its Kashin representation Y (ω) ∈ RN satisfying
N ≤ A is determined.
kY (ω)kl∞
3. To transmit the message ω, the transmitter sends
Y (ω).
4. Y (ω) + Z N ∈ RN is received.
5. Receiver multiplies Y (ω) + Z N by U to obtain
X(ω) + U Z N ∈ Rn .
6. Receiver decodes X(ω) + U Z N ∈ Rn .
We note that, in contrast to the approach of Smith [8], this
approach is still based on Gaussian codebooks, and, therefore, the extensive tools developed for Gaussian codebooks are still applicable.
matrix on Rn . One possible coefficient vector for equation (14) is a = U ∗ x. For this vector, we note
p
N
hU ∗ x, U ∗ xi (17)
≤ kakl2N
=
kakl∞
p
=
(18)
hIn x, xi = kxkl2n .
Consequently, for a tight frame, it is always possible to
N ≤ kxkln , and thus equafind a vector a satisfying kakl∞
2
tion (15) can
√be satisfied for every tight frame with Kashin
level K = N .
Of course the study of Kashin representations is concerned
with optimally small constants and their relation to the redundancy λ = N/n. We will be interested in the dependence of K = K(λ) on λ, but we postpone the discussion
of the constant K(λ) until the next section. Now, we show
a lower bound on the achievable
capacity when the ampli√
tude constraint is K(λ) P (or greater).
If we set any n orthonormal vectors in RN to be the rows
of a matrix U , then U U ∗ = In , and the columns of U
constitute a tight frame for Rn . Thus, a tight frame for
Rn can be constructed from any n-dimensional subspace
of RN . For U ∈ Cn×N , let UnN denote the subspace of
RN spanned by its rows. Then U (B2N ∩ UnN ) = B2n .
Therefore, for any x ∈ B2n , as long as the rows of U are
linearly independent there exists a y ∈ (B2N ∩ UnN ) such
that x = U y. In the higher dimensional space, we have
N -norm constraint. We thus want to find an nan k · kl∞
dimensional subspace of RN that can be mapped isometrically with respect to the k · kl2n -norm to Rn , and we must
be able to cover B2n in this way.
First results on the smallest constant C, such that a projecN
covers B2n was given by Kashin in
tion of the ball C · B∞
[3]. There he showed that the scaling is O(n−1/2 ), and the
exact optimal scaling was then determined in [2]. Since
the k · k2 -isometric projection is equivalent to the existence of a tight frame, we formulate their result in terms
of frames.
Theorem 1 ([3, 2]). For all positive integers N and n,
N > n, there exists a tight frame for Rn consisting of N
vectors such that every vector in Rn has a Kashin representation of level:
K(λ) := C
3.
Kashin Representations or Optimal Subspaces
Definition 2 (Kashin Representations). For a set of vecn
tors {ui }N
i=1 ⊂ R , N > n, the expansion
x=
N
X
ai ui
(15)
i=1
is a Kashin representation with level K of the vector x ∈
Rn if
Kkxkl2n
√
N ≤
kakl∞
, i = 1, ..., N.
(16)
N
See [3, 4, 5]. We denote by U the n × N dimensional
matrix with columns {ui }N
i=1 . If these vectors constitute a
tight frame, then U U ∗ = In , where In denotes the identity
SAMPTA'09
1/2
λ
λ
log 1 +
,
λ−1
λ−1
(19)
where λ = N/n with respect to this frame.
See also [4, 5] for further discussion of this result. In
[5] Lyubarskii and Vershynin have recently given an algorithm for determining a Kashin representation. In the
same paper they discuss various ways to generate the required frames and determine their Kashin constants.
Theorem 2. For a given amplitude constraint A, there
exists a constant λmin such that the capacity CP,A of the
discrete Gaussian channel with average power constraint
P , amplitude constraint A and noise variance σ 2 is lower
bounded by
λmin P
1
log 1 +
.
(20)
CP,A ≥
2λmin
σ2
105
Proof Theorem 1 shows the existence of a frame with the
necessary properties, as discussed in the communication
strategy in Section 2. Denoting the matrix corresponding
to this frame by U , for each codeword X(ω) ∈ Rn , there
exists a codeword Y (ω) ∈ RN such that X(ω) = U Y (ω),
and
N
kY (ω)kl∞
K(λmin )
√
kX(ω)kl2n
N
√
≤ K(λmin ) P .
≤
(21)
(22)
Lastly, λmin is the solution to
C
1/2
A
λ
λ
=√ ,
log(1 +
)
λ−1
λ−1
P
which exists and is unique since
is monotone increasing.
4.
λ
λ−1
log(1 +
(23)
1/2
λ
λ−1 )
Conclusion
We have considered an application of the redundant representations found in frame theory and geometric functional
analysis to a fundamental question in information theory.
References:
[1] T. M. Cover and J. A. Thomas. Elements of Information Theory. Wiley, New York, 1991.
[2] A. Garnaev and E. D. Gluskin. The widths of euclidean balls. Doklady An. SSSR., 277:1048–1052,
1984.
[3] B. S. Kashin. Diameters of some finite-dimensional
sets and classes of smooth functions. Izv. Akad. Nauk
SSSR Ser. Mat., 41(2):334–351, 478, 1977. English
transl. in Math. USSR IZV. 11 (1978), 317-333.
[4] B. S. Kashin and V. N. Temlyakov. A remark on compressed sensing. Mathematical Notes, 82(5):748–755,
Nov 2007.
[5] Y Lyubarskii and R. Vershynin. Uncertainty principles and vector quantization. preprint.
[6] W. Oettli. Capacity-achieving input distributions for
some amplitude-limited channels with additive noise
(corresp.). IEEE Transactions on Information Theory,
20(3):372–374, May 1974.
[7] C.E. Shannon. A mathematical theory of communication. Bell System Technical Journal, 27:379–423,623–
656, 1948.
[8] Joel G. Smith. The Information Capacity of Amplitude and Variance Constrained Scalar Gaussian Channels. Information and Control, 18:203–219, 1971.
SAMPTA'09
106
Erasure-proof coding with fusion frames
Bernhard G. Bodmann(1) , Gitta Kutyniok(2) and Ali Pezeshki(3)
(1) Department of Mathematics, University of Houston, Houston, TX 77204, USA
(2) Institute of Mathematics, University Osnabrueck, 49069 Osnabrueck, Germany
(3) Electrical and Computer Engineering Department, Colorado State University, Fort Collins, CO 80523, USA
bgb@math.uh.edu, kutyniok@uni-osnabrueck.de, pezeshki@engr.colostate.edu
Abstract:
The main goal of this paper is the design of frames for
transmitting vectors through a memoryless analog erasure
channel. The channel transmits the frame coefficients perfectly or discards them, depending on the outcomes of
Bernoulli trials with a failure probability q. For sufficiently small q, we construct frames which encode above a
fixed non-zero rate and allow the receiver to recover part of
the erased coefficients so that the remaining mean-square
error vanishes as the frame size increases. We give examples for which the mean-square reconstruction error remaining after corrections are applied decays faster than
any inverse power of the number of frame vectors.
1.
Introduction
We are concerned with the linear transmission of vectors
through a memoryless channel that either transmits a coefficient perfectly or discards it, in accordance with the
outcomes of independent, identically distributed Bernoulli
trials. The problem of reconstructing a vector in a finitedimensional real or complex Hilbert space when not all
of its frame coefficients are known has already received
much attention in the literature [1–9]. However, many results focus on optimal performance for the smallest possible number of erased coefficients [4, 7–9], which is not
typical for transmissions via a memoryless erasure channel. Other results on so-called maximally robust frames
guarantee recovery from a certain fraction of lost frame
coefficients [10], but this may involve inverting an arbitrarily ill-conditioned matrix.
The notion of a memoryless analog erasure channel is
simply one that transmits each frame coefficient independently with a given success probability q and otherwise
erases it, meaning it does not let the receiver access the
coefficient. Within this error model for transmissions, we
investigate the performance of fusion frames [11–13], previously also referred to as frames of subspaces [14] or
weighted projections resolving the identity [15], which
lend themselves to various methods of error correction.
What makes the fusion frames useful for error correction
purposes is that they have many subsets which are frames
for their span. Thus, one can design hierarchical methods
SAMPTA'09
for error correction which make error estimates feasible.
The main result presented here is that for a fixed, sufficiently small erasure probability q, we design fusion
frames such that their associated coding rate is bounded
away from zero and the mean-square error remaining after
error correction is applied decays faster than any polynomial in terms of the number of frame vectors.
The techniques for our results involve combinatorial elements similar to the construction of product codes initially investigated by Elias [16], together with some framespecific arguments.
2. Preliminaries
Throughout the paper, we let H be a real or complex
Hilbert space. Instead of expanding vectors in Hilbert
spaces with orthonormal bases, many applications nowadays use frames, stable, non-unique (redundant) expansions, for various purposes. We first briefly recall the basic
terminology, and refer the reader to [17] for further details.
Definition 1. We call a family of vectors F = {fj }j∈J
in H a frame if there exist constants
PA, B > 0 such that
for all x ∈ H with kxk = 1, A ≤ j∈J |hx, fj i|2 ≤ B.
If we can choose A = B, then we say that the frame is
A-tight. In case A = B = 1 we call F a Parseval frame.
A frame is called equal-norm if there is a c > 0 such that
all vectors have the norm kfj k = c. With each frame F,
we associate the analysis operator V : H → ℓ2 (J), which
maps a vector to its frame coefficients, (V x)j = hx, fj i.
The fact that a vector is over-determined by its frame coefficients helps correct errors which may occur in the course
of a transmission, or when frame coefficients are stored
in an unreliable medium. A main goal of frame design is
to optimize the performance of a frame given certain constraints. This could be, for example, the dimension of the
Hilbert space and the number of frame vectors, or their ratio. In analogy with binary codes, we define a coding rate
for a given frame.
Definition 2. Let H be a Hilbert space of dimension d and
F a frame for H consisting of n vectors. We say that F
has a coding rate of R = d/n.
The coding and error correction method we discuss hereafter relies on frames arising from tensor product constructions. These frames are a special type of a fusion
107
frame, see e.g. [12–15].
ily of vectors F = {fj1 ⊗ fj2 ⊗ · · · ⊗ fjm : ji ∈
Ji for all i} is a tight frame for H = H1 ⊗H2 ⊗· · ·⊗Hm .
We call this frame F a tight product frame.
Remark 1. If F is a Parseval frame then (V ∗ EV −
I)x = V ∗ (E − I)V x and the inverse can be obtained
from
Neumann series (V ∗ EV )−1 =
P∞ the norm-convergent
∗
n
n=0 (V (I − E)V ) . Applying this operator to the output of blind reconstruction gives perfect reconstruction of
the input vector.
Remark 1. We note that if we fix all but one index, say the
(1)
(2)
(m−1)
last, then the resulting set fj1 ⊗fj2 ⊗· · ·⊗fjm−1 ⊗F (m)
is a tight frame for its span. Therefore, F has a natural
fusion frame architecture.
Similarly, fixing only the first m − k indices of the frame
vectors in the tensor product would provide a tight frame
for a subspace for any 0 ≤ k < m. Moreover, there
is a partial ordering on these tight frames for subspaces
induced by the partial ordering of the subspaces they span.
Next, we define a measure for average reconstruction performance when probabilities for erasures are known. To
this end, we average the square of the reconstruction error
with the distribution of erasures and input vectors. Here
and herafter, we denote the expectation of any random
variable η with respect
to the underlying probability meaR
sure P by E[η] = ηdP.
Definition 3. Given Hilbert spaces H1 , H2 , . . . Hm and
(i)
tight frames F (i) = {fj }j∈Ji for each Hi , then the fam(1)
(2)
(m)
3. Erasures and the mean-square error
A communication system is given by a frame F for a
Hilbert space H, and an error model for the transmission of frame coefficients. Our main error model assumes memoryless erasures, that is, the values of randomly selected frame coefficients become unknown in
the course of transmission, in accordance with the outcomes of Bernoulli trials. In brief, frame coefficients are
erased, independently of each other, with a fixed probability q ≥ 0.
Depending on the implementation of decoding, the performance of a frame can be measured in different ways; we
generally distinguish active error correction and blind reconstruction. When actively correcting erasures, one tries
to fill in the values for the erased coefficients, and aims
for a high probability of successfully restoring all lost coefficients. When blind reconstruction is used, one sets the
missing coefficients to zero and reconstructs always in the
same way. In this case, the usual goal is obtaining a small
error norm, such as the mean-square error or the worstcase error.
In the present work we consider a combination of the two
approaches. We measure the quality of error correction by
the mean-square error that results from using the corrected
coefficients with the possibly remaining, uncorrected erasures set to zero. The average in this mean-square error
is taken over the random erasures and over random unitnorm input vectors. For simplicity, we consider input vectors which are independent of the erasures and uniformly
distributed on the unit sphere of the Hilbert space.
Definition 4. Let F = {f1 , f2 , . . . fn } be a Parseval frame for a real or complex Hilbert space H. The
blind reconstruction error for an input vector x ∈ H
and an erasure of frame coefficients with indices K =
{j1 , j2 , . . . jm }, m ≤ n, is given by
kV ∗ EV x − xk = k(V ∗ EV − I)xk
where E is the diagonal n × n matrix with Ej,j = 1 if
j 6∈ K and Ej,j = 0 else. If the positive operator V ∗ EV
has a bounded inverse, then we say that the corresponding
SAMPTA'09
erasure is correctible.
Definition 5. Let {βj }j∈J be a family of binary ({0, 1}valued) random variables governed by a probability measure P, and let ∆ be the random diagonal matrix with entries ∆j,j = βj . Moreover, let ξ be a random variable
with values in the unit sphere {x ∈ H : kxk = 1} which
is independent of the family {βj }, and assume that the distribution of U ξ is identical to that of ξ for any fixed unitary
U . Given a Parseval frame F for a Hilbert space H with
analysis operator V , we define the mean-square error by
σ 2 (V, β) = E[kV ∗ ∆V ξk2 ] .
There is a simple expression for the mean square error as
the square of a weighted Frobenius norm of the Grammian
V V ∗.
Lemma 1. Let {βj }j∈J be as above, assume the family is
identically distributed with probability P(β1 = 1) = q,
and assume the joint distribution is such that P(βj =
βj ′ = 1) = r for all j 6= j ′ . Let ∆ be the random diagonal matrix with entries ∆j,j = βj . If V is the analysis
operator of a Parseval frame F = {fj }j∈J containing
n = |J| vectors in a Hilbert space of dimension d, then
σ 2 (V, β) =
n
n
X
X
1
(q − r)
kfj k4 + r
|hfj , fl i|2 .
d
j=1
j,l=1
4. Bounding the mean-square error for iterative decoding
This section describes how product frames can be used to
trade an increase in block length of encoding for better
error correction capabilities.
We first consider the simplest case in which H has two factors, H = H1 ⊗H2 . Also, as preparation for our main theorem, we first consider packet erasures [15] instead of erasures for single frame coefficients. This means, we have
a frame F = F (1) ⊗ F (2) and a two-parameter family of
random variables {βj,j ′ } which govern erasures of frame
coefficients in such a way that either all coefficients belonging to some j ′ are erased or all of them are left intact.
We compute the mean-square error for this error model.
Proposition 1. Let H = H1 ⊗H2 and let V1 and V2 be the
(1)
analysis operators of Parseval frames F (1) = {fj }j∈J1
(2)
108
and F (2) = {fj ′ }j ′ ∈J2 for H1 and H2 having dimension
d1 and d2 , respectively. Let {βj,j ′ : j ∈ J1 , j ′ ∈ J2 } be
a two-parameter family of binary random variables which
have probabilities P(βj,j ′ = 1) = q and are distributed
(2)
(2)
such that there is a family {βj ′ }j ′ ∈J2 and βj,j ′ = βj ′
almost surely, regardless of j. The mean-square error for
the frame F and this type of packet erasures reduces to
that of F (2) ,
σ 2 (V1 ⊗ V2 , β) = σ 2 (V2 , β (2) ) .
Next, we continue with three combinatorial lemmata.
They prepare the main result which concerns the error
correction capabilities of tight product frames. The main
problem we wish to address with this result is the following: Given a fixed, sufficiently small erasure probability
q, find frames such that their associated coding rate is
bounded away from zero and the mean-square error remaining after error correction is applied decays fast in
terms of the number of frame vectors.
We show hereafter that product frames of the form F =
F (1) ⊗ · · · ⊗ F (m) , for which each factor F (i) can correct up to two erased frame coefficients, satisfy the desired
properties.
Lemma 2. Let n1 ≥ 3 and let {β1 , β2 , . . . βn1 } be a family of independent, identically distributed random variables which take values
Pn1 in {0, 1}. Suppose q0 = P(β1 =
βj ≥ 3), then
1) and let q1 = P( j=1
q1 ≤
m
Lemma 3. Let {ni }m
i=1 be the sizes of index sets {Ji }i=1 ,
with ni ≥ 3 for all i ∈ {1, 2, . . . m}. Assume there is
an m-parameter family of binary, independent identically
distributed random variables {βj1 ,j2 ,...jm } and associated
(1)
(2)
(m−1)
families {βj2 ,j3 ,...jm }, {βj3 ,j4 ,...jm }, . . . {βjm
} which
(0)
are iteratively defined by βj1 ,j2 ,...jm ≡ βj1 ,j2 ,...jm and
(
Pn
(k−1)
1, if jkk=1 βjk ,jk+1 ,...jm ≥ 3 ,
(k)
βjk+1 ,jk+2 ,...jm =
0, else.
(m−1)
If P(β1,1,...1 = 1) = q0 , then the family {βj
} is inde(m−1)
pendent, identically distributed with qm−1 = P(βj
1) having the bound
1
m−1
P(γj = 1) ≤
2
−1) 31
nm−1 n3m−2
· · · n13
m−1
q03
m−1
=
P(γj1 = γj2 = 1) ≤ n2 q 4 .
These lemmata allow us to formulate an error bound for
the remaining mean-square error for blind reconstruction
after the error correction protocol has been applied.
Theorem 1. Let V = V1 ⊗ V2 ⊗ · · · ⊗ Vm be the analysis
operator of a Parseval product frame F = F (1) ⊗ F (2) ⊗
· · · ⊗ F (m) for a Hilbert space H = H1 ⊗ H2 ⊗ · · · ⊗ Hm .
Denote the dimension of each Hi by di and the number of frame vectors in F (i) by ni . Let {βj1 ,j2 ,...jm } be
an m-parameter family of binary independent, identically
(m−1)
distributed random variables, define {βj
} as above,
Pnm
(m−1)
(m−1)
≥ 3 and
if jm =1 βjm
and let γj1 ,j2 ,...jm = βjm
γj1 ,j2 ,...jm = 0 otherwise, then
σ 2 (V, γ) ≤
The probability computed in the above lemma is the probability of an erased block after applying erasure correction
iteratively. The next lemma considers what happens when
the error correction is applied to packets at the final level.
Here, we deviate from the strategy of only reconstructing nontrivially when at most two packets are missing.
Instead, we correct for missing packets and compute the
SAMPTA'09
probabilities for the residual mean-square error.
nm
X
1
(m)
(qm − rm )
kfj k4
dm
j=1
+ rm
nm
X
j,l=1
(m)
|hfj
(m)
, fl
i|2
with
qm = 61−2·3
m−1
1
2
4·3
4·3
n4·3
n3m nm−1
m−2 · · · n1
and
rm ≤
m−1
q04·3
m−1
6
qm .
nm
Corollary 1. If V = V1 ⊗ V2 ⊗ · · · ⊗ Vm and all Vi belong to equal-norm Parseval frames, then it is well known
(i)
that kfj k2 = ndii and by the Cauchy Schwarz inequality
(i)
(i)
|hfj , fl i|2 ≤ d2i /n2i . Thus, we have
σ 2 (V, γ) ≤ qm
dm
dm
+ rm dm ≤ 7qm
nm
nm
with
qm = 61−2·3
.
1 3 4
n q ,
6
and for j1 6= j2 , we have
1 3 3
n q .
6 1 0
The probability estimated in this lemma is that of a packet
of n1 coefficients remaining corrupted after an error correction protocol has been applied which can correct any
two erased coefficients.
By iteration, we obtain a simple consequence.
qm−1 ≤ 6− 2 (3
Lemma 4. Let {β1 , β2 , . . . , βn }, n ≥ 1, be independent, identically distributed binary random variables with
probability P(β1 = 1) = q. Let the P
random variables
n
γ1 , γ2 , . . . , γn be defined by γj = βj if j=1 βj ≥ 3, and
otherwise γj = 0 for all j ∈ {1, 2, . . . n}. Then, for any j,
m−1
1
2
4·3
4·3
n3m n4·3
m−1 nm−2 · · · n1
m−1
q04·3
m−1
.
Example 1. Assume that an equal-norm product frame
F = F (1) ⊗ · · · ⊗ F (m) has F (i) with ni = i2 n1 vectors
for each i ∈ {1, 2, . . . m} and n1 ≥ 3. Let the dimension
of the Hilbert space Hi spanned by F (i) be
dim(Hi ) = i2 n1 − 2 ,
and assume the frame can correct any two erased coefficients. Examples of such frames are the harmonic ones,
109
see e.g. [2].
The tensor product of these m Hilbert spaces, H =
⊗m
i=1 Hi , has dimension
dim(H) = (m!)2 nm
1
m
Y
i=1
2
1− 2
.
i n1
This means, the coding rate R is bounded, independently
of m, by
m
Y
∞
2 X 1
2
2
)
1
−
>
(1
−
i2 n1
n1
n1 i=2 i2
i=1
2 π2
2
= (1 −
) 1−
−1 .
n1
6n1 6
R>
1−
It is straightforward to check that n1 ≥ 3 ensures R > 0.
The preceding theorem then states that after correcting
erasures, the probability of an uncorrected block at the
final level is
qm ≤ m6 n31 61−2·3
m−1
q04·3
m−1
e4
Pm−1
k=1
3m−k ln(k2 n1 )
and upon estimating the sum in the exponent with Jensen’s
inequality,
2
m−1
X
k=1
3−k ln k ≤ 2
∞
X
k=1
3−k ln k ≤ ln
3
,
2
we have
qm ≤ m6 n31 61−2·3
m−1
q04·3
m−1
e2(3
m
−1) ln n1 4·3m ln
e
3
2
.
To achieve exponential decay of qm in 3m requires
−2 ln 6 + 4 ln q0 + 6 ln n1 + 12 ln
3
< 0,
2
which amounts to
27
√ q0 n13/2 < 1 .
8 6
Since n1 = 3 is the smallest dimension to start the iteration,
fast decay of the mean-square error needs q0 <
√
8 2/81 ≈ 0.14.
The number of transmitted frame coefficients is (m!)2 nm
1 ,
1
so by Stirling’s approximation O(e(m+ 2 ) ln m+m ln n1 ),
whereas by the preceding corollary the decay of the meanm
square error is of order O(e−c3 ), for a suitable c > 0.
This implies that the mean-square error decays faster than
any inverse power of the number of transmitted coefficients.
Acknowledgment
This work was partially supported by National Science
Foundation grant DMS 08-07399 and by the Deutsche
Forschungsgemeinschaft under Heisenberg Fellowship
SAMPTA'09
KU 1446/8-1.
References:
[1] V. K. Goyal, M. Vetterli, and N. T. Thao, Quantized
overcomplete expansions in Rn : analysis, synthesis,
and algorithms. IEEE Trans. Inform. Theory, 44(1):
16–31, 1998.
[2] V. K. Goyal, J. Kovačević, and J. A. Kelner, “Quantized frame expansions with erasures,” Appl. Comp.
Harm. Anal., 10:203–233, 2001.
[3] J. Kovačević, P. L. Dragotti, and V. K. Goyal, “Filter
bank frame expansions with erasures,” IEEE Trans.
Inform. Theory, 48:1439–1450, 2002.
[4] P. Casazza and J. Kovačević, “Equal-norm tight
frames with erasures,” Adv. Comp. Math., 18:387–
430, 2003.
[5] G. Rath and C. Guillemot, Performance analysis
and recursive syndrome decoding of DFT codes for
bursty erasure recovery, IEEE Trans. on Signal Processing, 51 (5):1335–1350, 2003.
[6] G. Rath and C. Guillemot, Frame-theoretic analysis
of DFT codes with erasures, IEEE Transactions on
Signal Processing, 52 (2):447–460, 2004.
[7] R. Holmes and V. I. Paulsen, “Optimal frames for
erasures,” Lin. Alg. Appl., 377:31–51, 2004.
[8] B. G. Bodmann and V. I. Paulsen, “Frames, graphs
and erasures,” Linear Algebra Appl., 404: 118–146,
2005.
[9] D. Kalra, Complex equiangular cyclic frames and
erasures, Linear Algebra Appl., 419:373–399, 2006.
[10] M. Püschel and J. Kovačević, “Real, tight frames
with maximal robustness to erasures”, Proc. Data
Compr. Conf., Snowbird, UT, 63–72, March 2005.
[11] P. G. Casazza and G. Kutyniok, Robustness of fusion frames under erasures of subspaces and of local
frame vectors, Contemp. Math., 464, Amer. Math.
Soc., Providence, RI, 149–160, 2008.
[12] P. G. Casazza, G. Kutyniok, and S. Li, “Fusion
Frames and Distributed Processing,” Appl. Comput.
Harmon. Anal., 25:114–132, 2008.
[13] G. Kutyniok, A. Pezeshki, A. R. Calderbank,
and T. Liu, “Robust Dimension Reduction, Fusion
Frames, and Grassmannian Packings,” Appl. Comput. Harmon. Anal., 26:64–76, 2009.
[14] P. G. Casazza and G. Kutyniok, “Frames of subspaces,” in: “Wavelets, frames and operator theory,” Contemp. Math., 345, Amer. Math. Soc., Providence, RI, 87–113, 2004.
[15] B. G. Bodmann, “Optimal linear transmission by
loss-insensitive packet encoding,” Appl. Comput.
Harmon. Anal., 22:274–285, 2007.
[16] P. Elias, Error-free coding, IRE Trans. IT, 4:29–37,
1954.
[17] O. Christensen, “An Introduction to Frames and
Riesz Bases,” Birkhäuser, Boston, 2003.
110
Representation of operators by sampling in the
time-frequency domain
Monika Dörfler (1) and Bruno Torrésani(2)
(1) ARI, Austrian Academy of Science, Wohllebengasse 12-14, A-1040 Vienna, Austria.
(2) LATP, Centre de Mathématique et d’Informatique, 39 rue Joliot-Curie, 13453 Marseille cedex 13, France.
Monika.Doerfler@oeaw.ac.at, Bruno.Torresani@cmi.univ-mrs.fr
Abstract:
Gabor multipliers are well-suited for the approximation of
certain time-variant systems. However, this class of systems is rather restricted. To overcome this restriction, multiple Gabor multipliers allowing for more than one synthesis windows are introduced. The influence of the choice of
the various parameters involved on approximation quality
is studied for both classical and multiple Gabor multipliers.
efficient, e.g. in the sense of sparsity, to use several side
diagonals, but a lower redundancy in the Gabor system
used.
The aim of this contribution is the description of error estimates for the approximation of operators by generalized
Gabor multipliers, based on the operator’s spreading function. From this description guidelines for the choice of
good parameters for the approximation are deduced and
illustrated by various numerical experiments.
1.
2.
Introduction
In a recent paper [1], the authors describe the representation of operators in the time-frequency domain by means
of a twisted convolution with the operator’s spreading
function. Although not suitable for direct discretization,
the spreading representation provides a better understanding of certain operators’ behavior: it reflects the operator’s
action in the time-frequency domain. This motivates an
approach that uses the spreading representation of timefrequency multipliers [1], in order to optimize the parameters involved. More specifically, in the one-dimensional,
continuous-time case, given an operator H with integral
kernel κH and spreading function ηH :
Z ∞
ηH (b, ν) =
κH (t, t − b)e−2iπνt dt,
−∞
we aim at modeling the operator by its action on the sampled short-time Fourier transform (STFT) or Gabor coefficients, given for any f ∈ L2 (R) by
Vg f (mb0 , nν0 ) = hf, gmn i ,
m, n ∈ Z
(1)
where the gmn = Mnν0 Tmb0 g denote the Gabor atoms associated to g ∈ L2 (R) and the lattice constants b0 , ν0 ∈
R+ , see [3]1 . In the case of classical Gabor multipliers,
the modification consists of a pure multiplication. Thus,
the linear operator applied to the coefficients Vg f is diagonal, an approach that leads to accurate approximation
for so-called underspread operators [5]. The restriction to
diagonality may be relaxed in order to achieve better approximation for a wider class of operators at low cost. It
also appears, that in certain approximation tasks it is more
finite dimensional case H = CL is obtained similarly, replacing integrals with finite sums, and letting m = 0, . . . Nb − 1, n =
0, . . . Nν − 1, where Nb = L/b0 , Nν = L/ν0 and b0 , ν0 divide L.
1 The
SAMPTA'09
Approximation in the time-frequency domain: the parameters
Throughout this paper, H denotes a (finite or infinitedimensional) Hilbert space, equipped with an action of the
Heisenberg group of time-frequency shifts.
2.1
Time-frequency multipliers
Vg∗
Let
denote the adjoint of Vg . A Gabor multiplier [4] is
defined as
M : f ∈ H 7−→ Mf = V2∗ (m · V1 f ).
Here, m is the pointwise multiplication operator whose
symbol, defined on the lattice Λ will also be denoted by m.
We shall denote by Λo the adjoint lattice, o its fundamental domain, and Πo the corresponding periodization operator. In the infinite-dimensional situation H = L2 (R),
and for a product lattice of the form Λ = b0 Z × ν0 Z,
we have Λo = P
t0 Z × ξ0 Z with t0 = 1/ν0 , ξ0 = 1/b0 ,
and Πo f (ζ) = λo ∈Λo f (ζ + λo ), ζ ∈ o . In a finitedimensional setting H = CL , with Λ = ZNb × ZNν , with
Nb , Nν two divisors of L, we have Λo = ZNν × ZNb , and
the obvious form for the periodization operator.
In the definition of the multipliers, several parameters have
to be fixed: the analysis and synthesis windows g and h,
the lattice Λ, and the symbol m. For practical as well as
theoretical reasons, the windows should be well-localized
in time and frequency. As for the lattice, it is expected that
denser lattices will lead to better results in approximation,
but higher computational cost. However, it will be seen
that too dense lattices are not suitable.
Finally, the symbol m can be optimized to best approximate a given operator. In [1], an explicit expression for
the best approximation was obtainned in the spreading domain, yielding a very efficient algorithm (compare [2]).
111
The spreading function of Gabor multipliers takes the
form ηM (ζ) = M (ζ) · Vg h(ζ) , where M is the symplectic Fourier transform of m. Note, that this leads to a
periodic function with period o . Hence, good approximation by a classical Gabor multiplier is possible, if the
essential support of the spreading function is smaller than
1 and can then be contained in the fundamental domain o
of the adjoint lattice for a dense enough lattice Λ. Also,
to reduce aliasing as much as possible, the analysis and
synthesis windows must be chosen such that Vg h is small
outside o and positive on the support of the spreading
function, also see Section 4.1.
2.2
Generalized Gabor multipliers
Multiple Gabor multipliers are sums of Gabor multipliers
with different synthesis windows.
Definition 1 (Multiple Gabor Multiplier) Let g, h ∈ H
denote two window functions. Let Λ be a time-frequency
lattice. Let {µj , j ∈ J} denote a finite set of timefrequency shifts, and let {mj , j ∈ J} be a family of
bounded functions on Λ. Set h(j) = π(µj )h, then the
associated generalized Gabor multiplier M is defined, for
f ∈ H, as
XX
Mf =
m(λ, µj )hf, π(λ)giπ(λ)h(j) .
λ∈Λ j∈J
It is immediately obvious that in addition to the parameters
mentioned above, the window h as well as the sampling
points J must be chosen.
3.
Error analysis in L2 (R)
In [1], it was shown that the symbol m(λ, µj ) := mj (λ)
of the best approximation of a Hilbert-Schmidt operator
by a multiple Gabor multiplier with fixed sets Λ, J and
windows, is given by the symplectic Fourier transform
of the o -periodic functions Mj obtained via the vector
equation
M (ζ) = U(ζ)−1 · B(ζ) ,
ζ ∈ o ,
∞
The finite-dimensional situation is similar, replacing the
integral over o with a finite sum over the finite fundamental domain {0, . . . t0 − 1} × {0, . . . ξ0 − 1}.
4.
Choosing the parameters
For simlicity, we specialize the following discussion to the
infinite-dimensional case H = L2 (R), and rectangular lattice Λ = b0 Z × ν0 Z. The finite-dimensional situation is
handled similarly.
4.1
Gabor Multipliers
If an operator with known spreading function is to be
approximated by a Gabor multiplier, the lattice may be
adapted to the eccentricity of the spreading function according to the error expression obtained in Proposition 1,
which may be considerably simplified for the case of only
one synthesis window, see [1]. In order to choose the
eccentricity of the lattice accordingly and adapt the window to the chosen lattice as to avoid aliasing, assume,
that we may find b0 , ν0 , with b0 · ν0 < 1, such that
supp(ηH ) ⊆ Tz o , where o = [0, ν10 ] × [0, b10 ]. In
this case, the error resulting from best approximation by
a Gabor multiplier with respect to the lattice b0 Z × ν0 Z is
bounded by Ce · kηH k22 , with
|Vg h(t, ξ)|2
,
2
k,l |Vg h(t + kt0 , ξ + lξ0 )|
Ce = 1 − inf o P
t,ξ∈H
(3)
with oH = o ∩ Supp(ηH ), and becomes minimal for a
window that is optimally concentrated inside o . Heuristically as well as from numerical experiments we know,
that the tight window, [3], corresponding to the given lattice is usually a good choice to fulfill this requirement.
(2)
where the matrix and vector valued functions U and B are
given by the Λo -periodizations
′
Ujj ′ = Πo Vg h(j ) Vg h(j) , Bj = Πo ηH Vg h(j) ,
provided U is invertible a.e.
The case of one synthesis windows may be immediately
obtained from the above formula. Note that formula (2)
allows for an efficient implementation of the otherwise expensive calculation of the best approximation by multiple
Gabor multipliers.
We may now give an expression for the error in the approximation given above, in the case H = L2 (R)
Proposition 1 Let M denote the vector-valued function
obtained as in (2) and set, for the Hilbert-Schmidt operator H, ΓH =PΠo (|ηH |2 ). Then the approximation error
E = kηH − j Mj Vj k2 is given by
P
Z
−1
)ij (ζ)Bi (ζ)Bj (ζ)
i,j (U
E=
|ΓH (ζ)| 1 −
dζ
|ΓH (ζ)|
o
SAMPTA'09
Notice that this covers the multiplier case obtained in [1].
Notice also that this immediately yields
P
−1
)ij Bi Bj
i,j (U
2
E ≤ kηH k 1 −
|ΓH |
4.2
Generalized Gabor Multipliers
The main additional task in the generalized situation is the
choice of the sampling points µj for the synthesis windows. A good choice will again be guided by the behavior of the spreading function. The relevant areas in the
spreading domain should be covered as well as possible
with the smallest possible overlap by the cross-ambiguity
functions of the different synthesis windows with respect
to a given reference-window localized at (0, 0) e.g. the
Gaussian window. Motivated by the results from the Gabor multiplier situation, we choose a tight window with
respect to the analysis lattice and look for the most appropriate sampling points for the synthesis windows. Examples will be given in Section 5.2.
5.
Examples
We now turn to numerical experiments, in the finite case
H = CL . In the following examples, the relative approximation error for the best approximation H̃ of H is given
112
b0 = 2,3,4,5,6,9,10,12,15,18; ν0 = 2
Approximation error for redundancy 5 and different lattice eccentricities
b0 = 2,3,4,5,6,9,10,12,15,18; ν0 = 3
−0.5
−2
−4
−6
−8
−10
−12
−5
−10
−15
−1
−1.5
b0 = 6,ν0 = 6
b0=4,ν0 = 9
−2
5
10
15
20
25
30
5
b0 = 2,3,4,5,6,9,10,12,15,18; ν0 = 5
10
15
20
25
b0=9,ν0 =4
30
b0=8,ν0 = 2
b0=2,ν0 =8
−2.5
b0 = 2,3,4,5,6,9,10,12,15,18; ν0 = 6
b0=18,ν0 = 2
−3
−1
−1
−2
5
−3
−2
−4
−3
10
15
20
25
30
5
b = 2,3,4,5,6,9,10,12,15,18; ν = 9
0
10
15
20
25
30
25
30
.
b = 2,3,4,5,6,9,10,12,15,18; ν = 10
0
15
20
Support of Spreading function
Figure 2: Approximation error for different lattice-eccentricity
−5
5
10
0
Approximation error, b0 = ν0 = 6
0
−0.5
−0.5
−0.5
−1
−1
−1
−1.5
−1.5
−1.5
−2
5
10 15 20 25
Support of Spreading function
30
5
10 15 20 25
Support of Spreading function
30
Figure 1: Approximation error for different bandwidth of
spreading function and different values of b0 , ν0 .
1 window
2 windows
−2.5
−3
−3.5
5
by
E = kH̃ − Hk/kHk ,
the logarithm of which is represented in the next plots.
We display here the Fröbenius norm, the plots obtained
with the operator norm are almost identical.
5.1
Classical Gabor Multipliers
We generate operators with compact support in the spreading domain, in a square of side size between 3 and 61,
symmetric about 0. The values are random, the signal
length is L = 180. We then investigate the approximation
quality for various pairs of lattice constants, with b0 varying between 2 and 18 and ν0 between 2 and 10. The results
are presented in Figure 1. Note the two distinct regimes:
the error grows exponentially up to a certain value of the
support size, depending on the lattice density, and slower
thereafter. A possible explanation for this effect, to be further investigated, is the fact, that the error (see the bound
in (3)) is comprised of an aliasing error and the inherent
inaccuracy of Gabor multiplier approximation, even for
very high sampling density, of overspread operators.
In order to emphasize the importance of lattice adaptation to eccentricity, we show the results for different lattice
constants resulting in the same redundancy (5) in Figure 2.
The solid lines show the results for b0 = ν0 = 6, leading
to far better results than the lattice constants not adapted
to the (symmetric) support of the spreading function.
5.2
Generalized Gabor Multipliers
In order to illustrate the influence of additional synthesis
windows on the approximation quality, we first consider
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10
15
20
Support Spreading function
25
30
Figure 3: Spreading function of operator and best approximation with one or two synthesis windows, approximation error for
growing support of spreading function.
the same operators as in the previous section, but allow
for one additional synthesis window. Here, and in the subsequent examples, one window will always be a window
centered about 0, as above, with a time-shifted version of
the original window as additional window. Hence, only
the shift-parameter of the additional window has to be
considered. Figure 3 shows the improvement in approximation quality for shift-parameters of the additional window between −5 and 5 (solid), as opposed to the single
window approximation.
Next, we investigate the following situation: an operator
with two effectively disjoint components in the spreading
domain is, again, approximated by a multiple Gabor multiplier with 2 synthesis windows. For better comparison,
the two components are the component from the previous
examples plus a shifted version (by 90 samples) thereof.
Figure 4 shows the spreading functions of one of the operators and its best approximation with two synthesis windows, for the optimal additional window. Note the aliasing
effect. In this situation, using two appropriate synthesis
windows, the obtained results are similar to those in the
case of one spreading function component and one synthesis window, as discussed in the previous section. In
Figure 5, we display the results for 3 symmetric pairs of
lattice constants, the optimal window’s result being represented by the solid line, while the dashed lines show the
results of close but suboptimal synthesis windows. As the
operator was generated by a translation by 90 samples, the
113
0
−80
−60
−60
−40
−40
−20
−20
0
0
20
20
40
40
60
60
80
80
0
0
0
0
0
−0.2
−0.4
Approximation error
−80
−50
b =ν =6
Spreading function approximation, b = ν = 6
Spreading function operator, k = 12
−0.6
−0.8
−1
−1.2
−1.4
−1.6
−50
50
0
50
18
16
Figure 4: Spreading function of operator and best approxima-
14
tion.
12
Support Spreading function
Approximation error, b = ν = 4
0
10
0
0
20
10
30
40
50
60
Translation parameter g2
−2
Figure 6: Approximation error for growing support of spread-
−4
ing function and various additional synthesis windows.
−6
2
4
6
8
10
12
14
16
18
20
22
16
18
20
22
10
12
14
16
Support Spreading function
18
20
22
Approximation error, b0 = ν0 = 6
−1
−2
−3
2
4
6
8
10
12
14
Approximation error, b0 = ν0 = 10
−0.2
−0.4
−0.6
−0.8
−1
−1.2
2
4
6
8
Figure 5: Approximation error for varying support of two com-
going from |J| = 1 to larger index sets J involves inverting (generally small) matrices instead of computing a
point-wise ratio. Higher redundancy of the Gabor system
involved is more expensive in the sense of coefficients. In
many cases, using an additional window may be more favorable in improving approximation quality than a denser
lattice. Future work on this topic will include systematic
numerical experiments as well as the analytical investigation of the approximation quality of generalized and classical Gabor multipliers. Another goal is the development
of a method to determine an adapted sampling scheme for
the synthesis windows from an operator’s spreading function.
7.
Acknowledgments
ponents of spreading function and two synthesis windows.
tight window, shifted by 90 samples itself, is expected to
be the optimal additional window. This is confirmed by
the experiments.
In a last experiment, the two components in the spreading domain are close and, for growing bandwidth, overlapping. Figure 6 shows, as before, the results of approximation for growing support of both spreading function components, with b0 = ν0 = 6 and various additional synthesis windows. The additional window with shift-parameter
0 is, of course, the original window and yields the approximation result obtained for a single synthesis window. For
the optimal window, the result is close to the single window/single component case for the same lattice.
6.
Discussion and conclusions
The examples given in the previous section show that the
choice of various parameters has considerable influence
on the performance of approximation by (generalized) Gabor multipliers. While the situation is rather easily understood in the case of classical Gabor multipliers, it is much
more intricate in the generalized case. It should be noted
that, while yielding better results in the approximation, using a small number of additional synthesis windows does
not dramatically increase the computational cost: in (2),
SAMPTA'09
The first author was funded by project MA07-025 of
WWTF Austria. The second author was partly supported
by the CNRS programme PEPS/ST2I MTF&Sons.
References:
[1] Monika Dörfler and Bruno Torrésani. Representation
of operators in the time-frequency domain and generalized Gabor multipliers. arXiv:0809.2698, 2008, to
appear in Journal of Fourier Anal. and Appl.
[2] Hans G. Feichtinger, Mario Hampejs, and Günther
Kracher. Approximation of matrices by Gabor multipliers. IEEE Signal Proc. Letters, 11(11):883– 886,
2004.
[3] Hans G. Feichtinger and Thomas Strohmer. Gabor
Analysis and Algorithms. Theory and Applications.
Birkhäuser, 1998.
[4] Hans Georg Feichtinger and Kristof Nowak. A first
survey of Gabor multipliers. In H. G. Feichtinger
and T. Strohmer, editors, Advances in Gabor Analysis, Boston, 2002. Birkhauser.
[5] Werner Kozek. Adaptation of Weyl-Heisenberg
frames to underspread environments. In [3], 1998.
114
Operator Identification and Sampling
Götz Pfander (1) and David Walnut (2)
(1) School of Engineering and Science, Jacobs University Bremen, 28759 Bremen, Germany.
(2) Dept. of Mathematical Sciences, George Mason University, Fairfax, VA 22030 USA.
g.pfander@iu-bremen.de dwalnut@gmu.edu
Abstract:
Time–invariant communication channels are usually modelled as convolution with a fixed impulse–response function. As the name suggests, such a channel is completely
determined by its action on a unit impulse. Time–varying
communication channels are modelled as pseudodifferential operators or superpositions of time and frequency
shifts. The function or distribution weighting those time
and frequency shifts is referred to as the spreading function of the operator. We consider the question of whether
such operators are identifiable, that is, whether they are
completely determined by their action on a single function or distribution. It turns out that the answer is dependent on the size of the support of the spreading function,
and that when the operators are identifiable, the input can
be chosen as a distribution supported on an appropriately
chosen grid. These results provide a sampling theory for
operators that can be thought of as a generalization of the
classical sampling formula for bandlimited functions.
1. Letting ηH (t, ν) =
Hf (x) =
=
ZZ
ZZ
Z
hH (t, x) e−2πiν(x−t) dx gives
ηH (t, ν) e2πiν(x−t) f (x − t) dν dt
ηH (t, ν) Tt Mν f (x) dν dt.
ηH (t, ν) is the spreading function of H. If supp ηH ⊆
[0, a] × [−b/2, b/2] for some a, b > 0 then a is called the
maximum time-delay and b the maximum Doppler spread
of the channel.
Z
2. Letting σH (x, ξ) = hH (t, x) e2πitξ dt gives
Hf (x) =
Z
σH (x, ξ)fb(ξ) e2πixξ dξ.
σH (x, ξ) is the Kohn-Nirenberg (KN) symbol of H and
we have the relation
ZZ
ηH (t, ν) =
σH (x, ξ) e−2πi(νx−ξt) dx dξ.
The function hH (t, x) is referred to as the impulse response of the channel and is interpreted as the response of
the channel at time x to a unit impulse at time x − t, that
is, originating t time units earlier. If hH (t, x) = hH (t)
then the characteristics of the channel are time-invariant
and in this case the channel is modelled as a convolution
operator. Such channels are identifiable since hH (t) can
be recovered as the response of the channel to the input
signal δ0 (t), the unit-impulse at t = 0.
In other words, the spreading function ηH is the symplectic
Fourier transform of the KN symbol of H.
In 1963, T. Kailath [3, 4, 5] asserted that for time-variant
communication channels to be identifiable it is necessary and sufficient that the maximum time-delay, a, and
Doppler shift, b, satisfy ab ≤ 1 and gave an argument for
this assertion based on counting degrees of freedom. In
the argument, Kailath looks at the response of the channel
to a train of impulses separated by at least a time units,
so that in this sense the channel is being “sampled” by a
succession of evenly-spaced impulse responses. The condition ab ≤ 1 allows for the recovery of sufficiently many
samples of hH (t, x) to determine it uniquely.
Kailath’s conjecture was given a precise mathematical
framework and proved in [6]. The framework is as follows. Choose normed linear spaces D(R) and Y (R) of
functions or distributions on R, and a normed linear space
of bounded linear operators H ⊂ L(D(R), Y (R)). Each
fixed element g ∈ D(R) induces a map Φg : H −→
Y (R), H 7→ Hg. If for some g ∈ D(R), Φg is bounded
above and below, that is, there are constants 0 < A ≤ B
such that for all H ∈ H,
There are two representations of H that will be convenient
for our purposes.
AkHkH ≤ kHgkY ≤ B kHkH
1. Channel Models and Identification
A communications channel is said to be measurable or
identifiable if its characteristics can be determined by its
action on a single fixed input signal. A general model for
linear (time-varying) communication channels is as operators of the form
Hf (x) =
SAMPTA'09
Z
hH (t, x) f (x − t) dt.
115
then we say that H is identifiable with identifier g ∈
D(R).
Taking D = S0′ , Y = L2 , and HS = {H ∈
b
HS(L2 ) : ηH ∈ S0 (R × R),
supp ηH ⊆ S} where
2
b
S ⊆ R × R, HS(L ) is the class of Hilbert-Schmidt operators, and S0 is the Feichtinger algebra (defined below),
the following was proved in [6].
Theorem 1. If S = [0, a] × [−b/2, b/2] then HS is identifiable if and P
only if ab ≤ 1. In this case an identifier is
given by g = n δna .
2. Distributional Spreading Functions and
Operator Sampling
The requirement that ηH ∈ S0 excludes some very natural
operators from consideration in this formalism, for example the identity operator (ηH (t, ν) = δ0 (t)δ0 (ν)), convolution operators (ηH (t, ν) = h(t)δ0 (ν) giving Hf = f ∗
h), and multiplication operators, (ηH (t, ν) = δ0 (t)m(ν)
b
giving Hf = m · f ).
A more natural setting for operator identification is the
modulation spaces (see [2] for a full treatment of the subject). For convenience we give the definitions below for
modulation spaces on R, but all definitions and results can
be extended to Rd . For ϕ ∈ S(R) define for f ∈ S ′ (R)
the short-time Fourier transform (STFT) of f by
Vϕ f (t, ν)
= hf, Tt Mν ϕi
Z
=
f (x) e−2πiν(x−t) ϕ(x − t) dx.
For 1 ≤ p, q ≤ ∞ define the modulation space M p,q (R)
by
M p,q (R) = {f ∈ S ′ (R) : Vϕ f ∈ Lp,q (R)},
that is, for which
kVϕ kLp,q =
µZ µZ
p
|Vϕ f (t, ν)| dt
¶q/p ¶1/q
is finite. The usual modifications are made if p or q =
∞. M p,q is a Banach space with respect to the norm
kf kM p,q = kVϕ f kLp,q and different nonzero choices of
ϕ ∈ S define equivalent norms. The space M 1,1 is the
Feichtinger algebra denoted S0 and M ∞,∞ is its dual S0′ .
The space S0′ contains the Dirac impulses δx : f 7→ f (x)
for
P x ∈ R as well as distributions of the form g =
j cj δxj , xj ∈ R and {cj } ⊆ C a bounded sequence.
In our next step toward operator sampling we observe that
it is possible to take D = S0′ , Y = S0′ , and HS = {H ∈
L(D, Y ) : ηH ∈ S0′ , supp ηH ⊆ S} in the operator identification formalism. Indeed the following theorem was
shown in [10].
Theorem 2. The operator class HS (defined above) is
identifiable if S = [0, a] × [−b/2, b/2] and ab < 1, and is
not identifiable if ab > 1.
completely determined by its actions on a fixed input in
terms of a norm inequality. The next step is to find an
explicit reconstruction formula for the impulse response
of the channel operator directly from its response to the
identifier. Such formulas illustrate a connection between
operator identification and classical sampling theory and
lead to a definition of operator sampling.
If, in the operator identification formalism described earlier, an operator
Pclass H is identified by a distribution of
the form g = j cj δxj , then we call {xj } a set of sampling for H and g a sampling function for the operator
class H. In the results obtained so far, operator sampling
is possible only for operators with compactly supported
spreading function, and in order to interpret Theorem 1 in
this context we make the following definition.
Given a Jordan domain S ⊆ R2 , define the operator
Paley-Wiener space OP W 2 (S) by
OP W 2 (S) = {H ∈ HS(L2 ) : supp ηH ⊆ S}.
OP W 2 is a Banach space with respect to the HilbertSchmidt norm kHkOP W 2 = kηH kL2 . Then Theorem 1
can be extended as follows ([8]).
Theorem 3. Let Ω, T, T ′ > 0 with T ′ < T and ΩT < 1.
′
Then OP W 2 ([0,
P T ] × [−Ω/2, Ω/2]) is identifiable with
identifier g = n δnT and moreover we have the formula
X
hH (t, x) = r(t)
(Hg)(t + kT )ϕ(x − t − kT )
k∈Z
unconditionally in L2 (R2 ), where r ∈ S(R) is such that
r = 1 on [0, T ′ ] and vanishes outside a sufficiently small
neighborhood of [0, T ′ ], and where ϕ ∈ S(R) is such that
ϕ
b = 1 on [−Ω/2, Ω/2] and vanishes outside a sufficiently
small neighborhood of [−Ω/2, Ω/2].
In the more general modulation space setting we can define the operator Paley-Wiener space OP W p,q (S) by
OP W p,q (S)
= {H ∈ L(S0 , S0′ )
: supp ηH ⊆ S, σH ∈ M pq,11 }
where σH (x, ξ) ∈ M pq,11 means that the twodimensional STFT of σH satisfies
¶q/p ¶1/p
Z µZ µZ
|Vϕ⊗ϕ σH (t1 , t2 , ν1 , ν2 )|p dt1 dt2
dν1 dν2
is finite. Here
Vϕ⊗ϕ (t1 , t2 , ν1 , ν2 ) = hf, Tt1 Mν1 ϕ ⊗ Tt2 Mν2 ϕi.
3. A Theory of Operator Sampling
OP W p,q is a Banach space with respect to the norm
kHkOP W p,q = kσH kM pq,11 . In this case, Theorem 3 generalizes as follows ([8]).
Theorem 4. Let 1 ≤ p, q ≤ ∞, Ω, T, T ′ > 0 with T ′ <
′
T and ΩT < 1. Then OP W p,q ([0,
P T ] × [−Ω/2, Ω/2]) is
identifiable with identifier g = n δnT and moreover we
have the formula
X
hH (t, x) = r(t)
(Hg)(t + kT )ϕ(x − t − kT )
In discussing identifiability of operators in various settings, we have been content to show that an operator is
unconditionally in M 1p,q1 (R2 ) and in the weak-* sense if
p or q = ∞, where r and ϕ are as in Theorem 3.
SAMPTA'09
k∈Z
116
Example 1. If we take H to be ordinary convolution by
hH (t), this means that hH (t, x) depends only on t, that is,
hH (t, x) = hH (t). In this case H can be identified in principle by g = δ0 , the unit impulse, since Hg(x) = hH (x).
Translating this into our operator sampling formalism results in something slightly different.
Assume that h ∈ M 1,q is supported in the interval [0, T ′ ]
and that T > T ′ , and Ω > 0 are chosen so that ΩT < 1.
In this case, ηH (t, ν) = h(t) δ0 (ν) and σH (x, ξ) = b
h(ξ).
Therefore σH ∈ M ∞q,11 and H ∈ OP W ∞,q ([0, T ′ ] ×
{0}). P
If g = n δnT then Hg is simply the T –periodized impulse response h(t), and it follows that
X
r(t)
(Hg)(t + kT )ϕ(x − t − kT )
k∈Z
= r(t) h(t)
X
ϕ(x − t − kT ) = h(t)
area of the support of the spreading function. It is notable
that Kailath also asserted something along these lines.
This means that a time-variant channel whose spreading
function has essentially arbitrary support is identifiable as
long as the area of that support is smaller than one.
Using ideas from [6], Bello’s conjecture was proved in [9].
Theorem 5. HS is identifiable if vol+ (S) < 1, and not
identifiable if vol− (S) > 1. Here vol+ (S) is the outer
Jordan content and vol− (S) the inner Jordan content
of S.
P
In this case, the channel is identified by g = n cn δn/L
where L ∈ N and the L–periodic sequence {cn } is chosen
based on the geometry of S.
We next present a generalization of Theorem 4 to this case.
Before stating the result, a few preliminaries are required.
Definition 1. Given L ∈ N, let ω = e−2πi/L and define
the translation operator T on (x0 , . . . , xL−1 ) ∈ CL by
k∈Z
since r(t) = 1 on [0, T ′ ] and
P vanishes outside a neighborhood of [0, T ′ ] and since k ϕ(x − t − kT ) = 1 by the
Poisson Summation Formula and in consideration of the
support
b Indeed the theorem says that the
P constraints on ϕ.
sum k ϕ(x − t − kT ) converges to 1 in the M ∞,1 norm
and in particular uniformly on compact sets.
Example 2. If we take H to be multiplication by some
fixed function m ∈ M p,1 with supp m
b ⊆ [−Ω/2, Ω/2]
then ηH (t, ν) = δ0 (t)m(ν),
b
h(t, x) = δ0 (t) m(x − t),
and σH (x, ξ) = m(x). Therefore σH ∈ M p∞,11 and
H ∈ OPP
W p,∞ ({0} × [−Ω/2, Ω/2]).
If g =
n δnT , with
PT > 0 chosen small enough that
ΩT < 1, then Hg = n m(nT ) δnT , and it follows from
Theorem 4 that
δ0 (t) m(x − t)
X
= r(t)
(Hg)(t + kT )ϕ(x − t − kT )
k∈Z
=
=
r(t)
X
XX
m(nT ) δ(n−k)T (t)ϕ(x − t − kT )
k∈Z n∈Z
m(nT ) ϕ(x − nT )
n∈Z
by support considerations on the function r(t). Therefore
we have the summation formula
X
m(nT ) ϕ(x − nT )
m(x) =
n
where the sum converges unconditionally in M p,1 if 1 ≤
p < ∞ and weak-* if p = ∞, and moreover there are
constants 0 < A ≤ B such that for all such f ,
Akf kM p,1 ≤ k{f (nT )}kℓp ≤ Bkf kM p,1 .
Taking p = 2, this recovers the classical sampling formula
when the sampling is above the Nyquist rate.
4. Spreading functions with nonrectangular
support and Bello’s conjecture
In 1969, P. A. Bello [1] argued that what is important for
channel identification is not the product ab of the maximum time-delay and Doppler shift of the channel but the
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T x = (xL−1 , x0 , x1 , . . . , xL−2 ),
and the modulation operator M on CL by
M x = (ω 0 x0 , ω 1 x1 , . . . , ω L−1 xL−1 ).
Given a vector c ∈ CL the finite Gabor system with window c is the collection {T q M p c}L−1
q,p=0 .
Note that the discrete Gabor system defined above consists
of L2 vectors in CL so is necessarily overcomplete.
Definition/Proposition 2. The Zak Transform
is defined
X
for f ∈ S(R) by Zf (t, ν) =
f (t − n) e2πinν .
n
Zf (t, ν)
satisfies
the
quasi-periodicity
relations
Zf (t + 1, ν) = e2πiν Zf (t, ν)
and
Zf (t, ν + 1) = Zf (t, ν).
Z can be extended to a
unitary operator from L2 (R) onto L2 ([0, 1]2 ).
If the spreading function of H, ηH (t, ν), is supported in
b with vol+ (S) <
a bounded Jordan region S ⊆ R × R
1, then by appropriately shifting and scaling ηH we can
assume without loss of generality that for some L ∈ N,
S ⊆ [0, 1] × [0, L] and that S meets at most L of the L2
rectangles Rq,m = ([0, 1/L] × [0, 1]) + (q/L, m), 0 ≤
q, m < L whose union is [0, 1] × [0, L]. We can further
assume that S does not meet any of the rectangles Rq,m on
the “edge” of the larger rectangle, specifically it does not
meet Rq,m with q = 0, m = 0, q = L − 1 or m = L − 1.
The P
following Lemma connects the output Hg(x) where
g = n cn δn/L to the spreading function ηH (t, ν). From
this a reconstruction formula analogous to that in Theorem 4 can be derived.
Lemma
1. Given a period-L sequence (cn ) and g =
P
c
δ
n n n/L , then for (t, ν) in a sufficiently small neighborhood of [0, 1/L] × [0, 1],
e−2πiνp/L (Z ◦ H)g(t + p/L, ν)
=
L−1
X
X L−1
(T q M m c)p e−2πiνq/L ηH (t + q/L, ν + m).
q=0 m=0
In other words, the spreading function can be realized as
coefficients on the vectors of a finite Gabor system. The
system is in general underdetermined since there are L
117
equations and L2 unknowns. If, however, the support set
S of the spreading function ηH (t, ν) satisfies vol+ (S) < 1
and since S meets at most L of the rectangles Rq,m , there
are at most L nonzero unknowns in the above linear system. If the resulting L × L matrix is invertible, then ηH
can be determined uniquely from Hg. The vector c must
be chosen so that this matrix is invertible. It is shown in
[7] that if L is prime then such a c always exists.
We can prove the following theorem (cf. [8], [9]).
Theorem 6. Let 1 ≤ p, q ≤ ∞. If vol− (S) > 1 then
OP W p,q (S) is not identifiable. If vol+ (S) < 1 then
OP W p,q (S) is identifiable via
P operator sampling, and the
identifier is of the form g = n cn δn/L where L ∈ N and
(cn ) is an appropriately chosen period-L sequence. Moreover, we have the formula
hH (t, x)
=
L−1
X
j=0
rj (t)
X
vector spaces. J. Fourier Anal. Appl., 11(6):715–726,
2005.
[8] G. Pfander and D. Walnut. On the sampling of functions and operators with an application to Multiple–
Input Multiple–Output channel identification. In
Manos Papadakis Dimitri Van De Ville, Vivek
K. Goyal, editor, Proc. SPIE Vol. 6701, Wavelets XII,
pages 67010T–1 – 67010T–14, 2007.
[9] G.E. Pfander and D. Walnut. Measurement of
time–variant channels. IEEE Trans. Inform. Theory,
52(11):4808–4820.
[10] G.E. Pfander and D. Walnut. Operator identifcation
and Feichtinger’s algebra. Sampl. Theory Signal Image Process., 5(2):151–168, 2006.
bj,k (Hg)(t − qj /L + k/L)
k∈Z
× ϕj (x − t − qj /L − k/L)
unconditionally in M 1p,q1 and in the weak-* sense if p =
∞ or q = ∞. For 0 ≤ j < L, the rectangles Rqj ,mj are
precisely those that meet S. Also for each 0 ≤ j < L,
rj (t)ϕ
bj (ν) = 1 on Rqj ,mj and vanishes outside a small
neighborhood of Rqj ,mj , and bj,k is a period-L sequence
in k based on the inverse of the matrix derived from the
discrete Gabor system that appears in Lemma 1.
5. Conclusion
This paper contains a brief overview of some recent results
on the measurement and identification of communication
channels and the relation of these results to sampling theory. These connections provide explicit reconstruction
formulas for identification of operators modelling timevariant linear channels.
References:
[1] P.A. Bello. Measurement of random time-variant linear channels. 15:469–475, 1969.
[2] K. Gröchenig. Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis.
Birkhäuser, Boston, MA, 2001.
[3] T. Kailath. Sampling models for linear time-variant
filters. Technical Report 352, Massachusetts Institute
of Technology, Research Laboratory of Electronics,
1959.
[4] T. Kailath. Measurements on time–variant communication channels. 8(5):229–236, Sept. 1962.
[5] T. Kailath. Time–variant communication channels. IEEE Trans. Inform. Theory: Inform. Theory.
Progress Report 1960–1963, pages 233–237, Oct.
1963.
[6] W. Kozek and G.E. Pfander. Identification of operators with bandlimited symbols. SIAM J. Math. Anal.,
37(3):867–888, 2006.
[7] J. Lawrence, G.E. Pfander, and D. Walnut. Linear
independence of Gabor systems in finite dimensional
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118
Special session on
Sampling
and
Industrial Applications
Chair: Laurent FESQUET
SAMPTA'09
119
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120
An Event-Based PID Controller
With Low Computational Cost
Sylvain Durand and Nicolas Marchand
NeCS Project-Team, INRIA - GIPSA-lab - CNRS, Grenoble, France.
sylvain.durand@inrialpes.fr, nicolas.marchand@gipsa-lab.inpg.fr
Abstract:
In this paper, some improvements of event-based PID
controllers are proposed. These controllers, contrary to
a time-triggered one which calculates the control signal
at each sampling time, calculate the new control signal
only when the measurement signal sufficiently changes.
The contribution of this paper is a low computational cost
scheme thanks to a minimum sampling interval condition.
Moreover, we propose to reduce much more the error margin during the steady state intervals by adding some extra samples just after transients. A cruise control mechanism is used for simulations and a noticeable reduction
of the mean control computation cost is finally achieved
with similar closed-loop performances to the conventional
time-triggered ones.
1.
Introduction
The classical so-called discrete time framework of controlled systems consists in sampling the system uniformly
in the time with some constant sampling period hnom and
in computing and updating the control law every time instants t = khnom . This field, denoted time-triggered (or
synchronous in sense that all the signal measurements are
synchronous), has been widely investigated [6] even in the
case of sampling jitter or measure loss that can be seen
as some asynchronicity. However, some works addressed
more recently event-based sampling where the sampling
intervals are event-triggered (also called asynchronous), as
for example when the output crosses a certain level. Thus
the term sampling period denotes a time interval between
two consecutive level crossings and the sampling periods
are hence not equidistant in time anymore.
Event-triggered notion is taking more and more importance in the signal processing community with now various publications on this subject (see for instance [1] and
the references therein). In the control community, very
few works have been done. In [3], it is proved that such
an approach reduces the number of sampling instants for
the same final performance. In [8], it is shown that controlling an asynchronous sampled system or a continuous time
system with quantized measurements and a constant control law over sampling periods are equivalent problems.
Many reasons are motivating event-based systems and in
particular because more and more asynchronous systems
or systems with asynchronous needs are encountered. Ac-
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tually, the demand of low power electronic components
in all embedded applications encourages companies to develop asynchronous versions of the existing time-triggered
components, where a significant power consumption reduction can be achieved by decreasing the samplings and
consequently the CPU utilization: about four times less
power than its synchronous counterpart for the 80C51 microcontroller of Philips Semiconductors in [12]. Note
that the sensors and the actuators based on level crossing
events also exist, rendering a complete asynchronous control loop now possible. But the most important contributions come from the real-time control community. Indeed,
real-time synchronous control tasks are often considered
as hard tasks in term of time synchronization, requiring
strong real time constraints. Efforts are so carried on the
co-design between the controller and the task scheduler in
order to soften these constraints. The adopted approach
is often either to change dynamically the sampling period
related to the load [10, 11] or to use event-driven control
where the events are generated with a mix of level crossings and a maximal sampling period [9, 2].
This maximal sampling period seems to be added for stability reasons in order to fulfill the condition of NyquistShannon sampling theorem: a new control signal is performed when the time elapsed since the last sample exceeds a certain limit. We first proposed in [7] to remove it because, thanks to the level detection, the NyquistShannon sampling condition is no more consistent. The
CPU cost is hence considerably reduced without performance loss. We now focus on the improvement of eventbased control by reducing even more the computational
cost with a controller based on a fully asynchronous level
detection. The next two sections recall the conventional
time-triggered structure and the existing event-based algorithms. The main contribution is developed in section 4
where an event-driven controller with low computational
cost is detailed. All controllers are finally compared (in
terms of performances and CPU needs) in section 5.
Notations:
e− will denote the value of e at the last sampling time.
2.
Time-Based Control
The textbook PID controller is given as follows:
1
E(s) + Td sE(s)
U (s) = K E(s) +
Ti s
121
This equation can be divided into a proportional, an integral and a derivative parts, i.e. Up , Ui and Ud respectively,
which are then modified to improve performances [4].
First, set point weighting is applied on Up and Ud for a
more flexible structure, giving the PID two dimensions
of freedom. Moreover, a low-pass filter is added in the
derivative term to avoid problems with high frequency
measurement noise.
Up (s) = K (βYsp (s) − Y (s))
K
E(s)
Ui (s) =
Ti s
KTd s
Ud (s) =
(γYsp (s) − Y (s))
1 + Td s/N
A discrete time controller is finally obtained: the proportional part is straightforward and the backward difference
approximation is used for integral and derivative parts.
3.
Event-Based Control
The basic setup of an event-based PID controller, introduced in [2], consists of two parts: a time-triggered event
detector used for level crossings and an event-triggered
PID controller which calculates the control signal. The
first part runs with the sampling period hnom (that is the
same as for the corresponding conventional time-triggered
PID) whereas the second part runs with the sampling interval hact which depends on the requests sent by the event
detector when a new control signal has to be calculated.
This is required either when the relative error between
the measured signal and the desired one crosses a certain
level, i.e. abs(e − e− ) > elim , or if the maximal sampling
period is achieved, i.e. hact ≥ hmax .
We proposed in [7] to remove this maximal sampling period underlying a primordial fact in asynchronous control
that is that the Nyquist-Shannon sampling condition is no
more consistent thanks to the level detection. However,
the integral part, i.e. ui = u−
i + K/Ti · hact · e, leads
to important overshoots after the steady states since the
sampling period hact becomes huge due to the absence of
event. In fact, this time interval between the last sample
before the steady state and the first sample of the transient
can be divided into a “real” steady state interval which is
equal to hact −hnom , plus the detection time period hnom .
During the first part the error is very small (lower than elim
else the steady state is not achieved) and so is the product
he (lower than (hact − hnom ) elim ). As regards the second part, when the set point changes the error becomes
large but only during the event detection and therefore the
product is hnom e. From this observation, several control
algorithms were proposed in [7] and we will use the hybrid
one which gives good performances with the minimum of
samplings.
The hybrid algorithm is a mix between i) a controller with
a saturation of he which is bounded in (hact − hnom ) ·
elim + hnom · e when hact ≥ hmax and ii) a controller
with an exponential forgetting factor of hact to decrease
its impact after a long steady state interval, with hiact =
hact · exp (hnom − hact ) corresponding to the new sampling period used in the integral part. This mix leads to
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bound the exponential forgetting factor:
if hact ≥ hmax
he = hiact − hnom · elim + hnom · e
else
he = hact · e
end
ui = u−
i + K/Ti · he
(1)
A first improvement could be obtained by changing the
level crossing detection since only one level is really required. Indeed, the control signal needs to be calculated
when the measurement is too far from the set point, i.e. as
soon as abs(e) > elim . Of course, with this method the
number of samples increases during the transients but, at
least, the error between the system and the set point is now
sure to be lower than elim during the steady state intervals,
which was not the case before with the level detection of
the relative error abs(e − e− ) > elim .
A second improvement could be done on the timetriggered event detector which is currently a discrete time
system: an event could only be detected at the time instants t = khnom thereof several levels could miss if they
appear between two sampling instants. Thus we propose
to use a continuous time event detector which is in fact
closer to the real case since a sensor based on level crossing events will send a request as soon as a level is crossed.
Afterwards, the hybrid controller with these improvements is called the asynchronous event-based controller.
4.
Event-Based Control with Low Computational Cost
The asynchronous event-based controller is interesting but
the number of samples is still important during transients.
Indeed, a new request is sent as soon as the error is upper than the detection limit, i.e. abs(e) > elim , which
means (quasi)-continuously during the whole transient. To
avoid that, we propose to add a minimum sampling interval condition to lighten the transients in order that a
new control signal is performed only if a certain time was
elapsed since the last sample, i.e. hact > hmin . This minimum sampling interval could be chosen as the discrete
sampling period hnom corresponding to the conventional
time-triggered controller or not, but it does have to satisfy the Nyquist-Shannon sampling condition. The choice
hmin = hnom leads to a discrete-time event detector when
the dynamics is important and to a continuous-time event
detector when the dynamics is slow (quasi-steady state).
Thus, when an event occurs after a steady state configuration, a new control signal is instantaneously computed.
Whatever that may be the hmin value, an important reduction of the computational cost is achieved. Nevertheless, we propose to improve the event-based scheme
again by adding a few number of samples more. The idea
here is to decrease much more the error during the steady
state intervals. Currently, one could assure that the error
is lower than the limit elim but cannot know how much
lower. Moreover, one could not know if the measured
signal is going closer or moving away from the set point.
122
Therefore, we propose to add some extra samples after a
transient while an event-based controller would do not do
anything because the condition abs(e) > elim is wrong.
Thus, an extra event is sent to the controller if nothing appends after the last time a control signal was calculated
plus a certain sampling interval hextra . Then, this is repeated while the error is upper than a desired minimum
level emin . One only needs to define his desired error margin and some extra samples will be added to achieve that.
Note that the lower emin is chosen the higher the number
of extra samples will be.
5.
Simulation Results: Application to a
Cruise Control Mechanism
Event-based controller is a good solution, more especially
for all the systems which do not need to be constantly
controlled. We chose to illustrate our proposals with the
cruise control mechanism depicted in [5] because the desired speed of the car is constant most of the time and a
new control signal is so only required when the set point
changes or when the load (i.e. the slope of the road) varies.
The simulations run during 50s with the following test
bench: at time 0 the set point is set to 25m/s (90km/h),
then at time 2s it is changed to 30.6m/s (110km/h) and
changed again to 36.1m/s (130km/h) at time 30s. The
gear ratio is chosen accordingly to the speed range, i.e.
n = 5, and no disturbance is applied, i.e. θ = 0.
The first simulation results are shown on Figure 1 where
the conventional time-triggered PI controller is compared
to the asynchronous event-based one (see section 3). The
top plot shows the set point and both measured signals, the
bottom plot shows the sampling intervals (i.e. this signal
changes each time the controller calculates a new control
signal). The asynchronous event-based controller permits
to obtain a system response as quick as the time-triggered
one, by calculating a control signal about four time less
only (with this benchmark). However, the number of samples remains important during the transients. Our proposal, i.e. the event-based PI controller with a low computational cost, avoids that since the number of samples is
dropped by a ratio of 30, as shown on Figure 2.
The equation of motion of the car (ν is the velocity) is:
mν̇ = F − Fd
The force F is generated by the engine, whose torque is
proportional to a control signal 0 ≤ u ≤ 1 that controls
the throttle position and depends on engine velocity too.
2 !
αn ν
−1
F = αn uTm 1 − β
ωm
where αn depends on the gear ratio n.
The disturbance force Fd has three major components due
to the gravity Fg , to the rolling friction Fr and to the aerodynamic drag Fa .
Figure 1: A conventional time-triggered PI controller
(15000 sampling intervals) vs. the asynchronous eventbased one (3703 sampling intervals, that is 24.7%).
Fd = Fg + Fr + Fa
with
Fg = mgsin(θ)
Fr = mgCr sgn(ν)
Fa = 12 ρCd Aν 2
where θ is the slope of the road, i.e. the disturbance.
As regards the control law, an anti-windup mechanism is
added to consider the saturation of the control signal u.
Thus the integral part consists on the integral of the error
plus a reset based on the saturation of the actuator (in order
to prevent windup when the actuator is saturated).
ui = u −
i +
K
hact
x−
(u − usat )
Ti
Ta
where x = hact · e for the time-triggered controller and
x = he defined by (1) for the event-based controllers.
Parameter values are K = 0.8, Ti = 1.4 and Ta = 0.7.
The nominal and maximal sampling intervals used for the
hybrid algorithm are hnom = 0.1s and hmax = 0.5s and
those used for the low computational cost and the extra
samples ones are hmin = 0.1s and hextra = 0.5s. The
detection levels are elim = 0.1 and emin = 0.01 for crossing events and for extra samples respectively.
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Figure 2: The asynchronous event-based PI controller
(3703 sampling intervals) vs. the one with a low computational cost (126 sampling intervals, that is 3.4%).
Whatever the achieved gain with the low computational
cost controller, we propose to improve the error during the
steady state intervals by adding some samples just after the
transients. Results are shown on Figure 3 where one could
see that, by adding extra samples, the sampling number is
123
finally reduced and the steady state intervals are not oscillating anymore. These are thanks to a measurement signal
closer to the set point during the steady state intervals.
for more general types of control.
7.
Acknowledgments
This research has been supported by the GIPSA-lab,
CNRS and INRIA in the FeedNetBack project context.
The project aims to close the control loop over wireless
networks by applying a co-design framework that allows
the integration of communication, control, computation
and energy management aspects in a holistic way.
References:
Figure 3: The asynchronous event-based PI controller
with a low computational cost (126 sampling intervals)
vs. the one with extra samples (120 sampling intervals).
Finally the integral of the norm of the error are compared
for the whole controllers to verify if the responses are not
too far from the conventional time-triggered one. All measurements on Figure 4 have a similar behavior with some
differences during the steady state intervals because of the
allowed error margin elim . The final values are 74.67 for
the reference, 78.2 for the asynchronous event-based controller, 78.63 for the low computational cost one and 77.12
for the extra samples one. Moreover, as regards the last
one, it is possible to be much more closer to the time-based
value by reducing the minimum value emin .
Figure 4: Integral of the norm of the error.
6.
Conclusions and Future Works
In this paper we propose to improve the event-based PID
controllers depicted in [2] and [7]. The first improvement
consists on a minimum sampling interval condition used to
decrease the number of samples during the transients. The
second one comes from the wishing to reduce much more
the error margin during the steady state intervals. Based
on these ideas, event-based PID controllers with low computational cost and with extra samples are proposed.
A cruise control mechanism is used to compare them
(in simulation) with the conventional time-triggered and
with the classical event-based controllers. Both proposals
clearly give good performances with a minimum of sampling intervals and the controller with extra samples permits to reduce the error margin as low as desired to achieve
a response very closed to the conventional one.
Next steps in this research is naturally to test these controllers in practice and develop other event-based methods
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pages 118–127, 2005.
[12] H. van Gageldonk, K. van Berkel, A. Peeters,
D. Baumann, D. Gloor, and G. Stegmann. An asynchronous low-power 80C51 microcontroller. In Proceedings of the 4th International Sympsonium on Advanced Research in Asynchronous Circuits and Systems, pages 96–107, 1998.
124
A coherent sampling-based method for
estimating the jitter used as entropy source for
True Random Number Generators
Boyan Valtchanov, Viktor Fischer, Alain Aubert
Laboratoire Hubert Curien UMR CNRS 5516, Bât. F 18 Rue du Professeur Benoît Lauras , 42000 Saint Etienne, France.
{boyan.valtchanov,fischer,alain.aubert}@univ-st-etienne.fr
This paper was partially supported by the Rhône-Alpes
Region and Saint-Etienne Métropole, France
Abstract:
The paper presents a method, which can be employed to
measure the timing jitter present in periodic clock signals
that are used as entropy source in true random number
generators aimed at cryptographic applications in reconfigurable hardware. The method uses the principle of a coherent sampling and can be easily implemented inside the
chip in order to test online the jitter source. The method
was carefully validated in various simulations that have
shown that the measured jitter size corresponds perfectly
to that of the jitter injected to the model. While the primary aim of the proposed measuring technique was the
evaluation of the quality of jitter as an entropy source in
random number generators, we believe that the same principle can be used in order to characterize the jitter in fast
communication links as well.
1. Introduction
In the global communication era, more and more recent
industrial applications need to secure data and communications. Many cryptographic primitives and protocols that
are used to ensure confidentiality, integrity and authenticity use random number generators in order to generate
confidential keys, initial vectors, nonces, padding values,
etc. While random bit-stream generators can be easily implemented in analog or mixed-signal devices, the generation of random bit-streams is a challenging task when
the generator should be implemented in a logic device like
FPGAs (Field-Programmable Gate Arrays). Clearly, logic
devices are well suited for algorithmic (pseudo) random
number generators, but the true-random number generators need sources of randomness that are difficult to find
and explore in logic devices. A mathematical model of
the true random number generator (TRNG) is also a crucial element of the cryptographic application design since
the final entropy of the generated random bit-stream could
be characterized and thus certified if one is able to characterize the physical phenomenon that is used as the entropy source. If the model does not exist, there would be
no guarantee that the final entropy of the output stream
is true-random, pseudo-random or perhaps a mixture of
random and pseudo random phenomena. Characterizing
SAMPTA'09
and monitor the entropy source (the jitter) and proposing
a mathematical model is the main motivation of the paper.
2.
Jitter as an entropy source for TRNGs
Many of the TRNGs known up to date [1], [4], [5], use
the jitter present in clock signals (generated using ring oscillators, phase-locked loops or delay-locked loops) as a
source of entropy. The quality of the generated random
bits is related to the parameters of the clock jitter. In order
to avoid jitter manipulations and attacks, it is important to
measure these parameters on-line and, if possible, inside
the device.
The jitter can be defined as a short-term variation of an
event from its ideal position [6]. In general, it is expressed
as the variation in time of the zero crossing (rising or
falling edge) of the clock signal. The jitter can be a good
candidate for randomness generation, since its behavior
is closely related to the thermal noise inside semiconductors [2]. The advantage of the thermal noise employed as
a source of randomness is that it is relatively difficult to
manipulate it in order to realize an attack on the TRNG.
The method presented in this paper considers only a truerandom (Gaussian) jitter component and it does not take
into account the deterministic behavior of the jitter at this
stage of our research. For a deeper understanding of the
jitter behavior we recommend to read [9].
3.
Principle and theoretical background
Tro1
Tbeat
D
Counter
Tro2
or
Tvco
Figure 1: Random jitter component measurement based
on the coherent sampling.
The proposed method allows to accurately quantify the
random component of the jitter present in clock signals
generated inside logic devices. Although the technique
can be used to measure the jitter, it has been developed not
for measurement or testing purposes, but rather for modeling a TRNG that uses the jitter as a source of randomness.
125
3.1
Figure 2: Principle of the coherent sampling.
Measurement of the true-random jitter component
Let us assume that the two clock signals are derived
from two internal ring oscillators, and let Tro1Ideal and
Tro2Ideal be the two ideal jitter-free periods. We need to
achieve a small time period difference between Tro1Ideal
and Tro2Ideal , namely:
Tro2Ideal = Tro1Ideal + ∆Ideal .
(1)
This difference comes from the fact that even with the
same number of delay elements the two ring oscillators
differs due to process variations during manufacturing.
With a careful placement, one can obtain ∆ of several
tens of picoseconds. However the ∆ wont be reproducible
from one chip to another.
If a random jitter would be included in the previous equations, we obtain:
Figure 3: Experimental TBeat signal example.
The proposed measurement technique (see Figure 1) is
based on a coherent sampling: the sampling of a periodic signal by another periodic signal featuring similar
frequency [3]. The signal on the output of the sampler
is called a beat signal and it is a low-frequency signal depending on the frequency difference ∆ between the two
clock signals Tro1 and Tro2 .
Figure 2 shows the principle of the coherent sampling
using two (clock) signals having similar frequencies and
the resulting beat signal TBeat , representing the image of
Tro1 . An example of this TBeat signal captured on oscilloscope is given in Figure 3. Using the infinite persistence of the oscilloscope, we can clearly see the variations
of the period of the beat signal. These variations are the
consequence of the jitter present in Tro1 and Tro2 signals.
Because of the coherent relationship between the two frequencies, each ”half-period” of the beat signal is an integer number of the clock period Tro2 . A counter clocked
with this clock signal can thus be used in order to represent
these variations. In the next section, we will discuss how
we can compute the jitter present in Tro1 by observing the
variations in a population of several TBeat periods.
If the proposed technique would be used to measure precisely the jitter of the internal clock signal, one should use
an accurate external low phase-noise VCO (Voltage Controlled Oscillator) as a sampling clock and accurately tune
its period in relationship to the internal clock period in order to obtain a small ∆. Instead, in order to model the
TRNG behavior and to measure the jitter inside the device, we have used two ring oscillators, implemented in
the same FPGA. Both oscillators have the same number of
inverters. In order to guarantee a small difference between
clock periods (∆), the placement and routing have to be
done manually. The final period difference is thus caused
mainly by the different delays of the routing scheme selected by the placement and routing tool. Next, we will
analyze the case, when only random (Gaussian) jitter component is present in the generated clock signals.
SAMPTA'09
Tro1 = Tro1Ideal + N (0, σ1 ) = N (Tro1Ideal , σ1 )
(2)
Tro2 = Tro2Ideal + N (0, σ2 ) = N (Tro2Ideal , σ2 )
(3)
Where N (0, σ) denote a zero-mean Normal distribution
with standard deviation σ.
We can then express the difference ∆ by:
∆ = N (Tro2Ideal , σ2 ) − N (Tro1Ideal , σ1 )
(4)
q
∆ = N (∆Ideal , σ12 + σ22 )
(5)
If σ1 is the same as σ2 , what is the case when the two
signals are derived from internal ring oscillators, we get
√
(6)
∆ = N (∆Ideal , 2σ)
Otherwise one should make precise characterization of the
VCO used to match the frequencies in order to measure
the σV CO .
According to [8], we can express the length of TBeat as:
s
Tro1Ideal q 2
Tro1Ideal
TBeat
σ1 + σ22 ) (7)
= N(
,
∆Ideal
∆Ideal
∆Ideal
which, if σ1 equals σ2 , simplifies to:
s
Tro1Ideal
Tro1Ideal √
TBeat
2σ)
= N(
,
∆Ideal
∆
∆Ideal
(8)
The length of the resulting beat signal, TBeat can be then
expressed as a normal process:
TBeat
= N (µTBeat , σTBeat )
∆Ideal
with the mean and standard deviation:
r
Tro1Ideal
TRoIdeal √
2σ
, σTBeat =
µTBeat =
∆
∆Ideal
(9)
(10)
In consequence, if we measure the µTB eat and σTBeat using the principle presented in Figure 1, which is based on
126
the use of an 8-bit counter, we can precisely calculate the
amount of the random jitter, expressed in 1σ ps, i.e. the
RMS jitter (root mean square) present in the two clock
signals using equation (11).
∆Ideal
σT
σ = q Beat
TRoIdeal √
2
∆Ideal
(11)
4. Simulation results
In order to validate equation (11), we have used a simulation model presented in [8] and depicted in Figure 4. The
random jitter is generated in text files using Matlab and
then injected in VHDL simulation using the Textio package. We have injected different amounts of random jitter
(RMS) to the clock signals and analyzed the obtained values of the counter. The Tro1Ideal was set to 5 ns (200Mhz)
and ∆ to 40 ps. The results of the simulations and recalculated jitter values using equation 11 are presented in Table
4. As it can be seen, the measurement precision that can be
achieved is close to 1 ps RMS. Figure 5 present the case
for 7 ps RMS jitter present in both Tro1 and Tro2 signals.
Figure 4: Simulation setup.
Different Counter Values of TBeat Period
135
Histogram of TBeat: Mean=121.9694 TBeat Std dev=2.8127
1200
1000
130
800
125
600
120
400
115
110
200
0
2000
4000
6000
8000
0
115
120
125
130
Figure 5: Histogram of the simulated TBeat .
Injected 1σ
RMS jitter [ps]
10
9
8
7
6
Measured
µT beat
121.93
121.98
121.97
121.97
121.98
Measured
σT beat
4.03
3.64
3.24
2.81
2.47
Calculated
1σ RMS [ps]
10.19
9.20
8.19
7.10
6.24
Table 1: Simulation results of the random jitter quantification.
5. Discussion and conclusions
We have proposed a jitter measurement technique that can
be embedded in FPGA devices for evaluating and monitoring of the source of randomness employed in true random
SAMPTA'09
number generators. The measurement technique can be
used as well to characterize the jitter present in high-speed
clock signals, if an external VCO (Voltage Controlled Oscillator) is used. The use of an external and precise clock
source is necessary in order to closely match the period of
the signal under test to the period of the reference clock
signal. We have shown by simulation that the measurement error of the proposed method is less than 1 ps RMS
of the random component of the jitter.
However, in real world situations and especially inside
FPGAs, the jitter can exhibit a non negligible deterministic component due to various factors (power supply variations, cross-talks, R-F interference, etc...). In this case,
equation (11) cannot be used for random component jitter
quantification and the deterministic jitter has to be considered, too. However, we believe that it is possible to
integrate this deterministic behavior of the jitter in the proposed model. This integration is the objective of our current research.
References:
[1] V. Fischer, M. Drutarovsky, M. Simka, and
N. Bochard. High performance True Random Number Generator in Altera Stratix FPLDs. Lecture notes
in computer science, FPL’04, pages 555–564, 2004.
[2] A. Hajimiri and TH Lee. A general theory of phase
noise in electrical oscillators. Solid-State Circuits,
IEEE Journal of, 33(2):179–194, 1998.
[3] J.L. Huang and K.T. Cheng. An On-Chip Short-Time
Interval Measurement Technique for Testing HighSpeed Communication Links. Proceedings of the
19th IEEE VLSI Test Symposium, page 380, 2001.
[4] P. Kohlbrenner and K. Gaj. An embedded true random number generator for FPGAs. Proceedings of
the 2004 ACM/SIGDA 12th international symposium
on Field programmable gate arrays, pages 71–78.
[5] B. Sunar, W.J. Martin, and D.R. Stinson. A Provably Secure True Random Number Generator with
Built-In Tolerance to Active Attacks. IEEE TRANSACTIONS ON COMPUTERS, pages 109–119, 2007.
[6] T. Technologies. Synchronous Optical Network
(SONET) Transport Systems: Common Generic Criteria. Technical report, GR-253-CORE, 2000.
[7] K.H. Tsoi, K.H. Leung, and P.H.W. Leong. Compact FPGA-based true and pseudo random number
generators. Field-Programmable Custom Computing Machines, 2003. FCCM 2003. 11th Annual IEEE
Symposium on, pages 51–61, 2003.
[8] B. Valtchanov, A. Aubert, F. Bernard, and V. Fischer.
Modeling and observing the jitter in ring oscillators
implemented in FPGAs. In Design and Diagnostics
of Electronic Circuits and Systems, 2008. DDECS
2008. 11th IEEE Workshop on, pages 1–6, 2008.
[9] SW Wedge. Predicting random jitter-Exploring
the current simulation techniques for predicting the
noise in oscillator, clock, and timing circuits. Circuits and Devices Magazine, IEEE, 22(6):31–38,
2006.
127
SAMPTA'09
128
Orthogonal Exponential Spline Pulses with
Application to Impulse Radio
Masaru Kamada(1) , Semih Özlem(2) and Hiromasa Habuchi(1)
(1) Ibaraki University, Hitachi, Ibaraki 316-8511, Japan.
(2) Bogazici University, Bebek, Istanbul, Turkey.
kamada@mx.ibaraki.ac.jp, semozl@gmail.com, habuchi@mx.ibaraki.ac.jp
Abstract:
With application to the impulse radio communications in
mind, a locally supported and zero-mean pulse which is
orthogonal to its shifts by integers is sought among the
exponential splines having the knot interval 21 . An example pulse is obtained that complies with the regulation
imposed by the US Federal Communications Commission
and will potentially enable an impulse radio communications system as fast as 6G pulses per second.
1. Introduction
The M-shaped linear spline
√
0 ≤ t ≤ 12
√3t,
√3(2 − 3t), 12 ≤ t ≤ 1
3
M (t) =
√3(3t − 4), 13 ≤ t ≤ 2
3(2 − t),
2 ≤t≤2
0,
elsewhere
advantage that they can be shaped through linear dynamical systems [5] . The pulse functions, if they are found,
will work as practical pulses which carry information in
the impulse radio communications.
The problem is simple: we are to find a locally supported
and zero-mean exponential spline q(t) with the knot interval 12 that satisfies
{
∫ ∞
1, k = 0
q(t)q(t − k)dt =
(2)
0, k ̸= 0
−∞
for any ingeter k. This paper presents a procedure to find
such a pulse function and its application to the impulse
radio.
(1)
plotted in Fig. 1 is not a wavelet in the sense of muntiresolutional analysis because M (t) is not orthogonal to its
contracted version M (2t). But it has three remarkable
properties that (i) it is locally supported, (ii) its integration over the domain is zero, and (iii) its shifts by integers
are orthogonal to one another [2]. Those properties are exactly what is required of pulses for the impulse radio communications [6]. The three properties are required (i) for
the sake of real-time communications, (ii) for the pulse to
be feasible as a radio waveform, and (iii) for pulse detection to be robust against noise in the sense of least-square
estimation, respectively.
We shall look for this kind of pulse functions in the
broader family of exponential splines [4, 5] which have the
2. Construction of orthogonal pulses
Any exponential spline can be represented by a linear
combination of the exponential B-spline and its shifts
[4, 5]. An exponential B-spline with the knot interval 21
is the output
β(t) = S(b)(t)
(3)
of a linear dynamical system S having the transfer function
G(s) =
for the input being a series of delta functions
b(t) =
such that
B(z) =
n
∑
n
∑
l=0
bl δ(t − l/2)
(5)
l
bl z − 2
1
= (1−z − 2 e
1
1
2
-1
Figure 1: M-shaped linear spline.
SAMPTA'09
(4)
l=0
M(t)
0
µn−1 sn−1 + · · · + µ1 s + µ0
(s − λ0 )(s − λ1 ) · · · (s − λn−1 )
λ0
2
1
)(1−z − 2 e
λ1
2
1
) · · · (1−z − 2 e
λn−1
2
This exponential B-spline is locally supported as
( n)
β(t) = 0, t ∈
/ 0,
.
2
).(6)
(7)
In order to keep the splines zero-mean, instead of the original exponential B-spline β(t), we shall use
(
)
1
α(t) = β(t) − β t −
(8)
2
129
n−1
by {cl }l=0
, and prepare time-reversed functions
which has the zero mean
∫ ∞
α(t)dt = 0
(9)
ã(t) = a(−t), c̃(t) = c(−t), q̃(t) = q(−t)
(21)
−∞
and the “mirror” system S̃ having the transfer function
G(−s). Then we can express the correlation by
and is locally supported as
)
(
n+1
.
α(t) = 0, t ∈
/ 0,
2
r(k) = (q ∗ q̃)(k)
= (S ◦ S̃)(a ∗ ã ∗ c ∗ c̃)(k),
(10)
Another representation of this α(t) is the output
α(t) = S(a)(t)
(11)
where ∗ denotes the convolution integral, and we can write
D(z) = C(z)C(z −1 ) in the form
of S for the input
C(z)C(z −1 ) = d0 +
a(t) =
n
∑
l=0
where
A(z) =
n+1
∑
al z
n−1
∑
j
j
dj (z − 2 + z 2 )
(23)
j=1
al δ(t − l/2),
(12)
which implies
n−1
∑
(c ∗ c̃)(t) = d0 δ(t)+ dj (δ(t−j/2) + δ(t+j/2)) .(24)
− 2l
j=1
l=0
1
(22)
= (1−z − 2 e
λ0
2
1
) · · · (1−z − 2 e
λn−1
2
1
)(1 − z − 2 ).(13)
In the meantime, a locally supported exponential spline
ϕ(x) = (S ◦ S̃)(a ∗ ã)(x)
Let the desired pulse function be represented in the form
q(t) =
n−1
∑
l=0
cl α(t − l/2).
ϕ(x) = ϕ(−x).
n−1
∑
d0 ϕ(k)+ dj (ϕ(k − j/2) + ϕ(k + j/2))
(15)
j=1
q(t)dt = 0.
(16)
−∞
The remaining request is that its autocorrelation
∫ ∞
q(t)q(t − x)dt
r(x) =
(17)
−∞
should satisfy the orthogonality conditions
{
1, k = 0
r(k) =
0, k = ±1, ±2, · · ·
(18)
=
{
1, k = 0
0, k = 1, 2, · · · , n − 1.
We assume that (27) is solvable since we cannot proceed
unless this is the case. Then, C(z)C(z −1 ) determined by
n−1
(23) from {dj }j=0
can be factorized in the form
1
n−1
∑
l=0
cl δ(t − l/2) and C(z) =
SAMPTA'09
1
n−1
∑
l=0
l
that
cl z − 2 (20)
1
1
· · · (z − 2 −γn−1 )(z 2 −γn−1 ). (28)
Taking half the factors, we can find
1
1
1
√
C(z) = ± γ0 (z − 2 −γ1 )(z − 2 −γ2 ) · · · (z − 2 −γn−1 ) (29)
n−1
that gives the sought coefficients {cl }l=0
by (20). Exciting the system S with the input series of delta functions
v(t) =
Now we have only to find the coefficients
make (19) hold good. Define
1
C(z)C(z −1 ) = γ0 (z − 2 −γ1 )(z 2 −γ1 )(z − 2 −γ2 )(z 2 −γ2 )
reduced from (18) by (15) and the equality r(x) = r(−x).
n−1
{cl }l=0
(27)
n−1
Solvability of (27) for {dj }j=0
can be checked by numerical computation in practice. A simpler condition in terms
of dynamical parameters is yet to be established.
1
with respect to shift by integers. Here the number n of
n−1
{α(t − l/2)}l=0
employed for composing q(t) in (14) is
chosen so that the number n of the unknown coefficients
n−1
{cl }l=0
be the same as that of the essential conditions
{
1, k = 0
(19)
r(k) =
0, k = 1, 2, · · · , n − 1
c(t) =
(26)
By (22), (24), (25) and (26), we can reduce the orthogonality conditions (19) to the linear equations
Then it is automatic that q(t) is locally supported as
and has the zero mean
∫ ∞
associated with the composite system S ◦ S̃ satisfies
(14)
q(t) = 0, t ∈
/ (0, n)
(25)
n−1
∑
l=0
cl a(t − l/2),
(30)
we obtain the desired pulse function
q(t) = S(v)(t) =
n−1
∑
l=0
cl α(t − l/2).
(31)
130
In the case G(s) = 1s , the problem is trivial and the resulting pulse is the Haar function
1, 0 < t ≤ 12
−1, 12 < t ≤< 1
H(t) =
(32)
0, elsewhere.
as illustrated in Fig. 2. Since a good broadband antenna is
d
, the transmitted signal w(t)
well approximated [6] by dt
is differentiated once by the transmitter antenna to be the
radio signal
∞
∑
d
d
wl p(t − l)
w(t) =
dt
dt
1
s2
yields M (t) of (1) as expected. BeThe case G(s) =
√
cause it happens that M (t) = 3(H ∗ H)(t), we might
speculate that the pulse associated with G(s) = s13 could
be proportional to (H ∗H ∗H)(t). But that is not true since
(H ∗ H ∗ H)(t) is not orthogonal to (H ∗ H ∗ H)(t − 2).
It is interesting as well as disappointing that we obtain a
complex-valued pulse in the case G(s) = s13 . A nice example pulse will appear in the next section in the context
of its application to the impulse radio communications.
3.
Application to Impulse Radio
While the series of delta functions a(t) does not exist in
the real world, its integration
∫ t
t<0
0,
∑l
(33)
a(τ )dτ =
ak , 2l < t < l+1
2 , l = 0, 1, · · · , n
∑k=0
n+1
−∞
n+1
<
t
a
=
A(1)
=
0,
k=0 k
2
is a locally supported piecewise constant function that can
be easily generated by electric current switches.
The system S excited by the piecewise constant function
u(t) =
shapes the pulse
∫
t
v(τ )dτ =
−∞
n
∑
cl
∫
t−l/2
a(τ )dτ
(34)
−∞
l=0
and has the relationship
∫ t
n ∫ t−l/2
∑
p(t) =
cl α(τ )dτ = q(τ )dτ.
(36)
(37)
−∞
Besides the simple and practical system (35) to shape p(t)
from the piecewise constant seed u(t), the pulse p(t) has
the remarkable property
∫ ∞ 2
∫ ∞
d
p(t)
p(t
−
k)dt
=
−
q(t)q(t − k)dt
2
−∞ dt
−∞
{
−1, k = 0
=
(38)
0, k = ±1, ±2, · · ·
which follows from (17), (18), (36), (37) and the partial
integration formula. This property gives the foundation to
transmission and detection of the pulse p(t) in the impulse
radio communications.
Given data bits {wl }, we transmit the waveform
( ∞
)
∞
∑
∑
w(t) = S
wl u(t − l) =
wl p(t − l) (39)
SAMPTA'09
2
d
Correlating the received signal dt
2 w(t) with the template
pulse p(t−k), which is the same as the transmission pulse,
for its duration (k, k + n), we have the bit wk recovered
by
∫ k+n 2
∫ ∞ 2
d
d
w(t)
p(t−k)dt
=
w(t) p(t−k)dt
2
2
dt
k
−∞dt
∫ ∞ 2
∞
∑
d
p(t−l) p(t−k)dt
= wl
2
dt
−∞
l=−∞
= −wk
(42)
because of the property (38).
It should be noted that, because of (38), the detection formula (42) virtually performs the least-squares approximad
d
tion of the radio waveform dt
w(t) by dt
p(t−k) = q(t−k)
d
w(t)
to detect wk . Additive noises superimposed on dt
will then be most suppressed in the sense of least-squares
estimation.
G(s) =
p(t) = 0, t ∈
/ (0, n)
l=−∞
l=−∞
(35)
which is locally supported as
−∞
and again by the receiver antenna to arrive at the receiver
as
∞
∑
d2
d2
w(t)
=
(41)
wl 2 p(t − l).
2
dt
dt
An example pulse associated with the transfer function
p(t) = S(u)(t)
l=0
(40)
l=−∞
l=−∞
1
(43)
(s+18)(s+11.1i+10−13 )(s−11.1i+10−13 )
and its derivatives are plotted in Fig. 3. The correlation
in Fig. 4 becomes 1 and 0 at the origin and at the other
integers, respectively, to verify (38). The power spectral
d
density of the radio pulse dt
p(t) = q(t) is plotted in Fig. 5
along with the spectral mask (plotted by the boxy line) for
the indoor ultra-wideband communications systems [1]
imposed by the US Federal Communications Commission
received pulses
w
w dtd p
d2
dt2
2
2
d
dt
w
radio pulses
w dtd p
transmission pulses
w
wp
-w5 -w
S
-w4 -w
-w
template pulses
S
-w
w
w
w
w4
w
w5
S
S
receiver
transmitter
Figure 2: Schematic diagram of the transceiver.
131
p
q
0.2
0.2
0
1
3
3
2
0
1
q
3
2
d2
p
dt2
x
p
x
1
-0.2
-0.2
x
(a) pulse seed u(t)
1
3
2
0
(c) radio pulse
d
p(t)
dt
1
2
3
(d) received pulse
d2
p(t)
dt2
Figure 3: Pulses for impulse radio.
as the upper bound which no practical pulses are allowed
to exceed. The frequency axis of the mask is scaled down
by 6 GHz for the purpose of comparison, or equivalently,
the pulse repetition rate is assumed to be 6 G pulses per
second, which is much faster than the 1.32G pulses per
second of the high speed direct sequence ultra-wideband
protocol discussed in the IEEE 802.15.3a standard.
The fast transmission is possible because the pulses are orthogonal even though they are densely overlapping. But
dense pulses are prone to interfere with one another in
the situation that several reflected pulses arrive with various delays. Multipath compensation by digital filtering
is crucial in order to effectively exploit the dense pulses
we obtained. Transmitting a sounder pulse and digitizing the observed correlations, we have the end-to-end impulse response of the multipath channel. Digital filtering
by an FIR approximation of the inverse impulse response
will work as a kind of rake receiver. This compensation
requires an analog-to-digital converter and a digital filter
that work at the pulse rate and thus costs more hardware.
But this cost should be justified since all the pulse-based
systems cannot be faster without having denser pulses in
the first place. A detailed analysis of the multipath effects, channel modeling error, and pulse synchronization
is available in [3].
We may ignore the multipath effects and channel modeling error in the extreme situation that antennas are inductively coupled at a very short distance less than one inch.
TransferJet technology has been working in the same situation at the maximum transmission rate of 560Mbps since
2008. A faster system will hopefully be the first application of the dense pulses obtained in this paper.
4. Conclusions
Inspired by the M-shaped orthogonal pulse, we derived
a procedure to construct an exponential spline pulse with
the knot interval 12 that is locally supported, has its mean
zero, and is orthogonal to its shifts by integers. An exam-
SAMPTA'09
1
2
3
0
-5
-1
0
-1
Figure 4: Correlation of the pulse.
5
1
0
-2
d2
p
dt2
=q
Power Spectral Density (dB)
d
p
dt
-3
(b) transmission pulse p(t)
-10
-20
-30
-40
0
1
2
3
Relative Frequency
Figure 5: Power spectral density of the pulse and the FCC
spectral mask.
ple pulse was obtained that will potentially enable an impulse radio communications system as fast as 6G pulses
per second under the FCC regulation for the indoor ultrawideband communications.
5. Acknowledgment
This work was partially supported by JSPS grant-in-aid
No. 17560357.
References:
[1] Revision of Part 15 the Commission’s rule regarding ultra-wideband transmission systems. ET Docket
No.98-153, Federal Communications Commission,
Washington, D.C., 2002.
[2] A. J. Jerri. Wavelets – Detailed Treatment with Applications. Exercises of Chapter 3. Sampling Publishing,
Potsdam, NY, to appear in 2009.
[3] M. Kamada S. Özlem and H.Habuchi. Construction of
orthogonal overlapping pulses for impulse radio communications. IEICE Transactions on Fundamentals,
E91-A(11):3121–3129, Nov. 2008.
[4] M. Unser and T. Blu. Cardinal exponential splines:
Part I—Theory and filtering algorithms. IEEE Transactions on Signal Processing, 53(4):1425–1438, April
2005.
[5] M. Unser. Cardinal exponential splines: Part II—
Think analog, act digital. IEEE Transactions on Signal Processing, 53(4):1439–1449, April 2005.
[6] M. Z. Win and R. A. Scholtz. Impulse radio: how it
works. IEEE Commun. Lett., 2(2):36–38, 1988.
132
Special session on
Mathematical Aspects
of
Compressed Sensing
Chair: Holger RAUHUT
SAMPTA'09
133
SAMPTA'09
134
A short note on non-convex compressed sensing
Rayan Saab (1) and Özgür Yılmaz(2)
(1) Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, B.C. Canada V6T 1Z4.
(2) Department of Mathematics, University of British Columbia, Vancouver, B.C. Canada V6T 1Z2.
rayans@ece.ubc.ca, oyilmaz@math.ubc.ca
Abstract:
In this note, we summarize the results we recently proved
in [14] on the theoretical performance guarantees of the
decoders ∆p . These decoders rely on ℓp minimization
with p ∈ (0, 1) to recover estimates of sparse and compressible signals from incomplete and inaccurate measurements. Our guarantees generalize the results of [2] and
[16] about decoding by ℓp minimization with p = 1, to the
setting where p ∈ (0, 1) and are obtained under weaker
sufficient conditions. We also present novel extensions of
our results in [14] that follow from the recent work of DeVore et al. in [8]. Finally, we show some insightful numerical experiments displaying the trade-off in the choice
of p ∈ (0, 1] depending on certain properties of the input
signal.
1.
Introduction
Let ΣN
S be the set of all S-sparse vectors,
N
ΣN
S := {x ∈ R :
#supp(x) ≤ S},
and define, qualitatively, compressible vectors as vectors
that can be “well approximated” in ΣN
S . For p > 0, let
σS (x)ℓp denote the best S-term approximation error of x
in ℓp (quasi-)norm, i.e.,
σS (x)ℓp := min kx − vkp .
v∈ΣN
S
We are interested in recovering x from its possibly noisy
“encoding”
b = Ax + e,
(1)
where A is an M × N matrix with M < N . Equivalently,
we seek accurate, stable, and “implementable” decoders
∆ : RM 7→ RN such that k∆(Ax + e) − xk scales well
with the noise level kek, and is small whenever x is compressible.
In general, the problem of constructing decoders with such
properties is non-trivial (even if e = 0) as A is overcomplete. However, if A ∈ RM ×N is in general position, it
can be shown that there is a decoder ∆0 which satisfies
∆0 (Ax) = x for all x ∈ ΣN
S whenever S < M/2 [10].
This ∆0 can be explicitly computed via the optimization
problem
∆0 (b) := arg min kyk0 subject to b = Ay.
y
SAMPTA'09
(2)
Unfortunately, (2) is combinatorial in nature, thus its complexity grows extremely quickly as N becomes much
larger than M . Naturally, one then seeks to replace (2)
with a more tractable optimization problem.
1.1
Decoding by ℓp minimization
Define the decoders
∆ǫp (b) = arg min kxkp subject to kAx − bk2 ≤ ǫ, (3)
x
and
∆p (b) = arg min kxkp subject to Ax = b,
(4)
with 0 < p ≤ 1. [2, 4, 9, 10, 15], that in the noise-free
setting ∆1 recovers x exactly if x is sufficiently sparse
and if A has certain properties. Furthermore, one has error
guarantees even when x is not “exactly” sparse and when
the encoding is noisy, e.g., [2, 9].
In this note we focus on ∆p and ∆ǫp with 0 < p < 1. Early
work by Gribonval and co-authors (e.g. [12, 13]) presents
sufficient conditions for having ∆p (b) = ∆1 (b) = x and
stability conditions to deal with noisy encoding. However,
these conditions are pessimistic in the sense that they generally guarantee recovery of only very sparse vectors.
Recently, Chartrand [5] showed that in the noise-free setting, a sufficiently sparse signal can be recovered perfectly with ∆p , where p ∈ (0, 1), under less restrictive
requirements than those needed to guarantee perfect recovery with ∆1 . Moreover, in [6], Staneva and Chartrand
showed that if A is an M × N Gaussian matrix, recovery of x in ΣN
S is guaranteed provided M > C1 (p)S +
pC2 (p)S log(N/K). In other words, the dependence on
N of the required number of measurements M (that guarantees perfect recovery for all x ∈ ΣN
S ) disappears as p
approaches 0, unlike the case with p = 1. These results
motivate a more detailed study of the stability and robustness properties of the decoders ∆p .
In the remainder of the note, we summarize our recent results in [14] concerning the theoretical properties of ∆p
and ∆ǫp . In addition, we present some extensions of our results on the instance optimality in probability of ∆p when
the entries of A are drawn from any sub-Gaussian distribution. Finally, we present numerical results suggesting
scenarios where using ∆p , p ∈ (0, 1), is better than using
∆1 .
135
2.
Main Results
We begin with the relevant notation. Let δS , the Srestricted isometry constants of A (see, e.g., [2]), be the
smallest constants satisfying
(1 − δS )kck22 ≤ kAck22 ≤ (1 + δS )kck22
2.2
for every c ∈ ΣN
S . We say that a matrix satisfies RIP(S, δ)
if δS < δ. It has been shown that if A is an M × N
matrix the columns of which are i.i.d. random vectors with
any sub-Gaussian distribution, then A satisfies RIP (S, δ)
with S ≤ c1 M/log(N/M ), δ < 1 with probability >
1 − 2e−c2 M (see, e.g., [1], [3]). Following the notation
of [16], we say that a decoder ∆ is (q, p) instance optimal
if
(5)
k∆(Ax) − xkq ≤ CσS (x)ℓp /S 1/p−1/q
holds for all x ∈ RN . Moreover, a decoder ∆ is said to be
(q, p) instance optimal in probability if (5) holds for any
x with high probability on the draw of A. Note that the
stability results of Candès et al. [2] imply (2,1) instance
optimality of the decoder ∆1 , while the results of Wojtaszczyk in [16] show that ∆1 is (2,2) instance optimal in
probability if the entries of A are drawn from a Gaussian
distribution or if its columns are drawn uniformly from the
sphere.
2.1
Decoding with ∆p : stability and robustness
We consider the scenario where x is arbitrary and σS (x)ℓp
is its best S-term approximation error measured in ℓp
(qausi)-norm. In particular, we are interested in controlling the error k∆ǫp (b) − xkp2 .
Theorem 1 Let p ∈ (0, 1] and let x be arbitrary. Suppose
that
2
2
(6)
δkS + k p −1 δ(k+1)S < k p −1 − 1,
for some k > 1, kS ∈ Z+ . Let b = Ax + e where kek2 ≤
ǫ. Then ∆ǫp (b) satisfies
k∆ǫp (b) − xkp2 ≤ C (1) ǫp + C (2)
σS (x)pℓp
,
S 1−p/2
(7)
where C (1) and C (2) are given in [14].
Remark 2 This is a straightforward generalization of the
results of [2] regarding the performance of ∆1 . In fact, by
setting p = 1 in the above theorem, we obtain the main
theorem of [2], with precisely the same constants.
Remark 3 Using ǫ = 0 in the above theorem, we find
that the decoder ∆p is (2, p) instance optimal. Similarly,
ǫ
assuming x ∈ ΣN
S (hence σS (x)ℓp = 0), we see that ∆p
can stably recover sparse signals.
We can also compare Sp , the sparsity of vectors that are
guaranteed to be recovered with ∆p and S1 , the sparsity
of vectors that are guaranteed to be recovered with ∆1 .
This helps illustrate the possible benefits of using ∆p over
using ∆1 in recovering sparse signals.
SAMPTA'09
Corollary 4 (relationship between S1 and Sp ) Suppose
for some k and S1 , δ(k+1)S1 < k−1
k+1 . Then ∆1 recovers
S1 -sparse vectors and ∆p recovers Sp -sparse vectors
with
k+1
S
.
Sp ≥
1
k p/(2−p) + 1
Instance optimality in probability of ∆p
In [7], it was shown that no decoder, ∆ : RM 7→ RN , is
(2, 2) instance optimal unless M ∼ N . In this section, we
show that ∆p is (2, 2) instance optimal in probability. Our
approach is similar to that of [16], which we summarize
now. Denoting by BqK the unit ball of ℓq in K dimensions,
a matrix A is said to possess the LQ1 (α) property if and
only if
A(B1N ) ⊃ αB2M .
In [16], Wojtaszczyk shows that random Gaussian matrices of size M × N , as well as matrices whose
columns are drawn uniformly
q from the sphere posses the
)
with high probLQ1 (α) property, α = µ log (N/M
M
√
ability. Here µ < 1/ 2 is a constant. Noting that
such matrices also satisfy RIP ((k + 1)S, δ) with S <
M
c log(N/M
) with high probability, Wojtaszczyk proves that
∆1 , with these matrices, is (2,2) instance optimal in
probability. Our proof of the analogous result for ∆p ,
p ∈ (0, 1), relies on the non-trivial generalization of
the LQ1 property to an LQp (α) property with α =
(1/p−1/2)
)
1/Cp µ2 log (N/M
. Specifically, we say that
M
a matrix A satisfies LQp (α) if and only if
A(BpN ) ⊃ αB2M .
Below, we will use Aω to denote matrices whose entries
are drawn from a zero mean, normalized column variance
Gaussian distribution and Ãω to denote matrices drawn
uniformly from the sphere. The following lemma states
that the matrices Aω and Ãω satisfy the LQp property with
high probability.
Lemma 5 Ãω and Aω satisfy the LQp (α) property with
1/p−1/2
)
with probability ≥ 1 −
α = 1/Cp µ2 log (N/M
M
e−cM on the draw of the matrix.
√ Here, Cp is a constant
that depends only on p, µ < 1/ 2 is a constant, and c is
a constant that depends on µ.
Proving Lemma 5 is non-trivial and requires a result by
[11], relating the distances of p-convex bodies to their convex hulls. On the other hand, this lemma provides the machinery needed to prove the following theorem, which extends an analogous result of Wojtaszczyk [16].
Theorem 6 Let Aω ∈ RM ×N , ω ∈ Ω, be a random matrix with entries drawn independently from a zero-mean,
normalized column variance Gaussian distribution, and
let (Ω, P ) be the associated probability space. There
exists constants c1 , c2 , c3 > 0 such that for all S ≤
c1 M/ log (N/M ), the following are true.
(i) ∃Ω1 , with P (Ω1 ) ≥ 1 − e−c2 M , such that ∀x ∈ RN ,
∀e ∈ RM and ∀ω ∈ Ω1
k∆p (Aω (x)+e)−xk2 ≤ C(kek2 +
σS (x)ℓp
), (8)
S 1/p−1/2
136
(ii) ∀x ∈ RN , ∃Ωx , with P (Ωx ) ≥ 1 − e−c3 M , such that
∀e ∈ RM and ∀ω ∈ Ωx
1
0.9
0.8
k∆p (Aω (x)+e)−xk2 ≤ C (kek2 + σS (x)ℓ2 ) . (9)
0.7
p
0.6
eω .
The statement also holds for A
0.5
0.4
0.3
0.2
Note that the constants above (both denoted by C) rely on
the parameters of the particular LQp and RIP properties
that the matrix satisfies, and are omitted for ease of exposition. For the proofs of Lemma 5 and Theorem 6 see [14].
Finally, we present the following extension of Theorem 6.
0.1
10
20
30
40
50
S
(a)
1
Proposition 7 The conclusions of Theorem 6 also hold
when the entries of A are i.i.d., drawn from a subGaussian distribution.
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
p
0.5
0.5
Our proof of the above proposition, which we omit here,
relies on the recent work of [8] where the LQ1 (α) property was modified, allowing the authors to show the (2,2)
instance optimality of ∆1 when the entries of the matrix
A are drawn from any sub-Guassian distribution.
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
10
20
30
40
50
S
(b)
In this section, we present some numerical experiments to
highlight important aspects of sparse recovery using ∆p ,
0 < p ≤ 1. First, we are interested in the sufficient conditions under which decoding with ∆p can guarantee perfect
recovery of signals in ΣN
S for different values of p and S.
Our goal is to show empirically that with smaller values
of p ∈ (0, 1), ∆p allows recovery of less sparse signals
than would have been possible with larger values of p, as
Theorem 1 predicts.
To that end, we generate a 100 × 300 matrix whose
columns are drawn from a Gaussian distribution and estimate its RIP constants δS via Monte Carlo (MC) simulations. Under the assumption that the estimated constants
are the correct ones (while in fact they are only lower
bounds), Figure 1(a) shows the regions where (6) guarantees recovery for different (S, p)-pairs. On the other hand,
Figure 1(b) shows the empirical recovery rates using the
same matrix with fifty different instances of x ∈ ΣN
S , and
decoding by ∆p , where we choose the non-zero coefficients of x randomly from the Gaussian distribution. Here,
we compute ∆p (Ax), as a solution to the ℓp optimization
problem of (4) by using a projected gradient
algorithm
P
on a smoothed version of kxkpp , namely i (x2i + ǫ2 )p/2 ,
where the solution to each subproblem, starting with a
large ǫ is used as an initial estimate for the next subproblem with a smaller ǫ. Note that this approach is similar
to the one described in [5]. Clearly, the empirical results
show that ∆p is successful in a wider range of scenarios than those predicted by Theorem 1. This can be attributed to the fact that the conditions presented in this
paper are only sufficient. Moreover, what is observed in
practice is not necessarily a manifestation of uniform recovery. Rather, the practical results could be interpreted
as success of ∆p with high probability on either x or A.
In our second set of experiments, we wish to observe the
instance optimality of ∆p , i.e., the linear growth of the
SAMPTA'09
0.15
||"p(Ax)−x||2
Numerical Experiments
0.1
p=0.4
p=0.6
p=0.8
p=1
0.05
0
0
0.02
0.04
0.06
0.08
0.1
0.06
0.08
0.1
!
(a)
0.4
||"p(Ax)−x||2
3.
Figure 1: For a Gaussian matrix A ∈ R100×300 , whose
δS values are estimated via MC simulations, we generate the theoretical (a) and practical (b) phase-diagrams for
reconstruction via ℓp minimization. The lighter shading
indicates higher recoverability rates. .
0.3
p=0.4
p=0.6
p=0.8
p=1
0.2
0.1
0
0
0.02
0.04
!
(b)
Figure 2: Reconstruction error with compressible signals,
S = 5 (a), S = 35 (b). Observe the almost linear growth
of the error for different values of p, highlighting the instance optimality in probability of the decoders.
ℓ2 reconstruction error k∆p (Ax) − xk2 , as a function of
σS (x)ℓ2 . To that end, we generate scenarios that allude to
the conclusions of Theorem 6. We generate a signal composed of xT ∈ Σ300
S , supported on an index set T , and a
signal zT c supported on T c = {1, 2, ..., 300}\T , where all
the coefficients are drawn from the Gaussian distribution
and kxT k2 = kzT c k2 = 1. We then set xλ = xT + λzT c
with increasing values of λ starting from 0, i.e., xλ be-
137
comes less compressible as λ increases, and T is the “effective support” of xλ . Next, we choose our measurement
matrix A ∈ R100×300 by drawing its columns uniformly
from the sphere. For each value of λ we measure the reconstruction error k∆p (Axλ ) − xλ k2 , and we repeat the
process 50 times while randomizing the index set T but
preserving the coefficient values. We report the averaged
results for different values of p with S = 5 in Figure 2(a)
and S = 35 in Figure 2(b). Note that when S = 5, ∆1
provides the best performance, and the performance of ∆p
degrades monotonically as p decreases. On the other hand,
when S = 35, ∆p with p = 0.4 provides the best performance and the performance degrades as p increases.
We investigate this observation further by examining the
performance as a function of S ∈ {5, 10, ..., 35}. In
Figure 3, we plot the value of an “empirical effective constant” which we calculate as the maximum of
k∆p (Axλ ) − xλ k2 /λ as λ > 0 varies. This constant acts
as a surrogate for C in (9) assuming that such a constant
exists and that σS (x)ℓ2 = kλzT c k2 = λ. The behavior
gradually changes from favoring p = 1 when S, the size
of the effective support of xλ , is small to favoring p = 0.4
as S increases.
A closer look at the explicit value of the constant in Theorem 6 sheds some light on this behavior. Below, we use
the notation of [14]. The constant C in (9) behaves like
(2C (2) )1/p /γp (where C (2) and γp are explicitly given
in [14]). Specifically, 1/γp depends only on the matrix
A and increases exponentially as p decreases, while C (2) ,
the constant in Theorem 1, depends on p, as well as k and
δ(k+1)S (where k > 1 is a free parameter). When S is relatively small, the associated RIP constants remain small,
which consequently implies that [C (2) ]1/p remains small
provided p is isolated away from 0. In this case, the behavior of C is determined by that of γp , i.e., C is smallest
when p = 1. On the other hand, when S is large, [C (2) ]1/p
grows as p approaches 1 (this is a manifestation of the
more restrictive RIP requirements for larger p as stated in
(6)). This increase seems to be dominating the behavior
of C, thus for larger S we get better effective constants
with smaller p. Such a heuristic could be an interpretation
of the behavior we observe in Figure 3. For a rigorous
quantitative analysis, one needs to identify the s-restricted
isometry constants of the matrix A for every s. Such a
treatment is beyond the scope of this note.
Effective Constant
2
1.8
1.6
p=0.4
p=0.6
p=0.8
p=1
1.4
1.2
1
5
10
15
20
S
25
30
35
Figure 3: The empirical effective constant as a function
of S for different values of p. Note the gradual change
favoring p = 1 when S is small to p = 0.4 as S increases.
SAMPTA'09
References:
[1] R. Baraniuk, M. Davenport, R. DeVore, and
M. Wakin. A Simple Proof of the Restricted Isometry Property for Random Matrices. Constructive Approximation, 2008.
[2] E. J. Candès, J. Romberg, and T. Tao. Signal recovery from incomplete and inaccurate measurements.
Comm. Pure Appl. Math., 59(8):1207–1223, 2005.
[3] E. J. Candès and T. Tao.
Decoding by linear programming. IEEE Trans. Inform. Theory,
51(12):489–509, 2005.
[4] E. J. Candès and T. Tao. Near-optimal signal
recovery from random projections: universal encoding strategies? IEEE Trans. Inform. Theory,
52(12):5406–5425, 2006.
[5] R. Chartrand. Exact reconstructions of sparse signals
via nonconvex minimization. IEEE Signal Process.
Lett., 14(10):707–710, 2007.
[6] R. Chartrand and V. Staneva. Restricted isometry
properties and nonconvex compressive sensing. Inverse Problems, 24(035020), 2008.
[7] A. Cohen, W. Dahmen, and R. DeVore. Compressed
sensing and best k-term approximation. Journal
of the American Mathematical Society (to appear),
2008.
[8] R. DeVore, G. Petrova, and P. Wojtaszczyk.
Instance-optimality in probability with an ℓ1 minimization decoder. preprint, 2008.
[9] D. Donoho. Compressed sensing. IEEE Transactions on Information Theory, 52(4):1289–1306,
2006.
[10] D. Donoho and M. Elad. Optimally sparse representation in general (nonorthogonal) dictionaries
via ℓ1 minimization. Proc. Natl. Acad. Sci. USA,
100(5):2197–2202, 2003.
[11] Y. Gordon and N.J. Kalton. Local structure theory
for quasi-normed spaces. Bull. Sci. Math., 118:441–
453, 1994.
[12] R. Gribonval, R. M. Figueras i Ventura, and P. Vandergheynst. A simple test to check the optimality of
sparse signal approximations. EURASIP Signal Processing, special issue on Sparse Approximations in
Signal and Image Processing, 86(3):496–510, 2006.
[13] R. Gribonval and M. Nielsen. Highly sparse representations from dictionaries are unique and independent of the sparseness measure. Appl. Comput.
Harm. Anal., 22(3):335–355, May 2007.
[14] R. Saab and O. Yilmaz. Sparse recovery by nonconvex optimization – instance optimality. CoRR,
abs/0809.0745, 2008.
[15] J.A. Tropp. Recovery of short, complex linear combinations via l1 minimization. IEEE Transactions on
Information Theory, 51(4):1568–1570, April 2005.
[16] P. Wojtaszczyk. Stability and instance optimality
for gaussian measurements in compressed sensing.
Preprint, 2008.
138
Orthogonal Matching Pursuit with random
dictionaries
P. Bechler, and P. Wojtaszczyk
Institut of Applied Mathematics, University of Warsaw
P.Bechler@mimuw.edu.pl, wojtaszczyk@mimuw.edu.pl
Abstract:
In this paper we investigatet the efficiency of the Orthogonal Matching Pursuit for random dictionaries. We concentrate on dictionaries satisfying Restricted Isometry Property. We introduce a stronger Homogenous Restricted
Isometry Property which is satisfied with overwhelming
probability for random dictionaries used in compressed
sensing. We also present and discuss some open problems
about OMP.
1.
Introduction
In this paper we investigate
the efficiency of the Orthogo√
nal MatchinT = U T ∗ T g Pursuit for random dictionaries. Orthogonal Matching Pursuit is a well known greedy
algorithm widely used in approximation theory, statistical estimations and compressed sensing (for the review of
greedy algorithms see [6]). One of its main features is that
it can be applied for arbitrary dictionary. However the efficiency of the algorithm depend very strongly on properties
of the dictionary. We work in the context of a Hilbert space
H (which may be assumed to be finite dimensional). The
dictionary is a subset D ⊂ H such that span D = H. We
usually assume that kxk is close to 1 for x ∈ D. Generally
it is assumed that kxk = 1 for x ∈ D (see e.g. [6]). However for random dictionaries it is very rarely satisfied. On
the other hand for such dictionary kxk is close to 1 with
great probability.
The Orthogonal Matching Pursuit algorithm with respect to the dictionary D obtains iteratively a sequence
OMPn f ∈ H of approximants of a given element f ∈ H
and a sequence d1 , . . . , dn ∈ D in the following way:
define the set of m sparse vectors (with respect to the dictionary D) as
m
X
ΣD
=
Σ
=
{
aj dj : {dj }m
m
m
j=1 ⊂ D}.
(1)
j=1
For a given f ∈ H we define its best error of m–term
approximation as
σm (f ) = inf{kf − zk : z ∈ Σm }.
(2)
Clearly we always have σm (f ) ≤ kf − OMPm (f )k =
kfm k.
Obviously when our dictionary is an orthonormal basis
then σm (f ) = kf − OMPm (f )k for each f ∈ H. Unfortunately this is the only case when it is so. The fundamental, and still largely unanswered question is how close
OMPm (f ) can get to this optimal rate of approximation
given by σm (f ). It is to be expected that the answer to the
above question must depend on some extra properties of
the dictionary.
2.
Dictionaries
One of the commonly used quantitative parameters of the
dictionary is its mutual coherence. It is defined as
η=
sup
d1 6=d2 ∈D
|hd1 , d2 i|.
(3)
Recently, especially in the context of compressed sensing,
a restricted isometry property (RIP for short) became very
useful. Let us recall the following well known definition
(c.f. [1, 2]).
Definition 1 The dictionary Φ = {φj }N
j=1 has
RIP(k, δ), 0 < δ < 1 if for any set T ⊂ {1, . . . , N }
with #T = k and any sequence of numbers xj we have
• Given OMPn−1 f and d1 , . . . , dn−1 ∈ D choose
sX
sX
dn ∈ D such that
X
2 ≤k
(1
−
δ)
|x
|
x
φ
k
≤
(1
+
δ)
|xj |2 .
j
j j
j∈T
j∈T
j∈T
|hf −OMP f, dn i| = sup |hf − OMP f, di : d ∈ D
n−1
n−1
(4)
• Define OMP0 f = 0.
and define OMPn f as the orthogonal projection of
f onto span{d1 , . . . , dn }.
Generally we will write f − OMPs f := fs .
The standard measure of approximation power of a dictionary is the error of the best m–term approximation. We
SAMPTA'09
There are some easy relations between those notions. If
the dictionary D has mutual coherence η then it satisfies
RIP(k, 1−η) for k < η −1 . On the other hand if D satisfies
RIP(k, δ) then it has mutual coherence ∼ δ.
Usually dictionaries with RIP are exhibited as random dictionaries. To be more precise we define a dictionary in Rn
139
as Φ(ω) = {φj }N
j=1 where φj = (γj,1 , . . . , γj,n ) and γj,i
are idependent copies of a fixed subgaussian random variable normalised so that Ekφk k2 = 1.
In this context it is known (see e.g. [1]) that for a fixed
0 < δ < 1 there exists c > 0 such that the dictionary
Φ(ω) with overwhelming probability satisfies RIP(k, δ)
with k = ⌊cn/ log N ⌋. On the other hand it is also known
that such a dictionary with overwhelming probability has
mutual coherence of order k −1/2 . It is clear that when we
have two events each of them happening with big probability then they happen simultanously with big probability.
This leads to the following definition:
Definition 2 The dictionary Φ has homogenous restricted
isometry property HRIP(k,
δ), 0 < δ < 1 if for any l ≤ k
p
it satisfies RIP(l, δ l/k).
Following standard reasoning we obtain
Theorem 1 Suppose that integers n, N and numbers 0 <
δ < 1 and a > 0 are given and suppose that the random
dictionary Φ(ω) = {φ1 , . . . , φN } ⊂ Rn is as described
above. Then there exist c, c1 > 0 which depend only on the
subgaussian distribution involved, δ and a such that dictionary Φ(ω) satisfies HRIP(k, δ) for k = ⌊c1 n/ log N ⌋
with probability ≥ 1 − 3N −a
Basically this tells us that unless we are very unlucky
a randomly chosen dictionary satisfies HRIP, which is
clearly stronger property than RIP. We believe that HRIP
is a useful property. Theorem 4 is some indication of this.
3.
Main Result
Now we want to present a result on the approximation
power of OMP for dictionaries satisfying RIP. For dictionaries with incoherence analogous results were obtained
by D. Donoho, M. Elad and N.V. Temlyakov [3]. If we
are interested
p in random dictionaries results from [3] require S ≤ n/ log N while ours apply for the full range
S ≤ n/ log N .
Theorem 2 There exist constants C and c depending only
on ǫ > 0 such that for the dictionary Φ satisfying
RIP(2K, ǫ) and for 0 ≤ k ≤ S ≤ K we have
2
kfS k ≤ Ckfk k (σS−k (fk ) + Aǫkfk k) .
(5)
with A = c(1 + log K)).
Note that in particular seting k = 0 we get
kfS k2 ≤ Ckf k(σS (f ) + Aǫkf k).
(6)
The proof of Theorem 2 is rather complicated. It uses a
lot of geometry of Hilbert space, theory of Riesz bases
and ideas from [3] and [5]. The main new technical tool is
the following lemma on norm of matrices.
Lemma 1 Let 0 < ǫ < 1 and let A = [ai,j ] be an n × n
upper triangular matrix such that for any x ∈ Rn
(1 − ǫ)kxk ≤ kAxk ≤ (1 + ǫ)kxk
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(7)
and |ai,i | ≥ 1 − ǫ for i = 1, 2, . . . , n. Let B = [bi,j ] be
the off diagonal part of A i.e.
(
ai,j if i < j
bi,j =
0
if j ≤ i.
Then kBk ≤ 4ǫ⌈log2 n⌉.
The above inequailties (5) and (6) have some merit only if
ǫA < 1. Generally one would like to avoid the presence
of kfk k (or kf k) inside the brackets in (5), (6). The most
desirable would be to have direct estimates of the form
kfs k ≤ Cσs (f ). Unfortunatelly in full generality such
estimates are not true even when we replace the constant
by a function of s.
Pn
Here is an appropriate example. Let x = √1n j=1 ej ∈
R2n so kxk = 1. Let us consider the dictionary consisting
of vectors: e1 , . . . , en , ψj := kej + βn−1/2 xk−1 (ej +
βn−1/2 x) for j = n + 1, . . . , n + s plus orthonormal vectors which are orthonormal
to make a basis
√ to all those √
in R2n . We take β = 4 n and s = ⌊ǫ n⌋ . Then the
following are easy to check
• The mutual coherence is ≤ n−1/2 .
• The Riesz constant√of this basis is
nary has RIP(2n, ǫ)
√
ǫ so the dictio-
• Orthogonal Matching Pursuit for vector x in first s
iterations chooses vectors ψj and only later chooses
vectors ej .
Thus we see that σn (x) = 0 while x − OMPk (x) 6= 0 for
k = n + s − 1.
For dictionaries with mutual coherence η J. Tropp [7],
slightly improving estimate from [4], have proved
Theorem 3 If the dictionary has mutual coherence η then
√
(8)
kfm k ≤ 8 mσm (f ) for m < (3η)−1 .
Using this we obtain
Theorem 4 Let √
the dictionary Φ satisfies HRIP(k, δ).
Then for m ≤ c/ k we have
kf⌊m log m⌋ k ≤ Cσm (f ).
(9)
Let us give a sketch of a proof which
follows arguments
√
from [3]. We start with m ≤ c′ k for which (8) holds.
We set ml = m(2l − 1) and we fix K ∼ k 3/4 . Using
HRIP we get that dictionary Φ satisfies RIP(2K, ǫ) with
Aǫ ≤ δk −1/8 ≤ βm−1/4 . From Theorem 2 and (8) we
get
kfm2 k2
≤ Ckfm1 k(σm (f ) + Aǫkfm1 k)
≤ Ckfm1 k(σm (f ) + 8βm1/4 σm (f ))
2
(f )
≤ 8Cm1/2 (1 + 8βm1/4 )σm
′ 3/4 2
≤ C m σm (f ).
√
Thus we get kfm2 k ≤ C ′ m3/8 σm (f ). Repeating this
argument and carefully tracking constants we see that after
at most µ ∼ log log m steps we get the claim.
140
Analogous result from [3] uses only
√ mutual coherence and
in our case gives (9) for m ≤ c 3 k. The main drawback
of Theorem 4 is the limitation on m. It is clear from
the above sketch that this restriction is inherited from
Theorem 3. It is very unlikely that (8) can be substantially
improved using only mutual coherence. We believe
however that for dictionaries with RIP or HRIP one can
prove more. So let us state the following conjecture
Conjecture Assume that the dictionary satisfies
HRIP(k, δ). There exist constants C, c, α and β
(possibly depending on δ) such that for every f and for
m logα m ≤ ck we have
kf⌊m logα m⌋ k ≤ Cmβ σm (f ).
Let us note that it follows from Theorem 3 that there exists
a function ψ(k, δ) and constants C and β such that if the
dictionary satisfies HRIP(k, δ) then for every f ∈ H
kfm k ≤ Cmβ σm (f ).
for m ≤ ψ(k,
√ δ). (Clearly Theorem 3 gives β = 1/2 and
to know if ψ can
ψ(k, δ) ∼ k). It would be interesting
√
grow significantly faster than k.
References:
[1] R. Baraniuk, M. Davenport, R. DeVore, and M.
Wakin, A simple proof of the restricted isometry
property for random matrices, Constr. Approx., 28
(2008), no. 3, 253–263.
[2] E. Candés, The restricted isometry property and its
implications for compressed sensing, Compte Rendus de l’Academie des Sciences, Paris, Series I,
346(2008), 589–592.
[3] D.Donoho, M.Elad, V.N.Temlyakov, On Lebesguetype inequalities for greedy approximation J. Approx. Theory 147 (2007), no. 2, 185–195.
[4] A.Gilbert, S.Muthukrishnan, M.Strauss, Approximation of functions over redundant dictionaries using coherence, Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (Baltimore, MD, 2003), 243–252, ACM, New
York, 2003.
[5] S.Kwapień, A. Pełczyński, The main triangle projection in matrix spaces and its applications Studia
Math. 34 (1970) 43–68.
[6] V.N.Temlyakov, Greedy approximation, Acta Numerica 17 (2008) 235–409
[7] J.Tropp, Greed is good: Algorithmic results for
sparse approximation, IEEE Trans. Inform. Theory,
50 (2004), 2231-2242
SAMPTA'09
141
SAMPTA'09
142
Average Case Analysis of Multichannel Basis
Pursuit
Holger Rauhut (1) , Yonina C. Eldar (2)
(1) Hausdorff Center for Mathematics, and Institute for Numerical Simulation, University of Bonn
Endenicher Allee 62, 53115 Bonn, Germany.
(2) Department of Electrical Engineering, Technion, Israel Institute of Technology, Haifa, Israel 32000.
rauhut@hcm.uni-bonn.de, yonina@ee.technion.ac.il
Abstract:
We consider the recovery of jointly sparse multichannel
signals from incomplete measurements using convex relaxation methods. Worst case analysis is not able to provide insights into why joint sparse recovery is superior to
applying standard sparse reconstruction methods to each
channel individually. Therefore, we analyze an average
case by imposing a probability model on the measured signals. We show that under a very mild condition on the
sparsity and on the dictionary characteristics, measured
for example by the coherence, the probability of recovery failure decays exponentially in the number of channels. This demonstrates that most of the time, multichannel sparse recovery is indeed superior to single channel
methods.
1.
Introduction
Recovery of sparse signals from a small number of measurements is a fundamental problem in many different signal processing tasks such as image denoising [3], analogto-digital conversion [21, 11], radar, compression, inpainting, and many more. The recent framework of compressed
sensing (CS), founded in the works of Donoho [8] and
Candes [3], studies acquisition methods as well as efficient computational algorithms that allow reconstruction
of a sparse vector x from linear measurements y = Ax,
where A is referred to as the measurement matrix. The
key observation is that y can be relatively short, and still
contain enough information to recover x.
Determining the sparsest vector x consistent with the data
y = Ax is generally an NP-hard problem [7]. To determine x in practice, a multitude of efficient algorithms
have been proposed. The most extensively studied recovery method by now is the ℓ1 -minimization approach (Basis Pursuit). Greedy methods, such as simple thresholding
[23] or orthogonal matching pursuit (OMP) [26], are faster
in practice, but BP provides significantly better recovery
guarantees [10, 22].
The BP principle as well as greedy approaches have been
extended to the multichannel setup where the signal consists of several channels [29, 30, 15, 6, 5, 20, 12, 13, 18].
Here one assumes that each channel is sparse and in addition that the channels have a small common support set.
In this situation the signals are called jointly sparse. A variety of theoretical recovery results have been established
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already in this setting. In [5] a recovery result was derived
for a mixed ℓp /ℓ1 program (multichannel BP) in which
the objective is to minimize the sum of the ℓp -norms of
the rows of the estimated matrix whose columns are the
unknown vectors.
Recovery results for the more general problem of blocksparsity were developed in [13] based on the restricted
isometry property (RIP), and in [12] based on mutual coherence. In practice, multichannel reconstruction techniques perform much better than recovering each channel
individually. However, the theoretical equivalence results
predict no performance gain. The reason is that these recovery results apply to all possible input signals, and are
therefore worst-case measurements. Clearly, if we input
the same signal to each channel, then no additional information on the joint support is provided from multiple measurements. Therefore, in this worst-case scenario there is
no advantage for multiple channels.
In order to capture more closely the true underlying behavior of existing algorithms and observe a performance
gain when using several channels, we consider an average analysis. In this setting, the inputs are considered to
be random variables so that the case of identical inputs
in all channels has zero probability. The idea is to develop conditions on the measurement matrix A such that
the inputs can be recovered with high probability given
a certain input distribution. Most existing recovery results focus on worst-case analysis. Recently, there have
been several papers that consider random ensembles. In
[25] random sub-dictionaries of A are considered and analyzed. This allows to obtain results for BP with a single
input channel. In [23], average-case performance of single
channel thresholding was studied. These ideas were then
extended to two multichannel recovery algorithms: thresholding and simultaneous OMP (SOMP) [18, 17]. Under a
mild condition on the sparsity and on the matrix A, it was
shown that the probability of reconstruction failure decays
exponentially with the number of channels. In the present
paper we contribute to this line of research by adding an
average-case analysis of multichannel BP, that is mixed
ℓ2 /ℓ1 -minimization [30, 15, 13, 12].
We denote by AS the submatrix of A consisting of the
columns indexed by S ⊂ 1, . . . , N , while X S is the submatrix of X consisting of the rows of X indexed by S.
The ℓth column of A is denoted by aℓ or Aℓ . The ℓp -norm
is denoted by k · kp while k · kF is the Frobenius norm.
143
2.
Multichannel ℓ1 -minimization
We consider multichannel signal recovery where our goal
is to recover a jointly-sparse matrix X ∈ CN ×L from n
linear measurements per channel. Here N denotes the signal length and L the number of channels, i.e., the number
of signals. We assume that X is jointly k-sparse, meaning
that there are at most k rows in the matrix X that are not
identically zero. More formally,
SL we define the support of
the matrix X as supp X = ℓ=1 supp Xℓ , where the support of the ℓth column is supp Xℓ = {j, Xjℓ 6= 0}. Our
assumption is that kXk0 = k where kXk0 is equal to the
size of the support. The measurements are given by
Y = AX,
Y ∈ Cn×L ,
(1)
where A ∈ Cn×N is a given measurement matrix. Each
measurement vector Yℓ corresponds to a measurement of
the corresponding signal Xℓ .
The natural approach to determine X given Y is to solve
the problem
min kXk0
X
such that AX = Y.
N
X
j=1
kX j k2 ,
subject to AX = Y, (3)
which promotes joint sparsity, as argued for instance in
[15]. In the single channel case L = 1 this is the usual BP
principle.
3.
(1 − δ)kxk22 ≤ kAxk22 ≤ (1 + δ)kxk22 ,
for all vectors x that are 2k-sparse. Let X ∈ CN ×L ,
Y = AX, and let X be the minimizer of (3). Then
kX − XkF ≤ Ck −1/2 kX − X̂ (k) k1,2
(2)
However, (2) is NP hard in general [7]. Therefore, we
consider instead the convex relaxation [30, 15, 13] defined
by
min kXk2,1 =
Note that in both of the cited results the conditions do not
depend on the number of channels. Indeed, under the same
conditions as in Propositions 3..1 and 3..2, it is shown in
[26] that BP will recover a single k-sparse vector. Therefore, if (4) holds, then instead of solving (3) we may as
well use BP on each of the columns of Y .
q
N −n
The coherence is lower bounded by µ ≥
n(N −1)
√
[24]. The lower bound behaves like 1/ n for large N ,
which limits
√ the Proposition 3..2 to maximal sparsities
k = O( n). To improve on this we can generalize existing recovery results [3, 2] based on RIP to the multichannel setup. The next proposition follows from [13]:
√
Proposition 3..3 Assume X ∈ Cn×N with δ2k < 2 − 1,
where δ2k is the smallest constant δ such that
p
where C is a constant, kXkF =
Tr(X ∗ X) is the
PN
j
Frobenius norm of X, kXk1,2 =
j=1 kX k2 , and
X̂ (k) denotes the best k-term approximation of X, i.e.,
supp X̂ (k) consists of the indices corresponding to the k
largest row norms kX ℓ k2 . In particular, recovery is exact
if | supp X| ≤ k.
It is well known that Gaussian and
√ Bernoulli random matrices A ∈ Rn×N satisfy δ2k ≤ 2 − 1 with high probability as long as [1, 4]
n ≥ Ck log(N/k).
Worst Case Recovery Results
Recovery results for the program (3) were considered in
[5, 13, 12]. In particular, the lemma below is derived in
[5] and follows also from [12].
Proposition 3..1 Let S ⊂ 1, . . . , N and suppose that
kA†S aℓ k1 < 1
for all ℓ ∈
/ S,
(4)
with A†S = (A∗S AS )−1 A∗S denoting the pseudo-inverse of
AS . Then (3) recovers all X ∈ CN ×L with supp X = S
from Y = AX.
Assuming the columns of A are normalized, kaℓ k2 = 1,
we can guarantee that (4) holds as long as the coherence µ
of A is small enough, where [9]
µ = max |haj , aℓ i|.
j6=ℓ
(5)
The following result follows from Proposition 3..1 or from
[12] by noting that the block coherence in this setting is
equal to µ/d.
Proposition 3..2 Assume that
(2k − 1)µ < 1.
(6)
Then (3) recovers all X with kXk0 ≤ k from Y = AX.
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(7)
Therefore, Proposition 3..3 allows for a smaller number of
measurements. However, there is still no dependency on
the number of channels. Indeed, under the same RIP condition BP will recover a single k-sparse vector and therefore, as before, BP may as well be applied to each of the
columns of Y individually.
4.
Average Case Analysis
Intuitively, we would expect multichannel sparse recovery to perform better than single channel recovery. However, in the worst case setting this is not true as already
suggested by the results cited above. The reason is very
simple. If each channel carries the same signal, Xℓ = x
for ℓ = 1, . . . , L, then also the components of Y = AX
are all the same and we do not have more information on
the support of X than provided by a single component Yℓ .
This can indeed be proven rigorously.
Proposition 4..1 Suppose there exists a k-sparse vector
x ∈ RN that ℓ1 -minimization is not able to recover from
y = Ax. Then there exists a k-sparse multichannel signal
X ∈ RN ×L for which mixed ℓ2 /ℓ1 -minimization fails on
Y = AX.
144
For the simple proof we refer to the journal version [14].
Realizing that (3) is not more powerful than usual BP in
the worst case, we seek an average-case analysis. This
means that we impose a probability model on the k-sparse
X. In particular, as in [18], we will assume that on the ksparse support set S the coefficients of X are independent
and follow a normal distribution,
X S = ΣΦ
(8)
where Σ = diag(σj , j ∈ S) ∈ Rk×k is an arbitrary diagonal matrix with non-zero diagonal elements σj , while
Φ ∈ Rk×L is a Gaussian random matrix, i.e., all entries
are independent standard normal random variables. Note
that taking Σ to be the identity matrix results in a standard
Gaussian random matrix, while taking arbitrary non-zero
σj ’s on the diagonal of Σ allows for different variances.
The following recovery condition is instrumental in proving average case recovery results for multichannel BP. It
generalizes results of [27, 16] for the monochannel case.
In order to introduce we need to introduce the sign sgn(X)
of a signal matrix,
(
Xℓj
, kX ℓ k2 6= 0;
kX ℓ k2
sgn(X)ℓj =
0,
kX ℓ k2 = 0.
Proposition 4..2 Let X ∈ CN ×L with supp X = S and
assume AS to be non-singular. If
k sgn(X S )∗ A†S aℓ k2 < 1
for all ℓ ∈
/S
(9)
then X is the unique minimizer of (3).
Combining the above proposition with a concentration inequality for sums of independent random variables that are
uniformly distributed on the sphere [19], we arrive at the
following average case recovery result for multichannel
BP.
Theorem 4..3 Let S ⊂ {1, . . . , N } be a set of cardinality
k and let X ∈ RN ×L with supp X ⊂ {1, . . . , N } such
that the coefficients on S are given by (8) with some diagonal matrix Σ ∈ Rk×k . If
kA†S aℓ k2 ≤ α < 1
for all ℓ ∈
/ S,
then with probability at least
L
1 − N exp − (α−2 − log(α−2 ) − 1)
2
(10)
kA†S aℓ k2 ≤
δ
<1
1−δ
for all ℓ ∈
/ S.
Note that in contrast to the worst case result in Proposition
3..3 where a condition on δ2k is needed, we only require
that δk+1 is small, which is clearly weaker. For random
matrices A we have the following bound on kA†S aℓ k2 .
Proposition 4..5 Let S ⊂ {1, . . . , N } be a set of cardinality k and suppose that A ∈ Rn×N is drawn at random
according to a Gaussian or Bernoulli distribution. Then
kA†S aℓ k2 ≤ δ
for all ℓ ∈
/S
with probability at least 1 − ǫ provided that
n ≥ Cδ −2 [(k + 1) ln(1 + 12/δ) + ln(2N/ǫ)].
(12)
The constant C is no larger than 162/7 ≈ 23.1.
Note that the log-factor in (12) enters only as an additive
term, while in (7) it appears as multiplicative factor.
5.
Conclusion
Our main result is that under mild conditions on the sparsity and measurement matrix, the probability of failure of
multichannel BP (3) decays exponentially with the number of channels. To develop this result we assumed a
probability model on the non-zero coefficients of a jointly
sparse signal. This shows that multichannel BP outperforms single channel BP applied to each channel individually, on average. Proofs of our theorems, together with
improved results for simple thresholding and numerical
experiments will appear in [14].
6.
Acknowledgements
The work of YE was supported in part by the Israel Science Foundation under Grant no. 1081/07 and by the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications NEWCOM++ (contract no. 216715). HR acknowledges funding by the Hausdorff Center for Mathematics, University
of Bonn and the WWTF project SPORTS (MA 07-004).
(11)
(3) recovers X from Y = AX.
The proof of the theorem will appear in the journal version [14]. For α < 1 we are guaranteed that the exponent
has a negative argument, and therefore the error decays exponentially in L. We note that for the monochannel case
L = 1, Theorem 4..3 is contained implicitly in [28, Theorem 13]. The appearance of the 2-norm in (10) instead of
the 1-norm as in (4) makes the condition of the theorem
weaker than worst-case estimates.
Let us finally state conditions on the matrix A and the
sparsity level k ensuring that kA†S aℓ k2 is small, which is
needed in order to apply Theorem 4..3.
SAMPTA'09
Proposition 4..4 Suppose A has restricted isometry constant δk+1 ≤ δ < 1/2. If S ⊂ {1, . . . , N } has cardinality
k then
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Oct. 2008.
SAMPTA'09
146
Special session on
Sampling
Using
Finite Rate of Innovation Principles
Chairs: Pier-Luigi DRAGOTTI, Pina MARZILIANO
SAMPTA'09
147
SAMPTA'09
148
Sampling of Sparse Signals in Fractional
Fourier Domain
Ayush Bhandari (1) and Pina Marziliano (2)
(1)Temasek Labs @ NTU, 50 Nanyang Drive, Singapore - 637553
(2) School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore - 639798
{ayushbhandari,epina}@ntu.edu.sg
Abstract: In this paper, we formulate the problem of sampling sparse signals in fractional Fourier domain. The
fractional Fourier transform (FrFT) can be seen as a generalization of the classical Fourier transform. Extension
of Shannon’s sampling theorem to the class of signals
which are fractional bandlimited shows its association to
a Nyquist-like bound. Thus proving that signals that have
a non-bandlimited representation in FrFT domain cannot
be sampled. We prove that under suitable conditions, it is
possible to sample sparse (in time) signals by using the Finite Rate of Innovation (FRI) signal model. In particular,
we propose a uniform sampling and reconstruction procedure for a periodic stream of Diracs, which have a nonbandlimited representation in FrFT domain. This generalizes the FRI sampling and reconstruction scheme in the
Fourier domain to the FrFT domain.
where
q
2
2
1−j cot θ j t +ω
e 2
2π
def
Kθ (t, ω) =
δ(t − ω),
δ(t + ω),
cot θ−jωt csc θ
,
θ 6= pπ
θ = 2pπ
θ + π = 2pπ
(2)
is the transformation kernel, parametrized by the fractional order θ ∈ R and p is some integer.
The FrFT of a time-frequency representation e.g. Gabor
Transform results in rotation of the plane by the fractional
order of the FrFT [2]. Thus, we denote fractional order by
θ and from now on, we will use fractional order and angle
interchangeably. The inverse-FrFT with respect to angle θ
is the FrFT at angle −θ, given by,
Z ∞
x(t) =
x
bθ (ω) K−θ (t, ω) dω.
(3)
−∞
1.
Introduction
Shannon’s sampling theorem [1] provides access to the
digital world. Our understanding of this sampling theorem
together with the reconstruction formula is solely based
on the frequency content of the signal of interest. This is
where the indispensable Fourier transform comes into the
picture.
Almeida [2] introduced the fractional Fourier transform
or the FrFT–a generalization of the Fourier transform–to
the signal processing community in 1994. The generalization of the Fourier transform by FrFT has several interesting consequences from the signal processing perspective. For instance, non-bandlimited signals in the Fourier
domain can still have a compactly supported representation in FrFT domain [3], when dealing with non stationary
distortions, the FrFT based filters can perform better than
Fourier domain based filters (in sense of mean square error) [4] etc. To give the reader an idea about the growing
popularity of FrFT, it would be worth mentioning that on
at least eight occasions including, [3, 5, 6, 7, 8, 9, 10, 11],
Shannon’s sampling theorem [1, 12] was independently
extended to the class of fractional bandlimited signals. In
[13], the FrFT of a signal or a function, say x(t), is defined
by
x
bθ (ω) = FrFT{x(t)} =
SAMPTA'09
Z
x(t)Kθ (t, ω)dt
(1)
Whenever θ = π/2, (1) collapses to the classical Fourier
transform definition. A direct consequence of the generalization of the Fourier transform by the FrFT results in a
modification in the idea of bandlimitedness. Its impact is
visible in the change that manifests in Shannon’s sampling
theorem for fractional bandlimited signals [11], which is
stated in Theorem 1.
Theorem 1 (Shannon–FrFT). Let x(t) be a continuoustime signal. If the spectrum of x(t), i.e. x
bθ (ω) is fractional bandlimited to ωm which means, x
bθ (ω) = 0, when
|ω| > ωm , then x(t) is completely determined by giving its ordinates at a series of equidistant points spaced
T = ωπm sin θ seconds apart.
This theorem has an equivalence to the Shannon’s sampling theorem for θ = π/2. The reconstruction formula
for fractional bandlimited signals is given in [11],
X
x(t) = λ∗θ (t)
λθ (nT ) x(nT )sinc ((t − nT ) ωm csc θ)
n∈Z
(4)
is a domain independent chirp
where λθ (·) = e
modulation function and the ‘*’ in the superscript denotes
complex conjugation. If x
e (t) is the approximation of
2
x(t), then ke
x (t) − x(t)k = 0 when ωm 6 ω2s sin θ–the
Nyquist rate for FrFT–where ωs = 2π/T is the sampling
frequency. Note that all the aforementioned results are
equivalent to Shannon’s sampling theorem with respect to
def
θ
j(·)2 cot
2
149
Fourier domain for θ = π/2. Theorem 1 (for FrFT) has
a striking similarity with the Shannon’s sampling theorem
(for FT), in that, sampling non-bandlimited signals is impossible. Consider Dirac’s delta function or δ(t). Using
(2), we have,
q
(5)
δbθ (ω) = FrFT {δ (t)} = 1−j2πcot θ λθ (ω)
which is a non-bandlimited function (and least sparse
when compared to the time-domain counterpart) and thus,
Theorem 1 fails to answer the following question: If x(t)
is a fractional non-bandlimited signal, then, how can we
sample and reconstruct such a signal? To make this statement clear, we introduce the fractional convolution operator, which is denoted by ‘∗∗θ ’. Accordingly, filtering x(t)
by a filter, h(t), in ‘fractional sense’ 1 is equivalent to [14],
q
2.
2.1
2.2
Fractional Fourier Series (FrFS)
Periodic signals can be expanded in FrFT domain as a
fractional Fourier series or FrFS [19]. The FrFS of a periodic signal, say x(t), can be written as,
x(t) =
x
bθ [m]Φθ (m, t)
(8)
where,
We model our sparse signal as a periodic stream of K
Diracs, i.e.
K−1
X
k=0
=
r
sin θ − j cos θ j t2 +(2πm sin θ/τ )2
2
e
τ
ck
X
δ(t − tk − nτ )
(7)
n∈Z
x
bθ [m] =
Z
hτ i
x(t)Φ∗θ (m, t)dt = hx, Φθ (m, ·)i
(9)
where
hτ i denotes the integral width and ha, bi =
R
a(t)b∗ (t)dt denotes the inner product. The well-known
Fourier series (FS) is just a special case of FrFS for θ = π2 .
3.
Stream of Diracs in Fractional Fourier
Domain
In Fourier analysis, the Poisson summation formula (PSF)
plays an important role. It is a well-known fact that a
stream of Diracs (Dirac comb) in time-domain is another
stream of Diracs in Fourier domain. In this subsection, we
will derive the equivalent representation of Dirac comb in
FrFT domain. This can be seen as a generalization of the
Poisson summation formula for Dirac comb in FrFT domain.
Theorem 2. Let
P
n∈Z
δ(t − nτ ) be a Dirac comb, then
FrFT
δ (t − nτ ) ←→
−j
1 q 2π X b
δθ [kω0 sin θ] e
1−j
cot
θ
τ
t2 +
(kω0 sin θ)2
2
cot θ+jkω0 t
k∈Z
where ω0 =
2π
τ .
def P
Proof. Let s(t) =
n∈Z δ(t − nτ ). The proof is done
by expanding s(t) in FrFS basis or,
1 We
adhere to this modified definition of convolution operator as it inherits the fractional Fourier duality property, in that,
FrFT {x(t) ∗θ h(t)} = λ∗θ (ω) · x
bθ (ω)b
hθ (ω), which does not hold for
the FrFT of x(t) ∗ h(t) unless θ = π2 .
cot θ−j2πmt/τ
constitutes the basis for FrFS expansion for a τ -periodic
x(t). The FrFS coefficients are given by,
n∈Z
Sparse Signal Model
x(t) =
Φ∗θ (m, t)
X
Preliminaries
SAMPTA'09
X
m∈Z
1−j cot θ ∗
λθ (t)·([x(t)λθ (t)]
2π
∗ [h(t)λθ (t)])
(6)
where ‘∗’ denotes the usual convolution operator. In light
of this definition, we wish to address the problem of recovering parsimonious x(t) from the samples of its filtered version, i.e., y(nT ) = x(t) ∗θ h(t)|t=nT , n ∈ Z.
This problem has a natural/strong link with that of sparse
sampling [15, 16, 17]. The Heisenberg-Gabor uncertainty
principle for the FrFT [18] (a generalization of the Fourier
duality) asserts that the product of spreads of x
bθ (ω) and
2
x(t) has a lower bound which is proportional to sin4 θ (assuming that kxk = 1). This implies that sparsity in one
domain will lead to loss of compact support in canonically
conjugate domain.
Our contribution in this article is to propose a sampling
and reconstruction scheme for signals which have a sparse
representation in time domain and whose fractional spectrum is non-bandlimited. We model our sparse signal as
a continuous periodic stream of Diracs which is being
observed by an acquisition device which deploys a sincbased filter.
The paper is organized as follows: We assume that the
reader is familiar with basic ideas outlined in [12, 16, 17].
In Section II, we introduce our sparse signal model and
the definition of the fractional Fourier series (FrFS). Using these as preliminaries, in Section III, we derive an
equivalent representation of our signal in FrFT domain.
In Section IV, we discuss the sampling theorem and its
completeness and Section V is the conclusion.
x(t)∗θ h(t) =
K−1
with period τ , weights {ck }k=0 and arbitrary shifts,
K−1
{tk }k=0 ⊂ [0, τ ). In sense of [16], the signal has 2K
degrees of freedom per period and the rate of innovation
being ρ = 2K
τ . From now on, the signal x(t) will denote
the stream of Diracs.
s(t) =
X
k∈Z
hs, Φθ i Φθ (k, t).
| {z }
(10)
s
bθ [k]
150
Note that x(t) is non-bandlimited, however, it can be completely described by the knowledge of p[m] which in turn
can be expanded as a linear combination of K complex
exponentials.
The coefficients of this expansion are given by,
(9)
sbθ [k] = hs, Φθ (k, t)i
tZ
0 +τ
κ (θ)
= √
τ
s (t) Φ∗θ (k, t) dt,
∀t0 ∈ R
4.
t0
Zτ /2
κ (θ)
= √
τ
2
δ (t) ej (t
+(kω0 sin θ)2 /2) cot θ−jkω0 t
−τ /2
(since s(t + τ ) = s(t) and s(t) = δ (t) , t ∈
dt
We assume that a sinc–based kernel is used to prefilter x(t). In particular, we let the sampling ker-
, τ2 )
−τ
2
cot θ
inition in (6), prefiltering the input signal x(t) with the
kernel/low-pass filter ϕ(−t) and sampling can be written
as, y (nT ) = x(t) ∗θ ϕ(−t)|t=nT . The main result is in
the form of the following theorem.
Theorem 3. Let x(t) be a τ -periodic stream of Diracs
K−1
K−1
weighted by coefficients {ck }k=0 and locations {tk }k=0
2K
with finite rate of innovation ρ = τ . Let the sampling
kernel/prefilter ϕ(t) be an ideal low-pass filter which has
fractional bandwidth [−Bπ, Bπ], where B is chosen such
that B ≥ ρ. If the filtered version of x(t), i.e. y(t) =
x(t) ∗θ ϕ(−t) is sampled uniformly at locations t = nT ,
n = 0, . . . , N − 1 then the samples,
k∈Z
For
q sake of convenience, we will assume that the constant
1−j cot θ
has been absorbed in τ . Note that at θ = π2 ,
2π
P
1
s(t) = τ k∈Z ejkω0 t which is the result of applying the
PSF on s(t) in Fourier domain. Our immediate goal now
is to derive the FrFS equivalent of x(t) in (7). Since x(t)
is a linear combination of some s(t) delayed by some time
shift tk , it will be useful to recall shift property of FrFT [2]
which states that,
FrFT {s (t − tk )}
1
y (nT ) = x(t) ∗θ ϕ (−t)|t=nT , n = 0, . . . , N − 1,
are a sufficient characterization
of x(t), provided that
N ≥ 2Mθ + 1 and Mθ = Bτ 2csc θ .
Proof. Using the following FrFT pair,
q
FrFT
1−j cot θ ∗
ω
λθ (ω) · rect( 2πB
) ←→
2π
(B csc θ) λ∗θ (t) sinc (Bt csc θ)
2
(12)
= sbθ (ω − tk cos θ) ej 2 tk sin θ cos θ−jωtk sin θ .
PK−1
Therefore, call x(t) = k=0 ck · sk (t) where sk (t) is the
time-shifted version of s(t) with shift parameter tk . Using
Theorem 2 and the shift-property of FrFT, we have,
X
δ(t − tk − nτ )
sk (t) =
we define our sampling kernel as,
ϕB (t − nT ) = λ∗θ (t) ϕ (B csc θ (t − nT ))
which is compactly supported over [−Bπ, Bπ]. Prefiltering and sampling x(t) results in,
y (nT ) = x(t) ∗θ ϕ (−t)|t=nT , n = 0, . . . , N − 1
λ∗ (nT ) X
= θ
p[m]
τ
m∈Z
E
D 2πm
× ej τ t , (B csc θ) sinc ((B csc θ) (t − nT )) .
n∈Z
(8)
=
X
FrFT{ δ(t − tk )} |ω=mω0 sin θ Φθ (m, t)
m∈Z
(12)
=
2
2
cot θ
1X
ej 2 (tk −t )+jmω0 (t−tk ) .
m∈Z
{z
}
|τ
The inner product in the above step is further simplified
using the Fourier integral,
D 2πm
E
ej τ t , (B csc θ) sinc ((B csc θ) (t − nT )) =
PSF for Dirac Comb in FrFT
Having obtained the FrFT-version of sk (t), we can write,
x(t) =
K−1
X
k=0
=
K−1
X
ck ·
ck
k=0
X
n∈Z
X
X 1
τ
m∈Z
SAMPTA'09
cot θ
2
j
rect( Bτ m
csc θ )e
(t2k −t2 )+jmω0 (t−tk )
m∈Z
θ
−jt2 cot
2
=e
δ(t − tk − nτ )
ej
|
K−1
X
k=0
2
nel to be ϕn (t) = e−j 2 t sinc(t − nT ). Integer translates of ϕn (t) form an orthonormal basis and
the FrFT of
bθ (ω) =
ϕ(t)(= ϕ0 (t)) is given by ϕ
q
cot θ
−j 2 ω 2
1−j cot θ
rect(ω/2π). In light of the defe
2π
2
κ (θ)
= √ ej ((kω0 sin θ) /2) cot θ
τ
r
(5) κ (θ)
2π
= √
δbθ [kω0 sin θ]
(11)
1 − j cot θ
τ
√
where κ (θ) = sin θ − j cos θ. Back substitution of (11)
in (10) results in,
1 q 2π
s(t) =
τ 1−j cot θ
(kω0 sin θ)2
X
−j t2 +
cot θ+jkω0 t
2
δbθ [kω0 sin θ] e
×
.
This concludes the proof.
Sampling and Reconstruction of Sparse
Signals in Fractional Fourier Domain
2
θ
j cot
2 (tk )−jmω0 tk
ck e
{z
p[m]
!
ej
}
2πm
τ t
.
2πm
τ (nT )
.
We can therefore conclude that,
λ∗ (nT ) X
j 2πm
τ (nT )
p[m] rect( Bτ m
y (nT ) = θ
csc θ )e
τ
m∈Z
=
λ∗θ (nT )
τ
Mθ
X
m=−Mθ
p[m]ej
2πm
τ (nT )
, n = 0, . . . , N − 1
151
Figure 1: Sampling and reconstruction of periodic stream of Diracs in FrFT domain.
where Mθ =
Bτ csc θ
2
.
Signal reconstruction from its samples: Call p[m] =
PK−1
m
k=0 ak uk – a linear combination
of K-complex ex√
ω0 tk with weights ak =
ponentials, uk = λ∗π/2
K−1
ck · λθ (tk ). The problem of calculating {ak }k=0
K−1
and {uk }k=0 is based on finding a suitable polyno
QK−1
−1
mial A(z) =
whose inverse zk=0 1 − uk z
transform yields the annihilating filter coefficients, A[m]
which annihilate p[m]. In matrix notation, finding A[m]
is equivalent to finding a corresponding vector A that
forms a null space of a suitable submatrix of p[m]
i.e. P(2Mθ −K+1)×(K+1)
– which is essentially the set
Null(P) = A ∈ RK+1 : P · A = 0 . For details of this
computation, the reader is referred to (cf. Pg. 1427, [16]).
Figure 1 shows the layout of this algorithm.
5.
Conclusion
We presented a scheme for sampling and reconstruction
of sparse signals in fractional Fourier domain. A direct
consequence of modeling our signal of interest as a Finite
Rate of Innovation signal, is that, the outcome bears an
acute resemblance with the results previously derived, for
the Fourier domain case. This simplifies the problem to
the extent that reconstruction strategy remains unchanged
and as we have shown, one can obtain the precise locations
and amplitudes of the stream of Diracs using the annihilating filter method. Since time and frequency domains
are special cases of the FrFT domain, it turns out that the
number of values (Mθ ) required for exact reconstruction
of time domain signal depends on the chirp rate of transformation, i.e. θ.
References:
[1] C. E. Shannon. Communications in the presence of noise.
Proc. of the IRE, 37:10–21, January 1949.
[2] L. B. Almeida. The fractional Fourier transform and
time-frequency representations. IEEE Trans. Signal Proc.,
42(11):3084–3091, Nov 1994.
[3] X. G. Xia. On bandlimited signals with fractional Fourier
transform. IEEE Signal Proc. Letters, 3(3):72–74, Mar
1996.
SAMPTA'09
[4] A. Kutay, H. M. Ozaktas, O. Ankan, and L. Onural. Optimal filtering in fractional Fourier domains. IEEE Trans.
Signal Proc., 45(5):1129–1143, May 1997.
[5] A. I. Zayed. On the relationship between the Fourier and
fractional Fourier transforms. IEEE Signal Proc. Letters,
3(12):310–311, Dec 1996.
[6] T. Erseghe, P. Kraniauskas, and G. Carioraro. Unified
fractional Fourier transform and sampling theorem. IEEE
Trans. Signal Proc., 47(12):3419–3423, Dec 1999.
[7] A. I. Zayed and A. G. Garcı́a. New sampling formulae for
the fractional Fourier transform. Signal Proc., 77(1):111–
114, 1999.
[8] A. G. Garcı́a. Orthogonal sampling formulas: A unified
approach. SIAM Rev., 42(3):499–512, 2000.
[9] Ç. Candan and H. M. Ozaktas. Sampling and series expansion theorems for fractional Fourier and other transforms.
Signal Proc., 83(11):2455–2457, 2003.
[10] R. Torres, P. F. Pellat, and Y. Torres. Sampling theorem for
fractional bandlimited signals: A self-contained proof. application to digital holography. IEEE Signal Proc. Letters,
13(11):676–679, Nov. 2006.
[11] R. Tao, B. Deng, Z.-Q. Wei, and Y. Wang. Sampling and
sampling rate conversion of band limited signals in the
fractional Fourier transform domain. IEEE Trans. Signal
Proc., 56(1):158–171, Jan. 2008.
[12] M. Unser. Sampling-50 years after Shannon. Proc. IEEE,
88(4):569–587, 2000.
[13] H. M. Ozaktas and M. A. Kutay. Introduction to the fractional Fourier transform and its applications. Academic
Press, 1999.
[14] P. Kraniauskas, G. Cariolaro, and T. Erseghe. Method for
defining a class of fractional operations. IEEE Trans. Signal Proc., 46(10):2804–2807, Oct 1998.
[15] P. Marziliano. Sampling innovations. PhD thesis, EPFL,
Switzerland, 2001.
[16] M. Vetterli, P. Marziliano, and T. Blu. Sampling signals
with finite rate of innovation. IEEE Trans. Signal Proc.,
50(6):1417–1428, Jun 2002.
[17] T. Blu, P.-L. Dragotti, M. Vetterli, P. Marziliano, and
L. Coulot. Sparse sampling of signal innovations. IEEE
Signal Proc. Mag., 25(2):31–40, March 2008.
[18] S. Shinde and V. M. Gadre. An uncertainty principle for
real signals in the fractional Fourier transform domain.
IEEE Trans. Signal Proc., 49(11):2545–2548, Nov 2001.
[19] S. C. Pei, M. H. Yeh, and T. L. Luo. Fractional Fourier
series expansion for finite signals and dual extension to
discrete-time fractional Fourier transform. IEEE Trans.
Signal Proc., 47(10):2883–2888, Oct 1999.
152
Estimating Signals With Finite Rate of
Innovation From Noisy Samples:
A Stochastic Algorithm
Vincent Y. F. Tan and Vivek K Goyal
Massachusetts Institute of Technology, Cambridge, MA 02139 USA
vtan@mit.edu, vgoyal@mit.edu
Abstract:
As an example of the concept of rate of innovation, signals
that are linear combinations of a finite number of Diracs
per unit time can be acquired by linear filtering followed
by uniform sampling. However, in reality, samples are
not noiseless. In a recent paper, we introduced a novel
stochastic algorithm to reconstruct a signal with finite rate
of innovation from its noisy samples. Even though variants of this problem has been approached previously, satisfactory solutions are only available for certain classes of
sampling kernels, for example kernels which satisfy the
Strang–Fix condition. In our paper, we considered the
infinite-support Gaussian kernel, which does not satisfy
the Strang–Fix condition. Other classes of kernels can be
employed. Our algorithm is based on Gibbs sampling, a
Markov chain Monte Carlo (MCMC) method. This paper
summarizes the algorithm and provides numerical simulations that demonstrate the accuracy and robustness of our
algorithm.
1.
Introduction
The celebrated Nyquist–Shannon sampling theorem [4, 6]
states that a signal x(t) known to be bandlimited to Ωmax
Hz is uniquely determined by samples of x(t) spaced
1/(2Ωmax ) sec apart. The textbook reconstruction procedure is to feed the samples as impulses to an ideal lowpass
(sinc) filter. Furthermore, if x(t) is not bandlimited or the
samples are noisy, introducing pre-filtering by the appropriate sinc sampling kernel gives a procedure that finds the
orthogonal projection to the space of Ωmax -bandlimited
signals. Thus the noisy case is handled by simple, linear,
time-invariant processing.
Sampling has come a long way since the sampling theorem, but until recently the results have mostly applied only
to signals contained in shift-invariant subspaces [9]. Moving out of this restrictive setting, Vetterli et al. [10] showed
that it is possible to develop sampling schemes for certain
classes of non-bandlimited signals that are not subspaces.
As described in [10], for reconstruction from samples it is
necessary for the class of signals to have finite rate of innovation (FRI). The paradigmatic example is the class of
signals expressed as
x(t) =
X
k
SAMPTA'09
ck φ(t − tk )
(1)
where φ(t) is some known function. For each term in the
sum, the signal has two real parameters ck and tk . If the
density of tk s (the number that appear per unit of time) is
finite, the signal has FRI. It is shown constructively in [10]
that the signal can be recovered from (noiseless) uniform
samples of x(t)∗h(t) (at a sufficient rate) when φ(t)∗h(t)
is a sinc or Gaussian function. Results in [2] are based on
similar reconstruction algorithms and greatly reduce the
restrictions on the sampling kernel h(t).
In practice, though, acquisition of samples is not a noiseless process. For instance, an analog-to-digital converter
(ADC) has several sources of noise, including thermal
noise, aperture uncertainty, comparator ambiguity, and
quantization [11]. Hence, samples are inherently noisy.
This motivates our central question: Given the signal
model (i.e. a signal with FRI) and the noise model, how
well can we approximate the parameters that describe the
signal and hence the signal itself? In this work, we address this question by developing a novel algorithm to reconstruct the signal from the noisy samples. The main
contribution is to show that a stochastic approach can effectively circumvent the ill-conditioning of algebraic techniques.
This paper is an abridged version of [7], where many additional details can be found.
2. Problem Definition and Notation
The basic setup is shown in Fig. 1. As mentioned in the
introduction, we consider a class of signals characterized
by a finite number of parameters. In this paper, similar
to [2, 3, 10], the class is the weighted sum of K Diracs
x(t) =
K
X
ck δ(t − tk ).
(2)
k=1
(The use of a Dirac delta simplifies the discussion. It can
be replaced by a known pulse φ(t) and then absorbed into
the sampling kernel h(t), yielding an effective sampling
kernel φ(t) ∗ h(t).) The signal to be estimated x(t) is
filtered using a Gaussian lowpass filter
t2
(3)
h(t) = exp − 2
2σh
with width σh to give the signal z(t). Even though h(t)
does not have compact support, it can be well approximated by a truncated Gaussian, which does have compact
153
3. Presentation of the Gibbs Sampler
z(t)
x(t)
h(t)
z[n]
C/D
+
T
e[n]
y[n]
Figure 1: Block diagram showing our problem setup. x(t)
is a signal with FRI given by (2) and h(t) is the Gaussian
filter with width σh given by (3). e[n] is i.i.d. Gaussian
noise with standard deviation σe and y[n] are the noisy
samples. From y[n] we will estimate the parameters that
describe x(t), namely {(ck , tk )}K
k=1 , and σe , the standard
deviation of the noise.
support. The filtered signal z(t) is sampled at rate of 1/T
Hz to obtain z[n] = z(nT ) for n = 0, 1, . . . , N − 1.
Finally, additive white Gaussian noise (AWGN) e[n] is
added to z[n] to give y[n]. Therefore, the whole acqui−1
sition process from x(t) to {y[n]}N
n=0 can be represented
by the model M
M:
y[n] =
K
X
k=1
(nT − tk )2
+ e[n] (4)
ck exp −
2σh2
for n = 0, 1, . . . , N − 1. The amount of noise added is a
function of σe . We define the signal-to-noise ratio (SNR)
in dB as
!
PN −1
2
|z[n]|
△
dB. (5)
SNR = 10 log10 PN −1n=0
2
n=0 |z[n] − y[n]|
In the sequel, we will use boldface to denote vectors. In
particular,
y
=
c =
t =
[y[0], y[1], . . . , y[N − 1]]⊤ ,
⊤
[c1 , c2 , . . . , cK ] ,
[t1 , t2 , . . . , tK ]⊤ .
(6)
(7)
(8)
We will be measuring the performance of our reconstruction algorithms by using the normalized reconstruction error
R∞
2
△ −∞ |zest (t) − z(t)| dt
R∞
E=
,
(9)
|z(t)|2 dt
−∞
where zest (t) is the reconstructed version of z(t). By construction E ≥ 0 and the closer E is to 0, the better the reconstruction algorithm. The problem can be summarized
as: Given y = {y[n] | n = 0, . . . , N − 1} and the model
M, estimate the parameters {(ck , tk )}K
k=1 . Also estimate
the noise variance σe2 .
Ideally, we would like to minimize E in (9) directly, but
this does not seem to be tractable since the dependence
of y[n] on {tk }K
k=1 is highly nonlinear. Thus, we propose the use of a stochastic algorithm (known as the Gibbs
sampler) for the maximum likelihood (ML) estimation of
{tk }K
k=1 . The Gibbs sampler is a proxy for minimizing E.
This is followed by linear least squared error (LLSE) estimation of {ck }K
k=1 as a tractable and effective means for
approximate minimization of E.
SAMPTA'09
The algorithm introduced in [7] is a stochastic optimization procedure based on Gibbs sampling to estimate θ = {c, t, σe }. Detailed derivations and a
self-contained introduction to Gibbs sampling are given
in [7], and code written in MATLAB can be found at
http://web.mit.edu/∼vtan/frimcmc. Here, we merely summarize the main steps of the algorithm and the intuition
behind Gibbs sampling.
The overall procedure is given in Algorithm 1. The algorithm uses Gibbs sampling (Algorithm 2) to estimate the
set of Dirac positions {tk }K
k=1 . It then uses a least-squares
procedure to estimate the weights {ck }K
k=1 . The basic idea
of Gibbs sampling is to exploit the fact that it is easier to
compute samples drawn approximately according to the
posterior distribution of the parameters given the data than
it is to directly minimize E. This is true when one can
analytically determine the conditional distribution of one
parameter given the remaining parameters and the data.
(The required derivations are presented in [7].) After a
number of iterations Ib called the burn-in period, samples
drawn through Gibbs sampling can be treated as if they
are drawn from the true posterior. Thus, samples drawn
in I additional iterations can be averaged to obtain a good
approximation of the mean of the posterior distribution.
Algorithm 1 Parameter Estimation and Signal Reconstruction Algorithm
Require: Data y, Model M
1: Obtain estimates {t̂k }K
k=1 and σ̂e using the Gibbs
sampler detailed in Algorithm 2 with the data y and
the model M given in (4).
2: Obtain estimates {ĉk }K
k=1 using a linear least squares
estimation procedure and {t̂k }K
k=1 from the Gibbs
sampler.
3: Compute zest (t) = x̂(t) ∗ h(t) given the parameters
{(ĉk , t̂k )}K
k=1 and the known pulse h(t).
4: Compute reconstruction error E given in (9).
Algorithm 2 The Gibbs Sampling Algorithm
(0)
Require: y, I, Ib , θ (0) = {c(0) , t(0) , σe }
1: for i ← 1 : I + Ib do
(i)
(i−1) (i−1)
(i−1)
(i−1)
2:
c1 ∼ p(c1 |c2
, c3
, . . . , cK , t(i−1) σe
)
(i)
(i) (i−1)
(i−1) (i−1) (i−1)
3:
c2 ∼ p(c2 |c1 , c3
, . . . , cK , t
σe
)
..
..
4:
. ∼ .
5:
6:
7:
8:
9:
10:
11:
12:
13:
(i)
(i)
(i)
(i)
(i−1)
cK ∼ p(cK |c1 , c2 , . . . , cK−1 , t(i−1) , σe
(i)
t1
(i)
t2
∼
∼
)
(i−1) (i−1)
(i−1)
(i−1)
p(t1 |c , t2
, t3
, . . . , tK , σe
)
(i)
(i−1)
(i−1)
(i−1)
(i)
p(t2 |c , t1 , t3
, . . . , tK , σe
)
(i)
..
.
. ∼ ..
(i)
(i) (i)
(i)
(i−1)
tK ∼ p(tK |c(i) , t1 , t2 , . . . , tK−1 , σe
)
(i)
σe ∼ p(σe |c(i) , t(i) )
end for
Compute θ̂MMSE using least squares
return θ̂MMSE
154
Sampling ck .
tion given by
ck is sampled from a Gaussian distribu-
1
βk
,
,
p(ck |θ−ck , y, M) = N ck ; −
2αk 2αk
AF/RF (Fig. 2(a))
GS (Fig. 2(b))
N −1
1 X
(nT − tk )2
αk =
exp −
,
2σe2 n=0
σh2
N
30
30
σe
10−6
2.5
SNR
137 dB
10.2 dB
(10)
Table 1: Parameter values for comparing annihilating filter
and root-finding (AF/RF) against Gibbs sampling (GS).
where
△
K
5
5
(11)
N −1
(nT − tk )2
1 X
exp −
βk = 2
σe n=0
2σh2
K
X
(nT − tk′ )2
ck′ exp −
. (12)
×
−
y[n]
2σh2
k′′ =1
△
k 6=k
4. Numerical Results and Experiments
In this section, the annihilating filter and root-finding algorithm [10] provides a baseline for comparison. After
exhibiting its instability, we provide simulation results to
validate the accuracy of the algorithm we proposed in Section 3. More extensive experimentation, including comparisons to [3] and applications to an audio signal, is reported in [7].
It is easy to sample from Gaussian densities when the parameters (αk , βk ) have been determined.
4.1 Annihilating Filter and Root-Finding
Sampling tk .
form
tk is sampled from a distribution of the
"
p(tk |θ−tk , y, M) ∝ exp −
1
2σe2
N
−1
X
γk
n=0
#
(nT − tk )2
(nT − tk )2
× exp −
+ νk exp −
σh2
2σh2
(13)
where
△
γk = c2k ,
(14)
(nT − tk′ )2
△
′
νk = 2ck
− y[n] .
ck exp −
2
2σh
k′′ =1
K
X
k 6=k
(15)
It is not straightforward to sample from this distribution.
(i)
We used rejection sampling [5, 8] to generate samples tk
from p(tk |θ−tk , y, M). The proposal distribution q̃(tk )
was chosen to be an appropriately scaled Gaussian, since
it is easy to sample from Gaussians.
Sampling σe . σe is sampled from the ‘Square-root
Inverted-Gamma’ [1] distribution IG −1/2 (σe ; ϕ, λ),
p(σe |θ−σe , y, M) =
−(2ϕ+1)
2λϕ σe
Γ(ϕ)
λ
exp − 2 I[0,+∞) (σe ),
σe
N
,
2
"
#2
K
X
(nT − tk )2
△ 1
λ=
y[n] −
ck exp −
2
2σh2
△
(17)
(18)
k=1
Thus the distribution of the variance of the noise σe2 is Inverted Gamma, which corresponds to the conjugate prior
of σe2 in the expression of N (e; 0, σe2 ) [1] and thus it is
easy to sample from.
SAMPTA'09
4.2 Gibbs Sampling Algorithm
Initial Demonstration. To demonstrate the evolution
the Gibbs sampler, we performed an initial experiment
with parameters as above, with the exception that the noise
standard deviation was increased to σe = 2.5, giving an
SNR of 10.2 dB. We plot the iterates of the most challenging parameters—the tk s—in Fig. 3. We observe that
the sampler converges in fewer than 20 iterations for this
run, even though the parameter values were initialized far
from their optimal values. The true filtered signal z(t) and
its estimate zest (t) are plotted in Fig. 2(b). Note the close
similarity between z(t) and zest (t).
(16)
where
ϕ=
In [10], for signals of the form (2) and certain sampling
kernels, the annihilating filter was used as a means to locate the tk values. Subsequently a least squares approach
yielded the weights ck . It was shown that in the noiseless
scenario, this method recovers the parameters exactly. In
the same paper, a method for dealing with noisy samples
is suggested. Unfortunately, this method seems to be inherently ill-conditioned. In Fig. 2, we show a pair of simulations with the parameters as tabulated in Table 1. We
observe from Fig. 2(a) that (even with an oversampling
factor of N/(2K) = 3) the annihilating filter and rootfinding method is not robust to even a miniscule amount
of added noise.
Further Experiments on Simulated Data. To further
validate our algorithm, we performed extensive simulations on different problem sizes with different levels of
noise [7]. These experiments support the conclusion that
the Gibbs sampler algorithm is insensitive to initialization. It always finds approximately optimal estimates from
any starting point because the Markov chain provably converges to the stationary distribution [8]. We also find that
the noise standard deviation σe can be estimated accurately; this may be important in some applications.
155
5. Concluding Comments
σe = 1e−6, E = 0.2721
20
z(t)
zest(t)
15
10
5
0
−5
−20
−10
0
10
20
(a) The reconstruction using annihilating filter and rootfinding completely breaks down when noise of a small standard deviation σe = 10−6 (SNR = 137 dB) is added.
20
z(t)
z (t)
est
15
References:
10
5
0
−5
0
5
10
15
20
25
(b) The Gibbs sampling technique gives a much better reconstruction even at a higher noise level σe = 2.5 (SNR = 10.2
dB).
Figure 2: Demonstration of the instability of annihilating filter/root-finding approach and the improvement from
Gibbs sampling.
t
k
15
10
5
0
We addressed the problem of reconstructing a signal with
FRI given noisy samples. We showed that it is possible to
circumvent some of the problems of the annihilating filter and root-finding approach [3, 10]. We introduced the
Gibbs sampling algorithm to find the locations and augmented with a least squares approach to find the weights.
The success of the Gibbs sampling algorithm does not
depend on the choice of kernel h(t), but rather the i.i.d.
Gaussian noise assumption. The formulation of the Gibbs
sampler does not depend on the specific form of h(t). In
fact, we used a Gaussian sampling kernel to illustrate that
our algorithm is not restricted to the classes of kernels considered in [2].
A natural extension to our work here is to assign structured
priors to c, t and σe . These priors can themselves be dependent on their own set of hyperparameters, giving a hierarchical Bayesian formulation. In this way, there would
be greater flexibility in the parameter estimation process.
We can also seek to improve on the computational load of
the algorithms introduced here and in particular the sampling of tk via rejection sampling.
0
20
40
60
Iteration
80
100
Figure 3: Evolution of the tk s in the GS algorithm. The
true values are indicated by the broken red lines.
SAMPTA'09
[1] J. M. Bernardo and A. F. M. Smith. Bayesian Theory.
Wiley, 1st edition, 2001.
[2] P. L. Dragotti, M. Vetterli, and T. Blu. Sampling
moments and reconstructing signals of finite rate of
innovation: Shannon meets Strang–Fix. IEEE Trans.
Signal Processing, 55(5):1741–1757, 2007.
[3] I. Maravic and M. Vetterli. Sampling and reconstruction of signals with finite rate of innovation in the
presence of noise. IEEE Trans. Signal Processing,
53(8):2788–2805, 2005.
[4] H. Nyquist. Certain topics in telegraph transmission
theory. Trans. American Institute of Electrical Engineers, 47:617–644, April 1928.
[5] C. P. Robert and G. Casella. Monte Carlo Statistical
Methods. New York: Springer-Verlag, 2nd edition,
2004.
[6] C. E. Shannon. Communication in the presence of
noise. Proc. Institute of Radio Engineers, 37(1):10–
21, January 1949.
[7] V. F. Y. Tan and V. K. Goyal. Estimating signals
with finite rate of innovation from noisy samples: A
stochastic algorithm. IEEE Trans. Signal Process.,
56(10):5135–5146, October 2008.
[8] L. Tierney. Markov chains for exploring posterior
distributions. Technical Report 560, School of Statistics, Univ. of Minnesota, March 1994.
[9] M. Unser. Sampling–50 years after Shannon. Proc.
IEEE, 88(4):569–587, 2000.
[10] M. Vetterli, P. Marziliano, and T. Blu. Sampling signals with finite rate of innovation. IEEE Trans. Signal Processing, 50(6):1417–1428, 2002.
[11] R. H. Walden. Analog-to-digital converter survey
and analysis. IEEE J. Selected Areas of Communication, 17(4):539–550, April 1999.
156
The Generalized Annihilation Property
A Tool For Solving Finite Rate of Innovation Problems
Thierry Blu
The Chinese University of Hong Kong, Shatin N.T., Hong Kong
thierry.blu@m4x.org
Abstract:
We describe a property satisfied by a class of nonlinear
systems of equations that are of the form F(Ω)X = Y.
Here F(Ω) is a matrix that depends on an unknown Kdimensional vector Ω, X is an unknown K-dimensional
vector and Y is a vector of N ≥ K) given measurements. Such equations are encountered in superresolution
or sparse signal recovery problems known as “Finite Rate
of Innovation” signal reconstruction.
We show how this property allows to solve explicitly for
the unknowns Ω and X by a direct, non-iterative, algorithm that involves the resolution of two linear systems of
equations and the extraction of the roots of a polynomial
and give examples of problems where this type of solutions has been found useful.
At first sight, solving such a nonlinear system of equations is a daunting task. Fortunately, if the matrix F(Ω)
satisfies a property that we shall call “Generalized Annihilation Property” (GAP), this reduces to solving two linear systems of equations sandwiching a nonlinear step that
amounts to polynomial root extraction in practical cases.
The filters ϕ(t) that satisfy the GAP are thus especially interesting, since the related FRI problems enjoy a straight
non-iterative solution.
2.
The Generalized Annihilation Property
(GAP)
We carry on with the previously identified general nonlinear problem, namely
F(Ω) X = Y,
1. Introduction
We consider the signal resulting from the convolution between a window ϕ(t) and the sum of K Diracs with amplitude xk located at time tk . Given the N uniform samples
yn (T = sampling step )
yn =
K
X
xk ϕ(nT −tk )
where n = 1, 2, . . . , N, (1)
(3)
where the unknowns are Ω = [ω1 , ω2 , . . . ωK ] and X =
[x1 , x2 , . . . xK ], and where the measurements are Y =
[y1 , y2 , . . . yN ].
This system is said to satisfy the Generalized Annihilation
Property whenever there exist K + 1 constant matrices,
Ak , and K + 1 scalar functions of Ω, hk (Ω), such that we
have the identity
k=1
then FRI problems (see [1, 2]) consist in retrieving the parameters tk and xk . Solving such problems is conceptually interesting because it shows how to break the standard
Nyquist-Shannon bandlimitation rule for the exact reconstruction of signals from their uniform samples [3].
The system of consistent equations (1) can be expressed
under the generic form of a nonlinear problem as shown
in Fig. 1 (see next page), where the parameters Ω =
[ω1 , ω2 , . . . ωK ] are related unambiguously to the unknowns tk ’s. Because of the variety of settings adapted
to this general approach, it happens to be necessary to distinguish between the parameters ωk —which we shall call
“abstract parameters”—and the locations tk : typically, the
ωk ’s will be the zeros of some polynomial and from these
ωk ’s, we will be able to retrieve the tk ’s using a functional
relation of the form ωk = λ(tk ) for some invertible function λ(t).
SAMPTA'09
K
X
hk (Ω) Ak F(Ω) = 0.
(4)
k=0
for any vector of parameters Ω. By right multiplying with
X, the above equation implies that any solution Ω of (3) is
also a solution of the (generalized) annihilation equation
K
X
hk (Ω)Ak Y = 0.
(5)
k=0
This equation can be expressed in a matrix form AH = 0
where the unknown is H = [ h0 (Ω), h1 (Ω), . . . , hK (Ω) ]T
and the matrix A = A0 Y, A1 Y, . . . , AK Y . Thus, in
order to solve (3) for Ω and X, the idea consists in finding
the scalar coefficients hk (Ω) that satisfy (5), then retrieving ω1 , ω2 , . . . , ωK from the knowledge of hk (Ω), and finally finding X such that F(Ω) X = Y. Without elaborating on the conditions that make this solution unique, a
157
|
ϕ(T − t2 )
ϕ(2T − t2 )
..
.
···
···
ϕ(T − tK )
ϕ(2T − tK )
..
.
ϕ(N T − t1 ) ϕ(N T − t2 )
{z
···
ϕ(N T − tK )
xK
} | {z }
ϕ(T − t1 )
ϕ(2T − t1 )
..
.
x1
x2
..
.
=
X
F(Ω)
y1
y2
..
.
(2)
yN
| {z }
Y
Figure 1: Algebraic equivalent of the consistency equations (1).
minimal requirement is that the matrices Ak have at least
K rows.
In the simple case where the hk (Ω)’s are related to the
ωk ’s through a polynomial relation
K
X
hk (Ω)z −k =
k=0
K
Y
(1 − ωk z −1 ),
(6)
k=1
solving (3) boils down to a three-step algorithm that can
be summarized as follows:
1. Compute a solution H = [ 1, h1 , . . . , hK−1 , hK ]T of
A0 Y, A1 Y, . . . , AK Y H = 0;
2. Compute the roots ωk of the z-transform H(z) =
PK
−k
;
k=0 hk z
3. Compute a solution X of F(Ω) X = Y.
3.
The GAP is actually shared by many interesting filters
that can be used in sampling schemes, resulting in easily solvable FRI problems. Among them, the first ones
to be identified were the periodized sinc, the infinite (i-e.,
not periodized) sinc and the Gaussian kernels [1]. Even
more interestingly, recent research indicates that this property may somewhat be related to the Strang-Fix conditions
which makes a very intriguing connection with approximation theory [12], and considerably broadens the class
of FRI-admissible kernels. In all cases investigated so far,
the scalar coefficients hk (Ω) satisfy (6).
3.1
F(Ω) =
ω1
ω12
..
.
ω2
ω22
..
.
···
···
ω1N
ω2N
···
ωK
2
ωK
N
ωK
where the frequencies to retrieve, fk , are related to ωk
through ωk = ej2πfk . This problem satisfies the GAP
for band-diagonal matrices Ak which are more precisely
given by:
Ak = 0N −K,k IN −K 0N −K,K−k ,
where 0m,n is the m × n zero matrix and In is the n × n
identity matrix. A minimal—yet not sufficient—condition
for the unicity of the solution is N ≥ 2K. Since the Ak
can be seen as shifting operators by k samples, the annihilation equation is analogous to a filtering equation—with
an annihilating filter. The annihilation algorithm is then
equivalent to Prony’s method [4]. Of course, spectral estimation in the presence of noise has been addressed by numerous researchers since the 1970’s [5, 6, 7, 8, 9, 10, 11].
SAMPTA'09
Periodized sinc (Dirichlet) filter
Solving the FRI problem in the case of a periodic stream
of Diracs is equivalent to considering (1) where ϕ is a periodized sinc kernel, e.g., a Dirichlet kernel
Example—Spectral estimation problems boil down to a
nonlinear problem of the form (3) involving the Vandermonde matrix:
Some GAP Kernels
ϕ(t) =
X
sinc(B(t − k ′ τ )) =
k′ ∈Z
sin(πBt)
Bτ sin(πt/τ )
where τ is the period of the Dirac stream and B some
bandwith (chosen so that Bτ is an odd integer) [2]. This
problem can be reformulated using the annihilation equation (4) by defining the following annihilation matrices
Ak = 0Bτ −K,k IBτ −K 0Bτ −K,Bτ −k W
where W = [e−j2πmn/N ] for |m| ≤ ⌊Bτ /2⌋ and 1 ≤
n ≤ N , is the N -DFT submatrix of size Bτ × N . Then,
the abstract parameters ωk are related to the locations tk
through ωk = e−j2πtk /τ . This kernel has been found useful for the estimation of UWB channels [13] and for image
superresolution [14].
3.2
Infinite sinc filter
The filter ϕ(t) is given by ϕ(t) = sinc Bt with B = 1/T .
When ϕ ∗ x (t) is sampled uniformly at frequency B, the
nonlinear system of equations satisfies the GAP. The abstract parameters ωk are related to the locations tk through
158
ωk = tk and the annihilation matrices are given by
K
K
K
···
0
···
K
K−1
0
.
.
.
..
0
..
..
..
.
Ak = .
..
..
..
.
.
.
.
.
K
K
0
···
···
···
K
K−1
1 0 ··· ···
0 2k . . .
. .
. . 3k . . .
×
..
.
..
..
.
.
.
.
0 ··· ···
0
Additionally, there is a constraint on the minimal number
of samples N for the GAP to hold, which is that N be
0
larger than ⌈(S + maxk {tk })/T ⌉.
..
.
0 4. Conclusion
K
0
0
..
.
..
.
0
Nk
3.3 Gaussian filter
The filter ϕ(t) is given by ϕ(t) = exp(−t2 /σ 2 ). When
ϕ ∗ x (t) is sampled uniformly at frequency T −1 , the
nonlinear system of equations satisfies the GAP. The abstract parameters ωk are related to the locations tk through
ωk = exp(2tk T /σ 2 ) and the annihilation matrices are
given by
Ak = 0N −K,k IN −K 0N −K,K−k
T2
e σ2
0
··· ···
0
(2T )2
..
..
2
0
σ
.
.
e
×
..
..
..
.
.
.
0
(N T )2
2
0
···
···
0 e σ
A version of this solution (actually, for a Gabor kernel)
was used in Optical Coherence Tomography, showing the
possibility to resolve slices of a microscopic sample below the coherence length of the illuminating reference
light [15].
3.4 Finite Support Strang-Fix filters
Through linear combinations of its shifts, the finite support filter ϕ(t) is assumed to reconstruct polynomials up
to some degree L − 1 (standard Strang-Fix condition [16])
or exponentials eal t where al − a0 is linear with l =
0, 1, . . . , L − 1. More precisely, in the standard StrangFix case, we denote by cl,n the coefficients such that
X
cl,n ϕ(nT − t) = tl where l = 0, 1, . . . , L − 1,
n∈Z
by T the sampling step, and by [0, S] the support of ϕ(t).
Then, the abstract parameters ωk are related to the locations tk through ωk = tk and the annihilation matrices are
given by
Ak =
ck,1
ck,2
ck−1,1
..
.
ck−1,2
..
.
ck−L+1,1
ck−L+1,2
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···
···
···
ck,N
ck−1,N
..
.
ck−L+1,N
.
We have shown how to unify the different techniques used
in FRI signal reconstruction through an algebraic property that we call the Generalized Annihilation property. In
essence, this property allows to solve nonlinear system of
equations within two noniterative steps. We hope that this
property can be used to solve other FRI problems (i.e, with
new kernels) in particular in dimensions higher than 1 (for
instance, like in [17]), and maybe to solve other types of
problems not directly related to sampling.
References:
[1] M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Transactions on Signal Processing, vol. 50, pp. 1417–1428,
June 2002.
[2] T. Blu, P.-L. Dragotti, M. Vetterli, P. Marziliano, and
L. Coulot, “Sparse sampling of signal innovations,”
IEEE Signal Processing Magazine, vol. 25, no. 2,
pp. 31–40, 2008.
[3] C. E. Shannon, “A mathematical theory of communication,” Bell System Technical Journal, vol. 27,
pp. 379–423 and 623–656, July and October 1948.
[4] R. Prony, “Essai expérimental et analytique,” Annales de l’École Polytechnique, vol. 1, no. 2, p. 24,
1795.
[5] P. Stoica and R. L. Moses, Introduction to Spectral
Analysis. Upper Saddle River, NJ: Prentice Hall,
1997.
[6] S. M. Kay, Modern Spectral Estimation—Theory
and Application. Englewood Cliffs, NJ: Prentice
Hall, 1988.
[7] D. W. Tufts and R. Kumaresan, “Estimation of frequencies of multiple sinusoids: Making linear prediction perform like maximum likelihood,” Proceedings of the IEEE, vol. 70, pp. 975–989, September
1982.
[8] S. M. Kay and S. L. Marple, “Spectrum analysis—a
modern perspective,” Proc. IEEE, vol. 69, pp. 1380–
1419, November 1981.
[9] Special Issue on Spectral Estimation, Proceedings of
the IEEE, vol. 70, September 1982.
[10] V. F. Pisarenko, “The retrieval of harmonics from a
covariance function,” Geophysical Journal, vol. 33,
pp. 347–366, September 1973.
[11] R. Roy and T. Kailath, “ESPRIT–estimation of signal parameters via rotational invariance techniques,”
IEEE Transactions on Acoustics, Speech, and Signal
Processing, vol. 37, pp. 984–995, July 1989.
159
[12] P.-L. Dragotti, M. Vetterli, and T. Blu, “Sampling moments and reconstructing signals of finite
rate of innovation: Shannon meets Strang-Fix,”
IEEE Transactions on Signal Processing, vol. 55,
pp. 1741–1757, May 2007. Part 1.
[13] I. Maravić, J. Kusuma, and M. Vetterli, “Lowsampling rate UWB channel characterization and
synchronization,” Journal of Communications and
Networks, vol. 5, no. 4, pp. 319–327, 2003.
[14] L. Baboulaz and P.-L. Dragotti, “Exact feature extraction using finite rate of innovation principles
with an application to image super-resolution,” IEEE
Transactions on Image Processing, vol. 18, pp. 281–
298, February 2009.
[15] T. Blu, H. Bay, and M. Unser, “A new highresolution processing method for the deconvolution
of optical coherence tomography signals,” in Proceedings of the First IEEE International Symposium
on Biomedical Imaging: Macro to Nano (ISBI’02),
vol. III, (Washington DC, USA), pp. 777–780, July
7-10, 2002.
[16] G. Strang and G. Fix, “A Fourier analysis of the finite
element variational method,” in Constructive Aspects
of Functional Analysis (G. Geymonat, ed.), pp. 793–
840, Rome: Edizioni Cremonese, 1973.
[17] D. Kandaswamy, T. Blu, and D. Van De Ville, “Analytic sensing: reconstructing pointwise sources from
boundary Laplace measurements,” in Proceedings
of the SPIE Conference on Mathematical Imaging:
Wavelet XII, (San Diego CA, USA), August 26August 30, 2007. To appear.
SAMPTA'09
160
An “algebraic” reconstruction of
piecewise-smooth functions from integral
measurements
Dima Batenkov, Niv Sarig, Yosef Yomdin
Department of Mathematics, Weizmann institute of science, Rehovot, Israel.
{dima.batenkov, niv.sarig, yosef.yomdin}@weizmann.ac.il
1.
Introduction
This paper presents some results on a well-known problem
in Algebraic Signal Sampling and in other areas of applied
mathematics: reconstruction of piecewise-smooth functions from their integral measurements (like moments,
Fourier coefficients, Radon transform, etc.). Our results
concern reconstruction (from the moments) of signals in
two specific classes: linear combinations of shifts of a
given function, and “piecewise D-finite functions” which
satisfy on each continuity interval a linear differential
equation with polynomial coefficients.
Let us start with some general remarks and a conjecture.
It is well known that the error in the best approximation of
a C k -function f by an N -th degree Fourier polynomial is
of order NCk . The same holds for algebraic polynomial
approximation and for other basic approximation tools.
However, for f with singularities, in particular, with discontinuities, the error is much larger: its order is only √CN .
Considering the so-called Kolmogorow N -width of families of signals with moving discontinuities one can show
that any linear approximation method provides the same
order of error, if we do not fix a priori the discontinuities’
position (see [7], Theorem 2.10). Another manifestation
of the same problem is the “Gibbs effect” - a relatively
strong oscillation of the approximating function near the
discontinuities. Practically important signals usually do
have discontinuities, so the above feature of linear representation methods presents a serious problem in signal
reconstruction. In particular, it visibly appears near the
edges of images compressed by JPEG, as well as in the
noise and low resolution of the CT and MRI images.
Recent non-linear reconstruction methods, in particular,
Compressed Sensing ([2, 3]) and Algebraic Sampling
([4, 12, 14, 6, 9]), address this problem in many cases.
Both approaches appeal to an a priori information on the
character of the signals to be reconstructed, assuming
their “simplicity” in one or another sense. Compressed
sensing assumes only a sparse representation in a certain (wavelets) basis, and thus it presents a rather general
and “universal” approach. Algebraic Sampling usually requires more specific a priori assumptions on the structure
of the signals, but it promises a better reconstruction accuracy. In fact, we believe that ultimately the Algebraic
Sampling approach has a potential to reconstruct “simple
signals with singularities” as good as smooth ones. In par-
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ticular, the results of [5, 11, 8, 17, 14] strongly support
(also apparently do not accurately formulate and prove)
the following conjecture:
There is a non-linear algebraic procedure reconstructing
any signal in a class of piecewise C k -functions (of one
or several variables) from its first N Fourier coefficients,
with the overall accuracy of order NCk . This includes the
discontinuities’ positions, as well as the smooth pieces
over the continuity domains.
At present there are many approaches available to a robust
detection of discontinuities from Fourier data (see [8, 5,
11] and references therein). The remaining problem seems
to be an accurate estimate of the accuracy of the solution
of the nonlinear systems arising. Our results below can be
considered, in particular, as a step in this direction. On
the other hand, they have been motivated by the results in
[4, 12, 14], and in [9, 6].
2.
Linear combinations of shifts of a given
function
Reconstruction of this class of signals from sampling has
been described in [4, 12]. We study a rather similar problem of reconstruction from the moments. Our method is
based on the following approach: we construct convolution kernels dual to the monomials. Applying these kernels, we get a Prony-type system of equations on the shifts
and amplitudes.
Let us restate a general reconstruction problem, as it appears in our specific setting. We want to reconstruct signals of the form
F (x) =
N X
X
(l)
ai,j,l fi (x + xj )
(1)
i=1 j,l
where the fi ’s are known functions of x = (x1 , . . . , xd ),
and the form (1) of the signal is known a priori. The parameters ai,j,l , xj = (xj1 , . . . , xjd ) are to be found from a
finite number of “measurements”, i.e. of linear (usually
integral) functionals like polynomial moments, Fourier
moments, shifted kernels, evaluation over some grid and
more.
In this paper we consider only linear combinations of
shifts of one known function f (although the method
of “convolution dual” can be extended to several shifted
functions and their derivatives - see [16]). First we consider general integral “measurements” and then restrict
161
ourselves to the moments and Fourier coefficients. In what
follows x = (x1 , . . . , xd ), t = (t1 , . . . , td ), j is a scalar
index, while k = (k1 , . . . , kd ), i = (i1 , . . . , id ) and n =
(n1 , . . . , nd ) are multi-indices. Partial ordering of multiindices is given by k ≤ k ′ ⇔ kp ≤ kp′ , p = 1, . . . , d. So
we have
s
X
F (x) =
aj f (x + xj ).
(2)
j=1
RLet the measurements µk (F ) be given by µk (F ) =
F (t)ϕk (t)dt, for a certain (multi)-sequence of functions
ϕk (t), k ≥ 0 = (0, . . . , 0).
Given f and ϕ = {ϕk (t)}, k ≥ 0 we now try to find
certain “triangular” linear combinations
X
ψk (t) =
Ci,k ϕi (t)
(3)
0≤i≤k
forming, in a sense, some “f -convolution dual” functions
(similar to a bi-orthogonal set of function) with respect to
the system ϕk (t). More accurately, we require that
Z
(4)
f (t + x)ψk (t) = ϕk (x).
We shall call a sequence ψ = {ψk (t)} satisfying (3), (4)
f - convolution dual to ϕ. Below we find convolution dual
systems to the usual and exponential monomials.
We consider a general problem of finding convolution dual
sequences to a given sequence of measurements as an important step in the reconstruction problem. Notice that it
can be generalized by dropping the requirement of a spePk
cific representation (3): ψk (t)R =
i=0 Ci,k ϕi (t). Instead we can require only that f (t)ψk (t) be expressible
in terms of the measurements sequence µk . Also ϕk in
(4) can be replaced by another a priori chosen sequence
ηk . This problem leads, in particular, to certain functional equations, satisfied by polynomials and exponents
(as well as exponential polynomials and some kinds of elliptic functions).
Now we have the following result:
Theorem 1. Let a sequence ψ =
Pψk (t) be f -convolution
dual to ϕ. Define Mk by Mk = 0≤i≤k Ci,k µi . Then the
parameters aj and xj in (2) satisfy the following system of
equations (“generalized Prony system”):
s
X
aj ϕk (xj ) = Mk , k = 0, . . . .
(5)
j=1
P
=
Ci,k µi
=
Proof We have Mk
0≤i≤k
R
R
P
F (t) 0≤i≤k Ci,k ϕi (t)dt
=
F (t)ψk (t)
=
R
Ps
Ps
j
j
f
(t
+
x
a
a
ϕ
(x
).
)ψ
(t)dt
=
k
j=1 j
j=1 j k
In specific examples we can find the minimal number of
equations in (5) necessary to uniquely reconstruct the parameters aj and xj in (2).
So here ϕn (x) = xn1 1 ∙ ∙ ∙ xnd d for each multi-index n =
(n1 , . . . , nd ). We look for the dual functions ψn satisfying
the convolution equation
Z
f (t + x)ψn (t)dt = xn
(7)
for each multi-index n. To solve this equation we apply Fourier transform to both sides of (7). Assuming that
fˆ(ω) ∈ C ∞ (Rd ), fˆ(0) 6= 0 we find (see [16]) that there is
a unique solution to (7) provided by
X
ϕn (x) =
Cn,k xk ,
(8)
k≤n
where
Cn,k
"
1
n
∂ n−k
n+k
= √
(−i)
∂ω n−k
( 2π)d k
1
ω=0
fˆ(ω)
#
.
This calculation is symbolic and works for more general
cases. The actual calculation in our polynomial case is
done using straightforward matrix calculations. We set the
generalized polynomial moments as
X
Mn =
Cn,k mk
(9)
k≤n
and obtain, as in Theorem 1, the following system of equations:
s
X
aj (xj )n = Mn , n ≥ 0.
(10)
j=1
This system can be solved explicitly in a standard way
(see, for example, [13, 4, 15]). In one-dimensional case
it goes as follows (see [13]): from (10) we get that for
z = (z1 , . . . , zd ) the generalized moments generating
function (d = 1 yet, notice that the formulas are still multidimensional)
I(z) =
X
Mn z n =
s
X
j=1
n∈Nd
aj
d
Y
l=1
1
1 − xjl zl
(11)
is a rational function. Hence its Taylor coefficients satisfy linear recurrence relation, which can be reconstructed
through a linear system with the Hankel-type matrix
formed by an appropriate number of the moments Mn ’s.
This is, essentially, a procedure of the diagonal Padé approximation for I(z) (see [13]). The parameters aj , xj are
finally reconstructed as the poles and the residues of I(z).
For several variables, although the formulas are the same
as above, the generalization of the solution of the Prony
system is more involved and should be addressed separately.
In one dimensional case with the derivatives f (l) included
we have
F (x) =
r
s X
X
aj,l f (l) (x + xj ).
(12)
j=1 l=0
2.1
Reconstruction from moments
We are given a finite number of moments of a signal F as
in (2) in the form
Z
mn = F (t)tn dt.
(6)
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The corresponding moment-generating function in this
case takes the form
s X
l
r X
X
l (−1)q+l aj,l /(xj )l
I(z) =
.
(13)
q
(1 − xj z)q+1
q=0
j=1
l=0
162
which is still a rational function (d-dimensional case with
derivatives is similar). We would like to stress that in this
case the dual polynomials ψk are not changed and they
are given as in (8). Therefore also the formula for the
generalized moments Mn is the same as in (9).
2.2
b
Fourier case
fˆ(k) −ikx
= ϕ−k (x).
e
fˆ(k)
(14)
Here the triangular system of equations (3) is actually not
1
triangular any more but still since ψk (x) = fˆ(k)
ϕ−k (x)
we can express the generalized moments through the orig1
inal ones via Mk = fˆ(k)
µ−k [F ]. Now exactly as before
we can find a generalized Prony system in the form
X
X
1
aj e−ikxj =
aj ρkj (15)
µ−k [F ] = Mk =
ˆ
f (k)
j
j
where ρj = e−ixj . In this case we get a rational exponential generating function and we can find its poles and
residues on the unit complex circle as we did in the polynomial case.
2.3 Further extensions
The approach above can be extended in the following directions: 1. Reconstruction of signals built from several
functions or with the addition of dilations also can be investigated (a perturbation approach where the dilations are
approximately 1 is studied in [15]). 2. Further study of
“convolution duality” can significantly extend the class of
signals and measurements allowing for a closed - form signal reconstruction.
Reconstruction of piecewise D-finite functions from moments
Let g : [a, b] → R consist of K+1 “pieces” g0 , . . . gK with
K ≥ 0 jump points
a = ξ0 < ξ1 . . . < ξK < ξK+1 = b
D=
j=0
i
ai,j x
i=0
dj
dxj
N
X
i=1
SAMPTA'09
αi,n ui (x),
Piecewise D-finite Reconstruction Problem. Given
N, {ki }, K, a, b and the moment sequence {mk } of a
piecewise D-finite function g, reconstruct all the parameters {ai,j }, {ξi }, {αi,n }.
Below we state some results (see [1] for detailed proofs)
which provide explicit algebraic connections between the
above parameters and the measurements {mk }.
The first two theorems assume a single continuity interval
(compare [10]).
Theorem 2. Let K = 0 and D g ≡ 0 with D given by (16).
Then the moment sequence {mk (g)} satisfies a linear recurrence relation
N
N
(E −a I) (E −b I) ·
kj
N X
X
(i,j)
ai,j Π
(k, E) mk = 0
j=0 i=0
(18)
where E is the discrete forward shift operator and
Π(i,j) (k, E) are monomials in E whose coefficients are
(i+k)!
polynomials in k: Π(i,j) (k, E) = (−1)j (i+k−j)!
Ei−j .
Theorem 3. Denote
def
(i,j) def
E(E) = (E −a I)N (E −b I)N , vk = E(E) · Π(i,j) (k, E) mk ,
∞
dj
def X (0,j) k
def
hj (z) =
vk z ,
Gj (x) = E(x) j g(x)
dx
k=0
Assume the conditions of Theorem 2. Then
(1) The vector of the coefficients a = (ai,j ) satisfies a linear homogeneous system
(0,0)
(1,0)
(k ,N )
v0
. . . v0 N
v0
a0,0
(0,0)
(1,0)
(kN ,N )
v1
. . . v1
a1,0
v1
.
Ha =
=0
.
.
.
.
.
.
.
.
.
.
. ..
.
vc
(1,0)
M
vc
...
(k ,N )
M
v cN
akN ,N
(19)
c ∈ N.
for all M
(i,j)
(aij ∈ R)
(16)
Each gn may be therefore written as a linear combination of functions {ui }N
i=1 which are a basis for the space
ND = {f : D f ≡ 0}:
gn (x) =
a
We subsequently formulate the following
(0,0)
M
Furthermore, let g satisfy on each continuity interval some
linear homogeneous differential equation with polynomial
coefficients: D gn ≡ 0, n = 0, . . . , K where
kj
N X
X
xk g(x)dx
mk (g) =
In the same manner as in section 2.1 we now choose
1
e−ikx .
ϕk (x) = eikx . We get immediately ψk (x) = fˆ(k)
Indeed,
Z
Z
1 ikt
e dt =
f (t + x)ψk (t)dt = f (t + x)
ˆ
f (k)
3.
We term such functions g “piecewise D-finite”. Many
real-world signals may be represented as piecewise Dfinite functions, in particular: polynomials, trigonometric
functions, algebraic functions.
The sequence {mk = mk (g)} is given by the usual moments
Z
n = 0, 1, . . . , K
(17)
(2) vk = mi+k (Gj (x)). Consequently, hj (z) is the
moment generating function of Gj (x).
Pkj
ai,j xi . Then the functions
(3) Denote pj (x) = i=0
Φ = {1, h0 (z), . . . hN (z)} are polynomially depen
PN
max kj
dent:
pj (z −1 ) = Q(z) where
j=0 hj (z) z
Q(z) is a polynomial with deg Q < max kj . The system of polynomials {z max kj pj (z −1 )} is called the
Padé-Hermite form for Φ.
163
To handle the piecewise case, we represent
( the jump dis0 x<0
def
continuities by the step function H(x) =
and
1 x≥0
write g as a distribution
g(x) = ge0 +
K
X
n=1
gf
n (x)H(x − ξn )
(20)
Theorem 4. Let K > 0 and let g be as in (20) with operator D annihilating every piece gf
n . Then the operator
Y
K
def
N
b
D=
(x − ξi ) I · D
(21)
n=1
annihilates the entire g as a distribution. Consequently,
conclusions of Theorems 2 and 3 hold with D replaced by
b as in (21).
D
Proposition 5. Let K ≥ 0 and {ui }N
i=1 be a basis for the
space ND , where D annihilates every piece of g. Assume
Rξ
(17) and denote cni,k = ξnn+1 xk ui (x) for n = 0, . . . , K.
f ∈ N:
A straightforward computation gives ∀M
α1,0
.
0
m0
c1,0 . . . c0N,0 . . . cK
..
N,0
m1
..
..
..
..
..
= .
α
.
N,0
.
.
.
.
.. ..
K
0
c01,M
.
.
.
c
.
.
.
c
.
f
f
f
N,M
N,M
mM
f
αN,K
(22)
The above results can be combined as follows to provide
a solution of the Reconstruction Problem:
(a) Let N, {ki }, K, a, b and {mk (g)} be given. If K > 0,
b according to (21).
replace D (still unknown) with D
(b) Build the matrix H as in (19). Solve Ha = 0 and
obtain the operator D∗ = Da which annihilates g.
(c) If K > 0, factor out all the common roots of the polynomial coefficients of D∗ with multiplicity N . These
are the locations of the jump points {ξn }. The remaining part is the operator D† which annihilates every gn .
(d) By now D† and {ξn } are known. So compute the basis
for ND† and solve (22).
c and M
f determine the minimal required
The constants M
size of the corresponding linear systems (19) and (22) in
order for all the solutions of these systems to be also solutions of the original problem. It can be shown that:
c without any ad1. There exists no uniform bound on M
ditional information on the nature of the solutions.
Explicit bounds may be obtained for simple function
classes such as piecewise polynomials of bounded
degrees or real algebraic functions.
f=M
f(D)
2. For every specific D, an explicit bound M
may be computed for the system (22).
The above algorithm has been tested on exact reconstruction of piecewise polynomials, piecewise sinusoids and rational functions.
SAMPTA'09
References:
[1] D.Batenkov, Moment inversion problem for
piecewise D-finite functions, arXiv:0901.4665v2
[math.CA].
[2] E. J. Candes̀. Compressive sampling. Proceedings
of the International Congress of Mathematicians,
Madrid, Spain, 2006. Vol. III, 1433–1452, Eur.
Math. Soc., Zurich, 2006.
[3] D. Donoho, Compressed sensing. IEEE Trans. Inform. Theory 52 (2006), no. 4, 1289–1306.
[4] P.L. Dragotti, M. Vetterli and T. Blu, Sampling Moments and Reconstructing Signals of Finite Rate of
Innovation: Shannon Meets Strang-Fix, IEEE Transactions on Signal Processing, Vol. 55, Nr. 5, Part 1,
pp. 1741-1757, 2007.
[5] K. Eckhoff, Accurate reconstructions of functions of
finite regularity from truncated Fourier series expansions, Math. Comp. 64 (1995), no. 210, 671–690.
[6] M. Elad, P. Milanfar, G. H. Golub, Shape from
moments—an estimation theory perspective, IEEE
Trans. Signal Process. 52 (2004), no. 7, 1814–1829.
[7] B. Ettinger, N. Sarig. Y. Yomdin, Linear versus
non-linear acqusition of step-functions, J. of Geom.
Analysis, 18 (2008), 2, 369-399.
[8] A. Gelb, E. Tadmor, Detection of edges in spectral
data II. Nonlinear enhancement, SIAM J. Numer.
Anal. 38 (2000), 1389-1408.
[9] B. Gustafsson, Ch. He, P. Milanfar, M. Putinar, Reconstructing planar domains from their moments. Inverse Problems 16 (2000), no. 4, 1053–1070.
[10] V. Kisunko, Cauchy type integrals and a D-moment
problem. C.R. Math. Acad. Sci. Soc. R. Can. 29
(2007), no. 4, 115–122.
[11] G. Kvernadze, T. Hagstrom, H. Shapiro, Locating
discontinuities of a bounded function by the partial
sums of its Fourier series., J. Sci. Comput. 14 (1999),
no. 4, 301–327.
[12] I. Maravic and M. Vetterli, Exact Sampling Results
for Some Classes of Parametric Non-Bandlimited 2D Signals, IEEE Transactions on Signal Processing,
Vol. 52, Nr. 1, pp. 175-189, 2004.
[13] E. M. Nikishin, V. N. Sorokin, Rational Approximations and Orthogonality, Translations of Mathematical Monographs, Vol 92, AMS, 1991.
[14] P. Prandoni, M. Vetterli, Approximation and compression of piecewise smooth functions, R. Soc.
Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci.
357 (1999), no. 1760, 2573–2591.
[15] N. Sarig, Y. Yomdin, Signal Acquisition from Measurements via Non-Linear Models, C. R. Math. Rep.
Acad. Sci. Canada Vol. 29 (4) (2007), 97-114.
[16] N. Sarig and Y. Yomdin, Reconstruction of “Simple”
Signals from Integral Measurements, in preparation.
[17] E. Tadmor, High resolution methods for time dependent problems with piecewise smooth solutions.
Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 747–757, Higher
Ed. Press, Beijing, 2002.
164
Distributed Sensing of Signals Under a Sparse
Filtering Model
Ali Hormati , Olivier Roy , Yue M. Lu and Martin Vetterli
Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.
We consider the task of recovering correlated vectors at
a central decoder based on fixed linear measurements obtained by distributed sensors. Two different scenarios are
considered: In the case of universal reconstruction, we look
for a sensing and recovery mechanism that works for all
possible signals, whereas in the case of almost sure reconstruction, we allow to have a small set (with measure zero)
of unrecoverable signals. We provide achievability bounds
on the number of samples needed for both scenarios. The
bounds show that only in the almost sure setup can we effectively exploit the signal correlations to achieve effective
gains in sampling efficiency. In addition, we propose an
efficient and robust distributed sensing and reconstruction
algorithm based on annihilating filters.
1. Introduction
Consider two signals that are linked by an unknown filtering operation, where the filter is sparse in the time domain.
Such models can be used, e.g., to describe the correlation between the transmitted and received signals in an unknown multi-path environment. We sample the two signals
in a distributed setup: Each signal is observed by a different sensor, which sends a certain number of non-adaptive
and fixed linear measurements of that signal to a central decoder. We study how the correlation induced by the above
model can be exploited to reduce the number of measurements needed for perfect reconstruction at the central decoder, but without any inter-sensor communication during
the sampling process.
Our setup is conceptually similar to the Slepian-Wolf problem in distributed source coding [6], which consists of
correlated sources to be encoded separately and decoded
jointly. While communication between the encoders is precluded, correlation between the measured data can be taken
into account as an effective means to reduce the amount of
information transmitted to the decoder. The main difference between our work and this classical distributed source
coding setup is that we study a sampling problem and hence
are only concerned about the number of sampling measurements we need to take, whereas the latter is about coding
and hence uses bits as its “currency”. From the sampling
perspective, our work is closely related to the problem of
distributed compressed sensing, first introduced in [1] (see
also [4, 5]). In that framework, jointly sparse data need
to be reconstructed based on linear projections computed
SAMPTA'09
M1
x1
A1
Dec
h
Abstract:
x̂1 , x̂2
M2
x2
A2
Figure 1: Distributed sensing setup. Signals x1 and x2
are connected through an unknown sparse filter h. The ith
sensor (i = 1, 2) provides a Mi -dimensional observation of
the signal xi via a non-adaptive and fixed linear transform
Ai to a central decoder.
by distributed sensors. In this paper, we first introduce in
Section 2. a novel correlation model for distributed signals.
Instead of imposing any sparsity assumption on the signals
themselves (as in [1]), we assume that the signals are linked
by some unknown sparse filtering operation. Such models
can be useful in describing the signal correlation in several
practical scenarios (e.g. multi-path propagation and binaural audio recoding). In Section 3., we introduce two strategies for the design of the sampling system: In the universal
strategy, we seek to successfully sense and recover all signals, whereas in the almost sure strategy, we allow to have
a small set (with measure zero) of unrecoverable signals.
We establish the corresponding achievability bounds on the
number of samples needed for the two strategies mentioned
above. These bounds indicate that the sparsity of the filter
can be useful only in the almost sure strategy. Since the algorithms that achieves the bounds are computationally prohibitive, we introduce in Section 4., a concrete distributed
sampling and reconstruction scheme that can recover the
original signals in an efficient and robust way. Finally, Section 5. presents an application of our results in the area of
binaural hearing aids. A preliminary version of this work
was also presented at ICASSP 2009. In this paper, we add
results on the achievability bound for the almost sure setup
as well as a new section on applications.
2. The Correlation Model
Consider two signals x1 (t) and x2 (t), where x2 (t) can be
obtained as a filtered version of x1 (t). In particular, we
assume that
x2 (t) = (x1 ∗ h)(t) ,
(1)
165
x1 (t)
h(t)
x2 (t)
...
(x2 [0], . . . , x2 [N − 1])T , linked to each other through a
circular convolution
...
...
...
x2 [n] = (x1 ⊛ h)[n] for n = 0, 1, . . . , N − 1,
A/D
A/D
...
...
...
where h = (h[0], . . . , h[N − 1])T ∈ RN is an unknown
K-sparse vector, that is, khk0 = K.
...
windowing
windowing
(4)
3. Bounds
3.1 Universal Recovery
x1 [n]
x2 [n]
h[n]
Figure 2: The continuous-time sparse filtering operation
and its discrete-time counterpart.
P
where h(t) = K
k=1 ck δ(t − tk ) is a stream of K Diracs
K
with unknown delays {tk }K
k=1 and coefficients {ck }k=1 .
In this work, we study a finite-dimensional discrete version of the above model. As shown in Figure 2, we assume
that the original continuous signal x1 (t) is bandlimited to
[−σ, σ]. Sampling x1 (t) at uniform time interval T leads
def
to a discrete sequence of samples xs1 [n] = x1 (nT ), where
the sampling rate 1/T is set to be above the Nyquist rate
σ/π. To obtain a finite-length signal, we subsequently apply a temporal window to the infinite sequence xs1 [n] and
get
def
x1 [n] = xs1 [n] wN [n],
for n = 0, 1, ..., N − 1,
where wN [n] is a smooth temporal window of length N .
Note that when N is large enough, we can neglect the windowing effect, since w
bN (ω)/(2π) approaches a Dirac function δ(ω) as N → ∞.
Applying the above procedure to x2 (t) and using (1), we
have
2πm
1
b2
≈ X1 [m]H[m],
(2)
X2 [m] ≈ x
T
NT
where
def
H[m] =
K
X
ck e−j2πmtk /(N T ) .
(3)
k=1
The above relationship implies that the finite-length signals
x1 [n] and x2 [n] can also be approximately modeled as the
input and output of a discrete-time filtering operation1. In
general, the location parameters {tk } in (3) can be arbitrary
real numbers, and consequently, the discrete-time filter h[n]
is no longer sparse (see Figure 2 for a typical impulse response of h[n]). However, when the sampling interval T
is small enough, we can assume that the real-valued delays
{tk } are close enough to the sampling grid, i.e., tk /T ≈ nk
for some integers {nk }. We will follow this assumption2
throughout the paper.
Definition 1 (Correlation Model) The signals of interest
are two vectors x1 = (x1 [0], . . . , x1 [N − 1])T and x2 =
1 Note that in order to be unambiguous in the positions {t }, we need
k
to ensure that N T > max {tk }.
k
2 We
introduce this assumption (i.e. tk /T = nk for some nk ∈ Z)
mainly for the simplicity it brings to the theoretical analysis in later parts
of this paper. It is however not an inherent limitation of our work.
SAMPTA'09
Let A1 and A2 be the sampling matrices used by the two
sensors, and A be the block-diagonal matrix with A1 and
A2 on the main diagonal. We first focus on finding those
A1 and A2 such that every xT = (xT1 , xT2 ) is uniquely
determined by its sampling data Ax. Here we denote by X
the set of all stacked vectors x such that its components x1
and x2 satisfy (4) for some K-sparse vector h.
Definition 2 (Universal Achievability) We say a sampling pair (M1 , M2 ) is achievable for universal reconstruction if there exists fixed measurement matrices A1 ∈
RM1 ×N and A2 ∈ RM2 ×N such that the set
def
B(A1 , A2 ) = {x ∈ X : ∃ x′ ∈ X with x 6= x′
(5)
′
but Ax = Ax }
is empty.
Intuition suggests that, due to the correlation between
the vectors x1 and x2 , the minimum number of samples
needed to perfectly describe all possible vectors x can
made smaller than the total number of coefficients 2N . The
following proposition shows that, surprisingly, this is not
the case.
Proposition 1 A sampling pair (M1 , M2 ) is achievable for
universal reconstruction if and only if M1 ≥ N and M2 ≥
N.
Proof Let us consider two stacked vectors xT = (xT1 , xT2 )
′T
and x′T = (x′T
1 , x2 ), each following the correlation
model (4). They can be written under the form
IN
I
′
x=
x1 and x = N′ x′1 ,
C
C
where C and C ′ are circulant matrices with vectors h and
h′ as the first column, respectively. It holds that
I
−I N x1
x − x′ = N
.
C −C ′ x′1
Moreover, we have that
I
−I N
rank N
= N + rank (C − C ′ ) .
C −C ′
When C − C ′ is of full rank, the above matrix is of
rank 2N . This happens, for example, when K = 1 with
C = 2I N and C ′ = I N . In this case, x − x′ can take
any possible values in R2N . Hence, a necessary (and sufficient) condition for the set (5) to be empty is that the blockdiagonal matrix A is a M × 2N -dimensional matrix of full
rank, with M ≥ 2N . In particular, A1 and A2 must be full
rank matrices of size M1 × N and M2 × N , respectively,
with M1 , M2 ≥ N . Note that, in the centralized scenario,
the full rank condition would still require to take at least
2N measurements.
166
M2
3.2 Almost Sure Recovery
As shown in Proposition 1, universal recovery is a rather
strong requirement to satisfy since we have to take at least
N samples at each sensor, without being able to exploit the
existing correlation. In many situations, however, it is sufficient to consider a weaker requirement, which aims at finding measurement matrices that permit the perfect recovery
of almost all signals from X .
Definition 3 (Almost Sure Achievability) We say a sampling pair (M1 , M2 ) is achievable for almost sure reconstruction if there exist fixed measurement matrices A1 ∈
RM1 ×N and A2 ∈ RM2 ×N such that the set B(A1 , A2 ),
as defined in (5), is of probability zero.
The above definition for the almost sure recovery depends
on the probability distribution of the signal x1 and the
sparse filter h. In what follows, it is sufficient to assume
that the signal x1 and the non-zero coefficients of the filter h have non-singular3 probability distributions over RN
and RK , respectively. The following proposition gives an
achievability bound of the number of samples needed for
the almost sure reconstruction.
Proposition 2 A sampling pair (M1 , M2 ) is achievable for
almost sure reconstruction if
N
2K + 1
K +2
K + 2 2K + 1
N
M1
Figure 3: Achievable sampling region for universal reconstruction (shaded area), sampling pairs achieved for almost
sure reconstruction for K odd (solid line) and sampling
pairs achieved for almost sure reconstruction by the proposed algorithm based on annihilating filters (dashed line).
⌊N/2⌋ − K
K +1
X1
X2
Figure 4: Sensors 1 and 2 both send the first K + 1 DFT
coefficients of their observation, but only complementary
subsets of the remaining frequency components.
M1 ≥ min {K + r, N } ,
M2 ≥ min {K + r, N } ,
and M1 + M2 ≥ min {N + K + r, 2N } ,
(6)
where r = 1 + mod (K, 2).
Proof Due to space limitations, we just provide the sketch
of the proof which is constructive in nature. First, let the
two sensors take the Fourier transform of their signals and
send the first (K + r + 1)/2 frequency components to the
central decoder. By dividing the two sets of measurements
(Note that the denominator should not be zero, which is
guaranteed almost surely), the decoder calculates the necessary Fourier elements of the K-sparse filter h in order
to reconstruct it almost surely. Then, the sensors transmit complementary subsets of frequency indices up to the
Nyquist frequency. Knowing the filter h and the frequency
content of one of the signals at some index, the decoder
computes the corresponding frequency content of the other
signal using (4).
Proposition 2 shows that, in contrast to the universal scenario, the correlation between the signals by means of the
sparse filter provides a big saving in the almost sure setup,
especially when K ≪ N . This is depicted as the solid line
in Figure 3.
Unfortunately, the algorithm that attains the bound in (6)
is combinatorial in nature and thus, computationally prohibitive [1]. In the following, we propose a novel distributed sensing algorithm based on annihilating filters.
This algorithm needs effectively K more measurements
with respect to the achievability region for the almost
sure reconstruction but exhibits polynomial complexity of
O(KN ).
3 By a non-singular distribution, we mean any continuous distribution
such that the probability that the random variables lie in a low-dimensional
subspace is zero.
SAMPTA'09
4. Distributed Sensing Algorithm
The proposed distributed sensing scheme is based on a
frequency-domain representation of the input signals. Let
us denote by X 1 ∈ CN and X 2 ∈ CN the DFTs of the
vectors x1 and x2 , respectively. The circular convolution
in (4) can be expressed as
X2 = H ⊙ X1 ,
(7)
where H ∈ CN is the DFT of the filter h and ⊙ denotes the
element-wise product. Our approach consists of two main
steps:
1. Finding filter h by sending the first K + 1 (1 real and
K complex) DFT coefficients of x1 and x2 .
2. Sending the remaining frequency indices by sharing
them among the two sensors.
The decoder first finds the filter h using only the first K + 1
DFT coefficients of x1 and x2 . To this end, the decoder first
computes
H[m] =
X2 [m]
X1 [m]
and H[−m] = H ∗ [m]
(8)
provided that X1 [m] is non-zero for m = 0, 1, . . . , K. This
happens almost surely if the distribution of x1 is, for example, non-singular. Then, it finds the K-sparse filter with an
annihilating filter approach; see [7] for details. The senors
also transmit complementary subsets (in terms of frequency
indexes) of the remaining DFT coefficients of their signals
(N − 2K − 1 real values in total). This is illustrated in
Figure 4. The first K + 1 DFT coefficients allow to almost
surely reconstruct the filter h. The missing frequency components of x1 (resp. x2 ) are then recovered from the available DFT coefficients of x2 (resp. x1 ) using the relation (7).
167
α
αmax
d
t
αmin
αmax
ω
αmin
(a)
(b)
Figure 5: Audio Experiment Setup. (a) A sound source
travels at a distance of d meter in front of the head. (b)
Angular position of the sound source with respect to time.
relative delay between the two received signals, which can
be used to localize the source.
Figure 6 demonstrates the localization performance of the
algorithm. Figure 6(a) shows the evolution of the original
binaural impulse response over time. Figures 6(b)- 6(d) exhibits the sparse approximation to the filter, using different
number of measurements. This clearly demonstrates the
effect of the over-sampling factor on the robustness of the
reconstruction algorithm.
0.8
0.8
2
0.6
1.5
0.4
1
0.2
0.5
0
−0.2
delay (msec)
delay (msec)
0.4
Note that in order to compute X1 [m] from X2 [m], the frequency components of the filter H[m] should be nonzero.
This is insured almost surely with our assumption that the
nonzero elements of the filter h are chosen according to a
non-singular distribution in RK . In terms of achievability,
we have thus shown the following result.
0
−0.4
1
0.2
0.5
0
−0.2
0
−0.4
−0.5
−0.5
−0.6
−0.6
−1
0
2
4
6
8
−1
10
0
2
4
Time (s)
6
8
10
Time (s)
(a) Original
(b) L = 5
0.8
0.8
2
0.6
2
0.6
1.5
1.5
0.4
1
0.2
0.5
0
−0.2
0
−0.4
delay (msec)
0.4
delay (msec)
Proposition 3 A sampling pair (M1 , M2 ) is achievable for
almost sure reconstruction using the efficient annihilating
filter method if
2
0.6
1.5
1
0.2
0.5
0
−0.2
0
−0.4
−0.5
M1 ≥ min {2K + 1, N } ,
−0.6
0
M2 ≥ min {2K + 1, N } ,
and M1 + M2 ≥ min {N + 2K + 1, 2N } .
In the presence of noise or model mismatch, we add robustness to the system by sending L + 1 DFT coefficients of xi
(i = 1, 2) with L ≥ K to the decoder. We denoise the measurements by using the denoising algorithm due to Cadzow;
for details see [3]. Then the annihilating filter method uses
the denoised measurements to estimate the sparse filter.
−0.5
−0.6
−1
2
4
6
8
10
−1
0
Time (s)
2
4
6
8
10
Time (s)
(c) L = 15
(d) L = 25
Figure 6: Tracking the binaural impulse response. Each
column in the image corresponds to the binaural impulse
response at the time mentioned on the x axis. (a) Original
binaural filter. (b)-(d) Tracking the evolution of the main
peak with different values of the oversampling factor L.
6. Conclusions
5. Application
In a practical scenario, we consider the signals recorded by
two hearing aids mounted on the left and right ears of the
user. We assume that the signals of the two hearing aids
are related thorough a filtering operation. We refer to this
filter as binaural filter. In the presence of a single source
in far field, and neglecting reverberations and the headshadow effect [2], the signal recorded at hearing aid 2 is
simply a delayed version of the one observed at hearing aid
1. Hence, the binaural filter can be assumed to have sparsity
factor K = 1. In the presence of reverberations and head
shadowing, the filter from one microphone to the other is
no longer sparse which introduces model mismatch. Despite this model mismatch, the transfer function between
the two received signals should be approximately sparse,
with the main peak indicating the desired relative delay.
In our setup, a single sound source located at distance d = 1
meter from the head of a KEMAR mannequin, moves back
and forth between two angles αmin = −45◦ and αmax =
45◦ . The angular speed of the source is ω = 18 deg/sec.
The sound is recorded by the microphones of the two hearing aids, located at the ears of the mannequin. We want to
retrieve the binaural filter between the two hearing aids at
hearing aid 1, from limited data transmitted by hearing aid
2. Then, the main peak of the binaural filter indicates the
SAMPTA'09
A general formulation of the distributed sensing problem
has been proposed where the two signals are connected
through an unknown sparse filter. In this context, both universal and almost sure reconstruction were addressed together with their corresponding achievable bounds. In addition, a distributed sensing scheme was presented, together
with a method to make it robust to model mismatch. Our
future research will focus on investigating more the applications of the proposed methods in the distributed sensing
context.
References:
[1] D. Baron, M. B. Wakin, M. F. Duarte, S. Sarvotham, and R. G. Baraniuk. Distributed compressed sensing. Technical Report ECE-0612,
Electrical and Computer Engineering Department, Rice University,
Dec. 2006.
[2] J. Blauert. Spatial Hearing: The Psychophysics of Human Sound
Localization. MIT Press, Cambridge, MA, 1997.
[3] J. A. Cadzow. Signal enhancement – A composite property mapping
algorithm. IEEE Trans. Acoust., Speech, Signal Process., 36(1):49–
67, Jan. 1988.
[4] E. J. Candès, J. Romberg, and T. Tao. Robust uncertainty principles:
Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory, 52(2):489–509, Feb. 2006.
[5] D. L. Donoho. Compressed sensing. IEEE Trans. Inf. Theory,
52(4):1289–1306, Apr. 2006.
[6] D. Slepian and J. K. Wolf. Noiseless coding of correlated information
sources. IEEE Trans. Inf. Theory, 19:471–480, Jul. 1973.
[7] M. Vetterli, P. Marziliano, and T. Blu. Sampling signals with finite
rate of innovation. IEEE Trans. Signal Process., 50(6):1417–1428,
Jun. 2002.
168
A method for generalized sampling and
reconstruction of finite-rate-of-innovation
signals
Chandra Sekhar Seelamantula and Michael Unser
Biomedical Imaging Group
Ecole polytechnique fédérale de Lausanne
Switzerland
{chandrasekhar.seelamantula, michael.unser}@epfl.ch
signals that have a finite rate of innovation (FRI). SpecifiAbstract:
cally, consider the stream of time-ordered Dirac impulses:
We address the problem of generalized sampling and reconstruction of finite-rate-of-innovation signals. SpecifL
X
ically, we consider the problem of sampling streams of
x(t) =
aℓ δD (t − tℓ ),
(1)
Dirac impulses and propose a two-channel method that enℓ=1
ables fast, local reconstruction under some suitable condiwhere δD (·) denotes the Dirac impulse. The problem is
tions. We also specify some acquisition kernels and give
to compute the parameters {aℓ , tℓ ; 1 ≤ ℓ ≤ L} based
the associated reconstruction formulas. It turns out that
on
some measurements on x(t). The parametric nature
these kernels can also be combined into one kernel, which
of
the
problem has resulted in the development of techcan be employed in the single-channel sampling scenario.
niques
that are quite different from those that sampling
The single-kernel approach carries over all the advantages
theorists
have been familiar with. Typically, the reconof the two-channel counterpart. Simulation results are prestruction
techniques developed by Vetterli et al. [5] and
sented to validate the theoretical calculations.
Dragotti et al. [6] have a flavor of parametric spectral estimation [7]. They also employ in a novel fashion spline
1. Introduction and prior art
kernels [8, 9] that reproduce polynomials or exponentials.
It is remarkable that these kernels, which play a vital role
Sampling theory is the foundation on which digital sigin wavelet theory, are also quite useful for sampling FRI
nal processing has been built. The popular flavor of the
signals.
sampling theory is due to Shannon [1] and deals excluIn the techniques mentioned above, the focus is exsively with bandlimited signals. Shannon’s theory was
clusively on the single-channel case. Recently, some
generalized in several ways, the most prominent one benew multichannel approaches have also been developed.
ing the theory of multichannel sampling developed by PaKusuma and Goyal proposed a new technique for reconpoulis [3]—his theory is known as the Generalized Samstructing an unknown number of impulses over a finite inpling Theory. Papoulis’ formalism, however, deals only
terval of time by using a successive approximation critewith bandlimited signals. To accommodate the more genrion [10]. Their technique can be implemented using a
eral class of finite-energy signals, Unser and Zerubia [4]
bank of integrators and B-splines. Baboulaz and Dragotti
developed a theory, which does not rely on the bandlimproposed a distributed acquisition scheme for FRI sigiting constraint. Another important extension is the samnals and demonstrated applications to image registration
pling and reconstruction of signals that lie in some shiftand super-resolution image restoration [11]. In [12], we
invariant subspace spanned by the integer-shifted versions
have proposed a two-channel sampling method for the
of a generator kernel (see [2] and the references therein).
FRI problem (cf. Fig. 1). We have employed first-order
The specific case of bandlimited sampling corresponds to
resistor-capacitor networks to sample streams of Dirac ima sinc kernel and is subsumed by this formalism.
pulses and piecewise-constant functions. The reconstrucRecently, Vetterli et al. [5] extended sampling theory in
tion technique boils down to solving a system of two equaa new direction to answer a question that has not been
tions containing the unknown parameters in decoupled
addressed before—that of sampling and reconstructing
form. The key result in [12] is given below:
streams of Dirac impulses and signals derived therefrom.
These signals are not constrained to lie in the space of
Proposition 1 The stream of Dirac impulses in (1) is
finite-energy functions nor in the space of bandlimited
uniquely specified by the samples yα (nT ) = (x∗hα )(nT )
functions. They may also not lie in some shift-invariant
and yγ (nT ) = (x ∗ hγ )(nT ), n ∈ Z, where hα (t) =
subspace generated by a kernel. Typically, such signals
α e−α t u(t), hγ (t) = γ e−γ t u(t), and α 6= γ, provided
are specified by a set of discrete parameters per time unit,
that min {tℓ − tℓ−1 } ≥ T .
2≤ℓ≤L
also known as their rate of innovation. We are interested in
SAMPTA'09
169
Motivation for the present work
The above proposition relies on causal exponential functions for sampling. Working with exponentials has the
practical advantage that they can be easily generated by
employing first-order resistor-capacitor circuits. From a
mathematical viewpoint, however, exponentials are probably not the only class of functions that enable accurate
reconstruction. The main motivation behind the present
paper is the quest for alternative kernels hα (t) and hγ (t)
that would fit into the framework of the above proposition (also cf. Fig. 1). To that end, we first reformulate the
method proposed in [12] in a more general framework and
specify some kernels that enable exact reconstruction.
2.
Generalized sampling formulation
Consider the two-channel sampling scenario shown in
Fig. 1. Let hα (t) and hγ (t), α, γ ∈ C, denote the impulse responses of two causal linear shift-invariant systems, compactly supported on [0, T ] and nonzero over that
interval. Consider the stream of Dirac impulses in (1),
where the impulses are separated by at least T ; i.e.,
min {tℓ − tℓ−1 } ≥ T.
2≤ℓ≤L
(2)
Deviations from this condition shall be addressed later.
The output of the system to the input x(t) is given by
∆
yα (t) = (x ∗ hα )(t) =
L
X
aℓ hα (nT − tℓ ) δK [nT − r(tℓ )] ,
ℓ=1
where r(tℓ ) = tTℓ T is the operator that performs the
ceiling of tℓ with respect to the sampling grid and δK denotes the Kronecker impulse. The sequence yα (nT ) comprises Kronecker impulses, each corresponding to a Dirac
impulse in x(t) under the condition (2). Note that the sampling period T equals the support of the kernel. Similarly,
corresponding to a system with impulse response hγ (t),
γ 6= α, we have
yγ (nT ) =
L
X
aℓ hγ (nT − tℓ ) δK [nT − r(tℓ )] .
ℓ=1
Note that these sampling instants correspond to the
nonzero values in the sequences yα (nT ) and yγ (nT ) and
are therefore known. Consider the ℓth nonzero samples in
the sequences yα (nT ) and yγ (nT ):
yα (r(tℓ ))
= aℓ hα [r(tℓ ) − tℓ ] and
(3)
yγ (r(tℓ ))
= aℓ hγ [r(tℓ ) − tℓ ].
(4)
SAMPTA'09
hα (t)
T
ℓ=1
hγ (t)
T
{aℓ , tℓ }
Figure 1: Two-channel sampling of a stream of dirac impulses.
In (3) and (4), the indices r(tℓ ) and the values on the left
hand side are known. The impulse responses hα and hγ
are also known; their design shall be explained below. The
amplitude and position parameters {tℓ , aℓ } are unknown
and have to be determined. The amplitude of the ℓth Dirac
impulse appears as a multiplicative factor. The position
of the Dirac impulse is encoded in the amplitude of the
Kronecker impulse. Dividing (3) by (4) eliminates aℓ and
gives rise to an equation in the unknown tℓ , which can
be computed if and only if (hα /hγ )(t) is invertible on its
range. The value of tℓ thus obtained can then be substituted in (3) or (4) to obtain the value of aℓ . Some specific
functions that fit into the above reconstruction paradigm
are presented next.
3.
Let us next consider the samples of yα (t) taken on a uniform grid with a sampling step T . Note that we have
chosen the sampling period to be equal to the support
of hα (t); otherwise, we are likely to miss some closelyspaced impulses as the following analysis shows. The
samples of yα (t) are given by
L
X
aℓ δD (t − tℓ )
Kernels for two-channel sampling
aℓ hα (t − tℓ ).
ℓ=1
yα (nT ) =
L
!
Parameter
computation
1.1
We specify only the kernel hα (t); unless otherwise mentioned, hγ (t) is obtained by replacing α with γ; i.e., both
kernels have the same functional form. The kernels involve gating by the B-spline of order zero, at scale T :
β(t) = u(t) − u(t − T ), where u(t) is the unit step function. We specify the kernel definitions and give the expressions for {tℓ , aℓ } directly. The intermediate calculations are omitted but it is straightforward to supply them
starting from the definition of the kernel.
1. Exponential spline (E-spline) kernels [9]: hα (t) =
e−α t β(t), α ∈ R, where u(t) is the unit-step function. The parameters of ℓth impulse are given by
1
yα (r(tℓ ))
tℓ = r(tℓ ) +
log
and
α−γ
yγ (r(tℓ ))
α
yα (r(tℓ ))
log
.
aℓ = yα (r(tℓ )) exp −
α−γ
yγ (r(tℓ ))
This kernel choice has been analyzed in sufficient
detail in [12]. The specific kernel given above is
a first-order E-spline kernel. One could, in principle, also employ higher-order kernels. The advantage
of first-order E-spline kernels over the higher-order
ones, however, is that they always give rise to closedform solutions. The higher-order kernels exhibit this
property only for certain values of the spline parameters. For further discussion on this issue, we refer the
reader to [12].
2. Power functions: hα (t) = tα β(t), α ∈ R. Corre-
170
spondingly, the parameters of x(t) are given by
tℓ
aℓ
1
yα (r(tℓ )) α−γ
= r(tℓ ) −
and
yγ (r(tℓ ))
−α
yα (r(tℓ )) α−γ
= yα (r(tℓ ))
.
yγ (r(tℓ ))
For α ∈ Z+ , the power function becomes a monomial of degree α. Since B-splines of order α can reproduce polynomials (and naturally, monomials too)
up to degree α, they are included as special elements
of this class. Therefore, power functions, which play
a vital role in moment-based sampling approaches
[6, 11] for the FRI problem, are also useful in the
generalized sampling approach. Also, note that fractional powers are admissible in the kernel definition.
2
3. Gaussian functions: hα (t) = e−α t β(t), where α ∈
R. Correspondingly, we have that
s
1
yγ (r(tℓ ))
log
, and
tℓ = r(tℓ ) −
α−γ
yα (r(tℓ ))
α
yγ (r(tℓ ))
log
.
aℓ = exp
α−γ
yα (r(tℓ ))
4. Complex E-splines: hα (t) = e−jα t β(t), α ∈ R.
This kernel cannot be treated as a special case of the
E-spline kernels with an imaginary parameter. The
reason is that there is an issue related to parameter
identifiability that deserves special attention. The potential problem is that this kernel may give rise to
more than one solution for tℓ ; there is, however, no
ambiguity in the solution for aℓ . We further explain
this issue and also state a condition that helps overcome the non-uniqueness hurdle.
The cause of ambiguity is essentially the quasiperiodicity of the complex exponential over the support [0, T ]:
e−jα (r(tℓ )−tℓ ) = e−jα (r(tℓ )−tℓ +
2mπ
α
),
for m ∈ Z such that 0 ≤ (r(tℓ )−tℓ + 2mπ
α ) ≤ T . The
restriction on m is due to the fact that we are considering a truncated complex exponential. The inequality gives rise to multiple solutions for tℓ . The solution
to this problem lies in tying up the choices of the values of α and T such that m = 0 is the only possibility in the above inequality. This amounts to requiring
that the complex exponential have at maximum one
2π
> T . Unperiod within a sampling interval; i.e.,
α
der this condition, we have the reconstruction formulae:
yα (r(tℓ ))ejα r(tℓ )
, and
tℓ = −j log
yγ (r(tℓ ))ejγ r(tℓ )
aℓ = yα (r(tℓ )) exp (jα(r(tℓ ) − tℓ )) .
Similarly, a truncated Fresnel kernel can be employed
by considering purely imaginary parameters in the
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definition of the Gaussian above. For complex parameters, the E-spline and Fresnel kernels have an
exponential and Gaussian decay, respectively.
5. Hybrid sampling kernels: In the kernels considered
above, we have enforced the same functional form for
both hα (t) and hγ (t). By relaxing this property, we
can make the reconstruction technique more efficient.
For example, if we set one of the parameters (but not
both), say α to zero, the kernel reduces to a causal
B-spline of order 0: hα (t) = β(t). The second kernel can be taken from any of the choices listed above.
The samples from the zeroth-order B-spline channel
then directly yield aℓ = yα (r(tℓ )). Using the samples from the second channel, we can compute the
positions of the Dirac impulses. For example, if we
employ the truncated power function in the second
1
yγ (r(tℓ )) γ
.
channel, we have that tℓ = r(tℓ ) −
aℓ
Note that r(tℓ ) and yγ (r(tℓ )) are known.
Having listed a few kernel choices, we reiterate that, in
the present formalism, the condition stated in (2) is crucial for the super-resolution localization of impulses. If
two successive Dirac impulses are spaced closer apart than
the sampling period, then they give rise to overlapping
Kronecker impulses and resolving them is not possible
within the proposed formulation. The existing approaches
[5, 6, 10, 11] do not suffer from this limitation.
4.
Kernels for single-channel sampling
The principal advantage offered by the two-channel
method equipped with the choice of a proper kernel is
the decoupling between the amplitudes and positions of
the impulses. As shown next, this advantage can be carried over to the single-channel case by suitably integrating the previously listed kernels into a single function.
For example, consider the kernel: hα,γ (t) = e−α t β(t) +
e−γ (t−T ) β(t − T ), which has the same properties as the
hybrid kernel in the two-channel case (kernel (1) in Section 3.). This choice would give rise to two nonzero samples per Dirac impulse, which can be used to solve for aℓ
and tℓ . Again, if α = 0, the first sample would straightaway give the amplitude, which can then be used together
with the second sample to compute the position. Thus, we
have a similar algorithm as in the two-channel case, the
only difference being that, in the two-channel case, these
samples are acquired one per channel whereas in the onechannel case, they are acquired in the same channel—the
overall sampling rate, however, is the same in both the
cases. In general, the kernels for the single-channel case
can be defined as: hα,γ (t) = hα (t) + hγ (t − T ). Since
the support of the kernel hα,γ (t) is double that of hα (t)
or
hγ (t), impulses that are farther apart by at least 2T i.e.,
min {tℓ − tℓ−1 } ≥ 2T only can be resolved. The ker-
2≤ℓ≤L
nels defined in this paper are shown in Fig. 2.
171
1
1
0
0
1
Time
0.5
0
2
Complex exponential spline
Amplitude
1
0.5
0
1
Time
5
Truncated Gaussian
1.5
0.5
0
2
Hybrid kernel (B!spline & E!spline)
1.5
Amplitude
Truncated power function
1.5
Amplitude
Amplitude
Exponential spline
1.5
0
1
Time
0
2
Hybrid kernel (Two E!splines)
!1
1
Amplitude
Amplitude
Imaginary part
!5
0
0.5
(a)
1.5
1
0.2
0.4
0.6
0.8
Time (seconds)
1
0.2
0.4
0.6
0.8
Time (seconds)
1
1
0.5
5
1
Real part
!1
0
1
2
0
Time
0
1
Time
2
0
0
1
Time
2
Figure 2: Sampling kernels. The parameters α = 2, γ =
1, and T = 1, are chosen for the sake of illustration.
Amplitude
0
0
!5
5.
Simulations
We next validate the theoretical findings by numerical experiments. We simulate the two-channel sampling of nine
Dirac impulses shown in Fig. 3(a); the amplitudes and positions are chosen for the purpose of illustration. The minimum spacing between two impulses is 0.0076 seconds.
The sampling period T is chosen to be 0.0038 seconds to
ensure that (2) is satisfied. The impulses are sampled using
the power function kernels with parameters α = 3, γ = 2,
and T = 0.0038 seconds. These values are chosen for
the purpose of illustration. The reconstructed stream of
Dirac impulses is shown in Fig. 3(b). The reconstruction
is accurate to numerical precision. Identical results were
obtained with the other kernel choices.
6.
Conclusions
We have extended the results developed in [12] and
proposed new kernels for both single-channel and twochannel sampling scenarios. The kernels are built using
functions known in system theory such as the exponential, power function, Gaussian, etc. The main advantage
of the proposed formulation is that, under the condition
of minimum separation between consecutive impulses, a
fast local reconstruction algorithm can be developed. This
advantage, however, comes with the shortcoming that impulses spaced farther apart than the sampling period only
can be resolved. It would be a challenging task to develop local reconstruction algorithms without imposing
constraints on the minimum/average separation between
impulses or groups thereof.
Acknowledgments
This work was supported by the Swiss National Science
Foundation (SNSF) Grant 200020-101821.
References:
[1] C. E. Shannon, “Communication in the presence of
noise,” Proc. IRE, vol. 37, no. 1, pp. 10-21, Jan. 1949.
SAMPTA'09
(b)
Figure 3: (a) Ground truth, (b) Reconstructed signal.
[2] M. Unser, “Sampling—50 years after Shannon,”
Proc. IEEE, vol. 88, no. 4, pp. 569-587, Apr. 2000.
[3] A. Papoulis, “Generalized sampling expansion,”
IEEE Trans. Circuits Syst., vol. 24, no. 11, pp. 652–
654, 1977.
[4] M. Unser and J. Zerubia, “A generalized sampling theory without band-limiting constraints,” IEEE
Trans. Circuits Syst. II, Analog and Digit. Signal Process., vol. 45, no. 8, pp. 959–969, Aug. 1998.
[5] M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. Signal
Process., vol. 50, no. 6, pp. 1417–1428, Jun. 2002.
[6] P.L. Dragotti, M. Vetterli, and T. Blu, “Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets Strang-Fix,” IEEE Trans.
Signal Process., vol. 55, no. 5, pp. 1741–1757, May
2007, Part 1.
[7] P. Stoica and R. Moses, Introduction to Spectral Analysis, Englewood Cliffs, NJ: Prentice-Hall, 2000.
[8] M. Unser, “Splines: A perfect fit for signal and image
processing,” IEEE Signal Process. Mag., vol. 16, no.
6, pp. 22–38, Nov. 1999.
[9] M. Unser and T. Blu, “Cardinal exponential splines:
Part I—Theory and filtering algorithms,” IEEE Trans.
Signal Process., vol. 53, no. 4, pp. 1425–1438, Apr.
2005.
[10] J. Kusuma and V. K. Goyal, “Multichannel sampling
of parametric signals with a successive approximation
property,” in Proc. IEEE Intl. Conf. on Imag. Proc.,
2006, pp. 1265–1268.
[11] L. Baboulaz and P. L. Dragotti, “Distributed acquisition and image super-resolution based on continuous
moments from samples,” in Proc. IEEE Intl. Conf. on
Imag. Proc., 2006, pp. 3309–3312.
[12] C. S. Seelamantula and M. Unser, “A generalized
sampling method for finite-rate-of-innovation-signal
reconstruction,” IEEE Signal Process. Lett., vol. 15,
pp. 813-816, 2008.
172
MULTICHANNEL SAMPLING OF TRANSLATED, ROTATED AND SCALED BILEVEL
POLYGONS USING EXPONENTIAL SPLINES
Hojjat Akhondi Asl and Pier Luigi Dragotti
Imperial College London
Department of Electrical and Electronic Engineering
hojjat.akhondi-asl@imperial.ac.uk, p.dragotti@imperial.ac.uk
ABSTRACT
Recently there has been an interest in single and multichannel sampling of certain parametric signals based on rate of
innovation using exponential reproducing kernels. In [5] it
was shown that, using exponential reproducing kernels, we
can achieve a fully symmetric multichannel sampling system
where different channels receive translated versions of the input signal. For the case of bilevel polygons as the input signal
considered in [5], having only translations is not practical and
one may want to look at the cases of more complicated geometric transformations, such as rotation and scaling. In this
paper we present a sampling theorem for multichannel sampling of translated, rotated and scaled bilevel polygons using
Radon projections and generalized exponential splines.
1. INTRODUCTION
Recently, it was shown [1, 2] that it is possible to sample and
perfectly reconstruct some classes of non-bandlimited signals
using suitable sampling kernels. Signals that can be reconstructed using this framework are called signals with Finite
Rate of Innovation (FRI) as they can be completely defined
by a finite number of parameters. Stream of weighted Dirac
impulses and bilevel polygons are some examples of FRI signals.
There has been a recent interest in sampling FRI signals
using exponential spline [3] (E-spline) kernels. Dragotti et
al. [2] showed that E-splines can be used as the sampling
kernel to sample and perfectly reconstruct 1-D FRI signals.
Extensions to the multidimensional case were considered in
[5, 14] where we proposed sampling theorems for a stream of
2-D Dirac impulses (based on the ACMP algorithm [11]) and
bilevel polygons (based on Radon projections [10]). Apart
from the sampling kernels used in [5, 14], the reconstruction
algorithms are also different from the ones used in the conventional multidimensional sampling theories [12, 13].
An advantage of E-spline sampling kernels over polynomial reproducing kernels such as B-splines is that, they can be
employed in a fully symmetric multichannel sampling environment. By symmetric sampling, we mean that the sampling
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process can be evenly distributed between different acquisition devices. The inspiration and development of multichannel sampling of FRI signals is very recent and it has been
looked at in [5, 6, 7, 8].
In [6] Seelamantula and Unser, by using simple RC filters, propose a simple acquisition and reconstruction method
within the framework of multichannel sampling, where 1-D
FRI signals such as an infinite stream of nonuniformly-spaced
Dirac impulses and piecewise-constant signals can be sampled and perfectly reconstructed. In [7] Kusuma and Goyal
proposed new ways of sampling 1-D Dirac impulses using a
bank of integrators or B-splines. Their proposed scheme is
closely related to previously known cases [1, 2] but provides
a successive approximation property, which could be useful
for detecting undermodelling when the number of Dirac impulses are unknown. In [8] Baboulaz and Dragotti use a multichannel sampling setup for sampling FRI signals and utilize
that for image registration based on continuous moments and
image super-resolution.
In [5] we illustrate that symmetric multichannel sampling
of bilevel polygons can be achieved with the geometric transformations being a 2-D translation between the different signals. In practice, this is usually not the case, and in this paper
we want to look at the cases of more complicated geometric transformations, such as rotation and scaling. The paper
is organised as follows: In Section II we will briefly discuss
the sampling setup needed for sampling 2-D FRI signals (single channel) and based on that we describe our multichannel
sampling setup. In Section III we present our algorithm for
sampling and perfectly reconstructing translated, rotated and
scaled bilevel polygons with the use of generalized E-splines
and Radon projections. In Section IV we provide simulation
results to support our proposed theory.
2. MULTICHANNEL SAMPLING SETUP
Before describing the multichannel sampling framework, let
us first, for the sake of clarity, show how a general 2-D sampling setup (single channel) for FRI signals is represented.
Figure 1 shows a general 2-D sampling setup for FRI signals
173
where g(x, y) represents the input signal, ϕ(x, y) the sampling kernel, sj,k the samples and T x, T y are the sampling
intervals. From the setup shown in Figure 1, the samples sj,k
Fig. 1. 2-D sampling setup
are given by:
Z ∞ Z
sj,k =
−∞
∞
g(x, y) ϕ(
−∞
x
y
− j,
− k) dx dy (1)
Tx
Ty
where the kernel ϕ(x, y) is the time reversed version of the
filter response h(x, y). ϕ(x, y) can easily be produced by
the tensor product between ϕ(x) and ϕ(y), that is ϕ(x, y) =
ϕ(x) ⊗ ϕ(y). As mentioned before, ϕ(x, y) is chosen to be
an exponential reproducing kernel. The theory of exponential
reproducing kernels is quite recent and is based on the notion
of exponential splines (E-splines) [3]. A function βα~ (x) with
Fourier transform
β̂α~ (ω) =
N
Y
1 − eαn −jω
jω − αn
n=0
is called E-spline of order N where α
~ = (α0 , α1 , . . . , αN )
can be real or complex. The produced spline has a compact support and can reproduce any exponential in the subspace spanned by (eα0 t , eα1 t , . . . , eαN t ) which is obtained by
successive convolutions of lower order E-splines ((N+1)-fold
convolution). Exponential spline kernels can therefore reproduce, with their shifted versions, real or complex exponentials. That is, in 2-D form, any kernel satisfying:
X X m,n
cj,k ϕ(x − j, y − k) = eαm x eβn y
(2)
j∈Z k∈Z
is an E-spline for a proper choice of the coefficients cm,n
j,k .
Here, m = 0, 1, . . . , M , n = 0, 1, . . . , N , αm = α0 + mλ1
and βn = β0 + nλ2 . The values of (α0 , β0 ) and (λ1 , λ2 ) can
be chosen arbitrarily, but too small or too large values could
lead to unstable results for the reproduction of exponentials.
E-splines are biorthogonal functions and the coefficients cm,n
j,k
can be found using the dual of βα~ (x). An important property
of E-splines is that they are a generalized version of B-splines.
This is because, if the α
~ parameters are set to zero, then the
produced spline would result in a B-spline, a polynomial reproducing spline. This property will be used to estimate the
transformation parameters in Section III. The reader can refer
to [5, 14] for sampling theories on single-channel sampling
and perfect reconstruction of 2-D Dirac impulses and bilevel
polygons using exponential splines.
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We can now describe our multichannel sampling setup.
A multichannel sampling system can be thought of multiple acquisition devices observing an input signal. In order
to perfectly reconstruct the input signal using only one acquisition device, we normally require expensive acquisition
devices with high sampling rates. By using a bank of acquisition devices (filters) and synchronizing the different channels exactly, we are able to reduce the number of samples
needed from each device, resulting in a cheaper and more
efficient sampling system. To model our multichannel system, consider a bank of E-spline filters to acquire FRI signals where each filter has access to a geometrically transformed version of the input signal. Figure 2 shows the described multichannel sampling scenario where the bank of filters ϕ1 (x, y), ϕ2 (x, y), . . . , ϕN −1 (x, y) receive different versions of the input signal g0 (x, y). Here, the geometric transformations (e.g. translation, rotation and scaling ) are denoted
by T1 , T2 , . . . , TN −1 .
Fig. 2. Multichannel sampling setup
In [4] Baboulaz considered the use of E-splines for sampling a stream of 1-D Dirac impulses in a multichannel sampling setup described in Figure 2. He showed that if two 1-D
signals are just shifted version of the other, then by setting
one parameter to be common between the exponents of the
E-spline sampling kernels for the two signals, one can not
only estimate the shifts between the two signals, but also can
linearly relate the exponential moments of the two signals
(the reader can refer to [4, 5, 14] for more detailed discussion). Because of the direct relationship between the exponential moments of the two signals, we can achieve perfect
reconstruction of the reference signal with fewer exponential moments required. Since less moments are required from
each signal, a lower order E-spline sampling kernel would
be needed, which in turn less samples from each signal are
required to achieve perfect reconstruction. This is because,
from [2] we know that a stream of Dirac impulses is uniquely
determined from the samples if there are at most K Dirac impulses in an interval size of 2KLT where L is the support of
the sampling kernel. Since the support of the sampling kernels is reduced in the multichannel case, we can achieve the
same performance with a smaller sampling rate T .
174
3. ALGORITHM
Unfortunately we can not estimate the more complicated geometric transformations like the way it was done for the simple
translation case in [5] with exponential reproducing kernels.
Also, even if we assume that the transformation parameters
are known and given, we still can not use the sampling algorithm shown in [5] for the multichannel framework. This is
because introducing more complicated transforms such as rotation and/or scaling for example, would result in a non-linear
relationship between the exponential moments of the different
signals.
The first question we need to answer is that, assuming
an oracle gives us the values of the transformation parameters, can we sample and perfectly reconstruct translated, rotated and scaled bilevel polygons in a symmetric multichannel framework? It is known that for an N-sided bilevel polygon, with N+1 projections, perfect reconstruction of the polygon can be achieved. That is points that have N+1 line intersections from the N+1 back-projections correspond to the
N vertices of the polygon [9]. We also know that a Radon
projection at an angle φ of a rotated image with respect to
its reference image with an angle θ, is the same projection,
but scaled and translated, on the reference image at the angle φ + θ. Therefore, if all the transformation parameters
are known, and assuming that the rotation angle is not zero
that is, θ 6= 0, then the N + 1 projections needed could be
separated between the different channels, in order to sample
and perfectly reconstruct the reference image in a symmetric
manner.
The next question would be, how can we estimate the
transformation parameters? We know that with the use of
polynomial reproducing kernels, we can obtain the geometric moments of a signal, and geometric moments up to order
2 from two signals are enough to estimate translation, rotation and scaling parameters between the two signals. We also
know that, as E-splines are a generalized version of B-splines
[3], we can reproduce a combination of polynomials and exponentials from E-splines. From the polynomials moments up
to order 2, we can estimate all the transformation parameters.
4. RESULTS
As an example, in [5] we showed that to achieve perfect reconstruction for a 4-sided bilevel polygon, a 2-D E-spline
order of 12 is required to produce 5 projections at the angles 0, 45, 90, tan−1 (2) and tan−1 ( 21 ). With 2-D E-spline
order of 7 however we can produce 3 projections at the angles
0, 45, 90 on the reference signal, and a 2-D E-spline order of
7 on the second signal would give 3 projections for the reference signal at the angles θ, 45 + θ, 90 + θ where θ is the
rotation parameter. Assuming θ is not zero, we would have
enough projections to perfectly reconstruct the reference signal. Therefore an spline order of 7+2 = 9 (2 is needed for es-
SAMPTA'09
timating the transformation parameters) on each signal would
give us enough projections to perfectly reconstruct the reference signal. An example for a 4-sided bilevel polygon with
two acquisition devices is shown in Figure 3 where the reference signal, its translated, rotated and scaled version, their
samples, the E-spline sampling kernel, and the reconstructed
reference signal are all shown.
5. CONCLUSION
In this paper we showed that with the use of Radon projections and generalized E-splines, symmetric multichannel
sampling of translated, rotated and scaled bilevel polygons
can be achieved. For estimating the geometrical transformations, we showed that as E-splines are a generalized version
of B-splines, we can reproduce combination of polynomials
and exponentials from E-splines. Therefore from the polynomial moments up to order 2, we can estimate all the unknown
transformation parameters. For symmetric multichannel sampling of geometrically transformed bilevel polygons, we illustrated that the N+1 Radon projections needed for perfect
reconstruction of an N-sided bilevel polygon, can be separated between the different channels, assuming that the rotation parameter is not zero. Our sampling and reconstruction algorithm is based on noise-free communication between
the transmitter and receiver which is rather not very practical.
The future research of this work is to test the stability and
performance of our method in the presence of noise.
6. REFERENCES
[1] M. Vetterli, P. Marziliano and T. Blu, “Sampling Signals
with Finite Rate of Innovation”, IEEE Transactions on
Signal Processing, vol. 50, pp. 1417-1428, June 2002.
[2] P.L. Dragotti, M. Vetterli and T. Blu, “Sampling Moments and Reconstructing Signals of Finite Rate of Innovation: Shannon meets Strang-Fix”, IEEE Transactions on Signal Processing, vol. 55, pp. 1741-1757, May
2007.
[3] M. Unser and T. Blu, “Cardinal Exponential Splines:
Part I - Theory and Filtering Algorithms”, IEEE Transactions on Signal Processing, vol. 53, pp. 1425, 2005.
[4] L. Baboulaz, “Feature Extraction for Image Superresolution using Finite Rate of Innovation Principles”,
PhD thesis, Department of Electrical and Electronic
Engineering, Imperial College London, 2008. URL:
http://www.commsp.ee.ic.ac.uk/~lbaboula/
[5] H. Akhondi Asl and P.L. Dragotti, “Single and Multichannel Sampling of Bilevel Polygons Using Exponential Splines”, To Appear on IEEE International Conference on Acoustics, Speech, and Sig-
175
nal Processing, Taipei, Taiwan, April 2009. URL:
http://cspserver2.ee.ic.ac.uk/~Hojakndi/
[6] C. S. Seelamantula and M. Unser, “A Generalized Sampling Method for Finite-Rate-of-Innovation-Signal Reconstruction”, IEEE Signal Processing Letters, vol.15,
pp. 813-816, August 2008.
(a)
(b)
[7] J. Kusuma and V. K. Goyal, "Multichannel Sampling
of Parametric Signals with a Successive Approximation Property," IEEE International Conference on Image
Processing, pp. 1265-1268, October 2006.
[8] L. Baboulaz and P. L. Dragotti, "Distributed Acquisition
and Image Super-Resolution Based on Continuous Moments from Samples," IEEE International Conference
on Image Processing, pp. 3309-3312, October 2006.
(c)
(d)
[9] I. Maravic and M. Vetterli, “Exact sampling results for
some classes of parametric non-bandlimited 2-D signals", IEEE Transactions on Signal Processing, vol.52,
no.1, pp. 175-189, January 2004ation Principles”, PhD
thesis, Imperial College London, 2008.
[10] G. T. Herman, “Image Reconstruction from Projections: The Fundamentals of Computerized Tomography”, Academic Press, New York, 1980.
(e)
[11] F. Vanpoucke, M. Moonen and Y. Berthoumieu, “An
Efficient Subspace Algorithm for 2-D Harmonic Retrieval", IEEE International Conference on Acoustics,
Speech, and Signal Processing, vol.4, pp. 461-464, April
1994.
[12] P. Shukla and P.L. Dragotti, “Sampling Schemes for
Multidimensional Signals with Finite Rate of Innovation”, IEEE Transactions on Signal Processing, vol. 55,
pp. 3670-3686, July 2007.
[13] I. Maravic and M. Vetterli, “Exact sampling results for
some classes of parametric non-bandlimited 2-D signals", IEEE Transactions on Signal Processing, vol.52,
no.1, pp. 175-189, January 2004.
(f)
Fig. 3. Symmetric multichannel sampling of translated, rotated and
scaled bilevel polygons using E-spline sampling kernels. (a) The reference signal in a frame data size of 256 × 256. (b) The translated
(△x = −100, △y = 150), rotated (θ = 35) and scaled (a = 1.1) ver-
[14] H. Akhondi Asl, “Single and Multichannel Sampling of Signals With Finite Rate of Innovation
Using E-Splines”, MPhil to PhD Transfer Report, Department of Electrical and Electronic Engineering, Imperial College London, 2008. URL:
http://cspserver2.ee.ic.ac.uk/~Hojakndi/
sion of the reference signal. (c) & (d) The 16 × 16 samples of both signals. (e) 2-D generalized E-spline of order 9 (f) The reconstructed vertices of the reference signal with 6 back-projections, the crosses are the
actual vertices of the polygon. [Not to scale]
SAMPTA'09
176
Special session on
Sampling
and
Quantization
Chair: Özgür Yilmaz
SAMPTA'09
177
SAMPTA'09
178
Quantization for Compressed Sensing
Reconstruction
John Z. Sun and Vivek K Goyal
Massachusetts Institute of Technology, Cambridge, MA 02139 USA
johnsun@mit.edu, vgoyal@mit.edu
Abstract:
Quantization is an important but often ignored consideration in discussions about compressed sensing. This paper studies the design of quantizers for random measurements of sparse signals that are optimal with respect to
mean-squared error of the lasso reconstruction. We utilize
recent results in high-resolution functional scalar quantization and homotopy continuation to approximate the optimal quantizer. Experimental results compare this quantizer to other practical designs and show a noticeable improvement in the operational distortion-rate performance.
parallel, we present only fixed rate. To concentrate on the
central ideas, we choose signal and sensing models that
obviate discussion of quantizer overload.
2. Background
In our notation, a random vector is always lowercase and
in bold. A subscript then indicates an element of the vector. Also, an unbolded vector y corresponds to a realization of the random vector y.
2.1 Distributed functional scalar quantization
1.
Introduction
In practical systems where information is stored or transmitted, data must be discretized using a quantization
scheme. The design of the optimal quantizer for a given
stochastic source has been well studied and is surveyed
in [6]. Here, optimal means the quantizer minimizes the
error as measured by some distortion metric. In this paper, we explore optimal quantization for an emerging nonadaptive compression paradigm called compressed sensing (CS) [1, 4]. Several authors have studied the asymptotic reconstruction performance of quantized random
measurements assuming a mean-squared error (MSE) distortion metric [3, 5]. Other previous work presented modifications to existing reconstruction algorithms to mitigate distortion resulting from standard quantizers [3, 7] or
modified quantization that can be viewed as the binning of
quantizer output indexes [10].
Our contribution is to reduce distortion due to quantization
through design of the quantizer itself. The key observation is simply that the random measurements are used as
arguments in a nonlinear reconstruction function. Thus,
minimizing the MSE of the measurements is not equivalent to minimizing the MSE of the reconstruction. We
use the theory for high-resolution distributed functional
scalar quantization (DFSQ) recently developed in [9] to
design optimal quantizers for random measurements. To
obtain concrete results, we choose a particular reconstruction function (lasso [11]) and distributions for the source
data and sensing matrix. However, the general principle
of obtaining improvements through the use of DFSQ theory holds more generally, and we address the conditions
that must be satisfied for sensing and reconstruction. Also,
rather than develop results for fixed and variable rate in
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In standard fixed-rate scalar quantization [6], one is asked
to design a quantizer Q that operates separably over its
components and minimizes MSE between a probabilistic
source vector y ∈ RM and its quantized representation
ŷ = Q(y). The resulting optimization is
min E ky − Q(y)k2 ,
Q
subject to the constraint that the maximum number of
codewords or quantization levels for each yi is less than
2Ri . We can use high-resolution theory to find the quantizer point density of the optimal quantizer.
In DFSQ [9], the goal is to create a quantizer that minimizes distortion for some scalar function g(y) of the
source vector y rather than the vector itself. Hence, the
optimization is now
min E |g(y) − g(Q(y))|2
Q
such that the maximum number of codewords or quantization levels representing each yi is less than 2Ri . To apply
the following model, we need g(·) and fy (·) to satisfy certain conditions:
C1. g(y) is smooth and monotonic for each yi .
C2. The partial derivative gi (y) = ∂g(y)/∂yi is defined
and bounded for each i.
C3. The joint pdf of the source variables fy (y) is smooth
and supported in a compact subset of RM .
For valid g(·) and fy (·) pairs, we define a set of functions
h
i1/2
γi (t) = E |gi (y)|2 | yi = t
.
(1)
We call γi (t) the sensitivity of g(y) with respect to the
source variable yi . The optimal point density is then
1/3
λi (t) = C γi2 (t)fyi (t)
,
(2)
179
for some normalization constant C, which leads to a total
operational distortion-rate
2
X
γi (yi )
2−2Ri E
.
(3)
D({Ri }) =
12λ2i (yi )
i
The sensitivity γi (t) serves to reshape the quantizer, giving better resolution to regions of yi that have more impact
on g(y), thereby reducing MSE.
Similar results for variable-rate quantizers are also presented in [9]. However, we will only consider the fixedrate case in this paper. The theory of DFSQ can be extended to a vector of functions, where xj = g (j) (y) for
1 ≤ j ≤ N . Since the cost function is additive in its components, we can show that the overall sensitivity for each
component yi is
N
1 X (j)
γi (t) =
γ (t),
N j=1 i
(4)
(j)
where γi (t) is the sensitivity of the function g (j) (y) with
respect to yi .
2.2
Compressed Sensing
CS refers to estimation of a signal at a resolution higher
than the number of data samples, taking advantage of sparsity or compressibility of the signal and randomization in
the measurement process [1, 4]. We will consider the following formulation. The input signal x ∈ RN is K-sparse
in some orthonormal basis Ψ, meaning the transformed
signal u = Ψ−1 x ∈ RN contains only K nonzero elements. Consider a length-M measurement vector y = Φx,
where Φ ∈ RM×N with K < M < N is a realization of Φ. The major innovation in CS (for the case of
sparse u considered here) is that recovery of x from y
via some computationally-tractable reconstruction method
can be guaranteed asymptotically almost surely.
Many reconstruction methods have been proposed including a linear program called basis pursuit [2] and greedy
algorithms like orthogonal matching pursuit (OMP) [12].
In this paper, we focus on a convex optimization called
lasso [11], which takes the form
x̂ = arg min ky − Φxk22 + µkΨ−1 xk1 .
(5)
x
As one sample result, lasso leads to perfect sparsity pattern
recovery with high probability if M ∼ 2K log(N − K) +
K under certain conditions on Φ, µ, and the scaling of the
smallest entry of u [13]. Unlike in [5], our concern in this
paper is not how the scaling of M affects performance,
but rather how the accuracy of the lasso computation (5)
is affected by quantization of y.
A method for understanding the set of solutions to (5) is
the homotopy continuation (HC) method [8]. HC considers the regularization parameter µ at an extreme point
(e.g., very large µ so the reconstruction is all zero) and
slowly varies µ so that all sparsities and the resulting reconstructions are obtained. It is shown that there are N
values of µ where the lasso solution changes sparsity, or
equivalently N + 1 intervals over which the sparsity does
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Figure 1: A compressed sensing model with quantization
of measurement vector y. The vector ynl denotes the
noiseless random measurements.
not change. For µ in the interior of one of these intervals,
the reconstruction is determined uniquely by the solution
of a linear system of equations involving a submatrix of Φ.
In particular, for a specific choice µ∗ and observed random
measurements y,
2ΦTJµ∗ ΦJµ∗ x̂ + µ∗ v = 2ΦTJµ∗ y,
(6)
where v = sgn(x̂) and ΦJµ∗ is the submatrix of Φ with
columns corresponding to the nonzero elements Jµ∗ ⊂
{1, 2, . . . , N } of x̂.
3. Problem Model
Figure 1 presents a CS model with quantization. Assume
without loss of generality that Ψ = IN and hence the
(random) signal x = u is K-sparse. Also assume a random matrix Φ is used to take measurements, and additive
Gaussian noise perturbs the resulting signal, meaning the
continuous-valued measurement vector is y = Φx + η.
The sampler wants to transmit the measurements with total rate R and encodes y into a transmittable bitstream by
using encoder Q. Next, a decoder Q̂ produces a quantized
signal ŷ from by . Finally, a reconstruction algorithm G
outputs an estimate x̂. The function G is a black box that
may represent lasso, OMP or another CS reconstruction
algorithm.
We now present a probabilistic model for the input source
and sensing matrix. It is chosen to guarantee finite support
on both the input and measurement vectors, and prevent
overload errors for quantizers with small R. However, we
emphasize that the following theory is general, and other
choices for x and Φ are possible for large enough R.
Assume the K-sparse vector x has random sparsity J chosen uniformly from all possibilities, and each nonzero
component xi is distributed iid U(−1, 1). Also assume
the additive noise vector η is distributed iid Gaussian with
zero mean and variance σ 2 . Finally, let Φ correspond to
random projections such that each column φj ∈ RM has
unit energy (kφj k2 = 1). The columns of Φ thus form
a set of N random vectors chosen uniformly on the unit
(M − 1)-hypersphere. Since y = Φx,
yi =
N
X
j=1
Φij xj =
X
j∈J
Φij xj .
| {z }
zij
The distribution of each zij is found using derived distributions. The resulting pdfs can be shown to be iid fz (z),
where z is a scalar random variable that is identical in distribution to each zij . The distribution of yi is then the
K − 1 convolution cascade of fz (z) with itself. Thus,
fy (y) is smooth and supported for {|yi | ≤ K}, satisfying
180
compute γcs (·). To simplify our notation, let A = ΦJµ∗ .
The resulting differentials can be expressed as
3
2.5
∂G(j) (y, Φ) h T −1 T i
= A A
A
.
∂yi
ji
2
We now present the sensitivity through the following theorem:
1.5
1
0.5
0
−5
0
t
5
Figure 2: Distribution fyi (t) for (K, M, N ) =
(5, 71, 100). The support of yi is the range [−K, K],
where K is the sparsity of the input signal. However, the
probability is only non-negligible for small yi .
condition C3 for DFSQ. Figure 2 illustrates the distribution of yi for a particular case.
The reconstruction algorithm G is a function of the measurement vector y and sampling matrix Φ. We will show
that if G(y, Φ) is lasso with a proper relaxation variable
µ, then conditions C1 and C2 are met. Using HC, we see
G(y, Φ) is a piecewise smooth function that is also piecewise monotonic with every yi for a fixed µ. Moreover,
for every µ the reconstruction is an affine function of the
measurements through (6), so the partial derivative with
respect to any element yi is piecewise defined and smooth
(constant in this case). Conditions C1 and C2 are therefore
satisfied.
4.
Optimal Quantizer Design
We now pose the optimal fixed-rate quantizer design as
a DFSQ problem. For a given noise variance σ 2 , choose
an appropriate µ∗ to form the best reconstruction x̂ from
the unquantized random measurements y. We produce M
quantizers to transmit the elements of y such that the decoded message ŷ will minimize the distortion between
x̃ = G(y, Φ) and x̂ = G(ŷ, Φ) for a total rate R.
Note G can be visualized as a set of N scalar functions
x̂j = G(j) (ŷ, Φ) that are identical in distribution due to
symmetry in the randomness of Φ. Since the sparse input
signal is assumed to have uniformly distributed sparsity
and Φ distributes energy uniformly to all measurements
yi in expectation, we argue by symmetry that each measurement is allotted the same number of bits and that every measurement’s quantizer is the same. Moroever, since
the functions representing the reconstruction are identical,
we argue using (4) that the overall sensitivity γcs (·) is the
same as the sensitivity of any G(j) (ŷ, Φ). Computing (2)
yields the point density λcs (·).
This is when the homotopy continuation method becomes
extremely useful. For a given realization of Φ and η, we
can use HC to determine how many elements in the reconstruction are nonzero for µ∗ , denoted Jµ∗ . Equation (6)
is then used to find ∂G(j) (y, Φ)/∂yi , which is needed to
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(7)
Theorem 1 Let the noise variance be σ 2 and choose an
appropriate µ∗ . Define y\i to be all the elements of a
vector y except yi . The sensitivity of each element yi ,
(j)
which is denoted γi (t), can be written as
21
fyi |Φ (t|Φ) h T −1 T i
EΦ,y\i
A A
A
| yi = t
,
fyi (t)
ji
where A is the submatrix of Φ as described in HC for
µ∗ and some observation y. Moreover, for any Φ and its
corresponding J, fyi |Φ (t|Φ) is the convolution cascade of
{zj ∼ U(−Φij , Φij )} for j ∈ J. By symmetry arguments,
(j)
γcs (t) = γi (t) for any i and j.
This expectation is difficult to calculate but can be approached through L Monte Carlo trials on Φ, η, and x.
For each trial, we can compute the partial derivative using (7). We denote the Monte Carlo approximation to that
(L)
function to be γcs (·). Its form is
L
(L)
(t) =
γcs
1X
L
ℓ=1
fyi |Φ (t|Φℓ ) h T −1 T i2
Aℓ Aℓ
Aℓ
fyi (t)
ji
12
,
(8)
with i and j arbitrarily chosen. By the weak law of large
numbers, the empirical mean of L realizations of the random parameters should approach the true expectation for
L large.
We now substitute (8) into (2) to find the Monte Carlo approximation to the optimal quantizer for compressed sensing. It becomes
1/3
(L)
(t)f
(t)
,
λ(L)
(t)
=
C
γ
y
cs
cs
i
(9)
for some normalization constant C. Again by the weak
p
(L)
law of large numbers, λcs (t) −
→ λcs (t) for L large.
5. Experimental Results
We compare the CS-optimized quantizer, called the “sensitive” quantizer, to a uniform quantizer and “ordinary”
quantizer λord (t) which is optimized for the distribution
of y. This means the ordinary quantizer would be best
if we want to minimize distortion between y and ŷ, and
hence has a flat sensitivity curve over the support of y.
The sensitive quantizer λcs (t) is found using (9) and the
uniform quantizer λuni (t) = c, where c is a normalization
constant.
Using 1000 Monte Carlo trials, we estimate γcs (t). The
resulting point density functions for the three quantizers
are illustrated in Figure 3.
Experimental results are performed on a Matlab testbench.
Practical quantizers are designed by extracting codewords
181
1.5
Sensitive
Ordinary
Uniform
λ(t)
1
0.5
0
−5
0
t
5
Figure 3: Estimated point density functions λcs (t),
λord (t), and λuni (t) for (K, M, N ) = (5, 71, 100).
10
Sensitive
Ordinary
Uniform
i
log D(R )
5
0
−5
−10
2
3
4
5
component rate Ri
6
Figure 4: Results for distortion-rate for the three quantizers with µ = 0.01 and σ 2 = 0.3. We see that the sensitive
quantizer has the least distortion.
from the cdf of the normalized point densities. In the
approximation, the ith codeword is the point t such that
Z
t
λcs (t′ )dt′ =
−∞
i − 1/2
,
2Ri
where Ri is the rate for each measurement. The partition
points are then chosen to be the midpoints between codewords.
We compare the sensitive quantizer to uniform and ordinary quantizers using the parameters µ = 0.1 and
σ 2 = 0.3. Results are shown in Figure 4.
We find the sensitive quantizer performs best in experimental trials for this combination of µ and σ 2 at sufficiently high rates. This makes sense because λcs (t) is a
high-resolution approximation and should not necessarily
perform well at very low rates.
6.
Conclusion
We present a high-resolution approximation to an optimal
quantizer for the storage or transmission of random measurements in a compressed sensing system with lasso re-
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construction. Using DFSQ and HC, we find a sensitivity
function γcs (·) that determines the optimal point density
function λcs (·) of such a quantizer. Experimental results
show that the operational distortion-rate is best when using this so called “sensitive” quantizer.
We conclude that proper quantization in compressed
sensing is not simply a function of the distribution of
the random measurements themselves (using either a
high-resolution approximation or practical algorithms like
Lloyd-Max). Rather, quantization adds a non-constant effect, called functional sensitivity [9], on the distortion between the the lasso reconstructions of the random measurements and its quantized version.
A significant amount of work can still be done in this
area. Parallel developments could be made for variablerate quantizers. Also, this theory can be extended to other
probabilistic signal and sensing models, and CS reconstruction methods.
References:
[1] E. J. Candès, J. Romberg, and T. Tao. Robust uncertainty principles: Exact signal reconstruction from
highly incomplete frequency information. IEEE
Trans. Inform. Theory, 52(2):489–509, 2006.
[2] S. Chen, D. L. Donoho, and M. A. Saunders. Atomic
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[3] W. Dai, H. Vinh Pham, and O. Milenkovic. Quantized compressive sensing.
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[cs.IT]., 2009.
[4] D. L. Donoho. Compressed sensing. IEEE Trans.
Inform. Theory, 52(4):1289–1306, 2006.
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Process. Mag., 25(2):48–56, 2008.
[6] R. M. Gray and D. L. Neuhoff. Quantization. IEEE
Trans. Inform. Theory, 44(6):2325–2383, 1998.
[7] L. Jacques, D. K. Hammond, and M. J. Fadili. Dequantized compressed sensing with non-Gaussian
constraints. arXiv:0902.2367v2 [math.OC]., 2009.
[8] D. M. Malioutov, M. Cetin, and A. S. Willsky. Homotopy continuation for sparse signal representation. In Proc. IEEE ICASSP, pp. 733–736, 2006.
[9] V. Misra, V. K. Goyal, and L. R. Varshney.
Distributed functional scalar quantization: High-resolution analysis and extensions.
arXiv:0811.3617v1 [cs.IT]., 2008.
[10] R. J. Pai. Nonadaptive lossy encoding of sparse signals. Master’s thesis, Massachusetts Inst. of Tech.,
Cambridge, MA, 2006.
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via the lasso. J. Royal Stat. Soc., Ser. B, 58(1):267–
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[12] J. A. Tropp. Greed is good: Algorithmic results for
sparse reconstruction. IEEE Trans. Inform. Theory,
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[13] M. J. Wainwright. Sharp thresholds for highdimensional and noisy recovery of sparsity. Department of Statistics, UC Berkley, Tech. Rep 709, 2006.
182
Finite Range Scalar Quantization for
Compressive Sensing
Jason N. Laska (1) , Petros Boufounos (2) , and Richard G. Baraniuk(1)
(1) Rice University, 6100 Main St., Houston, TX 77005
(2) Mitsubishi Electric Research Laboratories, 201 Broadway Cambridge, MA 02139
laska@rice.edu, petrosb@merl.com, richb@rice.edu
Abstract:
Analog-to-digital conversion comprises of two fundamental discretization steps: sampling and quantization. Recent results in compressive sensing (CS) have overhauled
the conventional wisdom related to the sampling step, by
demonstrating that sparse or compressible signals can be
sampled at rates much closer to their sparsity rate, rather
than their bandwidth. This work further overhauls the
conventional wisdom related to the quantization step by
demonstrating that quantizer overflow can be treated differently in CS and by exploiting the tradeoff between
quantization error and overflow.
We demonstrate that contrary to classical approaches that
avoid quantizer overflow, a better finite-range scalar quantization strategy for CS is to amplify the signal such that
the finite range quantizer overflows at a pre-determined
rate, and subsequently reject the overflowed measurements from the reconstruction. Our results further suggest a simple and effective automatic gain control strategy
which uses feedback from the saturation rate to control the
signal gain.
1.
Introduction
Analog-to-digital converters (ADCs) are an essential part
of most modern sensing and communications systems.
They are the interface between the analog physical world
and the digital processing world that extracts the information we are interested in. Ever-increasing demands for information has pushed the requirements on ADCs to their
current physical limits. Fortunately, recent theoretical developments in the area of compressive sensing (CS) enable
us to significantly extend the capabilities of current ADCs
to keep pace with demand.
CS is a framework that allows signals that have sparse representation, i.e., few non-zero elements, or few non-zero
coefficients in some basis, to be sampled at a rate close to
the sparsity rate, rather than the Nyquist rate. CS employs
linear measurement systems and a non-linear reconstruction algorithms to acquire and recover sparse signals.
Most of the CS literature to-date focuses on one particular
aspect of ADCs, namely sampling. In this paper we reexamine the other significant aspect, quantization. Specifically, we show that the core tenets of CS enable us to
reduce the error due to quantization by allowing the quantizer to saturate more often than usual and removing the
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saturated measurements from the reconstruction process.
The organization of this paper is as follows. Section 2.
presents a brief background on analog-to-digital conversion, compressive sampling, and finite-range quantization.
Section 3. presents a brief analysis of finite-range quantization for CS. We show that CS measurements and the
quantization error are i.i.d. Gaussian, and analyze the proposed reconstruction strategy. Section 4., presents numerical results that validate our analysis. We conclude with a
brief discussion in Sec. 5.
2.
2.1
Background
Analog-to-digital conversion
Analog-to-digital conversion consists of two discretization
steps: sampling, which converts an analog signal to a set
of discrete measurements, and quantization, which converts each real-valued measurement to a discrete one chosen from a pre-determined set. Although both steps are
necessary to represent a signal in the discrete digital world,
classical results due to Shannon and Nyquist demonstrate
that the sampling step is information preserving if a sufficient number of samples, i.e., measurements, are obtained.
On the other hand quantization always degrades the signal.
The system design to goal is to take enough measurements
such that the signal does not alias, and to acquire enough
bits to limit the quantization distortion.
2.2
Finite-range quantization
Scalar quantization is the process of converting the continuous value of the measurements to one of several discrete
values through a non-invertible function R(·). In this paper we focus on uniform quantizers with quantization interval ∆. Thus, the quantization points are qk = q0 + k∆,
and every scalar a is quantized to the nearest quantization point R(a) = argminqk |a − qk |. For an infiniterange quantizer this implies that the quantization error is
bounded by |a − R(q)| ≤ ∆/2.
In practice quantizers have finite range, dictated by hardware constraints such as the voltage limits of the devices and the finite bit-rate of the quantized representation. Without loss of generality we assume a midrise Bbit quantizer that represents a symmetric range of values |a| < T , where T > 0 is the quantization threshold. The corresponding quantization points are at qk =
183
∆/2 + k∆, k = −2B−1 , . . . , 2B−1 − 1. This assumption implies a quantization interval ∆ = 2−B+1 T . Any
measurement with magnitude greater than T saturates the
quantizer and “clips” to magnitude T , i.e., it quantizes to
the quantization point T − ∆/2.
Most classical quantization error analysis assumes that
the measurements are scaled such that the quantizer never
clips. This is a sensible quantization strategy for classical approaches using linear reconstruction. In that context,
saturation events cause significant signal distortion and are
undesirable. For that reason, extreme attention is often devoted to pre-ADC automatic gain control (AGC) systems
to ensure that the quantizer saturates only rarely. Under
this assumption the analysis of a finite or an infinite range
quantizer is equivalent in terms of the quantization error.
Thus, an infinite-range quantizer is often assumed for its
mathematical simplicity.
2.3
Compressive sampling (CS)
The theory of compressive sampling (CS) overhauls the
conventional wisdom on the sampling process. Specifically, [2] and the references therein show that the number
of measurements that are sufficient to exactly reconstruct a
sampled signal are significantly fewer than the ShannonNyquist rate as long as the signal is sparse, i.e., can be
represented with very few non-zero components in some
basis.
The key components of CS are randomized measurements
and non-linear reconstruction. Specifically, a Nyquistrate sampled discrete-time signal x can be sampled at a
lower rate by using a random matrix Φ, of dimension
M × N:
y = Φx,
(1)
and reconstructed exactly, if the signal is K-sparse, i.e.,
only has K non-zero components in some basis and
the matrix Φ satisfies the Restricted Isometry Property
(RIP) [2]:
p
p
1 − δ2K kxk2 ≤ kΦxk2 ≤ 1 + δ2K kxk2 (2)
for all 2K-sparse signals x, where δ2K is the RIP constant of Φ. RIP guarantees that the norm of the measurements does not deviate significantly from the norm of the
K-sparse signal x.
b from y+n, where n is noise with knk2 =
To reconstruct x
η, we perform the optimization
b = Ψb
α (3)
α
b = min kαk1 s.t. kΦΨα − yk2 < η, x
α
P
where Ψ is a basis and kαk1 = i |αi | is the ℓ1 norm of
the coefficient vector. Reconstructing using (3) guarantees
that the norm of the reconstruction error is bounded by cη,
where c is a system-dependent constant [2].
In this paper we use the two key components of CS,
namely randomized measurements and non-linear reconstruction, to overhaul the conventional wisdom on scalar
quantization. In the next sections we demonstrate that the
CS measurement process makes the quantization error a
white noise process. We use that result demonstrate that
in the context of non-linear reconstruction it is advantageous to scale the signal such that the quantizer saturates
at a positive rate and reject the saturated measurements
from the reconstruction.
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3.
Finite-range quantization for CS
The non-linear reconstruction methods used in CS and the
democratic nature of the measurements, suggests that with
only a small performance penalty, we can choose to ignore measurements. Specifically, in this work we choose
to deliberately saturate the quantizer and ignore the measurements that saturated. In the analysis that follows we
demonstrate the advantages of this approach compared to
scaling the measurements such that they do not saturate
or incorporating the saturated measurements in the reconstruction.
The analysis is based on three distinct results:
1. CS measurements approximately follow an i.i.d.
Gaussian distribution, making the quantization error
a well characterized white noise process.
2. Clipping without quantization followed by dropping the saturated measurements preserves the signal
norm and the RIP.
3. Once quantization is introduced, the signal-toquantization noise ratio can be minimized by selecting a positive saturation rate and rejecting the saturated measurements.
The subsequent sections state and sketch the proofs for
these results and their consequences. Due to space limitations, we defer complete proofs and extended analysis to
future publications.
3.1
Distribution of CS measurements
We assume the measurement matrix Φ in (1) is randomly
generated using a zero-mean sub-Gaussian distribution
with variance 1/M
P . Under this assumption, all the measurements yi = j (Φ)i,j xj are i.i.d. zero-mean random
variables with variance kxk22 /M . Using the Lyapunov
variant of the Central Limit Theorem, it is also straightforward to show that as the dimension N of the signal x
increases the yi become normally distributed. The statement becomes non-asymptotic if the elements of Φ are
themselves distributed as a Gaussian. Our initial experiments show that commonly used CS matrix families reach
asymptotic behavior even for small N .
The implications of this statement are threefold:
1. The expected number of measurements
exceeding in
√
magnitude a threshold T kxk2 / M is 2Q(T ), where
R +∞
2
Q(x) = √12π x e−t /2 dt is the tail integral of the
standard Gaussian distribution.
√
2. The ratio of T kxk2 / M determines the saturation
rate. Thus, scaling the signal such that a specific saturation rate is achieved provides a very effective gain
control strategy.
3. The quantization error is a white process, although it
is correlated to the measurements.
We
√ should note that in the sequel only the ratio
T M /kxk2 is relevant. This ratio is the
√ threshold we select by varying the parameter T . The M factor reflects
that in practical systems the variance of the elements of
the measurement matrix is not a function of the number of
measurements. The normalization by kxk2 reflects that in
practice automatic gain control or prior signal knowledge
is used to determine the proper gain in the input.
184
3.2
Analysis of finite-range CS measurements
In this √section we introduce clipping at threshold
T kxk2 / M , without quantization. We reject the clipped
measurements and demonstrate that if the remaining meae , are sufficient in number, the
surements, denoted using y
measurement process still satisfies the RIP and preserves
f to
the norm of K-sparse signals. We use the notation (·)
denote the relevant quantities after the saturated measuref is the number of remaining meaments are dropped: M
e
surements and Φ the mutilated measurement matrix corresponding to the remaining measurements.
Assuming the result of Sec. 3.1, the expected number of
f
saturated measurements is 2M Q(T ). The remaining M
measurements
follow
a
truncated
Gaussian
distribution:
(
kxk2
T kxk2
N yi ; 0, M 2 , |yi | < √M 2
(4)
yei ∝
0, otherwise.
e is equal to:
Thus, the expected norm of y
E{ke
yk22 } = M (1 − 2Q(T ))σT2 ,
(5)
2
where σT is the variance of (4). Thus, the scaled system
e
Ge
y = GΦx
(6)
1/2
kxk22
G=
(7)
M (1 − 2Q(T ))σT2
!1/2
√
2π
= √
(8)
2π(1 − 2Q(T )) − 2T e−T 2 /2
preserves the expected value of the norm of the signal. It is
also straightforward to demonstrate that the density of the
norm of the signal concentrates around its expected value
with very high probability, in manner similar to [1, 3].
e
It is also possible to demonstrate that the resulting Φ,
which is now signal-dependent, preserves the RIP for all
f = O(K log (N/K)), or
K-sparse signals, as long as M
equivalently M = O(K log (N/K)/(1 − 2Q(T )). The
proof is beyond the scope of this paper [5]. However, it
is important since it guarantees recovery of the signal, and
the robustness to noise we need in the next section.
3.3
Quantization noise
In this section we quantize the thresholded measurements
√
using quantization interval ∆ = 2−B+1 T kxk2 / M :
e + ǫf
(9)
R(e
y) = y
Q,
where ǫf
Q is the vector of the quantization error. From
the results of Sec. 3.1 and the distribution of the measurements after thresholding it follows that ǫQ is a white random vector with elements distributed as a wrapped truncated Gaussian random variable and bounded by ±∆/2.
For small quantization intervals the distribution is well approximated by a uniform distribution in the same interval,
with variance ∆2 /12 [6]. Assuming a unit norm input x
the expected squared norm of the quantization error is:
(10)
E{kf
ǫQ k22 } = M (1 − 2Q(T ))∆2 /12
= 2−2B (1 − 2Q(T ))T 2 /3.
(11)
It can also be shown that for large M the measure of this
norm concentrates around its mean. When properly scaled
with the G in (8), the√quantization error becomes:
2π2−2B
T2
, (12)
E{kGf
ǫQ k22 } =
√
−T 2 /2
3
2π − T e
(1−2Q(T ))
SAMPTA'09
which suggests an optimal threshold T that minimizes the
error.
If the RIP is guaranteed, the norm of reconstruction error
can be bounded by ckGǫeq k22 with very high probability
[2]. For most practical applications, the minimizing T in
(12) is not sufficient to guarantee RIP, and therefore we
select the smallest T that does.
A similar analysis can be performed if we keep all the saturated measurements. In this case the RIP always holds
and the measurement error is equal to:
(13)
E{kǫQ k22 } =
2
2
2Q(T )kxk2 2
∆
+
σtrunc , (14)
= M (1 − 2Q(T ))
12
M
2−2B
2
, (15)
= kxk22 (1 − 2Q(T ))
+ 2Q(T )σtrunc
3
2
where σtrunc
is the variance of the tail distribution for a
standard Gaussian random variable, as truncated by the
saturation. Detailed analysis of this can be found in [4].
At T decreases, both σtrunc and Q(T ) increases, which
means the error due to the saturated measurements increases at the error due to the unsaturated measurements
decreases. The optimal T in this case minimizes (15).
The two strategies can be compared to select the optimal given the operating conditions. Especially in low-bit
conditions, reducing the quantization interval pays off in
terms of the error. However, the tail effects cause a significant penalty if we keep the measurements, and the better strategy is to discard them. As we discuss in the next
section in our extensive simulations under a large variety
of practical conditions discarding the measurements performs better than using them.
4.
4.1
Experimental validation
Experimental setup
Signal model: We study the performance of our approach
using signals sparse in the frequency domain: in each trial
K non-zero Fourier coefficients αn are drawn from an
i.i.d. Gaussian distribution, normalized to have unit norm,
and randomly assigned to K frequency bins out of the N dimensional space. The sampled signal x is the DFT of the
generated Fourier coefficients. Beyond quantization we do
not include additional noise sources. In addition to exactly
sparse signals, we have performed extensive simulations
with compressible signals and confirmed similar results.
However, compressible signals are beyond the scope of
this paper.
Measurement matrix: For each trial a measurement matrix is generated using a Rademacher distribution: each element is drawn independently to be +1 or −1 with equal
probability. Our extended experimentation, not shown
here in the interest of space, shows that our results are
robust to large variety of measurement matrix classes.
Reconstruction metric: We report the reconstruction
signal-to-noise ratio (SNR) in decibels (dB):
kxk22
,
(16)
SNR , 10 log
bk22
kx − x
b denotes the reconstructed signal.
where x
185
SNR (dB)
30
30
25
25
20
20
15
15
10
10
M/N=1/16
M/N=3/16
M/N=5/16
30
25
Keep
20
M/N=7/16
M/N=9/16
M/N=11/16
Discard
15
M/N=3/16
M/N=15/16
10
M/N=13/16
5
5
0
0
0
0
0.05
0.1
0.15
Quantizer Threshold (T)
(a)
M/N=15/16
5
0
0
0.05
0.1
0.15
Quantizer Threshold (T)
(b)
0.05
0.1
0.15
Quantizer Threshold (T)
(c)
Figure 1: Reconstruction SNR (dB) vs. quantizer saturation threshold (T ) using a 4-bit quantizer and downsampling rate
M
N
=
...
when (a) the saturated measurements are used for reconstruction and (b) the saturated measurements are discarded before
3
= 16
and 15
: by lowering the threshold T and rejecting saturated
reconstruction. (c) Side-by-side comparison of (a) and (b) for M
N
16
measurements, we achieve the highest reconstruction SNR.
1
16
13
16
4.2
Experimental results
We performed extensive simulations with a variety of signal parameters. Due to space limitations, we present here
the results for N = 2048, K = 60, and B = 4 which
are typical of the system performance. In our experiments
1
15
we vary M such that M
N = 16 . . . 16 and the threshold T
in the range [0, 0.18]. For each parameter combination we
repeat 100 trials, each trial with a different signal x and
matrix Φ as described in Sec. 4.1.
For each trial we quantize the measurements using a finiterange quantizer and use them to reconstruct the signal (a)
by incorporating the saturated measurements in the reconstruction and (b) by discarding the saturated measurements before reconstruction. Both cases use the linear program (3) with the appropriate value for η. We denote the
bdiscard , respectively.
bkeep and x
reconstructed signal with x
The results are shown in Fig. 1, which plots the average
reconstruction SNR versus the quantizer dynamic range T
for a variety of M
N . In particular, Figs. 1 (a) and (b) display
bdiscard , respectively. Figure 1 (c)
bkeep and x
the SNR of x
compares the two approaches for the two extreme cases of
M
3
M
15
N = 16 and N = 16 .
The plots demonstrate that lowering the threshold T such
that the saturation rate is non-zero achieves a higher reconstruction SNR compared to scaling such that no measurements clip. Furthermore, rejecting saturated measurements performs better than incorporating them in the reconstruction. This is best illustrated in Fig. 1 (c): the
optimal point on the dashed line, which corresponds to
discarding saturated measurements, exhibits better SNR
than the optimal point on the solid line, which corresponds
to incorporating saturated measurements. As expected,
the curves coincide when the saturation rate is effectively
zero.
We also performed this experiment for larger values of K
and B. As expected with higher B, we achieve less performance gain. As B grows, the quantization error goes
down and thus reducing the quantization interval by dropping measurements is less effective. As K increases, rejecting measurements remains an optimal strategy. However, when K is large enough such that the non-saturated
measurements do not satisfy RIP, our method performs
worse than incorporating the saturated measurements.
SAMPTA'09
5.
Discussion
Our results demonstrate that CS overthrows the conventional wisdom on finite range quantization. Specifically
the common practice of scaling the signal such that the
ADC does not overflow is not optimal in light of the nonlinear reconstruction. Our results demonstrate that allowing the signal to saturate is advantageous because it decreases the quantization interval in the unsaturated measurements. The non-linear reconstruction methods allow
us to discard saturated measurements and prevent the saturation error from affecting the reconstruction process.
Our results further suggests a simple automatic gain control (AGC) strategy, in which the deviation of the average
clipping rate from the desired one is used as a feedback
to modify the gain. Since the desired clipping rate is nonzero, the feedback is symmetric and increases the gain if
the clipping rate is too low. In comparison, classical AGC
systems rely on the clipping rate only when the gain is too
high and should be reduced. Since in such systems a zero
clipping rate is the desired behavior, the AGC needs to rely
on other signal features to ensure the gain is sufficient to
provide a good signal-to-quantization noise ratio.
6.
Acknowledgments
The work was supported by grants NSF CCF-0431150, CCF-0728867,
CNS-0435425, and CNS-0520280, DARPA/ONR N66001-08-1-2065,
ONR N00014-07-1-0936, N00014-08-1-1067, N00014-08-1-1112, and
N00014-08-1-1066, AFOSR FA9550-07-1-0301, ARO MURI W311NF07-1-0185, and the Texas Instruments Leadership University Program.
References:
[1] R. G. Baraniuk, M. A. Davenport, R. DeVore, and M. Wakin. A simple proof of the Restricted Isometry Property for random matrices.
In Constructive Approximation, volume 28(3), pages 253–263, Dec
2008.
[2] E. Candes. Compressive sampling. In Int. Congress of Mathematics,
volume 3, pages 1433–1452, 2006.
[3] S. Dasgupta and A. Gupta. An elementary proof of the JohnsonLindenstrauss lemma. In U.C. Berkeley Tech. Rep., volume TR-99006, 1999.
[4] G. A. Gray and G. W. Zeoli. Quantization and saturation noise due
to analog-to-digital conversion. In IEEE Trans. on Aerospace and
Electronic Systems, pages 222–223, Jan 1971.
[5] J. N. Laska, P. Boufounos, M. A. Davenport, and R. G. Baraniuk.
Democracy in action: finite-range scalar quantization for compressive sensing. In To be submitted, 2009.
[6] A. B. Sripad and D. L. Snyder. A necessary and sufficient condition
for quantization errors to be uniform and white. In IEEE Trans. on
Acoustics, Speech, and Signal Processing, volume ASSP-25, pages
442 – 448, 1977.
186
Special session on
Sampling
and
(In)Painting
Chair: Massimo FORNASIER
SAMPTA'09
187
SAMPTA'09
188
Report on Digital Image Processing
for Art Historians
Bruno Cornelis (1,2) , Ann Dooms (1,2) , Ingrid Daubechies (2,3) and Peter Schelkens (1)
(1) Dept. of Electronics and Informatics (ETRO), Vrije Universiteit Brussel (VUB) Interdisciplinary Institute for Broadband Technology (IBBT), Pleinlaan 2, B-1050 Brussels, Belgium.
(2) Computational and Applied Mathematics Program (CAMP), VUB
(3) Princeton University, Program in Applied and Computational Mathematics, Princeton, NJ 08544
bruno.cornelis@vub.ac.be
Abstract:
As art museums are digitizing their collections, a crossdisciplinary interaction between image analysts, mathematicians and art historians is emerging, putting to use
recent advances made in the field of image processing (in
acquisition as well as in analysis). An example of this is
the Digital Painting Analysis (DPA) initiative [2], bringing together several research teams from universities and
museums to tackle art related questions such as artist authentication, dating, etc. Some of these questions were
formulated by art historians as challenges for the research
teams. The results, mostly on van Gogh paintings, were
presented at two workshops. As part of the Princeton team
within the DPA initiative we give an overview of the work
that was performed so far.
1. Introduction - Penetrating the art world
Determining the authenticity of a painting can be a daunting task for art historians, requiring extensive art historical
research as well as the analysis of pigments, fabrics etc.
However much insight chemical analysis yields [4], it requires the destruction of a sample from the painting and is
therefore seldom allowed by conservators. Digital image
processing for analyzing paintings could thus prove a useful addition to the art experts’ toolbox, even beyond the
purpose of authentication. We expect that art historians
will gradually learn to use and trust these tools; a similar
emergence and eventual success took place in the medical
world in the mid 80s, with the advent of computed tomography. Subsequently, reconstruction algorithms played a
significant role in creating other medical imaging technologies, including MRI, PET and SPECT.
To stimulate the interaction between the art historical
world and branches of digital image processing, the Digital Painting Analysis (DPA) initiative organized two workshops in Amsterdam (IP4AI or Image Processing for
Artist Identification) and a symposium (celebrating the
inauguration of TiCC, Tilburg centre for Creative Computing) to facilitate a dialog between the two communities. The Van Gogh Museum (Amsterdam) and the Kröller
Müller Museum (Otterlo) made it possible for participating teams to work with high resolution digital images of
paintings (mostly van Goghs) in their collections.
SAMPTA'09
2.
Challenges - Convincing the art expert
To jumpstart the IP4AI workshops, art historians formulated challenges for the research teams, asking them to
provide convincing arguments in favor of digital image
processing. These included the following:
• Authentication: distinguish an original van Gogh
painting from a copy or forgery. This was the main
focus of the first workshop; preliminary results of the
participating research teams can be found in [7].
• Dating: classify works by van Gogh that were either
painted in his early Paris phase (1886 − 1888) or in
his later Arles period. Art historians noticed changes
in van Gogh’s way of painting throughout his career.
Small brushstrokes seem to be more prominent in his
Paris period while broader ones prevail in Arles.
• Identifying distinguishing features: can an artist’s
hand be characterized and features be found that distinguish him from other painters?
• Image enhancement: fuse information obtained by
different modalities (x-ray, infrared, visual, etc.) to
(virtually) enhance damaged paintings, or underpaintings. A first challenge here is detailed and precise registration.
• Inpainting: digitally reconstruct missing pieces from
a painting when only limited data is at hand.
The purpose of this paper is to provide an overview of
the tools and general methodology used by the Princeton
research team in order to tackle these challenges. Detailed
results can be found in [6, 10].
2.1
Classification - Authentication, dating and
identifying features
For the analysis of paintings it is crucial to extract distinguishing features/statistics that truly characterize the style
of an artist. It is obvious that simple image statistics such
as mean or variance of an image will not suffice by themselves. To take an extreme example: reordering by increasing grayscale the pixels in every row of a digital image of a natural scene, and then doing the same in every
column, produces an image with same mean and variance
as the original, but bereft of (almost) all other information.
More complex models that provide additional information
189
are needed. The approach taken for the first three challenges built such models. The analysis consists of three
main steps: transform, modeling and classification.
Transform. A multiresolution transform is performed
on patches of the image. We used the Dual-Tree Complex
Wavelet Transform (DTCWT) [9]; it provides approximate shift invariance and directional selectivity (properties standard wavelet transforms lack). The DTCWT uses
two parallel filter banks and produces six subbands of coefficients that let us analyze changes in the image in six
directions (±15◦ , ±45◦ and ± 75◦ ) at different scales.
Modeling. A large number of pixels, and thus also of
transform coefficients, combined with noise on the pixel
values (due to the acquisition process) impose robust dimensionality reduction and feature extraction techniques.
We used Hidden Markov Trees (HMT) [3]. It is possible
to describe the wavelet coefficients for a large class of images in terms of two key properties [11]:
• 2Population: smooth image regions are represented
by wavelet coefficients with a narrow probability distribution function (pdf); edges, ridges or other singularities by wavelet coefficients with a wide pdf.
• Persistence: the classification into narrow/wide pdfcoefficients tends to propagate across scales.
performed with WEKA [1], a collection of machine learning algorithms for data mining tasks.
2.1.1
Authentication challenge results
The authentication challenge was the main research topic
for the first IP4AI workshop [7]. To validate their earlier results the Princeton team asked Dutch art conservation student Charlotte Caspers to make original paintings on different materials, with different kinds of paint
and brushes, and to create a faithful copy for each of
these originals. The dataset provided ground truth: we
knew which paintings were original and which ones were
copies. We considered both HMT features and thresholding features [10]. The aim was to recognize the difference between a fluid and a more hesitant (copying) stroke
through machine learning. For this kind of classification
problem the SVM with polynomial kernel machine learning algorithm was the best classifier.
The images were subdivided into patches, some of which
were used for training the machine learning algorithm.
The best results were obtained by using only patches from
the painting under investigation and its copy (see Figure
1). The results can be found in Table 1; they show that
when both soft and hard brushes are used, the algorithm
achieves a succes rate similar to that obtained by state-ofthe-art authentication algorithms for handwriting.
These two properties are used to design a statistical model
to represent images. Due to the multiresolution nature
of the wavelet transform, the wavelet coefficients can be
arranged into a quadtree (one coefficient from a coarser
scale corresponds to four wavelet coefficients at the next
finer scale). At each scale, hidden variables control the
wavelet coefficients. They can have two states: L (large,
for edge-like structures) and S (small, for smooth regions).
The wavelet coefficients are modeled as samples from a
mixture of two Gaussian distributions, one with a large
variance for the coefficients corresponding to an edge and
one with a small variance for coefficients from a smooth
region. HMT model the statistical dependencies between
wavelet coefficients at different scales. The parameters of
the HMT we used as features are:
• αT : a 2 × 2 transition probability matrix, that depicts
the probabilities that a child node is in a particular
state, given the state of the parent node.
• µi : the means of the narrow and wide Gaussian distribution (i = 1, 2) for each subband.
• σi : variance of the narrow and wide Gaussian distribution (i = 1, 2) for each subband.
For example, if we apply a 4-level DTCWT transform on
a patch of an image, then the features extracted from that
patch would be the following: 6 × 4 × 2 means, 6 × 4 × 2
variances and 6 × 4 × (2 × 2) probabilities, adding up to
a total of 192 features. These HMT features are grouped
into a model parameter vector and are determined using
the expectation maximization algorithm.
Classification. The model parameters vectors extracted
in the previous step are used as the input for classification algorithms. We used several types of machine learning algorithms: Support Vector Machines, Adaboost, Decision Stump and Random Forest. All experiments were
SAMPTA'09
Figure 1: Four sets of patches without overlap.
2.1.2
Dating challenge results
For the dating challenge a set of 66 high resolution paintings (90 pixels per linear inch) were put at the disposal
of all teams. All the classifiers listed above were trained
with 256 × 256 patches using 10-fold cross validation. As
can be seen in Table 2, the Random Forest (RF) classifier
was the most accurate. Three paintings for which art historians are not sure when they were painted needed to be
attributed to one of two periods. Figure 2 shows the resulting classification success rate for patches of paintings
from the training set. The RF algorithm was then used on
the patches of the three paintings to be attributed, and a
majority vote of the patches was determined.
190
Pair
1
2
3
4
5
6
7
Ground
CP Canvas
CP Canvas
Smooth CP Board
Bare linen canvas
Chalk and Glue
CP Canvas
Smooth CP Board
Paint
Oils
Acrylics
Oils
Oils
Oils
Acrylics
Oils
Brushes
Soft&Hard
Soft&Hard
Soft&Hard
Soft
Soft
Soft
Soft
Style
TI
TI
TI
Sm,Bl
Total
78%
72%
78%
75%
50%
38%
55%
Copy
67%
55%
78%
50%
0%
75%
22%
Original
89%
89%
78%
100%
100%
0%
88%
Table 1: Accuracy for each test on the Caspers data set. Abbreviations: Sm=Smooth, Bl=Blended, TI=Thick Impasto.
SVM
61.2%
AB
63.2%
DS
63.1%
RF
70.5%
Table 2: Accuracy of different classifiers.
Abbreviations: SVM=Support Vector Machines, AB=
AdaBoost, DS=Decision Stump, RF=Random Forest.
Figure 3: Distinguishing feature challenge.
Left: “Still Life: Vase with Gladioli” by V. van Gogh.
Right: “Vase with Flowers” by G. Jeannin.
2.2
Arles
Tie
Paris
Figure 2: Classification results for three paintings.
2.1.3 Extracting Distinguishing Features results
The test set consisted of floral still lifes painted by van
Gogh, Monticelli and other contemporary artists. The goal
was to quantify to what extent van Gogh and Monticelli
share features, in their brushwork and color schemes, absent in the style of the others. The purpose here was thus
to distinguish styles instead of painters (as in authentication). The same methodology described above was used.
Results show that wavelet coefficients in direction −45◦ ,
scale 6 characterize the style of van Gogh and Monticelli
whereas wavelet coefficients in the 15◦ , scale 4 subband
are more prominent in the other paintings. Examples of
these distinguishing features are highlighted in Figure 3.
More detailed results are in [6].
SAMPTA'09
Using Different Image Acquisitions
Art museums typically have x-ray and infrared photographs in their collections, which can reveal much about
what is below the visible surface of a painting. These can
also be digitized (or acquired digitally, in the future), and
be studied with digital image processing tools. In order to
combine the different modes of image acquisition, the first
task is to register the images (we used x-ray, infrared and
color images of the same painting) to enhance and detect
hidden features. Figure 4 shows a woman’s face emerging (horizontally) from underneath the grass in the painting “Patch of Grass”. Because x-rays and photographs
are acquired by different modalities, the matching is not
as straightforward as it seems initially. Both images were
divided into patches and reference points in both images
were picked in order to define a smooth warping that gave
acceptable results. Another example is the counting of
threads/inch in the canvas, visible on x-rays, to determine
a painting’s authenticity and date [8].
2.3
Inpainting
An important aspect for art historians and conservators is
the preservation of works of art. When paintings become
damaged, all the available information (grayscale photographs, low resolution color photographs, ektachromes,
191
4.
Acknowledgments
We would like to thank Sina Jafarpour, Gungor Polatkan,
Andrei Brasoveanu, Eugene Brevdo and Shannon M.
Hughes for letting us report briefly on some of their results, and the Van Gogh and Kröller Müller Museums for
letting us use their high resolution data set. Special thanks
go to Massimo Fornasier for his help with the inpainting
challenge.
Research was partially supported by The Fund for Scientific Research Flanders (project G.0206.08 and postdoctoral fellowship of Peter Schelkens).
References:
Figure 4: Registered x-ray on “Patch of Grass”.
. . . ) is called upon to help art conservators in their reconstruction or restoration. In [5] techniques were proposed
to mathematically reconstruct the original colors of frescoes (reduced to rubble in a wartime bombing) by making use of the information given by preserved fresco fragments and gray level pictures of the intact frescoes taken
before the damage occurred.
We investigated whether such techniques would also work
on van Gogh pictures. With the help of M. Fornasier, one
of the authors of [5], we applied these algorithms to a
high resolution color image of the “Lemons on a Plate”
painting. A patch of 200 × 200 pixels was digitally removed; Figure 5 shows its mathematical reconstruction,
using only a low resolution color image (with faithful colors) and a high resolution grayscale image of that painting.
The results are quite satisfying and prove that these techniques could be used for restoration purposes.
Figure 5: Inpainting.
3. Conclusions
The results obtained for the first and second IP4AI workshop in Amsterdam were promising. It is clear however,
that these digital techniques on their own are not sufficient
to provide conclusive answers to questions of interest to
art historians. Nevertheless, they will likely be a worthy
addition to the toolbox of art historians and conservators;
they have the great advantage of not being invasive. There
is also still room for improvement in the different steps of
the analysis of paintings. It is worth pointing out, however, that in order to apply such techniques, the quality
of the acquired dataset (i.e. high resolution images) is of
utmost importance. Only images of equal quality can be
compared with each other.
SAMPTA'09
[1] http://www.cs.waikato.ac.nz/ml/weka/.
[2] http://www.digitalpaintinganalysis.org/.
[3] Matthew Crouse, Robert Nowak, and Richard Baraniuk. Wavelet-based statistical signal processing using hidden markov models. IEEE Transactions on
Signal Processing, 46:886–902, 1997.
[4] Joris Dik, Koen Janssens, Geert Van Der Snickt,
Luuk van der Loeff, Karen Rickers, and Marine
Cotte. Visualization of a lost painting by vincent van
gogh using synchrotron radiation based x-ray fluorescence elemental mapping. Anal. Chem., 80:6436–
6442, 2008.
[5] Massimo Fornasier, Ronny Ramlau, and Gerd
Teschke. A comparison of joint sparsity and total
variation minimization algorithms in a real-life art
restoration problem. to appear in Advances in Computational Mathematics, 2008.
[6] S. Jafarpour, G. Polatkan, E. Brevdo, S. Hughes,
A. Brasoveanu, and I. Daubechies. Stylistic analysis of paintings using wavelets and machine learning.
submitted to EUSIPCO 2009.
[7] C. R. Johnson, Ella Hendriks, Igor Berezhnoy, Eugene Brevdo, Shannon Hughes, Ingrid Daubechies,
Jia Li, Eric Postma, and James Z. Wang. Image
processing for artist identification - computerized
analysis of vincent van gogh’s painting brushstrokes.
IEEE Signal Processing Magazine, July 2008.
[8] D. H. Johnson, L. Sun, J. Guo, C. R. Johnson Jr.,
and E. Hendriks. Matching canvas weave patterns
from processing x-ray images of master paintings.
submitted to 16th IEEE International Conf. on Image
Processing, 25:37–48, November 2009.
[9] Nick Kingsbury. Complex wavelets for shift invariant analysis and filtering of signals. Applied and
Computational Harmonic Analysis, 10(3):234–253,
May 2001.
[10] G. Polatkan, S. Jafarpour, A. Brasoveanu, S. Hughes,
and I. Daubechies. Detection of forgery in paintings
using supervised learning. Submitted to IEEE International Conference on Image Processing 2009.
[11] Justin K. Romberg, Hyeokho Choi, Richard G. Baraniuk, and Nick Kingsbury. A hidden markov tree
model for the complex wavelet transform. IEEE
Transactions on Signal Processing, pages 133–136,
2001.
192
Edge Orientation Using Contour Stencils
Pascal Getreuer (1)
(1) Department of Mathematics, University of California Los Angeles
getreuer@math.ucla.edu
Abstract:
Many image processing applications require estimating the orientation of the image edges. This estimation
is often done with a finite difference approximation
of the orthogonal gradient. As an alternative, we apply contour stencils, a method for detecting contours
from total variation along curves, and show it more
robustly estimates the edge orientations than several
finite difference approximations. Contour stencils are
demonstrated in image enhancement and zooming applications.
1.
structure tensor J(∇u) = ∇u ⊗ ∇u. The structure tensor satisfies J(−∇u) = J(∇u) and ∇u is an
eigenvector of J(∇u). The structure tensor takes into
account the orientation but not the sign of the direction, thus solving the antipodal cancellation problem.
As developed by Weickert [9], let
Jρ (∇uσ ) = Gρ ∗ J(Gσ ∗ u)
where Gσ and Gρ are Gaussians with standard deviations σ and ρ. The eigenvector of Jρ (∇uσ ) associated
with the smaller eigenvalue is called the coherence direction, and is an effective approximation of edge orientation.
Introduction
2.
A fundamental and challenging problem in image
processing is estimating edge orientations. Accurate
edge orientations are important for example in edgeoriented inpainting methods [2], and optical character
recognition features [8].
1.1
(1)
∇u⊥ for Estimating Edge Orientation
A starting point to edge orientation estimation is to
approximate ∇u⊥ with finite differences. Finite difference estimation alone is typically too noisy to be
reliable, especially near edges, so the gradient is often
regularized by a convolution ∇u ≈ ∇(G ∗ u) where
G is for example a Gaussian. However, there is a serious problem in that ∇u⊥ and −∇u⊥ both describe
the same edge orientation, so linear smoothing tends
to cancel the desired edge information.
Introduced by Bigün and Granlund [1] and Forstner
and Gulch [3], a better approach is to use the 2 × 2
SAMPTA'09
Contour Stencils
Numerical implementation of J(∇u) yet involves estimating ∇u. Since numerical estimates of ∇u are
sensitive to noise and unreliable near edges, significant amounts of smoothing is still needed for acceptable results. We abandon ∇u⊥ and approach the estimation of edge orientation from an entirely different
principle.
Given a smooth curve C and a parameterization γ :
[0, T ] → C, consider measuring the total variation of
u along C,
Z T
∂t u γ(t) dt.
TV(C) =
(2)
0
Edge orientations can be estimated by comparing
TV(C) with various candidate curves. Contour stencils [4, 5] is a numerical implementation of this idea.
Let u : Λ → R be a discrete image. Denote by ui,j ,
(i, j) ∈ Λ, the value of u at the (i, j)th pixel, and let
xi,j ∈ R2 denote its spatial location.
193
+i
+j
S(α, β) =
8
1 α = (i, j), β = (i − 1, j + 1),
>
>
>
>1 α = (i, j), β = (i + 1, j − 1),
>
<
4 α = (i, j + 1), β = (i + 1, j),
1 α = (i + 1, j + 1), β = (i, j + 2),
>
>
>
>
>1 α = (i + 1, j + 1), β = (i + 2, j),
:
0 otherwise
1
(i, j)
1
4
1
1
1
Figure 1: An example contour stencil S for detecting
a 45◦ orientation.
A contour stencil is a function S : Λ × Λ → R+ describing weighted edges between pixels (see Figure 1).
These edges approximate several parallel curves localized over a small neighborhood. As a discretization of
(2), the total variation of S is
X
1
S(α, β) |uα − uβ | , (3)
TV(S) := |S|
α,β∈Λ
P
and |S| := α,β S(α, β) |xα − xβ |. For the contour
√
stencil in Figure 1, |S| = (1 + 1 + 4 + 1 + 1) 2 and
TV(S) =
1
|S|
|ui,j − ui−1,j+1 | + |ui,j − ui+1,j−1 |
+ 4 |ui,j+1 − ui+1,j |
1
1
1
2
1
2
2
1
1
1
1
2
2
1
1
1
1
2
1
1
1
2
1
1
1
1
1
1
1
2
4
2
1
1
2
1
1
2
1
1
2
1
1
2
2
1
1
2
1
2
1
2
2
1
1
1
2
2
1
1
2
1
1
1
1
2
1
2
1
1
2
1
1
2
1
2
1
1
2
1
2
1
2
1
Figure 3: A node-centered stencil set.
candidate stencil, and then determining the best-fitting
stencil S ∗ . For efficient implementation, define
H
= |vi,j − vi+1,j | ,
Di,j
V
Di,j
= |vi,j − vi,j+1 | ,
A
Di,j
= |vi,j − vi+1,j+1 | ,
B
Di,j
= |vi,j+1 − vi+1,j | ,
then the TV(S) can be computed as sums of these differences, and the differences may be reused between
successive cells. For the proposed stencil sets, contour
stencils cost a few dozen operations per pixel [4].
Input
Estimated Orientations
1
1
1
2
1
1
1
2
1
1
1
1
Figure 2: The proposed cell-centered contour stencils.
The contours of u are estimated by finding a stencil
with low total variation,
S ∗ = arg min TV(S)
S∈Σ
2
1
1
1
1
1
1
2
1
2
1
4
1
1
1
1
2
1
1
2
2
1
1
2
1
2
2
+ |ui+1,j+1 − ui,j+2 | + |ui+1,j+1 − ui+2,j | .
1
1
(4)
Figure 4: Edge orientation estimation with contour
stencils (using the cell-centered stencils in Figure 2).
Contour stencils extend naturally to nonscalar data by
replacing the absolute value in (3) with a metric. On
color images for example, a suitable choice is the ℓ1
vector norm in YCb Cr color space.
where Σ is a set of candidate stencils (see Figures 2
and 3). The best-fitting stencil S ∗ provides a model of
the underlying contours.
3.
In summary, contour stencil orientation estimation is
done by first computing the TV estimates (3) for each
Here we compare contour stencils and several finite
difference methods for estimating edge orientation.
SAMPTA'09
Comparison
194
As a test image with fine orientations, we use a small
image of straw (Figure 5).
u
square whose corners correspond to ui,j , ui+1,j ,
ui,j+1 , ui+1,j+1 . Cell-centered methods compute orientation estimates logically located in the center of the
cells. With node-centered methods, the edge orientation estimates are centered on the pixels.
Let Dx+ denote the forward difference operator
Dx+ ui,j = ui+1,j − ui,j and similarly in the other coordinate Dy+ . An estimate of ∇u symmetric over the
cell is
+
(Dx ui,j + Dx+ ui,j+1 )/2
.
(5)
∇ui,j ≈
(Dy+ ui,j + Dy+ ui+1,j )/2
Figure 6 compares ∇u⊥ estimated using (5) with contour stencils using the cell-centered stencil set shown
in Figure 2.
Figure 5: The test image.
Sobel filter (6)
As is done with coherence direction (1), any orientation field θ~ can be smoothed by filtering its tensor
~ But for easier comparison, all
product: Gρ ∗ (θ~ × θ).
methods are shown without smoothing.
∇u⊥ with (5)
Contour Stencils (Σ as in Figure 3)
Contour Stencils (Σ as in Figure 2)
Figure 7: Comparison of node-centered methods.
Figure 6: Comparison of cell-centered methods.
We consider two categories of methods: cell-centered
and node-centered. Define the (i, j)th cell as the
SAMPTA'09
The Sobel filter [7] is a node-centered approximation
of ∇u,
−1 0 1
∂x u ≈ −2 0 2 ∗ u
(6)
−1 0 1
and similarly for ∂y u. Figure 7 compares the Sobel filter with contour stencils using the node-centered stencil set from Figure 3.
195
4.
Applications
Input
Zooming (4×)
Contour stencils are useful in applications where
edges are significant.
Input
Contour Stencil Enhancement
Figure 9: (This is a color image.) Edge-adaptive
zooming using contour stencils [5].
References:
Figure 8: Simultaneous sharpening and denoising using contour stencils [4].
Contour stencils can be useful in discretizing image
diffusion processes. Figure 8 demonstrates image enhancement using a combination of the Rudin-Osher
shock filter [6] and TV-flow that has been discretized
with contour stencils.
As another application, Figure 9 shows an image
zooming result using contour stencils. The method
approaches zooming as an inverse problem using a
least-squares graph regularization. The regularization
is adapted according to the edge orientations estimated
from the contour stencils.
5.
Conclusions
[1] J. Bigün and G. H. Granlund. Optimal orientation
detection of linear symmetry. In IEEE First International Conference on Computer Vision, pages
433–438, London, Great Britain, June 1987.
[2] Folkmar Bornemann and Tom März. Fast image
inpainting based on coherence transport. J. Math.
Imaging Vis., 28(3):259–278, 2007.
[3] W. Förstner and E. Gulch. A fast operator for detection and precise location of distinct points, corners, and centers of circular features. pages 281–
305, 1987.
[4] Pascal Getreuer.
Contour stencils for edgeadaptive image interpolation. volume 7257, 2009.
[5] Pascal Getreuer. Image zooming with contour
stencils. volume 7246, 2009.
[6] S. J. Osher and L. I. Rudin. Feature-oriented image enhancement using shock filters. SIAM Journal on Numerical Analysis, 27:919–940, 1990.
[7] Irwin Sobel and Jerome A. Feldman. A 3x3
isotropic gradient operator for image processing. Presented at a talk at the Stanford Artificial
Project in 1968.
[8] Øivind Due Trier, Anil K. Jain, and Torfinn
Taxt. Feature-extraction methods for characterrecognition: A survey. Pattern Recognition,
29(4):641–662, April 1996.
[9] Joachim Weickert. Anisotropic diffusion in image processing. ECMI Series, Teubner-Verlag,
Stuttgart, Germany, 1998.
Contour stencils provide reliable orientation estimates
at low computational cost, enabling better results in
image processing applications.
SAMPTA'09
196
Smoothing techniques for convex problems.
Applications in image processing.
Pierre Weiss (1) , Mikaël Carlavan (2) , Laure Blanc-Féraud (2) and Josiane Zerubia (2)
(1) Institute for Computational Mathematics, Kowloon Tong, Hong Kong.
(2) Projet Ariana - CNRS/INRIA/UNSA, 2004 route des Lucioles, 06902 Sophia-Antipolis, France.
(1) pierre.armand.weiss@gmail.com, (2) firstname.lastname@sophia.inria.fr
Abstract:
In this paper, we present two algorithms to solve some
inverse problems coming from the field of image processing. The problems we study are convex and can be expressed simply as sums of lp -norms (p ∈ {1, 2, ∞}) of
affine transforms of the image. We propose 2 different
techniques. They are - to the best of our knowledge - new
in the domain of image processing and one of them is new
in the domain of mathematical programming. Both methods converge to the set of minimizers.
¡ ¢Additionally, we
show that they converge at least as O N1 (where N is the
iteration counter) which is in some sense an “optimal” rate
of convergence. Finally, we compare these approaches to
some others on a toy problem of image super-resolution
with impulse noise.
1. Introduction
Many image processing tasks like reconstruction or segmentation can be done efficiently by solving convex optimization problems. Recently these models received considerable attention and this led to some breakthrough.
Among them are the new sampling theorems [5] and the
impressive results obtained using sparsity or regularity assumptions in image reconstruction (see e.g. [4]).
These results motivate an important research to accelerate
the convergence speed of the minimization schemes. In
the last decade, many algorithms like iterative thresholding or dual approaches were reinvented by the “imaging
community” (see for instance [2, 3] for old references).
Recently, the “mathematical programming community”
got interested in those problems and it led to some drastic
improvements. As examples let us cite the papers by Y.
Nesterov [9, 10] and M. Teboulle [1] which improve by
one order of magnitude most first order approaches.
In this paper, we mainly follow the lines of Y. Nesterov
[9]. We consider the problem of minimizing the sum of
lp -norms (p ∈ {1, 2, ∞}) of affine transforms of the image. The general mechanism of the algorithms we propose
consists in smoothing the problem and solve it with an efficient first order scheme. Our contribution is mainly to
extend the results of [9] to a more general setting and to
propose a dual variant which behaves better in all problems we tested. We also give convergence rates for the
proposed algorithms. We believe, this gives some insight
on the important factors that influence the algorithms efficiency and helps designing solvable problems.
SAMPTA'09
2.
The problems considered
In this paper, we consider the following seminal model of
image deterioration:
u0 = Du + b
(1)
where u is an original, neat image, D : Rn → Rm is some
known linear transform, b ∈ Rm is some additive noise
and u0 ∈ Rm is a given observed image. This simple
formalism actually models many real situations. For instance, D can be an irregular sampling and a convolution.
In this case recovering u from u0 is a super-resolution
problem [7]. Other applications include image inpainting,
compression noise reduction, texture+cartoon decompositions, reconstruction from noisy indirect measurements...
Finding u from the observation u0 is an inverse problem.
There exists many ways to solve it. In this paper, we concentrate on two variational models. The first one consists
in solving the following convex problem:
min ||Bx||1 + λ||Dx − u0 ||p .
|
{z
}
x∈X
(2)
Ψ(x)
The second one consists in solving:
¡
¢
min ||y||1 + λ||DB ∗ y − u0 ||p .
y∈Y
(3)
In both problems, B : Rn → Ro is a linear transform, || ·
||p denotes the standard lp -norm and X and Y are simple
convex sets (like Rn or [0, 1]n ).
The interpretation of the first model is as follows: we look
for an image x which minimizes ||Bx||1 such that Dx is
close to u0 . The function x 7→ ||Bx||1 can be seen as a
regularity a priori on the image. For instance, if B is the
discrete gradient, then it corresponds to the total variation.
If B is some wavelet transform, it is equivalent to a Besov
semi-norm [6]. p must be chosen depending on the statistics of the additive noise. For instance, p should be equal
to 2 for Gaussian noise, to 1 for impulse noise and to ∞
for uniform noise.
The interpretation of the second model is the following:
we look for a decomposition y of the restored image in
some dictionary B ∗ such that its reconstruction B ∗ y is
close to u0 . Minimizing the l1 -norm of y is known to
favor sparse structures. The underlying assumption is thus
that the original image u is sparse in the dictionary B ∗ .
197
From a numerical point of view, both problems are very
similar. However, the first one is slightly more general and
complicated than the second. We will thus give a detailled
analysis of its resolution and only provide numerical results for the second one.
The remaining of the paper is as follows. We first present
an algorithm based on a regularization of the primal problem (2). Then we present a technique to regularize a dual
version of (2). Finally we propose theoretical and numerical comparisons of both techniques on a problem of image
super-resolution. Due to space limitations, we only provide the main ideas in this paper. We refer the reader to
[12] (in French), for the proofs of the propositions.
3. Smoothing of the primal problem
In this section, we propose a method to minimize (2). Its
principle is exactly the same as the method proposed by Y.
Nesterov in [9]:
1. Smooth the non-differentiable terms in (2).
2. Solve the regularized problem using an accelerated
gradient method.
The only difference is that we do not require the set X to
be bounded, which requires a slightly different analysis.
Now let us present some details of this approach. A key
observation to solve (2) is that it can be rewritten as a so
called
Let p′ denote the conjugate of
´
³ min-max problem.
p i.e. p1′ + p1 = 1 . We can rewrite problem (2) as follows:
µ
¶
¡
¢
min max hBx, y1 i + λhDx − u0 , y2 i
(4)
x∈X
y∈Y
=
min max (hAx − h, yi)
x∈X y∈Y
|
{z
}
(5)
Ψ(x)
where h·, ·i denotes the canonical scalar product,
¸
·
·
¸
0
B
, h=
A=
and
λu0
λD
(6)
Y = {y = (y1 , y2 ) ∈ Ro ×Rm , ||y1 ||∞ ≤ 1, ||y2 ||p′ ≤ 1}.
(7)
The function Ψ is a conjugate function and the set Y is
bounded. It can thus be smoothed using a Moreau regularization. Let us denote:
´
³
µ
(8)
Ψµ (x) = max hAx − h, yi − ||y||22 .
y∈Y
2
This function can be shown to be L-Lipschitz differentiable:
||∇Ψµ (x1 ) − ∇Ψµ (x2 )||2 ≤ L||x1 − x2 ||2
with L =
|||A|||2
µ
and |||A||| =
max
x∈Rn ,||x||2 ≤1
(9)
(||Ax||2 ).
Furthermore, it is a good uniform approximation of Ψ in
the following sense:
µ
0 ≤ Ψ(x) − Ψµ (x) ≤ D.
(10)
2
SAMPTA'09
where D =
µ
¶
¢
¡
max ||y||22 . Thus, we can make the dify∈Y
ference between Ψ and Ψµ as small as desired by decreasing µ. The approximation Ψµ is actually very common
in image processing. For instance, when p = 1, it corresponds to the approximation of the absolute value by a
Huber function. When p = ∞ it is slightly more difficult,
but it can still be computed in closed form.
The smoothed problem writes:
min (Ψµ (x)) .
x∈X
(11)
It consists in minimizing a differentiable function over a
simple set. We can thus apply projected gradient like algorithms to solve it. Unfortunately, µ has to be chosen
small in order to get a good approximate solution. This
requires to use small step sizes in the gradient descent and
thus results in a very slow convergence rate. The main
observation of Y. Nesterov in [9] is that using an accelerated version of the projected gradient methods can actually compensate the approximation
error. This results
¡ ¢
in a convergence rate in O N1 (where N is the iteration
counter), while other first order approaches
³
´ like projected
1
subgradient descents converge as O √N .
Now let us write down the complete algorithm to solve
(11). Let x∗µ denote a solution of (11) (it is not unique in
general). We propose the following algorithm:
Algorithm 1 (Primal)
Choose a number of iterations N .
Set a starting point x0 (as close as possible to x∗µ ).
|||A|||·||x0 −x∗ ||2
µ
Set µ = µ(N ) =
N
0
Set A = 0, g = 0 and x = x .
for k = 0 toq
N do
1
a = L + L12 + L2 A
¡
¢
v = ΠX x0 − g
y = Ax+av
A+a
³
´
∇Ψ (y)
x = ΠX y − Lµ
.
g = g + a∇Ψµ (x)
A=A+a
end for
Set xN = x.
Our main convergence results are as follows. Let x∗ denote a solution of (2).
Proposition 1 xN converges to the set of minimizers of
(2).
Proposition 2 The worst case convergence rate is:
√
2|||A||| · ||x0 − x∗µ ||2 D
N
∗
. (12)
Ψ(x ) − Ψ(x ) ≤
N
Note that the distance ||x0 − x∗µ ||2 is unknown in general, so that Algorithm 1 might not seem implementable.
In the case where X is a compact set, this quantity can
be bounded above by the diameter of X. When X is not
bounded, it actually suffices to choose µ of order |||A|||
to
N
198
¡ ¢
get a precision of order O N1 . Algorithm
(1) is thus im¡ ¢
plementable and converges as O N1 . This convergence
rate is neatly sublinear and might seem bad at first sight.
Actually, it is somehow optimal. Indeed, A. Nemirovski
shows in [8] that some instances of problems like (5) ¡can-¢
not be solved with a better rate of convergence than O N1
using first order methods.
4. Smoothing of the dual problem
In this section, we propose an approach alternative to the
previous one. Its flavor is similar to a proximal-method.
One way to understand this scheme is that we smooth the
“dual” problem instead of the primal problem. Note that
the min and the max in equation (5) cannot be inverted as
we do not suppose X to be compact. So we cannot use properly speaking - the term dual problem.
Instead of solving (2), we solve:
´
³
²
min ||Bx||1 + λ||Dx − u0 ||p + ||x − x0 ||22
x∈X
2
(13)
0
where ² ∈ R+
∗ and x should be chosen close to the set
of minimizers of (2). It can be shown that as ² goes to
0, the unique solution of (13) converges to the Euclidean
projection of x0 onto the set of minimizers of (2). We can
rewrite (13) as a min-max problem:
¶
²
0 2
min max (hAx − h, yi) + ||x − x ||2 (14)
x∈X
y∈Y
2
µ
´
³
²
= max min hAx − h, yi + ||x − x0 ||22 (15)
.
y∈Y x∈X
2
|
{z
}
Algorithm 2 (Dual)
Choose a number of iterations N .
Set a point x0 (as close as possible to X ∗ ).
Set a starting point y 0 (as close as possible to y²∗ ).
|||A|||·||x0 −x∗
² ||2
Set ² = ²(N ) =
.
N
Set A = 0, g = 0, x̄ = 0 and y = y 0 .
for k = 0 toq
N do
a = L1 + L12 + L2 A
¡
¢
v = ΠY y 0 − g
z = Ay+av
A+a
´
³
y = ΠY z + ∇ΨL² (z)
x̄ = x̄ + ax(y) (cf. equation (17))
g = g − a∇Ψ² (y)
A=A+a
end for
x̄
Set x̄N = A
.
This algorithm can be shown to have the following properties.
Proposition 3 x̄N converges to the projection of x0 onto
the set of minimizers of (2).
Proposition 4 The worst case convergence rate is:
√
2|||A||| · ||x0 − x∗² ||2 D
N
∗
Ψ(x̄ ) − Ψ(x ) ≤
. (18)
N
Rate (18) is actually very similar to (12). It is thus natural
to wonder if there is an interest in using this dual approach.
Let us present some interesting aspects of this scheme:
• In the dual approach, the solution of the regularized
problem is unique. This guarantees a certain stability
of the iterates.
Ψ² (y)
Note that we can invert the min and the max only because
the term 2² ||x − x0 ||22 makes the problem coercive in x.
Now, the important observation is that the function Ψ² is
the conjugate of a strongly convex function. It is thus concave and Lipschitz differentiable:
||∇Ψ² (y1 ) − ∇Ψ² (y2 )||2 ≤ L||x1 − x2 ||2
• We can show an additional convergence rate in norm
to the regularized solution. Namely, for a fixed ², we
have for all k:
||x̄k − x∗² ||22 ≤
(16)
³
´
²
x(y) = arg min hAx − h, yi + ||x − x0 ||22 . (17)
2
x∈X
Actually, a slight modification of Nesterov’s scheme (an
ergodic version) can be shown to ensure convergence of
xN with the desired convergence rate. In the following,
we detail briefly our main results.
Let x∗² denote the solution of (13) and y²∗ denote a solution
of (15). Let X ∗ denote the set of minimizers of (2) and let
us consider the following algorithm:
SAMPTA'09
(19)
• In practical experiments, model (13) with a small ²
leads to slightly better SNR than model (2) for some
restoration purposes in image processing.
2
∀(y1 , y2 ) ∈ Y × Y with L ≤ |||A|||
. Problem (15)
²
consists in maximizing a Lipschitz differentiable concave
function over a convex set. It thus seems interesting to use
a scheme similar to Algorithm 1 on this problem. Unfortunately we will get a convergence rate on the dual variable
y and not on the variable of interest:
D|||A|||
² · k2
• The practical convergence rates of the dual approach
were better than those of the primal approach in all
our experiments.
To conclude the theoretical part of this paper, let us precise that problem (3) can be solved with the same algorithms. However, it is preferable not to regularize the term
y 7→ ||y||1 which can be minimized using accelerated softthresholding algorithms [1, 10, 12].
5.
Numerical results
In this section we present some comparisons for a problem of image zooming with impulse noise. To solve this
problem, we simply set:
199
4
10
3
10
2
Ψ(xk) − Ψ(x*)
10
1
10
0
10
(a)
(b)
−1
10
Nesterov Primal
Nesterov Dual
−2
10
Projected Gradient Primal
Projected Gradient Dual
−3
10
0
500
1000
1500
2000
2500
3000
Number of iterations
3500
4000
4500
5000
Figure 1: Cost function w.r.t. the number of iterations.
(c)
• D: convolution by a low-pass filter followed by a
down sampling of factor d in the x and y directions.
• p = 1 (which is adapted to impulse noise).
• B: a redundant wavelet transform. We set B to be
the Dual-Tree Complex Wavelet Tranform (DTCW)
[11].
In that case |||A|||2 can be computed explicitly. For the
general case, let us point out that iterated power algorithms
provide good approximations of |||A∗ A||| = |||A|||2 .
In Figure 1, we chose ² = 0.045 and µ = 10−5 . This
ensures that both methods lead to the same asymptotic accuracy (measured in terms of objective function). We can
see that the dual approach seems to have a better behavior. For this problem reducing Ψ(x0 ) − Ψ(x∗ ) by a factor
103 is enough for visual purposes. The dual Nesterov approach requires 450 low cost iterations. The smoothing
method proposed by Y. Nesterov requires 1700 iterations.
The classical Cauchy steps requires much more than 5000
iterations to reach this goal. We can thus see the major
improvement of Y. Nesterov’s scheme on these problems.
We carried out many other experiments which led to the
same conclusion. Figure 2 shows the solution of the problem. The DTCW transform allows to retrieve thin details
but slightly blurs the image. Further investigations will be
led to address this issue.
6. Acknowledgments
The authors would like to thank the CS Compagny in
Toulouse (France) for partial funding of this research
work.
References:
[1] A. Beck and M. Teboulle. Fast iterative shrinkagethresholding algorithm for linear inverse problems. SIAM J. on Imaging Science, to appear.
[2] A. Bermudez and C. Moreno. Duality methods for
solving variational inequalities. Comp. and Maths.
with Appls., 7:43-58, 1981.
SAMPTA'09
Figure 2: Restoration of a down-sampled and noised image. (a) Original image, (b) down-sampled (by a factor 2)
and noised image by 10% of ”Salt & Pepper” noise and
finally (c) result of the Nesterov dual approach.
[3] R.J. Bruck. On the weak convergence of an ergodic
iteration for the solution of variational inequalities
for monotone operators in hilbert space. J. Math.
Anal. Appl, 61:159-164, 1977.
[4] J.F. Cai, R. Chan, Z.W. Shen, and L.X. Shen. Convergence analysis of tight framelet approach for
missing data recoverys. Advances in Computational
Mathematics, to appear.
[5] E. Candes, J. Romberg, and T. Tao. Robust uncertainty principles: Exact signal reconstruction from
highly incomplete frequency information. IEEE Inf.
Theory, 2006.
[6] A. Chambolle, R. Devore, N.Y. Lee, and B.J. Lucier.
Nonlinear wavelet image processing: Variational
problems, compression, and noise removal through
wavelet shrinkage. IEEE Trans. Image Processing,
7:319-335, 1998.
[7] G. Facciolo, A. Almansa, J.-F. Aujol, and Vicent
Caselles. Irregular to regular sampling, denoising
and deconvolution. SIAM Journal on Multiscale
Modeling and Simulation, in press.
[8] A. Nemirovski. Information-based complexity of
linear operator equations. Journal of Complexity,
8:153-175, 1992.
[9] Y. Nesterov. Smooth minimization of non-smooth
functions. Math. Program., 103(1):127-152, 2005.
[10] Y. Nesterov. Gradient methods for minimizing composite objective function. CORE Discussion Paper
2007/76, 2007.
[11] I. W. Selesnick, R. G. Baraniuk, and N. G. Kingsbury. The dual-tree complex wavelet transform.
IEEE Signal Processing Magazine, 22(6), Nov.
2005.
[12] P. Weiss. Algorithmes rapides d’optimisation convexe. Applications à la restauration d’images et à la
détection de changements. PhD thesis, Université de
Nice Sophia Antipolis, Dec. 2008.
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Image Inpainting Using a Fourth-Order Total Variation Flow
Carola-Bibiane Schönlieb∗
Andrea Bertozzi†
Martin Burger‡
Lin He§
April 19, 2009
Abstract
i.e., λ(x) = λ0 >> 1 in Ω \ D and 0 in D. The corresponding
steepest descent for the total variation inpainting model reads
We introduce a fourth-order total variation flow for image
ut = −p + λ(f − u), p ∈ ∂ |Du| (Ω),
(1)
inpainting proposed in [5]. The well-posedness of this new
inpainting model is discussed and its efficient numerical real- where p is an element in the subdifferential of the total variization via an unconditionally stable solver developed in [15] ation ∂ |Du| (Ω). The steepest-descent approach is used to
is presented.
numerically compute a minimizer of J , whereby it is iteratively solved until one is close enough to a minimizer of J . For
the numerical computation an element p of the subdifferential
1 Introduction
is approximatedpby the anisotropic diffusion ∇ · (∇u/|∇u|ǫ ),
where
|∇u|ǫ = |∇u|2 + ǫ.
An important task in image processing is the process of filling
in missing parts of damaged images based on the information Now, TV inpainting, while preserving edge information in
obtained from the surrounding areas. It is essentially a type of the image, fails in propagating level lines (sets of image points
interpolation and is referred to as inpainting. Given an image with constant grayvalue) smoothly into the damaged domain,
f in a suitable Banach space of functions defined on Ω ⊂ R2 , and in connecting edges over large gaps in particular. In an
an open and bounded domain, the problem is to reconstruct attempt to solve these issues from second order image diffuthe original image u in the damaged domain D ⊂ Ω, called sions, a number of third and fourth order diffusions have been
inpainting domain. In the following we are especially inter- suggested for image inpainting, e.g., [7, 9].
In this paper we present a fourth-order variant of total variested in so called non-texture inpainting, i.e., the inpainting
ation
inpainting, called TV-H−1 inpainting. The inpainted
of structures, like edges and uniformly colored areas in the
image
u of f ∈ L2 (Ω), shall evolve via
image, rather than texture.
In the pioneering works of Caselles et al. [6] (with the term
disocclusion instead of inpainting) and Bertalmio et al. [2]
partial differential equations have been first proposed for digital non-texture inpainting. In subsequent works variational
models, originally derived for the tasks of image denoising, deblurring and segmentation, have been adopted to inpainting.
The most famous variational inpainting model is the total
variation (TV) model, cf. [8, 10, 13, 14]. Here, the inpainted
image u is computed as a minimizer of the functional
ut = ∆p + λ(f − u),
p ∈ ∂T V (u),
(2)
with
(
|Du| (Ω)
T V (u) =
+∞
if |u(x)| ≤ 1 a.e. in Ω
otherwise.
(3)
This inpainting approach has been proposed by Burger, He,
and Schönlieb in [5] as a generalization of the sharp interface
limit of Cahn-Hilliard inpainting [3, 4] to grayvalue images.
The L∞ bound in the definition (3) of the total variation
1
2
functional T V (u) is motivated by this sharp interface limit
J (u) = |Du| (Ω) + kλ(f − u)kL2 (Ω) ,
2
and is part of the technical setup, which made it easier to
where |Du| (Ω) is the total variation of u (cf. [1]), and λ is the derive rigorous results for this scheme. A similar form of this
indicator function of Ω \ D multiplied by a (large) constant, higher-order TV approach already appeared in the context
∗ Department
of
Applied
Mathematics
and
Theoretical of decomposition and restoration of grayvalue images, see for
Physics (DAMTP), Centre for Mathematical Sciences, Wilber- example [12]. In the following we shall recall the main rigorforce Road, Cambridge CB3 0WA, United Kingdom.
Email: ous results obtained in [5], present an unconditionally stable
c.b.s.schonlieb@damtp.cam.ac.uk
solver for (2), and show a numerical example emphasizing the
† Department of Mathematics,UCLA (University of California Los
Angeles), 405 Hilgard Avenue, Los Angeles, CA 90095-1555, USA. superiority of the fourth-order TV flow over the second-order
Email: bertozzi@math.ucla.edu
one.
‡ Institut für Numerische und Angewandte Mathematik, Fachbereich
Mathematik und Informatik, Westfälische Wilhelms Universität (WWU)
Münster, Einsteinstrasse 62, D 48149 Münster, Germany. Email:
martin.burger@wwu.de
§ Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrasse 69, A4040 Linz, Austria. Email: lin.he@oeaw.ac.at
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2
Well-Posedness of the Scheme
In contrast to its second-order analogue, the well-posedness
of (2) strongly depends on the L∞ bound introduced in (3).
1
201
Acknowledgments
This is because of the lack of maximum principles which, in
the second-order case, guarantee the well-posedness for all
smooth monotone regularizations of p.
The existence of a steady state for (2) is given by the following theorem.
CBS acknowledges the financial support provided by project WWTF
Five senses-Call 2006,
Mathematical Methods for Image Analy-
sis and Processing in the Visual Arts, by the Wissenschaftskolleg
(Graduiertenkolleg, Ph.D. program) of the Faculty for Mathematics at
Theorem 1 [5, Theorem 1.4] Let f ∈ L2 (Ω). The stationary
equation
∆p + λ(f − u) = 0, p ∈ ∂T V (u)
(4)
the University of Vienna (funded by FWF), and by the FFG project
Erarbeitung neuer Algorithmen zum Image Inpainting, projectnumber
813610. Further, this publication is based on work supported by Award
No. KUK-I1-007-43, made by King Abdullah University of Science and
admits a solution u ∈ BV (Ω).
Technology (KAUST). For the hospitality and the financial support dur-
Results for the evolution equation (2) are a matter of future
research. In particular it is highly desirable to achieve asymptotic properties of the evolution. Note that additionally to the
fourth differential order, a difficulty in the convergence analysis of (2) is that it does not follow a variational principle.
ing parts of the preparation of this work, CBS thanks IPAM (UCLA),
the US ONR Grant N000140810363, and the Department of Defense,
NSF Grant ACI-0321917. The work of MB has been supported by the
DFG through the project Regularization with singular energies.
References
3
Unconditionally Stable Solver
[1] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded
Variation and Free Discontinuity Problems, Mathematical
Monographs, Oxford University Press, 2000.
[2] M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester, Image inpainting, Computer Graphics, SIGGRAPH 2000, July,
2000.
[3] A. Bertozzi, S. Esedoglu, and A. Gillette, Inpainting of Binary
Images Using the Cahn-Hilliard Equation. IEEE Trans. Image
Proc. 16(1) pp. 285-291, 2007.
[4] A. Bertozzi, S. Esedoglu, and A. Gillette, Analysis of a twoscale Cahn-Hilliard model for image inpainting, Multiscale
Modeling and Simulation, vol. 6, no. 3, pages 913-936, 2007.
[5] M. Burger, L. He, C. Schönlieb, Cahn-Hilliard inpainting and
a generalization for grayvalue images, UCLA CAM report 0841, June 2008.
[6] V. Caselles, J.-M. Morel, and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Processing,
7(3):376386, 1998.
[7] T.F. Chan, S.H. Kang, and J. Shen, Euler’s elastica and
curvature-based inpainting, SIAM J. Appl. Math., Vol. 63,
Nr.2, pp.564–592, 2002.
[8] T. F. Chan and J. Shen, Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math., 62(3):10191043,
2001.
[9] T. F. Chan and J. Shen, Non-texture inpainting by curvature driven diffusions (CDD), J. Visual Comm. Image Rep.,
12(4):436449, 2001.
[10] T. F. Chan and J. Shen, Variational restoration of non-flat
image features: models and algorithms, SIAM J. Appl. Math.,
61(4):13381361, 2001.
[11] D. Eyre, An Unconditionally Stable One-Step Scheme for Gradient Systems, Jun. 1998, unpublished.
[12] S. Osher, A. Sole, and L. Vese. Image decomposition and
restoration using total variation minimization and the H -1
norm, Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Vol. 1, Nr. 3, pp. 349-370, 2003.
[13] L. Rudin and S. Osher, Total variation based image restoration with free local constraints, Proc. 1st IEEE ICIP, 1:3135,
1994.
[14] L.I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation
based noise removal algorithms, Physica D, Vol. 60, Nr.1-4,
pp.259-268, 1992.
[15] C.-B. Schönlieb, and A. Bertozzi, Unconditionally stable
schemes for higher order inpainting, in preparation.
Motivated by the idea of convexity splitting schemes, e.g.,
[11], Bertozzi and Schönlieb propose in [15] the following timestepping scheme for the numerical solution of (2):
Uk+1 −Uk
∆t
+ C1 ∆∆Uk+1 + C2 Uk+1 = C1 ∆∆Uk
∇Uk
)) + C2 Uk + λ(f − Uk ),
−∆(∇ · ( |∇U
k|
(5)
ǫ
with C1 > 1/ǫ, C2 > λ0 . Here, Uk is the kth iterate of the
time-discrete scheme, which approximates a solution u of the
continuous equation at time k∆t, ∆t > 0. It can be shown
that (5) defines a numerical scheme that is unconditionally
stable, and of order 2 in time, cf. [15].
4
Numerical Results
In Figure 1 a result of the TV-H −1 inpainting model computed via (5) and its comparison with the result obtained by
the second order TV-L2 inpainting model for a crop of the
image is presented. The superiority of the fourth-order TVH −1 inpainting model to the second order model with respect
to the desired continuation of edges into the missing domain
is clearly visible.
Figure 1:
First row: TV-H−1 inpainting (2): u(1000) with λ0 = 103 .
Second row: (l.) u(1000) with TV-H−1 inpainting, (r.) u(5000) with
TV-L2 inpainting (1)
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Image Segmentation Through Efficient
Boundary Sampling
Alex Chen(1) , Todd Wittman(1) , Alexander Tartakovsky(2) , Andrea Bertozzi(1)
(1) Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095
(2) Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., KAP 108, Los Angeles, CA 90089
achen@math.ucla.edu, wittman@math.ucla.edu, tartakov@math.usc.edu, bertozzi@math.ucla.edu
Abstract:
This paper presents a combined geometric and statistical
sampling algorithm for image segmentation inspired by a
recently proposed algorithm for environmental sampling
using autonomous robots [1].
1.
Introduction
Segmentation is one of the most important problems in
image processing. Partitioning an image into a small number of homogeneous regions highlights important features,
allowing a user to analyze the image more easily. Applications include medical imaging, computer vision, and
geospatial target detection. Image segmentation methods
can be subdivided into region-based vs. edge-based methods. Region-based methods include the Mumford-Shah
[2] and related Chan-Vese [3] methods which both involve
energy minimization with a least squares fit of the data
and a partition, between regions, whose length is minimized. Edge-based methods include the well-known image snakes [4] and Canny edge detector [5]. Other approaches to segmentation have also been effective. Statistical methods such as region competition rely on the fact
that images have repetitive features that can be learned
and exploited to obtain a segmentation [6]. A more recent fast statistical method called DistanceCut [7] is semisupervised (the user identifies segments in each region)
and is based on weighted distances and kernel density estimation.
All of these methods involve, at some level, sampling all
the pixels in an image. For applications involving highdimensional or large data sets, it makes sense to subsample the image. This is especially important for high resolution data where it can be prohibitive to perform calculations on every pixel in the image. The proposed segmentation method is designed for this kind of application and
is based on ideas for cooperative environmental sampling
with robotic vehicles.
The UUV-gas algorithm [8] utilizes robots that “walk” in
a sinusoidal path along the boundary between two regions,
changing directions as they cross from one region into
another. This tracking method theoretically utilizes only
those points that are near the boundary in question, resulting in substantial savings in run-time. The sinusoidal
pattern has also been suggested as an efficient method
for atomic force microscopy scanning [9]. Interestingly,
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the same idea of tracking is behind the sinusoidal walking pattern in ants following pheromone trails [10]. As
with curve evolution methods in image processing, noise
can cause problems, since the tracking is done as a local search. It was proposed [1] that the use of a changepoint detection algorithm, e.g., Page’s cumulative sum
(CUSUM) algorithm [11] could improve tracking performance in noisy images. Testbed implementations of the
boundary tracking algorithm exploiting change-point detection methods suggested that robots can indeed track
boundaries efficiently in the presence of moderately intense noise [12]. We propose to adapt the above tracking
algorithms to the problem of segmentation, with the goal
of computational efficiency. Further improvements can be
made that are not practical in the environmental tracking
case. Many of these improvements are based on hypothesis testing for two regions, with the use of the CUSUM
algorithm as a special case.
2.
A two level sampling algorithm
The algorithm has two levels, namely a global searching method, which locates a boundary point, and a local sampling algorithm, which tracks the boundary using
the global method as an initial point. Occasionally, if the
tracker strays too far from the boundary, additional uses of
the global algorithm are needed. We briefly discuss several options for the global search and then focus on the
local sampling algorithm.
2.1
Global searching method – Locate an edge
Initialized at some point, the global search looks for some
instance of the boundary. This can be done in a few ways.
One method is simply to move out in a spiral pattern until
a boundary point is detected (see Figure 1). However, if
the boundary is small and far away from the initial point,
it may be positioned between revolutions of the spiral and
missed. Other options include deterministic paths that
do not have the tendency to miss boundaries or stochastic paths using a random walk. These searching methods assume no prior knowledge of the boundary location,
but they can be easily modified when some information is
known. Another possibility is to implement a coarse segmentation of the data first and use the resulting boundary
detection as an initialization for the local sampling. More
203
details on the last option are given later. Once a boundary point has been detected, the local sampling algorithm
begins.
2.2
Local sampling algorithm – Track an edge
In the environmental tracking problem [1, 8], a robot
tracks the boundary between two regions. The local sampling step is initialized at a point near the boundary, obtained from the global search. The robot then steers using
a bang-bang steering controller, travelling in circular arcs,
changing its direction of movement when it crosses into a
different region.
It is relatively straightforward to adapt the algorithm for an
image with domain Ω. As before, the problem is to find the
boundary B between two regions, which will be labelled
Ω1 and Ω2 , so that Ω = Ω1 ∪ Ω2 ∪ B and Ω1 ∩ Ω2 =
∅. Define an initial starting point ~x0 = (x10 , x20 ) for the
boundary tracker and an initial value θ0 , representing the
angle from the +x1 direction, so that the initial direction
vector is (cos θ0 , sin θ0 ). Also define the step size V and
angular increment ω, which depend on estimates for image
resolution and characteristics of the edge to be detected.
In general, V is chosen smaller for greater detail, and ω is
chosen smaller for straighter edges. A decision function
between Ω1 and Ω2 must also be specified and has the
following form:
if ~x ǫ Ω1 ,
1,
0,
if ~x ǫ B,
(1)
d(~x) =
−1, if ~x ǫ Ω2 .
The simplest example is thresholding of the image intensity I(~x) at a given spatial location ~x (in the case of a
grayscale image):
if I(~x) > T ,
1,
0,
if I(~x) = T ,
(2)
d(~x) =
−1, if I(~x) < T ,
where T is a fixed threshold value. Later in this section
we use statistical information about prior points sampled
along the path to modify d(~x). At each step k, the direction θk and current location ~xk are updated recursively.
Specifically, ~xk = ~xk−1 + V ∗ (cos θk−1 , sin θk−1 ) and θk
is updated according to the location of the new tracking
head ~xk . A simple update for θ is the bang-bang steering
controller, defined by
θk = θk−1 + ωd(~xk ).
(3)
An angle-correction modification [1] can be used for (3) if
step k is a region crossing:
θk = θk−1 + d(~xk )(tω − 2θref )/2,
(4)
where t is the number of steps since the last region crossing, and θref is a small fixed reference angle chosen based
on the expected curvature of the edge being tracked.
One stopping condition for the tracking of finite regions
is termination if the tracker comes within some range of
the first boundary point detected, given some minimum
number of iterations. Midpoints of line segments formed
from region crossings are labelled boundary points.
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Figure 1: Left: Global search via a spiral-like pattern. The
initial point is in blue, the final point (after a few iterations of local sampling) is in green, and the path is in red.
Right: Basic procedure for the boundary tracking (local
sampling) algorithm. The object is in cyan, the path of the
tracking head is in red, and the detected boundary points
are in yellow. Each small square represents one pixel. The
tracker travels at fractional spatial values but samples at
integral values.
While the local sampling method works well for clean images, it is susceptible to unavoidable errors in noisy images. Averaging readings from nearby pixels can minimize errors in the decision due to noise. In particular,
sequential change-point detection methods are well-suited
for detecting and tracking image edges in noise.
2.3
Decision algorithm
Change-point problems deal with detecting anomalies or
changes in statistical behavior of data. The observations
are obtained sequentially and, as long as their behavior is consistent with the normal state, one is content
to let the process continue. If the state changes, then
one is interested in detecting the change as soon as possible while minimizing false detections. More specifically, given a sequence of independent observations s1 =
I(x1 ), . . . , sn = I(xn ) and two probability density functions (pdf) f (pre-change) and g (post-change), determine
whether there exists N such that the pdf of si is f for
i < N and g for i ≥ N .
One of the most efficient change-point detection methods
is the CUSUM algorithm proposed by Page in 1954 [11].
Write Zk = log[g(sk )/f (sk )] for the log-likelihood ratio
and define recursively
Uk = max (Uk−1 + Zk , 0) , U0 = 0
(5)
the CUSUM statistic and the corresponding stopping time
τ = min{k | Uk ≥ U }, where U is a threshold controlling
the false alarm rate. Then τ is a time of raising an alarm.
In our applications, assuming that f is the pdf of the data
in Ω1 and g is the pdf in Ω2 , the value of τ may be interpreted as an estimate of the actual change-point, i.e., the
boundary crossing from Ω1 to Ω2 .
Note that if the pre-change and post-change densities f
and g are completely specified, then the CUSUM algorithm performs optimally with respect to certain performance metrics [14]. However, in our applications these
densities are usually unknown (while a Gaussian approximation may work well in certain scenarios). For this reason, the log-likelihood ratio Zk in (5) should be replaced
204
Figure 2: A 100 × 100 image was corrupted with additive Gaussian noise, N(0,0.5). Left: Boundary tracking without a
change-point detection modification. Middle: Boundary tracking with the CUSUM algorithm. Right: Threshold dynamics
[13].
Figure 3: A hybrid level set – boundary tracking segmentation on a 1000 × 1000 image. Left: Initial segmentation by
threshold dynamics. The image is subsampled by a factor of 10 on each axis. Right: Final segmentation by boundary
tracking, using points from the connected components of the initial segmentation as starting points for trackers.
by a score function Gk sensitive to expected changes.
Since we expect a change in the mean value, the appropriate score is Gk = sk − (θ1 + θ2 )/2, where θj is the
mean of the previous observations si in Ωj . The resulting score-based CUSUM test is not guaranteed to be optimal anymore. Note, however, that this score is optimal for
Gaussian distributions (i.e., when sensor noise and residual clutter may be well approximated by Gaussian processes) and can be easily adjusted to cover any member
of the exponential family of distributions (Bernoulli, Poisson, double exponential, etc.). For further details, see [15].
Changes from Ω2 to Ω1 can also be tracked in this manner.
Analogously to (5) define recursively the decision statistic
Lk = max(Lk−1 − Gk , 0), L0 = 0 and the stopping time
τ = min{k | Lk ≥ L}, where Gk is the score introduced
above, which is taken to be equal to Zk if the distributions
are known and where L is a threshold associated with a
given false detection rate.
Only one of the statistics Uk or Lk is used at a time,
i.e., when the tracking head is in Ω1 , the change-detection
statistic Uk is used for detecting a transition to Ω2 . Similarly, when the tracking head is in Ω2 , only Lk is used for
detecting a change to Ω1 . Once the tracking head enters a
new region, the other statistic is used, initialized at 0.
Note that we have implicitly assumed that the intensity
values on the path are independent observations. This assumption of independence is not entirely accurate, since
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the samples are taken from the tracking path, which is not
a random sampling of an area. However, if noise levels are
large, independence of observations is a relatively accurate
assumption due to the spatial independence of noise, while
if noise levels are small, the use of a change-detection
algorithm is less important. Furthermore, the proposed
score-based CUSUM tests are robust with respect to prior
assumptions, including independence.
3.
Boundary Tracking Examples
As mentioned above, one option for the global search is to
run a coarse segmentation on a subsampled version of the
image to obtain an initialization for the objects to be segmented. This “hybrid” method has an additional benefit of
being able to detect mutiple objects and of giving a priori
estimates for parameters in the decision function. The proposed two-stage hybrid boundary tracking algorithm that
combines the UUV-gas algorithm with the CUSUM-based
change-point detection identifies the true boundaries of an
object accurately even in high levels of noise, as seen from
Figure 2. The run-time and storage costs are minimal,
compared to most other segmentation methods.
An example of a noisy image is shown in Figure 3. The
original image is 1000 × 1000. Threshold Dynamics [13]
was first applied to a heavily subsampled version (100 ×
100) of the image. Then one pixel from each connected
205
component was taken as the starting point for a boundary
tracker. An example using multispectral data is shown in
Figure 4.
The hybrid method may be applied to more complicated
images, but some problems arise. In the first step, when
a coarse segmentation is applied to a subsampled image,
small features may not be detected accurately. These small
features will thus not be located by the boundary tracker
either. Similarly, if some features are close in space, they
may be placed in the same connected component class. In
the boundary detection step, only one feature will thus be
tracked. One solution is to use multiple initial points for
each connected component. This will result in a decrease
in efficiency but allow more objects to be tracked. Another
problem is that different objects in the image may require
different parameters to be chosen in the change-point detection algorithm. While some objects are detected accurately with certain parameters, often, some objects are
not detected completely. Multichart CUSUM tests can be
used effectively for this purpose.
Figure 4: Boundary tracking of the San Francisco Bay
coastline. A threshold of the Normalized Difference Vegetation Index (NDVI), commonly used for water detection
[16], was taken as the decision function.
4.
Discussion
The boundary tracking algorithm provides a fast alternative to many traditional segmentation methods due to its
local nature. With the addition of a change-point detection
method, the combined hybrid algorithm allows for accurate boundary tracking and, therefore, segmentation even
in highly noisy images. Furthermore, the algorithm can
operate efficiently even in data of large size or high resolution, scaling only with the size of the boundary rather than
the size of the image. While presented as a novel segmentation method, the boundary tracking algorithm can also
be used in conjunction with other segmentation methods
in a two-stage algorithm.
Acknowledgments
The authors thank C. Bachmann, Z. Hu and V. Meija. This
work was supported by the Department of Defense, ONR
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grant N000140810363, NSF ACI-0321917, ARO MURI
50363-MA-MUR.
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The Class of Bandlimited Functions with
Unstable Reconstruction under Thresholding
Holger Boche and Ullrich J. Mönich
Technische Universität Berlin, Heinrich-Hertz-Chair for Mobile Communications,
Einsteinufer 25, 10578 Berlin, Germany.
{holger.boche,ullrich.moenich}@mk.tu-berlin.de
Abstract:
The reconstruction of PW 1π -functions by sampling series
is not possible in general if the samples are disturbed by
the non-linear threshold operator which sets all samples
whose absolute value is smaller than some threshold to
zero. In this paper we characterize the set of functions for
which the sampling series diverges as the threshold goes
to zero and show that this set is a residual set.
1.
Notation
Before we start our discussion, we introduce some notations and definitions [4]. Let fˆ denote the Fourier transform of a function f , where fˆ is to be understood in the
distributional sense. Lp (R), 1 ≤ p < ∞, is the space of
all measurable, pth-power Lebesgue integrable functions
on R, with the usual norm k · kp , and L∞ (R) is the space
of all measurable functions for which the essential supremum norm k · k∞ is finite.
For σ > 0 and 1 ≤ p ≤ ∞ we denote by PW pσ the
Paley-Wiener space
R σ of functions f with a representation
f (z) = 1/(2π) −σ g(ω) eizω dω, z ∈ C, for some
g ∈ Lp (−σ, σ). If f ∈ PW pσ then g(ω) = fˆ(ω). The
norm for PW pσ , 1 ≤ p < ∞, is given by kf kPW pσ =
Rσ
(1/(2π) −σ |fˆ(ω)|p dω)1/p .
Furthermore, we need the threshold operator. For complex numbers z ∈ C, the threshold operator κδ , δ > 0, is
defined by
(
z |z| ≥ δ
κδ z =
0 |z| < δ.
For continuous functions f : R → C, we define the
threshold operator Θδ , δ > 0, pointwise, i.e., (Θδ f )(t) =
κδ f (t), t ∈ R.
2.
A well known fact [1, 2, 3] about the convergence behavior
of the Shannon sampling series for f ∈ PW 1π is expressed
by the following theorem.
Theorem (Brown). For all f ∈ PW 1π and T > 0 fixed
we have
max
N →∞ t∈[−T,T ]
SAMPTA'09
f (t) −
(Aδ f )(t) =
N
X
k=−N
f (k)
sin(π(t − k))
= 0.
π(t − k)
(1)
∞
X
f (k)
k=−∞
|f (k)|≥δ
sin(π(t − k))
.
π(t − k)
(2)
Since f ∈ PW 1π we have limt→∞ f (t) = 0 by the
Riemann-Lebesgue lemma, and it follows that the series
in (2) has only finitely many summands, which implies
Aδ f ∈ PW 2π ⊂ PW 1π . In general, Aδ f is only an approximation of f , and we want the function Aδ f to be
close to f if δ is sufficiently small.
The operator Aδ has several properties which complicate
its analysis. Aδ , δ > 0, is non-linear. Furthermore, for
each δ > 0, the operator Aδ : (PW 1π , k · kPW 1π ) →
(PW 1π , k · k∞ ) is discontinuous. This implies that Aδ :
(PW 1π , k · kPW 1π ) → (PW 1π , k · kPW 1π ) is discontinuous.
For some f ∈ PW 1π , the operator Aδ is also discontinuous
with respect to δ.
Of course (2) can be written as
∞
X
(Θδ f )(k)
k=−∞
Motivation
lim
This theorem plays a fundamental role in applications, because it establishes the uniform convergence on compact
subsets of R for a large class of functions, namely PW 1π ,
which is the largest space within the scale of Paley-Wiener
spaces.
The truncation of the series in (1) is done in the domain of the function f because only the samples f (k),
k = −N, . . . , N are taken into account. In contrast, it
is also possible to control the truncation of the series in
the codomain of f by considering only the samples f (k),
k ∈ Z, whose absolute value is larger than or equal to
some threshold δ > 0. This leads to the approximation
formula
sin(π(t − k))
,
π(t − k)
(3)
where Θδ denotes the threshold operator. Wireless sensor
networks are one possible application where the threshold
operator Θδ and the series (3) are important. The sensors sample some bandlimited signal in space and time
and then transmit the samples to the receiver. In order to
save energy, it is common to let the sensors transmit only
if the absolute value of the current sample exceeds some
threshold δ > 0. Thus, the receiver has to reconstruct the
signal by using only the samples whose absolute value is
larger than or equal to the threshold δ.
209
In addition to the sensor network scenario, the threshold
operator can be used to model non-linearities in many
other applications. For example, due to its close relation
to the quantization operator, the threshold operator can be
employed to analyze the effects of quantization in analog
to digital conversion.
3.
Problem Formulation and Main Result
Since the series in (2) uses all “important” samples of the
function, i.e., all samples that are larger than or equal to
δ, one could expect Aδ f to have an approximation behavior similar to the Shannon sampling series. In particular
the approximation error should decrease as the threshold
δ goes to zero. But, we will see that Aδ f exhibits a significantly different behavior.
In this paper we are interested in the structure of the set
and the set
D2 = {f ∈ PW 1π : lim sup|(Āδ f )(t)| = ∞ ∀ t ∈ R \ Z}.
δ→0
Both threshold operators and thus Aδ and Āδ are meaningful in practical applications, and one would expect the
difference being not important. However, as we will see,
Āδ can be analyzed more easily.
For t̂ ∈ R \ Z we furthermore define the sets
D1 (t̂) = {f ∈ PW 1π : lim sup|(Aδ f )(t̂)| = ∞}
δ→0
and
D2 (t̂) = {f ∈ PW 1π : lim sup|(Āδ f )(t̂)| = ∞}.
δ→0
Lemma 1 shows that we do not have to distinguish between the sets D1 and D1 (t̂) and between D2 and D2 (t̂).
D1 = {f ∈ PW 1π : lim sup|(Aδ f )(t)| = ∞ ∀ t ∈ R \ Z},
Lemma 1. For all t̂ ∈ R \ Z we have D1 = D1 (t̂) and
D2 = D2 (t̂).
i.e., in the structure of the set of functions for which the
approximation error |f (t) − (Aδ f )(t)| grows arbitrarily
large for all t ∈ R \ Z as δ → 0.
Proof. The inclusion D1 ⊂ D1 (t̂) is obvious. It remains
to show that D1 (t̂) ⊂ D1 . Let f ∈ D1 (t̂). For all t1 ∈
R \ Z and δ > 0 a short calculation shows that
Remark 1. The analysis of the operator Aδ is difficult because it is non-linear and discontinuous, and therefore the
standard theorems of functional analysis, like the BanachSteinhaus theorem, cannot be used.
1
1
(Aδ f )(t1 ) −
(Aδ f )(t̂)
sin(πt1 )
sin(π t̂)
∞
|t̂ − t1 | X
1
= C1 (t1 , t̂, f ),
≤ kf kPW 1π
π
|t − k||t̂ − k|
k=−∞ 1
δ→0
For the further discussion we need the following concepts
from metric spaces [5]. A subset M of a metric space X
is said to be nowhere dense in X if the closure [M ] does
not contain a non-empty open set of X. M is said to be of
the first category (or meager) if M is the countable union
of sets each of which is nowhere dense in X. M is said to
be of the second category (or nonmeager) if is not of the
first category. The complement of a set of the first category is called a residual set. Sets of first category may be
considered as “small”. According to Baire’s theorem [5]
we have that in a complete metric space, the residual set is
dense and a set of the second category. One property that
shows the richness of residual sets is the following: The
countable intersection of residual sets is always a residual set. In particular we will use the following fact in our
proof. In a complete metric space an open and dense set is
a residual set because its complement is nowhere dense.
Theorem 1 will show that the set D1 it is a residual set.
Thus the threshold operator destroys the good reconstruction behavior of the Shannon sampling series for “almost
all” functions in PW 1π .
4.
Proof of the Main Result
In addition to the threshold operator that sets all samples
whose absolute value is smaller than δ to zero, we consider
the threshold operator that sets all samples whose absolute
value is smaller than or equal to δ to zero. This operator
gives rise to the sampling series
(Āδ f )(t) :=
∞
X
k=−∞
|f (k)|>δ
SAMPTA'09
sin(π(t − k))
f (k)
π(t − k)
(4)
where C1 (t1 , t̂, f ) < ∞ is a constant that only depends on
t1 , t̂, and f . It follows that
|(Aδ f )(t1 )| ≥ |(Aδ f )(t̂)|
sin(πt1 )
− C2 (t1 , t̂, f ). (5)
sin(π t̂)
Taking the limit superior on both sides of (5) gives
lim sup|(Aδ f )(t1 )| = ∞.
(6)
δ→0
Since (6) is valid for all t1 ∈ R \ Z, it follows that f ∈ D1 .
The same calculation shows that D2 = D2 (t̂).
According to Lemma 1 it is sufficient to restrict the analysis to the sets D1 (t̂) and D2 (t̂) for some t̂ ∈ R \ Z. Furthermore, we can concentrate on one of both sets, because
of the following lemma.
Lemma 2. We have D1 = D2 .
Proof. Let f ∈ D2 (t̂) be arbitrary but fixed. By the definition of D2 (t̂), we have lim supδ→0 |(Āδ f )(t̂)| = ∞.
Thus, for every M > 0 there exists a δM > 0 such that
|(ĀδM f )(t̂)| > M . Let T (M ) = {k ∈ Z : |f (k)| > δM }
and f M = mink∈T (M ) |f (k)|. Then it follows that f M >
δM . Moreover, for all δ with f M > δ > δM we have
(Aδ f )(t̂) =
∞
X
f (k)
sin(π(t̂ − k))
π(t̂ − k)
f (k)
sin(π(t̂ − k))
= (ĀδM f )(t̂).
π(t̂ − k)
k=−∞
|f (k)|≥δ
=
∞
X
k=−∞
|f (k)|>δM
210
and consequently
Consequently,
sup|(Aδ f )(t̂)| > M.
(7)
|(Āδ̃M f )(t̂)| ≥ |(Āδ̃M f1 )(t̂)| − ǫ̃|T (M )| > M,
δ>0
Since (7) is valid for all M > 0, it follows that
supδ>0 |(Aδ f )(t̂)| = ∞, and, as a consequence,
lim supδ→0 |(Aδ f )(t̂)| = ∞, because |(Aδ f )(t̂)| < ∞
for all δ > 0. This shows that f ∈ D1 (t̂), which implies that D2 (t̂) ⊂ D1 (t̂). The converse D2 (t̂) ⊃ D1 (t̂) is
shown similarly. Hence D1 (t̂) = D2 (t̂), and the statement
D1 = D2 follows from Lemma 1.
In order to prove our main result, we need the important
Lemma 3.
Lemma 3. For all M ∈ N and t̂ ∈ R \ Z,
D2 (t̂, M ) = {f ∈ PW 1π : sup|(Āδ f )(t̂)| > M }
where the last inequality is due to (9). Therefore
sup|(Āδ f )(t̂)| > M,
δ>0
i.e., f ∈ D2 (t̂, M ), for all f ∈ PW 1π with kf1 −
f kPW 1π < ǫ̃.
Second, we show that D2 (t̂, M ) is dense in PW 1π . Let f ∈
PW 1π be arbitrary. We have to show that for every ǫ > 0
there exists a fǫ ∈ D2 (t̂, M ) such that kf − fǫ kPW 1π < ǫ.
Let ǫ > 0 be arbitrary but fixed. Since PW 2π is dense in
(1)
PW 1π , there exists a fǫ ∈ PW 2π with
δ>0
kf − fǫ(1) kPW 1π <
is a residual set.
Proof. Let M ∈ N and t̂ ∈ R \ Z be arbitrary but fixed.
First, we show that D2 (t̂, M ) is an open set. Let f1 ∈
D2 (t̂, M ) be arbitrary. We have to show that there exists an ǫ > 0 such that, given any f ∈ PW 1π with
kf − f1 kPW 1π < ǫ, f ∈ D2 (t̂, M ). By assumption, there
exists a δM > 0 such that
|(ĀδM f1 )(t̂)| > M.
Furthermore, let T (M ) = {k ∈ Z : |f1 (k)| > δM } and
f 1,M = mink∈T (M ) |f1 (k)|. Next, we choose δ̃M = δM +
(f 1,M − δM )/2. Then we have that
{k ∈ Z : |f1 (k)| > δ̃M } = T (M ).
(2)
.
(9)
For all f ∈ PW 1π with kf1 −f kPW 1π < ǫ̃ we have |f1 (k)−
f (k)| < ǫ̃, k ∈ Z. It follows, for all k ∈ Z with |f (k)| >
δ̃M , that
|f1 (k)| ≥ |f (k)| − |f (k) − f1 (k)| > δ̃M − ǫ̃ > δM ,
i.e., k ∈ T (M ). Conversely, k ∈ T (M ) implies f1 (k) ≥
f 1,M , and it follows that
2L−1
X
h(t, η, L) :=
g(t, η, L) :=h(t, η, L) −
−
k=−∞
|f (k)|>δ̃M
≤
X
k∈T (M )
SAMPTA'09
N
X
(−1)k sin(π(t − k))
|
|
π(t − k)
{z
(1 − η)
=:u1
}
(−1)k sin(π(t − k))
.
π(t − k)
{z
}
=:u2
Note that g(k, η, L) = 0 for |k| ≤ N . We have
∞
sin(π(t̂ − k))
sin(π(t̂ − k)) X
f1 (k)
−
π(t̂ − k)
π(t̂ − k)
k=−∞
|f1 (k) − f (k)|
−1
X
k=−N
(10)
|(Āδ̃M f )(t̂) − (Āδ̃M f1 )(t̂)|
f (k)
−2L < k < −L,
−L ≤ k < 0,
0 ≤ k ≤ L,
L < k < 2L,
and
Moreover, using (8) and (10), we obtain that
∞
X
sin(π(t − k))
,
π(t − k)
(−1)k (2(1 − η)+ 1−η
L k),
(−1)k (1 − η),
h(k, η, L) =
(−1)k ,
(−1)k (2 − L1 k),
> f 1,M − δ̃M + δM = δ̃M .
=
h(k, η, L)
k=0
Thus we have
(12)
where
|f (k)| ≥ |f1 (k)| − |f (k) − f1 (k)| > f 1,M − ǫ̃
{k ∈ Z : |f (k)| > δ̃M } = T (M ).
ǫ
.
3
Let N denote the smallest natural number such that N > t̂
(2)
and fǫ (k) = 0 for all |k| > N . Furthermore, let T2 =
(2)
(2)
= mink∈T2 |fǫ (k)|.
{k ∈ Z : |fǫ (k)| 6= 0} and f (2)
ǫ
For 0 < η < 1 and L ∈ N, L > N , consider the functions
h and g defined by
k=−2L+1
|(Āδ̃M f1 )(t̂)| − M
, δ̃M − δM
|T (M )|
(2)
kfǫ(1) − fǫ(2) kPW 1π <
We choose
ǫ̃ < min
(11)
Moreover, there exists a fǫ ∈ PW 2π such that fǫ (k) 6=
0 only for finitely many k ∈ Z and
(8)
!
ǫ
.
3
kg(t, η, L)kPW 1π
≤ kh( · , η, L)kPW 1π + ku1 kPW 1π + ku2 kPW 1π . (13)
|f1 (k)|>δM
sin(π(t̂ − k))
≤ ǫ̃|T (M )|
π(t̂ − k)
The norm ku1 kPW 1π is upper bounded by
ku1 kPW 1π <
π
+ log(N + 1),
2
(14)
211
Observing that N − t̂ > 0, we obtain
because
1
2π
ku1 kPW 1π =
Z
Z
N
X
π
e−iωk (−1)k dω
−π k=0
iω(N
+1)
e
Z
1 π sin( N2+1 ω)
1−
1
=
dω
=
dω
2π −π
1 − eiω
π 0
sin( ω2 )
Z N +1
Z π
sin( N2+1 ω)
sin( π2 ω)
dω =
dω
≤
ω
ω
0
0
Z
Z 1
N +1
sin( π2 ω)
π
1
dω +
dω < + log(N + 1).
≤
ω
ω
2
0
1
π
A similar calculation gives
ku2 kPW 1π <
(15)
kh( · , η0 (L), L)kPW 1π < 4.
(16)
Combining (13)–(16) gives, that for all L ∈ N, L > N
there exists an 0 < η0 (L) < 1 such that
kg( · , η0 (L), L)kPW 1π < 4 + π + 3 log(N + 1) =: C3 .
It is important that the constant C3 does not depend on L.
Next, we analyze
Gǫ (t, L) =
+ µg(t, η0 (L), L),
where µ > 0 is some real number that satisfies µ <
). By the choice of µ we have
min(ǫ/(3C3 ), f (2)
ǫ
kfǫ(2) − Gǫ ( · , L)kPW 1π = µC3 <
ǫ
3
(17)
for all L > N . Combining (11), (12), and (17), we see
that
kf − Gǫ ( · , L)kPW 1π < ǫ
(18)
for all L > N , i.e., Gǫ ( · , L) lies in the ǫ-ball around f .
Furthermore, for any L > N we can find a δ0 (L) that
fulfills
1
µ < δ0 (L) < µ.
max (1 − η0 (L))µ, 1 −
L
Since δ0 (L) < f (2)
, by the definition of µ, it follows that
ǫ
(Āδ0 (L) Gǫ ( · , L))(t̂)
N
X
=
Gǫ (k, L)
k=−N
|Gǫ (k,L)|>δ0 (L)
L
X
+
Gǫ (k, L)
k=N +1
|Gǫ (k)|>δ0 (L)
=
N
X
fǫ(2)(k)
k=−N
SAMPTA'09
sin(π(t̂ − k))
π(t̂ − k)
sin(π(t̂ − k))
π(t̂ − k)
L
X
(−1)k sin(π(t̂ − k))
sin(π(t̂ − k))
+µ
π(t̂ − k)
π(t̂ − k)
k=N +1
= fǫ(2) (t̂) + µ
L
sin(π t̂) X 1
.
π
t̂ − k
k=N
k=N
L−N
X
1
1
=
t̂ − k
k
+
N
− t̂
k=0
L−N
X Z k+1
1
≥
dτ
τ
+
N
− t̂
k
k=0
Z L−N +1
1
=
dτ
τ
+
N
− t̂
0
L − t̂
,
> log
N − t̂
and consequently
π
+ log(N ).
2
In addition we have kh( · , 0, L)kPW 1π ≤ 3, which
can be proven easily, and limη→0 kh( · , η, L) −
Therefore, there exists an
h( · , 0, L)kPW 1π = 0.
0 < η0 (L) < 1 such that
fǫ(2) (t)
L
X
|(Āδ0 (L) Gǫ ( · , L))(t̂)|
≥µ
|sin(π t̂)|
log
π
L − t̂
N − t̂
− |fǫ(2) (t̂)|. (19)
The right-hand side of (19) can be made arbitrarily large
by choosing L large. Let L1 > N be the smallest L such
that the right hand side of (19) is larger than M . It follows
that fǫ (t) = Gǫ (t, L1 ) is the desired function, because
supδ>1 |(Āδ fǫ )(t̂)| ≥ |(Āδ0 (L1 ) fǫ )(t̂)| > M , i.e., fǫ ∈
D2 (t̂, M ), and because kf − fǫ kPW 1π < ǫ, according to
(18).
Theorem 1. D1 and D2 are residual sets.
Proof. Since D2 = D1 , by Lemma 2, it is sufficient to
show that D2 is a residual set.
Let t̂ ∈ R \ Z be arbitrary but fixed. We have
\
D2 (t̂, M ).
D2 (t̂) =
M ∈N
From Lemma 3 we know that all D2 (t̂, M ), M ∈ N,
are residual sets. It follows that D2 (t̂) is a residual set,
because the countable intersection of residual sets is a
residual set. The application of Lemma 1 completes the
proof.
References:
[1] J. L. Brown, Jr. On the error in reconstructing a nonbandlimited function by means of the bandpass sampling theorem. Journal of Mathematical Analysis and
Applications, 18:75–84, 1967. Erratum, ibid, vol. 21,
1968, p. 699.
[2] P. L. Butzer, W. Splettstößer, and R. L. Stens. The
sampling theorem and linear prediction in signal analysis. Jahresber. d. Dt. Math.-Verein., 90(1):1–70, January 1988.
[3] P. L. Butzer and R. L. Stens. Sampling theory for
not necessarily band-limited functions: A historical
overview. SIAM Review, 34(1):40–53, March 1992.
[4] John R. Higgins. Sampling Theory in Fourier and Signal Analysis – Foundations. Oxford University Press,
1996.
[5] Kôaku Yosida. Functional Analysis. Springer-Verlag,
1971.
212
On Subordination Principles for Generalized
Shannon Sampling Series
Andi Kivinukk (1) and Gert Tamberg (2)
(1) Dept. of Math., Tallinn University, Narva Road 25, 10120 Tallinn, Estonia
(2) Dept. of Math., Tallinn University of Technology, Ehitajate tee 5 19086 Tallinn, Estonia
andik@tlu.ee, gert.tamberg@mail.ee
Abstract:
This paper provides some subordination equalities and
their applications for the generalized Shannon sampling
series.
Concerning some direct (Jackson-type) approximation
theorems we present certain subordination equalities,
which show that the sampling operators, like Rogosinski,
Zygmund, and Hann, are in some sense basic.
1.
2.
Introduction
For the uniformly continuous and bounded functions f ∈
C(R) the generalized Shannon sampling series (see [3]
and references cited there) are given by (t ∈ R; W > 0)
(SW f )(t) :=
∞
X
k=−∞
f(
k
)s(W t − k),
W
(1)
where the condition for the operator SW : C(R) → C(R)
to be well-defined is that for the kernel function s = s(t)
we assume
∞
X
|s(u − k)| < ∞
(u ∈ R).
k=−∞
Let be given an even window function λ ∈ C[−1,1] ,
λ(0) = 1, λ(u) = 0 (|u| > 1,) then in our approach the
kernel function will be defined by the equality
s(t) := sλ (t) :=
Z1
λ(u) cos(πtu) du.
(2)
Many window functions have been used in applications
(see, e.g. [1], [2], [4], [8]), in Signal Analysis in particular. Next window functions are important for our subordination equalities.
1) λ(r) (u) = 1 − ur , r ≥ 1 defines the Zygmund
(or Riesz) kernel, denoted by zr = zr (t), which special case r = 1, the Fejér (or Bartlett, see [8]) kernel
sF (t) = 21 sinc 2 2t , is well-known; the special case r = 2
is called also as the Welch [8] kernel;
2) λj (u) := cos π(j + 1/2)u, j = 0, 1, 2, . . . defines the
Rogosinski-type kernel (see [5]) in the form
j
3) λH (u) := cos2
kernel (see [6])
(−1) (j + 1/2) cos πt
;
π (j + 1/2)2 − t2
πu
2
(3)
= 21 (1 + cos πu) defines the Hann
1 sinct
.
sH (t) :=
2 1 − t2
SAMPTA'09
Subordination equalities state some relations between two
sampling operators.
2.1 Subordination by the Rogosinski-type sampling series
Let consider the Rogosinski-type sampling operators
RW,j defined by the kernel functions rj in (3). These
kernel functions are deduced by the window functions
λj (u) := cos π(j + 1/2)u, (j ∈ N ) and as a family of
functions it forms an orthogonal system on [0, 1]. Therefore, we may represent a quite arbitrary window function
λ by its Fourier series. But the Fourier representation allows us to prove for a given kernel function s the sampling
series
∞
X
s(t) = 2
s(j + 1/2) rj (t).
j=0
0
rj (t) :=
Subordination equalities
(4)
Bσp
In following
stands for the Bernstein class, it consists
of those bounded functions f ∈ Lp (R) (1 6 p 6 ∞),
which can be extended to an entire function f (z) (z ∈ C)
of exponential type σ. For s ∈ Bπ1 the sampling series
above is absolutely convergence and by (1) we get formally the equalities
SW f = 2
∞
X
s(j + 1/2)RW,j f,
j=0
f − SW f = 2
∞
X
s(j + 1/2)(f − RW,j f ),
j=0
calling as the subordination equalities, since the approximation properties of the general sampling operators (1)
can be described via the approximation properties of the
Rogosinski-type sampling operators RW,j : C(R) →
C(R). We have proved that [5]
kRW,j k =
2j
2
4X 1
= log(j + 1) + O(1),
π
2ℓ + 1
π
ℓ=0
213
2.3 Subordination by the Zygmund sampling series
thus the subordination equalities are valid, when
∞
X
|s(j + 1/2)| log(j + 1) < ∞.
j=0
Similar subordination equalities can be deduced for some
interpolating sampling series, i.e. for which the equation
k
k
(S̃W f )( W
) = f(W
) (k ∈ Z) is valid. In [7] we have
proved that the interpolating sampling operators will be
defined by (1) using the kernel s̃(t) := 2s(2t), where the
kernel s is generated by (2) with a window function λ for
which λ(u) + λ(1 − u) = 1 (u ∈ [0, 1]).
α
Let the operator SW
: C(R) → C(R) be defined by the
1
kernel sα := α s(α·) ∈ Bαπ
(0 < α ≤ 2), where s ∈
1
α
Bπ , and the modified Hann operator HW,j
is defined by
the kernel
(5)
Then here we have (see [7], Th. 2.3 and 2.4)
α
SW
f =4
∞
X
λ(u) = 1 −
α
s(2j + 1)HW,j
f,
∞
X
cj uj .
j=r
Then the formal subordination equalities are in the shape
SW f =
∞
X
j
cj ZW
f,
j=r
f − SW f =
2
(2j + 1)
α
sα
sinc(αt).
H,j (t) :=
2 (2j + 1)2 − (αt)2
r
The Zygmund sampling operator ZW
will be defined by
the window function λ(r) (u) = 1 − ur , r ≥ 1. Let us
consider the kernel s in (2), for which the corresponding
window function has the power series representation
∞
X
j
cj (f − ZW
f ).
j=r
Several other subordination equalities and their applications will be presented.
3. Acknowledgments
j=0
α
f − SW
f =4
∞
X
α
f ).
s(2j + 1)(f − HW,j
j=0
2.2 Subordination by the Rogosinski-type sampling series: 2D case
The two-dimensional generalized sampling series has the
form
(SW f )(x, y)
∞
X
k l
f ( , )s(W x − k, W y − l),
:=
W W
k,l=−∞
in particular, the multiplicative Rogosinski-type sampling
series we define as
(RW ;i,j f )(x, y)
∞
X
k l
:=
f ( , )ri (W x − k)rj (W y − l),
W W
k,l=−∞
where the Rogosinski-type kernel rj is defined by (3).
Here our subordination equalities read as
SW f = 4
∞
X
s(i + 1/2, j + 1/2)RW ;i,j f,
i,j=0
f − SW f = 4
∞
X
s(i + 1/2, j + 1/2)(f − RW ;i,j f ),
i,j=0
provided
∞
X
|s(i + 1/2, j + 1/2)| log i log j < ∞.
i,j=1
By given subordination equalities we see that the nonmultiplicative sampling series may be studied by the multiplicative Rogosinski-type sampling series.
SAMPTA'09
This research was partially supported by the Estonian
Sci. Foundation, grants 6943, 7033, and by the Estonian
Min. of Educ. and Research, projects SF0132723s06,
SF0140011s09.
References:
[1] H. H. Albrecht. A family of cosine-sum windows for
high resolution measurements. In IEEE International
Conference on Acoustics, Speech and Signal Processing, Salt Lake City, Mai 2001, pages 3081–3084. Salt
Lake City, 2001.
[2] R. B. Blackman and J. W. Tukey. The measurement of
power spectra. Wiley-VCH, New York, 1958.
[3] P. L. Butzer, G. Schmeisser, and R. L. Stens. An introduction to sampling analysis. In F Marvasti, editor,
Nonuniform Sampling, Theory and Practice, pages
17–121. Kluwer, New York, 2001.
[4] F. J. Harris. On the use of windows for harmonic analysis. Proc. of the IEEE, 66:51–83, 1978.
[5] A. Kivinukk and G. Tamberg. On sampling series
based on some combinations of sinc functions. Proc.
of the Estonian Academy of Sciences. Physics Mathematics, 51:203–220, 2002.
[6] A. Kivinukk and G. Tamberg. On sampling operators
defined by the Hann window and some of their extensions. Sampling Theory in Signal and Image Processing, 2:235–258, 2003.
[7] A. Kivinukk and G. Tamberg. Interpolating generalized Shannon sampling operators, their norms and approximation properties. Sampling Theory in Signal
and Image Processing, 8:77–95, 2009.
[8] E. H. W. Meijering, W. J. Niessen, and M. A.
Viergever. Quantitative evaluation of convolutionbased methods for medical image interpolation. Medical Image Analysis, 5:111–126, 2001.
214
Linear Signal Reconstruction
from Jittered Sampling
Alessandro Nordio (1) , Carla-Fabiana Chiasserini (1) and Emanuele Viterbo (2)
(1) Dipartimento di Elettronica, Politecnico di Torino1 , I-10129 Torino, Italy.
(2) DEIS, Università della Calabria, via P. Bucci, Cubo 42C, 87036 Rende (CS), Italy
alessandro.nordio@polito.it, carla.chiasserini@polito.it, viterbo@deis.unical.it
Abstract:
This paper presents an accurate and simple method to
evaluate the performance of AD/DA converters affected
by clock jitter, which is based on the analysis of the
mean square error (MSE) between the reconstructed signal and the original one. Using an approximation of the
linear minimum MSE (LMMSE) filter as reconstruction
technique, we derive analytic expressions of the MSE.
Through asymptotic analysis, we evaluate the performance of digital signal reconstruction as a function of the
clock jitter, number of quantization bits, signal bandwidth
and sampling rate.
1.
Introduction
A significant problem in Analog Digital Conversion
(ADC) of wide-band signals is clock jitter and its impact
on the quality of signal reconstruction. Indeed, even small
amounts of jitter can measurably degrade the performance
of analog to digital and digital to analog converters.
Clock jitter is typically detrimental because the analog to
digital process relies upon a sample clock to indicate when
a sample or snapshot of the analog signal is taken. The
sample clock must be evenly spaced in time; any deviation will result in a distortion of the digitization process.
If one had a perfect ADC and a perfect DAC and used the
same clock to drive both units, then jitter would not have
any impact on the reconstructed signal. In a real world
system, however, a digitized signal travels through multiple processors, usually it is stored on a disk or piece of
tape for a while, and then goes through more processing
before being converted back to analog. Thus, during reconstruction, the clock pulses used to sample the signal
are replaced with newer ones with their own subtle variations. Jitter may have different probability distributions
which may have different effects on the quality of the reconstructed signal.
While several results are available in the literature on jittered sampling [4, 5] as well as on experimental measurements and instruments performance [1, 3, 6, 7], an analytical methodology for the performance study of the AD/DA
conversion is still missing.
In this paper we fill this gap and propose a method for evaluating the performance of AD/DC converters affected by
This work was supported by Regione Piemonte through the VICSUM
project.
SAMPTA'09
jitter, which is based on the analysis of the mean square error (MSE) between the reconstructed signal and the original one [7].
As reconstruction technique, we consider linear filtering
methods, which typically have low complexity and are
used in a wide variety of fields. If jitter were known exactly, the linear minimum MSE (LMMSE) reconstruction
technique would be optimal, since it minimizes the MSE
of the reconstructed signal. In practice this is not the case,
hence we apply a reconstruction filter with the same structure of the LMMSE filter, where we let the jitter vanish.
Then, we apply asymptotic analysis to derive analytical
expressions of the MSE on the quality of the reconstructed
signal. We then show that our asymptotic expressions provide an excellent approximation of the MSE even for small
values of the system parameters, with the advantage of
greatly reducing the computation complexity. We apply
our method to study the performance of the AD/DA conversion system as a function of the clock jitter, number of
quantization bits, signal bandwidth and sampling rate.
2.
System model
Throughout the paper we use the following notations. Column vectors are denoted by bold lowercase letters and matrices are denoted by bold upper case letters. The (k, q)th entry of the generic matrix Z is denoted by (Z)k,q .
The n × n identity matrix is denoted by In , while I is
the generic identity matrix. (·)T is the transpose operator, while (·)† is the conjugate transpose operator. We denote by fx (z) the probability density function (pdf) of the
generic random variable x, and by E[·] the average operator.
2.1
Signal sampling and reconstruction
We consider an analog signal s(t) sampled at constant rate
fs = 1/Ts over the finite interval [0, M Ts ). Ts is the
sample spacing. When observed over a finite interval, s(t)
admits an infinite Fourier series expansion. Let N ′ denote
the largest index of the non-negligible Fourier coefficients,
then N ′ /Ts can be considered as the approximate onesided bandwidth of the signal. We therefore represent the
signal by using a truncated Fourier series with N = 2N ′ +
215
2.2
1 complex harmonics as
′
N
t
1 X
aℓ exp j2πℓ
,
s(t) = √
M Ts
N ℓ=−N ′
(1)
0 ≤ t < M Ts . The vector a = [a−N ′ , . . . , a0 , . . . , aN ′ ]T
represents the complex discrete spectrum of the signal.
Observe that the signal representation given in (1) includes
sine waves of any fractional frequency f0 = fs N ′ /M
(when aℓ = 0 for −N ′ < ℓ < N ′ and a−N ′ = a∗N ′ ),
which are frequently used as reference signal for calibration of ADC [1, 2]. We note that when the signal s(t)
is observed in the frequency domain through its M samples, the spectral resolution is given by ∆f = 1/(M Ts ).
Therefore, considering the expression in (1), the signal
= 2MNTs . By defining
bandwidth is given by B = N ∆f
2
the parameter
M
β=
(2)
N
as the oversampling factor of the signal s(t) with respect
to the Nyquist rate, we can also write:
B=
fs /2
β
(3)
In this work, we consider that sampling locations suffer
from jitter, i.e., the m-th sampling location is given by
tm = mTs + dm ,
(4)
m = 0, . . . , M − 1, where dm is the associated independent random jitter whose distribution is denoted by fd (z).
Typically, we have |dm | ≪ Ts .
Let the signal samples be s = [s0 , . . . , sM −1 ]T where
sm = s(tm ), 0 ≤ m ≤ M − 1. Using (1), the set of
signal samples can be written as
s = V† a
where V is an N × M random Vandermonde matrix defined as
1
tm
(V)ℓ,m = √ exp −j2πℓ
(5)
M Ts
N
ℓ = −N ′ , . . . , N ′ , and m = 0, . . . , N − 1. Note that V
accounts for the jitter in the AD/DA conversion process,
and that the parameter β defined in (2) also represents the
aspect ratio of matrix V.
Furthermore, in addition to jittered sampling, we assume
that signal samples are affected by some additive noise and
are therefore given by
y =s+n
where n is a vector of M noise samples, modeled as zero
mean i.i.d. random variables. In practice, the dominant
additive noise error is due to the n-bit quantization process
[10].
In order to reconstruct the signal we consider a reconstruction technique that provides an estimate â of the discrete
spectrum a. The reconstruction ŝ(t) of s(t) obtained from
â is given by
N′
t
1 X
âℓ exp j2πℓ
ŝ(t) = √
M Ts
N
′
ℓ=−N
SAMPTA'09
Reconstruction error
We consider as performance metric of the AD/DA conversion process the mean square error (MSE) associated to
the estimate. The MSE, evaluated in the observation interval [0, M Ts ), can be equivalently computed in both time
and frequency domains as:
#
"Z
M Ts
E ka − âk2
2
MSE = E
|s(t) − ŝ(t)| dt =
N
0
More specifically, we consider the MSE relative to the signal average power, i.e.,
J=
MSE
σa2
which can be thought of as a noise to signal ratio and will
be plotted using a dB scale in our results.
Among the possible techniques that can be applied to reconstruct the original signal, we focus on linear filters
that provide an estimate of a through the linear operation
â = By where B is an N × M matrix.
3.
Jittered AD/DA conversion with linear filtering
Let us assume kak2 = σa2 N and E[nn† ] = σn2 I, then we
define the signal to noise ratio (SNR) in absence of jitter
as
σ2
γ = a2
σn
Under the assumption that E[aa† ] = σa2 I, the linear filter
that provides the best performance in terms of MSE is the
linear minimum mean square error (LMMSE) filter, which
is given by
−1
1
Bopt = VV† + I
V
(6)
γ
In [8], it has been shown that, by applying the LMMSE
filter, we obtain:
h n
−1 oi
1
J = 2 E ka − âk2 = E tr γVV† + I
σa N
where tr{·} is the normalized matrix trace operator and
the average is over the randomness in V.
Note, however, that the filter in (6) cannot be employed in
practice, since the jitters dm (hence the matrix V) are unknown (see the definition of V in (5)). We therefore resort
to an approximation of the optimum filter Bopt , based on
the assumption that jitter has a zero mean.
In particular, we approximate V with the matrix F defined
as,
F = V|dm =0
with the generic√element of F given by, (F)ℓ,m =
m
/ N , ℓ = −N ′ , . . . , N ′ , and m =
exp −j2πℓ M
0, . . . , N − 1. We observe that F is such that: FF† = βI
and it is related to the discrete Fourier transform matrix.
Substituting the approximation of V in (6), we obtain:
−1
1
F
(7)
B= β+
γ
216
Notice that the filter in (7) is the LMMSE filter adapted
to the linear model y = F† a + n. By letting ω =
(β + 1/γ)−1 , the noise to signal ratio J provided by the
approximate filter (7) is given by
1
2
=
E ka − ωFyk
σa2 N
†
†
2
†
= tr ω E FV VF − 2ωℜ{E FV }
J
where, from (3), we used the fact that 1/βTs = fs /β =
2B. Similarly, we define
′
µ2
2
N
1 X
2πℓ
=
lim
Cd
N,M →+∞ N
M Ts
′
ℓ=−N
β
=
Z
1/2
2
|Cd (4πBx)| dx
(11)
−1/2
d
d
(8)
By using (10) (11), and (9), the asymptotic expression of
J is given by
where the operator E[·] averages over the random jitters
(β,γ)
J∞
= 1+ω 2 β(1+1/γ)−2ωβµ1 +ω 2 β(β−1)µ2 (12)
2
+1 +
ω β
γ
d
dm , m = 0, . . . , M − 1.
Assuming the jitters to be independent [1] and with characteristic function Cd (w) = E[exp(jwz)], the first two
d
terms in (8) are given by
It is worth mentioning that for large SNRs (i.e., in absence
(β,γ)
of measurement noise), J∞ reduces to
1
1
(β)
(β,γ)
J∞
= lim J∞
= 1 + − 2µ1 + 1 −
µ2 (13)
γ→∞
β
β
′
N
2πℓ
β X
†
Cd
tr E FV =
N
M Ts
d
′
Equation (13) provides us with a floor that represent the
best quality of the reconstructed signal (minimum MSE)
we can hope for.
ℓ=−N
′
N
2πℓ
(β − 1) X
†
†
C
FV
VF
=
β
+
β
E
d
N
M Ts
d
′
2
4.1
ℓ=−N
Hence, we can write:
J
=
1
1+ω β 1+
γ
2
+ω 2 β
(β − 1)
N
′
N
β X
2πℓ
− 2ω
Cd
N
M Ts
′
ℓ=−N
N′
X
Cd
ℓ=−N ′
2πℓ
M Ts
2
(9)
In order to reduce the complexity of the computation of
the reconstruction error and provide simple but accurate
analytical tools, in the next section we let the parameters
N and M go to infinity, while the ratio β = M/N is kept
constant. We therefore derive an asymptotic expression of
J, which we will show well approximates the expression
in (9) even for small N and M .
4.
Asymptotic analysis
When N and M grow to infinity while β is kept constant,
we define the asymptotic noise to signal ratio J as:
(β,γ)
J∞
=
lim
N,M →+∞
β
(β,γ)
′
=
N
1 X
2πℓ
Cd
N,M →+∞ N
M Ts
′
=
Z
lim
β
SAMPTA'09
ℓ=−N
1/2
−1/2
Let us now assume the jitter to be uniformly distributed
with pdf given by
1
−dmax ≤ z ≤ dmax
2dmax
fd (z) =
0
elsewhere
where dmax is the maximum jitter, independent of
the sampling frequency fs . In this case, the characteristic function of the jitter is given by Cd (w) =
sin(dmax w)/(dmax w). Then,
µ1 =
and
µ2 =
Si(2πηu )
2πηu
cos2 (2πηu ) + 2πηu Si(4πηu ) − 1
4π 2 ηu2
where Si(·) is the integral sine function and ηu = dmax B
is a dimensionless parameter which relates maximum jitter
and signal bandwidth.
5.
Results
J
In [8], it has been shown that J∞
provides an excellent approximation of MSE/σa2 even for small values of
N and M , with the advantage of greatly simplifying the
computation.
In the limit N, M → ∞ with constant β, we compute
µ1
Example: uniform jitter distribution
Cd (4πBx) dx
(10)
For the ease of representation, we assume that the dominant component of the additive noise is due to quantization, and we express the SNR in absence of jitter, γ,
as a function of the number of quantization bits n of the
ADC [9]:
(γ)dB = 6.02n + 1.76
Then, in the following plots we show the value of J as a
function of γ or, equivalently, of the number of quantization bits n.
Figure 1 compares the value of J obtained through its
asymptotic expression against the performance of a system with finite parameters values (i.e., the value of J
computed using (9)). The results are derived for ηu =
217
10−1 , 10−2 , 10−3 , and β = 10. Solid lines refer to the
asymptotic expression (12), while markers represent the
values of J computed through (9), with N ′ = 100. We
observe an excellent matching between our approximation
(β,γ)
of J∞ and the results computed through (9), even for
small values of N and M . We point out that this tight
match can be observed for any β > 1 and ηu ≪ 1.
We also notice that J shows a floor, whose expression is
given by (13). This floor is due to the mismatch between
the filter F employed in the reconstruction and the matrix
V characterizing the sampling system.
10
20
30
40
γ [dB]
50
60
0
80
90
Approx.
-1
ηu = 10
ηu = 10-2
ηu = 10-3
Floor
-10
-20
J [dB]
70
Conclusions
We studied the performance of AD/DA converters, in presence of clock jitter and quantization errors. We considered
that a linear filter approximating the LMMSE filter is used
for signal reconstruction, and evaluated the system performance in terms of MSE. Through asymptotic analysis, we
derived analytical expressions of the MSE which provide
an accurate and simple method to evaluate the behavior of
AD/DA converters as clock jitter, number of quantization
bits, signal bandwidth and sampling rate vary. We showed
that our asymptotic approach provides an excellent approximation of the MSE even for small values of the system parameters. Furthermore, we derived the MSE floor,
which represents the best reconstruction quality level we
can hope for and gives useful insights for the design of
AD/DA converters.
References:
-30
-40
-50
-60
-70
2
4
6
8
10
12
14
16
n [bit]
Figure 1: Comparison between the reconstruction error J
(β,γ)
derived through (9), the approximation of J∞ and the
(β)
floor J∞ in (13).
Furthermore, in the case of unknown jitter, and, thus, of a
floor in the behavior of J, there exists a number of quantization bits n = n∗ beyond which a further increase in
the ADC precision does not provide a noticeable decrease
in the reconstruction error J. The relation between ηu , β,
and n∗ is shown in Figure 2. Note that n∗ is lightly affected by an increase of β, provided that β > 1, and a
good compromise for choosing the oversampling rate is
β = 5.
-2
10
-3
10
10-4
ηu
6.
10-5
-6
10
β=1
β=2
β=5
β=10
β=100
10-7
10-8
6
8
10
12
14
16
18
20
22
24
[1] Project DYNAD, SMT4-CT98, Draft Standard Version 3.4, Jul. 12, 2001.
[2] IEEE Standard for Terminology and Test Methods
for Analog-to-Digital Converters, IEEE Std. 1241,
2000.
[3] P. Arpaia, P. Daponte, and S. Rapuano, “Characterization of digitizer timebase jitter by means of the
Allan variance,” Computer Standards & Interfaces,
Vol. 25, pp. 15–22, 2003.
[4] B. Liu, and T. P. Stanley, “Error bounds for jittered
sampling,” IEEE Transactions on Automatic Control, Vol. 10, No. 4, pp. 449–454, Oct. 1965.
[5] J. Tourabaly, and A. Osseiran, “A jittered-sampling
correction technique for ADCs,” IEEE International
Workshop on Electronic Design, Test and Applications, pp. 249–252, Los Alamitos, CA, USA, 2008.
[6] E. Rubiola, A. Del Casale, and A. De Marchi, “Noise
induced time interval measurement biases,” 46th
IEEE Frequency Control Symposium, pp. 265–269,
May 1992.
[7] J. Verspecht, “Accurate spectral estimation based on
measurements with a distorted-timebase digitizer,”
IEEE Trans. on Instrumentation and Measurement
Vol. 43, pp. 210–215, Apr. 1994.
[8] A. Nordio, C.-F. Chiasserini, and E. Viterbo “Performance of linear field reconstruction techniques with
noise and uncertain sensor locations,” IEEE Trans.
on Signal Processing, Vol. 56, No. 8, pp. 3535–3547,
Aug. 2008.
[9] G. Gielen, “Analog building blocks for signal processing,” ESAT-MICAS, Leuven, Belgium, 2006.
[10] S. C. Ergen, and P. Varaiya, “Effects of A-D conversion nonidealities on distributed sampling in dense
sensor networks,” IPSN ’06, Nashville, Tennessee,
Apr. 2006.
*
n [bit]
Figure 2: Minimum number of bits n∗ required to reach
(β,γ)
the floor of J∞ as a function of β and ηu .
SAMPTA'09
218
Uniform Sampling and Reconstruction of
Trivariate Functions
Alireza Entezari
E301 CSE Building, University of Florida, Gainesville, FL, USA.
entezari@cise.ufl.edu
Abstract:
The Body Centered Cubic (BCC) and Face Centered Cubic (FCC) lattices have been known to outperform the
commonly-used Cartesian sampling lattice due to their
improved spectral sphere packing properties. However,
the Cartesian lattice has been widely used for sampling
of trivariate functions with applications in areas such as
biomedical imaging, scientific data visualization and computer graphics. The widespread use of Cartesian lattice is
partly due to the availability of tensor-product approach
that readily extend the univariate reconstruction methods
to trivariate setting. In this paper we report on recent advances on non-separable reconstruction algorithms, based
on box splines, for reconstruction of data sampled on the
BCC and FCC lattices. It turns out that these box spline
reconstructions are faster than the corresponding tensorproduct B-spline reconstructions on the Cartesian lattice.
This suggests that not only the BCC and FCC lattices are
more accurate sampling patterns, their respective reconstruction methods are also more computationally efficient
than the tensor-product reconstructions – a fact which is
contrary to the common assumption among practitioners.
1.
Introduction
Sampling and reconstruction play a vital role in visualization and computer graphics. Various volume rendering
algorithms rely on accurate reconstruction as a key step
since the quality and fidelity of the rendered image heavily depends on reconstruction. In image processing reconstruction is used in resampling, resizing, conversion, and
manipulation of sampled data.
In the realm of sampling, the term regular is often used to
refer to the case that the sampling grid is uniform. Although there has been significant research, recently, in
non-uniform sampling (e.g., sparse sampling, compressed
sensing), the regular sampling is the most commonly-used
sampling scheme in practice [21].
When it comes to sampling multivariate functions, the
tensor-product of uniform sampling, which forms a Cartesian lattice, is almost always the choice. The simple structure of the Cartesian lattice and its separable nature allows
one to readily apply a tensor-product paradigm to many
problems in a multi-dimensional setting. The power of
the dimensionality reduction will remain the major reason
that the Cartesian lattice is the preferred tool in numerical
SAMPTA'09
algorithms. The other attraction of the Cartesian lattice is
that it simply exists in any dimension and often tools and
theory extend to problems in a higher dimensional setting
in a trivial manner.
However, the Cartesian lattice has been known to be an inefficient lattice from the sampling-theoretic point of view.
Miyakawa [12] and then Petersen and Middleton [16]
were among the first people to discover the superiority of
sphere-packing and sphere-covering lattices for sampling
multivariate functions. In particular they have demonstrated that Cartesian lattice is very inefficient for sampling multivariate functions.
2.
Optimal Sampling Lattices
When sampling a multivariate function with a lattice, generated by (integer linear combinations of the columns of)
a sampling matrix, M , the spectrum of the signal is contained in the Brillouin zone. Brillouin zone is the Voronoi
cell of the reciprocal lattice. The reciprocal lattice to
the lattice M is generated by the columns of the matrix
2πM −⊤ . The multivariate version of the Nyquist frequency is the boundary of the Brillouin zone.
Without a priori knowledge when sampling multivariate
functions, one often assumes that the underlying function
has features possibly in all directions. Therefore, without
knowledge about particular orientations of high-frequency
features, we need to capture an isotropic spectrum during
the sampling process. Therefore, the objective of optimal
sampling is to maximize the isotropic content of the Brillouin zone. In other words, the sampling lattice whose
Brillouin zone has the largest inscribing (hyper) sphere is
the best sampling lattice. Therefore, the optimal sampling
lattice in any dimension is the lattice whose reciprocal lattice allows for the densest packing of spheres.
In the bivariate setting the hexagonal lattice is the best
sampling lattice since its reciprocal lattice, which happens
to be the dual hexagonal lattice, allows for the best packing of 2-D with disks. When compared to the commonlyused Cartesian lattice with the same sampling density, the
hexagonal lattice allows for about 14% more information
to be captured in the spectrum of the underlying signal.
This is illustrated in Figure 1 as the area of inscribing
disc to the Brillouin zone of the hexagonal lattice (i.e.,
hexagon) is larger than the area of inscribing disc to the
Brillouin zone of the Cartesian lattice (i.e., square), even
219
rH
rH
rC
Figure 1: A square and a hexagon with unit area corresponding to the Brillouin zone of Cartesian and hexagonal sampling. The area of inscribing disk to a square is
about 14% less than the area of the inscribing disk to the
hexagon.
though the two Brillouin zones have the same area.
In the trivariate setting, the optimal sampling lattice is
the BCC lattice whose reciprocal lattice (i.e., the FCC
lattice) is the densest sphere packing lattice. The sampling efficiency of the BCC lattice, when compared to
the commonly-used Cartesian lattice is about 30% higher.
Appendix A in [6] presents a thorough comparison of the
Brillouin zone of the Cartesian, BCC and FCC lattices.
The FCC lattice, is also superior to the Cartesian lattice
as its efficiency compared to the Cartesian lattice is about
27% higher. Although among the FCC and BCC lattices
the BCC wins, by a small margin, for optimal sampling,
the FCC lattice appears to have good resistance to aliasing. This can be justified since its reciprocal lattice (i.e.,
the BCC lattice) allows for the best sphere covering of the
space. The best covering of the space translates to replication of isotropic spectrum with minimal overlap between
them– minimizing the aliasing for that sampling resolution.
These facts about comparison of the Cartesian, BCC
and FCC lattices together with their higher-dimensional
counter parts are discussed for sampling stationary
isotropic random processes [10]. The arguments of the optimal sampling (BCC) and resilience to aliasing (FCC) is
generalized to the notion that the reciprocal lattice for optimal sphere-packing lattice is the best choice for sampling
functions at relatively high resolutions, while the spherepacking lattice is the best option for sampling functions at
relatively low resolutions [10].
3.
Reconstruction
There is abundant research on reconstruction (i.e., interpolation or approximation) of data based on univariate filtering methods [15]. Various 1-D filters have a low-pass
behavior and approximate the ideal kernel (i.e., sinc) for
reconstruction into the space of band-limited functions. Bsplines, offer a framework for representation of piecewise
polynomial functions and thus are widely used in reconstruction of univariate functions [3].
There are two common methods for extending the univariate reconstruction ‘kernels’ to multivariate setting. The
separable approach builds the multivariate kernel by a
simple tensor-product of univariate kernels. The separable approach is obviously suitable for reconstruction of
data on the Cartesian lattice since the lattice itself is also
separable. The radial basis approaches construct the multivariate reconstruction kernel by spherical extension of
SAMPTA'09
univariate kernel. Due to the spherical extension, the radial basis approach ignores the underlying geometry of the
sampling lattice and is often used for scattered data interpolation/approximation.
Splines have been widely accepted for image processing [20]. In the context of image processing, splines are
often constructed as a tensor-product of two univariate
splines. Mitchell and Netravali [11], demonstrated the
advantages of using splines for image processing. Recently, Van De Ville [22], developed the so called Hexsplines that are used for reconstruction of hexagonal images. Hex-splines can not be constructed as a tensorproduct of univariate splines. Due to the non-separable
structure of hexagonal lattice, the tensor-product splines
can not be applied for processing of hexagonal data.
3.1
Reconstruction of trivariate functions
In the visualization community reconstruction filters have
received a lot of attention since accurate reconstruction
of trivariate functions and their gradients is crucial in fidelity of rendering algorithms [14, 1, 5, 13]. Similar to
image processing, in volume visualization algorithms, often the tensor-product approach is used for reconstruction
of Cartesian sampled data.
Theußl [18] introduced the BCC sampling in volume
rendering. However, since the BCC lattice is a nonseparable lattice, various ad-hoc tensor-product [17] and
radial basis [18] algorithms fail to provide satisfactory reconstruction algorithms and they exhibit blurry artifacts.
Csébfalvi [2] proposed a global pre-processing algorithm
(based on generalized interpolation [19]) that reconstructs
the BCC lattice based on its two Cartesian sub-lattices.
This approach is computationally inefficient and does not
guarantee approximation order.
The author’s recent work in this area establishes the relationship between box splines and the above-mentioned
sampling lattices. The box splines have been developed
as a generalization of B-splines to the multivariate setting.
While box splines have been considered as non-separable
basis functions for approximation based on their shifts on
the Cartesian lattice [4], here their shifts on BCC and FCC
lattices are considered. The interesting fact about these
box splines is that while their shifts on the Cartesian lattice do not form a linearly independent set of functions,
their shifts on the FCC and BCC lattices are linearly independent – a rare and useful property for the spline space!
3.2
Four direction box splines on BCC
The relation of box splines with the BCC lattice was established based on the fact that the immediate neighborhood of a lattice point on the BCC pattern forms a rhombic dodecahedron (see Figure 2). This polyhedron has the
special property that is a projection of a four-dimensional
hypercube (tesseract). This makes it a perfect match to be
the support of a box spline since the geometric definition
of box splines precisely amounts to projecting hypercubes
(i.e., box) down to lower dimensional spaces. Generally,
the class of polytopes that are the shadow of higher dimensional hypercubes are referred to as zonotopes. This
linear box spline is defined by the four direction and is
220
z
z
y
y
x
x
Figure 5: The neighborhood of a FCC lattice point forms
a truncated octahedron. This polyhedron is another zonohedron which is the support of a six-direction box spline.
Figure 2: The neighborhood of a BCC lattice point forms a
rhombic dodecahedron. This polyhedron is a zonohedron
which is the support of a linear box spline.
Figure 3: Benchmark example dataset. The CT dataset of
a carp fish at a high resolution of 256 × 256 × 256.
a C 0 kernel. The shifts of this box spline on the BCC
lattice generate a spline space whose approximation order is two. By convolving this box spline by itself, one
obtains a smoother, C 2 , quintic box spline that is specified by a repetition of the four principal directions. The
shifts of this box spline generate a spline space whose approximation order is four [7, 8]. This smoothness and approximation order match that of the tricubic B-spline on
the Cartesian lattice and hence we compare the two on a
Carp fish dataset in first row in Figure 4. The piecewise
polynomial representation of these box splines along with
efficient evaluation methods can be found in [8].
3.3
The six direction box spline on FCC
Unlike the BCC lattice, the immediate neighborhood in
the FCC lattice is not a zonohedron. However, by enlarging the neighborhood one finds the truncated octahedron
which is a zonohedron Figure 5. This polyhedron is a
projection of a six-dimensional hypercube and the corresponding box spline is a cubic six-direction box spline [6].
The spline space that is generated by shifts of this cubic
box spline on the FCC lattice is a C 1 space whose approximation order is three. These characteristics match
the triquadratic B-spline on the Cartesian lattice which is
the base for our comparisons in second row in Figure 4.
The piecewise polynomial representation of the cubic box
spline along with efficient spline evaluation method on the
FCC lattice is demonstrated in [9].
SAMPTA'09
3.4
Computational advantages
Once efficient evaluation algorithms are derived for the
four-direction box splines [8] and the six direction box
spline [9], one can compare these box spline reconstructions to the commonly-used tensor-product B-spline reconstructions on the Cartesian lattice.
For the C 2 , fourth-order method the tricubic B-spline uses
a neighborhood of 4 × 4 × 4 = 64 points for reconstruction, while the quintic box spline only uses a total
of 32 points for reconstruction. Therefore as documented
in [8] the BCC non-separable box spline approach outperforms the comparable tensor-product B-spline approach
by a factor of two. Similarly the triquadratic B-spline
uses a neighborhood of 3 × 3 × 3 = 27 Cartesian data
points, while the cubic box spline only requires a total
of 16 FCC data points for the reconstruction. Therefore,
the non-separable box spline reconstruction outperforms
the comparable tensor-product B-spline approach as documented in [9].
4.
Conclusions
The recent research on optimal sampling lattices suggests
that not only the FCC and BCC lattices offer higherfidelity sampling schemes, but also their reconstruction
algorithms outperform the corresponding tensor-product
reconstructions on the traditionally-popular Cartesian lattice. These encouraging results are crucial for acceptance
of these efficient lattices in practical applications.
5.
Acknowledgments
The author would like to thank Dimitri Van De Ville,
Torsten Möller and Carl de Boor for valuable insight and
advice at various stages of the work.
References:
[1] I. Carlbom. Optimal Filter Design for Volume Reconstruction and Visualization. In Proc. IEEE Conf
on Visualization, pages 54–61, October 1993.
[2] B. Csébfalvi. Prefiltered gaussian reconstruction for
high-quality rendering of volumetric data sampled
221
Cartesian, C 2 , fourth order
BCC, C 2 , fourth order
Cartesian, C 1 , third order
FCC, C 1 , third order
Figure 4: The Carp dataset at 6% resolution on Cartesian, BCC and FCC subsampled from the ground truth volume data
of Figure 3. Top row: the Cartesian dataset is reconstructed by the tricubic B-spline and the BCC dataset is reconstructed
by the quintic box spline. Bottom row: the Cartesian dataset is reconstructed with the triquadratic B-spline, while the
FCC dataset is reconstructed with the cubic box spline. Superiority of the FCC and the BCC sampling is demonstrated
since their images offer more accurate reconstruction than the Cartesian specially on the ribs and tail area.
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
on a body-centered cubic grid. In IEEE Visualization, pages 311–318, 2005.
C. de Boor. A practical guide to splines, volume 27
of Applied Mathematical Sciences. Springer-Verlag,
New York, revised edition, 2001.
C. de Boor, K. Höllig, and S. Riemenschneider. Box
Splines. Springer Verlag, 1993.
S. C. Dutta Roy and B. Kumar. Handbook of
Statistics, volume 10, chapter Digital Differentiators,
pages 159–205. Elsevier Science Publishers B. V., N.
Holland, 1993.
A. Entezari. Optimal Sampling Lattices and Trivariate Box Splines. PhD thesis, Simon Fraser University, Vancouver, Canada, July 2007.
A. Entezari, R. Dyer, and T. Möller. Linear and Cubic Box Splines for the Body Centered Cubic Lattice.
In Proceedings of the IEEE Conference on Visualization, pages 11–18, October 2004.
A. Entezari, D. Van De Ville, and T. Möller. Practical box splines for volume rendering on the body
centered cubic lattice. IEEE Trans. on Visualization
and Comp Graphics, 14(2):313 – 328, 2008.
M. Kim, A. Entezari, and J. Peters. Box Spline
Reconstruction on the Face Centered Cubic Lattice.
IEEE Trans. on Visualization and Computer Graphics, 14(6):1523–1530, 2008.
HR Kunsch, E. Agrell, and FA Hamprecht. Optimal lattices for sampling. Information Theory, IEEE
Transactions on, 51(2):634–647, 2005.
D. P. Mitchell and A. N. Netravali. Reconstruction
Filters in Computer Graphics. In Computer Graphics
(Proceedings of SIGGRAPH 88), volume 22, pages
221–228, August 1988.
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stochastic variables in multidimensional space. Journal of the Institute of Electronic and Communication
Engineers of Japan, 42:421–427, 1959.
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[13] T. Möller, R. Machiraju, K. Mueller, and R. Yagel. A
Comparison of Normal Estimation Schemes. In Proceedings of the IEEE Conference on Visualization,
pages 19–26, October 1997.
[14] T. Möller, K. Mueller, Y. Kurzion, R. Machiraju, and
R. Yagel. Design of Accurate and Smooth Filters
for Function and Derivative Reconstruction. Proceedings of the Symposium on Volume Visualization,
pages 143–151, Oct 1998.
[15] A.V. Oppenheim and R.W. Schafer. Discrete-Time
Signal Processing. Prentice Hall Inc., Englewoods
Cliffs, NJ, 1989.
[16] D. P. Petersen and D. Middleton. Sampling and Reconstruction of Wave-Number-Limited Functions in
N -Dimensional Euclidean Spaces. Information and
Control, 5(4):279–323, December 1962.
[17] T. Theußl, O. Mattausch, T. Möller, and E. Gröller.
Reconstruction schemes for high quality raycasting
of the body-centered cubic grid. TR-186-2-02-11,
Institute of Computer Graphics and Algorithms, Vienna University of Technology, December 2002.
[18] T. Theußl, T. Möller, and E. Gröller. Optimal Regular Volume Sampling. In Proc of the IEEE Conf on
Visualization, pages 91–98, Oct 2001.
[19] P. Thévenaz, T. Blu, and M. Unser. Interpolation
revisited. IEEE Transactions on Medical Imaging,
19(7):739–758, July 2000.
[20] M. Unser. Splines: A perfect fit for signal and image processing. IEEE Signal Processing Magazine,
16(6):22–38, November 1999. IEEE Signal Processing Society’s 2000 magazine award.
[21] M. Unser. Sampling—50 Years after Shannon. Proceedings of the IEEE, 88(4):569–587, April 2000.
[22] D. Van De Ville, T. Blu, M. Unser, W. Philips,
I. Lemahieu, and R. Van de Walle. Hex-Splines: A
Novel Spline Family for Hexagonal Lattices. IEEE
Trans. on Img Proc., 13(6):758–772, June 2004.
222
1
An Efficient Algorithm for the Discrete Gabor
Transform using full length Windows
Peter L. Søndergaard
Abstract—This paper extends the efficient factorization of the
Gabor frame operator developed by Strohmer in [17] to the
Gabor analysis/synthesis operator. The factorization provides a
fast method for computing the discrete Gabor transform (DGT)
and several algorithms associated with it. The factorization
algorithm should be used when the involved window and signal
have the same length. An optimized implementation of the
algorithm is freely available for download.
I. I NTRODUCTION
The finite, discrete Gabor transform (DGT) of a signal f of
length L is given by
c (m, n, w) =
L−1
X
l=0
f (l, w)g (l − an)e−2πiml/M .
II. D EFINITIONS
We shall denote the set of integers between zero and some
number L by
hLi = 0, . . . , L − 1.
(2)
The Discrete Fourier Transform (DFT) of a signal f ∈ CL
is defined by
(1)
Here g is a window (filter prototype) that localizes the signal
in time and in frequency. The DGT is equivalent to a Fourier
modulated filter bank with M channels and decimation in time
a, [2].
Efficient computation of a DGT can be done by several
methods: If the window g has short support (consists of
relatively few filter taps), a filter bank based approach can
be used. We shall instead focus on the case when g and f are
equally long. The main advantage of the algorithm presented
is its ease of use: The running time is guaranteed to be small
even for long windows. This allows for the practical use of
non-compactly supported windows like the Gaussian and its
tight and dual windows without truncating them.
In the case when the window and signal have the same
length, a factorization of the frame operator matrix was found
by Zibulski and Zeevi in [19]. The method was initially
developed in the L2 (R) setting, and was adapted for the
finite, discrete setting by Bastiaans and Geilen in [1]. They
extended it to also cover the analysis/synthesis operator. A
simple, but not so efficient, method was developed for the
Gabor analysis/synthesis operator by Prinz in [15]. Strohmer
[17] improved the method and obtained the lowest known
computational complexity for computing the Gabor frame
operator. This paper extends Strohmer’s method to also cover
the Gabor analysis and synthesis operators.
The advantage of the method developed in this paper as
compared to the one developed in [1], is that it works with
FFTs of shorter length, and does not require multiplication by
complex exponentials caused by the quasi-periodicity of the
Zak transform. The two methods have the same asymptotic
complexity, O (N M log M ), where M is the number of channels and N is the number of time steps. A more accurate flop
count is presented later in the paper.
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We shall study the DGT applied to multiple signals at once.
This is for instance a common subroutine in computing a
multidimensional DGT. The DGT defined by (1) works on
a multi-signal f ∈ CL×W , where W ∈ N is the number of
signals.
(FL f ) (k) =
L−1
1 X
√
f (l)e−2πikl/L .
L l=0
(3)
We shall use the · notation in conjunction with the DFT to
denote the variable over which the transform is to be applied.
To denote all elements indexed by a variable we shall use
the : notation. As an example, if C ∈ CM ×N then C:,1 is a
M × 1 column vector, C1,: is a 1 × N row vector and C:,: is
the full matrix. This notation is commonly used in Matlab
and FORTRAN programming and also in some prominent
textbooks, [8].
The convolution f ∗ g of two functions f, g ∈ CL and the
involution f ∗ is given by
(f ∗ g) (l) =
L−1
X
k=0
f (k) g (l − k) ,
f ∗ (l) = f (−l),
l ∈ hLi
l ∈ hLi .
(4)
(5)
It is well known how convolution can be computed efficiently
using the discrete Fourier transform. We shall use a variant of
this result
√ −1
LFL (FL f ) (·) (FL g) (·) (l) . (6)
(f ∗ g ∗ ) (l) =
The Poisson summation formula in the finite, discrete setting is given by
!
b−1
X
√
FM
g(· + kM ) (m) =
b (FL g) (mb), (7)
k=0
where g ∈ CL , L = M b with b, M ∈ N.
A family of vectors ej , j ∈ hJi of length L is called a
frame if constants 0 < A ≤ B exist such that
2
A kf k ≤
J−1
X
j=0
2
2
|hf, ej i| ≤ B kf k ,
∀f ∈ CL .
223
(8)
2
Algorithm 1 Window factorization
WFAC (g, a, M )
1) for r = hci k = hpi, l = hqi
2)
for s = hdi
3)
tmp (s) ←
g (r + c · (k · q − l · p + s · p · q mod d · p · q))
4)
end for
5)
P hi (r, k, l, :) ←DFT(tmp)
6) end for
7) return Phi
The constants A and B are called lower and upper frame
bounds. If A = B, the frame is called tight. If J > L,
the frame is redundant (oversampled). Finite- and infinite
dimensional frames are described in [4].
A finite, discrete Gabor system (g, a, M ) is a family of
vectors gm,n ∈ CL of the following form
gm,n (l) = e2πilm/M g (l − na) ,
l ∈ hLi
(9)
for m ∈ hM i and n ∈ hN i where L = aN and M/L ∈ N. A
Gabor system that is also a frame is called a Gabor frame. The
analysis operator Cg : CL 7→ CM ×N associated to a Gabor
system (g, a, M ) is the DGT given by given by (1). The Gabor
synthesis operator Dγ : CM ×N 7→ CL associated to a Gabor
system (γ, a, M ) is given by
f (l) =
N
−1 M
−1
X
X
n=0 m=0
c (m, n) e2πiml/M γ (l − an) .
(10)
In (1), (9) and (10) it must hold that L = N a = M b for some
M, N ∈ N. Additionally, we define c, d, p, q ∈ N by
c = gcd (a, M ) , d = gcd (b, N ) ,
(11)
a
b
M
N
p= =
, q=
= ,
(12)
c
d
c
d
where GCD denotes the greatest common divisor of two
natural numbers. With these numbers, the redundancy of the
transform can be written as L/ (ab) = q/p, where q/p is an
irreducible fraction. It holds that L = cdpq. The Gabor frame
operator Sg : CL 7→ CL of a Gabor frame (g, a, M ) is given
by the composition of the analysis and synthesis operators
Sg = Dg Cg . The Gabor frame operator is important because it
can be used to find the canonical dual window g d = Sg−1 g and
−1/2
g of a Gabor frame.
the canonical tight window g t = Sg
The canonical dual window is important because Dgd is a left
inverse of Cg . This gives an easy way to construct an inverse
transform of the DGT. Similarly, then Dgt is a left inverse of
Cgt . For more information on Gabor systems and properties
of the operators C, D and S see [9], [6], [7].
III. T HE ALGORITHM
We wish to make an efficient calculation of all the coefficients of the DGT. Using (1) literally to compute all
coefficients c (m, n, w) would require 8M N LW flops.
To derive a faster DGT, one approach is to consider the
analysis operator Cg as a matrix, and derive a faster algorithm
SAMPTA'09
Algorithm 2 Discrete Gabor transform
DGT (f, g, a, M )
1) P hi =WFAC(g, a, M )
2) for r = hci
3)
for k = hpi, l = hqi, w = hW i
4)
for s = hdi
5)
tmp (s) ←
f (r + (k · M + s · p · M − l · ha · a mod L) , w)
6)
end for
7)
P sitmp (k, l + w · q, ·) ←DFT(tmp)
8)
end for
9)
for s = hdi
10)
G ← P hi (:, :, r, s)
11)
F ← P sitmp (:, :, s)
12)
Ctmp (:, :, s) ← GT · F
13)
end for
14)
for u = hqi, l = hqi, w = hW i
15)
tmp ←IDFT(Ctmp (u, l + w · q, :))
16)
for s = hdi
17)
coef (r + l · c, u + s · q − l · ha mod N, w)
← tmp (s)
18)
end for
19)
end for
20) end for
21) for n = hN i,w = hW i
22)
coef (:, n, w) ←DFT(coef (:, n, w))
23) end for
24) return coef
through unitary matrix factorizations of this matrix. This is
the approach taken by [17], [16]. Unfortunately, this approach
tends to introduce many permutation matrices and Kronecker
product matrices. Another approach is the one taken in [1]
where the Zak transform is used. This approach has the
downside that values outside the fundamental domain of the
Zak transform require an additional step to compute. In this
paper we have chosen to derive the algorithm by directly
manipulating the sums in the definition of the DGT.
To find a more efficient algorithm than (1), the first step is
to recognize that the summation and the modulation term in
(1) can be expressed as a DFT:
√
c (m, n, w) = LFL f (·, w)g (· − an) (mb) .
(13)
We can improve on this because we do not need all the
coefficients computed by the Fourier transform appearing in
(13), only every b’th coefficient. Therefore, we can rewrite by
the Poisson summation formula (7):
c (m, n, w)
!
b−1
X
√
=
M FM
f (· + m̃M, w)g (· + m̃M − an) (m)
m̃=0
=
(FM K (·, n, w)) (m) ,
(14)
where
K (j, n, w) =
√
M
b−1
X
m̃=0
f (j + m̃M, w) g (j + m̃M − na) , (15)
224
3
for j ∈ hM i and n ∈ hN i. From (14) it can be seen that
computing the DGT of a signal f can be done by computing
K followed by DFTs along the first dimension of K.
To further lower the complexity of the algorithm, we wish
to express the summation in (15) as a convolution.
We split j as j = r + lc with r ∈ hci, l ∈ hqi and introduce
ha , hM ∈ Z such that the following is satisfied:
c = hM M − ha a.
(16)
The two integers ha , hM can be found by the extended Euclid
algorithm for computing the GCD of a and M .
Using (16) and the splitting of j we can express (15) as
K (r + lc, n, w)
b−1
√ X
=
M
f (r + lc + m̃M, w) ×
(18)
K (r + lc, n − lha , w)
b−1
√ X
=
M
f (r + lc + (m̃ − lhM ) M, w) ×
m̃=0
×g (r + m̃M − na)
(19)
b−1
√ X
M
f (r + m̃M + l (c − hM M ) , w) ×
=
m̃=0
(20)
We split m̃ = k + s̃p with k ∈ hpi and s̃ ∈ hdi and n =
u + sq with u ∈ hqi and s ∈ hdi and use that M = cq, a = cp
and c − hM M = −ha a:
K (r + lc, u + sq − lha , w)
p−1 X
d−1
X
M
f (r + kM + s̃pM − lha a, w) ×
=
√
k=0 s̃=0
×g (r + kM − ua + (s̃ − s) pM )
(21)
After having expressed the variables j, m̃, n using the variables r, s, s̃, k, l, u we have now indexed f using s̃ and g
using (s̃ − s). This means that we can view the summation
over s̃ as a convolution, which can be efficiently computed
using a discrete Fourier transform. Define
Ψfr,s (k, l + wq) = Fd f (r + kM + ·pM − lha a, w) , (22)
√
Φgr,s (k, u) = M Fd g (r + kM + ·pM − ua) , (23)
Using (6) we can now write (21) as
K (r + lc, u + s̃q − lha , w)
p−1
√ X
d
Fd−1 Ψfr,· (k, l + wq) Φgr,· (k, u) (s̃) (24)
=
k=0
=
√
dFd−1
p−1
X
k=0
SAMPTA'09
!
[1]
Alg. 2
L
8L ag + 4N M log2 (M )
”
“
”
q
+ 4L 1 + pq log2 N + 4M N log2 (M )
p
“
”
L (8q) + 4L 1 + pq log2 d + 4M N log2 (M )
“
L 8q + 1 +
Flop counts for 4 different way of computing the DGT: By the linear algebra
definition (1), by the method based on Poisson summation (14), by the method
of Bastiaans and Geilen from [1] and by Algorithm 2. The term Lg denotes the
length of the window used so Lg /a is the overlapping factor of the window.
Note for comparison that log2 N = log2 d + log2 q
IV. RUNNING TIME
We substitute m̃ + lhM by m̃ and n + lha by n and get
×g (r + m̃M − na)
Eq. (14)
Flop count
8M N L
(17)
m̃=0
×g (r + (m̃ + lhM ) M − (n + lha ) a)
Alg.:
Eq. (1)
If we consider Ψfr,s and Φgr,s as matrices for each r and s, the
sum over k in the last line can be written as matrix products.
Algorithm 2 follows from this.
m̃=0
×g (r + l (hM M − ha a) + m̃M − na)
b−1
√ X
M
f (r + lc + m̃M, w) ×
=
Table I
F LOP COUNTS
Ψfr,· (k, l + wq) Φgr,· (k, u) (s̃) (25)
When computing the flop count of the algorithm, we will
assume that a complex FFT of length M can be computed
using 4M log2 M flops. A nice review of flop counts for
FFT algorithms is presented in [14]. Table I shows the flop
count for Algorithm 2 and compares it with the definition
of the DGT (1), with the algorithm for short windows using
Poisson summation (14) and with the algorithm published
in [1]. The algorithm by Prinz presented in [15] has the
same computational complexity as the Poisson summation
algorithm. For simplicity we assume that both the window and
signal are complex valued. In the common case when both f
and g are real-valued, all the algorithms will see a 2 to 4 times
speedup.
The flop count for definition (1) is that of a complex
matrix multiplication. All the other algorithms share the
4M N log2 M term coming from the application of an FFT
to each ’block’ of coefficients and only differ in how the application of the window is performed. The Poisson summation
algorithm is very fast for a small overlapping factor Lg /a, but
turns into an O L2 algorithm for a full length window. In
this
algorithms have an advantage. The term
case the other
L 8q + 1 + pq in the [1] algorithm comes from calculation
of the needed Zak-transforms, and the 4L 1 + pq log2 N
term comes from the transform to and from the Zak-domain.
Compared to (22) and (23) this transformation uses longer
FFTs. Algorithm 2 does away with the multiplication with
complex exponentials in the [1] algorithm, and so the first
term reduces to L (8q).
Both the Poisson summation based algorithm and Algorithm
2 can do a DGT with L ≈ 2000000 in less than 1 second on
a standard PC at the time of writing. We have not created
an efficient implementation of the algorithm from [1] in C so
therefore we cannot reliably time it.
V. E XTENSIONS
The algorithm just developed can also be used to calculate
the synthesis operator Dγ . This is done by applying Algorithm
225
4
Algorithm 3 Canonical Gabor dual window
GABDUAL (g, a, M )
1) P hi =WFAC(g, a, M )
2) for r = hci, s = hdi
3)
G ← P hi (:, :, r, s)
−1
·G
4)
P hid (:, :, r, s) ← G · GT
5) end for
6) g d =IWFAC P hid , a, M
7) return g d
2 in the reverse order and inverting each line. The only lines
that are not trivially invertible are lines 10-12, which becomes
10) Γ ← P hid (:, :, r, s)
11) C ← Ctmp (:, :, s)
12) P sitmp (:, :, s) ← Γ · C
where the matrices P hid (:, :, r, s) should be left inverses of
the matrices P hi (:, :, r, s) for each r and s.
The matrices P hid (:, :, r, s) can be computed by Algorithm
1 applied to a dual Gabor window γ of the Gabor frame
(g, a, M ). It also holds that all dual Gabor windows γ of a
Gabor frame (g, a, M ) must satisfy that P hid (:, :, r, s) are
left inverses of the matrices P hi (:, :, r, s). This criterion was
reported in [11], [12].
A special left-inverse in the Moore-Penrose pseudo-inverse.
Taking the pseudo-inverses of P hi (:, :, r, s) yields the factorization associated with the canonical dual window of
(g, a, M ), [3]. This is Algorithm 3. Taking the polar decomposition of each matrix in Φgr,s yields a factorization of the
canonical tight window (g, a, M ). For more information on
these methods, as well as iterative methods for computing the
canonical dual/tight windows, see [13].
VI. S PECIAL CASES
We shall consider two special cases of the algorithm:
The first case is integer oversampling. When the redundancy
is an integer then p = 1. Because of this we see that c = a
and d = b. This gives (16) the appearance
a = hM qa − ha a,
(26)
indicating that hM = 0 and ha = −1 solves the equation for
all a and q. The algorithm simplifies accordingly, and reduces
to the well known Zak-transform algorithm for this case, [10].
The second case is the short time Fourier transform. In this
case a = b = 1, M = N = L, c = d = 1, p = 1, q = L and
as in the previous special case hM = 0 and ha = −1. In this
case the algorithm reduces to the very simple and well known
algorithm for computing the STFT.
VII. I MPLEMENTATION
The reason for defining the algorithm on multi-signals, is
that the multiple signals can be handled at once in the matrix
product in line 12 of Algorithm 2. This is a matrix product
of two matrices size q × p and p × qW , so the second matrix
grows when multiple signals are involved. Doing it this way
reuses the Φgr,s matrices as much as possible, and this is an
SAMPTA'09
advantage on standard, general purpose computers with a deep
memory hierarchy, see [5], [18].
The benefit of expressing Algorithm 2 in terms of loops (as
opposed to using the Zak transform or matrix factorizations)
is that they are easy to reorder. The presented Algorithm 2
is just one among many possible algorithms depending on in
which order the r, s, k and l loops are executed. For a given
platform, it is difficult a priory to estimate which ordering of
the loops will turn out to be the fastest. The ordering of the
loops presented in Algorithm 2 is the variant that uses the least
amount of extra memory.
Implementations of the algorithms described in this paper
can be found in the Linear Time Frequency Toolbox (LTFAT)
available from http://ltfat.sourceforge.net. The implementations are done in both the Matlab/Octave scripting language
and in C. A range of different variants of Algorithm 2 has been
implemented and tested, and the one found to be the fastest
on a small range of computers is included in the toolbox.
R EFERENCES
[1] M. J. Bastiaans and M. C. Geilen. On the discrete Gabor transform and
the discrete Zak transform. 49(3):151–166, 1996.
[2] H. Bölcskei, F. Hlawatsch, and H. G. Feichtinger. Equivalence of DFT
filter banks and Gabor expansions. In SPIE 95, Wavelet Applications in
Signal and Image Processing III, volume 2569, part I, San Diego, july
1995.
[3] O. Christensen. Frames and pseudo-inverses. J. Math. Anal. Appl.,
195:401–414, 1995.
[4] O. Christensen. An Introduction to Frames and Riesz Bases. Birkhäuser,
2003.
[5] J. Dongarra, J. Du Croz, S. Hammarling, and I. Duff. A set of level 3
basic linear algebra subprograms. ACM Trans. Math. Software, 16(1):1–
17, 1990.
[6] H. G. Feichtinger and T. Strohmer, editors. Gabor Analysis and
Algorithms. Birkhäuser, Boston, 1998.
[7] H. G. Feichtinger and T. Strohmer, editors. Advances in Gabor Analysis.
Birkhäuser, 2003.
[8] G. H. Golub and C. F. van Loan. Matrix computations, third edition.
John Hopkins University Press, 1996.
[9] K. Gröchenig. Foundations of Time-Frequency Analysis. Birkhäuser,
2001.
[10] A. J. E. M. Janssen. The Zak transform: a signal transform for sampled
time-continuous signals. Philips Journal of Research, 43(1):23–69,
1988.
[11] A. J. E. M. Janssen. On rationally oversampled Weyl-Heisenberg frames.
pages 239–245, 1995.
[12] A. J. E. M. Janssen. The duality condition for Weyl-Heisenberg frames.
In Feichtinger and Strohmer [6], chapter 1, pages 33–84.
[13] A. J. E. M. Janssen and P. L. Søndergaard. Iterative algorithms to
approximate canonical Gabor windows: Computational aspects. J.
Fourier Anal. Appl., published online, 2007.
[14] S. Johnson and M. Frigo. A Modified Split-Radix FFT With Fewer
Arithmetic Operations. IEEE Trans. Signal Process., 55(1):111, 2007.
[15] P. Prinz. Calculating the dual Gabor window for general sampling sets.
IEEE Trans. Signal Process., 44(8):2078–2082, 1996.
[16] S. Qiu. Discrete Gabor transforms: The Gabor-gram matrix approach.
J. Fourier Anal. Appl., 4(1):1–17, 1998.
[17] T. Strohmer. Numerical algorithms for discrete Gabor expansions. In
Feichtinger and Strohmer [6], chapter 8, pages 267–294.
[18] R. C. Whaley, A. Petitet, and J. Dongarra. Automated empirical
optimization of software and the ATLAS project. Technical Report UTCS-00-448, University of Tennessee, Knoxville, TN, Sept. 2000.
[19] Y. Y. Zeevi and M. Zibulski. Oversampling in the Gabor scheme. IEEE
Trans. Signal Process., 41(8):2679–2687, 1993.
226
Nonstationary Gabor Frames
Florent Jaillet (1) , Peter Balazs (1) and Monika Dörfler (1)
(1) Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, A-1040 Vienna,Austria
florent@kfs.oeaw.ac.at, peter.balazs@oeaw.ac.at, monid@kfs.oeaw.ac.at
Abstract:
Introduction
Gabor analysis [7] is widely used for applications in signal
processing. For some of these applications, which include
processing of signals using Gabor frame multipliers [6, 1],
the rigid construction of the Gabor atoms results in important limitations on the signal analysis and processing
ability of the associated schemes. The Gabor transform
uses time-frequency atoms built by translation over time
and frequency of a unique prototype function, leading to
a signal decomposition having a fixed time-frequency resolution over the whole time-frequency plane. This can be
very restricting when dealing with signals with characteristics changing over the time-frequency plane. For example, this led some people to prefer the use of alternative
decompositions with time-frequency resolution evolving
with frequency in some applications, to better fit the feature of interest of the signal. Examples of such decompositions are the wavelet transform [5] or the decompositions
using filter banks based on perceptive frequency scales for
processing of audio signals, as for example gammatone
filters [9].
A case for which the limitation induced by the constant
time-frequency resolution of the Gabor transform can be
seen is shown on the didactic example of Figure 1. On this
figure, two spectrograms of the same glockenspiel signal
are represented. These spectrograms are obtained by plotting the square absolute value of the Gabor coefficients
using a color scale with a level coding in dB. Both spectrograms are obtained from the Gabor coefficients using
the same type of window, but using two different window
lengths. We see that the signal contains two very contrasting types of components:
• at the beginning of the notes, the signal presents
sharp attacks which are spread in frequency, but very
SAMPTA'09
• during the resonance of the notes, the signal contains quasi-sinusoidal components which are spread
in time, but very localized in frequency.
20000
15000
Frequency (Hz)
1.
localized in time,
10000
5000
0
0.2
0.4
0.6
0.8
1
1.2
0.8
1
1.2
Time (s)
20000
15000
Frequency (Hz)
To overcome the limitation induced by the fixed timefrequency resolution over the whole time-frequency plane
of Gabor frames, we propose a simple extension of the Gabor theory leading to the construction of frames with timefrequency resolution evolving over time or frequency. We
describe the construction of such frames and give the explicit formulation of the canonical dual frame for some
conditions. We illustrate the interest of the method on a
simple example.
10000
5000
0
0.2
0.4
0.6
Time (s)
Figure 1: Two spectrograms of the same glockenspiel signal obtained using two different window lengths. On the
top plot, a narrow window of 6 ms is used, on the bottom
plot, a wide window of 93 ms is used.
We see that the use of the narrow window is well suited for
the analysis and processing of the attacks, leading to a very
sparse decomposition for these components, but gives an
unsatisfying representation of the resonance, as the different sinusoidal components are not resolved. On the other
hand, the wide window gives a good representation of the
resonance part, but a blurred representation of the attacks.
For this example, it appears that if we want to build an
227
2.
2.1
Frequency
optimised scheme for processing of both attacks and the
resonances at the same time, it would be suitable to be
able to adapt the time-frequency resolution locally for the
different types of components.
The purpose of this paper is to describe one way to achieve
this goal. For this, we show that, while staying in the context of frame theory [2, 4], the standard Gabor theory can
be easily extended to provide some freedom of evolution
of the time-frequency resolution of the decomposition in
either time or frequency. Furthermore, this extension is
well suited for applications as it can easily be implemented
using fast algorithm based on fast Fourier transform [12].
We first describe the construction of the frames in Section
2., and then illustrate in Section 3. the potential of the approach on the preceding example of Figure 1.
Time
Figure 2: Example of sampling grid of the time-frequency
plane when building a decomposition with time-frequency
resolution evolving over time.
Construction of the frames
Resolution evolving over time
As opposed to standard Gabor analysis, we replace time
translation for the construction of atoms by the use of
different windows for the different sampling positions in
time. For each time position we still build atoms by regular frequency modulation. So using a set of functions
{gn }n∈Z of L2 (R), for m ∈ Z and n ∈ Z, we define
atoms of the form:
relations:
K(t, s)
=
XX
=
X
m
gn (t)gn (s)ei2πmbn (s−t)
n
gn (t)gn (s)
X
ei2πmbn (s−t)
m
n
X 1
X
k
=
gn (t)gn (s)
δ s−t−
bn
bn
n
k
i2πmbn t
gm,n (t) = gn (t)e
.
In practice we will choose each window gn centered
around a time an , and it will typically be constructed by
translating a well localized window centered around 0 by
an , as in the standard Gabor scheme, but with the possibility to vary the window gn for each position an . Thus
the sampling of the time-frequency plane is done on a grid
which is irregular over time, but regular over frequency.
Figure 2 shows an example of such a sampling grid. It can
be noted that some results exist in Gabor theory for semiregular sampling grids, as for example in [3]. Our study
here uses a more general setting, as the sampling grid is in
general not separable, and more importantly, the window
can evolve over time.
In this case, the coefficients of the decomposition are
given by:
cm,n = hf, gm,n i ,
and the frame operator is given by:
Sf =
XX
m
hf, gm,n igm,n .
Sf (s) =
XX 1
k
k
gn (s) f s −
gn s −
bn
bn
bn
n
k
In general, the inversion of S is not obvious. However we
can identify a special case, which is analog to the “painless” case in standard Gabor analysis [8], for which the
expression of S simplifies.
More precisely, we suppose from now on that for every n ∈ Z, the function gn has a limited time support
supp gn = [cn , dn ] such that dn − cn < b1n . Due to this
support condition, the terms of the summation over k in
the preceding equation are 0 for k 6= 0 and the frame operator S becomes a multiplication operator:
X 1
Sf (s) =
|gn (s)|2 f (s).
b
n
n
In this case the invertibility of the frame operator is easy
to check and the system of functions
P gm,n forms a frame
for L2 (R) if and only if ∀t ∈ R, n b1n |gn (t)|2 ≃ 1.
When this condition is fulfilled, the canonical dual frame
elements are given by:
n
The frame operator can be described by its kernel K given
the following relation, which holds at least in a weak
sense:
Z
Sf (s) = K(t, s)f (t)dt.
Here the kernel K simplifies according to the following
SAMPTA'09
thus,
g̃m,n (t) = P
gn (t)
ei2πmbn t ,
1
2
|g
(t)|
bk k
k
and the associated canonical tight frame elements can be
calculated by:
ġm,n (t) = qP
gn (t)
1
2
k bk |gk (t)|
ei2πmbn t .
228
2.2
Resolution evolving over frequency
Frequency
An analog construction is possible with a sampling of the
time-frequency plane irregular over frequency, but regular
over time. An example of the sampling grid in such a case
is given on Figure 3.
In this case, we introduce a family of functions {hm }m∈Z
of L2 (R), and for m ∈ Z and n ∈ Z, we define atoms of
the form:
hm,n (t) = hm (t − nam ).
2.3
For the practical implementation, we have developed the
equivalent theory in a finite discrete setting, that is to say
working with complex vectors as signals. This theory
won’t be described here due to lack of space, but the construction is very similar to the one described in 2.1 and 2.2
and leads to a frame matrix which simplifies to a diagonal
matrix in the “painless” case, suitable for applications.
The implementation is then very similar to the implementation of the standard Gabor case and can exploit fast
Fourier transform algorithms for efficiency. The only differences compared to standard Gabor implementation are
due to the fact that the storage of coefficients requires
more advanced storage structures due to the irregularity
of the time-frequency sampling grid, and that the computation of the dual window must be performed for every time position resulting in a slight increase in computational cost.
3.
Time
Figure 3: Example of sampling grid of the time-frequency
plane when building a decomposition with time-frequency
resolution evolving over frequency.
In practice we will choose each function hm as a well
localized pass-band function having a Fourier transform
centered around some frequency bn .
In this case the frame operator is given by:
XX
Tf =
hf, hm,n ihm,n ,
m
n
and the problem is completely analog to the preceding up
to a Fourier transform, as we have:
XX
c =
[
Tf
hfb, h[
m,n ihm,n ,
m
n
−i2πnam ν
c
and h[
. So the preceding compum,n = hm (ν)e
tation can be done, working on the Fourier transforms of
the involved functions instead of directly on the functions.
Now the “painless” case appears when we suppose that
cn has a limited frequency
for every m ∈ Z, the function h
cn = [en , fn ] such that fn − en < 1 . Then
support supp h
an
the following expression holds:
c (ν) =
Tf
X 1
2b
|hc
m (ν)| f (ν),
a
m
m
and the system of functions hm,n forms a frame of L2 (R)
P
2
if and only if ∀ν ∈ R, n a1m |hc
m (ν)| ≃ 1.
The associated canonical dual and tight frame can be computed as preceding, with the addition of an inverse Fourier
transform.
SAMPTA'09
Implementation
Example
The possibility to build a decomposition with timefrequency resolution evolving over time can be exploited
to solve the problem described in example of Section 1. illustrated by Figure 1. For the corresponding glockenspiel
signal, as we have seen before, the use of narrow window
is suitable for the attacks of the notes, while a wide window should be used for the resonances. Figure 4 shows
a representation built with our approach using a narrow
window of 6 ms for the attacks and a wide window of 93
ms for the resonance. The frame used for this figure is
a tight frame. It should be noticed that the evolution of
the window size between the two target window lengths
is smoothed in order to ensure that the atoms used for the
decomposition maintain a “nice” shape, in the sense of
having a good time-frequency concentration. This ensures
the easy interpretability of the decomposition, especially
for processing using frame multipliers.
This figure gives an idea of the type of decompositions that
can be constructed with our approach and should be compared to the decomposition obtained using standard Gabor
analysis on Figure 1. With our approach, it becomes possible to have a simultaneous good representation of both
types of components of this signal while keeping the same
processing ability than with standard Gabor.
We see that our approach allows to build decompositions
with better time-frequency localization of the signal energy. This can be helpful for many processing tasks, in
particular to reduce artifacts in component extraction or
denoising.
4.
Conclusion
Our approach enables the construction of frames with flexible evolution of time-frequency resolution over time or
frequency. The resulting frames are well suited for applications as they can be implemented using fast algorithms,
at a computational cost close to standard Gabor frames.
Exploiting evolution of resolution over time, the proposed approach can be of particular interest for applica-
229
20000
Frequency (Hz)
15000
10000
5000
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
Figure 4: Spectrogram of the same glockenspiel signal as
in Figure 1 using a nonstationary Gabor decomposition.
tions where the frequency characteristics of the signal are
known to evolve significantly with time. Order analysis
[11], in which the signal analyzed is produced by a rotating machine having evolving rotating speed, is an example
of such application.
Exploiting evolution of resolution over frequency, the presented approach could be valuable for applications requiring the use of a tailored non uniform filter bank. In particular, it can be used to build filter banks following some
perceptive frequency scale.
One difficulty when using our approach is to adapt the
time-frequency resolution to the evolution of the signal
characteristics. If prior knowledge is available, this can
be done by hand, as for the example of Figure 4. But to
go further, our approach could be extended to construct
an adaptive decomposition of the signal by automatically
adapting the resolution to the signal. To achieve this, we
plan to investigate the possibility to couple our approach
with the use of sparsity criterion as proposed in [10]. The
general idea would then be to consider time segments of
the signal, and for each time segment compare the sparsity
criterion obtained for Gabor transforms computed with
different possible windows. We would then use in our
decomposition the window corresponding to the best criterion for each time segment, leading to a decomposition
optimizing the sparsity of the decomposition over time.
[4] O. Christensen. An Introduction To Frames And
Riesz Bases. Birkhäuser, 2003.
[5] I. Daubechies. Ten Lectures On Wavelets. CBMSNSF Regional Conference Series in Applied Mathematics. SIAM Philadelphia, 1992.
[6] H. G. Feichtinger and K. Nowak. A first survey
of Gabor multipliers. In H. G. Feichtinger and
T. Strohmer, editors, Advances in Gabor analysis,
chapter 5, pages 99–128. Birkhäuser Boston, 2003.
[7] H. G. Feichtinger and T. Strohmer. Gabor Analysis and Algorithms - Theory and Applications.
Birkhäuser Boston, 1998.
[8] K. Gröchenig. Foundations of Time-Frequency Analysis. Birkhäuser Boston, 2001.
[9] W. M. Hartmann. Signals, Sounds, and Sensation.
Springer, 1998.
[10] F. Jaillet and B. Torrésani. Time-frequency jigsaw puzzle: adaptive multiwindow and multilayered gabor representations. International Journal for
Wavelets and Multiresolution Information Processing, 5(2):293–316, 2007.
[11] H. Shao, W. Jin, and S. Qian. Order tracking by
discrete Gabor expansion. IEEE Transactions on
Instrumentation and Measurement, 52(3):754–761,
2003.
[12] J. S. Walker. Fast Fourier Transforms. CRC Press,
1991.
Acknowledgment
This work was supported by the WWTF project MULAC
(“Frame Multipliers: Theory and Application in Acoustics”, MA07-025).
References:
[1] P. Balazs. Basic definition and properties of Bessel
multipliers. Journal of Mathematical Analysis and
Applications, 325(1):571585, January 2007.
[2] P. G. Casazza. The art of frame theory. Taiwanese J.
Math., 4(2):129–202, 2000.
[3] P. G. Casazza and O. Christensen. Gabor frames over
irregular lattices. Adv. Comput. Math., 18(2-4):329–
344, 2003.
SAMPTA'09
230
A Nonlinear Reconstruction Algorithm from
Absolute Value of Frame Coefficients for Low
Redundancy Frames
Radu Balan
Department of Mathematics, CSCAMM and ISR, University of Maryland, College Park, MD 20742, USA
rvbalan@math.umd.edu
Abstract:
In this paper we present a signal reconstruction algorithm
from absolute value of frame coefficients that requires a
relatively low redundancy. The basic idea is to use a nonlinear embedding of the input signal Hilbert space into a
higher dimensional Hilbert space of sesquilinear functionals so that absolute values of frame coefficients are associated to relevant inner products in that space. In this space
the reconstruction becomes linear and can be performed in
a polynomial number of steps.
1.
2. Assume e ∈ E , e = 1 is so that Kx e = 0. Then:
Let us denote by E n the n-dimensional space of signals
(e.g. E n = Rn or E n = Cn ), and assume we are given
a frame of m vectors {f 1 , . . . , fm } ⊂ En that span E n .
Thus necessarily m ≥ n. In this paper we look at the
following problem: Given c l = |x, fl |, 1 ≤ l ≤ m,
reconstruct the original signal x ∈ E n up to a constant
phase ambiguity, that is, obtain a signal y ∈ E n such that
y = eiϕ x for some ϕ ∈ [0, 2π).
This problem arises in several areas of signal processing (see [BCE06] for a more detailed discussion of these
issues). In particular, in X-Ray Crystallography (see
[LFB87]) it is known as the phase retrieval problem. In
speech processing it is related to the use of cepstral coefficients in Automatic Speech Recognition as well as direct
reconstruction from denoised spectogram (see [NQL82]).
By the same token the solution posed here can be viewed
as a new, nonlinear signal generating model.
Recently ([BBCE09]) we proposed a quasi-linear reconstruction algorithm that requires the frame to have high redundancy (m = O(n 2 )). The algorithm works as follows.
First note that two vectors x, y ∈ E n that are equivalent
(i.e. equal to one another up to a constant phase) generate
the same rank-one operators K x = Ky , where
(1)
with u = x or u = y. Conversely, if K x = Ky then
necessarily there exists a phase ϕ so that y = e iϕ x. Thus
the reconstruction problem reduces to obtaining first K x ,
and then a representative of the class x̂. Next notice that
the absolute value of frame coefficient |x, f l | is related to
the Hilbert-Schmidt
SAMPTA'09 inner product between K x and Kfl :
Kx , Kfl := trace(Kx Kf∗l ) = |x, fl |2
l=1
n
Introduction
Ku : En → En , Ku (z) = z, uu
Hence, if {Kfl , 1 ≤ l ≤ m} form a frame for the set
of Hilbert-Schmidt operators (this is the same as the set of
quadratic forms), then K x can be reconstructed from d 2l
with a linear algorithm, from where a vector y ∈ x̂ can
be obtained. Explicitely, the algorithm is as follows: First
l : En → En , 1 ≤ l ≤ m} the canonical
denote by { K
dual frame of {K fl , 1 ≤ l ≤ m}.
1. Compute:
m
l
Kx =
c2l K
(2)
1
Kx (e)
y=
Kx (e), e
(3)
is a vector in En equivalent to x.
While very appealing from a computational perspective,
this algorithm requires the set {K fl , 1 ≤ l ≤ m} to
be complete (spanning) in the Hilbert space of n × n
quadratic forms. In the real case (E = R) this latter
Hilbert space is of dimension n(n + 1)/2. In the complex
case (E = C) the dimension becomes n 2 . Thus the algorithm requires the original frame set {f l , 1 ≤ l ≤ m}
to have m = O(n2 ) vectors. In practice this requirement may not be feasible. Furthermore, in [BCE06] we
obtained that generically m ≥ 4n − 2 should suffice in the
complex case, and n ≥ 2n − 1 should suffice in the real
case. In this paper we present an algorithm that applies to
a generic frame set of m = 5.394n − 4.394 vectors in the
complex case, and m = 2n − 1 in the real case. The main
ingredient of this algorithm is the nonlinear embeding of
En into a linear space Λ d,d of (d, d)-sesquilinear symmetric forms where the absolute value of frame coefficients
provide the inner products with a frame set.
2. Nonlinear Embeddings
Let En be the signal n-dimensional Hilbert space. Let
F = {f1 , . . . , fm } be a spanning set of m vectors in E n .
Its redundancy is r = m/n ≥ 1. Fix an integer d ≥ 1
which is going to measure the embedding depth. Let
Λd,d (En ) denote the linear space of (d, d)-sesquilinear
functionals, that is
231
Λd,d (En ) = { α : En ×
· · · En → C }
2d
(4)
3. The Reconstruction Algorithm
where α(y1 , . . . , yd , z1 , . . . , zd ) is linear in y1 , . . . , yd ,
and antilinear in z 1 , . . . , zd . Note Λd,d (En ) is a vector
space of dimension n 2d . Let {ek , 1 ≤ k ≤ n} be an orthonormal basis of E n . For each 2d-tuple (k 1 , . . . , k2d ) of
integers from 1, . . . , n (repetitions are allowed) define
Under Assumption A, let us denote by { ψ
j1 ,...,jd , 1 ≤
j1 ≤ · · · ≤ jd ≤ m} the canonical dual frame to
P Ψ. This dual frame allows us to recover Φ(x). Recall
n
{e
δk1 ,...,k2d (y1 , . . . , yd , z1 , . . . , zd ) = y1 , ek1 · · · yd , ekd · 1 , . . . , en } is an orthonormal basis of E . Notice the
following relations:
(5)
ekd+1 , z1 · · · ek2d , zd
Φ(x)(ek , . . . , ek ) = |x, ek |2d
(11)
Note ∆ = {δk1 ,...,k2d ; 1 ≤ kl ≤ n, 1 ≤ l ≤ 2d}
n
1/d
forms a basis in Λd,d (En ). We define an inner product
(Φ(x)(ek , . . . , ek ))
=
x 2
(12)
n
on Λd,d (E ) so that this basis is orthonormal. Consider
k=1
two sesquilinear functionals in Λ d,d (En ):
Φ(x)(ej , . . . , ej , ek ) = |x, ej |2d−2 ej , xx, ek
α(y , . . . , y , z , . . . , z ) = y , a · · · y , a b , z · · · b , z
1
d
1
d
1
1
d
d
1
1
d
d
2d−1
β(y1 , . . . , yd , z1 , . . . , zd ) = y1 , g1 · · · yd , gd h1 , z1 · · · hd , zd
Then their inner product is defined as
α, β := g1 , a1 · · · gd , ad b1 , h1 · · · bd , hd
From (11) and (13) we obtain:
(6)
Extend this binary operation to an inner product on
Λd,d (En ). With this inner product ∆ becomes an orthonormal basis for the Hilbert space Λ d,d (En ).
Now we are ready to define the nonlinear embedding of
the input Hilbert space E n in Λd,d (En ). This is given by
the map Φ : En → Λd,d(En )
Φ(x)(y1 , . . . , yd , z1 , . . . , zd ) =
·x, z1 · · · x, zd
(13)
y1 , x · · · yd , x ·
(7)
Let Ed = span(Φ(Λd,d (En ))) be the linear span of the
embedding. Note in general E d Λd,d (En ) unless d =
1. Let P denote the orthogonal projection onto E d , P :
Λd,d (En ) → Ed .
Define now the following sesquilinear functionals associated to the frame set F . Fix 1 ≤ j 1 , . . . , jd ≤ m.
Φ(x)(ej , . . . , ej , ek )
x, ej
|x, ej | (Φ(x)(ej , . . . , ej , ej ))(2d−1)/2d
(14)
The Reconstruction Algorithm is as follows.
Reconstruction Algorithm
Input: Coefficients c1 = |x, f1 |, ... cm = |x, fm |.
m
Step 0. If k=1 c2k = 0 then y = 0 and stop. Otherwise
continue.
Step 1. Construct the following sesquilinear functional
c2j1 · · · c2jd ψ
(15)
α=
j1 ,...,jd
x, ek =
1≤j1 ≤···≤jd ≤m
Step 2. Find a 1 ≤ j0 ≤ n so that α(ej0 , · · · , ej0 ) > 0.
This is possible due to (12). Set
ν=
2d
α(ej0 , . . . , ej0 )
(16)
ψj1 ,...,jd (y1 , . . . , yd , z1 , . . . , zd ) =
y1 , fj1 · · · yd , fjd · Step 3. Set
·fj1 , z1 · · · fjd , zd
(8)
n
1
α(ej0 , . . . , ej0 , ek )ek
(17)
y
=
d
Note there are m distinct such functionals, however the
ν 2d−1
k=1
number of distinct projections onto E d is much smaller.
2d−1
Notice
Summarizing all results obtained so far we obtain:
2
2
Φ(x), ψj1 ,...,jd = |x, fj1 | · · · |x, fjd |
(9)
Theorem 3..1 For every x ∈ E n there is z ∈ C so that
Thus if (k1 , . . . , kd ) is a permutation of (j 1 , . . . , jd ) then
|z| = 1 and the output of the Reconstruction Algorithm
x,e
P ψk1 ,...,kd = P ψj1 ,...,jd . For converse we need to assatisfies x = zy. Specifically z = |x,ejj0 | , with j0 ob0
sume first that frame vectors belong to distinct equivatained in Step 2.
lence classes (that is, for any two 1 ≤ l < j ≤ m
and any a ∈ [0, 2π), f l = eia fj ). Then we get that
4. Redundancy Constraint
P ψk1 ,...,kd = P ψj1 ,...,jd if and only if (k 1 , . . . , kd ) is
a permutation of (j 1 , . . . , jd ). Thus we obtain that for
In this section we analyse the necessary condition |Ψ| ≥
frames with frame vectors in distinct equivalence classes
dim(Ed ).
the set
Ψ = {ψj1 ,...,jd , 1 ≤ j1 ≤ j2 ≤ · · · ≤ jd ≤ m}
(10)
is a maximal set of sesquilinear functionals of type (8) that
have distinct projections through P .
For our algorithm to work we need to assume:
Assumption A. The set P Ψ := {P ψ , ψ ∈ Ψ} is spanning
in Ed .
In section
4. we analyze the dimensionality constraint
SAMPTA'09
|P Ψ| ≥ dim(Ed ), and in section 5. we present numerical results supporting Assumption A for a generic frame.
4.1 The Cardinal of Set Ψ
The set Ψ given in (10) has the same cardinal as
{(k1 , . . . , kd ) , 1 ≤ k1 ≤ · · · ≤ kd ≤ m}
(18)
Let us denote this number by M m,d . In order to compute
it, consider the following cardinal equivalent set:
232
{(n1 , . . . , nm ) , 0 ≤ n1 , . . . , nm ≤ d, n1 +· · ·+nm = d}
(19)
The bijective correspondence between d-tuples of (18) and
m-tuples of (19) is given by the following interpretation:
nl is the number of times l is presented in the d-tuple
(k1 , . . . , kd ). Then, one can obtain the following recursion:
d
Mm,d
Mm+1,d =
r=0
where we set Mm,0 = 1. Since M1,d = 1, one obtains by
induction that:
Mm,d =
m+d−1
m−1
=
m(m + 1) · · · (m + d − 1)
d!
(20)
4.2 The Dimension of Ed
d-tuples l:
Nn,d = (Mn,d )2 =
n(n + 1) · · · (n + d − 1)
d!
We shall group together terms containing same t k terms.
The real case will be treated separately from the complex
case.
To simplify the exposition, we introduce notation common to both cases. Let us denote by k = (k 1 , . . . , kr )
an ordered r-tuple of integers each from 1 to n, where
the length r is equal to 2d (in the real case), or d (in
the complex case). Let us denote by P r the set of rpermutations, and by P k the quotient set Pk = P/ ∼k
where π ′ , π ′′ ∈ Pr are equivalent π ′ ∼k π ′′ if and only if
π ′ (k) = π ′′ (k). Note
|Pk | =
r!
m1 ! · · · mn !
where ml denotes the number of repetitions of l in k.
The Complex Case
In the complex case, t k and tk can be treated as indepedent (real) variables. Then terms in (21) are grouped
using two independent d-tuples, j = (j 1 , . . . , jd ) and
l = (l1 , . . . , ld ) as follows
tj1 · · · tjd tl1 · · · tld ×
1≤j1 ≤···≤jd ≤n 1≤l1 ≤···≤ld ≤n
×
δπ(j1 ),...,π(jd ),ρ(l1 ),...,ρ(ld )
π∈Pj ρ∈Pl
Then the following sesquilinear functionals are orthonormal and form a basis in E d :
1
δπ(j1 ),...,π(jd ),ρ(l1 ),...,ρ(ld )
|Pj | |Pl | π∈Pj ρ∈Pl
(22)
SAMPTA'09
Their number (and hence dimension of E d ) is equal to the
number of ordered d-tuples j times the number of ordered
dj,l =
(23)
where we used (20). Note N n,1 = n2 and we recover the
complex case considered in [BBCE09].
The Real Case
In the real case, tk and tk are the same variables. Then the
independent terms in (21) are indexed by 2d-tuples k =
(k1 , . . . , k2d ) as follows:
tk1 · · · tk2d
δπ(k1 ),...,π(k2d )
1≤k1 ≤k2d ≤n
π∈Pk
and an orthonormal basis of E d is given by the following
vectors indexed by ordered 2d-tuples k:
Recall Ed is the linear span of vectorx Φ(x) in Λ d,d(En ).
1
Recall also that ∆ whose n2d vectors are defined in (5) is
dk =
δπ(k1 ),...,π(k2d )
n
an orthonormal basis in Λ d,d (E ). Let us denote by N n,d
|Pk | π∈P
k
the dimension of E d . We will describe an orthonormal
basis in Ed . Fix t1 , . . . , tn ∈ C and expand:
The dimension of E d in real case is then:
tk1 · · · tkd tkd+1 · · · tk2d · Nn,d = Mn,2d = n(n + 1) · · · (n + 2d − 1)
Φ(t1 e1 + · · · tn en ) =
(2d)!
1≤k1 ,...,k2d ≤n
·δk1 ,...,k2d
2
(21)
Note Nn,1 =
[BBCE09].
n(n+1)
2
(24)
(25)
and this recovers the real case in
4.3 The Optimal Depth and Redundancy Condition
For given n we would like to find the minimum m = m ∗
so that Mm,d ≥ Nn,d for some d ≥ 1.
The Complex Case
We need to solve
m(m + 1) · · · (m + d − 1)
≥
d!
n(n + 1) · · · (n + d − 1)
d!
or, completing the factorials:
(m + d − 1)! d! ((n − 1)!)2 ≥ (m − 1)! ((n + d − 1)!)2
Let us denote
R(n, m, d) =
(m + d − 1)!d!((n − 1)!)2
(m − 1)!((n + d − 1)!)2
(26)
Ideally we would like to solve:
(1)
d∗ (n, m) = argmaxd R(n, m, d)
(2)
m∗ (n) = minR(n,m,d∗ (n,m))≥1 m
Instead we make the following choices for d = d(n) and
m = m(n), and then optimize using Stirling’s formula:
d = n−1
(27)
m = A(n − 1) + 1.
(28)
√
Using Stirling’s formula n! = 2πnnn e−n we obtain for
R(n + 1, An + 1, n),
R(n+1, An+1, n) =
233 n
8π(A + 1)n A + 1
1
(1 + )A
A
16
A
2
F(j,l),k = ψk , dj,l . Explicitely this becomes
1.8
1.6
F(j,l),k
1.4
1.2
1
eπ(j1 ) , fk1 · · ·
|Pj | |Pl | π∈Pj ρ∈Pl u
·eπ(jd ) , fkd fk1 , eρ(l1 ) · · · fkd , eρ(ld )
q(A)
1
0.8
0.6
0.4
0.2
0
=
0
1
2
3
4
5
A
6
7
8
9
10
Figure 1: The plot of q = q(A) from (29).
To obtain R ≥ 1 for large n, we need
q(A) =
A+1
1
(1 + )A ≥ 1
16
A
(29)
In Figure 1 we plot the function q = q(A). Numerically
we obtain A = 5.394. The √
remaining factor in R(n +
1, An + 1, n) becomes 5.376 n ≥ 1 for all n. Thus we
obtain as sufficient conditions:
d =
m =
n−1
5.394n − 4.394
(35)
Thus P Ψ is frame for E d if and only if the N n,d × Mm,d
matrix F is of full rank. The frame operator is given by
S = F F ∗.
We considered the complex case (E = C) with the following parameters n = 5 and d = 3. For m = 21 the ratio
function (26) takes the value R(5, 21, 3) = 1.4457 > 1.
Note for the algorithm in [BBCE09] to work m has to
be greater than or equal to n 2 , that is m ≥ 25. For a
frame with 21 vectors in dimension 5 whose vectors are
obtained as realizations of complex valued normal random
variables of zero mean and variance 2 (each real and imaginary part is i.i.d. N (0, 1)), the distribution of eigenvalues
of its frame operator is plotted in Figure 2. Note the conditioning number is cond(S) = 6267.7. While relatively
large, the important thing to note is that the realization
P Ψ is frame (spanning) for E d . While this result is by no
(30)
(31)
The Real Case
In the real case we need to solve
m(m + 1) · · · (m + d − 1)
n(n + 1) · · · (n + 2d − 1)
≥
d!
(2d)!
Following the same approach we obtain the following ratio function that we need to make supraunital:
R(n, m, d) =
(m + d − 1)!(n − 1)!(2d)!
(m − 1)!(n + 2d − 1)!d!
(32)
Figure 2: Distribution of eigenvalues for a random frame..
means a proof, or even an exhaustive experiment, it suggests the Assumption A might be generically true whenever R(n, m, d) > 1.
It follows:
R(n + 1, 2n + 1, n) = 1
Hence a possible choice is
d =
m
=
n−1
2n − 1
(33)
(34)
It is interesting to note that in the real case we recover the
critical case m ≥ 2n − 1.
5.
Numerical Evidence Supporting Genericity of the Assumption A.
While the previous section computed necessary conditions
for Assumption A to hold true, we still need to prove (or
check) that P Ψ is frame in E d . In this section we plot the
distribution of eigenvalues of the frame operator associated toSAMPTA'09
P Ψ for one randomly generated example.
Using (22), each vector P ψ k is represented by a N n,d vector whose components are indexed by a pair (j, l),
References:
[BCE06] R. Balan, P. Casazza, D. Edidin, On
signal
reconstruction
without
phase,
Appl.Comput.Harmon.Anal.
20
(2006),
345–356.
[BBCE09] R. Balan, B. Bodman, P. Casazza, D. Edidin,
Painless reconstruction from magnitudes of
frame coefficients, to appear in the Journal of
Fourier Analysis and Applications, 2009.
[LFB87] R. G. Lane, W. R. Freight, and R. H. T. Bates,
Direct Phase Retrieval, IEEE Trans. ASSP 35,
no. 4 (1987), 520–526.
[NQL82] H. Nawab, T. F. Quatieri, and J. S. Lim,
Signal Reconstruction from the Short-Time
234
Fourier Transform Magnitude, in Proceedings
of ICASSP 1984.
Matrix Representation of Bounded Linear
Operators By Bessel Sequences, Frames and
Riesz Sequence
Peter Balazs
Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, 1040 Wien, Austria.
peter.balazs@oeaw.ac.at
Abstract:
In this work we will investigate how to find a matrix representation of operators on a Hilbert space H with Bessel
sequences, frames and Riesz bases as an extension of the
known method of matrix representation by ONBs. We
will give basic definitions of the functions connecting infinite matrices defining bounded operators on l2 and operators on H. We will show some structural results and give
some examples. Furthermore in the case of Riesz bases we
prove that those functions are isomorphisms. We are going to apply this idea to the connection of Hilbert-Schmidt
operators and Frobenius matrices. Finally we will use this
concept to show that every bounded operator is a generalized frame multiplier.
1.
Introduction
From practical experience it became apparent that the concept of an orthonormal basis is not always useful. This led
to the concept of frames, which was introduced by Duffin
and Schaefer [12] and today it is one of the most important
foundations of sampling theory [1].
The standard matrix description [8] of operators O using an ONB (ek ) is by constructing an matrix M with
the entries Mj,k = hOek , ej i. In [6] a concept was
presented,
where
Ean operator R is described by the maD
trix
Rφj , φ̃i
with (φi ) being a frame and (φ̃i ) its
g ∈ H2 then define the inner tensor product as an operator
from H2 to H1 by (f ⊗i g) (h) = hh, gi f for h ∈ H2 .
2.1.1 Hilbert Schmidt Operators
A bounded operator T ∈ B(H1 , H2 ) is called a HilbertSchmidt (HS) [18] operator s
if there exists an ONB (en ) ⊆
∞
P
2
H1 such that kT kHS :=
kT en kH2 < ∞. Let
n=1
HS(H1 , H2 ) denote the space of Hilbert Schmidt operators from H1 to H2 .
2.2
Frames
A sequence Ψ = (ψk |k ∈ K) is called a frame [5, 7] for
the Hilbert space H, if constants A, B > 0 exist, such that
X
2
2
2
A · kf kH ≤
|hf, ψk i| ≤ B · kf kH ∀ f ∈ H (1)
k
A sequence Ψ = (ψk ) is called a Bessel sequence with
Bessel bound B if it fulfills the right inequality above. The
index set will be omitted in the following, if no distinction
is necessary.
A complete sequence (ψk ) in H is called a Riesz basis if
there exist constants A, B > 0 such that the inequalities
2
2
A kck2
≤
i,j
canonical dual. Such a kind of representation is used for
the description of operators in [15] using Gabor frames
and [19] using linear independent Gabor systems. In this
work we are presenting the main ideas for Bessel sequences, frames and Riesz sequences and also look at the
dual function which assigns an operator to a matrix. For
proofs and details we refer to [3].
X
2
ck ψk
k∈K
H
≤ B kck2
hold for all finite sequences (ck ).
3.
Representing Operators with Frames
Let (ψk ) be a frame in H1 . An existing operator U ∈
B(H1 , H2 ) is uniquely determined
by its images of the
P
frame elements. For f = ck ψk
k
2.
2.1
Notation and Preliminaries
Hilbert spaces and Operators
Let B(H1 , H2 ) denote the set of all linear and bounded
operators from the Hilbert space H1 to H2 . Furthermore
we will denote the range of an operator A by ran(O) and
its kernel by ker(A).
Let X, Y, Z be sets, f : X → Z, g : Y → Z be arbitrary
functions. The Kronecker product ⊗o : X × Y → Z
is defined by (f ⊗o g) (x, y) = f (x) · g(y). Let f ∈ H1 ,
SAMPTA'09
X
X
U (f ) = U (
ck ψk ) =
ck U (ψk ).
k
k
On the other hand, contrary to the case for ONBs, we
cannot just choose a Bessel sequence (ηk ) and define
an P
operator just by
P choosing V (ψk ) := ηk and setting
V ( ck ψk ) =
ck ηk . This is in general not wellk
k
defined. Only if
X
X
X
X
ck ψk =
dk ψk =⇒
ck ηk =
dk ηk
k
k
k
k
235
this definition is non-ambiguous, i.e. if ker (Dψk ) ⊆
ker (Dηk ). This condition is certainly fulfilled, if Dψk
is injective, i.e. for Riesz bases.
This problem can be avoided by using the following definition
E
XD
(2)
V (f ) :=
f, ψ̃k ηk .
k
As (ηk ) forms a Bessel sequence, the right hand side of
Eq. (2) is well-defined. It is clearly linear, and it is
bounded. The Bessel condition is necessary in the case
of ONBs to get a bounded operator, too [8]. But contrary
to the ONB case, here, in general, V (ψk ) 6= ηk . So this
option does not seem very useful. Instead of changing the
sequence with which the coefficients are resynthezised,
an operator can also be described by changing the coefficients, as presented in the following sections.
4.
4.1
Matrix Representation
Motivation: Solving Operator Equalities
⇐⇒
X
k
X
k
1. Let O : H1 → H2 be a bounded, linear operator.
Then the infinite matrix
M(Φ,Ψ) (O)
= hOψn , φm i
m,n
defines a bounded
operator from l2 to l2 with
√
kMkl2 →l2 ≤ B · B ′ · kOkH1 →H2 . As an operator l2 → l2
M(Φ,Ψ) (O) = CΦ ◦ O ◦ DΨ
This means the function M(Φ,Ψ) : B(H1 , H2 ) →
B(l2 , l2 ) is a well-defined bounded operator.
2. On the other hand let M be an infinite matrix defin2
ing
to l2 , (M c)i =
P a bounded operator from l(Φ,Ψ)
Mi,k ck . Then the operator O
defined by
X X
Mk,j hh, ψj i φk ,
O(Φ,Ψ) (M ) h =
O(Φ,Ψ) (M )
This gives us an algorithm for finding an approximative
solution to the inverse operator problem Of = g.
1. Set M = M(Φ,Φ̃) (O).
2. Find a good finite dimensional approximation MN of
M by using the finite section method [14, 16] and
3. then apply an algorithm like e.g. the QR factorization
[21] to find a solution for the operator equation.
4. and synthezise with the dual frame Φ̃.
H1 →H2
≤
√
O(Φ,Ψ) (M ) = DΦ ◦M ◦CΨ =
B · B ′ kM kl2 →l2 .
XX
k
j
Mk,j ·φk ⊗i ψ j
This means the function O(Φ,Ψ) : B(l2 , l2 ) →
B(H1 , H2 ) is a well-defined bounded operator.
O(Φ,Ψ) (M )
H1
E
D
hf, φk i Oφ̃k , φk = hg, φk i
It can be easily seen that this is equivalent to projecting c
on ran(C), solving M CΦ DΦ̃ c = d, which is a common
idea found in many algorithms, for example for a recent
one see [20].
j
k
hf, φk i Oφ̃k = g ⇐⇒
⇐⇒ M(Φ,Φ̃) (O) · CΦ f = CΦ g.
SAMPTA'09
Theorem 4.2.1 Let Ψ = (ψk ) be a Bessel sequence in H1
with bound B, Φ = (φk ) in H2 with B ′ .
(3)
for example using the pseudoinverse [7]. Still, if using
frames, we can not expect to find a true solution for the
operator equality just by applying DΦ̃ on c as in general c
is not in ran(CΦ ) even if d is. But we see the following:
Of = g ⇐⇒
Bessel sequences
k
Given an operator equality O · f = g it is natural to
discretize it to find a solution. Let Φ = (φk ) be a frame.
Let us suppose that for a given g with coefficients d =
(dk ) = (hg, φk i) and a matrix representation M of O there
is an algorithm to find the least square solution of
M ·c=d
4.2
✻
DΨ
CΨ
❄
l2
✲
O
H2
✻
DΦ
M(Ψ,Φ) (O)
M
CΦ
❄
✲
l2
Figure 1: The operator induced by a matrix M and the
matrix induced by an operator O.
If we do not want to stress the dependency on the frames
and there is no change of confusion, the notation M(O)
and O(M ) will be used.
In the above theorem we have avoided the issue, when an
infinite matrix defines a bounded operator from l2 to l2 . A
criterion has been proved in [9]:
236
Theorem 4.2.2 An infinite matrix M defines a bounded
n
operator from l2 to l2 , if and only if (M ∗ M ) is defined
i1/n
h
n
<
for all n = 1, 2, 3, . . . and sup sup (M ∗ M )l,l
n
l
∞.
For similar conditions see [17].
5.
Matrix Representation of HS Operators
We now have the adequate tools to state that HS operators correspond exactly to the Frobenius matrices, as expected. Let A be an m by n matrix, then kAkf ro =
s
n−1
P m−1
P
2
|ai,j | is the Frobenius norm. Let us denote
i=0 j=0
4.3
the set of all matrices with finite Frobenius norm by l(2,2) ,
the set of Frobenius matrices.
Frames
Proposition 4.3.1 Let Ψ = (ψk ) be a frame in H1 with
bounds A, B, Φ = (φk ) in H2 with A′ , B ′ . Then
1. O(Φ,Ψ) ◦ M (Φ̃,Ψ̃) = Id = O(Φ̃,Ψ̃) ◦ M (Φ,Ψ) .
And therefore for all O ∈ B(H1 , H2 ):
O=
XD
k,j
E
Oψ̃j , φ̃k φk ⊗i ψ j
2. M(Φ,Ψ) is injective and O(Φ,Ψ) is surjective.
3. Let H1 = H2 , then O(Ψ,Ψ̃) (Idl2 ) = IdH1
4. Let Ξ = (ξk ) be any frame in H3 , and O : H3 → H2
and P : H1 → H3 . Then
M(Φ,Ψ) (O ◦ P ) = M(Φ,Ξ) (O) · M(Ξ̃,Ψ) (P )
As a direct consequence we get the following corollary:
Corollary 4.3.2 For the frame Φ = (φk ) the function
M(Φ,Φ̃) is a Banach-algebra monomorphism between the
algebra of bounded operators (B(H1 , H1 ), ◦) and the infinite matrices of B(l2 , l2 ), · .
Lemma 4.3.3 Let O : H1 → H2 be a linear and bounded
operator, let Ψ = (ψk ) and Φ = (φk ) be frames in
H1 resp. H2 . Then M(Φ,Ψ̃) (O) maps ran (CΨ ) into
ran (CΦ ) with
(hf, ψk i)k 7→ (hOf, φk i)k .
Proposition 5.0.2 Let Ψ = (ψk ) be a Bessel sequence in
H1 with bound B, Φ = (φk ) in H2 with B ′ . Let M be
a matrix in l(2,2) . Then O(Φ,Ψ) (M ) ∈ HS(H1 , H2 ), the
Hilbert Schmidt√class of operators from H1 to H2 , with
kO(M )kHS ≤ BB ′ kM kf ro .
(Φ,Ψ)
(O) ∈ l(2,2) with
Let O ∈ HS,
√ then M
′
kM(O)kf ro ≤ BB kOkHS .
5.1
Matrices and the Kernel Theorems
For L2 (Rd ) the HS operators are exactly those integral
operators with kernels in L2 R2d [18]. This means that
there exists a κO ∈ L2 (R2d ) such an operator can be described as
Z
(Of ) (x) = κO (x, y)f (y)dy
Or in weak formulation
Z Z
hOf, gi =
κO (x, y)f (y)g(x)dydx = hκO , f ⊗o gi .
(4)
From 4.2.1 we know that
E
XD
O=
Oψ̃j , φ̃k φk ⊗i ψ j
j,k
and so
Corollary 5.1.1 Let O ∈ HS L2 Rd . Let Ψ = (ψj )
and Φ = (φk ) be frames in L2 Rd . Then the kernel of
O is given as:
If O is surjective, then M(Φ,Ψ̃) (O) maps ran (CΨ ) onto
ran (CΦ ). If O is injective, M(Φ,Ψ̃) (O) is also injective.
κO =
X
j,k
M(Ψ̃,Φ̃) (O)k,j · φk ⊗o ψ j
The other function O is in general not so “well-behaved”.
It is, if the dual frames are biorthogonal. In this case these
functions are isomorphisms, see the next section.
This directly leads to the next concept.
4.4
Let m be a sequence and diag(m) the matrix that has this
sequence as diagonal. Then define
Riesz sequences
Theorem 4.4.1 Let Φ = (φk ) be a Riesz basis for H1 ,
Ψ = (ψk ) one for H2 . The functions M(Φ,Ψ) and O(Φ̃,Ψ̃)
between B(H1 , H2 ) and the infinite matrices in B(l2 , l2 )
are bijective. M(Φ,Ψ) and O(Φ̃,Ψ̃) are inverse to each
other. For H1 = H2 the identity is mapped on the identity by M(Φ,Ψ) and O(Φ̃,Ψ̃) . If furthermore Ψ = Φ then
M(Φ,Φ̃) and O(Φ,Φ̃) are Banach algebra isomorphisms,
respecting the identities idl2 and idH .
SAMPTA'09
6.
Generalized Bessel Multipliers
Mm,Φ,Ψ := O(Φ,Ψ) (diag(m)) =
X
k
mk · φk ⊗ ψk
This means we have arrived quite naturally at the definition of frame multipliers as introduced in [2].
It is a very natural idea to extend this definition to include
more side-diagonals:
237
Definition 6.0.2 Let H1 , H2 be Hilbert-spaces, let
(ψk )k∈L ⊆ H1 and (φk )k∈K ⊆ H2 be Bessel sequences.
Let M be a (K × L)-matrix that defines a bounded operator from l2 to l2 . Define the operator MM,(φk ),(ψk ) :
H1 → H2 , the generalized Bessel multiplier for the Bessel
sequences (ψk ) and (φk ), as the operator
XX
Mm,(φk ),(ψk ) (f ) =
Ml,k hf, ψk i φl .
l
k
The sequence m is called the symbol of M. If the sequence is a frame, we call the operator a ’generalized
frame multiplier’.
For Gabor frames, this is a particular case of the ’generalized Gabor multipliers’ as found in [10] or [11] in this
volume.
Using the results above we can write
Proposition 6.0.3 For two frames (ψk ) ⊆ H1 and
(φk } ⊆ H2 every operator O : H1 → H2 can be
written as
frame multiplier with the symbol
D a generalized
E
Ml,k = Oψ̃k , φ̃l .
Further results as the following are easy to prove:
Theorem 6.0.4 Let M = Mm,φk ,ψk be a Bessel multiplier for the Bessel sequences (ψk ) ⊆ H1 and (φk } ⊆ H2
with the bounds B and B ′ . Then
1. If M, M ∗ ∈ l1,∞ with kM k1,∞ = K1 and
kM ∗ k1,∞ = K2 then M is a well defined bounded
√
operator with kMkOp ≤ B ′ BK1 K2 .
2. If sup M (n)
= K < ∞ then M is a well de√
fined bounded operator with kMkOp ≤ B ′ BK.
n
Op
n
3. If (M ∗ M ) is defined for n = 1, 2, . . . and
h
i1/n
n
= K < ∞ then
sup sup hM ∗ M )i,i
n
i
√
kMkOp ≤ B ′ BK.
4. If φk = ψk and M ∈ B(l2 ) is a positive matrix, M is
positive.
∗
5. Let M ∈ B(l2 ), then MM,(φk ),(ψk )
=
MM ∗ ,(ψk ),(φk ) . Therefore if M is self-adjoint and
φk = ψk , M is self-adjoint.
6. Let M
∈ B(l2 ) be a matrix such that
(n)
lim M
− M Op = 0, then M is compact.
n
7. If M ∈ l2,2√
, M√is a Hilbert Schmidt operator with
kM kHS ≤ B ′ B kM k2,2 .
Here for an operator A we denote A(n) = Pn APn , where
Pn (x0 , x1 , x2 , . . . ) = (x1 , x2 , . . . , xn−1 , 0, 0, . . . ), see
[14] (finite sections).
7.
Perspectives
In this work we have investigated the basic idea of matrix
representations using frames. An interesting question, as
discussed in Section 4.1, is how to find a good finite approximation matrix. For first ideas in the Gabor case see
[13, 10, 11, 22, 4].
SAMPTA'09
8.
Acknowledgments
The author would like to thank Jean-Pierre Antoine for
many helpful comments and suggestions.
This work was partly supported by the WWTF project
MULAC (Frame Multipliers: Theory and Application in
Acoustics, MA07-025).
References:
[1] A. Aldroubi and K. Gröchenig. Non-uniform sampling
and reconstruction in shift-invariant spaces. SIAM Review,
43:585–620, 2001.
[2] P. Balazs. Basic definition and properties of Bessel multipliers. Journal of Mathematical Analysis and Applications,
325(1):571–585, January 2007.
[3] P. Balazs. Matrix-representation of operators using frames.
Sampling Theory in Signal and Image Processing (STSIP),
7(1):39–54, Jan. 2008.
[4] J. Bendetto and G. Pfander. Frame expansions for Gabor
multipliers. Applied and Computational Harmonic Analysis (ACHA)., 20(1):26–40, Jan. 2006.
[5] P. G. Casazza. The art of frame theory. Taiwanese J. Math.,
4(2):129–202, 2000.
[6] O. Christensen. Frames and pseudo-inverses. J. Math.
Anal. Appl, 195(2):401–414, 1995.
[7] O. Christensen. An Introduction To Frames And Riesz
Bases. Birkhäuser, 2003.
[8] J. B. Conway. A Course in Functional Analysis. Graduate
Texts in Mathematics. Springer New York, 2. edition, 1990.
[9] Lawrence Crone. A characterization of matrix operator on
l2 . Math. Z., 123:315–317, 1971.
[10] M. Dörfler and B. B. Torrésani. Spreading function representation of operators and gabor multiplier approximation.
In Proceedings of SAMPTA’07, 2007.
[11] M. Dörfler and B. B. Torrésani. Representation of operators
by sampling in the time frequency domain. In Proceedings
of SAMPTA’09, 2009.
[12] R. J. Duffin and A. C. Schaeffer. A class of nonharmonic
Fourier series. Trans. Amer. Math. Soc., 72:341–366, 1952.
[13] H. G. Feichtinger, M. Hampejs, and G. Kracher. Approximation of matrices by Gabor multipliers. IEEE Signal Procesing Letters, 11(11):883–886, 2004.
[14] I. Gohberg, S. Goldberg, and M. Kaashoek. Basic Classes
of Linear Operators. Birkhäuser, 2003.
[15] K. Gröchenig. Time-frequency analysis of Sjöstrand’s
class. Rev. Mat. Iberoam., 22:(to appear), 2006.
[16] O.Christensen and T.Strohmer. The finite section method
and problems in frame theory. Journal of Approximation
Theory, 133(2):221–237, 2005.
[17] W. H. Ruckle. Sequence spaces. Research Notes in Mathematics 49. Pitman London, 1981.
[18] R. Schatten. Norm Ideals of Completely Continious Operators. Springer Berlin, 1960.
[19] T. Strohmer. Pseudodifferential operators and Banach algebras in mobile communications. Appl.Comp.Harm.Anal.,
20(2):237–249, 2006.
[20] G. Teschke. Multi-frame representations in linear inverse
problems with mixed multi-constraints. Applied and Computational Harmonic Analysis, 22(1):43–60, Jan. 2007.
DFG-SPP-1114 preprint 90.
[21] L. N. Trefethen and D. Bau III. Numerical Linear Algebra.
SIAM Philadelphia, 1997.
[22] P. Wahlberg and P Schreier. Gabor discretization of the
Weyl product for modulation spaces and filtering of nonstationary stochastic processes. Appl.Comp.Harm.Anal.,
26:97–120, 2009.
238
Quasi-Random Sequences for
Signal Sampling and Recovery
Mirosław Pawlak (1) and Ewaryst Rafajłowicz (2)
(1) Dept. of Electrical & Computer Eng., University of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2
(2) Institute of Computer Eng., Control and Robotics, Wrocław University of Technology, Wroclaw, Poland
pawlak@ee.umanitoba.ca, ewaryst.rafajlowicz@pwr.wroc.pl
Abstract:
The problem of reconstruction of band-limited signals
from sampled and noisy observations is studied. It is
proposed to sample a signal at quasi-random points,
that form a deterministic sequence with properties resembling a random variable being uniformly distributed.
Such quasi-random points can be easily and efficiently
generated yielding signal reconstruction algorithms with
the improved accuracy. In fact, in this paper we propose
a reconstruction method based on the modified orthogonal sampling formula where the sampling rate and the reconstruction rate are treated separately. This distinction
is necessary to ensure consistency of the reconstruction
algorithm in the presence of noise. Asymptotical properties of the algorithm are evaluated including its convergence to the true signal and the corresponding rate. It
is shown that the rate of convergence is better than that
for reconstructions algorithms that utilize the traditional
uniform sampling. Similar results are also obtained for
the case of multivariate signals.
1.
Introduction
Signal sampling is an inherent part of the modern signal
processing theory and as such it has attracted a great deal
of research activities lately [9], [10]. In particular, the
problem of signal sampling and recovery from imperfect
data has been addressed in a number of recent works [1],
[2], [7]. In this case, one assumes that the signal samples
{f (kτ )} are observed with noise, i.e., we have
yk = f (kτ ) + zk ,
where zk is uncorrelated noise process with E zk = 0,
var(zk ) = σ 2 < ∞. Throughout the paper we assume
that f (t) has a bounded spectrum and that f (t) is a finite
energy type signal. Any signal with such a property is
referred to as band-limited and will denote this class of
signals as BL(Ω), where Ω is the bandwidth of f (t).
The celebrated Whittaker-Shannon theorem says that
SAMPTA'09
any band-limited signal f (t) can be perfectly recovered
from its discrete values {f (kτ )} provided that τ ≤ π/Ω.
Application of the resulting interpolation formula to
noisy data would lead to the following reconstruction
scheme based on 2n + 1 random samples
X
fn (t) =
yk sinc πτ −1 (t − kτ ) ,
(1)
|k|≤n
where sinc(t) = sin(t)/t, and τ ≤ π/Ω. The fundamental question, which arises is whether fn (t) can
be a consistent estimate of f (t) for any f ∈ BL(Ω).
Hence, whether ̺(fn , f ) → 0 as n → ∞, in a certain
probabilistic sense, for some distance measure ̺. Since
f (t) is assumed to be square integrable, then the natural
measure between fn (t) and f (t) is the mean integrated
square error
Z ∞
(fn (t) − f (t))2 dt.
(2)
M ISE(fn ) = E
−∞
It can be easily shown, see [6], that M ISE(fn ) → ∞
as n → ∞ for any fixed τ ≤ π/Ω. This unpleasant
property of the estimate fn (t) is caused by the presence
of the noise process in the observed data and the fact
that fn (kτ ) = yk , i.e., fn (t) interpolates the noisy observations. It is clear that one should avoid interpolation
schemes in the presence of noise since they would retain random errors. The aim of this paper is to propose
a consistent estimate of f (t) being a smooth correction
of the naive algorithm fn (t) . This task is carried out
by sampling a signal at irregularly spaced quasi-random
points and by carefully selecting the number of terms in
the sampling series. The conditions for consistency of
our estimate are established and the corresponding rate
of convergence is evaluated.
The statistical aspects of signal sampling and recovery
have been examined first in [5], and next in [6], [7], [2],
[1]. In [2], [1] the sampling rate τ has been assume to be
a fixed constant. This assumption, however, cannot lead
to consistent estimates of the true signal of the bandlimited type. On the other hand, in [5], [6], [7] τ = τn
239
such that τn → 0 as n → ∞ with a controlled rate. Such
a choice of τ allows us to design a signal recovery algorithm for which the reconstruction error M ISE tends
to zero with a certain rate. In this paper, we propose a
nonlinear sampling scheme based on the theory of quasirandom sequences, i.e., we observe the following noisy
samples
yk = f (τk ) + zk ,
where {τk } is a sequence of quasi-random points. We
show that a proper choice of {τk } leads to the reconstruction algorithm with the improved convergence rate.
2.
Reconstruction Algorithms with QuasiRandom Points
The notion of quasi-random sequences has been originally established in the theory of numerical integration
[4]. A sequence of real numbers {xj } is said to be a
quasi-random sequence in [0, 1] if for every continuous
function b(x) on [0, 1] we have
n
1X
b(xj ) =
n→∞ n
j=1
lim
Z
1
b(x) dx.
(3)
0
Quasi-random sequences are also called equidistributed
sequences, since (3) means that the sequence {xj } behaves like uniformly distributed random variables. Nevertheless, an important property of quasi-random sequences is that they are more uniform than random uniform sequences which tend to clump. A consequence of
this fact is that the accuracy of approximating integrals
based on quasi-random sequences is superior to the accuracy obtained by random sequences. In fact, the celebrated Koksma-Hlawka inequality [4] says that for any
function of bounded variation on [0, 1] we have
n−1
n
X
j=1
f (xj ) −
Z
0
1
f (t)dt ≤ V(f )Dn∗ ,
where V(f ) is the total variation of f on [0, 1], and Dn∗
denotes the so-called discrepancy of the quasi-random
sequence {xj }. The discrepancy measures the strength
of the sequence to approximate the uniform distribution
on [0, 1]. There are quasi-random sequences with discrepancy of order O(log(n)/n) [4]. This should be contrasted with a random sequence of uniformly distributed
points√on [0, 1] that possesses the discrepancy of order
O(1/ n). This basic observation plays a key role in
our developments concerning the signal recovery problem from quasi-random points. Numerous quasi-random
sequences have been constructed that have the aforementioned property of approximating the uniform distribution. The simplest, and sufficient for our purposes, way
SAMPTA'09
of generating a quasi-random sequence is the following
xj = frac(j ϑ),
(4)
where ϑ is an irrational number and frac(.) denotes the
fractional part of
√ a number in the parenthesis. A good
choice of ϑ is ( 5 − 1)/2, see [8] for an extensive discussion on the choice of ϑ.
Since band-limited signals are defined on the whole real
line we need a rescaled version of quasi-random sequences. Thus, let us define the following sampling
points on the interval [−T, T ]
τj = T sgn(j) frac(|j| ϑ), j = 0, ±1, ±2, . . . , n, (5)
where sgn(.) is the sign of a number. The observation
horizon T must increase with n such that T (n) → ∞
as n → ∞. In order, however, to establish the consistency result of our reconstruction algorithm we must
control the growth of T (n). The approximation property of quasi-random sequences applied to the sequence
defined in (5) reads now as
Z T
2T X
f (τj ) ≈
f (t) dt.
2n + 1
−T
(6)
|j|≤n
It has been known since the work of Hardy [3] that the
cardinal expansion can be viewed as the orthogonal expansion in BL(Ω). Using this fact and then the reasoning as in [5] we can define the following estimate of f (t)
f˜n (t) =
X
c̃k sk (t) ,
(7)
|k|≤N
c̃k
=
X
2T
yj sk (τj ) ,
(2n + 1) h
(8)
|j|≤n
where {τj } is the quasi-random sequence defined in (5).
Here {sk (t) = sinc(πh−1 (t − kh)), k = 0, ±1, . . .}
forms the orthogonal and complete system in BL(Ω)
provided that h ≤ π/Ω
R ∞. The corresponding Fourier
coefficient is ck = h−1 −∞ f (t)sk (t)dt. It is also clear
that for f ∈ BL(Ω) we have ck = f (kh). The parameter h is called the reconstruction rate. In (7) the parameter N defines the number of terms in the expansion
which are taken into account and 2n + 1 is the sample
size. The truncation parameter plays important role in
our asymptotic analysis, i.e., N depends on n such that
N (n) → ∞ with the controlled rate. It is also worth noting that the sampling rate is nonuniform (defined by the
discrepancy of the quasi-random sequence in (5)) and
different than the reconstruction rate h. We assume that
h is constant and not greater than π/Ω.
Throughout the paper we use the worst localized base
system utilizing the sinc function. The methodology
240
presented in this paper can be extended to the windowed
version of the estimate f˜n (t) of the form
fˆn (t) =
X
wk c̃k sk (t),
|k|≤n
where {wk , |k| ≤ n} is a sequence of numbers such
that 0 ≤ wk ≤ 1. The proper choice of this window
sequence yields an estimate with better time-localized
properties and consequently better convergence rates.
The case when wk = 1 for |k| ≤ N and wk = 0 otherwise corresponds to the estimate f˜n (t).
3.
The MISE Consistency and Rate
This assumption can be also expressed in the frequency domain by requiring that the Fourier transform
of f (t) has r derivatives on [−Ω, Ω]. A further analysis
of the reconstruction error leads to the following bound
M ISE(f˜n ) ≤ (2 N + 1) C1 T −(2r+1)
C2 T 3 log2 (n) C3 T
+
+
(9)
n2
n
+C4 N −(2r+1) ,
for some constants C1 , C2 , C3 , C4 . By optimizing the
above bound we can obtain the following asymptotically
optimal choice of T (n) and N (n).
1
In this section we summarize the results concerning the
convergence of MISE(f˜n ) to zero as n → ∞ for any
signal f ∈ BL(Ω). Also the rate of convergence is established.
Due to Pareseval’s formula we can decompose the
M ISE(f˜n ) as follows:
M ISE(f˜n ) = h
X
var(c̃k ) + h
|k|≤N
+h
X
|k|≤N
X
(Ec̃k − ck )2
c2k .
T ∗ (n) = an 2r+3
1
N ∗ (n) = bn 2r+3 ,
subject to the condition a > bh. Plugging these values
of T (n) and N (n) back into the bound for M ISE(f˜n )
we obtain the following rate
2r+1
M ISE(f˜n ) = O(n− 2r+3 ).
It is worth noting that under Assumption (F) the best
possible rate obtained for the reconstructionr algorithms
discussed in [6] and [7] is of order O(n− r+1 ). This is
clearly a slower rate than the one obtained in this paper.
|k|≥N
The first term of the decomposition controls the stochastic part of the estimate, whereas the the remaining term
describe the systematic error (bias) of the estimate. A
careful examination of these terms lead to the following
result on the consistency of our estimate.
Theorem 1 Let f ∈ BL(Ω) and let the reconstruction
rate h be constant such that h ≤ π/Ω. Suppose that
N (n) < T (n)/h. Assume T (n) → ∞, N√(n) → ∞
such that T (n) does not grow faster than n/log(n).
Let, moreover,
N (n)T (n)
→ 0.
n
Then
M ISE(f˜n ) → 0
as n → ∞.
The conditions required on the parameters T (n) and
N (n) in Theorem 1 impose some general restrictions on
their growth. In order further see how to choose T (n)
and N (n) let us assume the following condition on the
decay of band-limited signals.
(F) There exists r ≥ 0 and a constant Cf > 0 such
that for |t| sufficiently large we have |f (t)| ≤ Cf /|t|r+1 .
SAMPTA'09
4.
Concluding Remarks
In this paper we have proposed an algorithm for recovering a band-limited signal observed under noise. Assuming that the signal is a square integrable function
the sufficient conditions for the convergence of the mean
integrated square error have been established. The distinguishing feature of the proposed approach is its utilization of nonuniform samples taken at quasi-random
points. When quasi-random sequences are applied to
the problem of numerical evaluation of integrals they
reveal the approximation rate O(log(n)/n) for a class
of bounded variation
functions. This rate is superior to
√
the rate O(1/ n) that characterizes usual numerical algorithms and classical Monte Carlo methods. This advantage of quasi-random sequences seems to be carried
out to the problem of signal sampling and recovery. In
our consistency results we assume that the reconstruction rate h is constant and could be chosen as large as
π/Ω. One could also consider the case when h = h(n)
and h(n) → 0 as n → ∞. The estimates with variable
h would be needed for the problem of recovering not
necessarily band-limited signals. Finally, let us mention
that the results of this paper can be extended to the ddimensional case, where the orthogonal system can be
obtained in the form of the product of sinc functions,
Qd
i.e., sk (t) = i=1 ski (ti ), where k = (k1 , k2 , . . . , kd ),
241
t = (t1 , t2 , . . . , td ). We should mention that multidimensional quasi-random sequences can be generated in
a relatively straightforward way. Moreover, they exhibit
the favorite discrepancy of order O(n−1 (log(n))d ) for
any d. This fact may have important consequences for
sampling problems of two-dimensional objects like images.
5.
Acknowledgements
The work of E. Rafajłowicz was supported by the Research and Development Grant from the Ministry of Science and Higher Education of Poland.
References:
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analysis of frame reconstruction from noisy samples. IEEE Trans. Signal Processing, 56:2311–
2315, 2008.
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and interpolation from noisy observations in shiftinvariant spaces. IEEE Trans. Signal Processing,
54:2636–2651, 2006.
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Whittaker’s cardinal series. Proc. Camb. Phil. Soc.,
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of band-limited signals. IEEE Trans. Information
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from noisy samples. IEEE Trans. Information Theory, 49:3195–3212, 2003.
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and recovery under dependent noise. IEEE Trans.
Information Theory, 53:2526–2541, 2007.
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designes in nonparametric regression. Statistica
Sinica, 13:129–142, 2003.
[9] M. Unser. Sampling – 50 years after Shannon. Proceedings of the IEEE, 88:569–587, 2000.
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Theory and Applications, 48:1094–1109, 2001.
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242
On the incoherence of noiselet and Haar bases
Tomas Tuma, Paul Hurley
IBM Research, Zurich Laboratory 8803 Rüschlikon, Switzerland
E-mail: {uma,pah}@zurich.ibm.com
Abstract:
Noiselets are a family of functions completely uncompressible using Haar wavelet analysis. The resultant perfect incoherence to the Haar transform, coupled with the
existence of a fast transform has resulted in their interest and use as a sampling basis in compressive sampling.
We derive a recursive construction of noiselet matrices and
give a short matrix-based proof of the incoherence.
whenever the products AC, BD exist. This property is
sometimes called the mixed product property.
Definition 2. Let A be a m × n matrix. A(k, ∗) denotes the (row) vector (A(k, 1) A(k, 2) . . . A(k, n))
while, A(∗, l) similarly denotes the (column) vector
(A(1, l) A(2, l) . . . A(m, l))T .
2.2
1.
Introduction
The noiselet basis, originally described in [2], has garnered interest recently because noiselets (1) are maximally
incoherent to the Haar basis and (2) have a fast algorithm
for their implementation. Thus, they have been employed
in compressive sampling to sample signals that are sparse
in the Haar domain [1].
The work presented here was motivated by the observation
that it had not been previously shown in a straightforward
way that the discrete Haar transform is maximally incoherent to a discretized version of the noiselet transform.
Additionally, the exact form of a noiselet matrix needed to
be inferred from the original work.
The main contributions are the derivation of a recursive,
tensor product-based, construction of noiselet matrices,
the unitary matrices that result from the noiselet transform
for discrete input, and an intuitive proof showing its incoherence to the corresponding Haar matrix.
2.
2.1
Noiselets
Noiselets [2] are functions that are completely uncompressible under the Haar transform. The family of noiselets is constructed on the interval [0, 1) as follows:
f1 (x) = χ[0,1) (x),
f2n (x) = (1 − i)fn (2x) + (1 + i)fn (2x − 1)
f2n+1 (x) = (1 + i)fn (2x) + (1 − i)fn (2x − 1)
Here, χ[0,1) (x) = 1 on the definition interval [0, 1) and 0
otherwise. It is shown in [2] that {fj } is a basis:
Theorem 1. The set {fj |j = 2N , . . . , 2N +1 − 1} is an
orthogonal basis of the vector space V2N , which is the
space of all possible approximations at the resolution 2N
of functions in L2 [0, 1).
2.3
Haar Transform
Haar wavelet transform can be described by a real square
matrix. For our purposes, it is advantageous to recursively
build the Haar matrix using the Kronecker product [3]:
Preliminaries
General definitions
Definition 1. Let A be an m×n matrix, and B be a matrix
of an arbitrary size. The Kronecker product of A and B is
a11 B · · · a1n B
.. .
..
A ⊗ B = ...
.
.
am1 B
···
amn B
The Kronecker product (see e.g. [4]) is a bilinear and associative operator which is not generally commutative. It
can be combined with a standard maxtrix multiplication as
follows:
(A ⊗ B)(C ⊗ D) = AC ⊗ BD
SAMPTA'09
1 Hn/2 ⊗ (1 1)
Hn = √
.
2 In/2 ⊗ (1 −1)
The iteration starts with H1 = 1 . The normalization
constant √12 ensures that HnT Hn = I. Haar wavelets are
the rows of Hn .
3.
Matrix construction of noiselets
First we extend and discretize the noiselet functions.
Definition 3. The extensions of noiselets to the interval
[0, 2m − 1] sampled at points 0, . . . , 2m − 1 is the series
243
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
1
2
3
4
5
6
7
1
8
(a) Real part of 8x8 noiselet matrix
2
3
4
5
6
7
8
(b) Imaginary part of 8x8 noiselet matrix
10
10
20
20
30
30
40
40
50
50
60
60
10
20
30
40
50
60
10
(c) Real part of 64x64 noiselet matrix
20
30
40
50
60
(d) Imaginary part of 64x64 noiselet matrix
Figure 1: Noiselet matrix: graphical view. In figures (a) and (b), the black and white colors denote values of −0.25
and 0.25 respectively. In figures (c) and (d), the black, gray and white colors denote values of −0.125, 0 and 0.125
respectively. .
of functions fm (k, l)
(
1 l = 0, . . . , 2m − 1
fm (1, l) =
0 otherwise
Lemma 1. Let m > 0.
The noiselet matrices
N1 , N2 , N4 , . . . , N2m are built up from a series of discretised and extended noiselets fm :
m
fm (2k, l) = (1 − i)fm (k, 2l) + (1 + i)fm (k, 2l − 2 )
fm (2k + 1, l) = (1 + i)fm (k, 2l) + (1 − i)fm (k, 2l − 2m )
where m denotes the range of extension, k = 1, . . . , 2m+1
is the function index and l = 0, . . . , 2m − 1 is the sample
index.
Starting with a 1 × 1 matrix N1 , a sequence of noiselet
matrices N1 , N2 , N4 , . . . , N2m of sizes 1 × 1, 2 × 2, 4 × 4,
. . . , 2m × 2m , respectively, is generated. The rows of the
Nn matrix are noiselets which form an orthonormal basis
for the space Cn .
Definition 4. For n = 1, N1 = 1 . Then the n × n
noiselet matrix Nn is built up recursively according to:
Nn (k, ∗) =
1
k
(1 − i 1 + i) ⊗ Nn/2 ( , ∗)
2
2
when k = 0, 2, 4, . . . , n − 2 and
Nn (k, ∗) =
1
k−1
(1 + i 1 − i) ⊗ Nn/2 (
, ∗)
2
2
when k=1,3,. . . ,n-1.
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Nn (k, l) = fm (n + k,
2m
l), k, l = 0, . . . , n − 1.
n
Proof. Let m > 0 be fixed. For n = 1
N1 (0, 0) = fm (1, 0) = 1.
By induction, for a matrix of size n = 2p , p = 1, . . . , m,
its basis vector k = 0, 2, 4, . . . , n − 2 and vector indices
l = 0, . . . , n2 − 1
k
Nn (k, l) = (1 − i)Nn/2 ( , l)
2
n k 2m
2m
l).
= (1 − i)fm ( + , n l) = fm (n + k,
2
2 2
n
For the same n, k and l =
n
2,...,n
− 1,
k
n
Nn (k, l) = (1 + i)Nn/2 ( , l − )
2
2
n k 2m l
2m
= (1 + i)fm ( + , 2
− 2m ) = fm (n + k,
l).
2
2
n
n
To see this, observe that fm is zero outside of [0, 2m − 1]
and therefore, the first half of samples of fm (k, l) are
defined exclusively by the expression (1 ± i)fm (k, 2l)
244
whereas the second half of the samples are defined exclusively by (1 ± i)fm (k, 2l − 2m ).
For k odd (k = 1, 3, . . . , n − 1) the proof is similar.
Specially, the noiselet matrix Nn for n = 2m can be found
as the “tail” of the function series fm . Indeed, the expression in Theorem 1 becomes N (k, l) = fm (n + k, l) for
n = 2m .
4.
Incoherence of noiselets and Haar
In what follows, we adhere to the terminology of basis coherence which is common in the field of compressive sampling. See for example [1] for details on these definitions
and related literature.
Mutual coherence of two bases is defined as the maximum
scalar product of any pair of their basis vectors:
Definition 5. Mutual coherence between two orthonormal
bases Ψ, Φ is
µ(Ψ, Φ) = max|hψk , φj i|.
k,j
The minimal coherence is usually termed maximal or perfect incoherence, which means that µ(Ψ, Φ) = O(1). In
other words, the matrix of scalar products ΨΦ∗ is “flat”.
As Candès and Romberg suggest [1], we will show the
perfect incoherence of Haar and noiselets in the following
setting. Given an orthonormal n × n Haar matrix H, we
compute the matrix of scalar products for a corresponding
noiselet matrix N normalized such that N ∗ N = nI. By
doing so, the product will be flat with all values having the
magnitude of 1.
For clarity of the main proof, it saves some technical work
to define a “twisted” noiselet basis.
Definition 6. The twisted noiselet matrix N̂1 = 1 .
Then the n × n twisted noiselet matrix N̂n is built up recursively by
N̂n (k, ∗) =
k
1
N̂n/2 ( , ∗) ⊗ (1 − i 1 + i)
2
2
when k = 0, 2, 4, . . . , n − 2 and
N̂n (k, ∗) =
1
k−1
N̂n/2 (
, ∗) ⊗ (1 + i 1 − i)
2
2
when k = 1, 3, . . . , n − 1.
The difference between this and the definition of the noiselet matrix N (Definition 4) is that the order of operands in
the Kronecker product is changed. In fact, each one is just
a permutation of the other.
Lemma 2. For n = 2m , the bases Nn , N̂n consist of the
same set of basis vectors.
The claim holds for n = 1. For n = 2, 4, 8, . . . , 2m ,
Pn Nn (k, l) = Pn (k, ∗)Nn (l, ∗)T
as it can easily be shown that Nn is symmetric. Using the
recurrent equations for Pn and Nn and applying the mixed
product rule, we get, for k = 0, 2, 4, . . . , n − 2,
1
k
l
(1 − i)Pn/2 ( , ∗)Nn/2 (∗, )
2
2
2
when l = 0, 2, 4, . . . , n − 2 and
Pn Nn (k, l) =
1
k−1
l
(1 + i)Pn/2 (
, ∗)Nn/2 (∗, )
2
2
2
when l = 1, 3, . . . , n − 1. By induction,
Pn Nn (k, l) =
1
k
N̂n/2 ( , ∗) ⊗ (1 − i 1 + i)
2
2
for even k indices. This situation for odd k is similar.
Pn Nn (k, ∗) =
Now the main result can be shown.
Theorem 2. Let n = 2m where m is a non-negative integer. Let Nn be the noiselet matrix of size n × n and let
Hn be the Haar matrix of size n × n. Then Hn and Nn
are maximally incoherent.
Proof. Without loss of generality, assume the bases are
normalized such that HnT Hn = I and Nn∗ Nn = nI. For
the case of n = 1,
H1 N1∗ = 1 · 1 = 1
For n = 2m , m > 1, the incoherence is shown by induction. Suppose we know maximal incoherence holds for n2
and we want to show it for n. In the induction step, we use
the iterative construction of the Haar matrix by means of
Kronecker product. By computing the product
Hn N̂n∗ = H(Nn∗ Pn∗ ) = (Hn Nn∗ )PnT
we will still be able to conclude on magnitude of the elements of (Hn Nn∗ ), since permutation matrices do not
change magnitudes.
The product Hn N̂n∗ can be computed per-column; we take
the j-th column of N̂n∗ , j = 0, 2, 4, . . . , n − 2 and transform it by Hn , getting
1 Hn/2 ⊗ (1 1)
∗
Hn N̂n (∗, j) = √
2 In/2 ⊗ (1 −1)
j
1 ∗
(∗, ) ⊗ (1 − i 1 + i)∗
· √ N̂n/2
2
2
Note the altered normalization factor of noiselets. Now
the mixed product property can be applied to get
1+i
j
∗
Hn/2 N̂n/2 (∗, 2 ) ⊗ (1 1)
1
1 − i =
1
+
i
2
∗
In/2 N̂n/2
(∗, 2j ) ⊗ (1 −1)
1−i
"
#
∗
(∗, 2j ) ∗ 2
1 Hn/2 N̂n/2
.
∗
2 In/2 N̂n/2
(∗, 2j ) ∗ 2i
Proof. Indeed, we can write N̂n = Pn Nn where P is the
permutation matrix:
∗
(i, 2j )| = 1 and
By induction, it follows that |Hn/2 N̂n/2
(
∗
(i, 2j )| = 1 for i = 1, . . . , n2 . The Kronecker
(1 0) ⊗ Pn/2 ( k2 , ∗)
k = 0, 2, 4, . . . , n − 2 |In/2 N̂n/2
P (k, ∗) =
multiplication is only by entries with magnitude 2, thus the
(0 1) ⊗ Pn/2 ( k−1
2 , ∗) k = 1, 3, . . . , n − 1
resulting magnitudes are 12 ∗2 = 1. The proof is equivalent
for j = 1, 3, . . . , n − 1.
starting with P = [1].
SAMPTA'09
245
References:
[1] Emmanuel Candès and Justin Romberg. Sparsity and
incoherence in compressive sampling. Inverse Problems, 23(3):969–985, 2007.
[2] R. Coifman, F. Geshwind, and Y. Meyer. Noiselets. Applied and Computational Harmonic Analysis,
10:27–44, 2001.
[3] B.J. Falkowski and S. Rahadja. Walsh-like functions
and their relations. In IEE Proceedings on Vision, Image and Signal Processing, volume 143, pages 279 –
284, 1996.
[4] Alan J. Laub. Matrix Analysis for Scientists and Engineers. SIAM, 2005.
SAMPTA'09
246
Adaptive compressed image sensing based on
wavelet modeling and direct sampling
Shay Deutsch (1), Amir Averbuch (1) and Shai Dekel (2)
(1) Tel Aviv University. Israel
(2) GE Healthcare, Israel
shayseut@post.tau.ac.il, Shai.dekel@ge.com, amir@math.tau.ac.il
1.2 The “single pixel” camera
Abstract:
We present Adaptive Direct Sampling (ADS), an
algorithm for image acquisition and compression which
does not require the data to be sampled at its highest
resolution. In some cases, our approach simplifies and
improves upon the existing methodology of Compressed
Sensing (CS), by replacing the ‘universal’ acquisition of
pseudo-random measurements with a direct and fast
method of adaptive wavelet coefficient acquisition. The
main advantages of this direct approach are that the
decoding algorithm is significantly faster and that it
allows more control over the compressed image quality,
in particular, the sharpness of edges.
1. Introduction
Compressed Sensing (CS) [1, 3, 4, 6] is an approach to
simultaneous sensing and compression which provides
mathematical tools that, when coupled with specific
acquisition hardware architectures, can perhaps reduce
the acquired dataset sizes, without reducing the
resolution or quality of the compressed signal. CS builds
on the work of Candès, Romberg, and Tao [4] and
Donoho [6] who showed that a signal having a sparse
representation in one basis can be reconstructed from a
small number of non-adaptive linear projections onto a
second basis that is incoherent with the first. The
mathematical framework of CS is as follows:
Consider a signal x N that is k -sparse in the basis
for N . In terms of matrix representation we have
x f , in which f can be well approximated using
only k N non zero entries and is called the sparse
basis matrix. Consider also an n N measurement
matrix , where the rows of are incoherent with the
columns of . The CS theory states that such a good
approximation of signal x can be reconstructed by taking
only n O ( k log N ) linear, non adaptive measurements
as follows: [1, 3]:
(1.1)
y x ,
where y represents an n 1 sampled vector. Working
under this ‘sparsity’ assumption an approximation to x
can be reconstructed from y by ‘sparsity’ minimization,
such as l1 minimization
SAMPTA'09
1 f y
min
f
l1
(1.2)
For imaging applications, the CS framework has been
applied within a new experimental architecture for a
‘single pixel’ digital camera [10]. The CS camera
replaces the CCD and CMOS acquisition technologies by
a Digital Micro-mirror Device (DMD). The DMD
consists of an array of electrostatically actuated micromirrors where each mirror of the array is suspended
above an individual SRAM cell. In [10] the rows of the
CS sampling matrix are a sequence of n pseudorandom binary masks, where each mask is actually a
‘scrambled’ configuration of the DMD array (see also
[2]). Thus, the measurement vector y , is composed of
dot-products of the digital image x with pseudo-random
masks. At the core of the decoding process, that takes
place at the viewing device, there is a minimization
algorithm solving (1.2). Once a solution is computed,
one obtains from it an approximate ‘reconstructed’ image
by applying the transform to the coefficients. The CS
architecture of [10] has few significant drawbacks:
1. Poor control over the quality of the output
compressed image: the CS architecture of [10] is not
adaptive and the number of measurements is
determined before the acquisition process begins,
with no feedback during the acquisition process on
the progressive quality.
2. Computationally intensive sampling process: Dense
measurement matrices such as the sampling operator
of the random binary pattern are not feasible because
of the huge space and multiplication time
requirements. Note that in the one single pixel
camera, the sampling operator is based on the
random binary pattern, which requires a huge
memory and a high computation cost. For example,
to get 512 512 image with 64k measurements
(25% sampling rate) a random binary operator
requires nearly a gigabyte of storage and Giga-flop
operations, which makes the recovery almost
impossible [14]. The designing of an efficiently
measurement basis was proposed [14, 16] by using
highly sparse measurements operators, which solve
the infeasibility of Gaussian measurement matrix or
a random binary masks such as in the one pixel
camera. Note, however, in [16], the trade-off
between acquisition time and visual quality. To
obtain good visual quality, when using TV
minimization (which significantly increase the
decoding time, compared to LP decoding time)
247
3.
recovery times of a 256 256 ‘boat’ image are
around 60 min.
Computationally intensive reconstruction algorithm:
It is known that all the algorithms for the
minimization (1.2) are very computationally
intensive.
2. Direct and adaptive image sensing
Our proposed architecture aims to overcome the
drawbacks of the existing CS approach and achieve the
following design goals:
1. An acquisition process that captures n measurements,
with n N and n O k , where N is the dimension
of the full high-resolution image, assumed to be ‘ k sparse’. The acquisition process is allowed to adaptively
take more measurements if needed to achieve some
compressed image target quality.
2. A decoding process which is not more
computationally intensive than the existing algorithm in
use today such as JPEG or JPEG2000 decoding.
We now present our ADS approach: Instead of
acquiring the visual data using a representation that is
incoherent with wavelets, we sample directly in the
wavelet domain. We use the DMD array architecture in a
very different way than in [10]:
1. Any wavelet coefficient is computed from two
measurements of the DMD array.
2. We take advantage of the ‘feedback’ architecture of
the DMD where we make decisions on future
measurements based on values of existing measurements.
This adaptive sampling process relies on a well-known
modeling of image edges using a wavelet coefficient
tree-structure and so decisions on which wavelet
coefficients should be sampled next are based on the
values of wavelet coefficients obtained so far [8, 9]. First
we explain how the DMD architecture can be used to
calculate a wavelet coefficient from two DMD
measurements.
Modeling
an
image
as
a
we
have
the
wavelet
function f L2 2 ,
representation f x f , ej ,l ej ,l , where e 1, 2,3
is the subband, j the scale and l 2 the shift. For
measurements. Moreover, there exist DMD arrays with
micro-mirrors that can produce a grayscale value, not
just 0 or 1 (contemporary DMD can produce 1024
grayscale value). We can use these devices for
computation of arbitrary wavelet transforms, where the
computation of each coefficient requires only two
measurements, since the result of any real-valued
functional g acting on the data can be computed as a
difference of two ‘positive’ g , g ‘functionals’, i.e.
,where
the
coefficients
are
positive:
g g g ,g , g 0 .
3. Modeling of image edges by wavelet treeStructures and the ADS algorithm
Most of the significant wavelet coefficients are located in
the vicinity of edges. Wavelets can be regarded as multiscale local edge detectors, where the absolute value of a
wavelet coefficient corresponds to the local strength of
the edge. We impose the tree-structure of the wavelet
coefficients. Due to the analysis properties of wavelets,
coefficient values tend to persist through scale. A large
wavelet coefficient in magnitude generally indicates the
presence of singularity inside its support. A small
wavelet coefficient generally indicates a smooth region.
We use this nesting property and acquire wavelet
coefficients in the higher resolutions if their parent is
found to be significant. For further detection of
singularities at fine scales, we estimate the Lipschitz
exponent.
3.1 The Lipschitz exponent
Our goal is to estimate the significance of wavelet
coefficients that were not sampled yet, using values of
coefficients that were already sampled. To this end we
use the well known characterization of local Lipschitz
smoothness by the decay of wavelet coefficients across
scales [12]. A function f is said to be Lipschitz in
the neighborhood of ( x1 , x2 ) if there exists 1 and 2 as
well as A 0 such that for any h1 1 and h2 2
f ( x1 h1 , x2 h2 ) f ( x1 , x2 ) A( h12 h22 ) / 2
(3.1)
e , j ,l
orthonormal wavelets ej ,l ej , l . If we consider the Haar
basis as an example, than a bivariate Haar wavelet
coefficient of type 1 can be computed as follows
2 l1 1
f , 1j ,l 2 j
2 j l
1
j
2 j l2 1 2
2 j l2
f x1 , x2 dx1dx2
2 j l1 1 2 j l2 1
2 j l1
2 j l2 1 2
f x1 , x2 dx1dx2 ,
(2.1)
i.e., the difference of pixel sums over two neighboring
dyadic rectangles multiplied by 2 j . By Similar
computation we can sample the Haar wavelet
coefficients of the second and third kinds with two
SAMPTA'09
We actually use a subtler, ‘directional’ notion of local
Lipschitz smoothness. So, for example, for the horizontal
subband, e 1 , we defined local 1 Horizontal
Lipschitz smoothness by the minimal A 0 satisfying
for h1 1
f ( x1 h1 , x2 ) f ( x1 , x2 ) Ah11 .
If the function is locally e Lipschitz at ( x1 , x2 ) then
for any wavelet ej,l whose support contains ( x1 , x2 ) ,
f , ej , l C 2 j . By taking the
e
we have that
logarithm we have
log 2 f , ej, l j log 2 (C ) .
(3.3)
248
Thus the Lipschitz exponents can be determined from
log 2 f ,
the slope of the decay of
e
j,l
across scales
(see also [15]).
These slopes are considered
measurements of local singularities, such that when
0 e 1 a function f has a directional singularity
which increases as e 0 . Thus we estimate the
existence of local directional singularities and the
significance of unsampled coefficients at high scales,
using estimates of local directional Lipschitz exponents
from wavelet coefficients that were already sampled.
3.2 The ADS Algorithm
Our adaptive CS algorithm works as follows:
1. Acquire the values of all low-resolution coefficients
up to a certain low-resolution J . Each computation is
done using two DMD array measurements as in (2.1). In
one embodiment the initial resolution J can be selected
log N
as 2 const . In any case, J should be bigger if
2
the image is bigger. Note that the total number of
2 1 J
coefficients at resolutions J is 2 N , which is a
small fraction of N .
2. Initialize a ‘sampling queue’ containing the indices of
each of the four children of significant coefficients at the
resolution J . Thus for a significant coefficient with
index e, J , l , we add to the queue the coefficients with
e, J 1, 2l , 2l , e, J 1, 2l , 2l 1 ,
e, J 1, 2l 1, 2l and e, J 1, 2l 1, 2l 1 .
indices:
1
1
2
2
1
1
2
2
3. Process the sampling queue until it is exhausted as
follows:
a. Sample the wavelet coefficient corresponding to
the index e, j , l at the beginning of the queue using
two DMD array measurements (see Section 2).
b. Add to the end of the queue the indices of the
coefficient’s four children, only if one of the following
holds:
(i) The coefficient is at a resolution j J 2 and
the coefficient’s absolute value is greater than a given
threshold tlow .
wavelet coefficients, which can be substantially smaller
than the number of pixels N . The number of samples is
influenced by the size of the thresholds used by the
algorithm in step 3.b. It is also important to understand
that the number of samples is influenced by the amount
of visual activity in the image. If there are more
significant edges in the image, then their detection at
lower resolutions will lead to adding higher resolution
sampling to the queue.
4. Experimental results
To evaluate our approach, we use the optimal k -term
wavelet approximation as a benchmark. It is well known
[5] that for a given image with N pixels, the optimal
orthonormal wavelet approximation using only
k coefficients is obtained using the k largest
coefficients
f , ej11, l1 f , ej22 ,l2 f , ej33,l3 ,
f f , ejii ,li ejii ,li
k
i 1
L2
2
min f
# k
e , j , l
f , ej ,l ej ,l
For biorthogonal wavelets this ‘greedy’ approach gives a
near-best result, i.e. within a constant factor of the
optimal k -term approximation. One can apply
thresholding and construct a k -term approximation
using only coefficients whose absolute value is above the
threshold, which still requires the order of N
computations. In contrast, our ADS algorithm is output
sensitive and requires only order of n computations. To
simulate our algorithm in software, we first pre-compute
the entire wavelet transform of a given image. However,
we strictly follow the recipe of our ADS algorithm and
extract a wavelet coefficient from the pre-computed
coefficient matrix only if its index was added to the
adaptive sampling queue. In fig 1(a) we see a
‘benchmark’ near-best 7000-term biorthogonal [9,7]
wavelet approximation of the Lena image, extracted
from the ‘full’ wavelet representation by thresholding.
In fig 1(b) we see a 6782-term approximation extracted
from an ADS adaptive sampling process with n =12796
sampled wavelet coefficient.
(ii) The coefficient is at resolution 1 j J 2 and
the corresponding estimated absolute value of its
children using the local Lipschitz exponent method (see
Section 3.1) is greater than a given threshold thigh .
c. Remove the processed index from the queue and
go to step (a).
In a way, our algorithm can be regarded as an adaptive
edge acquisition device where the acquisition resolution
increases only in the vicinity of edges! Observe that the
algorithm is output sensitive. Its time complexity is of
the order n where n is the total number of computed
SAMPTA'09
L2 2
249
.
REFERENCES
(a) 7000-term
(b) ADS 6782-term
Fig.1. (a) Near-best 7000-term [9,7] approximation
computed from the ‘full’ wavelet representation
N=262,144, PSNR=31 dB (b) ADS 6782-term [9,7]
approximation, extracted from n=12,796 adaptive
wavelet samples, PSNR=28.7 dB.
5. Conclusion
We present an architecture that acquires and compresses
high resolution visual data, without fully sampling the
entire data at its highest resolution. By sampling in the
wavelet domain we are able to acquire low resolution
coefficients within a small number of measurements. We
then exploit the wavelet tree structure to build an
adaptive sampling process of the detail wavelet
coefficients. Experimental results show good visual and
PSNR results with a small number of measurements. The
coefficients acquired by the ADS algorithm can be
streamed into a tree-based wavelet compression
algorithm whose decoding time is significantly faster
then the solution of (1.2).
SAMPTA'09
1. R. Baraniuk, Compressive Sensing, Lecture Notes in
IEEE Signal Processing Magazine, Vol. 24, No. 4, pp.
118-120, July 2007.
2. R. Baraniuk, M. Davenport, R. DeVore, and
M. Wakin, A simple proof of the restricted isometry
property
for
random
matrices,
Constructive
Approximation 28 (2008), 253-263.
3. E. Candès, Compressive sampling, Proc. International
Congress of Mathematics, 3 (2006), 1433-1452.
4. E. Candès, J. Romberg, and T. Tao, Robust
uncertainty principles: Exact signal reconstruction from
highly incomplete frequency information, IEEE Trans.
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(1998), 50-51.
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Information Theory, 52 (2006), 1289-1306.
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September 2005.
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image codec based on set partitioning in hierarchical
trees, IEEE Trans. Circuits Syst. Video Tech., 6 (1996),
243-250.
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of wavelet coefficients, IEEE Trans. Signal Process. 41
(1993), 3445-3462.
10 .D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron,
S. Sarvotham, K. Kelly and R. Baraniuk, A New
Compressive Imaging Camera Architecture using
Optical-Domain Compression, Proc. of Computational
Imaging IV , SPIE, 2006.
11. S. Dekel, Adaptive compressed image sensing based
on wavelet-trees, report 2008.
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and processing with wavelets,” IEEE Trans. Inf. Theory
38,617-642(1992).
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using scrambled Hadamard transform ensemble, preprint
2008.
15. Z. Chen and M. A. Karim, Forest representation of
wavelet transforms and feature detection, Opt. Eng. 39
(2000), 1194–1202.
16. R. Berinde, P. Indik, sparse recovery using sparse
random matrices, Tech. Report of MIT 2008.
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image blur in the wavelet domain, IEEE Benelux Signal
Processing Symposium (SPS-2002).
250
Asymmetric Multi-channel Sampling in Shift
Invariant Spaces
Sinuk Kang (1) and K.H. Kwon (1)
(1) KAIST, 335 Gwahangro, Yuseong-gu, Daejeon 305-701, S. Korea.
sukang@kaist.ac.kr, khkwon@kaist.edu
Abstract:
We consider a multi-channel sampling with asymmetric
sampling rates in shift invariant spaces, while related previous works have supposed that each channel has a symmetric(uniform) sampling rate. Motivated by the fact that
shift invariant spaces are isomorphic images of L2 [0, 2π],
we obtain a sampling expansion in shift invariant spaces
by using frame or Riesz basis expansion in L2 [0, 2π]. The
samples in the expansion are expressed in terms of frame
coefficients of an appropriate function with respect to a
certain frame in L2 [0, 2π]. The involved reconstruction
functions are given explicitly by using the frame operator. We also present relation between asymmetric multichannel sampling and symmetric one.
1.
reconstruction functions by means of the frame operator.
The theory contains both a frame and Riesz basis expansion as sampling formulas.
2.
Asymmetric multi-channel sampling
Assume that φ(t) is everywhere well defined complex valued square integrable function on R throughout the paper.
Moreover, let φ(t) be a Riesz generator with Cφ (t) < ∞
for any t ∈ R so that V (φ) is an RKHS(see Proposition
2.4 in [4]). We now are given a LTI system {Lj [·]}N
j=1
whose impulse response is {lj (t) : lj ∈ L2 (R), j =
1, 2, · · · , N }. The aim of this paper is to recover any
f (t) ∈ V (φ) via discrete samples from {Lj [f ]}N
j=1 as
Introduction
Reconstructing a band-limited signal f from samples
which are taken from several channeled versions of f is
called multi-channel sampling. The multi-channel sampling method goes back to the work of Shannon [6] and
Fogel [2], where the reconstruction of a band-limited signal from samples of the signal and of its derivatives was
suggested. Generalized sampling expansion for arbitrary
multi-channel sampling was introduced first by Papoulis
[5].
Papoulis’ result has been extended to a general shiftinvariant space [1, 7, 8]. Here, a shift invariant space V (φ)
with a generator φ ∈ L2 (R) is defined by the closed subspace of L2 (R) spanned by integer translates {φ(t − n) :
n ∈ Z} of φ. Recently Garcı́a and Pérez-Villalón [3] derived stable generalized sampling in a shift-invariant space
by using some special dual frames in L2 [0, 1].
The previous works related to the multi-channel sampling
have assumed that numbers of samples from each channel
are uniform, namely, sampling rates of channels are same.
In this paper we consider a multi-channel sampling with
asymmetric sampling rates in shift invariant spaces. We
find an expression for the samples as frame coefficients of
an appropriate function in L2 [0, 2π] with respect to some
particular frame in L2 [0, 2π] and present the sufficient and
necessary condition under which a sequence of functions
of particular form becomes a frame or a Riesz basis for
L2 [0, 2π]. Using isomorphism between a shift invariant
space V (φ) and L2 [0, 2π], we derive sampling theory in
V (φ) with some Riesz generator φ and find a formula of
SAMPTA'09
f (t) =
N X
X
Lj [f ](σj + rj n)sj,n (t),
(1)
j=1 n∈Z
where {sj,n (t) : j = 1, · · · , N and n ∈ Z} is a frame or
a Riesz bases of V (φ) and 0 ≤ σj < rj with a positive
integer rj for j ∈ {1, 2, · · · , N }.
2.1
An expression for the samples
Define an isomorphism J from L2 [0, 2π] onto V (φ) by
JF (t) =
1 X
hF (ξ), e−inξ iφ(t−n), F (ξ) ∈ L2 [0, 2π].
2π n∈π
By the isomorphism J : L2 [0, 2π] → V (φ), the reconstruction formula (1) is equivalent to the following one:
F (ξ) =
N X
X
Lj [f ](σj +rj n)Sj,n (ξ), F (ξ) ∈ L2 [0, 2π],
j=1 n∈Z
(2)
where f (t) = JF (t) and sj,n (t) = JSj,n (t). Notice further that Lj f (σj + rj n) is represented by an inner product
of F (ξ) and some function in L2 [0, 2π].
Lemma 2.1.1 Let L[·] be a LTI system with an impulse
response l(t) ∈ L2 (R) and ψ(t) = L[φ](t) = (φ ∗ l)(t).
(a) L is a bounded operator from L2 (R) into L∞ (R),
o
kf ∗ lk∞ ≤ kf k2 klk2 and Lf (t) ∈ C∞
(R),
(b) supR Cψ (t) < ∞,
251
(c) (cf. Lemma 2 in [3]) for any f (t) = (c ∗ φ)(t) with
c ∈ ℓ2 in V (φ), L[f ](t) = (c∗ψ)(t) converges absolutely and uniformly on R. For any f (t) = JF (t) ∈
V (φ) with F (ξ) ∈ L2 [0, 2π],
L[f ](t) = hF (ξ),
1
Zψ (t, ξ)iL2 [0,2π] .
2π
In particular,
L[f ](σj +rj n) = hF (ξ),
Theorem 2.2.2 Let φ(t) be a Riesz generator with
Cφ (t) < ∞, t ∈ R and {Lj [·]}N
j=1 be LTI systems with
∈
L2 (R) . Let {ψj (t) =
an impulse response {lj (t)}N
j=1
N
(φ ∗ lj )(t)}j=1 , rj ≥ 1 an integer and 0 ≤ σj < rj .
(a) If 0 < αG ≤ βG < ∞, i.e., 0 < αG and
Zψj (σj , ξ) ∈ L∞ [0, 2π], 1 ≤ j ≤ N , then there
is a frame {sj,n (t) : 1 ≤ j ≤ N, n ∈ Z } of V (φ)
for which
1
Zψ (σj , ξ)e−irj nξ iL2 [0,2π] .
2π
(3)
f (t) =
N X
X
Lj f (σj +rj n)sj,n (t), f (t) ∈ V (φ).
j=1 n∈Z
(4)
2.2
The sampling theorem
For a given LTI system {Lj [·]}N
j=1 , let Lj φ(t) = ψj (t),
1 ≤ j ≤ N . Using equation (3), the expansion (2) is
equivalent to
F (ξ) =
N
X
X
hF (ξ),
j=1 n∈Z
1
Zψ (σj , ξ)e−irj nξ iL2 [0,2π]
2π j
·Sj,n (ξ), F (ξ) ∈ L2 [0, 2π],
where f (t) = JF (t) and sj,n (t) = JSj,n (t).
For convenience, we introduce a few more notations.
Let gj (ξ) ∈ L2 [0, 2π] for 1 ≤ j ≤ N , gj,mj (ξ) :=
gj (ξ)eirj (mj −1)ξ for 1 ≤ mj ≤ rrj and
G(ξ) = [Dg1,1 (ξ), Dg1,2 (ξ), · · · , Dg1, rr (ξ),
Dg2,1 (ξ), · · · , DgN, rr (ξ)] ,
Remark 2.2.3 Asymmetric multi-channel sampling series
with LTI system {Lj [·]}N
j=1 whose impulse response is
{lj (t)}N
can
be
considered
as symmetric multi-channel
j=1
N,
N
where D is an unitary operator from L2 [0, 2π] onto L2 (I)r
defined by (DF )(ξ) = [F (ξ), F (ξ + 2π
r ), · · · , F (ξ +
T
2
(r − 1) 2π
)]
,
F
(ξ)
∈
L
[0,
2π].
Note
that G(ξ) is
PN r r
the j=1 rj × r matrix whose entries are in L2 [0, 2π
r ].
And define λM (ξ)(resp. λm (ξ)) as the largest(resp. the
smallest) eigenvalue of r × r matrix G(ξ)∗ G(ξ), βG as
kλM (ξ)k∞ and αG as kλm k0 .
Lemma 2.2.1 Let gj ∈ L2 [0, 2π] and rj be a positive integer for 1 ≤ j ≤ N . Define r as the least common multi−irj nξ
plier of {rj }N
: 1 ≤ j ≤ N, n ∈
j=1 . Then {gj (ξ)e
Z } is a
(a) Bessel sequence in L2 [0, 2π] if and only if
kλM (ξ)k∞ < ∞ if and only if gj ∈ L∞ [0, 2π]
for 1 ≤ j ≤ N . In this case, optimal bound is
2π
r kλM (ξ)k∞ ;
(b) frame of L2 [0, 2π] if and only if 0 < kλm (ξ)k0 ≤
PN
kλM (ξ)k∞ < ∞ so that r ≤ j=1 rrj and optimal
2π
bounds are 2π
r kλm (ξ)k0 ≤ r kλM (ξ)k∞ ;
2
(c) Riesz basis of L [0, 2π] if and only if frame of
PN 1
PN r
L2 [0, 2π] and r =
j=1 rj , i.e., 1 =
j=1 rj if
and only if gj (ξ) ∈ L∞ [0, 2π] for 1 ≤ j ≤ N ,
PN
1 = j=1 r1j and | det G(ξ)| ≥ ∃α > 0 a.e..
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In all cases, sampling series (4) converges in L2 (R), absolutely on R and uniformly on any subset of R on which
Cφ (t) is bounded.
j
sampling series with LTI system {L̃j,mj [·]}j=1,m
with
j =1
T
1
2π Zψj (σj , ξ)
(c) Assume that Zψj (σj , ξ) ∈ L∞ [0, 2π], 1 ≤ j ≤ N .
Then there is a Riesz basis {sj,n (t) : 1 ≤ j ≤
N, n ∈ Z } of V (φ) for which (4) holds if and only
PN
if 0 < αG and 1 = j=1 r1j .
N, rr
1
Appealing to the setting gj (ξ) =
j ≤ N , we have
(b) Assume that Zψj (σj , ξ) ∈ L∞ [0, 2π], 1 ≤ j ≤ N .
Then there is a frame {sj,n (t) : 1 ≤ j ≤ N, n ∈ Z }
of V (φ) for which (4) holds if and only if 0 < αG .
for 1 ≤
r
rj
, where ˜lj,mj (t) =
impulse response {˜lj,mj (t)}j=1,m
j =1
lj (rj (mj − 1) + t).
2.3
Reconstruction functions
Let S be a frame operator with frame {gj (ξ)e−irj nξ }j,n .
For any F (ξ) ∈ L2 [0, 2π],
r
SF (ξ) =
rj
N X
X
gj (ξ)e−irj (mj −1)ξ
j=1 mj =1
·
2π
gj,m (ξ)T DF (ξ)
r
so that
DSF (ξ) =
2π ∗
G G(ξ)DF (ξ).
r
Then, from Lemma 2.2.1 (b), there exists (G∗ G)−1 (ξ) a.e.
such that
r
D(S −1 (gj (ξ)e−irj nξ )) =
(G∗ G)−1 (ξ)D(gj (ξ)e−irj nξ )
2π
for 1 ≤ j ≤ N and n ∈ Z . Hence,
r
{sj,n }j,n = { JD−1 [(G∗ G)−1 (ξ)D(gj (ξ)e−irj nξ )]}j,n .
2π
Remark 2.3.1 One sufficient condition under which
{sj,n }j,n is translates of a single function in L2 [0, 2π] is
that r divides rj for all 1 ≤ j ≤ N . Since r is the least
common multiplier of {rj }N
j=1 , the condition holds if and
only if r = rj for all 1 ≤ j ≤ N .
252
References:
[1] I. Djokovic, P. P. Vaidyanathan, Generalized sampling theorems in multiresolution subspaces, IEEE
Trans. Signal Process., 45:583-599, 1997.
[2] L. J. Fogel, A note on the sampling theorem, IRE
Tran. Infor. Theory IT-1:47-48, 1995.
[3] A. G. Garcı́a and G. Pérez-Villarón, Dual frames
in L2 (0, 1) connected with generalized sampling in
shift-invariant spaces, Appl. Comput. Harmon. Anal.,
20:422-433, 2006.
[4] J. M. Kim, K. H. Kwon, Sampling expansion in shift
invariant spaces, Intern. J Wavelets, Multiresolution
and Inform. Processing, 6(2):223-248, 2008.
[5] A. Papoulis, Generalized sampling expansion, IEEE
Trans. Circuits Systems, 24(11), 652-654, 1977.
[6] C. E. Shannon, Communication in the presence of
noise, Proc. IRE, 37:10-21, 1949.
[7] M. Unser, J. Zerubia, Generalized sampling: Stability and performance analysis, IEEE trans. Signal
Process., 45(12):2941-2950, 1997.
[8] M. Unser, J. Zerubia, A generalized sampling theory
without band-limiting constraints, IEEE Trans. Circuits Syst. 2, 45(8):959-969, 1998.
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253
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254
Sparse Data Representation on the Sphere
using the Easy Path Wavelet Transform
Gerlind Plonka (1) and Daniela Roşca (2)
(1) Department of Mathematics, University of Duisburg-Essen, Campus Duisburg, 47048 Duisburg, Germany.
(2) Department of Mathematics, Technical University of Cluj-Napoca, 400020 Cluj-Napoca, Romania.
gerlind.plonka@uni-due.de, Daniela.Rosca@math.utcluj.ro
Abstract:
In this paper we consider the Easy Path Wavelet Transform
(EPWT) on spherical triangulations. The EPWT has been
introduced in [7] in order to obtain sparse image representations. It is a locally adaptive transform that works along
pathways through the array of function values and exploits
the local correlations of the data in a simple appropriate
manner. In our approach the usual one-dimensional discrete wavelet transform (DWT), orthogonal or biorthogonal, can be applied.
1.
Introduction
One important problem in data analysis is to construct efficient low-level representations using only a very small
part of the original data. However, these sparse approximations should provide a precise characterization of relevant features of the data like discontinuities (edges) and
texture components.
It is well-known that wavelets can represent piecewise
smooth signals efficiently. However, higher-dimensional
structures may not be represented suitably by sparse
wavelet decompositions based on tensor product wavelets,
because directional geometrical properties of the data cannot be adapted.
The last years have seen many attempts to construct locally adaptive wavelet-based schemes that take into account the special geometry of the data. In particular, for
sparse representation of images, different ideas, that try to
exploit the local correlations of the data, have been developed (see e.g. [1, 2, 3, 4, 5, 6, 7, 10]).
We will focus on the EPWT recently introduced in [7]
for sparse image representation. In this paper, we want
to adapt the EPWT to triangulations of the sphere.
For this purpose, we apply the idea used by Roşca [8, 9]
to obtain a suitable spherical triangulation. We employ a
polyhedral subdivision domain. The triangular faces of the
polyhedron are successively subdivided into four smaller
triangles. Each triangle can be transported radially to the
sphere. This approach has been used in [8, 9] for the construction of Haar wavelets and of locally supported rational spline wavelets on the sphere.
The idea of the EPWT on spherical triangulations is very
simple. First we fix a certain neighborhood of a triangle,
e.g. the three triangles that have common edges with the
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reference triangle. Next, we use a one-dimensional indexing of all triangles of the fixed triangulation and assume
that each function value of a given data vector is associated to one triangle, or rather to its corresponding (onedimensional) index.
In the first step we select a path through the complete index set in such a way that data points associated to neighbor indices in the path are strongly correlated. For this
purpose, for each index we choose “the best” neighbor index that has not been used in the path yet, such that the
absolute difference between neighboring data values is the
smallest. The complete path vector can be seen as a permutation of the original index vector. Then we apply a
suitable (one-dimensional) discrete wavelet transform to
the data vector along the path, and the choice of the path
will ensure that most wavelet coefficients remain small.
The same procedure can be successively applied to the
down-sampled data. After a suitable number of iterations,
we apply a shrinkage procedure to all wavelet coefficients
in order to find a sparse digital representation of the function. For reconstruction one needs the path vector at each
level in order to apply the inverse wavelet transform.
2. Spatial and spherical triangulations
Consider the sphere S 2 = {x ∈ R3 , x2 = 1} and let
Π be a convex polyhedron with triangular faces, containing O inside. For example we can take an icosahedron,
a cube with triangulated faces, an octahedron, etc. The
boundary of the polyhedron will be denoted by Ω. We denote by T 0 = {T1 , . . . , TM } the set of faces of Π. For
each triangle T ∈ T 0 we take the mid-points of its edges
and construct four triangles of equal area, as in Figure 1.
All these small triangles will form a refined triangulation
of T 0 , denoted T 1 . Continuing the refinement process in
the same manner, we obtain a triangulation T j of Ω, for
j ∈ N. For application of the EPWT we will stop the
refinement process at a suitable sufficiently high (fixed)
level j depending on the data set in the application. For
application of the EPWT we will need a one-dimensional
index set J = J j for the triangles in T j . Using the octahedron, this one-dimensional index set J can be as in Figure
1 (right). Observe that for the octahedron the number of
triangles at the jth level is given by #J = #T j = 22j+3 .
In order to obtain a spherical triangulation, for the given
255
A
B
F
16 28 27 32
B
A
4 15 14 26
E
17 5 6 1 2 3 12 13 25
D
C
E
29 18 19
B
F
7
C
8 9 10 11 23 24 31
20 21 22
D
F
30
F
F
Figure 1: Illustration of the octahedron with triangulation T 1
(left) and a fold apart version of the octahedron on the plane,
with a one-dimensional indexing of all triangles.
polyhedron Π we define the radial projection p : Ω → S 2 ,
p(x, y, z) = (x2 +y 2 +z 2 )−1/2 ·(x, y, z),
(x, y, z) ∈ Ω.
The set U j = {U = p(T ), T ∈ T j } will be a triangulation of the sphere S 2 . For indexing the spherical triangles
in U j , we use the same index set J as for the triangulation
T j of the polyhedron.
Decomposition
First level
We first determine a complete path vector p L through the
index set J = {1, 2, . . . , N } and then apply a suitable
discrete one-dimensional (periodic) wavelet transform to
the function values f L = (f L (j))j∈J along the path p L .
We start with pL (1) := 1. Next, for pL (2) we take
pL (2) := argmin {|f L (1) − f L (k)|, k ∈ N (1)}.
k
We proceed in this manner, thereby determining a path
vector through the index set J, that is locally adapted to
the function f (easy path). With the procedure described
above, we obtain a pathway such that the absolute differences between neighboring function values f L (l) along
the path are as small as possible. In general, for a given
the index pL (l), 1 ≤ l ≤ N − 1, the next value p L (l + 1)
is defined by
pL (l + 1) := argmin {|f L (pL (l)) − f L (k)|,
k
3.
Definitions and Notations for the EPWT
In order to explain the idea of the EPWT, where we want to
use the discrete one-dimensional wavelet transform along
path vectors through the data, we need some definitions
and notations.
Let us assume that a fixed refined spherical triangulation
U j is given.Let J be a one-dimensional index set for the
spherical triangles in U j .
We define a neighborhood of an index ν ∈ J as
N (ν) = {µ ∈ J\{ν} : Tµ and Tν have a common edge}.
Hence, each index ν ∈ J has exactly three neighbors. One
may also use a bigger neighborhood, e.g. N (ν) = {µ ∈
J \ {ν} : Tµ and Tν have a common edge or a common
vertex }, in which case each index has 12 neighbors.
We also need a definition of neighborhood of subsets of
an index set. We shall consider disjoint partitions of J of
the form
{J1 , J2 , . . . , Jr }, where Jµ ∩ Jν = ∅ for µ = ν
and rν=1 Jν = J. We then say that two different subsets
Jν and Jµ from the partition are neighbors, and we write
Jν ∈ N (Jµ ), if there exist the indices l ∈ J ν and l1 ∈
Jµ such that l ∈ N (l1 ). We consider a function f being
piecewise constant on the triangles of U j , i.e., we identify
each spherical triangle in U j with a value of f . Hence, f
is uniquely determined by the data vector (f ν )ν∈J .
We will look for path vectors through index subsets of J
and we apply a one-dimensional wavelet transform along
these path vectors. Any orthogonal or biorthogonal onedimensional wavelet transform can be used here.
4.
Description of the EPWT
In this section we give a summary of the idea of the EPWT,
described in more details in [7]. We start with the decomposition of the real data (f ν )ν∈J , and we assume that
N = #J is a multiple of 2L with L ∈ N. Then we will
be able to apply L levels of the EPWT. For the considered
octahedron we have N = 2 2j+3 .
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k ∈ N (pL (l)) \ {pL (ν), ν = 1, . . . , l}}.
It can happen that the choice of the next index value
pL (l + 1) is not unique, if the above minimum is attained
for more than one index. In this case, one may fix favorite
directions in order to determine a unique pathway.
Another situation which can occur during the procedure is
that all indices in the neighborhood of an index p L (l) have
already been used in the path p L . In this case we have an
interruption in the path vector. We need to choose one index pL (l+1) from the remaining indices in J, which have
not been taken yet in p L . There are different possibilities
for finding a suitable next index. One simple choice is to
take the smallest index from J that has not been used so
far. Another choice is to look for a next index, such that
again the absolute difference |f L (pL (l)) − f L (pL (l + 1))|
is minimal, i.e., we take in this case
pL (l + 1) =
argmin {|f L (pL (l)) − f L (k)|,
k
k ∈ J \ {pL (ν), ν = 1, . . . , l}}.
By proceeding in this manner, we finally obtain a path vector pL ∈ ZN , which is a permutation of (1, 2, . . . , N ).
After having constructed the path p L , we apply one
level of the 1-D Haar DWT (or any other orthogonal or
biorthogonal periodic DWT) to the vector of function valL
ues (f L (pL (l)))N
l=1 along the path p . We obtain the vecN/2
L−1
tor f
∈ R
, containing the low-pass part, and the
vector of wavelet coefficients g L−1 ∈ RN/2 . While the
wavelet coefficients will be stored in g L−1 , we further proceed with the low-pass vector f L−1 at the second level.
Further levels
If N = 2L r with r ∈ N being greater than or equal to
the lengths of low-pass and high-pass filters in the chosen
DWT, then we may apply the procedure L times. For a
given vector f L−j , 0 < j < L, at the (j + 1)-th level we
consider the index sets
L−j+1
j
JlL−j := JpL−j+1
L−j+1 (2l−1) ∪ JpL−j+1 (2l) , l = 1, . . . , N/2 ,
256
N/2j
with the corresponding function values (f L−j (l))l=1 . In
particular, the index sets at the second level are J lL−1 :=
{pL (2l−1), pL(2l)}, l = 1, . . . , N/2, determining a partition of J.
We repeat the procedure described in the first step, but
replacing the single indices with the new index sets J lL−j ,
and the corresponding function values with the smoothed
function values f L−j (l).
j
The new path vector p L−j ∈ ZN/2 should now be a permutation of (1, 2, . . . , N/2 j ). We start again with the first
index set J1L−j , i.e., pL−j (1) = 1. Having already found
pL−j (l), 1 ≤ l ≤ N/2j − 1, we determine the next value
pL−j (l + 1) as
B
A
5
17
18
29
F
1
6
7
19
2
9
8
C
4
3
10
20 22
14
13
12
11
16
14
6
B
E
A
1
2
11
16
31
D
13
F
F
21
30
F
10
3
E
5
14
25
24
23
F
15
32
27
26
16
15
B
B
F
28
6
7
4
12
9
C
12
9
D
F
8
F
Figure 2. Illustration of first path through the triangulation T 1
of the octahedron (left) and of the low-pass part after the first
level of EPWT with Haar DWT (right). Index sets at the second
level are illustrated by different gray values, and path vectors are
represented by arrows.
pL−j (l + 1) = argmin {|f L−j (pL−j (l)) − f L−j (k)|,
k
L−j
JkL−j ∈ N (JpL−j
(ν), ν = 1, . . . , l}}.
L−j (l) ) \ {p
If the new value p L−j (l+1) is not uniquely determined by
the minimizing procedure, we can fix favorite directions in
order to obtain a unique path. If for the set J pL−j
L−j (l) there
is no neighboring index set that has not been used yet in
the path vector p L−j , then we have to interrupt the path
and to find a new good index set (that has been not used
so far) to continue the path. As at the first level, we try to
keep the differences of function values along the path as
small as possible.
Finally, we apply the (periodic) wavelet transform to the
N/2j
vector (f L−j (pL−j (l)))l=1 along the path p L−j , thereby
j+1
obtaining the low-pass vector f L−j−1 ∈ RN/2
and the
N/2j+1
L−j−1
vector of wavelet coefficients g
∈R
.
Output
As output of the complete procedure after L iterations we
obtain the coefficient vector
g = (f 0 , g0 , g1 , . . . , gL−1 ) ∈ RN
and the vector determining the paths at each iteration step
L
p = (p1 , p2 , . . . , pL ) ∈ R2N (1−1/2 ) .
These two vectors contain the entire information about the
original function f .
In order to find a sparse representation of f , we apply a
shrinkage procedure to the wavelet coefficients in the vecj .
tors gj , j = 0, . . . , L − 1 and obtain the vectors g
Reconstruction
= (f 0 , g
0 , g
1 , . . . , g
L−1 )
The reconstruction of f L from g
and p is given as follows.
f 0 = f 0;
For j = 0 to L − 1
j
j ) ∈ Rr2
- Apply the inverse DWT to the vector (
fj, g
j+1
in order to obtain
fpj+1 ∈ Rr2 .
- Apply the permutation
f j+1 (pj+1 (k)) =
fpj+1 (k), for
j+1
k = 1, . . . , r2 .
5.
Example
sphere, where each function value corresponds to a spherical triangle that has been obtained by radial projection of
the triangulated octahedron in Figure 1 (left). The values
are given as a vector f = f 5 of length 32, corresponding
to the one-dimensional indexing of the triangles in Figure
1 (right),
f = (0.4492, 0.4219, 0.4258, 0.4375, 0.4141, 0.4531,
0.4180, 0.4258, 0.4375, 0.4292, 0.4219, 0.4219,
0.4219, 0.4258, 0.4023, 0.4141, 0.4219, 0.4219,
0.4297, 0.4375, 0.4141, 0.4023, 0.4258, 0.4219,
0.4258, 0.4180, 0.4531, 0.4141, 0.4375, 0.4258,
0.4219, 0.4492).
Starting with the index 1, with the function value 0.4492,
we determine the first path vector. This index has the
three neighbors 2, 4, and 6, with the corresponding values 0.4219, 0.4375 and 0.4531, respectively (see Figure
2). Hence, the second index in the path is 6. Proceeding
further according to Section 4 we obtain
p5 =(1, 6, 7, 8, 9, 10, 11, 12, 13, 14, 26, 25, 24, 31, 30, 21,
22, 23; 3, 2, 17, 18, 19, 20; 4, 15, 16, 5; 28, 27, 32, 29),
where the interruptions in the path are indicated by semicolons. This path has four interruptions and is illustrated
by arrows in Figure 2 (left). An application of the Haar
DWT (with unnormalized filter coefficients h 0 = h1 =
1/2, g0 = 1/2, g1 = −1/2) along this path gives (with
truncation after four digits) the low-pass coefficients
f 4 = (0.4512, 0.4219, 0.4334, 0.4219, 0.4238, 0.4219,
0.4219, 0.4200, 0.4140, 0.4238, 0.4219, 0.4336, 0.4199,
0.4141, 0.4336, 0.4434),
and the wavelet coefficients
g4 = (−0.0020, −0.0039, −0.0042, 0., −0.0020,
−0.0039, 0., 0.0058, −0.0118, 0.0020, 0., −0.0039,
0.0176, 0., −0.0195, 0.0058).
We illustrate the simple idea of function decomposition
with the EPWT on the sphere in the following small example. Let a set of 32 function values be given on the
SAMPTA'09
We now proceed to the second level. For the smoothed
vector of function values f 4 corresponding to the 16 index
257
B
B
F
3
B
A
7
6
B
E
7
1
5
2
C
F
4
D
E
1
3
2
4
6
A
4
3
8
F
2
8
F
F
4
C
2
D
F
4
4
Figure 3. Illustration of the third and fourth paths.
Figure 4. Approximation f 6 at level 6 of the original dataset topo
and the compressed version e
f 6 with threshold 2500.
sets that are illustrated by gray values in Figure 2 (right),
we obtain the next path
The results are contained in Table 1, where the mean of f 6
is −2329.
F
F
p4 = (1, 10, 4, 5, 6, 7, 8, 9, 3, 2, 12, 11, 14, 13; 15, 16),
illustrated by arrows in Figure 2 (right). An application of
the Haar DWT along p 4 gives
f 3 = (0.4375, 0.4229, 0.4219, 0.4170, 0.4276, 0.4278,
0.4170, 0.4385),
g = (0.0136, −0.0010, 0., 0.0030, 0.0057, 0.0058,
3
0.0029, −0.0049).
At the third level we start with the smoothed vector f 3
corresponding to the 8 index sets that are illustrated by
gray values in Figure 3 (left). We find now the path p 3 =
(1, 5, 6, 8, 3, 2, 4; 7), see Figure 3 (left). This leads to
f2
g2
= (0.4326, 0.4331, 0.4224, 0.4170),
= (0.0049, −0.0054, 0.0005, 0.).
At the fourth level we have only 4 index sets that correspond to the values in f 2 , see Figure 3 (right). Hence we
find p2 = (1, 2, 3, 4) and
f 1 = (0.4328, 0.4197),
g1 = (−0.0003, 0.0027).
Finally, with p1 = (1, 2), the last transform yields f 0 =
(0.4263) and g0 = (0.0066).
6.
Numerical experiments
To illustrate the efficiency of our method, we took the
dataset topo and we considered the regular octahedron
with triangulation T 6 , containing 32768 triangles. The approximation f 6 at level 6 is represented in Figure 4. We
applied the EPWT with different thresholds, obtaining the
compressed vector
f 6 , and we measured the SNR given as
SN R = 20 · log10
threshold
1
100
500
1000
1500
2000
2500
f 6 − mean(f 6 )2
.
f 6 −
f 6 2
number of remaining
wavelet coeff.
27732
14185
5230
3313
2699
2402
2265
l 2 -norm
of error
26.4031
5.34e+03
2.47e+04
3.97e+04
5.00e+04
5.79e+04
6.35e+04
SNR
84.72
38.59
25.30
21.17
19.18
17.89
17.10
Table 1: Compression results for the dataset topo.
SAMPTA'09
Acknowledgments
This research in this paper is supported by the project
436 RUM 113/31/0-1 of the German Research Foundation
(DFG). This is gratefully acknowledged.
References:
[1] R.L. Claypoole, G.M. Davis, W. Sweldens, and
R.G. Baraniuk. Nonlinear wavelet transforms for image coding via lifting. IEEE Trans. Image Process.
12:1449–1459, 2003.
[2] A. Cohen and B Matei. Compact representation of images by edge adapted multiscale transforms. In Proc.
IEEE Int. Conf. on Image Process. (ICIP), Thessaloniki, pages 8–11, 2001.
[3] S. Dekel and D. Leviatan. Adaptive multivariate approximation using binary space partitions and geometric wavelets. SIAM J. Numer. Anal. 43:707–732,
2006.
[4] W. Ding, F. Wu, X, Wu, S. Li, and H. Li. Adaptive directional lifting-based wavelet transform for image coding. IEEE Trans. Image Process. 16:416–427,
2007.
[5] D.L. Donoho. Wedgelets: Nearly minimax estimation
of edges. Ann. Stat. 27:859–897, 1999.
[6] S. Mallat. Geometrical grouplets. Appl. Comput. Harmon. Anal., 26 (2): 143–290, 2009.
[7] G. Plonka. The easy path wavelet transform: A new
adaptive wavelet transform for sparse representation
of two-dimensional data. Multiscale Model. Simul.
7:1474–1496, 2009.
[8] D. Roşca. Haar wavelets on spherical triangulations.
In Dodgson, N.A., Floater, M.S., Sabin, M.A., editors, Advances in Multiresolution for Geometric Modelling, Springer, pages 405–417, 2005.
[9] D. Roşca. Locally supported rational spline wavelets
on a sphere. Math. Comput. 74:1803–1829, 2005.
[10] R. Shukla, P.L. Dragotti, M.N. Do, and M. Vetterli.
Rate-distortion optimized tree structured compression
algorithms for piecewise smooth images. IEEE Trans.
Image Process. 14:343–359, 2005.
258
A fully non-uniform approach to FIR filtering
Brigitte Bidégaray-Fesquet (1) and Laurent Fesquet (2)
(1) LJK, CNRS / Grenoble University, B.P. 53, 38042 Grenoble Cedex 9, France.
(2) TIMA, 46 avenue Félix Viallet, 38031 Grenoble Cedex, France.
Brigitte.Bidegaray@imag.fr, Laurent.Fesquet@imag.fr
Abstract:
We propose a FIR filtering technique which takes advantage of the possibility of using a very low number of samples for both the signal and the filter transfer function
thanks to non-uniform sampling. This approach leads to
a summation formula which plays the role of the discrete
convolution for usual FIR filters. Here the formula is much
more complicated but it can be implemented and the evaluation of more elaborate expressions is compensated by
the very low number of samples to process.
1.
Introduction
Reducing the power consumption of mobile systems –
such as cell phones, sensor networks and many others
electronic devices – by one to two orders of magnitude is
extremely challenging but will be very useful to increase
the system autonomy and reduce the equipment size and
weight. In order to reach such a goal, this paper proposes
a solution applicable to FIR filtering which completely rethinks the signal processing theory and the associated system architectures.
Today the signal processing systems uniformly sample
analog signals (at Nyquist rate) without taking advantage
of their intrinsic properties. For instance, temperature,
pressure, electro-cardiograms, speech signals significantly
vary only during short moments. Thus the digitizing system part is highly constrained due to the Shannon theory,
which fixes the sampling frequency at least twice the input signal frequency bandwidth. It has been proved in [4]
and [6] that Analog-to-digital Converters (ADCs) using a
non equi-repartition in time of samples leads to interesting
power savings compared to Nyquist ADCs. A new class
of ADCs called A-ADCs (for Asynchronous ADCs) based
on level-crossing sampling (which produces non-uniform
samples in time) [2, 3] and related signal processing techniques [1, 5] have been developed.
This work suggests an important change in the FIR filter
design. As sampling analog signals is usually performed
uniformly in time, sampling the filter transfer function is
also done in a regular way with a constant frequency step.
Non-uniform sampling leads to an important reduction of
the weight-function coefficients. Combined with a nonuniform level-crossing sampling technique performed by
an A-ADC, this approach drastically reduces the compu-
SAMPTA'09
tation load by minimizing the number of samples and operations, even if they are more complex.
2.
Principle and notations
For a large class of signal, non-uniform sampling leads
to a reduced number of samples, compared to a Nyquist
sampling. This feature has already been used in [1] to
design non-uniform filtering techniques based on interpolation. In this work the authors however used a classical
(uniform) filter, that is a usual discretization in time of the
impulse response.
Here we want to go further and take advantage of the fact
that the filter transfer function (the Fourier transform of
the impulse response) is a very smooth function with respect to frequency. It can therefore be well approximated
by the linear interpolation of quite few samples.
2.1
Level crossing sampling
The initial signals are supposed to be analog ones. The
signal which we want to filter is given in the time domain
and is denoted by s(t). The filter transfer function is given
in the frequency domain and is denoted by H(ω). The
result of the filtering process x(t) is then theoretically the
convolution of s(t) with the impulse response h(t) which
is the inverse Fourier transform of H(ω):
Z +∞
x(t) =
h(t − τ )s(τ )dτ,
−∞
h(t)
=
1
2π
Z
+∞
H(ω)e−iωt dω.
−∞
These signal are sampled in their initial domain using a
level crossing scheme. This technique has to be adapted
for the filter transfer function. Indeed level crossing has
a sense if an order can be defined, for example for a real
valued function. The filter transfer function is complex
valued, therefore we can choose to sample either when the
amplitude crosses some predefined values, or the phase,
or both. The samples read (sn , δtn ) for the signal and
(Hk , δωk ) for the filter transfer function. These samples
are formed of a value and the (time or frequency) interval length ”elapsed” since the last sample. To give results or describe algorithms we will use
Pnthe sample times
or frequencies defined as tn = t0 + 1 δtn′ and ωk =
Pk
ω0 + 1 δωk′ but computations will be performed using
259
only the time and frequency intervals δtn and δωk . We
will also denote by In = [tn−1 , tn ] and Jk = [ωk−1 , ωk ]
the time and frequency intervals.
which has a compact support. The convolution reads
x̄(t)
Z
=
+∞
h̄(t − τ )s̄(τ )dτ
−∞
2.2
Linear interpolation
n
h(t − τ )sn (τ )dτ
tn−1
n
X
=
s̄(t)
=
[an + bn t]χIn ,
an
H̄(ω)
=
i(γk +δk ω)
(αk + βk ω)e
h0nk (t)
χJk ,
where χ denotes the indicator function of the set given
in index. The coefficients an and bn can be expressed in
terms of sn , sn−1 , tn and δtn . The coefficients αk , βk ,
γk and δk can be expressed in terms of Hk , Hk−1 , ωk and
δωk .
In fact these formulae cover the piecewise constant case
(only take bn = βk = δk = 0) in three possible forms:
constant on intervals In or nearest neighbor interpolation,
with a possible need to modify the definition of tn and δtn
in the algorithms. They also cover two ways to linearly interpolate the complex valued filter transfer function: either
interpolate separately the amplitude and the phase (αk and
βk are real) or interpolate in the complex plane (αk and βk
are complex, γk and δk are zero).
The digital filter then consists in computing (possibly) for
all time
x̄(t)
=
+∞
h̄(t − τ )s̄(τ )dτ,
h̄(t)
3.
3.1
=
Z
+∞
H̄(ω)e
−iωt
dω.
=
h1nk (t) =
A summation formula
The impulse response h̄(t) can be
P split in contributions for
each frequency sample h̄(t) = k hk (t) with
Z
ωk
(αk + βk ω)ei(γk +δk ω) e−iωt dω
ωk−1
for which we will give an explicit expression in Section
3.2. Although the piecewise linear function H̄(ω) has a
compact support (we only have a finite number of samples), the functions hk (t) have an infinite support. This
is not a problem since the convolution will involve s̄(t)
SAMPTA'09
!
h1nk (t)
k
tn
Z
hk (t − τ )dτ,
tn
Z
hk (t − τ )τ dτ.
tn−1
We obtain a summation formula as in the classical FIR
filtering case where it takes the form of a discrete convolution. To be closer to this classical case, we should write
this as
X X
sn
hnk (t),
x̄(t) =
n
k
which is possible but the effective expression depends on
the type of interpolation used (piecewise constant or linear).
There remains to make explicit these two types of elementary contributions.
3.2
Elementary impulse responses
A straightforward computation of the integral formulation
for hk (t) yields
hk (t)
=
αk eiγk
2π
−∞
Deriving a filtering formula in the general
context
1
hk (t) =
2π
X
tn−1
−∞
1
2π
h0nk (t) + bn
k
k
Z
X
where
n
X
hk (t − τ )(an + bn τ )dτ
tn−1
k
n
X
tn
XXZ
=
To derive the FIR algorithm and approximate the theoretical integral formula, we form new analog functions from
the previously described samples. To this aim we choose
linear interpolation and we have
tn
XZ
=
Z
ωk
ei(δk −t)ω dω
ωk−1
Z ωk
+
βk eiγk
2π
=
αk eiγk ei(δk −t)ωk − ei(δk −t)ωk−1
2πi(δk − t)
+
+
ei(δk −t)ω ωdω
ωk−1
βk eiγk ωk ei(δk −t)ωk − ωk−1 ei(δk −t)ωk−1
2πi(δk − t)
iγk
i(δk −t)ωk
βk e
e
− ei(δk −t)ωk−1
.
2π(δk − t)2
These formulae seem singular when t = δk . This is not
the case and has no reason to be since the function we integrate is smooth with respect to all parameters and variables. The limiting value for t = δk is clearly
hk (δk )
=
=
αk eiγk
2π
ωk
βk eiγk
dω +
2π
ωk−1
Z
Z
ωk
ωdω
ωk−1
1
eiγk
δωk (αk + βk (ωk−1 + ωk )).
2π
2
260
3.3
Elementary summation coefficients
A quick glance at the explicit expression of hk (t) clearly
provides the impression that the explicit formulae for
h0nk (t) and h0nk (t) will not fit in the columns here. We
will give only their flavor. Indeed we want to compute the
time integrals of of hk (t − τ ) and hk (t − τ )τ for τ ∈ In .
This leads to integrate the product of a rational function
with a complex exponential function. The results cannot
be given in terms of simple functions but only in terms of
the exponential integral function
Z ∞
dy
π
Ei(ix) = −
eiy
+i .
y
2
x
We give in the next section a simple example of elementary summation coefficient calculation in the piecewise
linear context.
4.
4.1
A simple and ideal example
Computation of the coefficients
Our sampling for the filter transfer function yields a particularly simple formulation for the ideal low-pass filter
which is 1 on the frequency interval [−ωc , ωc ] and zero
elsewhere. This yields a single sample (1, 2ωc ) and linearly interpolated coefficients α1 = 1, β1 = 0, γ1 = 0
and δ1 = 0. The expression for the elementary impulse
response is
e−iωc t − eiωc t
ωc
=
sinc(ωc t).
h1 (t) =
−2πit
π
Then we have to compute
Z tn
Z
0
h1 (t − τ )dτ = −
hn1 (t) =
tn−1
t−tn
h1 (τ )dτ
t−tn−1
1
= − (Si(ωc (t − tn )) − Si(ωc (t − tn−1 )),
π
where Si is the special function known as sine integral and
defined by
Z x
dy
1
π
sin(y)
= (Ei(ix) − Ei(−ix)) + ,
Si(x) =
y
2i
2
0
4.2
Numerical results
To illustrate this simple example we filter the signal
s(t) = 0.45 sin(2πt) + 0.45 sin(10πt) + 0.9
with the ideal low pass filter with the cutoff frequency
ωc = 4π. The theoretical result is therefore supposed to
be
x(t) = 0.45 sin(2πt) + 0.9.
This is not the typical sort of signal which is supposed to be addressed by our technique since it
is not a sporadic one and a relatively large number of samples are taken.
We perform the computations within the M ATLAB SPASS (Signal Processing for ASynchronous Systems) framework
(http://ljk.imag.fr/membres/Brigitte.Bidegaray/SPASS/).
This signal is sampled with a M -bit Asynchronous A/D
Converter (AADC) which leads to a level crossing sampling over the amplitude range [0, 1.8].
We can choose as we want the times at which the filtered
signal is computed. To display the results we choose the
sequence of times tm = .17m (m integer) to have sampling points dispatched irregularly over the obtained solution.
On Figure 1, you can see the result for a linear interpolation of the signal non-uniform samples and a 3-bit AADC.
We plot continuous functions with lines: the initial signal
s(t) (dashed line) and the theoretical filtered signal x(t)
(solid line). We plot the sampled results with markers: the
non-uniformly sampled initial signal sn (asterisk markers)
and the computed filtered samples xm (circle markers) at
times tm .
1.8
s(t)
x(t)
sn
1.6
xm
1.4
and
h1n1 (t)
of a numerical implementation of these algorithms. Moreover these functions are however very smooth: the Si function for example is almost linear in the neighborhood of 0
and tends to ±π/2 at ±∞ with very gentle oscillations.
This feature makes possible the construction of efficient
lookup tables in view of a hardware implementation.
=
tn
Z
1.2
h1 (t − τ )τ dτ
tn−1
Z t−tn
= −
1
h1 (τ )(t − τ )dτ
0.8
t−tn−1
Z
1 t−tn−1
+
= t
sin(ωc τ )dτ
π t−tn
1
= t h0n1 (t) −
(cos(ωc (t − tn ))
πωc
− cos(ωc (t − tn−1 ))).
h0n1 (t)
This case is simple due to its minimal number of samples
in the frequency domain, but it displays all the difficulties
of the general case, i.e. the need to evaluate special functions. These functions are built in many libraries in view
SAMPTA'09
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Figure 1: Filtering result. Initial signal (dashed line),
theoretical filtered signal (solid line), non-uniformly sampled initial signal (asterisk markers) and computed filtered
samples (circle markers).
261
This very simple test case has quite a low number of parameters compared to the full problem for which we can
finely tune the filter transfer function sampling for example. We compare here the results obtained for a zeroth and
a first order interpolation of the signal and for different
values (2, 3, 4 and 5) of the AADC resolution. On Table
1 we give the relative l1 error between computed filtered
samples xm at times tm = .01m (m integer) and the theoretical values x(tm ).
M
M
M
M
=2
=3
=4
=5
0th order
0.0608
0.0076
0.0052
0.0046
1st order
0.0584
0.0046
0.0045
0.0045
[3]
[4]
[5]
Table 1: l1 error of the filtering method for 0th and first
order interpolation of the signal and and M bit resolution
of the AADC (M = 2, 3, 4, 5).
In the case of the 2-bit AADC, there are 2.8 points per
wavelength for the highest frequency part of the signal.
This is a very low rate, and we are however able to have
only 6% error on the filtered result which is quite sufficient for a large range of applications. The other results
all show less than 1% error. The values displayed on Table 1 are very dependent on the choice of the function to
filter. Finer results (allowing less than .45% error) should
certainly be obtained by using a higher order interpolation
for the signal.
5.
[6]
converter. In 12th IEEE International Symposium
on Asynchronous Circuits and Systems (ASYNC’06),
pages 11–22, Grenoble, France, March 2006.
Emmanuel Allier, Gilles Sicard, Laurent Fesquet, and
Marc Renaudin. A new class of asynchronous A/D
converters based on time quantization. In 9th IEEE
International Symposium on Asynchronous Circuits
and Systems (ASYNC’03), pages 197–205, Vancouver,
Canada, May 2003.
Jon W. Mark and Terence D. Todd. A nonuniform sampling approach to data compression. IEEE
Trans. on Communications, COM-29(1):24–32, January 1981.
Saeed Mian Qaisar, Laurent Fesquet, and Marc Renaudin. Adaptive rate filtering for a signal driven
sampling scheme.
In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP07), pages III–1465–III–1468, Honolulu,
Hawaii, USA, April 2007.
N. Sayiner, H.V. Sorensen, and T.R. Viswanathan. A
level-crossing sampling scheme for A/D conversion.
IEEE Trans. on Circuits and Systems II, 43(4):335–
339, April 1996.
Conclusions
We have presented a novel approach to FIR filtering based
on the non-uniform sampling of the signal but also the
non-uniform sampling in frequency of the filter transfer
function. The final result is complex but is nonetheless
possible to implement in hardware devices and of course
in numerical codes. This complexity is balanced by the
very low number of samples and the relatively low number
of operations needed for each evaluation. This approach
is very promising to achieve a lower power consumption
in mobile systems.
6.
Acknowledgments
This work has been supported by a funding from the
Joseph Fourier-Grenoble 1 University: MSTIC project
TATIE.
References:
[1] Fabien Aeschlimann, Emmanuel Allier, Laurent Fesquet, and Marc Renaudin. Asynchronus fir filters, towards a new digital processing chain. In 10th IEEE International Symposium on Asynchronous Circuits and
Systems (ASYNC’04), pages 198–206, Crete, Greece,
April 2004.
[2] Filipp Akopyan, Rajit Manohar, and Alyssa B. Apsel.
A level-crossing flash asynchronous analog-to-digital
SAMPTA'09
262
ADAPTIVE TRANSMISSION FOR LOSSLESS IMAGE RECONSTRUCTION
Elisabeth Lahalle, Gilles Fleury, Rawad Zgheib
Department of Signal Processing and Electronic Systems, Supélec, Gif-sur-Yvette, France
E-mail : firstname.lastname@supelec.fr
tel: +33 (0)1 69 85 14 27, fax: +33 (0)1 69 85 14 29
ABSTRACT
This paper deals with the problem of adaptive digital transmission systems for lossless reconstruction. A new system,
based on the principle of non-uniform transmission, is proposed. It uses a recently proposed algorithm for adaptive stable identification and robust reconstruction of AR processes
subject to missing data. This algorithm offers at the same time
an unbiased estimation of the model’s parameters and an optimal reconstruction in the least mean square sense. It is an
extension of the RLSL algorithm to the case of missing observations combined with a Kalman filter for the prediction. This
algorithm has been extended to 2D signals. The proposed
method has been applied for lossless image compression. It
has shown an improvement in bit rate transmission compared
to the JPEG2000 as well as the JPEG-LS standards.
Index Terms— adaptive, lossless, compression
1. INTRODUCTION
Lossless compression methods are important in many medical applications where large data set need to be transmitted
without any loss of information. Actually, some lesions risk
becoming undetectable due to the effects of lossy compression. General lossless compression coders are considered to
be composed of two main blocks: a data decorrelation block
and an entropy coder for the decorrelated data. Two main tendencies may be noticed for the methods used for the decorrelation step: methods based on wavelet transforms and methods based on predictive coding. They have led to the main
compression standards : the JPEG2000 for the former group
of methods [1], the JPEG-LS for the latter [2]. Intensive attention is paid to transform based compression methods with
many algorithms which perform well regarding the bit rate
such as SPIHT [3], QT [4], etc.
All these coders use a uniform transmission of the binary elements to transmit. In a previous paper [5], the design of digital systems based upon non-uniform transmission of signal
samples was introduced. The idea behind is to avoid sending
a sample if it can be efficiently predicted, e.g. with a prediction error smaller than the quantization one, thus reducing the average transmission bit rate and increasing the signal
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to noise ratio (SNR). A speech coder based on the Adaptive
Pulse Code Modulation (ADPCM) principle and non-uniform
transmission of signals have already been proposed in [6]. It
uses the Least Mean Square (LMS)-like algorithm [7] for the
prediction of the samples that were not sent. However, this algorithm converges toward biased estimations of the model’s
parameters and does not use an optimal predictor in the least
mean square sense. Recently, we proposed a Recursive Least
Square Lattice (RLSL) algorithm for adaptive stable identification of non stationary Autoregressive (AR) processes subject to missing data, using a Kalman filter as a predictor [8].
This algorithm is fast, guarantees the stability of the model
identified and offers at the same time an optimal reconstruction error in the least mean square sense and an unbiased estimation of the model’s parameters in addition to the fast adaptivity to the variations of the parameters in the case of non stationary processes. Non stationnary AR processes can model a
large number of signals in practical situations, such as images
in the bi-dimensional case [9]. A new lossless image coder
based on a non-uniform transmission principle is proposed:
it is based on an adaptation of the algorithm proposed in [8]
for optimal prediction and identification of 2D AR processes
subject to missing observations.
In the following, begin by presenting the non-uniform transmission idea for lossless compression. In a second part, the
adaptive algorithm for reconstruction of AR processes with
missing observations [8] is described and extended to 2D AR
processes. Its integration into a non-uniform transmission
system is studied in the third section. Finally, an example
illustrates the performances of the proposed system. It is compared to a uniform digital transmission system : the JPEG2000.
2. NON-UNIFORM TRANSMISSION SYSTEM FOR
LOSSLESS RECONSTRUCTION
The proposed system uses predictive coding and non-uniform
transmission to reduce the bit rate transmission. An AR signal modeling is considered for the prediction. Let xn be the
amplitude of the signal at time n. The prediction of a sample
will be noted x̂n,P and the prediction error en,P = xn − x̂n,P .
In the receiver, a sample xn is predicted using the estimated
model parameters at time n − 1, ân−1 , and the available sam-
263
ples. The key ideas of the proposed system are the following.
If en,P ≈ 0, xn is replaced by x̂n,P in the receiver without
any loss, requiring only one bit flag to be transmitted for the
first and the last sample where en,P ≈ 0. If an efficient prediction method for non-uniformly sampled data is used, the
above situation occurs many times during the transmission.
This is the case for example outside the region of interest of
the image where the sample value is constant or null. The
whole number of transmitted samples is thus considerably reduced. As some of the samples are not transmitted, the receiver has to deal with the problem of online identification
and reconstruction of signals subject to missing samples. The
probability law of the prediction error of the image to transmit
is then used to adapt the number of bit coding the prediction
error in the case where it is non zero.
3. PREDICTION/RECONSTRUCTION FOR
NON-UNIFORMLY SAMPLED DATA
Let {xn } be an AR process of order L with parameters {ak },
and {ǫn } the corresponding innovation process of variance
σǫ2 . The loss process is modeled by an i.i.d binary random
variable {cn }, where cn = 1 if xn is available, otherwise
cn = 0. Let {zn } be the reconstruction of the process {xn }.
If xn is available zn = xn , otherwise, zn = x̂n , the prediction of xn . In order to identify, in real time, the AR process
subject to missing data, the algorithm proposed in [8] can be
summarised as follows. The reflection coefficients of the lattice structure are determined by minimizing the weighted sum
(l)
(l)
of the quadratic forward, ft , and backward, bt , prediction
errors :
n
³
´
X
wn−i fn(l)2 + bn(l)2 .
En(l) =
(1)
i=1
A Kalman filter provide an optimal prediction of the signal
using the AR estimated parameters. These parameters are
deduced from the estimated reflection coefficients using the
Durbin Levinson recursions. At time n + 1, the first line of
the matrix A of the state space representation of an AR process is built with â(L)⊤
, the vector of the parameters estimated
n
at time n. The matrix is then named An+1 .
(L)
(L)
. . . . . . âL,n
0
0
An+1
..
..
.
.
0
1
0
Pn+1|n = An+1 Pn|n A⊤
n+1 + Rǫ ,
x̂n+1|n = An+1 x̂n|n
ŷn+1|n = cn+1 x̂n+1|n
â1,n
1
=
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−1
Kn+1 = Pn+1|n cn+1 (c⊤
,
n+1 Pn+1|n cn+1 )
Pn+1|n+1 = (Id −
Kn+1 c⊤
n+1 )Pn+1|n ,
(3a)
(3b)
x̂n+1|n+1 = x̂n+1|n + Kn+1 (yn+1 − ŷn+1|n )
(3c)
The predictions of the previous missing data up to time n −
L + 1 are updated thanks to the filtering of the state in equation (3c). It is convenient now to calculate all the variables
of the lattice filter since the last available observation at time
n − h, where h ≥ 0 depends on the observation pattern. At
each time t, for n − h + 1 ≤ t ≤ n + 1, the recursive equations of the RLSL algorithm given by (5) are applied to es(l)
timate the different reflection coefficients k̂t and prediction
(l) (l)
errors fˆt , b̂t for 1 ≤ l ≤ L. The values of the forward and
backward prediction errors are initialized using the updated
estimates of the missing samples (those contained within the
(0)
(0)
filtered state x̂n+1|n+1 ), i.e. fˆt = b̂t = x̂t|n+1 .
Hence,
• For t = n − h + 1 to n + 1
3.1. Kalman RLSL algorithm
If xn+1 is available, i.e. cn+1 = 1,
,
(2)
– Initialize for l = 0
(0)
(0)
(0)
fˆt = b̂t = x̂t|n+1 , k̂t = 1,
(4)
– For l = 1 to min(L, n)
(l)
(l−1)
(l)
Ct = λCt−1 + 2fˆ(l−1)t b̂t−1 ,
(5a)
(l)
(l)
(l−1)2
(l−1)2
Dt = λDt−1 + fˆt
+ b̂t−1 ,
(5b)
(l)
k̂t =
(l)
fˆt =
(l)
b̂t
=
(l)
C
− t(l) ,
Dt
(l−1)
(l) (l−1)
fˆt
− k̂t b̂t−1 ,
(l−1)
(l) (l−1)
b̂t−1 − k̂t fˆt
,
(5c)
(5d)
(5e)
– end
• end.
(L)
The AR parameters at time n+1, (âi,n+1 )1≤i≤L , are deduced
(l)
from the reflection coefficients (k̂n+1 )1≤i≤L using the Durbin
Levinson recursions. owever if xn+1 is absent, cn+1 = 0, the
predicted state, x̂n+1|n , is not filtered by the Kalman filter,
and the parameters are not updated since the reflection coeffi(l)
cients (k̂n+1 )1≤l≤L are not yet calculated,
Kn+1 = 0,
(6a)
Pn+1|n+1 = Pn+1|n ,
(6b)
x̂n+1|n+1 = x̂n+1|n ,
(6c)
(L)
ân+1
=
â(L)
n .
(6d)
The cost function minimized by this algorithm is the weighted
mean of all quadratic prediction errors. When a sample is
264
missing, the prediction error can not be calculated, it is replaced by its estimation. Indeed, recall that in order to update
the reflection coefficients at a time n, the lattice filter variables
must have been calculated at all previous times. Therefore,
using this algorithm, the lattice filter variables are estimated
at all times even when a sample is missing. Consequently,
this algorithm presents an excellent convergence behavior and
have fast parameter tracking capability even for a large probability of missing a sample. The computational complexity of
this algorithm is found to be O((1 − q)N L2 ), where q is the
bernoulli’s probability of losing a sample, N is the size of the
signal and L the order of the AR model.
3.2. Adaptation to 2D signals
A first solution to use the previous algorithm for 2D signals
is to use the classical video scanning of the image in order to
get a 1D signal. However, only a 1D decorrelation is achieved
using this method.
In order to get a 2D decorrelation of the image, a 2D AR
predictor x̂i,j of the sample xi,j (7) must be used in addition
to the video scanning of the image.
m
S
n
Fig. 1. AR 2D: prediction support
X
ân,m xi−n,j−m
(7)
n,m∈S
In order to integrate this 2D AD predictor into the previous
algorithm, the first line of the A matrix is built with the ân,m
⊤
parameters, and the regressor vector [xn−1 . . . xn−L ] is re⊤
placed by [xi−1,j . . . xi−n,j−m . . . xi−p,j−q ] . The renumbering task excepted, to built the A matrix, the computational
time of these 2D algorithm is similar to the 1D one.
4. PROPOSED ADAPTATIVE TRANSMISSION
ALGORITHM
In this section, we propose to use the algorithms discussed
in section 3 as efficient predictors in the non uniform transmission system proposed in section 2 in order to minimize
the number of bit to transmit. At each time n, knowing all
transmitted samples and using the same identification and reconstruction method as the one used in the receiver, the transmitter evaluates the signal reconstruction performance in the
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• In the transmitter:
. en,P = xn − x̂n,P
. if (|en,P | = 1e−5 and |en−1,P | > 1e−5 or |en,P | >
1e−5 and |en−1,P | = 1e−5 ), one bit flag is transmitted,
. else if |en,P | < S2 ,
.
if |en,P | < S3 , B3 bits are transmitted,
.
else B2 bits are transmitted,
. else B1 bits are transmitted.
• In the receiver, the method described in 3 is used for
adaptive identification and reconstruction of a signal
subject to missing data: if a new sample is received,
the AR parameters are updated. Otherwise, the missing sample is predicted in terms of the past available
samples and the current estimation of the parameters.
5. SIMULATIONS
x(i, j )
x̂i,j =
receiver. This can be done by comparing the receiver prediction error, |en,P |, with different thresholds, S1 ≈ 0, S2 , ..., Si .
Thus, if the receiver is able to reconstruct the sample without error (error greatly smaller than the quantification error
(1e−5 )), only a one bit flag is transmitted to indicate the first
and the last missing sample. The number of thresholds Si and
their values are chosen according to the probability law of the
prediction error to transmit only the Bi bits required to code
the prediction error for each threshold. The proposed coding
decoding algorithm can be summarized, at a time n, as:
The performances of both proposed methods are compared to
the JPEG2000. The first method uses a 1D AR model of order
3 of the signal. In the second method, the image is modeled
by a 2D AR process of order (2, 2). The performances of the
different methods are evaluated in term of bit rate (in bpp) on
CT images. The PSNR is computed for the proposed methods
to show the lossless reconstruction of the image. The PSNR
which have been reached for all the simulations corresponds
to the infinity value. Table 1 shows the results for CT images
of (512x512x12) bits presented in figures 2, 3 and 4 (Images
courtesy of Dr Kopans, MGH Boston, USA. Tomosynthesis
investigational device from GE Healthcare (Chalfont St Giles,
UK)). In these images the prediction error is in most of the
case small (lower than 32), but for the pixels of the edge of
the ROI the prediction error requires 12 bits to be coded. Consequently, the following values are chosen for the number of
bit to code the prediction error : B1 = 13, B2 = 8, B3 = 6.
6. CONCLUSION
A new digital transmission system for lossless image reconstruction has been proposed. It is based on a non-uniform
transmission principle and on extensions to 2D of the algorithm proposed in [8] for real time identification and reconstruction of AR processes subject to missing data. The pro-
265
50
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100
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150
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200
200
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250
300
300
350
350
400
400
450
450
500
500
50
100
150
200
250
300
350
400
450
500
Fig. 2. CT1 image
50
100
150
200
250
300
350
400
450
500
Fig. 4. CT3 image
coding system, 2000.
50
100
[2] ISO/IEC 14495-1, “Information technology - lossless
and near-lossless compression of continuous-tone still
images,” JPEG-LS standard, Baseline, 2000.
150
200
250
[3] A. Said and W. A. Pearlman, “A new fast and efficient
image codec based on set partitionning in hierarchical
trees,” IEEE Trans. on Circuits and systems for Video
Technology, vol. 6, pp. 243–250, June 1996.
300
350
400
450
500
50
100
150
200
250
300
350
400
450
500
Fig. 3. CT2 image
posed methods, applied on CT images, has shown in their two
forms (2D as well as 1D) an improvement in bit rate comparing to the JPEG2000 and JGPEG-LS standards. Comparing
to the JPEG2000, significant gains for lossless compression
are reached: 3.4% for CT3 image up to 4.6% for CT1 image.
Comparing to the JPEG-LS, the most significant gains (2.7%
up to 3.6%) are reached for CT2 and CT1 images where the
RLE coding of the JPEG-LS is not used.
7. REFERENCES
[1] ISO/IEC 15444-1, “Information technology - jpeg2000
image coding system,” JPEG2000 standard, Part 1-Core
Table 1. Comparison of the three methods in bit rate (in bpp)
for CT images of (512x512x12) bits:
Method
CT 1 CT 2 CT 3
1
6.53 6.67 5.10
2
6.45 6.61 5.10
JPEG2000 6.76 6.89 5.28
JPEG-LS
6.69 6.79 5.15
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[4] A. Munteanu and J. Cornelis, “Wavelet based lossless
compression scheme with progressive transmission capability,” International Journal of Imaging Systems and
Tecnology, vol. 10, pp. 76–85, January 1999.
[5] S. Mirsaidi, G. Fleury, and J. Oksman, “Reducing quantization error using prediction/non uniform transmission,”
in Proc. International Workshop on Sampling Theory and
Applications. IEEE, 1997, pp. 139–143.
[6] E. Lahalle and J. Oksman, “ADPCM speech coder
with adaptive transmission and ARMA modelling of
non-uniformly sampled signals,” in 5th Nordic Signal
Processing Symposium, CD-ROM proceedings, Norway.
IEEE, 2002.
[7] S. Mirsaidi, G. Fleury, and J. Oksman, “LMS like AR
modeling in the case of missing observations,” IEEE
Transactions on Signal Processing, vol. 45, pp. 1574–
1583, June 1997.
[8] R. Zgheib, G. Fleury, and E. Lahalle, “Lattice algorithm
for adaptive stable identification and robust reconstruction of non stationary ar processes with missing observations,” IEEE Transactions on Signal Processing, vol. 56,
pp. 2746–2754, July 2008.
[9] N. S. Jayant and P. Noll, “Digital coding of waveform,
principles and applications to speech and video,” Prentice
Hall, 1984.
266
Geometric Sampling of Images, Vector
Quantization and Zador’s Theorem
Emil Saucan (1) , Eli Appleboim (2) and Yehoshua Y. Zeevi (2)
(1) Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel.
(3) Electrical Engineering Department, Technion - Israel Institute of Technology, Haifa 32000, Israel.
semil@tx.technion.ac.il, eliap@ee.technion.ac.il, zeevi@ee.technion.ac.il
Abstract:
We present several consequences of the geometric approach to image sampling and reconstruction we have previously introduced. We single out the relevance of the geometric method to the vector quantization of images and,
more important, we give a concrete and candidate for the
optimal embedding dimension in Zador’s Thorem. An additional advantage of our approach is that that this provides a constructive proof of the aforementioned theorem,
at least in the case of images. Further applications are also
briefly discussed.
1. Introduction
In recent years it became common amongst the signal processing community, to consider images and other signals
as well, as Riemannian manifolds embedded in higher dimensional spaces. Usually, the embedding manifold is
taken to be Rn , but other options can, and had been considered. Along with that, sampling is an essential preliminary step in processing of any continuous signal by a digital computer. This step lies at heart of any digital processing of any (presumably continuous) data/signal. It is
therefore natural to strive to achieve a sampling method
for images, viewed as such, that is as higher dimensional
dimensional objects (i.e. manifolds), rather than their representation as 1-dimensional signals. In consequence, our
sampling and reconstruction techniques stem from the the
fields of differential geometry and topology, rather than
being motivated by the traditional framework of harmonic
analysis. More precisely, our approach to Shannon’s Sampling Theorem is based on sampling the graph of the signal, considered as a manifold, rather than a sampling of
the domain of the signal, as is customary in both theoretical and applied signal and image processing. In this context it is important to note that Shannon’s original intuition
was deeply rooted in the geometric approach, as exposed
in his seminal work [14].
Our approach is based upon the following sampling theorem for differentiable manifolds that was recently presented and applied in the context image processing [12]:
Theorem 1 Let Σn ⊂ RN , n ≥ 2 be a connected, not
necessarily compact, smooth manifold, with finitely many
compact boundary components. Then, there exists a sampling scheme of Σn , with a metric density D = D(p) =
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³
´
1
D k(p)
, where k(p) = max{|k1 |, ..., |kn |}, and where
k1 , ..., kn are the principal curvatures of Σn , at the point
p ∈ Σn .
In particular, if Σn is compact, then there exists a sampling
of Σn having uniformly bounded density. Note, however,
that this is not necessarily the optimal scheme (see [12]).
The constructive proof of this theorem is based on the existence of the so-called fat (or thick) triangulations (see
[11]). The density of the vertices of the triangulation (i.e.
of the sampling) is given by the inverse of the maximal
principal curvature. An essential step in the construction
of the said triangulations consists of isometrically embedding of Σn in some RN , for large enough N (see [10]),
where the existence of such an embedding is guaranteed
by Nash’s Theorem ([9]). Resorting to such a powerful
tool as Nash’s Embedding Theorem appears to be an impediment of our method, since the provided embedding
dimension N is excessively high (even after further refinements due to Gromov [4] and Günther [5]). Furthermore,
even finding the precise embedding dimension (lower than
the canonical N ) is very difficult even for simple manifolds. However, as we shall indicate in the next section,
this high embedding dimension actually becomes an advantage, at least from the viewpoint of information theory.
The resultant sampling scheme is in accord with the classical Shannon theorem, at least for the large class of (bandlimited) signals that also satisfy the condition of being C 2
curves. In our proposed geometric approach, the radius of
curvature substitutes for the condition of the Nyquist rate.
To be more precise, our approach parallels, in a geometric setting, the local bandwidth of [7] and [16]. In other
words, manifolds with bounded curvature represent a generalization of the locally band limited signals considered
in those papers.
We concentrate here only on some of the consequences
of Theorem 1. More precisely, we present, in Sections 2
and 3, two applications of our geometric sampling method
and of the embedding technique employed in the proof,
namely to the vector quantization of images and to determining the embedding dimension in Zador’s Theorem,
respectively. Further directions of study are briefly discussed in the concluding section.
267
2. Vector Quantization for Images
A complementary byproduct of the constructive proof of
Theorem 1 is a precise method of vector quantization (or
block coding). Indeed, the proof of Theorem 1 consists
in the construction of a Voronoi (Dirichlet) cell complex
{γ̄kn } (whose vertices will provide the sampling points).
The centers ak of the cells (satisfying a certain geometric
density condition) represent, as usual, the decision vectors. An advantage of this approach, besides its simplicity, is entailed by the possibility to estimate the error in
terms of length and angle distortion when passing from
the cell complex {γ̄kn } to the Euclidean cell complex {c̄nk }
having the same set of vertices as {γ̄kn } (see [10]). Indeed, in contrast to other related studies, our method not
only produces a piecewise-flat simplicial approximation
of the given manifold, it also actually renders a simplicial complex on the manifold. Moreover, one can actually
compute the local distortion resulting by passing from the
Euclidean geometry of the piecewise-flat approximation
to the intrinsic geometry of its projection on the manifold.
If M = M n is a manifold without boundary, then locally,
for any triangulation patch the following inequality holds
[10]:
3
5
dM (x, y) ≤ deucl (x̄, ȳ) ≤ dM (x, y) ;
4
3
where deucl , dM denote the Euclidean and intrinsic metric (on M ) respectively, and where x, y ∈ M and x̄, ȳ
are their preimages on the piecewise-flat complex. For
manifolds with boundary, the same estimate holds (for the
intM and ∂M ), except for a (small) zone of “mashing”
triangulations (see [11]), where the following weaker distortion formula is easily obtained:
5
3
dM (x, y)−f (θ)η∂ ≤ deucl (x̄, ȳ) ≤ dM (x, y)+f (θ)η∂ ;
4
3
where f (θ) is a constant depending on the θ =
min {θ∂ , θint M } – the fatness of the triangulation of ∂M
and int M, respectively, and η∂ denotes the mesh of the triangulation of a certain neighbourhood of ∂M (see [11]).
In other words, the (local) projection mapping π between
the triangulated manifold M and its piecewise-flat approximation Σ is (locally) bi-lipschitz if M is open, but only
a quasi-isometry (or coarsely bi-lipschitz) if the boundary
of M is not empty.
But the main advantage of a geometric sampling of images resides in the fact that the sampling is done according
to the geometric, hence intrinsic, features of the image,
rather in the arbitrary (as far as features are concerned)
manner of classical approach that transforms the image
into a 1-dimensional array (signal). Therefore, the resulting sampling is adaptive, hence sparse in regions of low
curvature, and, as shown in [1], it is even compressive in
some special cases.
3. Zador’s Theorem
A more important application stems, however, from
Zador’s Theorem [15], implying that we can turn into an
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advantage the inherent “curse of dimensionality”. Indeed,
by of Zador’s Theorem, the average mean squared error
per dimension:
Z
1
E=
deucl (x, pi )p(x)dx ,
N RN
pi being the code point closest to x and p(x) denoting
the probability density function of x, can be reduced by
making avail of higher dimensional quantizers (see [2]).
Since for embedded manifolds it obviously holds that
p(x) = p1 (x)χM , we obtain:
Z
1
E=
deucl (x, pi )p1 (x)dx ,
N Mn
It follows that, if the main issue is accuracy, not simplicity,
then 1-dimensional coding algorithms (such as the classical Ziv-Lempel algorithm) perform far worse than higher
dimensional ones. Of course, there exists an upper limit
for the coding dimension, since otherwise one could just
code the whole data as one N -dimensional vector (albeit
of unpractically high dimension). The geometric coding
method proposed here provides a natural high dimension
for the quantization of M n – the embedding dimension N .
Moreover, it closes (at least for images and any other data
that can be represented as Riemannian manifolds) an open
problem related to Zador’s Theorem: finding a constructive method to determine the dimension of the quantizers
(Zador’s proof is nonconstructive). In fact, for a uniformly
distributed input (as manifolds, hence noiseless images,
can assumed to be, at least in first approximation) a better
estimate of the average mean squared error per dimension
can be obtained, namely:
E=
1
N
R
d
(x, pi )dx
M n R eucl
Mn
dx
=
1
N
R
deucl (x, pi )dx
,
Vn (M n )dx
Mn
where Vn denotes the n-dimensional volume (area) of M .
Whence, for compact manifolds one obtains the following
expression for E:
E=
1
N
R
d
(x, pi )dx
=
dx
eucl
n
MP
mR
i
Vi
1
N
R
d
(x, pi )dx
M n eucl
P
,
m
V
(V
n i )dx
i
where Vi represent the Voronoi cells of the partition.
Moreover, we have the following estimate for the quantizer problem, that is: Chose centers of cells such that the
quantity
R
1
d
(x, pi )dx
1 m
M n eucl
.
Q=
¢1+ N2
¡
P
m
N
1
V
m
i
n
is minimized. Here, again, the high embedding dimension
N furnishes us with yet an additional advantage. Indeed,
manifolds N increases dramatically, even for compact
manifolds and even taking into consideration Gromov’s
and Günther’s improvement of Nash’s original method
(see [4], resp. [5]). For instance, n = 2 requires embedding dimension N = 10 and n = 3 the necessitates
N = 14. Hence, for large enough n one can write the
following rough estimate:
268
1
Q≈
N
R
Mn
deucl (x, pi )dx
Pm
.
i Vn
4. Conclusions and Future work
As we have stressed above, our geometrical approach to
sampling lends itself to consideration of a much broader
range of topics in communications, for such problems
as Coding, Channel Capacity, amongst others (see [13]).
In particular, and almost as an afterthought of the ideas
presented in Section 2, it offers a new method for PCM
(pulse code modulation – see [2] for a brief yet lucid presentation) of images, considered as such and not as 1dimensional signals. This approach is endowed with an
inherent advantage in that the sampling points are associated with relevant geometric features (via curvature) of
the image, viewed as a manifold of dimension ≥ 2, and are
not chosen via the Nyquist rate of some rather arbitrarily
computed 1-dimensional signal. Moreover, the sampling
is in this case adaptive and, indeed, compressive, lending
itself to interesting technological benefits.
The implementation of the PCM method described above,
as well as experimenting with the geometric quantization
method, represent the applicative directions of study that
are natural and interesting to pursue further. A better understanding of the geometry of images, included color,
texture and other relevant features, in terms of curvature,
represent the theoretical directions to be pursued in future.
In particular, determining the lowest embedding dimension and finding global curvature constraints are, as we
have seen, important for a highly compressive sampling.
5. The role of curvature
We briefly discuss here the crucial role of curvature in determining the embedding dimension (and hence the Zador
dimension) by illustrating it on a “toy” example, namely
that of the torus.
For a “round” torus of revolution Tr2 in R3 , the embedding dimension is N = 3, since the metric of Tr2 is the
intrinsic one induced by the Euclidian one of the ambient space R3 , thus in this case our method does not depart
too much from standard ones. However, if one considers the flat torus Tf2 , i.e. of Gaussian curvature K ≡ 0,
then the minimal dimension needed for isometric embedding is N = 4 (see, e.g. [3]). (Before we proceed further,
let us note that such tori arise naturally when considering
planar rectangles with opposite sides identified – that is,
“glued” – via translations. In a practical context, these
would model 2-dimensional repetitive patterns on a computer screen, e.g. screen savers. Flat tori also appear in
another context relevant to Computer Graphics and Image Processing, namely as solutions for discrete curvature
flows (on triangular meshes), see e.g. [8].) In general,
given a 2-dimensional torus, equipped with generic Riemannian metric, the whole range of dimensions, up to,
and including, the one prescribed by the Nash-GromovGünther Theorem, is possible. There are huge differences
arising not only from the sign of the curvature, but from
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its “speed of change” as well – for a exhaustive treatment
of this subject see [6].
6.
Acknowledgments
The authors would like to thank Professor Peter Maass, for
his constructive critique and encouragement. The first author would also like to thank Professor Shahar Mendelson
– his warm support is gratefully acknowledged.
References:
[1] Eli Appleboim, Emil Saucan and Yehoshua Y. Zeevi.
Geometric Sampling For Signals With Applications to
Images. Proceedings of Sampta 2007, 2008.
[2] John H. Conway and Neil J. A. Sloane Sphere Packings, Lattices and Groups. Springer, New York, 1999.
[3] Manfredo P. do Carmo Differential Geometry of
Curves and Surfaces. Prentice-Hall, Englewood
Cliffs, N.J., 1976.
[4] Mikhail Gromov. Partial differential relations,
Springer-Verlag, Ergeb. der Math. 3 Folge, Bd. 9,
Berlin-Heidelberg-New-York, 1986.
[5] Matthias Günther. Isometric embeddings of Riemannian manifolds. Proc. ICM Kyoto, pages 1137–1143,
1990.
[6] Qing Han and Jia-Xing Hong Isometric embeddings
of Riemannian manifolds in Euclidean Spaces. AMS
MSM 130, Providnce, RI, 2006.
[7] K. Horiuchi. Sampling principle for continuous signals with time-varying bands. Information and Control, 13(1): 53-61, 1968.
[8] Miao Jin, J. Kim and David Gu. Discrete Surface
Ricci Flow: Theory and Applications. In Mathematics
of Surfaces, LNCS 4647, pages 209–232, 2007.
[9] John Nash. The embedding problem for Riemannian
manifolds. Ann. of Math. 63:20–63, 1956.
[10] Kirsi Peltonen. On the existence of quasiregular
mappings. Ann. Acad. Sci. Fenn., Series I Math., Dissertationes 1992.
[11] Emil Saucan. Note on a theorem of Munkres.
Mediterr. j. math. 2(2):215–229, 2005.
[12] Emil Saucan, Eli Appleboim, and Yehoshua Y Zeevi.
Sampling and Reconstruction of Surfaces and Higher
Dimensional Manifolds. Journal of Mathematical
Imaging and Vision 30(1):105–123, 2008.
[13] Emil Saucan, Eli Appleboim, and Yehoshua Y Zeevi.
Geometric Approach to Sampling and Communication. Technion CCIT Report #707, November 2008.
[14] Claude E. Shannon. Communication in the presence
of noise. Proceedings of the IRE 37(1):10–21, 1949.
[15] Paul G. Zador. Asymptotic Quantization Error of
Continuous Signals and the Quantization Dimension.
IEEE Trans. on Info. Theory, 12(1):23–86, 1982.
[16] Yehoshua Y. Zeevi and E. Shlomot. Nonuniform
sampling and antialiasing in image representation.
IEEE Trans. Signal Process., 41(3):1223–1236, 1993.
269
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270
On average sampling restoration of
Piranashvili–type harmonizable processes
Andriy Ya. Olenko† and Tibor K. Pogány‡
† Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia.
‡ Faculty of Maritime Studies, University of Rijeka, Studentska 2, HR-51000 Rijeka, Croatia.
a.olenko@latrobe.edu.au, poganj@pfri.hr
Abstract:
Such a process will be called Piranashvili process in the
sequel [11], [12].
The harmonizable Piranashvili – type stochastic processes are approximated by a finite time shifted average
sampling sum. Truncation error upper bound is established; various consequences and special cases are
discussed.
MSC(2000): 42C15, 60G12, 94A20.
Keywords: WKS sampling theorem; time shifted sampling; Piranashvili–, Loève–, Karhunen– harmonizable
stochastic process; weakly stationary stochastic process;
local averages; average sampling reconstruction.
1.
Introduction and preparation
Given a probability space Ω, F, P and the related
Hilbert–space L2 (Ω) := {X : E|X|2 < ∞}. Let us consider a non–stationary, centered stochastic L2 (Ω)–process
ξ : R × Ω 7→ R having covariance function (associated to
some domain Λ ⊆ R with some sigma–algebra σ(Λ)) in
the form:
Z Z
B(t, s) =
f (t, λ)f ∗ (s, µ)Fξ (dλ, dµ), (1)
Λ
Λ
with analytical exponentially bounded kernel function
f (t, λ), while Fξ is a positive definite measure on R2 provided the total variation kFξ k(Λ, Λ) of the spectral distribution function Fξ such that satisfies
Z Z
Fξ (dλ, dµ) < ∞.
kFξ k(Λ, Λ) =
Λ
Λ
(We mention that the sample function ξ(t) ≡ ξ(t, ω0 ) and
f (t, λ) possess the same exponential types [1, Theorem
4], [11, Theorem 3]). Then, by the Karhunen–Cramér theorem the process ξ(t) has the spectral representation as a
Lebesgue integral
Z
ξ(t) =
f (t, λ)Zξ (dλ);
(2)
Λ
in (1) and (2)
Fξ (S1 , S2 ) = EZξ (S1 )Zξ∗ (S2 )
SAMPTA'09
S1 , S2 ⊆ σ(Λ).
Being f (t,P
λ) entire, it possesses the Maclaurin expansion
∞
f (t, λ) = n=0 f (n) (0, λ)tn /n!. Put
q
γ := sup c(λ) = sup lim n |f (n) (0, λ)| < ∞ .
(3)
Λ
n
Λ
As the exponential type of f (t, λ) is equal to γ, for all
w > γ there holds
X nπ sin(wt − nπ)
ξ(t) =
ξ
,
(4)
w
wt − nπ
n∈Z
uniformly in the mean square and in the almost sure sense
[11, Theorem 1]. This result we call Whittaker–Kotel’nikov–Shannon (WKS) stochastic sampling theorem [12].
Specifying Fξ (x, y) = δxy Fξ (x) in (1) we conclude the
Karhunen–representation of the covariance function
Z
f (t, λ)f ∗ (s, λ)Fξ (dλ).
B(t, s) =
Λ
Also, putting f (t, λ) = eitλ in (1) one gets the Loèverepresentation:
Z Z
ei(tλ−sµ) Fξ (dλ, dµ).
B(t, s) =
Λ
Λ
Here is c(λ) = |λ|. Therefore, WKS–formula (4) holds
for all w > γ = sup |Λ|. Then, the Karhunen process
with the Fourier kernel f (t, λ) = eitλ we recognize as the
weakly stationary stochastic process having covariance
Z
eiτ λ Fξ (dλ),
τ = t − s.
B(τ ) =
Λ
Deeper insight into different kind harmonizabilities
present [5, 13, 14] and the related references therein. Finally, using Λ = [−w, w] for some finite w in this considerations, we get the band–limited variants of the same
kind processes.
By physical and applications reasons the measured samples in practice may not be the exact values of the measured process ξ(t), or its covariance B(t, s) itself, near to
the sample time tn , but only the local average of the signal
ξ near to tn . So, the measured sample values will be
Z
hξ, un iU =
ξ(x)un (x)dx, U = supp(un ) (5)
U
271
for a sequence u := un (t) n∈Z of non–negative, normalized, that is h1, un i ≡ 1, averaging functions such that
supp(un ) ⊆ tn − σn′ , tn + σn′′ .
(6)
The local averaging method was introduced by Gröchenig
[2] and developed by Butzer and Lei. Recently Sun
and Zhou gave some results in this direction, while the
stochastic counterpart of this average sampling was intensively studied in the last three–four years by He, Song,
Sun, Yang and Zhu in a set of articles [15], [16] and their
references therein; see for example the exhaustive references list in [4]. The listed, recently considered stochastic
average sampling results are restricted to weakly stationary stochastic processes, while the approximation average sampling sums are used around the origin.
Our intentions are to extend these results to time shifted
average sampling, considered for the very wide class of
Piranashvili processes.
Theorem 1 Let f (z) be entire, bounded on the real axis
and exponentially bounded having type γ < w. Denote
Lf := sup f (x) ,
L0 (z) :=
R
2wLf | sin(wz)|
.
π(w − γ) 1 − e−π
Then for all z ∈ int ΓN (x) and N ∈ N enough large it
holds
X π sin(wz − nπ)
f n
w
wz − nπ
Z\IN (x)
≤
L0 (z)e−(N +1/2)π(w−γ)/w
(N + 1/2) 1 −
|z−Nx |w
(N +1/2)π
<
L0 (z)
.
N
(7)
The proving method is contour integration, following Piranashvili’s traces [11]. Denote here and in what follows
X nπ sin(wt − nπ)
YN (ξ; t) :=
ξ
w
wt − nπ
IN (t)
2.
the time shifted truncated WKS restoration sum.
Time shifted average sampling
Now, instead to follow the approach used in [16] we
take time shifted [7], [8] finite average sampling sum
in approximating the initial stochastic signal ξ. First,
we consider weighted average over Jn (t) := nπ/w −
σn′ (t), nπ/w + σn′′ (t) for the measured value of ξ(t) at
nπ/w, n ∈ IN (t) where
IN (t) := {n ∈ Z : |tw/π − n| ≤ N },
N ∈ N.
Let Nt be the integer nearest to tw/π.
By obvious reasons we restrict the study to
π
.
σ := max sup max σn′ (t), σn′′ (t) ≤
2w
IN (t) R
Let us define the time shifted average sampling approximation sum in the form
X
sin(wt − nπ)
Au (ξ; t) =
hξ, un iJn (t) ·
,
wt − nπ
Z
and its truncated variant
X
sin(wt − nπ)
hξ, un iJn (t) ·
.
Au,N (ξ; t) =
wt − nπ
IN (t)
One defines mean–square, time shifted, average sampling
2
truncation error Tu,N (ξ; t) := E ξ(t) − Au,N (ξ; t) .
Now, we are interested in some reasonably simple efficient mean square truncation error upper bound appearing
in the approximation ξ(t) ≈ Au,N (ξ; t).
Let us introduce some auxiliary results. As Nx stands for
the integer nearest to xw/π, x ∈ R, let
n
πo
ΓN (x) := z ∈ C : |z − Nx | ≤ N + 12 w
, N ∈ N.
In what follows denote int(R) the interior of some R,
while the series
∞
X
1
λ(q) :=
(2n
−
1)q
n=1
stands for the Dirichlet lambda function.
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By simple use of (1), (2) and the Theorem 1 one deduces
the following modest generalization of [11, Theorem 2] to
time shifted case of sampling restoration procedure.
Theorem 2 Let ξ(t) be a Piranashvili process with exponentially bounded kernel function f (t, λ) and let
e f := sup sup |f (t, λ)|,
L
Λ
R
e
e 0 (t) := 2Lf w | sin(wt)| .
L
π(w − γ) 1 − e−π
Then for all t ∈ int ΓN (t) , we have
E ξ(t) − YN (ξ; t)
2
<
e 2 (t)
L
0
kFξ k(Λ, Λ) .
N2
(8)
(9)
Remark 1 Let us point out that the straightforward consequence of (9) is not only the exact L2 –restoration of the
initial Piranashvili–type harmonizable process ξ by a sequence of approximants YN (ξ; t) when N → ∞, but since
2
E ξ(t) − YN (ξ; t) = O N −2 ,
the perfect reconstruction is possible in the a.s. sense as
well (by the celebrated Borel–Cantelli Lemma).
Second, the first order difference ∆x,y B [3] of B(t, s) on
the plane satisfies
∆x,y B (t, s) = B(t + x, s + y) − B(t + x, s)
− B(t, s + y) + B(t, s)
Z xZ y
∂2
=
B t + u, s + v dvdu .
(10)
0
0 ∂u∂v
Theorem 3 Let ξ(t) be a Piranashvili process with the
covariance B(t, t) ∈ C 2 (R). Let (p, q) be a conjugated
Hölder pair of exponents:
1 1
+ = 1,
p q
p > 1.
272
Then we have
2
E YN (ξ; t) − Au,N (ξ; t)
N n
X
≤1+C
n=1
∞
X
2
≤
Cq π
sup B ′′ (t, t) · (2N + 1)2/p ,
4w2 R
< 1 + 2C
(11)
o
1
1
+
(n − ∆)q
(n + ∆)q
1
(n
−
1/2)q
n=1
< 1 + 2q+1 C λ(q) ,
where
2/q
2q+1 | sin(wt)|q
Cq = 1 +
λ(q)
.
q
π
(12)
where
| sin(wt)|q
.
πq
Collecting all these estimates, we deduce (11).
C=
P ROOF. Having on mind (1), the properties of averaging
functions sequence u and (10), we clearly derive
2
E YN (ξ; t) − Au,N (ξ; t)
X
sin(wt − nπ)
=E
hξ nπ
w − ξ(x), un iJn (t) ·
wt − nπ
2
IN (t)
XZ
=
I2N (t)
Z
x
′′
σn
(t)
′ (t)
−σn
Z
Z
′′
σm
(t)
′ (t)
−σm
π
π
un (x + n w
)um (y + m w
)
y
∂2
π
, v + m πv dvdu
B u + nw
∂u∂v
0
0
sin(wt − nπ) sin(wt − mπ)
·
wt − nπ
wt − mπ
X sin(wt − nπ) sin(wt − mπ)
≤
wt − nπ
wt − mπ
2
·
3.
Main result
We are ready to formulate our upper bound result for the
mean square, time shifted average sampling truncation error Tu,N (ξ; t). The almost sure sense restoration procedure has been treated too.
As we use average sampling sum Au,N (ξ; t) instead of
YN (ξ; t) to obtain asymptotically vanishing Tu,N (ξ; t), it
is not enough letting N → ∞ as in Remark 1. For average sampling we need additional conditions upon w or
σ to guarantee smaller average intervals for larger/denser
sampling grids.
IN (t)
· sup
x,y≤σ
Z
0
x
Z
0
y
∂2
π
, v + m πv dvdu
B u + nw
∂u∂v
being u normalized. For the sake of brevity let us denote
Hσ (n, m) the sup–term in the last display. Then, by the
Hölder inequality with conjugate exponents p, q; p > 1,
we get
2
E YN (ξ; t) − Au,N (ξ; t)
)2/q
(
)1/p (
X sin(wt − nπ) q
X
p
≤
Hσ (n, m)
.
wt − nπ
2
IN (t)
IN (t)
It is not hard to see that for all n, m ∈ IN (t) there holds
∂ 2 B(t, s)
Hσ (n, m) ≤ σ sup
∂t∂s
R2
Theorem 4 Assume the conditions of Theorems 2 and 3
have been fulfilled. Then, we have
Tu,N (ξ; t) ≤
+
e 2 (t)
2L
0
kFξ k(Λ, Λ)
N2
Cq π 2
sup B ′′ (t, t) · (2N + 1)2/p ,
2w2 R
(13)
e 0 , Cq are described by (8), (11) respectively.
where L
Moreover, when w = O N 1/2+1/p+ε , ε > 0, we have
(14)
P lim Au,N (ξ; t) = ξ(t) = 1
N →∞
for all t ∈ R.
2
∂ 2 B(t, s)
π2
≤
sup
.
4w2 R2
∂t∂s
PROOF. By direct calculation we deduce
Tu,N (ξ; t) = E ξ(t) − Au,N (ξ; t)
= E ξ(t) − YN (ξ; t) + YN (ξ; t) − Au,N (ξ; t)
Applying now the Cauchy–Bunyakovsky–Schwarz inequality to the covariance ∂ 2 B, we deduce
sup
R2
R
It remains to evaluate the sum of qth power of the sinc–
functions. As
sin(wt − Nt π)
≤1
wt − Nt π
we conclude
X sin(wt − nπ) q
IN (t)
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≤ 2E ξ(t) − YN (ξ; t)
wt − nπ
2
2
2
∂ 2 B(t, t)
∂ 2 B(t, s)
≤ sup
∂t∂s
∂t2
R
= sup |B ′′ (t, t)| .
2
+ 2E YN (ξ; t) − Au,N (ξ; t) .
Now, we get the asserted upper bound by (9) and (11).
To derive (14), we apply the Chebyshev inequality to evaluate the probability
PN := P ξ(t) − Au,N (ξ; t) ≥ η ≤ η −2 Tu,N (ξ; t) .
e 0 (t) = O(1) as N → ∞, we have
Accordingly, since L
X
N
PN ≤ K
X 1
(2N + 1)2/p
+
< ∞,
N2
w2
N
273
K being a suitable absolute constant. Therefore, by the
Borel–Cantelli Lemma, the the a.s. convergence result
(14) holds true.
Remark 2 Theorem 4 ensures the perfect time shifted average sampling restoration
in the mean square sense when
w = O N 1/p+ε , ε > 0:
lim Tu,N (ξ; t) = 0 .
N →∞
The a.s. sense restoration (14) requires stronger assumption, it holds when w = O N 1/2+1/p+ε .
Remark 3 In both cases we use the so called approximate
sampling procedure, that is, when in the restoration procedure w → ∞ in some fashion. The consequence of these
results is that we have to restrict ourselves to the case
Λ = R, such that we recognize as the non–bandlimited
Piranashvili type harmonizable process case.
The importance of approximate sampling procedures for
investigations of aliasing errors in sampling restorations
and different conditions on joint asymptotic behaviour of
N and w have been discussed in detail in [7].
4.
Conclusions
We have analyzed upper bounds on truncation error for
time shifted average sampling restorations in the stochastic initial signal case. The convergence of the truncation
error to zero was discussed. However, certain new questions immediately arise:
• to derive sharp upper bounds in Theorems 3 and 4;
• to obtain new results for Lp –processes using recent
deterministic findings [9], [10];
• to obtain similar results for irregular/nonuniform
sampling restoration using methods exposed in [6]
and [10].
Acknowledgements
The recent investigation was supported in part by Research
Project No. 112-2352818-2814 of Ministry of Sciences,
Education and Sports of Croatia and in part by La Trobe
University Research Grant–501821 ”Sampling, wavelets
and optimal stochastic modelling”.
References:
[3] Muhammed K. Habib and Stamatis Cambanis. Sampling approximation for non–band–limited harmonizable random signals. Inform. Sci 23:143–152,
1981.
[4] Gaiyun He, Zhanjie Song, Deyun Yang and Jianhua
Zhu. Truncation error estimate on random signals by
local average. In Y. Shi et al., editors. ICCS 2007,
Part II, Lecture Notes in Computer Sciences 4488,
pages 1075–1082, 2007.
[5] Yûichirô Kakihara. Multidimensional Second Order
Stochastic Processes. World Scientific, Singapore,
1997.
[6] Andriy Ya. Olenko and Tibor K. Pogány. Direct
Lagrange–Yen type interpolation of random fields.
Theor. Stoch. Proc. 9(25)(3–4): 242–254, 2003.
[7] Andriy Ya. Olenko and Tibor K. Pogány. Time
shifted aliasing error upper bounds for truncated
sampling cardinal series. J. Math. Anal. Appl.
324(1): 262–280, 2006.
[8] Andriy Ya. Olenko and Tibor K. Pogány. On sharp
bounds for remainders in multidimensional sampling
theorem. Sampl. Theory Signal Image Process. 6(3):
249–272, 2007.
[9] Andriy Ya. Olenko and Tibor K. Pogány. Universal
truncation error upper bounds in sampling restoration. (to appear)
[10] Andriy Ya. Olenko and Tibor K. Pogány. Universal
truncation error upper bounds in irregular sampling
restoration. (to appear)
[11] Zurab A. Piranashvili. On the problem of interpolation of random processes. Teor. Verojat. Primenen.
XII(4): 708–717, 1967. (in Russian)
[12] Tibor K. Pogány. Almost sure sampling restoration
of bandlimited stochastic signals. In John R. Higgins and Rudolf L. Stens, editors. Sampling Theory
in Fourier and Signal Analysis: Advanced Topics,
Oxford University Press, pages 203–232, 284–286,
1999.
[13] Maurice B. Priestley. Non–linear and Non–
stationary Time Series. Academic Press, London,
New York, 1988.
[14] Malempati M. Rao. Harmonizable processes: structure theory. Einseign. Math. (2) 28(3–4): 295–351,
1982.
[15] Zhanjie Song, Zingwei Zhu and Gaizun He. Error
estimate on non–bandlimited random signals by local averages. In V.N. Aleksandrov it et al., editors.
ICCS 2006, Part I, Lecture Notes in Computer Sciences 3991, pages 822–825, 2006.
[16] Zhan–jie Song, Wen–chang Sun, Shou–yuan Yang
and Guang–wen Zhu. Approximation of weak sense
stationary stochastic processes from local averages.
Sci. China Ser. A 50(4): 457–463, 2007.
[1] Yuri K. Belyaev. Analytical random processes. Teor.
Verojat. Primenen. IV(4): 437–444, 1959. (in Russian)
[2] Karlheinz Gröchenig. Reconstruction algorithms in
irregular sampling. Math. Comput. 59: 181–194,
1992.
SAMPTA'09
274
Sampling of Homogeneous Polynomials
Somantika Datta (1) , Stephen D. Howard (2) , and Douglas Cochran (1)
(1) Arizona State University, Tempe, Arizona 85287, USA.
(2) Defence Science & Technology Organisation, Edinburgh, South Australia.
somantika.datta@asu.edu, stephen.howard@dsto.defence.au, cochran@asu.edu
Abstract:
Conditions for reconstruction of multivariate homogeneous polynomials from sets of sample values are introduced, together with a frame-based method for explicitly obtaining the polynomial coefficients from the sample
data.
1. Introduction
Several authors have noted the importance of interpolation
and reconstruction of multivariate polynomials from sample data in applications. Zakhor [10], for example, considered the problem of interpolation of bivariate polynomials
from irregularly spaced sample values in connection with
two-dimensional filter design and image processing. The
case of multivariate polynomials presents significant difficulties not encountered with polynomials of one variable,
in particular due to the zeros of these entire functions of
several variables not being isolated as occurs in the univariate setting. Consequently, it is not surprising that, in
her work, Zakhor develops conditions in which suitable
sampling sets are constrained to lie on certain algebraic
curves.
Very recent work by Varjú [9] and Benko and Króo [1]
develops Weierstraß types of results for approximation
of smooth multivariate functions by homogeneous polynomials. This suggests the potential utility of interpolation and reconstruction of homogeneous polynomials
from sample values. It is well known that the linear space
Hk (Cn ) of homogeneous polynomials of degree k in n
complex variables is isomorphic to the space Symk (Cn )
of symmetric k-tensors over Cn . This fact was used by
the authors in [3] to develop results concerning frames
and grammians on Symk (Cn ). In this paper, a similar
perspective is used to derive conditions under which coefficients of a multivariate homogeneous polynomial of
known degree can be reconstructed explicitly from sets of
sample values. It is shown that a sampling set that suffices
for n-variate homogenous polynomials of degree k is also
suitable for reconstructing the coefficients of any homogeneous polynomial in n variables of degree 1 6 ℓ < k.
Further, it is noted that, modulo general position issues,
the number of samples is the crucial issue in determining
suitability of a sampling set. Nevertheless, some sampling
sets are “better” than others in that they provide snugger frames and hence the numerical advantages they en-
SAMPTA'09
tail. The relative merits of sampling sets in this respect
do not depend on the particular polynomial to be reconstructed, thus allowing generically good sampling sets to
be designed before any sampling is actually carried out.
Before beginning the mathematical sections of the paper,
a few comments on notation and terminology are in order.
For x = [x(1) · · · x(n) ]T and y = [y (1) · · · y (n) ]T in Cn ,
their inner product will be denoted by
hx, yi =
n
X
x̄(j) y (j)
j=1
where the bar denotes complex conjugate; i.e., the inner product is conjugate linear in its first argument and
linear in its second argument. The corresponding convention will be used for inner products in other complex
Hilbert spaces. Given a finite frame X = {x1 , ..., xm }
for an n-dimensional complex vector space V , the function F : V → ℓ2 ({1, . . . , m}) = Cm given by F (w) =
[hx1 , wi . . . hxm , wi]T will be called the frame operator
associated with X, while F = F ∗ F : V → V (i.e., the
composition of the adjoint of F with F ) will be called the
metric operator associated with X.
The k-fold tensor product V ⊗k of an n-dimensional vector space V is a vector space spanned by elements of the
form v1 ⊗ · · · ⊗ vk where each vi ∈ V [8]. The vector
(ℓ)
v1 ⊗ · · · ⊗ vk has nk coordinates {vi |i = 1, . . . , k; ℓ =
(ℓ)
1, . . . , n} where vi denotes the ℓth coordinate of the vector vi . The space of symmetric k-tensors associated with
V , denoted Symk (V ), is the subspace of V ⊗k consisting of those tensors which remain fixed under permutation
(see Chapter 10 of [8]). Symk (V ) is spanned by the tensor powers v ⊗k where
¡ v ∈¢ V . If kV has dimension n then
dim Symk (V ) = n+k−1
. Sym (V ) has a natural inner
k
product with the property
⊗k ⊗k ®
k
= hv, wiV .
(1)
v ,w
Symk (V )
2.
Sampling of Homogeneous Polynomials
It is well known (see, e.g., [8]) that Hk (Cn ), the linear
space of homogeneous polynomials of total degree k in
variables z̄ (1) , . . . , z̄ (n) is isomorphic to Symk (V ). This
section points out a connection between the condition that
k
⊗k
X (k) = {x⊗k
1 , . . . , xm } is a frame for Sym (V ) and the
n
reconstructability of polynomials in Hk (C ) from the values they take at sets of m points in Cn .
275
Beginning with k = 1, let w ∈ V = Sym1 (V ) and denote
by [w(1) · · · w(n) ]T ∈ Cn the coordinates of w in some
orthonormal basis for V . There is an obvious isomorphism
that takes w ∈ V to the polynomial pw ∈ H1 (Cn ) defined by pw (z (1) , . . . , z (n) ) = w(1) z̄ (1) + · · · w(n) z̄ (n) . If
X = {x1 , . . . , xm } is a frame for V , the associated frame
operator F : V → Cm is given by
(1)
(n)
pw (x1 , . . . , x1 )
hx1 , wi
..
..
. (2)
F (w) =
=
.
.
hxm , wi
(1)
(n)
pw (xm , . . . , xm )
In other words, F (w) is a vector of values obtained by
evaluating (i.e., “sampling”) pw at the points x1 , . . . , xm .
One may ask whether this set of m sample values is sufficient to uniquely determine pw .
To address this question, define a sampling function PX :
H1 → Cm by
(n)
(1)
p(x1 , . . . , x1 )
..
PX (p) =
.
(1)
(n)
p(xm , . . . , xm )
and note that (2) shows the frame operator is given by
F (w) = PX (pw ). Because the frame operator is invertible, w is uniquely determined by F (w). Hence any
pw ∈ H1 is uniquely determined by its samples PX (pw ).
Conversely, if X fails to frame V , the mapping F defined
by (2) is still well-defined, but has non-trivial kernel K.
In this case, PX (pw ) = PX (pw+u ) for all u ∈ K. So, in
particular, pw is not uniquely determined from its samples
at x1 , ..., xm .
A similar situation occurs for k > 1, where the
space of interest is Symk (V ) and the frame is X (k) =
⊗k
{x⊗k
1 , . . . , xm }. As in the k = 1 case, mapping a
polynomial to its coefficient sequence defines an isomorphism between Hk (Cn ) and Symk (V ) for k > 1. If
v = w⊗k ∈ Symk (V ) is a pure tensor power of w ∈ V ,
then
⊗k ⊗k ®
x1 , w
..
(k)
F (v) =
.
⊗k ⊗k ®
xm , w
k
pv (x1 )
hx1 , wi
..
..
=
=
.
.
k
hxm , wi
pv (xm )
k
where pv ∈ Hk is defined by pv (z) = hz, wi . Symk (V )
is spanned by pure tensor powers of elements in V [8].
Thus, for arbitrary v ∈ Symk (V ), F (k) (v) is a vector of m samples of a polynomial in Hk taken at points
x1 , ..., xm . Thus, as in the k = 1 case, polynomials in
(k)
Hk are uniquely determined by the samples PX (p) =
[p(x1 ), . . . , p(xm )]T if and only if X (k) frames Symk (V ).
Theorem 1 given below implies that if one can reconstruct
a polynomial in Hk (Cn ) from a certain sampling set then
the same set can be used to reconstruct polynomials in
Hℓ (Cn ) for all 1 6 ℓ < k. Conversely, almost every
SAMPTA'09
sampling set in Cn for H1 gives rise to a sampling set for
Hk where k > 1, provided there are enough vectors in the
set.
Theorem 1. (i) Given n and m with m > n, if X (k) =
k
⊗k
⊗k
(ℓ)
{x⊗k
1 , x2 , . . . , xm } is a frame for Sym (V ), then X
ℓ
is a frame for Sym (V ) for all 1 6 ℓ < k.
(ii)
Almost
Cn such that m >
¡n+k−1
¢ every set of m vectors in
k
results in a frame for Sym (Cn ) for k > 1.
k
Proof. (i) Suppose that X (ℓ) is not a frame for Symℓ (V ).
Then X (ℓ) does not span Symℓ (V ) and there exists
g ∈ (span(X (ℓ) ))⊥ ⊂ Symℓ (V ). Take some h ∈
⊗(k−ℓ)
Symk−ℓ (V ). Let h = x1
. Then
E
D
®
⊗(k−ℓ)
⊗ℓ
g ⊗ h, x⊗k
=
g
⊗
h,
x
⊗
x
i
i
i
E
D
®
⊗(k−ℓ)
h, xi
= g, x⊗ℓ
i
E
D
⊗(k−ℓ)
⊗(k−ℓ)
=0
= 0 · x1
, xi
(k)
which is a contradiction since g ⊗h ∈ Symk (V
) and X⊗k ®
k
is a frame for Sym (V ) so that for any i, g ⊗ h, xi
cannot be zero.
(ii) It has been shown in [7] that for almost every set
of vectors X = {x1 , . . . , xm } in Cn , the
rank¢of the
¡n+k−1
⊗k
when
grammian of X (k) = {x⊗k
,
.
.
.
,
x
}
is
m
1
k
¡n+k−1¢
. This means that the maximum number of
m>
k
¡
¢
linearly independent vectors in X (k) is n+k−1
which is
k
k
n
the same as the dimension of Sym (C ) and hence X (k)
is a frame for Symk (Cn ).
3.
Illustrative Examples
Example 1. Consider the space V = C2 over the field
C. Let x1 = [1, 0]T , x2 = [0, 1]T and x3 = [1, 1]T . The
set X = {x1 , x2 , x3 } is a frame for V with corresponding
frame operator
1 0
F = 0 1 .
1 1
The metric operator is
F = F ∗F =
·
2 1
1 2
¸
.
The eigenvalues of F are 1 and 3, which are the optimal
lower and upper frame bounds respectively.
·
¸
1
2 −1
F −1 =
2
3 −1
which is the metric operator of the dual frame The dual
frame is denoted by X̃ = {x̃1 , x̃2 , x̃3 } where
¸T
·
2 1
−1
,
,−
x̃1 = F x1 =
3 3
·
¸T
1 2
−1
x̃2 = F x2 = − ,
, and
3 3
·
¸T
1 1
−1
x̃3 = F x3 =
,
.
3 3
276
The minimum and maximum eigenvalues of F are .2679
and 3.7321, which are the optimal lower and upper frame
bounds respectively. The metric operator for the dual
frame is
−1 −1
0
3 −1
F −1 = −1
0 −1
1
Plane z = u+v
2
1
0
−1
−2
−1
−0.5
0
0.5
u
1
0.5
0
−0.5
−1
1
v
Figure 1: The plane z = u+v, a homogeneous polynomial
of degree one.
Consider reconstruction of the homogeneous polynomial
p of degree one in two variables defined by p(u, v) =
c(1) u + c(2) v from the three frame elements. Here k = 1,
n = 2 and m = 3. Any c = [c(1) , c(2) ]T ∈ C2 can be
reconstructed via the frame reconstruction formula
c=
3
X
hxi , ci x̃i .
(3)
i=1
Since p(xi ) = hxi , ci and for this example p(x1 ) = c(1) ,
p(x2 ) = c(2) , and p(x3 ) = c(1) + c(2) , the right side of (3)
is
c(1) x̃1 + c(2) x̃2 + (c(1) + c(2) )x̃3 = [c(1) , c(2) ]T .
This shows that the coefficients of p(u, v) can be reconstructed from its samples at the frame elements. The polynomial p(u, v) = u + v together with the sampling set is
shown in Figure 1.
Example 2. If the homogeneous polynomial to be reconstructed is of degree two as given by p(u, v) = c(1) u2 +
c(2) uv + c(3) v 2 then one considers the space Sym2 (C2 ) ⊂
C⊗2 . The dimension of Sym2 (C2 ) is three, which is
the same as the dimension of H2 (C2 ). Hence at least
three sampling points are needed. Consider the same set
of sampling points as in Example 1; i.e., x1 = [1, 0]T ,
x2 = [0, 1]T and x3 = [1, 1]T . One can extend this set
to C⊗2 by taking Kronecker products and restricting to
⊗2
T
T
Sym2 (C2 ) yields x⊗2
1 = [1, 0, 0] , x2 = [0, 0, 1] , and
⊗2
⊗2
⊗2
⊗2
T
(2)
x3 = [1, 1, 1] . Let X = {x1 , x2 , x3 }. The polynomial p can be uniquely determined from its sample values at x1 , x2 and x3 because c(1) = p(x1 ), c(3) = p(x2 ),
and c(2) = p(x3 ) − p(x2 ) − p(x1 ). This means that X (2)
is a frame for Sym2 (C2 ). The frame operator is
1 0 0
F = 0 0 1
1 1 1
making the metric operator
2
F = 1
1
SAMPTA'09
1
1
1
1
1 .
2
⊗2
= [1, −1, 0]T ,
= F −1 x⊗2
making the dual frame xg
1
1
⊗2
⊗2
xg
= F −1 x⊗2
= [0, −1, 1]T , and xg
= F −1 x⊗2
=
2
2
3
3
T
[0, 1, 0] .
The polynomial p(u, v) = c(1) u2 + c(2) uv + c(3) v 2 satisfies p(x1 ) = c(1) , p(x2 ) = c(3) , and p(x3 ) = c(1) + c(2) +
c(3) . The coefficients of p can be obtained from its samples at x1 , x2 , and x3 by the frame reconstruction formula
for Sym2 (C2 ); i.e.,
g
g
⊗2
⊗2
⊗2
[c(1) , c(2) , c(3) ]T = p(x1 )xg
1 + p(x2 )x2 + p(x3 )x3 .
Example 3. Consider now the frame for C2 formed by
x1 = [1, 0]T , x2 = [2, 0]T , and x3 = [0, 1]T . In this case,
reconstruction of p(u, v) = c(1) u2 + c(2) uv + c(3) v 2 from
samples p(x1 ), p(x2 ), and p(x3 ) is not generally possible,
even though the number of samples is the same as the dimension of H2 (C2 ). This is because x1 and x2 are scalar
multiples of each other and the corresponding vectors in
Sym2 (C2 ), {[1, 0, 0]T , [2, 0, 0]T , [0, 0, 1]T } do not constitute a frame for Sym2 (C2 ). This is an example where the
tensor powers of a frame for V do not frame Symk (V ),
even though the number of vectors is adequate.
Example 4. Reconstruction of homogeneous polynomials in H3 (C2 ) requires at least four points, since the
dimension of Sym3 (C2 ) and hence that of H3 (C2 ) is
four. Taking the frame X = {x1 , x2 , x3 , x4 } =
{[1, 0]T , [0, 1]T , [1, 1]T , [1, −1]T } for C2 , computing Kronecker products and restricting to Sym3 (C2 ) yields
X (3)
⊗3
⊗3
= {x⊗3
, x⊗3
2 , x3 , x4 }
1
1
0
1
0
0
, , 1
=
0 0 1
1
1
0
1
−1
.
,
1
−1
A homogeneous polynomial of the form p(u, v) =
c(1) u3 + c(2) u2 v + c(3) uv 2 + c(4) v 3 can be reconstructed
from its samples at these points as c(1) = p(1, 0), c(2) =
1
(3)
= 12 (p(1, 1) +
2 (p(1, 1) + p(1, −1) − 2p(1, 0)), c
(4)
p(1, −1) − 2p(1, 0)), and c = p(0, 1) so that X (3) constitutes a frame for Sym3 (C)2 . The frame operator is
1 0 0 0
0 0 0 1
F =
1 1 1 1
1 −1 1 −1
making the metric operator
3
0
F =
2
0
0
2
0
2
2
0
2
0
0
2
.
0
3
277
4.
z = u3 + u2v + uv2 + v3
4
3
2
1
0
−1
−2
−3
−4
−1
−0.5
0
0.5
1
0
−0.5
−1
0.5
1
v
u
Figure 2: A homogeneous polynomial of degree three.
The optimal lower and upper frame bounds are A =
0.4384 and B = 4.5616. The metric operator of the dual
frame is
1
0 −1
0
0 1.5
0 −1
.
F −1 =
−1
0 1.5
0
0 −1
0 −1
⊗3
= [1, 0, −1, 0]T ,
= F −1 x⊗3
The dual frame is xg
1
1
⊗3
⊗3
=
= F −1 x⊗3
= [0, −1, 0, 1]T , xg
= F −1 x⊗3
xg
3
3
2
2
g
⊗3
⊗3
T
−1
T
[0, .5, .5, 0] , and x
= F x
= [0, −.5, .5, 0] .
4
4
The coefficients of a degree-three polynomial p(u, v) =
c(1) u3 + c(2) u2 v + c(3) uv 2 + c(4) v 3 are given by
g
⊗3
⊗3
[c(1) , c(2) , c(3) , c(4) ]T = p(x1 )xg
1 + p(x2 )x2
g
⊗3
⊗3
+ p(x3 )xg
3 + p(x4 )x4 .
Such a polynomial and the sampling points are shown in
Figure 2.
Example 5. As the degree k or the dimension n gets
larger numerical issues arise in calculating the inverse of
the metric operator in order to get the dual frame that
is needed for the reconstruction [4]. Ideally, one would
like to construct tight frames for Symk (Cn ). Since the
upper and lower frame bounds determine the numerical merits of a particular frame, it is interesting to observe how starting with a fixed frame for C2 the frame
bounds change as this frame is extended to frames for
Symk (C2 ) as k increases. Taking the frame for C2 to
be {[1, 0]T , [0, 1]T , [1, 1]T , [1, −1]T }, the frame bounds for
C2 , Sym2 (C2 ), and Sym3 (C2 ) are tabulated below.
Space
C2
Sym2 (C2 )
Sym3 (C2 )
Optimal lower
frame bound A
3
1
.4384
Optimal upper
frame bound B
3
5
4.5616
B/A
1
5
10.4
In this particular case it appears that the ratio B/A increases as k, the degree of the polynomial increases.
SAMPTA'09
Conclusions and Future Work
It has been shown that homogeneous polynomials of degree k in n variables can be reconstructed from their samples at elements of a frame for Symk (Cn ). Such a set
can also be used to reconstruct n-variate homogeneous
polynomials of all degrees ℓ where 1 6 ℓ < k. In recent work [1], [9] conditions under which a smooth function can be approximated by homogeneous polynomials
have been established. Combining these results to approximately reconstruct smooth functions from sampled data
and a possible construction of tight frames for Symk (Cn )
will be given in a detailed version of this work. The
metric operator and the grammian of a frame have the
same non-zero eigenvalues. Also G ◦k , the grammian of
⊗k
⊗k
X (k) = {x⊗k
1 , x2 , . . . , xm }, is the k-fold Hadamard
product of G, the grammian of X = {x1 , x2 , . . . , xm }.
Relationship between the eigenvalues of G and G ◦k ([2],
[5], [6]) may be used to obtain information about the frame
bounds for a frame for Symk (Cn ) which comes from a
frame for Cn , see Example 5.
5.
Acknowledgments
The authors would like to thank John McDonald for useful
discussions on the topic of this paper.
References:
[1] D. Benko and A. Kroó. A Weierstrass-type theorem for homogeneous polynomials. Transactions of
the American Mathematical Society, 361(3):1645 –
1665, 2009.
[2] G. Cheng, X. Cheng, T. Huang, and T. Tam. Some
bounds for the spectral radius of the Hadamard product of matrices. Applied Mathematics E-Notes,
5:202–209, 2005.
[3] S. Datta, S. D. Howard, and D. Cochran. Geometry
of the Welch bounds. IEEE Transactions on Information Theory. In review.
[4] I. Daubechies. Ten Lectures on Wavelets. SIAM,
1992.
[5] M. Fang. Bounds on eigenvalues of the Hadamard
product and the Fan product of matrices. Linear Algebra and its Applications, 425:7–15, 2007.
[6] E. I. Im. Narrower eigenbounds for Hadamard products. Linear Algebra and its Applications, 264:141–
144, 1997.
[7] I. Peng and S. Waldron.
Signed frames and
Hadamard products of Gram matrices. Linear Algebra and its Applications, 347:131–157, 2002.
[8] R. Shaw. Linear algebra and group representations,
volume 2. Academic Press, 1983.
[9] P. Varjú. Approximation by homogeneous polynomials. Constructive Approximation, 26:317 – 337,
2007.
[10] A. Zakhor and G. Alvstad. Two-dimensional polynomial interpolation from nonuniform samples. IEEE
Transactions on Signal Processing, 40(1):169 – 180,
1992.
278
On sampling lattices with similarity scaling
relationships
Steven Bergner (1) , Dimitri Van De Ville(2) , Thierry Blu(3) , and Torsten Möller(1)
(1) GrUVi-Lab, Simon Fraser University, Burnaby, Canada.
(2) BIG, Ecole Polytechnique F édérale de Lausanne, Switzerland.
(3) The Chinese University of Hong Kong, Hong Kong, China.
sbergner@cs.sfu.ca, thierry.blu@m4x.org, Dimitri.VanDeVille@epfl.ch, torsten@cs.sfu.ca
Abstract:
R=
We provide a method for constructing regular sampling
lattices in arbitrary dimensions together with an integer
dilation matrix. Subsampling using this dilation matrix
leads to a similarity-transformed version of the lattice with
a chosen density reduction. These lattices are interesting candidates for multidimensional wavelet constructions
with a limited number of subbands.
0
1
−0.3307
−0.375
,K=
2
4
−1
−1
, θ = 69.3◦
1.5
1
0.5
0
1.
Primer on sampling lattices and related
work
A sampling lattice is a set of points {Rk : k ∈ Z n } ⊂ Rn
that is closed under addition and inversion. The nonsingular generating matrix R ∈ R n×n contains basis vectors in its columns. Lattice points are uniquely indexed
by k ∈ Zn and the neighbourhood around each sampling
point is identical. This makes them suitable sampling patterns for the reconstruction in shift-invariant spaces.
Subsampling schemes for lattices are expressed in terms
of a dilation matrix K ∈ Z n×n forming a new lattice with
generating matrix RK. The reduction rate in sampling
density corresponds to
|det K| = αn = δ ∈ Z+ .
(1)
Dyadic subsampling discards every second sample along
each of the n dimensions resulting in a δ = 2 n reduction
rate. To allow for fine-grained scale progression we are
particularly interested in low subsampling rates, such as
δ = 2 or 3.
As discussed by van de Ville et al. [8], the 2D quincunx subsampling is an interesting case permitting a twochannel relation. With the implicit assumption of only
considering subsets of the Cartesian lattice it is shown
that a similarity two-channel dilation may not extend for
n > 2.
Here, we show that by permitting more general basis vectors in Rn the desired fixed-rate dilation becomes possible for any n. Our construction produces a variety of lattices making it possible to include additional quality criteria into the search as they may be computed from the
Voronoi cell of the lattice [9] including packing density
and expected quadratic quantization error (second order
moment). Agrell et al. [1] improve efficiency for the computation by extracting Voronoi relevant neighbours. Another possible sampling quality criterion appears in the
SAMPTA'09
−0.5
−1
−1.5
−1.5
−1
−0.5
0
0.5
1
1.5
Figure 1: 2D lattice with basis vectors and subsampling
as given by R and K in the diagram title. The spiral
shaped points correspond to a sequence of fractional subsamplings RKs for s = 0..1 with the notable feature that
for s = 1 one obtains a subset of the original lattice sites
shown as thick dots. This repeats for any further integer
power of K, each time reducing the sample density by
|det K| = 2.
work of Lu et al. [4] in form of an analytic alias-free sampling condition that is employed in a lattice search.
2. Lattice construction
We are looking for a non-singular lattice generating matrix
R that, when sub-sampled by a dilation matrix K with reduction rate δ = αn , results in a similarity-transformed
version of the same lattice, that is, it can be scaled and rotated by a matrix Q with Q T Q = α2 I. An illustration of a
1
subsampling resulting in a rotation by θ = arccos 2√
in
2
2D is given in Figure 1. Formally, this kind of relationship
can be expressed as
QR = RK
(2)
leading to the observation that subsampling K and scaled
rotation Q are related by a similarity transform
R−1 QR = K.
(3)
279
1 j
it is possible to diago1 −j
nalize a 2D rotation matrix by the following similarity
transform
jθ
0
cos θ − sin θ
e
−1
= J2
J2 = J−1
2 ∆J2 .
0 e−jθ
sin θ
cos θ
(4)
Using this observation to replace the scaled rotation matrix
Q in Equation 3 leads to
Using a matrix J2 =
K = R−1 QR
−1
K = αR−1 J−1
Jn R
n S∆S
−1
K = αP∆P
with
R
Q
−1
= J−1
n SP
−1
= αJn ∆Jn .
(5)
(6)
Thus, given a matrix K that has an eigen-decomposition
corresponding to that of a uniformly scaled rotation matrix, we can compute the lattice generating matrix R as
in Equation 6. The elements of the diagonal matrix S inserted in the construction of R scale the otherwise unit
eigenvectors in the columns of P. Below, we will refer to
this construction as function formRQ(K, S) using S = I
by default.
2.1 Constructing suitable dilation matrices K
The eigenvalues of K, ∆ and Q impose restrictions on
their shared
polynomial d(λ) = det(K −
n characteristic
k
c
λ
as
discussed
in the appendix. For
λI) =
k=0 k
the case n = even with the only non-zero integer coefficients c0 = δ, c2n/2 < 4δ, cn = 1 this leaves a finite
number of different options for c n/2 . The case n = odd
permits a single possible polynomial with non-zero coefficients c0 = −δ, cn = 1. For these monic polynomials
it is possible to directly construct a candidate K via the
companion matrix ([6], p. 192)
⎤
⎡
0
−c0
⎢ 1 0
−c1 ⎥
⎥
⎢
⎥
⎢
..
⎥.
⎢
1 0
.
K=⎢
(7)
⎥
⎥
⎢
.
.
. . . . −c
⎦
⎣
n−2
1 −cn−1
This allows to construct a lattice fulfilling the self-similar
subsampling condition for any dimensionality n, one for
every possible characteristic polynomial.
With this starting point it is possible to construct additional
suitable dilation matrices via a similarity transform with a
unimodular matrix T
KT = TKT−1 = PT ∆P−1
T .
(8)
Using a unimodular rather than any non-singular T guarantees that T−1 is also unimodular following from the fact
that T−1 can be constructed from the adjugate (the transposed co-factor matrix) of T. Thus, K T remains an integer matrix by this transform. Possible generators for this
unimodular group are discussed in ([5], pp. 23). Our implementation, referred to as function genUnimodular(n),
SAMPTA'09
uses a construction of T = LU from several random integer lower and upper triangular matrices having ones on
their diagonal.
It is not guaranteed that all possible K for a given characteristic polynomial can be generated through a similarity
transform with some T. However, formRQ(K T ) provides
numerous non-equivalent R T lattice generators. Among
them it is possible to apply further criteria to select the
“best” lattice.
An alternative to transforming K is the eigenvector scaling by diagonal matrix S in Equation 6. Using non-unit
scaling allows to produce further lattices for any given K
resulting in an n-dimensional continuous search space.
2.2 Construction Algorithm
The steps for constructing lattices with the desired
subsampling matrices are summarized in algorithm 1.
The function compoly(n, α, C) is defined in the
Algorithm 1 genLattices(n, δ)
1: Llist ← {}
2: Ks ← genKompans(n, δ)
3: Ts ← genUnimodular(n) ∪{I}
4: for all K ∈ Ks do
5:
for all T ∈ Ts do
6:
KT = TKT−1
7:
(RT , QT ) ← formRQ(KT )
8:
Llist← Llist∪{(KT , RT , QT )}
9:
end for
10: end for
11: return Llist
Algorithm 2 genKompans(n, δ)
1: Ks = {}
2: if n is even then
3:
for all C ∈ Z : C 2 < 4δ do
1
4:
Ks ← Ks ∪ compoly(n, δ n , C)
5:
end for
6: else {n is odd}
1
7:
Ks ← {compoly(n, δ n )}
8: end if
9: return Ks
appendix. A possible implementation for the function genUnimodular(n) is described in Section 2.1 and
formRQ(K) is defined below Equation 6.
It should be noted that the list of lattices returned by
genLattices may contain several equivalent copies of the
same lattice. A Gram matrix implicitly represents angles
between basis vectors as A = R T R. Two lattices R1 and
R2 , scaled to have the same determinant, are equivalent if
their Gram matrices are related via A 1 = TT A2 T with
a unimodular matrix T ∈ Z n×n and |det T| = 1. Determining this unimodular matrix is known to be a difficult
problem, as it for instance also occurs when relating the
adjacency matrices of two supposedly isomorphic graphs.
Hence, our current method employs a simpler necessary
test for equivalence by comparing the first few elements
280
1: R = [0.71 −0;−0.71 1.4]
K = [2 −2;1 0] θ=45
2: R = [0 0.58;−1.7 0.65]
K = [2 −1;4 −1] θ=69.3
3: R = [0 0.84;−1.2 0]
K = [0 −1;2 0] θ=90
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
0
0
0
2
4
0
2
4
0
0
2
4
Figure 2: Three non-equivalent 2D lattices obtained for a design with dilation matrices having |det K| = 2. The lattice
on the left is the well known quincunx sampling with a θ = 45 ◦ rotation. The other two are new schemes with different
rotation angles. The black markers show the sample positions that are retained after subsampling by K.
1: R = [−0 0.93;1.1 −0.54] 2: R = [0 0.84;−1.2 0]
K = [1 −1;2 1] θ=54.74
K = [−1 2;−2 1] θ=90
5
5
3: R = [0 0.74;−1.3 0.22] 4: R = [0.66 0;1.1 −1.5]
K = [1 −1;3 0] θ=73.22
K = [−3 4;−3 3] θ=90
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
0
0
0
0
5
0
5
0
5
0
0
5
Figure 3: Three non-equivalent 2D lattices obtained for a design with dilation matrices having |det K| = 3. The lattice
on the left is the well known hexagonal lattice with a θ = 30 ◦ rotation. The other three are new schemes with different
rotation angles.
of the set q(A) = {kT Ak : k ∈ Zn } using the Gram matrices of the respective lattices. If the sorted lists q(A 1 )
and q(A2 ) disagree in any element, R 1 and R2 are not
equivalent ([5], p. 60). It is possible to restrict the set of
indices k ∈ Zn to the Voronoi relevant neighbours [1].
Further, since these neighbours determine the hyperplanes
bounding the Voronoi polytope of the lattice, they can also
be used for a sufficient test for equivalence.
kissing # = 2, # f = 14, # v = 24, G(P) = 0.081904, # zones = 6
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
0.5
−0.8
3.
Constructions for different dimensions
and subsampling ratios
For the 2D case we have created lattices permitting a reduction rate 2 in Figure 2 and rate 3 in Figure 3. In both
cases, familiar examples arise in the quincunx and the hex
lattice for the respective ratios.
A search of 3D lattices enjoying the self-similar subsampling property with rate 2 dilations resulted in 53 nonequivalent cases. These lattices were compared in terms of
their dimensionless second order moments, corresponding
to the expected squared vector quantization error ([2], p.
451). When performing the continuous optimization mentioned at the end of Section 2.1, all of these cases converged to the same optimum lattice shown in Figure 4.
The dimensionless second order moment for the Voronoi
Cell of this lattice is G = 0.081904. For comparison, the
Cartesian cube has Gcc = 0.0833 and the truncated octahedron of the BCC lattice has G bcc = 0.0785.
4.
Discussion and potential applications
The current formation of candidate matrices K based on
similarity transforms of one valid example is not guaran-
SAMPTA'09
0
−0.5
0
0.5
−0.5
Figure 4: The best 3D lattice obtained for a design with
dilation matrices having |det K| = 2. The letters f and v
in the title line indicate faces and vertices, respectively.
teed to produce all possible solutions. For 2D and 3D we
also employed an exhaustive search over a range of integer
matrices with values in [−3, 3] resulting in the same number of non-equivalent 2D cases as the construction via K T .
However, for dimensionality n > 3 the exhaustive search
had to be replaced by a random sampling of integer matrices ultimately rendering the method infeasible for n > 5.
In that light the current construction via scaled eigenvectors of the companion matrix is a significant improvement
as it allows to produce a large number of non-equivalent
lattices for any dimensionality.
Our subsampling schemes may have applications for multidimensional wavelet transforms [7]. Another direction
for possible investigation is the construction of sparse
grids that are employed in the context of high-dimensional
integration and approximation adapting to smoothness
conditions of the underlying function space [3].
281
Appendix: Characteristic polynomial of a
scaled rotation matrix in Rn
The similarity relationship between K and Q in Equation 2 implies that they share the same characteristic polynomial d(λ) = det(K − λI) = det(Q − λI) leading to
an agreement in eigenvalues d(λ k ) = 0 and determinant
d(0) ([6], p. 184). Further, since K is an integer matrix
the polynomial d(λ) ∈ Z[λ] has integer coefficients c k .
In order to find integer matrices K with the eigenvalues
of a scaled rotation matrix, it will be important to distinguish the two different forms of the diagonal matrix ∆ in
Equation 4 and 5 for the case n = even
∆ = diag[e
jθ1
e
−jθ1
...e
jθn/2
e
−jθn/2
]
and the case n = odd
Thus, if
n
ck λk
d(λ) =
k=0
n
=−
ck
k=0
n
α2
λ
k
λ
α
n
(13)
cn−k αn−2k λk
=−
k=0
2k
⇔ ck = −αn−2k cn−k = −δ 1− n cn−k .
By the same reasoning as for the even case, c k = 0 for all
k = 1, 2, . . . n−1
2 resulting in only one possible characteristic polynomial
∆ = diag[1 ejθ1 e−jθ1 . . . ejθ(n−1)/2 e−jθ(n−1)/2 ]
d(λ) = λn − αn .
with analogue block-wise constructions for J n .
For dimensionality n = even the characteristic polynomial of K and Q fulfills
To refer to the above procedure we will invoke a function
compoly(n, α, C) that returns a companion matrix (Equation 7) with a characteristic polynomial as in Equation 11
or 14.
n/2
(αejθk − λ)(αe−jθk − λ)
d(λ) =
(14)
References:
k=1
n/2
(α2 − 2λα cos θk + λ2 )
=
(9)
k=1
n/2
=
k=1
=d
4
(
3
2
α
λ
α
− 2 cos θk + α2 ) 2
2
λ
λ
α
α2
λ
n
λ
α
Thus, if
n
ck λk
d(λ) =
k=0
n
=
ck
k=0
n
k
α2
λ
n
λ
α
(10)
cn−k αn−2k λk
=
k=0
2k
⇔ ck = αn−2k cn−k = δ 1− n cn−k .
2k
If ck = 0 and ck , δ ∈ Z then δ 1− n ∈ Q. This is impossible for 0 < 2k < n, assuming small values of δ, such
as 2, 3 or any simple product of primes. This implies that
ck = cn−k = 0 for k = 1, 2, . . . n2 − 1. For k = n2 the ck
can be non-zero leading to
n
d(λ) = λn + Cλ 2 + αn
(11)
with the requirement that C 2 < 4αn so that the complex
eigenvalues d(λk ) = 0 are evenly distributed on the complex circle of radius |λ k | = α.
For dimensionality n = odd the polynomial fulfills
(n−1)/2
(αejθk − λ)(αe−jθk − λ)
d(λ) = (α − λ)
(12)
k=1
⇒ d(λ) = −
SAMPTA'09
λ
α
[1] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger. Closest point search in lattices. Information Theory, IEEE
Transactions on, 48(8):2201–2214, August 2002.
[2] J.H. Conway and N.J.A. Sloane. Sphere Packings,
Lattices and Groups. – 3rd ed. Springer, 1999.
[3] M. Griebel. Sparse grids and related approximation schemes for higher dimensional problems. In
L. Pardo, A. Pinkus, E. Suli, and M.J. Todd, editors, Foundations of Computational Mathematics
(FoCM05), Santander, pages 106–161. Cambridge
University Press, 2006.
[4] Y.M. Lu, M.N. Do, and R.S. Laugesen. A Computable
Fourier Condition Generating Alias-Free Sampling
Lattices. IEEE Transactions on Signal Processing,
57(5):(15 pages), May 2009.
[5] M. Newman. Integral Matrices. Academic Press,
1972.
See http://www.dleex.com/read/
?3907 for a digital copy.
[6] L.N. Trefethen and D. Bau III. Numerical Linear Algebra. SIAM, 1997.
[7] D. Van De Ville, T. Blu, and M. Unser. Isotropic
polyharmonic B-Splines: Scaling functions and
wavelets. IEEE Transactions on Image Processing,
14(11):1798–1813, November 2005.
[8] D. Van De Ville, T. Blu, and M. Unser. On the multidimensional extension of the quincunx subsampling
matrix. IEEE Signal Processing Letters, 12(2):112–
115, February 2005.
[9] E. Viterbo and E. Biglieri. Computing the Voronoi
cell of a lattice: The diamond-cutting algorithm. Information Theory, IEEE Trans. on, 42(1):161–171,
1996.
n
d
α2
λ
282
General Perturbations of Sparse Signals
in Compressed Sensing
Matthew A. Herman and Thomas Strohmer
Department of Mathematics, University of California, Davis, CA 95616-8633, USA.
{mattyh,strohmer}@math.ucdavis.edu
Abstract:
We analyze the Basis Pursuit recovery of signals when observing sparse data with general perturbations. Previous
studies have only considered partially perturbed observations Ax + e. Here, x is a K-sparse signal which we wish
to recover, A is a measurement matrix with more columns
than rows, and e is simple additive noise. Our model also
incorporates perturbations E (which result in multiplicative noise) to the matrix A in the form of (A + E)x + e.
This completely perturbed framework extends the previous work of Candès, Romberg and Tao on stable signal
recovery from incomplete and inaccurate measurements.
Our results show that, under suitable conditions, the stability of the recovered signal is limited by the noise level
in the observation. Moreover, this accuracy is within a
constant multiple of the best-case reconstruction using the
technique of least squares.
1. Introduction
Employing the techniques of compressed sensing (CS) to
recover signals with a sparse representation has enjoyed a
great deal of attention over the last 5–10 years. The initial
studies considered an ideal unperturbed scenario:
b = Ax.
(1)
Here b ∈ Cm is the observation vector, A ∈ Cm×n
(m ≤ n) is a full-rank measurement matrix or system
model, and x ∈ Cn is the signal of interest which has a Ksparse representation (i.e., it has no more than K nonzero
coefficients) under some fixed basis. More recently researchers have included an additive noise term e into the
received signal [1, 2, 4, 8], creating a partially perturbed
model:
b̂ = Ax + e
(2)
This type of noise generally models simple, uncorrelated
errors in the data or at the receiver/sensor.
As far as we can tell, practically no research has been
done yet on perturbations E to the matrix A. Our completely perturbed model extends (2) by incorporating a
perturbed sensing matrix in the form of
cally implementing the matrix A in a sensor. When A represents a system model, such as in the context of radar [7]
or telecommunications, then E can absorb errors in assumptions made about the transmission channel, as well as
quantization errors arising from the discretization of analog signals. In general, these perturbations can be characterized as multiplicative noise, and are more difficult to analyze than simple additive noise since they are correlated
with the signal of interest. To see this, simply substitute
A = Â − E in (2); there will be an extra noise term Ex.
(Note that it makes no difference whether we account for
the perturbation E on the “encoding side” (2), or on the
“decoding side” (7). The model used here was chosen so
as to agree with the conventions of classical perturbation
theory which we use in Section 4.)
1.1
Assumptions and Notation
Without loss of generality, assume the original data x to
(K)
be a K-sparse vector for some fixed K. Denote σmax (Y ),
(K)
kY k2 , and rank(K) (Y ) respectively as the maximum
singular value, spectral norm, and rank over all K-column
(K)
submatrices of a matrix Y . Similarly, σmin (Y ) is the minimum singular value over all K-column submatrices of Y .
Let the perturbations in (2) be relatively bounded by
(K)
kEk2
(K)
kAk2
It is important to consider this kind of noise since it can account for precision errors when applications call for physi-
SAMPTA'09
kek2
≤ εb
kbk2
(3)
(K)
with kAk2 , kbk2 6= 0. In the real world we are only
(K)
interested in the case where both εA , εb < 1.
2.
2.1
CS ℓ1 Perturbation Analysis
Previous Work
In the partially perturbed scenario (i.e., E = 0 in (2)) we
are concerned with solving the Basis Pursuit (BP) problem [3]:
z ⋆ = argmin kẑk1 s.t. kAẑ − b̂k2 ≤ ε′
ẑ
 = A + E.
(K)
≤ εA ,
(4)
for some ε′ ≥ 0.
The restricted isometry property (RIP) [2] for any matrix A ∈ Cm×n defines, for each integer K = 1, 2, . . . ,
283
the restricted isometry constant (RIC) δK , which is the
smallest nonnegative number such that
where
(1 − δK )kxk22 ≤ kAxk22 ≤ (1 + δK )kxk22
CBP
(5)
holds for any K-sparse vector x. In the context of the
√
(K)
(K)
RIC, we observe that kAk2 = σmax (A) = 1 + δK ,
√
(K)
and σmin (A) = 1 − δK .
√
Assuming K-sparse x, δ2K < 2 − 1 and kek2 ≤ ε′ ,
Candès has shown in Theorem 1.2 of [1] that the solution
to (4) obeys
kz ⋆ − xk2 ≤ CBP ε′
(6)
for some constant CBP .
2.2 Incorporating nontrivial perturbation E
Now assume the completely perturbed situation with
E, e 6= 0 in (2). In this case the BP problem of (4) can be
generalized to include a different decoding matrix Â:
z ⋆ = argmin kẑk1 s.t. kÂẑ − b̂k2 ≤ ε′A,K,b
ẑ
(7)
for some ε′A,K,b ≥ 0. The following two theorems summarize our results.
Theorem 1 (RIP for Â). For any K = 1, 2, . . . , assume
and fix the RIC δK associated with A, and the relative
(K)
perturbation εA associated with E in (3). Then the RIC
´2
¡
¢³
(K)
−1
δ̂K := 1 + δK 1 + εA
(8)
for matrix  = A + E is the smallest nonnegative constant such that
(1 − δ̂K )kxk22 ≤ kÂxk22 ≤ (1 + δ̂K )kxk22
(9)
holds for any K-sparse vector x.
Remark 1. The flavor of the RIP is defined with respect to
the square of the operator norm. That is, (1 − δK ) and
(1 + δK ) are measures of the square of minimum and
maximum singular values of A, and similarly for Â. In
keeping with the convention of classical perturbation the(K)
ory however, we defined εA in (3) just in terms of the
operator norm (not its square). Therefore, the quadratic
(K)
dependence of δ̂K on εA in (8) makes sense. Moreover,
in discussing the spectrum of Â, we see that it is really a
(K)
linear function of εA .
Theorem 2 (Completely perturbed observation). Fix the
(K)
(2K)
relative perturbations εA , εA
and εb in (3). Assume that the RIC for matrix A satisfies δ2K <
√ ¡
(2K) ¢−2
2 1 + εA
− 1. Set
³
(K)
ε′A,K,b := c εA
√
´
+ εb kbk2 ,
(10)
1+δK
where c = √1−δ
. If x is K-sparse, then the solution to
K
the BP problem (7) obeys
kz ⋆ − xk2 ≤ CBP ε′A,K,b ,
SAMPTA'09
(11)
´
³
√
(2K)
4 1 + δ2K 1 + εA
µ
¶.
:=
´2
³
√
(2K)
1 − ( 2 + 1) (1 + δ2K ) 1 + εA
−1
(12)
Remark 2. Theorem 2 generalizes of Candès’ results in [1]
for K-sparse x. Indeed, if matrix A is unperturbed, then
(K)
E = 0 and εA = 0. It follows that δ̂K = δK in (8),
and the RIPs for A and  coincide. √
Moreover, the condition in Theorem 2 reduces to δK < 2 − 1, and the total
perturbation (see (17)) collapses to kek2 ≤ ε′b := εb kbk2 ;
both of these are identical to Candès’ assumptions in (6).
Finally, the constant CBP in (12) reduces to the same as
outlined in the proof of [1].
It is also interesting to examine the spectral effects due
to the assumptions of Theorem 2. Namely, we want to be
assured that the rank of submatrices of A are unaltered by
the perturbation E.
Lemma 1. If the hypothesis of Theorem 2 is satisfied, then
for any k ≤ 2K
(k)
(k)
σmax
(E) < σmin (A),
(13)
and therefore
rank(k) (Â) = rank(k) (A).
This fact is necessary (although, not explicitly stated) in
the least squares analysis Section 4.
The utility of Theorems 1 and 2 can be understood
with two simple numerical examples. Suppose that measurement matrix A in (2) is designed to have an RIC of
δ2K = 0.100. Assume, however, that its physical implementation will experience a worst-case relative error
(2K)
of εA
= 5%. Then from (8) we can design a matrix  with RIC δ̂2K = 0.213 to be used in (7) which
will yield a solution whose accuracy is guaranteed by (11)
with CBP = 9.057. Note from (12), we see that if there
had been no perturbation, then CBP = 5.530.
Consider now a different example. Suppose instead
(2K)
that δ2K = 0.200 and εA
= 1%. Then δ̂2K = 0.224
and CBP = 9.643. Here, if A was unperturbed, then we
would have had CBP = 8.473.
These numerical examples show how the stability constant CBP of the BP solution gets worse with perturbations
to A. It must be stressed however, that they represent
worst-case instances. It is well-known in the CS community that better performance is normally achieved in practice.
2.3
Numerical Simulations
Numerical simulations were conducted as follows. Gaussian matrices of size 128 × 512 were randomly generated
in M ATLAB. The entries of matrix A were normally dis2
2
tributed N (0, σA
) where σA
= 1/128, while those of ma2
2
trix E were N (0, σE ) with σE
= ε2A /128. The parameter εA is a measure of the relative perturbation of matrix A
and took on values {0, 0.01, 0.05, 0.10}. Next, a random
284
||z* − x||2/||x||2
0.6
εA = 0.10
0.5
εA = 0.05
0.4
εA = 0.01
• Equality occurs in (15) whenever x is in the direction
of the vector associated with the value (1 + δK ) in
the RIP for A.
• Equality occurs in (16) since, in this hypothetical
case, we assume that E = βA for some 0 < β < 1.
(K)
Therefore, the relative perturbation εA in (3) no
longer represents a worst-case deviation (i.e., the ra-
εA = 0
0.3
(K)
tio
0.2
kEk2
(K)
kAk2
(K)
= β =: εA ).
The full details of this proof can be found in [6]
0.1
3.2
10
20
30
40
Sparsity K
50
60
Figure 1: Average (100 trials) relative error of BP solution
z ⋆ with respect to K-sparse x vs. Sparsity K for different
relative perturbations εA of A ∈ C128×512 (and εb = 0) .
vector x of sparsity K = 1, . . . , 64 was randomly generated (nonzero entries uniformly distributed with N (0, 1))
and b̂ = Ax in (2) was created (note, we set e = 0 so
as to focus on the effect of perturbation E). Given b̂ and
 = A + E, the BP program (7) was implemented with
cvx software [5]. For each value of εA and K, 100 trials
were performed.
Fig. 1 shows the average relative error kz ⋆ −xk2 /kxk2
as a function of K for each εA . As a reference, the ideal,
noise-free case can be seen for εA = 0. It is interesting
to notice that all perturbations, including εA = 0, experience significant jumps simultaneously at several places,
such as K = 31, 42, 43, 44, etc. Now fix a particular
value of K ≤ 30 and compare the relative error for the
three nonzero values of εA . It is clear that the error scales
roughly linearly with εA . This empirical study essentially
confirms the conclusion of Theorem 2, that the stability of
(K)
the BP solution scales linearly with εA (i.e., the singular
values of E).
Note that better performance in theory and in simulation can be achieved if BP is used solely to determine the
support of the solution. Then we can use least squares to
find a better result. This is similar to the the best-case,
oracle least squares solution discussed in Section 4.
3. Proofs
3.1 Proof Sketch of Theorem 1
From the triangle inequality, (5) and (3) we have
¡
¢2
kÂxk22 ≤ kAxk2 + kExk2
(14)
´2
³p
(K)
≤
1 + δK + kEk2
kxk22 (15)
³
´2
(K)
≤ (1 + δK ) 1 + εA
(16)
kxk22 .
Moreover, this inequality is sharp for the following reasons:
• Equality occurs in (14) if E is a multiple of A.
SAMPTA'09
¤
Bounding the perturbed observation
Before proceeding, we need some sense of the size of the
total perturbation incurred by E and e. We don’t know a
priori the exact values of E, x, or e. But we can find an
upper bound in terms of the relative perturbations in (3).
The main goal in the following lemma is to remove the
total perturbation’s dependence on the input x.
Lemma 2 (Total perturbation bound). Set ε′A,K,b :=
´
³
√
(K)
(K)
1+δK
, and εA and
cεA + εb kbk2 , where c = √1−δ
K
εb are defined in (3). Then the total perturbation obeys
kExk2 + kek2 ≤ ε′A,K,b
(17)
for all K-sparse x.
Proof. From (1), (5) and (3) we have
µ
¶
kExk2
kek2
kExk2 + kek2 =
kbk2
+
kAxk2
kbk2
!
Ã
(K)
kEk2 kxk2
kek2
√
kbk2
≤
+
kbk2
1 − δK kxk2
´
³
(K)
≤
c εA + εb kbk2
for all x which are K-sparse.
Note that the results in this paper can easily be expressed
in terms of the perturbed observation by replacing
kbk2 ≤
kb̂k2
.
1 − εb
This can be useful in practice since one normally only has
access to b̂.
3.3
Proof Sketch of Theorem 2
We duplicate the techniques used in Candès’ proof of Theorem 1.2 in [1], but with decoding matrix A replaced
by Â. Set the BP minimizer in (7) as z ⋆ = x+h. Here, h
is the perturbation from the true solution x induced by E
and e. Instead of Candès’ (9), we determine that the image
of h under  is bounded by
kÂhk2
≤ kÂz ⋆ − b̂k2 + kÂx − b̂k2
≤
2 ε′A,K,b
which follows from the BP constraint in (7) as well as x
being a feasible solution (i.e., it satisfies Lemma 2). The
rest of this proof can be found in [6]
¤
285
5.
3.4 Proof of Lemma 1
Assume the hypothesis of Theorem 2. It is easy to show
that this implies
p
√
4
(2K)
kEk2
< 2 − 1 + δ2K .
Simple algebraic manipulation then confirms that
p
p
√
4
(2K)
2 − 1 + δ2K <
1 − δ2K = σmin (A).
Therefore, (13) holds with k = 2K. Further, for
(k)
(2K)
any k ≤ 2K we have σmax (E) ≤ σmax (E) and
(2K)
(k)
σmin (A) ≤ σmin (A), which proves the lemma.
¤
4. Classical ℓ2 Perturbation Analysis
Let the subset T ⊆ {1, . . . , n} have cardinality |T | = K,
and note the following T -restrictions: AT ∈ Cm×K denotes the submatrix consisting of the columns of A indexed by the elements of T , and similarly for xT ∈ CK .
Suppose the “oracle” case where we already know the
support T of K-sparse x. By assumption, we are only
interested in the case where K ≤ m in which AT has full
rank. Given the completely perturbed observation of (2),
the least squares problem consists of solving:
z#
T = argmin kÂT ẑ T − b̂k2 .
ẑ T
Since we know the support T , it is trivial to extend z #
T to
z # ∈ Cn by zero-padding on the complement of T . Our
goal is to see how the perturbations E and e affect z # .
More discussion on the oracle least squares analysis
can be found in [6]. In the end, we find using the same
ε′A,K,b in (10) that its stability is
where CLS
kz # − xk2 ≤ CLS ε′A,K,b
√
:= 1/ 1 − δK .
(18)
4.1 Comparison of LS with BP
Now, we can compare the accuracy of the least squares
solution in (18) with the accuracy of the BP solution found
in (11). In both cases the error bound is of the form
C ε′A,K,b .
A detailed numerical comparison of CLS with CBP is not
entirely valid, nor illuminating. This is due to the fact
that we assumed the oracle setup in the the least squares
analysis, which is the best that one could hope for. In this
sense, the least squares solution we examined here can be
considered a “best, worst-case” scenario. In contrast, the
BP solution really should be thought of as a “worst, of the
worst-case” scenarios.
The important thing to glean is that the accuracy of the
BP solution, like the least squares solution, is on the order
of the noise level ε′A,K,b in the perturbed observation.
This is an important finding since, in general, no other
recovery algorithm can do better than the oracle least
squares solution. These results are analogous to the
comparison by Candès, Romberg and Tao in [2], although
they only consider the case of additive noise e.
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Conclusion
We introduced a general perturbed model for CS, and
found the conditions under which BP could stably recover
the original data. This completely perturbed model extends previous work by including a multiplicative noise
term in addition to the usual additive noise term. We only
considered K-sparse signals, however these results can be
extended to also include compressible signals (see [6]).
Simple numerical examples were given which demonstrated how the multiplicative noise reduced the accuracy
of the recovered BP solution. In terms of the spectrum
of the perturbed matrix Â, we showed that the penalty
on δ̂K was a graceful, linear function of the relative per(K)
turbation εA . Numerical simulations were performed
with εb = 0 and appear to confirm the conclusion of The(K)
orem 2, that the BP solution scales linearly with εA .
We also found that the rank of  did not exceed the
rank of A under the assumed conditions. This permitted an analysis of the oracle least squares solution which
showed that its accuracy, like the BP solution, was limited
by the total noise in the observation.
Acknowledgment
This work was partially supported by NSF Grant No.
DMS-0811169 and NSF VIGRE Grant No. DMS0636297.
References:
[1] E. J. Candès. The restricted isometry property and its
implications for compressed sensing. Académie des
Sciences, I(346):589–592, 2008.
[2] E. J. Candès, J. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math., 59:1207–1223,
2006.
[3] S. S. Chen, D. L. Donoho, and M. A. Saunders.
Atomic decomposition by basis pursuit. SIAM Journal Sci. Comput., 20(1):33–61, 1999.
[4] D. L. Donoho, M. Elad, and V. Temlyakov. Stable
recovery of sparse overcomplete representations in the
presence of noise. IEEE Trans. Inf. Theory, 52(1):6–
18, Jan. 2006.
[5] M. Grant, S. Boyd, and Y. Ye.
cvx: Matlab software for disciplined convex programming.
http://www.stanford.edu/∼boyd/cvx/.
[6] M. A. Herman and T. Strohmer. General Deviants:
An analysis of perturbations in compressed sensing.
http://www.math.ucdavis.edu/∼mattyh/
publications.html.
[7] M. A. Herman and T. Strohmer. High-resolution radar
via compressed sensing. To appear in IEEE Trans.
Signal Processing, Jun. 2009.
[8] J. A. Tropp. Just relax: Convex programming methods
for identifying sparse signals in noise. IEEE Trans.
Inf. Theory, 51(3):1030–1051, Mar. 2006.
286
Analysis of High-Dimensional Signal Data
by Manifold Learning and Convolutions
Mijail Guillemard (1) and Armin Iske (1)
(1) Department of Mathematics, University of Hamburg, D-20146 Hamburg, Germany.
guillemard@math.uni-hamburg.de, iske@math.uni-hamburg.de
Abstract:
A novel concept for the analysis of high-dimensional signal data is proposed. To this end, customized techniques
from manifold learning are combined with convolution
transforms, being based on wavelets. The utility of the
resulting method is supported by numerical examples concerning low-dimensional parameterizations of scale modulated signals and solutions to the wave equation at varying initial conditions.
1.
Introduction
Recent advances in nonlinear dimensionality reduction
and manifold learning have provided new methods for the
analysis of high-dimensional signals. In this problem, a
very large data set U ⊂ Rn of scattered points is given,
where the data points are assumed to lie on a compact
submanifold M of Rn , i.e. U ⊂ M ⊂ Rn . Moreover, the dimension k = dim(M) of M is assumed to
be much smaller than the dimension of the ambient space
Rn , k ≪ n. Now, the primary goal in the dimensionality
reduction is the construction of a low-dimensional representation of the data U .
In this paper, a novel concept for signal data analysis
through dimensionality reduction is proposed. To this end,
suitable techniques from manifold learning are combined
with convolution transforms. Moreover, another important
ingredient is a (suitable) projection map P : Rn → Rk
that finally outputs the desired low-dimensional representation for U . Note that for the sake of approximation quality, we need to preserve intrinsic geometrical and topological properties of the manifold M, and so the construction of the composite dimensionality reduction method requires particular care. In the proposed data analysis, the
geometric distortion of the manifold, being incurred by
the chosen convolution transform, plays a key role.
We remark that similar concepts from differential geometry are enjoying increasing interest in related applications of sampling theory, including surface reconstruction in reverse engineering and image analysis [5]. Further related concepts can be found in classical dimensionality reduction schemes, such as in principal component analysis and multidimensional scaling, while more
recent techniques are including Isomap and LLE methods [4, 7] Local Tangent Space Alignment (LTSA) [6],
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Sample Logmaps [1], and, most recently, Riemannian
Normal Coordinates [2, 3].
The outline of the paper is as follows. In the following
Section 2, the main ingredients of the proposed nonlinear
dimensionality reduction scheme, especially the construction of the convolution and projection map, are explained.
Then, in Section 3 relevant aspects concerning distortion
analysis are addressed. Finally, Section 4 shows the good
performance of the resulting nonlinear dimensionality reduction method. To this end, numerical examples concerning low-dimensional parameterization of scale modulated signals and solutions to the wave equation at varying
initial conditions are illustrated.
2.
Construction of the Data Analysis
Given a set of signals U = {ui }m
i=1 ⊂ M, that we assume
to lie in (or near) a low-dimensional Riemannian compact
submanifold M, of Rn , we wish to analyse the given data
for the purpose of dimensionality reduction. Therefore,
we assume that there is an embedding A : Ω → M, giving a parameterization of M, where the domain Ω ⊂ Rd
lies in a low-dimensional Euclidean space Rd , i.e., d ≪ n.
But the parameter domain Ω is unknown. Therefore, the
goal of dimensionality reduction is to find a sufficiently
accurate approximation Ω′ of Ω, through which the desired low-dimensional representation for U is obtained.
We remark that the construction of the data analysis is required to depend on intrinsic geometrical and topological properties of the manifold M. To this end, we apply a particular convolution transform T : M → MT ,
MT = {T (p) : p ∈ M}, to each of the data sites ui ,
followed by a suitable projection P : MT → Ω′ , yielding
a nonlinear data transformation for dimensionality reduction. The following diagram reflects our concept.
Ω ⊂ Rd
A
/ U ⊂ M ⊂ Rn
(1)
T
Ω′ ⊂ Rd o
P
UT ⊂ MT ⊂ Rn
Note that both the construction of the transformation T
and the projection need particular care. Indeed, in order to
maintain the intrinsic geometrical properties of the manifold M, it is required to investigate the curvature distortion of M under the transform T . For this purpose, convolution filters are powerful tools for the construction of
287
suitable signal transforms T . This is supported by our numerical results in Section 4., where wavelet transforms are
used for a customized construction of T .
Finally, let us remark that standard methods in signal processing rely on on special characteristics of a discrete-time
signal uk ∈ Rn , such as frequency content, time duration,
phase and amplitude information, etc. In typical application scenarios, signal data are not just isolated items of
information, but they are rather incorporating correlations
reflecting characteristic properties of the sampled object.
Therefore, when designing customized signal transforms,
one should exploit available context information on characteristic properties of the target object in order to improve
the quality of the data analysis. In our particular application scenario, special emphasis needs to be placed on intrinsic geometrical properties of the manifold M, where
a preprocessing distortion analysis of the curvature is of
vital importance.
3.
When considering the linear transformation T representing the convolution filter, an important case is when T is
represented by a Toeplitz matrix, with filter coefficients
H = (h1 , . . . , hm ), i.e.,
Curvature Distortion Analysis
Our main objective is to estimate the curvature distortion
in the geometry of the manifold M incurred by the application of the linear transformation T : M → MT ,
where T may, for instance, representing a wavelet or a
convolution filter. To this end, we first need to evaluate
relevant effects on the geometrical deformation of M under various specific transformations T . This then amounts
to constructing suitable transformations T which are welladapted to the characteristic properties of the specific data.
Preferable choices for T : M → MT are diffeomorphisms, in which case dim(M) = dim(MT ).
3.1
If T is invertible, then the Gaussian curvature KMT in
MT can be computed as a function of the metric g in M
by using a pullback of the curvature tensor R in M with
respect to the inverse map T −1 : MT → M, or, equivalently, by using a pushforward of the curvature tensor R in
M with respect to T : M → MT . An alternative strategy
is to consider the composition of T with a particular system of local coordinates (x1 , . . . , xn ) of M, along with
the metric tensor
∂
∂
.
,
gij (p) = gij (x1 , . . . , xm ) =
∂xi ∂xj
Sectional Curvature Distortions
In general, a fundamental invariant of a manifold with respect to its isometries are the sectional curvatures. This
concept is derived from the idea of the Gaussian curvature
in the setting of 2-manifolds, and is defined as
KM =
< R(X, Y )Y, X >
,
kXk2 kY k2 − < X, Y >2
for the curvature tensor R, defined for a triple of smooth
vector fields X, Y, Z as
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z.
We recall that the affine connection (a Levi-Cevita connection for our situation) is a bilinear map
h1
h2
..
.
T =
hm
0
.
..
0
3.2
T
DK
(p)
= KM (p) − KMT (T (p))
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for p ∈ M.
0
...
...
...
...
...
0
0
..
.
h1
h2
..
.
hm
.
Curvature Distortions for Curves
As for the special case of a curve r : I = [t0 , t1 ] → Rm ,
Rt
with arc-length parameterization s(a, t) = a kr′ (x)k dx,
′′
recall that the curvature of r is k(s) = kr (s)k. For an
arbitrary parameterizations of r, its curvature is given by
K2 =
kr̈k2 kṙk2 − < r̈, ṙ >2
.
(kṙk2 )3
In the remainder of this section, we briefly discuss the curvature distortion under linear maps (e.g. convolution transform) and under smooth maps. To compute the curvature
distortion of a curve r : I = [t0 , t1 ] → Rm under a linear
map T , we consider the curvature of rT = {T r(t), t ∈ I},
computed as follows.
ℓ=1
In order to estimate the distortion caused by the linear map
T : M → MT , we compare the Gaussian curvatures between M and MT , denoted respectively KM , and KMT ,
hm−1
hm
..
.
...
...
Note that the curvature distortion caused by the map T will
be controlled by the singular values of T , which due to the
Toeplitz matrix structure, are obtained from the Fourier
coefficients of H.
Now, our primary objective is to investigate the influence
of the filter coefficients in H on the curvature distortion
T
. Moreover, we study filters being required to obtain a
DK
given curvature distortion. The latter is particularly useful
for the adaptive construction of a low dimensional representation of U .
∇ : C ∞ (M, T M) × C ∞ (M, T M) → C ∞ (M, T M)
that can be expressed with the Christoffel symbols defined,
for a particular
Pnsystem of local coordinates (x1 , . . . , xn ),
as ∇∂i ∂j = k=1 Γkij ∂k . The Christoffel symbols can be
described with respect to the metric tensor via
m
∂giℓ
∂gij
1 X ∂gjℓ
k
+
+
Γij =
g ℓk .
2
∂xi
∂xj
∂xℓ
0
h1
..
.
KT2 ≡ KT2 (t) =
kT r̈k2 kT ṙk2 − < T r̈, T ṙ >2
.
(kT ṙk2 )3
(2)
As for the general case of smooth maps F : Rm → Rr ,
the curvature distortion can be approximated by using the
288
Jacobian matrix JF and its singular value decomposition,
JF (p)
=
∂f1
∂x1 (p)
..
.
∂fr
∂x1 (p)
...
..
.
...
∂f1
∂xm (p)
..
.
∂fr
∂xm (p)
= UF (p)DF (p)VFT (p)
for p ∈ M.
The curvature distortion of a curve r : [t0 , t1 ] → Rm under F can in this case be analyzed through the expression
kJF r̈k2 kJF ṙk2 − < JF r̈, JF ṙ >2
KF2 ≡ KF2 (p) =
,
(kJF ṙk2 )3
where, unlike in the linear case (2), the Jacobian matrices
JF depend on p ∈ M.
4.
Numerical Examples
This section presents three different numerical examples
to illustrate basic properties of the proposed analysis of
high-dimensional signal data. Further details shall be discussed during the conference.
4.1
Low-dimensional parameterization of scale
modulated signals
In this example, we illustrate the geometrical effect of a
convolution transform for a set of functions lying on a
curve embedded in a high dimensional space. More precisely, we analyze a scale modulated family of functions
U ⊂ R64 , parameterized by three values in Ω ⊂ R3 ,
(
)
3
X
2
U = fα(t) =
e−αi (t)(· −bi ) : α(t) ∈ Ω .
presents a curvature correction that recovers the original
geometry of Ω fairly well.
To explain the resulting curvature correction, we need to
analyze the singular values and singular vectors of the convolution map T . In fact, the singular values of T can be
viewed as scaling factors (stretching or shrinking) along
corresponding axis in the (local) embedding of U . Moreover, the spectrum of T depends on the particular filter
design.
4.2
Low dimensional parameterization of wave
equation solutions
In this second example, we regard the one-dimensional
wave equation
∂u
∂u
= c2 ,
∂t
∂x
Figure 1 (left) shows the parameter domain Ω, a star
shaped curve in R3 . A PCA projection in R3 , applied
to the set U ⊂ R64 , is also displayed in Figure 1 (middle). The projection illustrates the curvature distortion
caused by the nonlinear map A : Ω ⊂ R3 → U ⊂
R64 , A(α(t)) = fα(t) .
(3)
with initial conditions
u(0, x) = f (x),
∂u
(0, x) = g(x),
∂t
0 ≤ x ≤ 1.
(4)
We make use of the previous example to construct a set
of initial values (i.e. functions) parameterized by a star
shaped curve U0 = U . Our objective is to investigate the
distortion caused by the evolution Ut of the solutions on
given initial values U0 . Recall that the evolution of the
wave equation is constituted by the set of solutions
Ut
= {uα ≡ uα (t, x) : uα satisfying (3) with
initial condition f ≡ fα in (4) for α ∈ Ω}.
Now, the solution of the wave equation can numerically be
computed by using finite differences, yielding the iteration
u(j+1) = Au(j) + b(j) ,
i=1
The parameter set for the scale modulation is given by the
curve
Ω = α(t) = (α1 (t), α2 (t), α3 (t))T ∈ R3 , : t ∈ [t0 , t1 ] .
0 < x < 1, t ≥ 0,
where for µ = γ∆t/(∆x)2 , the iteration matrix is given
by
1 − 2µ
µ
µ
1 − 2µ
µ
µ
1
−
2µ
µ
A=
.
..
.. ..
.
.
.
0
µ 1 − 2µ
Recall that in the convergence analysis of the iteration,
which can be rewritten as,
u(j+2)
= Au(j+1) + b(j+1)
= A(Au(j) + b(j) ) + b(j+1)
= A(2) u(j) + Ab(j) + b(j+1) ,
Figure 1: Parameter set Ω ⊂ R3 , data U ⊂ R64 , and
wavelet correction T (U ) ⊂ R64 .
Finally, Figure 1 (right), shows the resulting data transformation T (U ) using a Daubechies wavelet w.r.t. a specific
band of the multiresolution analysis, resulting in a filtering process for each element in U . The resulting T (U ),
SAMPTA'09
the spectrum of the matrices Ak play a key role. In fact,
due to the decomposition Ak = U Dk U T , the geometrical
distortion in the evolution of Ut depends on the evolution
of the eigenvalues of A.
4.3
Topological Distortion via Filtering
In this final example, we illustrate one relevant phenomenon concerning the topological distortion caused by
289
utilized convolution involves a selection of suitable bands
from the corresponding wavelet multiresolution decomposition. Further details on this shall be explained during the
conference.
ftorus12
0.01
0.005
0
Figure 2: One solution of the wave equation u(t, x) and
one measurement u(tk , x), tk = 20.
−0.005
−0.01
−0.02
0
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
Figure 5: PCA projection of U ⊂ R64 onto R3 .
band4
band3
0.015
0.01
0.01
0.005
0.005
0
0
−0.005
−0.005
−0.01
−0.01
0.01
−0.02
0
Figure 3: Curvature distortion of the initial manifold under the evolution of the wave equation. The outer curve
represents the initial conditions U0 while the inner curve
reflects the corresponding solutions Ut for some time t.
the utilized convolution transformation. In this couple of
two test cases, we take one 1-torus Ω1 ⊂ R3 and one 2torus Ω2 ⊂ R3 as parameter space, respectively. As in
the previous examples, we generate a corresponding set
of scale modulation functions U1 and U2 (see Figure 4),
using Ω1 and Ω2 as parameter domains. This gives, for
j = 1, 2, two different data sets
(
)
3
X
j
j 2
e−αi (t)(· −bi ) : αj (t) ∈ Ωj .
Uj = fαj (t) =
i=1
ftorus2
ftorus1
0.015
0.01
0.01
0.005
0.005
0
0
−0.005
−0.005
−0.02
−0.01
−0.02
−0.01
−0.01
0
0
0.01
0.01
0.02
0.02
−0.01
0.01
0.02
0.01
0.005
0
−0.005
0
−0.01
−0.015
−0.01
Figure 4: PCA projections of U1 , U2 ⊂ R64 onto R3 , generated by Ω1 , Ω2 ⊂ R3 , two tori of genus 1 and 2.
Now we combine the set U1 and U2 by
U = ft = fα1 (t) + fα2 (t) : α1 (t) ∈ Ω1 , α2 (t) ∈ Ω2 .
The resulting projection of the data U is shown in Figure 5.
For the purpose of illustration, we recover the sets U1 and
U2 from U . Note that this is a rather challenging task,
especially since the genus of surfaces U1 and U2 are different. Figure 6 shows the reconstructions of the two surfaces U1 and U2 . Note that the both the geometrical and
topological properties of U1 and U2 are recovered fairly
well, which supports the good performance of our convolution transform yet once more. The reconstruction of the
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−0.01
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
−0.015
0.02
−0.01
0.01
0
0
−0.01
−0.02
0.01
Figure 6: Reconstruction of U1 (left), U2 (right) from U .
5.
Acknowledgments
The authors were supported by the priority program DFGSPP 1324 of the Deutsche Forschungsgemeinschaft.
References:
[1] A. Brun, C. Westin, M. Herberthsson, and
H. Knutsson.
Sample logmaps: Intrinsic processing of empirical manifold data. Proceedings of
the (SSBA) Symposium on Image Analysis, 1, 2006.
[2] A. Brun, C.-F. Westin, M. Herberthson, and
H. Knutsson. Fast manifold learning based on riemannian normal coordinates. In Proceedings of the
SCIA;05, pages 920–929, Joensuu, Finland, June
2005.
[3] T. Lin, H. Zha, and S.U. Lee. Riemannian Manifold Learning for Nonlinear Dimensionality Reduction. Lecture Notes in Computer Science, 3951:44,
2006.
[4] S.T. Roweis and L.K. Saul. Nonlinear Dimensionality
Reduction by Locally Linear Embedding, 2000.
[5] E. Saucan, E. Appleboim, and Y.Y. Zeevi. Sampling
and Reconstruction of Surfaces and Higher Dimensional Manifolds. Journal of Mathematical Imaging
and Vision, 30(1):105–123, 2008.
[6] H. Zha and Z. Zhang. Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space
Alignment. SIAM Journal of Scientific Computing,
26(1):313–338, 2004.
[7] H. Zha and Z. Zhang. Continuum Isomap for manifold
learnings. Computational Statistics and Data Analysis, 52(1):184–200, 2007.
290
Geometric Reproducing Kernels for Signal
Reconstruction
Eli Appleboim (1) , Emil Saucan (2) and Yehoshua Y. Zeevi (1)
(1) Technion, Dept. of Electrical Engineering
(2) Technion, Dept. of Mathematics
eliap@ee.technion.ac.il, semil@ee.technion.ac.il, zeevi@ee.technion.ac.il
Abstract:
In this paper we propose a smoothing method for non
smooth signals, which control the geometry of a sampled
signal. The signal is considered as a geometric object and
the smoothing is done using a smoothing kernel function
that controls the curvature of the obtained smooth signal
in a close neighborhood of a metric curvature measure of
the original signal.
1. Introduction
In [11], [12], a sampling scheme for signals that posses
Riemannian geometric structure was introduced.It turns
out that a variety of signals fall in this setting while gray
scale images is just one such example. Rather then some
Nyquist rate, the sampling scheme presented in [11], [12],
is based on geometric characteristics of the sampled signals. Being precise, the following sampling theorem was
proved.
Theorem 1 Let Σn , n ≥ 2 be a connected, not necessarily compact, smooth manifold, with finitely many compact boundary components. Then there exists a sampling
of Σn , with a proper density D = D(p) =
´
³ scheme
1
, where k(p) = max{|k1 |, ..., |k2n |}, and where
D k(p)
k1 , ..., k2n are the principal (normal) curvatures of Σn , at
the point p ∈ Σn .
While the assumed Riemannian structure relies on the assumption that the signal satisfies C 2 smoothness criteria,
the authors presented in [11], an extended version of Theorem 1 also for non smooth geometric signals, where the
proposed strategy uses smoothing of the original signal.
The following theorem was proved.
Theorem 2 Let Σ be a connected, non-necessarily compact surface of class C 0 . Then, for any δ > 0, there exists
a δ-sampling of Σ, such that if Σδ → Σ, then Dδ → D,
where Dδ and D denote the densities of Σδ and Σ, respectively.
In the above Theorem 2 Σδ is a smoothing of Σ obtained
by a convolution of Σ with a partition of unity kernel.
Such a kernel being very common for manifolds smoothing indeed guarantees that the resultant manifold is as
smooth as we wish however, in this process we do not have
any control on the curvature of the obtained manifold.
Some natural question raise in this context,
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1. To what extent can we smooth the original signal, using such a reproducing kernel while assuming a predefined bounds on the curvature of the resultant manifold?
2. Can the reproducing kernel be made local, namely,
can we have different kernel characteristics for different areas along the sampled signals, while being able
to glue the smoothed signal along common boundaries?
3. In what way if at any, we can give affirmative answers
to 1 and 2 that are adaptive to the signal? Meaning,
how can we have good prior estimates for the desired
curvature bounds?
This paper aims at answering the above questions. Note
that answering question 1 is analogous to smoothen a signal to have a predefined frequency band-pass, using a
band-pass filter as commonly done in signal processing
for decades. Answering 1, 2, 3 is equivalent to the use of
filter banks with different band-pass characteristics. In all,
giving affirmative answers to all above questions give rise
to an adaptive non uniform sampling scheme for a variety
of signals.
We will focus along the paper on signals that are do not admit a Riemannian structure but rather have a more general
geometric structure of the so called Alexandrov spaces.
We will term such signals as geometric-signals.
2.
Preliminaries
In this section we will give some basic preliminary definitions and notations.
2.1
Alexandrov spaces
Definition 3 (Alexandrov - Toponogov) [ [9]] A complete metric space X, satisfies the triangle comparison
condition w.r.t κ ∈ R if for every geodesic triangle
∆pqr ∈ X, there exists a comparison triangle, i.e. a triangle, ∆p′ q′ r′ ∈ M2κ , such that
pq = p′ q ′ ; qr = q ′ r′ ; rp = r′ p′
so that, for every point s ∈ pr we have that
dX (s, q) > dM2κ (s′ , q ′ )
291
where s′ ∈ p′ r′ such that
ps = p′ s′ ; sr = s′ r′
Where M2κ is a complete simply connected surface of constant curvature κ.
If X is an Alexandrov space then there exists a self-adjoint
operator ∆, called the Laplacian defined on L2 (X) so
that,
Z
Z
v∇udHn
< ∇u, ∇v > dHn =
X
X
n
th
where H is the n Hausdorff measure of X, u ∈
D(∆), v ∈ W 1,2 (X).
Theorem 7 ( [6]) 1. If X is compact then the spectrum
of ∆ is discrete.
2. There exists a continuous heat kernel ht (x, y) on X
so that,
Z
e−t∆ u(x) =
ht (x, y)u(y)dHn (y)
X
2.2
Figure 1: Comparison triangle.
Definition 4 A complete metric space X, is an Alexandrov space of curvature > κ iff
1. For all x, y, ∈ X there exists a length minimizing
curve γ joining x and y such that,
L(γ) = dX (x, y);
where L denotes the arc length of curves in X and
dX stands for the metric given on X. γ is called a
minimal geodesic.
2. X satisfies the triangle comparison condition for κ.
3.
dimH X < ∞;
dimH = Hausdorff dimension.
Remark 5 In a similar way, while reversing the direction
of inequalities, one can define Alexandrov space of curvature < κ. For instance, in the comparison triangle condition, we will demand,
dX (s, q) < dM2κ (s′ , q ′ )
Definition 6 (Gromov) If X is an Alexandrov space of
curvature < κ and κ ≤ 0 then X is called CAT (κ)space. CAT = Cartan-Alexandrov-Toponogov.
2.1.1 Examples:
1. Every complete Riemannian manifold of bounded
sectional curvature.
2. The boundary of convex set in Rn is an Alexandrov
space of curvature ≥ 0.
3. If Xi is a sequence of n-dimensional Alexandrov
spaces of curv. ≥ κ then their Gromov-Hausdorff
limit, if exists, is an Alexandrov space of curv. ≥ κ
and dimension ≤ n.
If the limit of the above sequence is of dimension < n
we say the sequence collapses.
SAMPTA'09
Approximations of manifolds
Let M be a complete Riemannian manifold of bounded
sectional curvature. Let p ∈ M be some point and let
φi be some C ∞ kernel function supported on some ǫi neighborhood of p. For example one can take φ to be
partition of unity, heat kernel and others. Let Mi be the
manifold obtained by convolution,
Z
φi ∗ M dµ;
Mi =
M
Note that Mi is smooth in a δi neighborhood of p even if
M fails to be smooth at p. Well known results (see for
instance, [7]) in differential topology assert that,
ǫi → 0 ⇒ Mj → M ;
where convergence of manifolds is considered in the
Gromov-Hausdorff topology. While the above result concerns the convergence on a topological level, in order to
have curvature control we have to account for geometric
convergence as well. This is guaranteed from the studies in [3], [4] and [10]. In [3], [4] it is proved that
similar convergence to the above also exist for Betti numbers which are generalizations of Euler characteristic to
all dimensions and are related to curvature through higher
dimensional of Gauss-Bonnet type theorems [2]. In [10]
the question of proper gluing of approximations in adjacent neighborhoods is addressed. It is shown that one can
obtain geometric convergence in different neighborhoods
V, U of the points p, q resp. so that, on the common boundary ∂V ∩ ∂U the approximations coincide. In addition, if
we write the heat operator on a manifold, N , as
e−t∆N f (x),
where f ∈ L2 (N ) and t > 0, x ∈ N , and ∆N , denotes the Laplace-Beltrami operator associated with N ,
then there is a smooth kernel function KN , such that,
Z
KN (t, x, y)f (y)dy;
e−t∆N f (x) =
N
In [3] convergence of the heat kernel is also achieved,
e−t∆Mi → e−t∆M
292
3. Smoothing geometric signals with curvature control
In this section we present the results concerning questions
1, 2 and 3 posed in the introduction. These results give us
the ability to smoothen a geometric signal while having an
adaptive control on obtained curvatures.
Definition 8 We say that a signal is a geometric signal
iff it admits a structure of an Alexandrov space for some
κ ∈ R.
Let Σ be a geometric signal of sectional curvature
bounded from below (above). Let p ∈ Σ be a point, and
U (p) ⊂ Σ some compact neighborhood of p. Let
κ = lim sup K
3. Smooth the signal while controlling the curvature of
the smoothed signal to suitably approximate the estimated curvature.
4. Sample the smoothed signal according to Theorem 1
4.1
Special case - images
It is common to regard images as surfaces embedded in
some Rn . For gray scale images R3 is considered while
for color images it is usual to take R5 . Figure 2 shows image re-sampled according to the geometric sampling proposed in Theorem1. In this example no smoothing was
applied prior to sampling and artifacts of this can be seen
in the reconstructed image. “Flat areas” of the image have
20 times reduced sampling resolution with respect to the
original resolution.
such that U (p) is an Alexandrov space of curvature > K.
3.1 Approximations of geometric signals
Theorem 9 ( [1]) Given a point p on Σ, there exists
smooth local kernel φi as above, yielding a sequence of
manifolds Mi , smooth inside an ǫi neighborhoods of p,
such that
1.
Mi =
Z
φi ∗ Σdµ → Σ,
Σ
as ǫ → 0.
2. If we further assume that while the Riemannian manifolds Mi converge to Σ, no collapse occurs i.e. the
Hausdorff dimension of Σ is the same as of Mi , then,
the sectional curvature Ki (p) of Mi at p satisfies,
lim Ki (p) = κ;
ǫ→0
The theorem above answers both questions 1 and 2. We
can control the curvature of the obtained smooth signals
in an adaptive way by making it converge to the lim sup
of Alexandrov curvature of the signal Σ.
3.2 Gluing
By arguments similar to those in [10] we have,
Theorem 10 ( [1]) Let the above smooth approximations
of Σ be given in neighborhoods of two points p, q.
Then they coincide as well as their sectional curvatures
Ki,Vi , Ki,Ui on the common boundary, if non empty.
4. Sampling of geometric signals
We propose the following scheme for sampling of a geometric signals.
1. Consider the signal as an Alexandrov space. This requires the representation of the signal as a tame metric space in a meaningful manner.
2. Assess the appropriate Alexandrov curvature bound.
This can be done by the use of discrete metric curvature measures.
SAMPTA'09
Figure 2: Geometric sampling of a gray scale image. Top
to bottom - original Lena; Lena resampled. The white
dots are the new sampling points. One can see the sparseness w.r.t the original; Lena reconstructed. Reconstruction
using linear interpolation over the sampling points. No
smoothing was done.
In order to estimate the curvature of an image as an
Alexandrov space we can take the set of discrete curvature
measures proposed in [5] where such measures are suggested for very general cell-complexes. It is shown in [5]
293
that the one-dimensional curvature measure resembles the
Ricci curvature of a cell-complex which, in the case of images (since they are 2-dimensional manifolds) coincides
with the Gaussian curvature. Figure 3 shows the combinatorial Ricci (= Gauss) curvature of the image in Figure
2, see [13] for details about the adoption of the curvature
measures introduced in [5] to images.
Figure 3: Discrete Ricci curvature of Lena. Apart from
giving an assessment for the curvature of the image as an
Alexandrov space, it also serves as an excellent edge detector as itself.
[4] Cheeger, J. and Gromov, M., Bounds on the Von Neumann dimension of L2 -cohomology and the GaussBonnet theorem for open manifolds, J. Diff. Geom. 21,
1985.
[5] Forman, R., Bochner‘s method for cell-complexes and
combinatorial Ricci curvature, Disc. Comp. Geom.,
29, 2003.
[6] Kuwae, K. Machigashira, Y. and Shioya, T., Sobolev
spaces, Laplacian and heat kernel on Alexandrov
spaces, Math. Z. 238, 2001.
[7] Munkres, J. Elementary Differential Topology, Ann.
Math. Stud. 54, 1966.
[8] Nash, J., The Imbedding problem for Riemannian
manifolds, Ann. Math. 63, 1956.
[9] Otsu, Y. and Shioya, T., The Riemannain stracture of
Alexandrov Spaces, J. Diff. Geom., 39, 1994.
[10] Petersen, P., Wei, G. and Ye, R., Controlled geometry
via smoothing, Comm. Math. Helv., 74, 1999.
[11] Saucan, E., Appleboim, E. and Zeevi, Y. Y. Sampling
and Reconstruction of Surfaces and Higher Dimensional Manifolds, J. Math. Imaging. Vis., 30, 2008.
[12] Saucan, E., Appleboim, E. and Zeevi, Y. Y. Geometric Sampling of Manifolds for Image Representation
and Processing LNCS, 4485, 2007.
[13] Saucan, E. Appleboim, E., Wolansky G. and Zeevi,
Y. Y., Combinatorial Ricci curvature for image processing, Midas Jour. Proc. MICCAI 2008
5. Further study
Current and future studies of geometric sampling of images and signals, focus on two aspects. First we wish to
modify the smoothing process introduced herein so it will
be done in the Fourier domain rather than the spatial domain. Namely, we wish to smooth the Fourier transform
of the signal while considering curvature in the Fourier
plane. This is inspired by the Nash embedding Theorem
[8] while the Fourier transform of a manifold is smoothen
prior to its embedding thus achieving a higher degree of
smoothness with respect to smoothing in the spatial domain.
Another direction of study is devoted to the development
of a geometric theory of sparse representations and geometric compress sensing.
References:
[1] Appleboim, E., Saucan, E. and Zeevi, Y. Y. Geometric
reproducing kernels for signals, preprint.
[2] Bochner, S. and Yano, K., Curvature and Betti numbers, Ann. Math. Stud. 32, 1953.
[3] Cheeger, J. and Gromov, M., On the characteristic
numbers of complete manifolds of bounded curvature
and finite volume, Diff. Geom. and Com. Anal. Chavel
Farkas Ed., Springer, 1985.
SAMPTA'09
294
Multivariate Complex B-Splines, Dirichlet
Averages and Difference Operators
Brigitte Forster (1,2) and Peter Massopust (2,1)
(1) Zentrum Mathematik, M6, Technische Universität München, Germany
(2) Institut für Biomathematik und Biometrie, Helmholtz Zentrum München, Germany
forster@ma.tum.de, massopust@ma.tum.de
Abstract:
For the Schoenberg B-splines, interesting relations between their functional representation, Dirichlet averages
and difference operators are known. We use these relations to extend the B-splines to an arbitrary (infinite) sequence of knots and to higher dimensions. A new Fourier
domain representation of the multidimensional complex
B-spline is given.
1.
Complex B-Splines
Complex B-splines are a natural extension of the classical Curry-Schoenberg B-splines [2] and the fractional
splines first investigated in [16]. The complex B-splines
Bz : R → C are defined in Fourier domain as
z
Z
1 − e−iω
−iωt
F(Bz )(ω) =
Bz (t)e
dt =
iω
R
for Re z > 1. They are well-defined, because of { 1−eiω |
ω ∈ R} ∩ {y ∈ R | y < 0} = ∅ they live on the main
branch of the complex logarithm. Complex B-splines are
elements of L1 (R) ∩ L2 (R). They have several interesting basic properties, which are discussed in [5]. Let
Re z, Re z1 , Re z2 > 1.
−iω
• Complex B-splines Bz are piecewise polynomials of
complex degree.
• Smoothness and decay:
– Bz ∈ W2r (R) for r < Re z − 12 . Here W2r (R)
denotes the Sobolev space with respect to the
L2 -Norm and with weight (1 + |x|2 )r .
– Bz (x) = O(x−m ) for m < Re z +1, |x| → ∞.
• Recursion formula: Bz1 ∗ Bz2 = Bz1 +z2 .
• Complex B-splines are scaling functions and generate multiresolution analyses and wavelets.
B-splines
Dirichlet averages
Difference operators
Figure 1: Relations between classical B-splines, difference operators and Dirichlet averages.
2. Representation in time-domain
We defined complex B-splines in Fourier domain, and
Fourier inversion shows that these functions are piecewise
polynomials of complex degree:
Proposition 1. [5] Complex B-splines have a timedomain representation of the form
z
1 X
z−1
(t − k)+
,
(−1)k
Bz (t) =
k
Γ(z)
k≥0
pointwise for all t ∈ R and in L2 (R)-norm. Here,
z
t = ez ln t , if t > 0,
z
t+ =
0,
if t ≤ 0,
is the truncated power function, and Γ : C \ Z−
0 → C
denotes the Euler Gamma function.
Compare: The cardinal B-spline Bn , n ∈ N, has the similar representation
n
X
1
n−1
(−1)k
(t − k)+
k
(n − 1)!
n
Bn (t)
=
k=0
n
1 X
(−1)k
(t − k)n−1
+ .
Γ(n)
k
∞
=
k=0
• But in general, they don’t have compact support.
• Last but not least: They relate difference and differential operators.
In this paper, we take closer look at this last relation and
the respective multivariate setting. To this end, we will
consider the known relations between classical B-splines,
difference operators and Dirichlet averages.
SAMPTA'09
3. Relations to Difference Operators
It is well-known that in the construction of the CurrySchoenberg B-splines difference operators are deeply involved. The same is true for complex B-splines. To establish the corresponding relation, let us first recall the definition of the backward difference operator ∇.
295
Let g : R → C be a function. Then the backward difference operator ∇ = ∇1 is recursively defined as follows:
∇g(t)
∇
g(t)
= g(t) − g(t − 1),
= ∇(∇n g(t)) for n ∈ N.
n+1
This definition yields the explicit representation
n
X
n
(−1)k g(t − k).
∇n g(t) =
k
k=0
For the cardinal B-splines Bn we can write:
n
X
1
n−1
k n
(−1)
(t − k)+
Bn (t) =
(n − 1)!
k
k=0
=
1
∇n tn−1
+ .
(n − 1)!
In comparison: For the complex B-splines, we have an
analog representation:
∞
1 X
z−1
k z
, Re z ≥ 1.
(t − k)+
Bz (t) =
(−1)
k
Γ(z)
k=0
This invites to define a complex difference operator:
Definition 2. [5, 6] The difference operator ∇z of complex order z is defined as
∞
X
z
∇z g(t) :=
(−1)k
g(t − k), z ∈ C, Re z ≥ 1.
k
k=0
Hence a second time domain representation of the complex B-spline is
Bz (t) =
1
z−1
.
∇ z t+
Γ(z)
In a similar way, we can establish a relation to divided differences. Recall that for a knot sequences {t0 , . . . , tn } ⊂
R, n ≥ 1, divided differences are recursively defined as
follows. Let g : R → C be some function.
[t0 ]g
[t0 , . . . , tn ]g
= g(t0 ),
[t0 , . . . , tn−1 ]g − [t1 , . . . , tn ]g
=
t0 − t n
n
X
g(tj )
Q
=
.
l6=j (tj − tl )
j=0
For the cardinal B-spline,
n
X
1
(−1)k
(t − k)n−1
+
(n − 1)!
k
Definition 3. Let g : R → C be some function. We define
the complex divided differences for the knot sequence N0
via
[z; N0 ]g :=
X
k≥0
(−1)k
g(k)
.
Γ(z − k + 1)Γ(k + 1)
Then the complex B-spline can be written as
z−1
.
Bz (t) = z[z, N0 ](t − •)+
Comparing “old” and “new” divided difference operator
for z = n ∈ N, yields
(−1)n [0, 1, . . . n] = [n, N0 ].
Proposition 4. [6, 7] Let Re z > 0 and g ∈ S(R+ ). Then
Z
1
Bz (t)g (z) (t) dt,
[z; N0 ]g =
Γ(z + 1) R
where g (z) = W z g is the complex Weyl derivative:
For n = ⌈Re z⌉, ν = n − z,
Z ∞
n
1
z
n d
ν−1
W g(t) = (−1)
(x − t)
g(x) dx .
dtn Γ(ν) t
Sketch of proof:
Z
1
Bz (t)g (z) (t) dt
Γ(z + 1) R
Z
1
z−1
W z g(t) dt
z[z, N0 ](t − •)+
=
Γ(z + 1) R
Z ∞
1
z−1
= [z, N0 ]
W z g(t) dt
(t − •)+
Γ(z) •
= [z, N0 ]W −z W z g = [z, N0 ]g.
R∞
z−1
1
f (t) dt is the complex
(t − •)+
Here, W −z f = Γ(z)
•
Weyl integral of the function f , i.e., the inverse operator
of W z .
Now we are able to establish a first relation between divided difference operators and Dirichlet averages.
Proposition 5. (Generalized Hermite-Genocchi-Formula:
Divided Differences and Dirichlet Averages) [6, 7]
Let ∆∞ be the infinite-dimensional simplex
N0
∆∞ := {u := (uj ) ∈ (R+
0) |
n
Bn (t)
=
k=0
= n
n
X
(−1)k
k=0
1
(t − k)n−1
+
k!(n − k)!
n
X
(t − k)n−1
+
Q
(k
− l)
l6=k
=
(−1)n n
=
(−1)n n[0, 1, . . . , n](t − •)n−1
+ .
k=0
(The factor (−1)n is due to our representation of the cardinal B-spline via backward difference operators.)
The same ideas give rise to the definition of complex divided differences.
SAMPTA'09
∞
X
j=0
uj = 1} = lim ∆n ,
←−
defined as the projective limit of the finite dimensional
simplices ∆n , and let µ∞
e be the generalized Dirichlet
measure defined by the projective limit
µ∞
lim Γ(n + 1)λn ,
e =←
−
where λn the Lebesgue measure on ∆n . Then
Z
1
g (z) (N0 · u)dµ∞
[z, N0 ]g =
e (u)
Γ(z + 1) ∆∞
Z
1
Bz (t)g (z) (t) dt
=
Γ(z + 1) R
for all real-analytic g ∈ S(R+ ).
296
Up to now we have considered complex B-splines with
knot sequence N0 and derived from there new difference
operators and finally the relation to Dirichlet averages, just
as indicated in the diagram in Fig. 1:
B-splines → Difference operators → Dirichlet averages.
Our next step will consist of generalizing the setting with
appropriate weights in travelling through the diagram another way round: Dirichlet averages for other knot sequences τ and with weights → Generalized B-splines with
knot sequence τ → Difference operators.
4. Splines and Dirichlet Averages
0
∈ RN
Let b ∈ R∞
+ be a weight vector and τ = {tk }k∈N0 √
+
an increasing sequence of knots with lim supk→∞ k tk ≤
ρ < e.
Definition 6. A complex B-spline Bz (• | b; τ ) with
weight vector b and knot sequence τ is a function satisfying
Z
Z
g (z) (τ · u)dµ∞
Bz (t | b; τ )g (z) (t) dt =
b (u) (1)
∆∞
R
for all real-analytic g ∈ S(R+ ). Here, µ∞
lim µnb is
b = ←
−
the projective limit of Dirichlet measures with densities
Γ(b0 ) . . . Γ(bn ) b0 −1 b1 −1
u
u1
. . . unbn −1 .
Γ(b0 + . . . + bn ) 0
Since both W z and W −z are linear operators mapping
S(R+ ) into itself [11, 15] and since the real-analytic functions in S(R+ ) are dense in S(R+ ) [13], (1) holds for all
g ∈ S(R+ ). Moreover, since S(R+ ) is dense in L2 (R+ ),
we deduce that Bz (• | b, τ ) ∈ L2 (R+ ).
Equation (1) means, we define the weighted version of the
complex B-spline in a weak sense via Dirichlet averages.
Referring again to the diagram in Fig. 1, we now move
from the generalized B-splines to generalized divided differences.
0
RN
+
and weight
Definition 7. For knot sequences τ ∈
vectors b ∈ R∞
+ as above, we define the generalized complex divided differences [z; τ ]b as follows. Let g : R → C
be some function.
Z
1
Bz (t|b; τ )g (z) (t) dt
[z; τ ]b g :=
Γ(z) R
for all g ∈ S(R).
This definition is compatible with the usual Dirichlet
and
splines. In fact, for all finite τ = τ (n) ∈ Rn+1
+
n+1
b = b(n) ∈ R+ , and for z = n ∈ N0 the Dirichlet
spline Dn (•|b; τ ) of order n is defined by
Z
Z
(n)
g (n) (τ · u) dµnb (u)
g (t)Dn (t|b; τ ) dt =
∆n
R
= G(n) (b; τ )
for all g ∈ C n (R). Here, G is the Dirichlet average of g:
Z
G(b; τ ) =
g(τ · u) dµnb (u).
∆n
SAMPTA'09
5. Multivariate Complex B-Splines
To define complex B-splines in a multivariate setting, we
consider ridge functions and define multivariate B-splines
on their basis. Then, we walk again through the diagram
in Fig. 1: Multivariate B-splines → Multivariate difference operators. Results on Dirichlet averages yield new
recurrence relations for multivariate B-splines: Dirichlet
averages → B-splines.
Note that the approach via ridge functions had already let
to an extension of the Curry-Schoenberg-splines to a multivariate setting, e.g. [3, 4, 10, 12]. However, some of
these approaches have certain restrictions on the knots and
none of them considers complex splines.
Given λ ∈ Rs \{0}, a direction, and g : R → C a function.
The ridge function gλ corresponding to g is defined via
gλ : Rs → C,
gλ (x) = g(hλ, xi) for all x ∈ Rs .
∈ (Rs )N0 a sequence
Definition 8. [9] Let τ = {τ n }n∈Np
0
n
s
of knots in R with lim supn→∞ kτ n k ≤ ρ < e. The
multivariate complex B-spline B z (•|b; τ ) with weights
0
b ∈ CN
+ and knots τ is defined on ridge functions via
Z
Z
g(hλ, xi)B z (x | b; τ ) dx =
g(t)Bz (t | b; λτ ) dt,
Rs
R
(2)
where g ∈ S(R+ ) and λ ∈ Rs \ {0}, such that λτ =
{hλ, τ n i}n∈N0 is separated.
Since ridge functions are dense in L2 (Rs ) [14], we deduce
that B z (• | b; τ ) ∈ L2 ((R+ )s ).
Example 9. (Divided differences in the multivariate case)
Given b = e := (1, 1, 1, . . .). Then for all g ∈ S(R∞ ):
[z; τ ]e gλ
=
=
=
[z; τ ]gλ = [z; τ ]g(hλ, •i)
Z
1
g (z) (hλ, xi)B z (x | e; τ ) dx
Γ(z) Rs
Z
1
g (z) (t)Bz (t | e; λτ ) dt = [z; λτ ]g.
Γ(z) R
for all λ ∈ Rs such that λτ is separated.
Example 10. (Multivariate cardinal B-splines) For n ∈ N
and a finite sequence of knots τ = {τ 0 , τ 1 , . . . , τ n }:
[τ 0 , . . . , τ n ]gλ := [n; τ ]g(hλ, •i)
Z
1
=
g (n) (hλ, xi)B n (x | e; τ ) dx
n! Rs
Z
1
g (n) (t)Bn (t | e; λτ ) dt
=
n! R
n
X
g(hλ, τ j i)
Q
= [n; λτ ]g =
.
j
l
l6=j hλ, τ − τ i
j=0
Given a sequence of knots τ ⊂ Rs and a weight vector b
as above. In addition, let b ∈ l1 (N0 ) such that kbk1 =: c.
(z+1)
Then the Dirichlet averages of g (z) ∈ D(R) and gj
:=
j
(z+1)
(hλ, τ i − •)g
, j ∈ N0 , satisfy:
(1+z)
(c−1)G(z) (b; λτ ) = (c−1)G(z) (b−ej ; λτ )+Gj
(b; λτ ).
297
For the finite dimensional case see [1, 12]. These and other
relations of similar type on Dirichlet averages yield new
results for multivariate complex B-splines. As a example,
we state:
Proposition 11. [9] Under the above conditions, for all
j ∈ N0 :
(c − 1)
Z
Rs
Z
(z)
gλ (x)B z (x | b; τ )dx =
(z)
gλ (x)B z (x | b − ej ; τ )dx
(c − 1)
Rs
Z
(1+z)
+
(x)B z (x | b; τ )dx.
hλ, τ j − xi gλ
=
Rs
More relations of this type are given in [8].
6. Fourier representation of multivariate
complex B-splines
We saw above that both the univariate and the multivariate complex B-splines are L2 -functions: Bz (• | b; τ ) ∈
L2 (R+ ) and B z (• | b; τ ) ∈ L2 ((R+ )s ) Therefore, we
can apply the Plancherel transform to both functions and
consider their frequency spectrum.
Let ω = (ω1 , . . . , ωs ) ∈ Rs and let λ ∈ Rs , kλk = 1, be
the direction of ω, i.e., ω = ωλ for some ω ≥ 0. For the
Fourier transform of the generalized complex B-spline we
have for x = (x1 , . . . , xs ) ∈ Rs :
bz (ω | b; λτ ) =
B
Z
e−iωt Bz (t | b; λτ ) dt
=
ZR
=
e−iωhλ,xi B z (x | b; τ ) dx
Rs
Z
=
e−iω(λ1 x1 +...+λs xs ) B z (x | b; τ ) dx
Rs
Z
=
e−i(ω1 x1 +...+ωs xs ) B z (x | b; τ ) dx
s
ZR
=
e−ihω,xi B z (x | b; τ ) dx
Rs
b z (ω | b; τ )
= B
=
b z (ωλ | b; τ ).
B
This shows that the frequency spectrum of the multivariate complex B-spline along directions λ is given by the
spectrum of the univariate spline with knots projected onto
these λ.
7. Summary
Complex B-splines allow to define difference and divided
difference operators of complex order for arbitrary knots
and weights. Via their relation to Dirichlet averages and
Dirichlet splines, they can be extended to higher dimensions via ridge functions. The Fourier transform of the
univariate and multivariate complex B-spline are also related on ridges.
SAMPTA'09
8. Acknowledgments
This work was partially supported by the grant MEXTCT-2004-013477, Acronym MAMEBIA, of the European
Commission.
References:
[1] B. C. Carlson. B-Splines, hypergeometric functions
and Dirichlet averages. J. Approx. Th., 67:311–325,
1991.
[2] H. B. Curry and I. J. Schoenberg. On spline distributions and their limits: The Pólya distribution functions. Bulletin of the AMS, 53(7–12):1114, 1947.
Abstract.
[3] W. Dahmen and C. A. Micchelli. Statistical Encounters with B-Splines. Contemporary Mathematics,
59:17–48, 1986.
[4] C. de Boor. Splines as linear combinations of Bsplines. In G. G. Lorentz et al., editor, Approximation
Theory II, pages 1–47. Academic Press, 1976.
[5] B. Forster, T. Blu, and M. Unser. Complex B-splines.
Appl. Comp. Harmon. Anal., 20:281–282, 2006.
[6] B. Forster and P. Massopust. Statistical encounters
with complex B-Splines. to appear in Constructive
Approximation.
[7] B. Forster and P. Massopust. Some remarks about the
connection between fractional divided differences,
fractional B-Splines, and the Hermite-Genocchi formula. International Journal of Wavelets, Multiresolution and Information Processing, 6(2):279–290,
2008.
[8] P. Massopust. Double Dirichlet averages and complex B-splines. Submitted to SAMPTA 2009.
[9] P. Massopust and B. Forster. Multivariate complex
B-splines and Dirichlet averages. Submitted to Journal of Approximation Theory.
[10] C. A. Micchelli. A constructive approach to Kergin
interpolation in Rk : Multivariate B-splines and Lagrange interpolation. Rocky Mt. J. Math., 10(3):485–
497, 1980.
[11] K. S. Miller and B. Ross. An introduction to the fractional calculus and fractional differential equations.
Wiley, 1993.
[12] E. Neuman and P. J. Van Fleet. Moments of Dirichlet splines and their applications to hypergeometric
functions. Journal of Computational and Applied
Mathematics, 53:225–241, 1994.
[13] O. V. Odinokov. Spectral analysis in certain spaces
of entire functions of exponential type and its applications. Izv. Math., 64(4):777–786, 2000.
[14] A. Pinkus. Approximating by ridge functions. In
A. Le Méhauté, C. Rabut, and L. L. Schumaker, editors, Surface Fitting and Multiresolution Methods,
pages 1–14. Vanderbilt University Press, 1997.
[15] S. G. Samko, A. A. Kilbas, and O. I. Marichev. Fractional Integrals and Derivatives. Gordon and Breach
Science Publishers, Minsk, Belarus, 1987.
[16] M. Unser and T. Blu.
Fractional splines and
wavelets. SIAM Review, 42(1):43–67, March 2000.
298
Concrete and discrete operator reproducing
formulae for abstract Paley–Wiener space
J.R. Higgins
I.H.P., 4 rue du Bary, 11250 Montclar, France.
rowlandhiggins@yahoo.com
Abstract:
The classical Paley–Wiener space possesses two reproducing formulae; a ‘concrete’ reproducing equation and
a ‘discrete’ analogue, or sampling series, and there is a
striking comparison between them. It is shown that such
analogies persist in the setting of Paley–Wiener spaces that
are more general than the classical case. In fact, there
are ‘operator’ versions of the reproducing equation and
of the sampling series that are also comparable, not ‘exactly’ but nearly so. Reproducing kernel theory and abstract harmonic analysis are brought together to achieve
this, then the special case of multiplier operators with respect to the Fourier transform is considered. The Riesz
transforms provide a two-dimensional example, with possibilities of extension to higher dimensions and to further
classes of operators.
1.
Introduction
It has often been remarked that the classical Paley–Wiener
space possesses two reproducing formulae; a ‘concrete’
reproducing equation
Z
f (t) sinc(s − t)dt, (s ∈ R),
(1)
f (s) =
2.1
The basic setting of this paper is that of the reproducing
kernel theory of Saitoh [8, Ch. 2, §1]. Very briefly the
background is as follows. Let E be an abstract set. For
each t belonging to E let Kt belong to H (a separable
Hilbert space with inner product denoted by h, iH ). Then
k(s, t) := hKt , Ks iH is defined on E ×E and is called the
kernel function of the map Kt . This kernel function is a
positive matrix [8, Ch. 2, §2] and as such it determines one
and only one Hilbert space for which it is the reproducing
kernel. This Hilbert space is denoted by R(K) since it
turns out to be the set of images of H under the transformation (Kg)(t) := hg, Kt iH , (g ∈ H).
Theorem 1 (Saitoh) With the notations established
above, R(K) (which is now abbreviated to just R) is a
Hilbert space which has the reproducing kernel k(·, ·),
and is uniquely determined by this kernel k. For f ∈ R
there exists α ∈ H such that
kf kR = kKαkR ≤ kαkH ,
(3)
and there exists a unique member, g say, of the class of all
α’s satisfying (3) such that
f (t) = hg, Kt iH ,
R
and a ‘discrete’ reproducing equation, or sampling series,
X
f (s) =
f (n) sinc(s − n), (s ∈ R),
(2)
The reproducing kernel theory
(t ∈ E),
and
kf kR = kgkH .
n∈Z
and that there is a striking analogy between the two (see,
e.g., [3, p. 58]). Here, sinc denotes the standard function
sinc x := (sin πx)/πx.
The purpose of the present lecture is to point out that concrete and discrete reproducing formulae and analogies between them persist in the setting of Paley–Wiener spaces
that are more general than the classical case. It will be
shown that for suitably chosen operators there are ‘operator’ versions of the reproducing equation and of the sampling series that are also comparable, in the same way as
in the classical case described above.
2.
The setting
Abstract theories that lead to reproducing formulae are
outlined in §2.1 and §2.2, and are brought together in §2.3.
SAMPTA'09
The reproducing equation for f ∈ R is
f (t) = hf, k(·, t)iR
(4)
The following theorem is simple but very useful.
Theorem 2 The convergence of a sequence in the norm
of R implies that it converges pointwise over E, and the
convergence is uniform over any subset of E on which
k(t, t) = kk(·, t)k2 is bounded.
The following Theorem is to be found in [8].
Theorem 3 With notations as above, let {sn }, (n ∈ X),
be points of E such that {Ksn } is an orthonormal basis
for H. Then the sampling series representation
X
f (t) =
f (sn )k(sn , t),
(5)
n∈X
299
holds, convergence being in the norm of R; and then of
course Theorem 2 applies.
2.2
Abstract harmonic analysis
A very brief introduction (mostly just notations) to the abstract harmonic analysis that will be needed is now given.
All necessary background, and much more, is to be found
in [1], [2].
Let G be a locally compact abelian (LCA) group (written
additively). Let (t, γ) be a character of G, that is, a continuous homomorphism of G into the circle group. Let
G∧ = Γ denote the group of continuous characters on G,
usually called the dual group of G. We assume that Γ has
a countable discrete subgroup Λ.
Haar measures on the various groups are normalised in the
standard way [1], and this means in particular that there is
a measurable transversal (i.e., a complete set of coset representatives) Ω ⊂ Γ of Γ/Λ, and it has finite Haar measure.
Now
H = Λ⊥ := {t ∈ G : (t, λ) = 1, (λ ∈ Λ)}.
is a subgroup of G and is called the ‘annihilator’ of Λ. We
assume that H is discrete; it follows that the quotient group
Γ/Λ is compact.
The Fourier transform on L2 (G) is defined in the usual
way:
Z
f ∧ (γ) = (Ff )(γ) :=
f (t)(t, γ) dt,
G
in the L2 sense, where dt denotes the element of Haar
measure on G (likewise, dγ denotes the element of Haar
measure on Γ). The inverse Fourier transform will be denoted by ∨ or by F −1 .
We shall need the ‘shift’ property of the Fourier transform:
f (· − x)∧ (γ) = (−x, γ)f ∧ (γ).
Abstract Paley Wiener space P WΩ (G) is defined as follows:
2
P WΩ (G) := {f : f ∈ L (G) ∩ C(G),
f ∧ (γ) = 0 (Haar) a.a. γ 6∈ Ω}
2.3
close association between sampling in the harmonic analysis setting and Saitoh’s theory.
The space R of §2.1 is now seen to be the Paley–Wiener
space defined in (6), and its reproducing equation is
(6)
Combining harmonic analysis with the reproducing kernel theory
The abstract set E of §2.1 is often taken to be R or C.
Here, however, we take it to be an LCA group G thus combining two abstract theories, harmonic analysis and the reproducing kernel theory. In the notations of §2.1 and §2.2
we also take Kt = (t, ·), H = L2 (Ω) and Kg = F −1 g,
g ∈ L2 (Ω). Then we have
Z
Z
k(s, t) = (t, γ)(s, γ) dγ = (t − s, γ) dγ
Ω
Ω
∨
χ
= Ω (t − s) =: ϕΩ (t − s),
(7)
where χS denotes the characteristic function of a set S. It
does not seem to have been recognised that ϕΩ (t − s) is
the reproducing kernel for P WΩ (G), and that this allows a
SAMPTA'09
f (t) = hf, ϕΩ (t − ·)iL2 (G)
(8)
Kluvánek’s sampling theorem [4, p. 45] is a consequence:
Theorem 4 Let f ∈ P WΩ .
With the assumptions of
§2.2,
X
f (h)ϕΩ (t − h)
(9)
f (t) =
h∈H
in norm, etc., (see Theorem 2).
Our concrete – discrete comparison is beween (8) and (9).
3.
Operator kernels and operator reproducing formulae
The presence of kernels and reproducing equations associated with operators on a reproducing kernel Hilbert space
add greatly to the richness of its structure, as we shall see
in this section.
3.1
Operator kernels and operator reproducing
equations
Let R be the separable Hilbert space of functions defined
on E with reproducing kernel k(s, t), as we have discussed it in §2.1. Let B be a bijection on R, and let B ∗
denote the adjoint operator. The action of B on R is governed by the action of B ∗ on the reproducing kernel k,
because for f ∈ R,
Bf (t) = hBf, k(·, t)iR = hf, B ∗ k(·, t)iR .
(10)
See, e.g., [5].
Definition 1 The kernel
h(s, t) := B ∗ k(·, t) (s),
s, t ∈ E
will be called the operator kernel of B.
In this notation (10) is
Bf (t) = hf, h(·, t)i.
(11)
∗ −1
Now from Definition 1 above, ((B ) h(·, t))(s) =
k(s, t), so that, using the ordinary reproducing formula
(4), we have
f (t) = hf, k(·, t)i = hf, (B ∗ )−1 h(·, t)i
∗
= h (B ∗ )−1 f, h(·, t)i.
Now using standard properties of operators and their adjoints (e.g., [6, p. 202]) we can summarise these calculations as:
f (t) = h(B −1 f )(·), h(·, t)i.
(12)
This formula tells us that f can be reproduced, not from its
own values as in the ordinary reproducing kernel theory,
but from the result of acting on it with an operator. We
can call this an operator reproducing equation in analogy
with the ordinary reproducing equation (4).
Similar formulae for B ∗ can be obtained in the same way.
First, we make the following
300
Definition 2
∗
h (s, t) := h(t, s)
((t, s) ∈ E × E)
will be called the adjoint operator kernel of B.
Kernels and their adjoints occur in important areas of
study such as the theory of integral equations (see, e.g.,
[6, p. 170] for basic information). We shall find series
expansions for such kernels and identify the action of h∗
explicitly in Theorem 5 below.
First, let {ϕn }, n ∈ X, be an orthonormal basis for R.
for a constant B which is consequent upon the fact that,
since B is bounded, B ∗ is bounded and by Banach’s
‘bounded inverse’ theorem (B ∗ )−1 is bounded.
Now N can be made to approach ∞. Since Fn (t) converges to 0 both in norm and pointwise on E (see Theorem 2), the expression in (21) approaches 0 for each fixed
t ∈ E. Finally, from (20) we obtain the following
Theorem 5 Let R, B and E be as above. Then we have
the adjoint operator reproducing formula
f (t) = h(B ∗ )−1 f, h∗ (·, t)i.
Lemma 1
h(s, t) =
X
n∈X
Bϕn (t) ϕn (s),
(s, t ∈ E).
(13)
Convergence is in the norm of R for each t ∈ E, and the
pointwise convergence is governed by Theorem 2.
Proof The coefficients for the expansion of h(·, t), t
fixed, in the basis {ϕn } are
hh(·, t), ϕn i = hϕn , h(·, t)i = Bϕn (t)
by (11), thus (13) is obtained.
It will be recalled that if we put
(
Bϕn = ψn
(B ∗ )−1 ϕn = ψn∗ ,
(14)
then {ψn } is a Riesz basis for R with dual basis {ψn∗ }. In
this notation (13) can be written
X
ψn (t)ϕn (s).
(15)
h(s, t) =
n∈X
Hence by Definition 2 we have
X
ϕn (t)ψn (s).
h∗ (s, t) =
(16)
n∈X
in the norm of R for each t ∈ E.
By uniqueness the coefficients {ϕn (t)} are such that
ϕn (t) =
hh∗ (·, t), ψn∗ i
∗ −1
= h(B )
∗
ϕn , h (·, t)i,
(17)
N
Consider
P
f (t) − h(B ∗ )−1 f, h∗ (·, t)i.
f (t) − h(B ∗ )−1 f, h∗ (·, t)i
(20)
X
∗ −1
∗
cn ϕn ), h (·, t)i
= Fn (t) − h(B ) (f −
N
≤ Fn (t) + h(B )
∗ −1
≤ Fn (t) + k(B )
(f −
≤ Fn (t) + Bkf −
SAMPTA'09
(f −
X
N
X
cn ϕn ), h∗ (·, t)i
N
X
cn ϕn )kkh∗ (·, t)k
N
cn ϕn kkh∗ (·, t)k
Operator sampling series
There are connections here to the theory of single channel
sampling (see, e.g., [3, Ch. 12]), but the present approach
is much more general.
In order to match the operator reproducing equation (12)
with a discrete analogue, some further assumption will
have to be made. In fact we shall assume the existence
of a sequence (sn ) ⊂ E, n ∈ X such that {h(sn , t)} is
an orthogonal basis for R with normalising factors νn , so
that {νn h(sn , t)} is orthonormal. This can sometimes be
traced back to the condition that {Ksn } be an orthogonal basis for H. Again, we could assume that {h(sn , t)}
is just a basis for R, or just a frame. However, weaker
assumptions demand more technicalities and we will not
pursue this kind of generality here.
Let f ∈ R. Its expansion in our assumed orthonormal
basis is
X
f (t) =
cn νn h(sn , t)
(22)
where
cn = hf, νn h(sn , ·)i = νn hf, h∗ (·, sn i = νn (B ∗ f )(sn )
by Theorem 5. So (22) is
X
f (t) =
|νn |2 (B ∗ f )(sn ) h(sn , t).
(23)
n∈X
Then (12) and (23) are concrete – discrete analogues of
each other.
(19)
Put f (t) − N cn ϕn (t) = Fn (t). Now inserting the right
and left hand sides of (18) we find from (19) that
∗ −1
3.2
n∈X
Since this relationship is true for every member ϕn of a
basis for R, it holds for every f ∈ R by the usual density
argument.
This argument runs as follows:
P
Let N cn ϕn be the Nth partial sum of the expansion for
f in the basis ϕn . Then taking linear combinations in (17),
X
X
cn ϕn (t) = h(B ∗ )−1
cn ϕn (t), h∗ (·, t)i. (18)
N
This shows the basic property of h∗ ; it reproduces f from
(B ∗ )−1 f .
(21)
4.
Multiplier operators with respect to the
Fourier transform
Take E to be an LCA group G with dual Γ (for notations
and references, see §2.2), and let R be a Paley–Wiener
space P WΩ . Let µ(γ) be a non-nul complex valued function on Γ such that
(
0 < α ≤ |µ(γ)| ≤ β < ∞, (Haar) a.a. γ ∈ Ω;
(24)
µ(γ) = 0,
γ 6∈ Ω.
Let M denote the operation of multiplication by
χΩ (γ)µ(γ).
301
Definition 3 Let f ∈ P WΩ . The operator T is defined by
(T f )(s) := (F −1 MFf )(s)
Lemma 2 The operator T of Definition 3 is a bijection on
P WΩ
Proof Clearly T is linear. Furthermore it is one-to-one,
since the null space of T is
{f : T f = θ} = {f : µ(γ)f ∧ (γ) = θ}
which implies that f = θ.
Again, T is “onto”. Let g ∈ P WΩ . Then if M−1 denotes
multiplication by [µ(γ)]−1 , f = F −1 M−1 Fg ∈ P WΩ .
Then from Definition 3, T f = g.
The boundedness of T follows from two applications of
Plancherel’s Theorem. Indeed, let f ∈ P WΩ . Then
kT f kL2 (G) = kF −1 MFf kL2 (G) = kMFf kL2 (Γ)
First we need to know the adjoint T ∗ . Let f1 , f2 ∈ P WΩ .
The defining equation is
hT f1 , f2 i = hf1 , T ∗ f2 i.
Suppose that T ∗ is of the same form as T of Definition 3,
that is,
T ∗ f = F −1 M∗ Ff,
(25)
where M∗ denotes multiplication by the multiplier µ∗
which is to be determined.
In the integral notation, and using the ‘hat’ notation for the
Fourier transform, the criterion is:
Z
Z
µ(·)f1 ∧ (·) ∨ (t)f2 (t) dt =
f1 (t) µ∗ (·)f2 ∧ (·) ∨ (t) dt.
G
By Plancherel’s theorem this is:
Z
Z
µ(γ)f1 ∧ (γ)f2 ∧ (γ) dγ =
f1 ∧ (γ)µ∗ (γ)f2 ∧ (γ) dγ,
Γ
Γ
from which we may choose µ∗ (γ) = µ(γ).
It may be noted that T is self-adjoint if µ is real-valued.
It is now evident that the assumption (25) leads to
Z
∗
(T f )(s) =
µ(γ)f ∧ (γ)(s, γ) dγ
(26)
Γ
The operator kernel for T can now be calculated. From
Definition 1 and (7) we have
h(s, t) = T ∗ ϕΩ (· − t) (s),
SAMPTA'09
= µ(·)∨ (s − t).
Hence
h(s, t) = µ(·)∨ (s − t) = µ∧ (s − t).
Now (12) becomes
f (t) = h(T −1 f )(·), µ ∨ (· − t)i,
and (23) becomes
X
f (t) =
|νn |2 (T ∗ f )(sn ) µ∧ (sn − t).
(27)
(28)
5.
Examples
Example 1 The classical case
The operator kernel for T
G
Ω
n∈X
≤ |µ|kFf kL2 (Γ) = |µ|kf kL2 (G) .
4.1
Therefore from (26), and using the ‘shift’ property of the
Fourier transform,
Z
µ(γ) χΩ ∨ (· − t) ∧ (γ)(s, γ) dγ
h(s, t) =
Z Γ
µ(γ)(−t, γ)χΩ (γ)(s, γ) dγ
=
Γ
Z
µ(γ)(s − t, γ) dγ
=
s, t ∈ G.
Naturally, we expect to recover the case of the classical reproducing equation and sampling formula as special cases
of the theory. To do this we pick G = R, Ω = [−π, π],
T = I = T ∗ = T −1 and µ = χ[−π,π] (y). Therefore we
have
Z π
√
1
∨
ei(s−t)y dy = 2π sinc(s − t).
µ (s − t) = √
2π −π
Here and in subsequent Examples the choice of Haar measure on G, Γ, etc., accounts for apparent anomalies in the
normalising constants in the formulae (e.g., Haar measure
on R is taken to be (2π)−1/2 times Lebesgue measure. See
[2, p. 257]). With these choices, (27) becomes (1).
The classical sampling series
√ (2) now follows the textbook proof. Since {e−iny / 2π : n ∈ Z} is an orthonormal (ON) basis of L2 (−π, π), Plancherel’s theorem shows
that the inverse Fourier transforms {sinc(n−t)} : n ∈ Z}
form an orthonormal basis of P W[−π,π] . Coefficients in
the expansion of f in this basis are obtained from (1) and
so, with sn = n, our choices for T and µ show that (28)
becomes (2).
Example 2 The Hilbert transform
Another well-known example illustrates the present theory; a member of P W[−π,π] can be sampled and reconstructed from samples of its Hilbert transform (see, e.g.,
[3, p. 126] and references there). This idea can be fitted
it into the theme of the present lecture by taking G = R,
Ω = [−π, π], T = H := F −1 MF where M denotes
multiplication by −i sgn(y). H is the Hilbert transform
on P W[−π,π] .
302
For (27) we need
π
i
1
µ ∨ (s − t) = √ √
sgn(y)ei(s−t)y dy
2π 2π −π
(29)
= − sinc 12 (s − t) sin π2 (s − t)
Z
after a simple calculation. Also we have H−1 = −H =
H∗ , therefore (27) is
Z
f (t) = −
Hf (τ ) sinc 12 (τ −t) sin π2 (τ −t) dτ. (30)
R
For (28) we need to find {sn } such that {µ∧ (sn − t)}, n ∈
Z, is an√ON basis of P Wπ . We can start with the ON basis
{einy / 2π}, (n ∈ Z), of L2 (−π, π), then multiply each
member by −i sgn(y). The result is again an ON basis, as
a consequence of | − i sgn(y)| = 1 a.e. on [−π, π]. The
inverse Fourier transform of a typical one of these basis
elements is
−i −1
sgn(·)e−in· (t) = − sinc 12 (n − t) sin π2 (n − t)
F
2π
by the same calculation as in (29). But, taking account
of Haar measure, this also gives µ∧ (n − t). Hence (28)
becomes
X
f (t) =
Hf (n) sinc 12 (n − t) sin π2 (n − t) (31)
n∈Z
Our concrete – discrete comparison is between (30) and
(31).
Since the multiplier is of unit modulus, a two dimensional
version of the construction that we used in the previous
example shows that
−i y1
y2
−ihk,yi
, (k ∈ Z2 ),
+i
e
2π |y|
|y|
is an ON basis of L2 ([−π, π]2 ). Then (28) becomes
X
R∗ f (k)m∧ (k − t).
f (t) =
(35)
k∈Z2
The comparison for this example lies between the concrete
(34) and the discrete (35).
Other combinations of the Riesz transforms are possible,
in two and higher dimensions, whose multipliers satisfy
(24) but are not always of unit modulus.
6.
Conclusions
The multiplier transforms treated in this study form a
rather restricted class of operators; nevertheless, the methods can be used in connection with the very important
Riesz transforms. It remains to investigate extensions to
other types of operator. Likely candidates are, for example, multiplier transforms with less restrictive conditions on the multiplier, the singular integral operators of
Calderón–Zygmund type (a class containing the Riesz
transforms, see, e.g., [9, Ch. VI]), and operators of the
Hankel and Toeplitz type (e.g., [7]).
Example 3 The Riesz transforms
References:
For background on the Riesz transforms see [9, p. 223]
Take G to be Rd , (d ∈ N). Let t = (t1 , . . . , td ) and
let y = (y1 , . . . , yd ) etc. Let the scalar product in Rd be
denoted by h , i.
Definition 4 Let f ∈ L2 (Rd ), and define
Rj f := F −1 Mj Ff,
j = 1, . . . , d,
(32)
Mj denoting multiplication by −iyj /|y| χ[−π,π]d (y).
We note that this multiplier is not bounded away from zero
when d ≥ 2 and y ∈ [−π, π]d and therefore does not
always satisfy the criterion (24). However, it is possible to
define operators involving the Riesz transforms which do
satisfy the criterion (24).
First we consider the case d = 2, and define the operator
R := R1 + iR2
(33)
acting on P W[−π,π]2 . Its multiplier is
y2
y1
+i
m(y) := (−i)
|y|
|y|
and clearly we have |m(y)| = 1 a.e. Hence m satisfies the
criterion (24) with respect to two-dimensional Lebesgue
measure. Now (27) becomes
Z
1
R−1 f (s)m∨ (s − t) ds.
(34)
f (t) =
2π R2
SAMPTA'09
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[6] F. Riesz and B. Sz.-Nagy. Functional analysis. Dover
Publications, New York, 1990.
[7] R. Rochberg. Toeplitz and Hankel operators on the
Paley–Wiener space. Integral Equations Operator
Theory, 10(2), 1987.
[8] S. Saitoh. Integral transforms, reproducing kernels
and their applications. Longman, Harlow, 1997.
[9] E.M. Stein and G. Weiss. Introduction to Fourier
analysis on Euclidean spaces. Princeton University
Press, Princeton, 1971.
303
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304
Explicit localization estimates for spline-type
spaces
José Luis Romero
Departamento de Matemática
Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires
Ciudad Universitaria, Pabellón I
1428 Capital Federal
ARGENTINA
and CONICET, Argentina.
jlromero@dm.uba.ar
Abstract:
given there are not explicit. We will derive a polynomial
decay condition for the dual basis {gk }k , giving explicit
We give some explicit decay estimates for the dual system
information on the resulting constants. This yields some
of a basis of functions that are polynomially localized in
qualitative information, like the dependence of theses conspace.
stants on A, C and s and the corresponding p-Riesz basis
bounds for the original basis.
1.
Introduction
A spline-type space S is a closed subspace of L2 (Rd ) possessing a Riesz basis of functions well localized is space.
That is, there exists a family of functions {fk }k ⊆ S and
constants 0 < A ≤ B < +∞ such that
X
ck fk kL2 ≤ Bkckℓ2 ,
(1)
Akckℓ2 ≤ k
k
holds for every c ∈ ℓ2 , and the functions {fk }k satisfy an
spatial localization condition.
In a spline-type space
P any function in f ∈ S has a unique
expansion f =
k ck fk . Moreover the coefficients are
given by ck = hf, gk i, where {gk }k ⊆ S is the dual basis,
a set of functions characterized by the relation hgk , fj i =
δk,j . These spaces provide a very natural framework for
the sampling problem.
The general theory of localized frames (see [6], [5] and
[2]) asserts that the functions forming the dual basis satisfy
a similar spatial localization. This can be used to extend
the expansion in (1) to other spaces, so that the family
{fk }k becomes a Banach frame for an associated family
of Banach spaces (see [4] and [6]). In the case of a splinetype space S, this means that the decay of a function in S
can be characterized by the decay of its coefficients and,
in particular, that the functions {fk }k form a so called pRiesz basis for its Lp -closed linear span, for the whole
range 1 ≤ p ≤ ∞.
We derive, in some concrete case, explicit bounds for the
localization of the dual basis. We will work with a set of
functions satisfying a polynomial decay condition around
a set of nodes forming a lattice. By a change of variables,
we can assume that the lattice is Zd . So, we will consider a
set of functions {fk }k ⊆ L2 (Rd ) satisfying the condition,
−s
|fk (x)| ≤ C (1 + |x − k|)
,
x ∈ Rd and k ∈ Zd ,
for some constant C. This type of spatial localization is
specifically covered by the results in [5], but the constants
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2.
Main result
Theorem 1 Let C ≥ 1, and let t > d be integers. Let
s > d + t be a real number. For k ∈ Zd let fk : Rd → C
be a measurable function such that
−s
|fk (x)| ≤ C (1 + |x − k|)
,
(x ∈ Rd ).
Suppose that {fk }k is a Riesz basis for its L2 closed linear
span S, with bounds 0 < A ≤ B < ∞. Let {gk }k ⊆ S be
its dual basis.
Then, the dual functions satisfy,
−t
|gk (x)| ≤ D (1 + |x − k|)
,
(x ∈ Rd ).
where D is given by,
D=
E st C 2t+1 1 + At−1
,
(s − t − d)t At+1
for some constant E > 0 that only depends on the dimension d.
Remark 1 The constant E can be explicitly determined
from the proof.
The results in [6] prescribe polynomial decay estimates
for the dual basis similar to those possessed by the original basis. As a trade-off for the explicit constants we will
not obtain the full preservation of these decay conditions.
Nevertheless, any degree of polynomial decay on the dual
system can be granted, provided that the original basis has
sufficiently good decay.
Finally observe that, although the basis {fk }k is assumed
to be concentrated around a lattice of nodes, the functions
fk are not assumed to be shifts of a single function. In
particular, Theorem 1 below allows for a basis of functions whose ‘optimal’ concentration nodes do not form a
lattice but are comparable to one. The ‘eccentricity’ of the
configuration of concentration nodes is, however, penalized by the constants modelling the decay.
305
3.
Sketch of a proof and comments
Now we sketch the proof of the main result, for a complete
proof see [11].
Consider the gram matrix of the basis {fk }k given by,
M ≡ (mk,j )k,j∈Zd ,
mk,j := hfk , fj i .
Since {fk }k is a Riesz sequence, M , as an operator on ℓ2 ,
has an inverse N ≡ (nk,j )k,j∈Zd . Moreover, kN kℓ2 →ℓ2 ≤
A−1 and nk,j = hgk , gj i, where {gk }k ⊆ S is the dual
basis of {fk }k .
The localization assumptions on the basis {fk }k yield a
polynomial decay estimate on the entries of M,
|mk,j | . (1 + |k − j|)−s .
If we can establish a similar estimate for the entries of N ,
|nk,j | . (1 + |k − j|)−t .
with all the constants given explicitly, then, using calculations similar to those in [5], we obtain the desired polynomial concentration conditions for the dual functions.
Let us first consider the case where the basis {fk }k consists of integer shifts of a single generator f (that is,
fk = f (· − k), k ∈ Zd ). In this case, the matrix M is
constant on its diagonals. That is,
mk,j = ak−j ,
for some sequence a. Similarly, N is given by
nk,j = bk−j ,
where the sequence b satisfies a ∗ b = δ.
Therefore, in this special case, M and N are convolution
operators. The off-diagonal decay of their entries is equivalent to the decay of their kernels a and b. Since the decay
of a sequence x can be characterized by the smoothness
of its Fourier transform x̂, the problem can be reformulated as the preservation of the smoothness of the function
â under pointwise inversion. This reasoning is present, for
example, in [1].
We can measure the smoothness of â by considering weakderivatives and use repeatedly a chain-rule argument for
Sobolev spaces to obtain similar smoothness conditions
for b̂.
In the general case, where M and N need not be convolution operators, we try to imitate this reasoning, but we
avoid using the Fourier transform.
Given a matrix L ≡ (lk,j )k,j∈Zd and 1 ≤ h ≤ d, we
consider the new matrix,
Dh (L)k,j := (kh − jh )lk,j .
Observe that, up to some multiplicative constant, the map
Dh acts on a convolution operator by taking a partial
derivative of its symbol (that is, the Fourier transform of
its kernel.) The domain of Dh consists of those matrices
L such that Dh (L) defines a bounded operator on ℓ2 . We
call Dh (L) the derivative of L (with respect to xh .)
Dh is a derivation in the sense that it satisfies the equation
Dh (AB) = Dh (A)B + ADh (B), provided that Dh (A)
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and Dh (B) are both defined. Derivations are a wellknown tool in operator-algebras theory (see [3], [9] and
[10].)
Since M N = I and Dh (I) = 0, we can formally express
the high-order derivatives of N in terms of its lower-order
ones and all the derivatives of M ,
u−1
X u
u
Dh (N ) = −
(2)
Dhl (N )Dhu−l (M )N.
l
l=0
Using the polynomial off-diagonal decay bounds on M
and the bound kN kℓ2 →ℓ2 ≤ A−1 we can obtain bounds
for the ℓ2 → ℓ2 norms of some derivatives of N . These
imply polynomial off-diagonal decay estimates for N , and
hence yield the desired spatial localization bounds for the
dual basis.
In the argument above we related the off-diagonal decay
of a matrix with the ℓ2 → ℓ2 norm of its derivatives. The
ℓ2 → ℓ2 norm of a matrix is not determined by the size of
its entries. However, there are some necessary and (other)
sufficient conditions on the size of the entries of a matrix
for it to be bounded on ℓ2 . This “gap” in the conditions accounts for the loss of some decay information in Theorem
1, when passing from the original basis to its dual system.
Finally we point out that the formal computations in the
above argument are not sufficient to prove the theorem.
Consider again the simple case of a basis of integer shifts.
With the notation of the discussion above, we have the
relation
a ∗ b = δ,
(3)
we have some decay estimate on a (that can be reformulated as a smoothness condition on â) and we want to
prove a similar decay condition for b. There may be various sequences x satisfying the relation a ∗ x = δ; b can
be singled out as the only one of them having a bounded
Fourier transform. For example, when a is finitely supported, equation 3 is a linear difference equation which
has other solutions besides b (that grow exponentially).
The decay of the sequence b can be rigorously proved by
resorting to some Sobolev-space smoothing argument.
In the general case, to derive equation (2), one needs to use
the associativity of the product of matrices. This is justified only if all the matrices involved represent bounded operators. In other words, we need to know a priori that the
derivatives of N that are involved in equation (2) define
bounded operators. This can be proved using the general
results on derivations on Banach algebras (see [3], [9]) or
Jaffard’s Theorem [8].
The use of derivations is somehow implicit in Jaffard’s paper [8]. Recently, Gröchenig and Klotz [7] have systematically studied the use of derivations in connection to various problems including the preservation under inversion
of various kinds of off-diagonal decay conditions.
4.
Application
From Theorem 1 we can derive the following qualitative
statement.
Theorem 2 Let F i i∈I be a family of Riesz sequences,
F i ≡ fki k∈Zd ⊆ L2 (Rd ), (i ∈ I).
306
sharing a uniform lower basis bound. Suppose that the
family F i i satisfies a uniform concentration condition,
−s
fki (x) ≤ C (1 + |x − k|)
, (x ∈ Rd , k ∈ Zd , i ∈ I),
for some constants C ≥ 1, s > d + t and t > d, with t an
integer.
Then the following holds.
(a) The respective family of dual systems Gi i - where
Gi ≡ gki k∈Zd - satisfies a uniform concentration
condition,
−t
gki (x) ≤ D (1 + |x − k|)
, (x ∈ Rd , k ∈ Zd , i ∈ I),
for some constant D ≥ 1.
(b) A uniform p-Riesz basis condition holds, for all 1 ≤
p ≤ ∞. More precisely, there exist constants q, Q >
0 such that for any p ∈ [1, ∞] and any i ∈ I, the
relation
X
qkckℓp ≤ k
ck fki kLp ≤ Qkckℓp
k
holds for all finitely supported sequences (ck )k∈Zd .
Statement (a) follows directly from Theorem 1. Examining the proofs in [5] we see that the uniformity of the
constants given in (a) yields statement (b).
This qualitative conclusion on Theorem 2 was the original
motivation for Theorem 1.
Finally, observe that the arguments given above are applicable to a generel intrinsically localized basis in the sense
of [5].
5.
[4] Hans G. Feichtinger and Karlheinz Gröchenig. Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct.
Anal., 86:307–340, 1989. reprinted in ’Fundamental Papers in Wavelet Theory’ Heil, Christopher and
Walnut, David F.(2006).
[5] Massimo Fornasier and Karlheinz Gröchenig. Intrinsic localization of frames. Constr. Approx.,
22(3):395–415, 2005.
[6] Karlheinz Gröchenig. Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator. J. Fourier Anal. Appl., 10(2):105–132, 2004.
[7] Karlheinz Gröchenig and Klotz Andreas. Noncommutative approximation: Inverse-closed subalgebras
and off-diagonal decay of matrices. Preprint, available at http://arxiv.org/abs/0904.0386, 2009.
[8] Stephane Jaffard. Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications. Ann. Inst. H. Poincaré Anal. Non Linéaire,
7(5):461–476, 1990.
[9] Edward Kissin and Victor Shulman. Dense qsubalgebras of banach and c*-algebras and unbounded derivations of banach and c*-algebras.
Proc. Edinburgh Math. Soc, 36:261–276, 1993.
[10] Edward Kissin and Victor Shulman. Differential
properties of some dense subalgebras of c*-algebras.
Proc. Edinburgh Math. Soc, 37:399–422, 1994.
[11] José Luis Romero. Explicit localization estimates
for spline-type spaces. Submitted, available at
http://arxiv.org/abs/0902.0557, 2008.
Acknowledgements
The author wishes to thank Karlheinz Gröchenig and Andreas Klotz for their comments and for sharing an early
draft of [7], and is indebted to Hans Feichtinger and Ursula Molter for some insightful discussions.
The author holds a fellowship from the CONICET and
thanks this institution for its support. His research is also
partially supported by grants: PICT06-00177, CONICET
PIP N 5650, UBACyT X149.
This note was partially written during a long-term visit
to NuHAG in which the author was supported by the
EUCETIFA Marie Curie Excellence Grant (FP6-517154,
2005-2009).
References:
[1] Akram Aldroubi and Karlheinz Gröchenig. Nonuniform sampling and reconstruction in shift-invariant
spaces. SIAM Rev., 43(4):585–620, 2001.
[2] Radu M. Balan, Peter G. Casazza, Christopher Heil,
and Z. Landau. Density, overcompleteness, and localization of frames I: Theory. J. Fourier Anal.
Appl., 12(2):105–143, 2006.
[3] Ola Bratteli and Derek W. Robinson. Unbounded
derivations of c*-algebras. Commun. math. Phys,
42:253–268, 1975.
SAMPTA'09
307
SAMPTA'09
308
A Fast Fourier Transform with Rectangular
Output on the BCC and FCC Lattices
Usman R. Alim (1) and Torsten Möller (1)
(1) School of Computing Science, Simon Fraser University, Burnaby BC V5A 1S6, Canada.
ualim@cs.sfu.ca, torsten@cs.sfu.ca
Abstract:
This paper discusses the efficient, non-redundant evaluation of a Discrete Fourier Transform on the three dimensional Body-Centered and Face-Centered Cubic lattices.
The key idea is to use an axis aligned window to truncate
and periodize the sampled function which leads to separable transforms. We exploit the geometry of these lattices and show that by choosing a suitable non-redundant
rectangular region in the frequency domain, the transforms
can be efficiently evaluated using the Fast Fourier Transform.
1.
Introduction
The Discrete Fourier Transform (DFT) is an important
tool used to analyze and process data in an arbitrary number of dimensions. Most applications of the DFT in higher
dimensions, however, rely on a tensor product extension of
a one-dimensional DFT, with the assumption that the underlying data is sampled on a Cartesian lattice. This extension has the advantage that it allows for a straightforward
application of the Fast Fourier Transform (FFT).
The Cartesian lattice is known to be sub-optimal when
it comes to sampling a band-limited function in two or
higher dimensions [6]. In 3D, for instance, the BodyCentered Cubic (BCC) lattice is the optimal sampling lattice and yields a 30% savings in samples as compared to
the Cartesian lattice [8]. The Face-Centered Cubic (FCC)
lattice, although not optimal, is still better than the Cartesian lattice and is also the lattice that yields the minimum
amount of Fourier-domain aliasing when sampling a general trivariate function [2].
From the perspective of continuous signal reconstruction,
both the BCC and FCC lattices have received considerable
attention because of their many applications in Visualization and Computer Graphics. Entezari et al. have devised
a set of Box-Splines that can be used for signal approximation on the BCC [3] as well as the FCC [5] lattices.
However, very little effort has gone into the development
of discrete processing tools that are suitable for these nonCartesian lattices.
The idea of a multidimensional DFT (MDFT) on nonCartesian lattices is not new. Mersereau provided a derivation of a DFT for a hexagonally periodic sequence and
designed other digital filters suitable for a 2D hexagonal lattice [6]. Later, the idea was extended to higher
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dimensions and a MDFT for arbitrary sampling lattices
was proposed [7]. Guessoum et al. proposed an algorithm
for evaluating the MDFT that has the same computational
complexity as the Cartesian DFT [4].
Recently, Csébfalvi et al. [1] applied the MDFT to the
BCC and FCC lattices by choosing a Cartesian periodicity in the spatial domain which leads to a Cartesian sampling of the Fourier transform. This allows the MDFT to
be written in a separable form that can be evaluated via
the FFT. However, their representation is redundant and
leads to inefficient transforms. The aim of this paper is to
revisit these transforms and show that they can be computed much more efficiently by exploiting the geometric
properties of the BCC and FCC lattices to eliminate the
redundancy.
The paper is organized as follows. We provide a basic review of multidimensional sampling in Section 2. which is
later used in the derivation of a fast DFT for BCC and FCC
lattices in Section 3. Some properties of these transforms
are discussed in Section 4. and a summary is presented in
Section 5.
2.
Optimal Trivariate Sampling
Let fc (x) be a continuous trivariate function and Fc (ξ) be
its Fourier transform defined as
Z
fc (x) exp[−2πjξ T x]dx
(1)
Fc (ξ) =
R3
where T denotes the transpose operation. Let f (n) be
the sampled sequence obtained by sampling the function
through
f (n) = fc (Ln)
(2)
where L is a 3 × 3 sampling matrix and n is an integer
vector. Sampling on the lattice defined by the matrix L
amounts to a periodization of the Fourier spectrum on a
reciprocal lattice generated by the matrix L−T . In particular, the spectrum of the sampled sequence is given by [9]
X
1
Fc (ξ − L−T r)
(3)
F̂ (ξ) =
| det L| r
where r is any integer vector.
If we assume that fc (x) is isotropically band-limited (i.e
Fc (ξ) = 0 for kξk > ξ0 for some band-limit ξ0 ), then one
of the lattices that achieves the tightest possible packing
of the spectrum replicas (spheres) in the Fourier domain is
309
the FCC lattice. Thus, in order to sample a trivariate bandlimited function optimally, the function should be sampled
on the reciprocal of the FCC lattice, i.e. the BCC lattice.
3.
Discrete Fourier Transform
If the sequence f (n) is non-zero within a finite region, it
can be periodically extended spatially and represented as a
Fourier series which is a sampled version of the transform
(3) [7]. The pattern with which the continuous transform
(3) is sampled in the Fourier domain depends on the periodicity pattern in the spatial domain. Merserau et al. [7]
used a periodicity matrix to define the periodic extension
of the finite sequence. Here, we use a somewhat different approach by splitting the sampled sequence into constituent Cartesian sequences [1].
The BCC and FCC lattices LB and LF are generated by
the integer sampling matrices
h 1 −1 1 i
h1 0 1i
LB = −1 1 1 and LF = 0 1 1
1
1 −1
choose a cuboid shaped fundamental region generated by
limiting n to the set N := {n ∈ Z3 : 0 ≤ n1 <
N1 , 0 ≤ n2 < N2 , 0 ≤ n3 < N3 } for some positive integers N1 , N2 and N3 . This region consists of 2N1 N2 N3
data points (i.e. Voronoi cells) and has a total volume of
8N1 N2 N3 h3 . If we define N to be the diagonal matrix
diag(N1 , N2 , N3 ), then the two subsequences f0 (n) and
f1 (n) contained within the fundamental region can be periodically extended on a Cartesian pattern such that they
satisfy
f0 (n + N r) = f0 (n) and
for all n and r in Z .
This Cartesian periodic extension in the spatial domain
amounts to a Cartesian sampling in the Fourier domain.
In particular, the continuous transform (3) is sampled at
1
N −1 k yielding the sequence
the frequencies ξ = 2h
F (k) =F̂ (ξ)
1
ξ= 2h N −1 k
−2πj T −1
k N 2hIn +
2h
n∈N
−2πj T −1
f1 (n) exp
k N (2hIn + ht)
2h
X
f0 (n) + f1 (n) exp −πjkT N −1 t ·
=
n∈N
(4)
exp −2πjkT N −1 n ,
110
respectively. Both these lattices are based on a cubic sampling pattern whereby, in addition to samples at the eight
corners of a cube, LB has an additional sample in the center of the cube and LF has six additional samples on the
faces. Both these lattices can also be built from shifts of a
Cartesian sublattice as shown in Fig. 1. In particular, samples that lie on the corners of cubes form the sublattice
2Z3 . The quotient group LB /2Z3 is isomorphic to Z2 and
the quotient group LF /2Z3 is isomorphic to Z4 . Therefore, LB can be partitioned into two Cartesian cosets while
LF has four Cartesian cosets (Fig. 1).
f1 (n + N r) = f1 (n),
3
=
X
f0 (n) exp
where k = (k1 , k2 , k3 )T ∈ Z3 is the frequency index
vector. The above equation defines a forward BCC DFT.
Since it is a sampled version of a continuous transform that
is periodic on an FCC lattice, it should be invariant under
translations that lie on the reciprocal lattice generated by
1
LF . This property is easily
the matrix (hLB )−T = 2h
demonstrated as follows. If r ∈ Z3 , then after substituting
1
ξ = 2h
(N −1 k + LF r) in (4) and simplifying, we get
1
(N −1 k + LF r)
2h
X
=
f0 (n) + f1 (n) exp −πj(kT N −1 + r T LF )t ·
n∈N
exp −2πj(kT N −1 + r T LF )n
1
=F̂ ( N −1 k),
2h
F̂
Figure 1: Left, the BCC lattice, a 16 point view. Right, the FCC lattice, a
32 point view. Lattice sites that are Voronoi neighbors are linked to each
other. Cosets are indicated by different colors.
3.1
BCC DFT
The BCC lattice with arbitrary scaling is obtained via the
sampling matrix hLB where h is a positive scaling parameter. The Voronoi cell is a truncated octahedron having a
volume of | det hLB | = 4h3 . The Voronoi cell of the reciprocal FCC lattice is a rhombic dodecahedron having a
volume of 4h1 3 . Since LB has two Cartesian cosets, a sampled sequence can be split up into two subsequences given
by
f0 (n) = fc (2hIn) and
f1 (n) = fc (2hIn + ht),
where I is the 3 × 3 identity matrix, t is the translation
vector (1, 1, 1)T and n = (n1 , n2 , n3 )T is any integer
vector. f0 (n) is the sequence associated with the first
coset while f1 (n) is associated with the second. Since
these sequences are sampled on a Cartesian pattern, a
straightforward truncation of the original sequence is to
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since r T LF n is always an integer and r T LF t is always
even.
One fundamental period of the BCC DFT is contained
within a rhombic dodecahedron of volume 4h1 3 . The
sampling density in the frequency domain is given by
1
| det 2h
N −1 | = (8N1 N2 N3 h3 )−1 . Thus, the fundamental period consists of a total of 2N1 N2 N3 distinct frequency samples which is the same as the number of distinct spatial samples.
The inverse BCC DFT is obtained by summing over all
the distinct sinusoids and evaluating them at the spatial
sample locations. This gives
1 X
f0 (n) =
F (k) exp 2πjkT N −1 n
(5a)
N
k∈K
1
1 X
F (k) exp 2πjkT N −1 (n + t) (5b)
f1 (n) =
N
2
k∈K
310
where N = 2N1 N2 N3 is the number of samples and
K ⊂ Z3 is any set that indexes all the distinct frequency
samples. It is easily verified that both the sequences (5a)
and (5b) are periodic with periodicity matrix N .
3.1.1 Efficient Evaluation
Since N is diagonal, the kernel in both equations (4)
and (5) is separable. This suggests that the transform
can be efficiently computed via the rectangular multidimensional FFT, provided that a suitable rectangular index set K can be found. Observe that the Cartesian sequence F (k) is periodic with periodicity matrix 2N , i.e
F (k + 2N r) = F (k) for all r ∈ Z3 . Therefore, one
way to obtain a rectangular index set is to choose K such
that it contains all the frequency indices within one period
generated by the matrix 2N . This consists of a total of
| det 2N | = 4N indices and hence contains four replicas
of the fundamental rhombic dodecahedron.
A non-redundant rectangular index set can be found by
exploiting the geometric properties of the FCC lattice. If
we consider the first octant only, 4N samples are contained within a cube formed by the FCC lattice sites that
have even parity. This cube also contains six face-centered
sites. By joining any two axially opposite face-centered
sites, we can split the cube into four rectangular regions
such that each region consists of non-redundant samples
only. Six rhombic dodecahedra contribute to such a region
as illustrated in Fig. 2. The non-redundant region shown
in Fig. 2b is obtained by limiting k to the index set given
by K = {k ∈ Z3 : 0 ≤ k1 < N1 , 0 ≤ k2 < N2 , 0 ≤
k3 < 2N3 }.
(a)
(b)
Figure 2: (a) Six rhombic dodecahedra contribute to a non-redundant
rectangular region. (b) Zoomed in view of the non-redundant rectangular
region that contains the full spectrum split into six pieces. ξ1 , ξ2 and ξ3
indicate the principal directions in the frequency domain.
This region can further be subdivided into two cubes
stacked on top of each other, each containing N1 ×N2 ×N3
samples. The forward transform (4) can then be evaluated
in the two cubes separately by appropriately applying the
Cartesian FFT to the two sequences f0 (n) and f1 (n) and
combining the results together. After rearranging terms in
(4), the forward transform in the bottom cube becomes
X
f0 (n) exp −2πjkT N −1 n +
F0 (k) = F (k) =
n∈N
X
(6)
exp[−πjkT N −1 t]
f1 (n) exp −2πjkT N −1 n ,
n∈N
where k is now restricted to the set N . Since this equation is valid for all k ∈ Z3 , the forward transform in
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the top cube can be computed from (6) by F1 (k) =
F0 (k + (0, 0, N3 )T ) which simplifies to
X
F1 (k) =
f0 (n) exp −2πjkT N −1 n −
n∈N
X
(7)
f1 (n) exp −2πjkT N −1 n ,
exp[−πjkT N −1 t]
n∈N
for k ∈ N . Equations (6) and (7) are now in a form that
permits a straightforward application of the Cartesian FFT.
Since the two equations are structurally similar, only two
N1 × N2 × N3 FFT computations are needed, one for the
sequence f1 (n) and one for f2 (n).
In a similar fashion, the inverse transform (5) can be computed using two inverse FFT computations. Splitting the
summations in (5) into the two constituent cubes gives
1 X
(F0 (k) + F1 (k)) exp 2πjkT N −1 n ,
f0 (n) =
N
k∈N
1 X
f1 (n) =
(F0 (k) − F1 (k)) exp[πjkT N −1 t] ·
N
k∈N
(8)
exp 2πjkT N −1 n .
3.2
FCC DFT
The FCC lattice with arbitrary scaling is generated by the
sampling matrix hLF . The rhombic dodecahedral Voronoi
cell has a volume of | det hLF | = 2h3 . The frequency
spectrum is replicated according to (3) on a reciprocal
BCC lattice that has a truncated octahedral Voronoi cell
having a volume of 2h1 3 .
A sequence sampled on the FCC lattice can be split up into
four Cartesian subsequences corresponding to the four
Cartesian cosets. Each subsequence is given by
fi (n) = fc (2hIn + hti ),
where i ∈ {0, 1, 2, 3} and ti are the integer shift vectors
(0, 0, 0)T , (1, 0, 1)T , (0, 1, 1)T and (1, 1, 0)T respectively.
Analogous to the BCC case, let us choose a rectangular
truncation of the original sequence by limiting n to the set
N and extend the sequences periodically so that they satisfy fi (n+N r) = fi (n). This truncation yields a rectangular fundamental region in the spatial domain consisting
of a total of N = 4N1 N2 N3 distinct samples. Therefore,
each truncated octahedron in the frequency domain tessellation will consist of N distinct points that are sampled
1
in a Cartesian fashion at the frequencies ξ = 2h
N −1 k
3
where k ∈ Z . The sampled sequence in the frequency
domain is thus given by
1
F (k) = F̂ ( N −1 k)
2h
3
(9)
XX
1
fi (n) exp −2πjkT N −1 (n + ti ) .
=
2
i=0
n∈N
This defines a forward FCC DFT. Like the BCC case, it is
1
(N −1 k + LB r)
invariant under shifts of the type ξ = 2h
making it periodic on a BCC lattice with one fundamental
period contained in a truncated octahedron.
The inverse FCC DFT is obtained by summing over all the
distinct sinusoids evaluated at the spatial sample locations
1 X
F (k) exp 2πjkT N −1 (n+ 21 ti ) , (10)
fi (n) =
N
k∈K
where K ⊂ Z3 is any set that indexes all the N distinct
sinusoids.
311
3.2.1 Efficient Evaluation
Since N is diagonal, the key to efficiently evaluating the
FCC DFT pair (9) and (10) is to choose a suitable rectangular region in the frequency domain that contains N distinct samples. Similar to the BCC DFT, the sequence (9)
is 2N periodic with one complete rectangular period containing | det 2N | = 2N samples and hence two complete spectrum replicas. These 2N samples are contained
within a cube, the corners of which lie at the even parity
points of the BCC lattice. This cubic region can be split
into two by halving along any of the three principal directions yielding a rectangular region that contains only
non-redundant samples as illustrated in Fig. 3. The index
set that spans the region depicted in Fig. 3b is given by
K = {k ∈ Z3 : 0 ≤ k1 < 2N1 , 0 ≤ k2 < 2N2 , 0 ≤ k3 <
N3 }.
computes only two N1 × N2 × N3 FFTs for the BCC case
and four N1 × N2 × N3 FFTs for the FCC case.
Any operation in the frequency domain must respect the
arrangement of the different portions of the spectrum. The
BCC DFT splits the spectrum into six parts as illustrated
by the six pieces (two lunes and four spherical triangles)
of the sphere in Fig. 2b. The FCC transform splits the frequency spectrum into five parts as indicated by the hemisphere and the four spherical triangles in Fig. 3b.
5.
Summary
In this paper, we have shown that a MDFT of a Cartesian
periodic sequence sampled on the BCC or FCC lattices
can be efficiently evaluated using the FFT. The BCC lattice can be represented as two shifted Cartesian lattices.
This representation leads to a separable transform that is
efficiently computed via two non-redundant FFT evaluations of the Cartesian subsequences. Similarly, the FCC
lattice consists of four shifted Cartesian lattices and the
MDFT requires four non-redundant FFT evaluations.
References:
(a)
(b)
Figure 3: (a) Five truncated octahedra contribute to a non-redundant rectangular region. (b) Zoomed in view of the rectangular region that contains the full spectrum.
The non-redundant region can be split into four N1 ×N2 ×
N3 cubic subregions and the forward transform (9) can be
evaluated in each of the subregions separately by appropriately applying the FFT to each of the subsequences fi (n)
and combining the output. The derivation is very similar
to the BCC case and we leave the details to the reader. The
forward transform in each subregion can be written as
Fm (k) =
3
X
Him exp[−πjkT N −1 ti ]·
i=0 X
(11)
fi (n) exp −2πjkT N −1 n ,
n∈N
where m ∈ {0, 1, 2, 3}, k ∈ N and
element of
1 H1im 1is an
1
1 −1 1 −1
the 4 × 4 Hadamard matrix H = 1 1 −1 −1 . The four
1 −1 −1
1
subregions Fm (k) have their bottom left corners at the frequency index vectors (0, 0, 0)T , (N1 , 0, 0)T , (0, N2 , 0)T
and (N1 , N2 , 0) respectively.
Likewise, the inverse transform (10) can be evaluated using four inverse FFT evaluations, one for each of the subsequences. This yields
fi (n) =
4.
3
X
1 X
Him Fm (k) ·
exp[πjkT N −1 ti ]
N
k∈N
m=0
(12)
exp 2πjkT N −1 n .
Discussion
The decomposition of the non-redundant region in the frequency domain into cubes leads to transforms that are
much more efficient. Both the BCC and FCC DFTs proposed by Csébfalvi et al. [1] are redundant and require the
FFT of a 2N1 ×2N2 ×2N3 sequence. In contrast, our proposed evaluation strategy eliminates the redundancy and
SAMPTA'09
[1] B. Csébfalvi and B. Domonkos. Pass-Band Optimal
Reconstruction on the Body-Centered Cubic Lattice.
In Vision, Modeling, and Visualization 2008: Proceedings, October 8-10, 2008, Konstanz, Germany,
page 71. IOS Press, 2008.
[2] A. Entezari. Optimal Sampling Lattices and Trivariate Box Splines. PhD thesis, Simon Fraser University,
Vancouver, Canada, July 2007.
[3] A. Entezari, D. Van De Ville, and T. Möller. Practical
box splines for volume rendering on the body centered
cubic lattice. IEEE Transactions on Visualization and
Computer Graphics, 14(2):313 – 328, 2008.
[4] A. Guessoum and R. Mersereau. Fast algorithms
for the multidimensional discrete Fourier transform.
IEEE Transactions on Acoustics, Speech, and Signal
Processing, ASSP-34(4):937–943, 1986.
[5] M. Kim, A. Entezari, and J. Peters. Box Spline Reconstruction on the Face Centered Cubic Lattice. IEEE
Transactions on Visualization and Computer Graphics (Proceedings Visualization/Information Visualization 2008), 14(6):1523–1530, 2008.
[6] R. Mersereau. The Processing of Hexagonally Sampled Two-dimensional Signals. Proceedings of the
IEEE, 67(6):930–949, June 1979.
[7] R. Mersereau and T. Speake. The processing of periodically sampled multidimensional signals. IEEE
Transactions on Acoustics, Speech, and Signal Processing, (1):188–194, 1983.
[8] T. Theußl, T. Möller, and M. Gröller. Optimal regular
volume sampling. In Proceedings of the conference on
Visualization’01, pages 91–98. IEEE Computer Society Washington, DC, USA, 2001.
[9] P. Vaidyanathan.
Fundamentals of multidimensional multirate digital signal processing. Sadhana,
15(3):157–176, 1990.
312
Daubechies Localization Operator in
Bargmann - Fock Space and Generating
Function of Eigenvalues of Localization
Operator
Kunio Yoshino, Tamazutsumi, 1-28-1, Setagaya-ku, Tokyo, Japan,158-8557.
yoshinok@tcu.ac.jp.
Abstract:
We will express Daubechies localization operators in
Bargmann - Fock space. We will prove that the Hermite functions are eigenfunctions of Daubechies localization operator. By making use of generating function of
eigenvalues of Daubechies localization operator, we will
show some reconstruction formulas for symbol function
of Daubechies localization operator with rotational invariant symbol.
1. Introduction
Daubechies localization operator was introduced in
I. Daubechies : A Time Frequency Localization Operator: A Geometric Phase Space Approach, IEEE. Trans.
Inform. theory. vol.34, pp.605-612(1988)
She obtained following results.
Theorem(Daubechies)([2])
Suppose that symbol function of Daubechies localization
operator is rotational invariant. Then
(i) Eigenfunctions of Daubechies localization operator
are Hermite functions.
(ii) Eigenvalues are given by Mellin transform of symbol
function.
In this paper we realize Daubechies localization opeartor
in Bargamann - Fock space. We will consider the eigenvalue problem of Daubechies localization opeartor in
Bargmann - Fock space. By making use of Bargamann
- Fock space we will give a new proof of above theorem.
We will establish reconstruction formula of symbol function of Daubechies localization operator with rotational invariant symbol by generating function of eigenvalues of
Daubechies localization operator. For the simplicity, we
will confine ourselves to 1-dimensional case.
2. Bargmann Transform
Put
½
¾
√
1 2
2
A(z, x) = π
exp − (z + x ) + 2z · x ,
2
where z ∈ C and x ∈ R.
−1/4
SAMPTA'09
Bargmann transform B(ψ) is defined as follows :
def
B(ψ)(z) =
Z
R
ψ(x)A(z, x)dx, (ψ ∈ L2 (R)).
Put
BF = {g ∈ H(C) :
Z
C
2
|g(z)|2 e−|z| dz ∧ dz̄ < ∞}
where H(C) denotes the space of entire functions in the
complex plane.
BF is called Bargmann-Fock space.
Theorem 1([1])
Bargmann transform is a unitary mapping from L2 (R) to
Bargmann-Fock space BF .
For the details of Bargmann transform and Bargmann Fock space, we will refer the reader to [1] and [3].
3.
Hermite Functions
Definition 1([1],[3]) Hermite functions hm (x) is defined
by :
√
dm
hm (x) = (−1)m (2m m! π)−1/2 exp(x2 /2) m exp(−x2 ),
dx
(m ∈ N).
Hermite functions has following generating function
expansion :
¾
√
1 2
2
π
exp − (z + x ) + 2z · x
2
∞
X zm
√ hm (x),
=
m!
m=0
(z ∈ C, x ∈ R).
−1/4
½
We recall some well known facts about Hermite functions.
Proposition 1([1],[3])
313
(i) {hm (x)}∞
m=0 is complete orthonormal basis in
L2 (R).
∂2
+ x2 − 1)hm (x) = mhm (x),
∂x2
zm
B(hm )(z) = √ , (z ∈ C)
m!
(ii) (−
5.
A Realization of Daubechies Localization
Operator in Bargmann Fock space
where F is Fourier transform.
In this section we will express Daubechies Localization
Operator in Bargmann - Fock space.
First we need following lemmas.
Lemma 1
2
p + iq
B(φp,q )(z) = ezw−1/2|w| +1/2ipq , (w = √ )
2
Proposition 2([1],[3])
Lemma 2([1])
(iii)
(iv)
F(hm )(x) = (−i)m hm (x),
(i) (B ◦ L ◦ B −1 )g(z) = z
∂2
where L = − 2 + x2 − 1.
∂x
∂
g(z),
∂z
(ii) (B ◦ F ◦ B −1 )g(z) = g(−iz),
where F is Fourier transform and g(z) ∈ BF .
4.
Daubechies Localization Operator
Put
2
φp,q (x) = π −1/4 eipx e−(x−q) /2 .
< φp,q , f >=
Z
φp,q (x)f (x)dx.
R
This is so called Short time Fourier transform (or Windowed Fourier transform, or Gabor transform).
Definition 2([2])
Suppose that F (p, q) ∈ L1 (R2 ) and f (x) ∈ L2 (R).
We put
PF (f )(x) =
1
2π
Z Z
F (p, q)φp,q (x) < φp,q , f > dpdq,
R2
We call PF (Daubechies) localization operator F (p, q) is
called symbol function.
Daubechies obtained following results.
Theorem([2]). Suppose that F (p, q) ∈ L1 (R2 ) and
g(z) =
1
2πi
Z Z
C
2
ewt g(t)e−|t| dt ∧ dt,
(g ∈ BF )
Theorem 2 Under the same assumptions in Prop. 3, we
have
(B ◦ PFZ ◦ZB −1 )(g)(z)
2
1
=
F (w, w)ezw g(w)e−|w| dw ∧ dw,
2πi
C
(∀g ∈ BF )
(Proof)
Since Bargmann transform
is unitary operator, we have
Z Z
1
F (p, q)φp,q (x) < φp,q , f > dpdq,
PF (f )(x) =
Z Z 2π
1
=
F (p, q)φp,q (x) < Bφp,q , Bf > dpdq,
2π
So by lemma 1,
B ◦ PFZ(fZ)(x)
1
F (p, q)Bφp,q (z) < Bφp,q , Bf > dpdq,
=
2π Z Z
2
1
F (p, q)ezw−1/2|w| +1/2ipq < Bφp,q , Bf >
=
2π
dpdq,
Hence we have
(B ◦ PFZ ◦ZB −1 )(g)(z)
2
1
=
F (p, q)ezw−1/2|w| +1/2ipq < Bφp,q , g >
2π
dpdq,
On the other hand
< Bφp,q
Z ,Zg >
2
2
1
=
et̄w̄−1/2|w| −1/2ipq g(t)e−|t| dtdt̄,
2π
By Lemma 2,
2
= e−1/2|w| −1/2ipq g(w̄)
Thus we obtained our desired result.
Proposition 3([2]). Suppose that F (p, q) ∈ L1 (R2 ) and
F (p, q) is rotational invariant function,
i.e. F (p, q) = F̃ (r2 ), (r2 = p2 + q 2 ).
Then
(i) Hermite functions hm (x) are eigenfunctions of
Daubechies operator PF .
F (p, q) is rotational invariant function,
i.e. F (p, q) = F̃ (r2 ), (r2 = p2 + q 2 ).
Then
(i) Functions z m are eigenfunctions of operator
B ◦ PF ◦ B −1 .
PF (hm )(x) = λm hm (x), (m ∈ N),
Z ∞
1
(ii) λm =
e−s sm F̃ (2s)ds, (m ∈ N).
m! 0
(B ◦ PF ◦ B −1 )(z m ) = λm z m , (m ∈ N),
Z
1 ∞ −s n
e s F̃ (2s)ds, (n ∈ N).
(ii) λn =
n! 0
(Proof)
SAMPTA'09
314
By Theorem 2, we have
(B ◦ PFZ ◦ZB −1 )(z m )
2
1
=
F (2|w|2 )ezw wm e−|w| dw ∧ dw,
2πi
C
PFa =
As a corollary of Proposition 3, we obtained following
Daubechies’s results in section 4.
Proposition 4([8]) Let {λm } be eigenvalues of PF .
Then
there exists a positive constant C such that
C
, (m ∈ N).
|λm | ≤ p
|m|
Λ(w) =
∞
X
λm w m .
m=0
We call Λ(w) generating function of eigenvalues of
Daubechies Localization Operator.
Theorem 3 Under the same assumptions in Prop. 3, we
have
(B ◦ PF ◦ B −1 )(g)(z) = (2πi)−n
I
z dt
g(t)Λ( ) ,
t t
(∀g ∈ BF )
(Proof)
Suppose that g(z) ∈ BF . We consider Taylor expansion
of g(z) at the origin.
Put
∞
X
am z m
g(z) =
m=0
By Proposition 3, we have
(B ◦ PF ◦ B −1 )(z m ) = λm z m .
So
(B ◦ PF ◦ B −1 )(g)(z) = (B ◦ PF ◦ B −1 )(
=
∞
X
am λm z m = (2πi)−n
m=0
I
∞
X
am z m )
m=0
z dt
g(t)Λ( )
t t
Hence we have
I
z dt
−1
−1
(B ◦ PF ◦ B )(g)(z) = (2πi)
g(t)Λ( ) .
t t
6. An Example of Daubechies Localization
Operator
In this section we will consider following special
Daubechies localization operators.
Put
a−1 2
a−1
2
2
Fa (p, q) = e 2a (p +q ) = e 2a r ,
Then
a
λm = am+1 , Λ(w) =
.
1 − aw
m+1
hm (x).
PFa (hm )(x) = a
SAMPTA'09
am+1 hm (x)hm (y).
m=0
Employing polar coordinte transform w = reiθ and s =
r2 ,
we have Z ∞
1
= zm
e−s sm F̃ (2s)ds.
m! 0
Put
∞
X
(0 < a < 1).
valids in operator sense.
∞
X
am+1 |m >< m|,
(PFa =
in Dirac’s Notation.)
m=0
−1
If a = 2 , this is Schatten decomposition of PFa and
PFa is called density operator in quantum statistical mechanics.
Proposition 5 (Mehler’s formula [3],[5])
∞
X
am+1 hm (x)hm (y)
m=0
−1 1−a
1+a
2
2
a
e 4 ( 1+a (x+y) + 1−a (x−y) ) ,
=p
2
π(1 − a )
(|a| < 1).
Corollary 3
(i) Z PFa (f )
1+a
−1 1−a
2
2
a
p
e 4 ( 1+a (x+y) + 1−a (x−y) ) f (y)dy,
=
2)
π(1
−
a
R
(f ∈ L2 ).
(ii) If a ∈ C, |a| < 1, then
PFa : L2 −→ L2 is bounded linear operator.
(Proof)
1−a 1+a
If a ∈ {a ∈ C : |a| < 1}, then real part of
+
1+a 1−a
is positive. So PFa is bounded linear operator from L2
to L2 . Namely, we obtained analytic continuation of PFa
under the condition (a ∈ C, |a| < 1).
7.
Realization of PFa in Bargmann - Fock
space
In this section we will consider PFa in Bargmann - Fock
space.
Proposition 6
∞
X
z m w̄m
am+1 √ √ .
m! m!
m=0
valids in operator sense.
(i) B ◦ PFa ◦ B −1 =
−1
(ii) (B
Z Z◦ PFa ◦ B )(g)(z)
2
ia
=
eazw̄ g(w)e−|w| dw ∧ dw̄,
2
C
(g ∈ BF )
(Proof)
zm
Since √ . are eigenfunctions of B ◦ PFa ◦ B −1 ,
m!
we have
∞
X
z m w̄m
B ◦ PFa ◦ B −1 =
am+1 √ √ .
m! m!
m=0
Proposition 7 Suppose that |a| < 1, (a ∈ C). Then we
have
(B ◦ PFa ◦ B −1 )(g)(z) = ag (az) ,
(g ∈ BF ).
315
(Proof)
−1
−1
I
z dt
g(t)Λ( )
t t
(B ◦ PFa ◦ B )(g)(z) = (2πi)
I
a
dt = ag(az).
= (2πi)−1 g(t)
t − az
Proposition 8
(i)
lim PFa (f ) = (−i)Ff,
a→−i
lim PFa (f ) = iF−1 f,
m=0
a→i
is an asymptotic expansion of G(t).
where F is Fourier transform.
Remark Λ(w) is the Borel transform of formal power
∞
X
series
m!λm t−m−1 .
(Proof) By Prop.7, we have
m=0
(B ◦ PFa ◦ B −1 )(g)(z) = ag (az) ,
(i)
(g ∈ BF ).
lim (B ◦ PFa ◦ B −1 )(g) = lim ag(az) = g(z).
a→1
a→0
This means that lim PFa = Identity operator.
a→1
(ii)
m!λm t−m−1
In general this series is divergent series. We put
Z ∞
F̃ (2s)e−s
G(t) =
ds, (t ∈ C\[0, ∞]).
t−s
0
We have
Proposition 10([8])
Formal power series
∞
X
m!λm t−m−1
For f ∈ L2 , we have
lim PFa (f ) = f,
(iii)
∞
X
m=0
a→1
(ii)
Λ(w) is called generating function for eigenvalues of PF
Now we consider following formal power series :
lim (B ◦ PFa ◦ B −1 )(g) = lim ag(az)
a→−i
a→−i
= (−i)g(−iz).
By (ii) in Proposition 2, this means that
lim PFa = (−i)F .
Since G(t) is Hilbert transform of F̃ (2s)e−s , we have
Theorem 5
−1
F̃ (2s) = es lim
(G(s + it) − G(s − it))
t→0 2πi
We also have
Theorem 6([8])
F̃ (2s) = (2π)−n es F(Λ(iv))(s),
valids in distribution sense.
where F is Fourier transform.
a→−i
Proof of (iii) is same as that of (ii).
Proposition 9
(i)
G = {P Fa : a ∈ C, |a| < 1} ∪ {Id }
is semigroup.
(ii)
PFa ◦ PFa = PFab .
(Proof) By Proposition 7,
(B ◦ PFa ◦ B −1 )(g)(z) = ag(az), g(z) ∈ BF
So, we have
(B ◦ PFa ◦ PFb ◦ B −1 )(g)(z) = bag (baz)
Hence we have
PFb ◦ PFa = PFab .
In these cases, Fa (p, q) ∈
/ L1 . But these operators still
define bounded operators from L2 to L2 .
As seen in Proposition 8, these operators are obtained as
limit of P Fa , (Fa ∈ L1 ).
8. Reconstruction formulas
We assume that F (p, q)p
is rotational invariant L1 function.
Namely, F (p, q) = F̃ ( p2 + q 2 ).
References:
[1] V. Bargmann : On a Hilbert Space of Analytic Functions and an Associated Integral Transform Part I,
Comm.Pure.Appl.Math, pp. 187-214(1961)
[2] I. Daubechies : A time frequency localization operator; A geometric phase space approach, IEEE.
Trans. Inform. theory. vol.34, pp.605-612(1988)
[3] G. B. Folland : Harmonic Analysis in Phase Space,
Princeton Univ. Press (1989)
[4] K. Gröhenig: Foundations of Time-Frequency Analysis, Birkhäuser-Verlag, Basel, Berlin, Boston(2000)
[5] M.W. Wong : Weyl Transforms, Springer-Verlag.
New York. (1998)
[6] M.W. Wong : Localization Operators on the WeylHeisenberg Group, Geometry, Analysis and Applications, Proceedings of the International Conference
(editor:P.S.Pathak) 303-314(2001)
[7] M.W. Wong : Wavelet Transforms and Localization
Operator, Birkhäuser-Verlag. Basel, Berlin, Boston.
(2002)
[8] K. Yoshino : Localization operators in Bargmann Fock space and reconstruction formula for symbol
functions, preprint (2009)
In section 5, we introduced following generating function:
∞
X
Λ(w) =
λm w m
m=0
SAMPTA'09
316
Signal-dependent sampling and reconstruction
method of signals with time-varying bandwidth
Modris Greitans and Rolands Shavelis
Institute of Electronics and Computer Science, 14 Dzerbenes str., Riga LV-1006, Latvia.
greitans@edi.lv, shavelis@edi.lv
Abstract:
The paper describes the sampling method of nonstationary signals with time-varying spectral bandwidth. The reconstruction procedure exploiting the low-pass filter with
time-varying cut-off frequency is derived. The filter application in signal reconstruction from its level-crossing
samples is shown. The results of computer simulations
are presented.
1.
Introduction
The spectral characteristics of signals of practical interest often change with time. Generally, a signal with
time-varying spectral bandwidth can be approximated
with fewer samples per interval using appropriate nonequidistantly spaced samples than using uniform sampling
procedure, where the sampling rate is chosen taking into
account the highest signal frequency. For example, let us
inspect a signal with wide bandwidth regions and narrow
spectral bandwidth in the rest of signal observation. It is
more efficient to sample the narrow bandwidth regions at
a lower rate than the regions, where spectral bandwidth is
wide. Solving this problem correctly requires the knowledge of the function of the instantaneous maximum frequency of signal. The paper will show two typical situations. First, information about the time-varying bandwidth
is known a priori. In this case the deliberately non-uniform
sampling instants can be calculated in advance, and reconstruction is based on application of filter with appropriate time-varying impulse response function. Second, the
signal-dependent sampling scheme - level crossing sampling (LCS) is used for analog-to-digital (A/D) conversion. The idea of level-crossing sampling is based on the
principle that samples are captured when the input signal
crosses predefined levels. Such a sampling strategy has
quite long history and is exploited for various applications
[1, 2]. It has been shown that LCS has several interesting
properties and is more efficient than traditional sampling
in many respects [3]. In particular, it can be related to the
processing of non-stationary signals, because if a waveform is changing rapidly, the samples are spaced more
closely, and conversely – if a signal is varying slowly, the
samples are spaced sparsely [4]. This property allows to
calculate the estimate of the function of the instantaneous
maximum frequency of signal from the positions of samples. In this case to reconstruct the waveform of signal,
SAMPTA'09
an additional resampling procedure is needed before the
use of time-varying reconstruction filter, which will be described in next section.
Note that in both cases the local sampling density reflects
the local bandwidth of the signal, therefore samples are
spaced non-uniformly and advanced algorithms are required for digital signal processing.
2. Reconstruction of signal with timevarying bandwidth
There are several methods used for reconstruction of nonuniformly sampled band-limited signals. For correct recovery, they typically require that the maximal length of
the gaps between the sampling instants does not exceed
the Nyquist rate [5]. If the signal is non-stationary with
time-varying spectral bandwidth, satisfying globally this
requirement is not an appropriate decision, because this
provides redundant data. The use of level-crossing sampling scheme can reduce the amount of samples, because
the intervals between samples are determined by signal local properties and by the number of quantization levels.
The quality of processing can be improved if the recovery
procedure takes into account the local bandwidth of the
signal [6]. In the following subsections the proposed idea
and methods for reconstruction using filters with timevarying bandwidth and for the estimation of local maximum frequency of signal from its level-crossing samples
will be discussed.
2.1 Idea of signal-dependent reconstruction
functions
The sampling theorem states that every bandlimited signal s(t) can be reconstructed from its equidistantly spaced
samples if the sampling rate equals or exceeds the Nyquist
rate 2Fmax , where Fmax is the maximum frequency in the
signal spectrum. The reconstruction in time domain can be
expressed as
ŝ(t) =
N
−1
X
s(tn )h(t − tn ),
(1)
n=0
where sb(t) denotes reconstructed signal, N is the number
of the original signal samples s(tn ) and h(t) is an appropriate impulse response of the reconstruction filter, classi-
317
cally, sinc-function
h1 (t) = sinc(2πFmax t)
As the sampling instants tn =
response
n
2Fmax ,
(2)
then the impulse
h1 (t − tn ) = h1 (t, tn ) = sinc(2πFmax t − nπ),
(3)
where h1 (t − tn ) = h(t, tn ) is written as the function of
two arguments. The reconstructed signal becomes
ŝ(t) =
N
−1
X
s(tn )h1 (t, tn )
(4)
n=0
If the signal with time-varying frequency bandwidth
fmax (t) is considered, then the sampling rate of the signal according to Nyquist must be at least 2Fmax , where
Fmax = max(fmax (t)). In this case any information
about the local spectral bandwidth is ignored during the
sampling process. To take it into account, it is proposed
instead of h1 (t, tn ) to use more general function
h2 (t, tn ) = sinc(Φ(t) − Φ(tn )) = sinc(Φ(t) − nπ), (5)
Rt
where Φ(t) = 2π 0 fmax (t)dt is the phase of the sinusoid, whose frequency changes in time as fmax (t),
t ≥ 0 and sampling instants tn are chosen such that
Φ(tn ) = nπ. If the signal is stationary and band-limited
fmax (t) = const = Fmax , Eq. (3) and (5) become
equivalent. In case of non-constant fmax (t) waveform
of the reconstruction function h2 (t, tn ) and the desired
sampling instants tn are determined by fmax (t). Samples
are spaced non-equidistantly and the mean sampling frequency can be less than it is required by Nyquist criterion,
which, in this case, should be satisfied rather in local than
in global sense.
2.2
Reconstruction algorithm
To reconstruct the non-uniformly sampled signal according to equation (1), the reconstruction procedure involves
signal resampling to the equidistantly spaced sampling set
1
.
{tn } with sampling period ∆t = tn − tn−1 = 2Fmax
The estimation of ŝ(tn ) is possible according to the simple iterative algorithm [5] the idea of which is to interpolate the sampled
band-limited signal s(t) by the sum
P
šs(tm ) (t) = m s(tm )ψm and filter it in order to remove
high frequencies. Piecewise linear interpolation, which is
well suited to level-crossing samples, uses ψm consisting
of the triangular functions
t−tm−1
tm −tm−1 for tm−1 ≤ t < tm ,
tm+1 −t
ψm (t) = tm+1
for tm ≤ t < tm+1 , (6)
−tm
0
elsewhere.
It is proved [5] that if the maximum length of the gaps
1
, then evbetween the sampling instants τmax ≤ 2Fmax
ery s(t) can be reconstructed from the values s(tm ) of an
arbitrary τmax -dense sampling set {tm } iteratively. The
recovery algorithm can be written as:
ŝ0 (tn ) = šs(tm ) (tn );
ŝ0 (t) = C [ŝ0 (tn )] ;
ŝi (tn ) = ŝi−1 (tn ) + š(s−si−1 )(tm ) (tn );
ŝi (t) = C [ŝi (tn )] ,
SAMPTA'09
(7)
Figure 1: Piecewise polynomial p1k (t) approximation.
where i indicates the number of iteration. The linear operator C denotes filtering as the convolution of samples
s(tn ) with impulse response h1 (t, tn ) of the filter according to Eq. (4)
C [s(tn )] =
N
−1
X
s(tn )h1 (t, tn )
(8)
n=0
The sampling of non-stationary signal using levelcrossing scheme does not ensure the satisfaction of the
1
. Direct application of the
requirement τmax ≤ 2Fmax
above described algorithm leads to a considerable reconstruction error, therefore two substantial enhancements are
introduced to the algorithm - performing resampling to the
non-equidistantly spaced values and the use of filter with
impulse response h2 (t, tn ) instead of classical h1 (t, tn ).
The resampling instants tn are determined by Φ(t), which
depends on fmax (t), that in general case is not known
in advance. To solve this problem, an algorithm is developed, which estimates the time-varying instantaneous
maximum frequency using information about locations of
level-crossings.
2.3 Estimation of instantaneous maximum frequency
The obvious ways to estimate the local bandwidth of
the signal is by finding its time-frequency representation
(TFR) using, for example, short-time Fourier transform,
wavelet transform or Wigner-Ville distribution. These
methods are developed for uniformly sampled signals,
however, there are some modifications in order to find
the TFR of non-uniformly sampled signals [7]. The use
of such approach is time consuming, therefore a simpler
method is considered that is based on empirical evaluations.
To estimate the function fbmax (t) from samples s(tm ),
starting with the initial index value m = 0 two pairs of
successive level-crossing samples s(tm′j ) = s(tm′j +1 ) and
s(tm′′j ) = s(tm′′j +1 ) are found such that m′′j > m′j and the
difference m′′j − m′j is minimal. Thereafter the next two
pairs are found considering that m′j+1 = m′′j . For each
j = 1, 2, . . . the value f (tj ) is calculated as
−1
f (tj ) = tm′′j + tm′′j +1 − tm′j − tm′j +1
, (9)
where
tj =
1
tm′′j + tm′′j +1 − tm′j − tm′j +1
4
(10)
318
a)
b)
1,2
f, [Hz]
s(t), [V]
1.25
0
0,8
0,4
−1.25
0
50
100
t, [s]
150
0
0
200
50
100
t, [s]
150
200
Figure 2: (a) Test signal sampled by Φ(tn ) = nπ and (b) frequency traces of its components.
a)
b)
1.2
f, [Hz]
s(t), [V]
1.25
0
0.8
0.4
−1.25
0
50
100
t, [s]
150
200
0
0
50
100
t, [s]
150
200
Figure 3: (a) Test signal sampled by level-crossings and (b) estimated instantaneous maximum frequency fbmax (t) as solid
line, true instantaneous maximum frequency as dashed line and f (tj ) as black points.
If a single sinusoid is sampled, then f (tj ) = f (tj+1 ) for
all j and it equals the frequency of the sinusoid. If the
signal consists of more harmonics, then f (tj ) for different
PJ
j vary around the average value of f¯ = J1 j=1 f (tj ),
where J is the total number of detected pairs within the
observation time of the signal. Experiments show that f¯ is
close to the frequency of the highest component. Thus, the
estimate of function of instantaneous maximum frequency
fbmax (t) can be obtained by {f (tj )} approximation with
piecewise polynomials prk (t) of order r. By choosing the
number L > 1 the observation interval of signal is divided
into subintervals
∆Tk : t ∈ [tk,1 ; tk,2 ] ,
(11)
where k = 0, 1, . . . is the number of subinterval and
tj=kL + tj=kL+1
,
2
tj=(k+1)L + tj=(k+1)L+1
=
2
tk,1 =
tk,2
ensure
prk−1 (tk,1 )(0) = prk (tk,1 )(0) , prk (tk,2 )(0) = prk+1 (tk,2 )(0)
prk−1 (tk,1 )(1) = prk (tk,1 )(1) , prk (tk,2 )(1) = prk+1 (tk,2 )(1)
..
.
prk−1 (tk,1 )(r) = prk (tk,1 )(r) , prk (tk,2 )(r) = prk+1 (tk,2 )(r)
and the value of expression
K−1
X (k+1)L
X
2
[f (tj ) − prk (tj )] = min
is minimal. The denotation (. . .)(r) means the derivative
of order r and K is the total number of subintervals. After
solving the minimization task using the method of least
squares, the coefficients of polynomials prk (t) are obtained
and the estimate of instantaneous maximum frequency
fbmax (t) = prk (t), if tk,1 ≤ t ≤ tk,2
(12)
(13)
k=0 j=kL+1
(14)
depends on the number L of samples f (tj ) per subinterval.
To reduce the dependency the final frequency estimate is
obtained by averaging fbmax (t) calculated for different L
values. The example of piecewise polynomial of order
r = 1 approximation when L = 7 is shown in Fig. 1
3. Simulation results
For
each
subinterval
∆Tk
the
coefficients
ak,r , ak,r−1 , . . . , ak,1 , ak,0 of polynomial prk (t) =
ak,r tr + ak,r−1 tr−1 + · · · + ak,1 t + ak,0 are found to
SAMPTA'09
The methods described in previous section are applied to
reconstruct nonstationary signal from its nonuniform sam-
319
a)
b)
reconstruction error, [V]
e(t), [V]
1.25
0
−1.25
0
50
100
t, [s]
150
200
0.2
0.15
0.1
0.05
0 0
10
1
10
number of iteration i
2
10
Figure 4: (a) The difference between original and recovered signal from its 349 level-crossing samples after 10 iterations
and (b) reconstruction error (solid lines - reconstruction from level-crossings using h2 (t, tn ), dashed lines - reconstruction
from level-crossings using h1 (t, tn ), dotted line - reconstruction from samples obtained by Φ(tn ) = nπ).
ples s(tn ) obtained in two different ways. The first one is
when fmax (t) is given and sampling instants tn satisfy
Φ(tn ) = nπ (Fig. 2). The second way is by level-crossing
sampling and fmax (t) is not known in advance (Fig. 3).
In the first case 239 nonequidistantly spaced samples were
obtained during 200 seconds of the test signal, which consists of three sinusoids with constant amplitudes and timevarying frequencies as shown in Fig. 2b. As the reconstructed signal according to Eq. (4) using h2 (t, tn ) differs
insignificantly from the original one, it is not illustrated
here. In order to obtain similar result in uniform sampling case, at least 360 samples would be required since
the maximum frequency of the signal is Fmax = 0.9 Hz.
In the level-crossing sampling case 349 samples were captured using 6 quantization levels (Fig. 3a). To recover the
signal the first task was to find the values f (tj ) according
to Eq. (9) in order to estimate the instantaneous maximum
frequency (14). In Fig. 3b f (tj ) are shown as black points,
true fmax (t) as dashed line and calculated fbmax (t) as
solid line. The similarity between frequency traces is obvious. The second step was to recover the original signal
according to Eq. (7) using level-crossing samples and estimated fbmax (t). The difference signal ei (t) = s(t) − si (t)
after 10 iterations q
i = 10 is illustrated in Fig. 4a. The reRT
construction error T1 0 ei (t)2 dt reduces as the number
of iterations i increases. It is shown in Fig. 4b as a grey
solid line. The grey dashed line corresponds to reconstruction error, when instead of time-varying bandwidth
filter h2 (t, tn ) the filter with constant cut-off frequency of
Fmax = 0.9 Hz and impulse response h1 (t, tn ) is used. In
this case the achieved result is not so good as the reconstruction quality remains only in intervals, where the sampling density is sufficient. The reconstruction error can be
reduced by decreasing the distance between quantization
levels giving 437 level-crossing samples. It is shown in
Fig. 4b as black solid and dashed lines. The dotted line
corresponds to the first case when fmax (t) is given and
sampling instants tn satisfy Φ(tn ) = nπ.
4.
Conclusions
The proposed approach for non-stationary signal processing uses signal dependent techniques: level crossing sam-
SAMPTA'09
pling for data acquisition and filtering with time-varying
bandwidth for signal reconstruction. The information carried by level-crossing samples is employed in two ways –
time instants of samples are used to estimate the instantaneous maximum frequency of the signal, while the amplitude values of samples are used in reconstruction algorithm. The reconstruction procedure is based on the
use of iterative filtering with time-varying bandwidth filter. The enhancement of classical signal reconstruction
approach is made by introducing signal-dependent, ”nonstationary” impulse response and resampling to the corresponding, nonuniform sampling set.
Speech signal processing can be quoted as one of the potential application areas of the proposed algorithm. The
level-crossing sampling technique reduces the number of
samples and leads to effective signal coding approaches.
References:
[1] P. Ellis. Extension of phase plane analysis to quantized systems. IRE Transactions on Automatic Control, 4(2):43–54, 1959.
[2] M. Miskowicz. Send-on-delta concept: An eventbased data reporting strategy. Sensors, 6:49–63, 2006.
[3] E. Allier and G. Sicard. A new class of asynchronous
a/d converters based on time quantization. In Proc.
of International Symposium on Asynchronous Circuits
and Systems ASYNC’03, pages 196–205, 2003.
[4] M. Greitans. Processing of non-stationary signal using level-crossing sampling. In Proc. of the International Conference on Signal Processing and Multimedia Applications SIGMAP’06, pages 170–177, 2006.
[5] H. G. Feichtinger and K. Grochening. Theory and
practice of irregular sampling. 1994.
[6] M. Greitans and R. Shavelis. Speech sampling by
level-crossing and its reconstruction using splinebased filtering. In Proceedings of the 14th International Conference IWSSIP 2007, pages 305–308,
2007.
[7] M. Greitans. Time-frequency representation based
chirp-like signal analysis using multiple level crossings. In Proceedings of the 15th European Signal Processing Conference EUSIPCO 2007, 2007.
320
Optimal Characteristic of Optical Filter for
White-Light Interferometry based on
Sampling Theory
Hidemitsu Ogawa (1) and Akira Hirabayashi (2)
(1) Toray Engineering Co., Ltd., 1-45, Oe 1-chome, Otsu, Shiga, 520-2141, Japan.
(2) Yamaguchi University, 2-16-1, Tokiwadai, Ube City, Yamaguchi 755-8611, Japan.
hidemitsu-ogawa@kuramae.ne.jp, a-hira@yamaguchi-u.ac.jp
Abstract:
White-light interferometry is a technique of profiling surface topography of objects such as semiconductors, liquid crystal displays (LCDs), and so on. The world fastest
surface profiling algorithm utilizes a generalized sampling
theorem that reconstructs the squared-envelope function
r(z) directly from an infinite number of samples of the
interferogram f (z). In practical measurements, however, only a finite number of samples of the interferogram
g(z) = f (z) + C with a constant C are acquired by an
interferometer. We have to estimate the constant C and to
truncate the infinite series in the sampling theorem. In order to reduce both the truncation error and the estimation
error for C, we devise an optimal characteristic of the optical filter installed in the interferometer in the sense that
the second moment of the square of the interferogram is
minimized. Simulation results confirm the effectiveness
of the optimal characteristic of the optical filter.
CCD camera
Optical Filter
Beam splitter A
White-light
source
Beam splitter B
O
L (fixed)
Reference
mirror
L
E
Object
z
Stage
1.
Introduction
White-light interferometry is a technique of profiling surface topography of objects such as semiconductors, liquid crystal displays (LCDs), and so on. It is attractive because of its advantages including non-contact measurement and unlimited measurement range in principle
[1, 2, 3, 5, 6, 8, 9]. From the viewpoint of sampling theory,
white-light interferometry has the following two interesting features. First, a signal to be processed, a white-light
interferogram, f (z), is a bandpass signal. Second, a signal to be reconstructed from sampled values of f (z) is not
the interferogram itself, but its squared-envelope function
r(z). This type of sampling theorem is called a generalized sampling theorem [4, 10, 11].
The present authors also derived such a sampling theorem [9]. Based on the theorem, the world fastest surface profiling algorithm were proposed and installed in
commercial systems [5]. The sampling theorem is expressed in a form of infinite series and uses samples of
the interferogram f (z). In practical measurements, however, only a finite number of samples of the interferogram
g(z) = f (z) + C with a constant C are acquired by an
interferometer. Hence, in the algorithm, the constant C is
estimated from the samples, and the infinite series is truncated with the number of samples. If both the truncation
error and the estimation error for C were reduced, we can
SAMPTA'09
Figure 1: Basic setup of an optical system used for surface
profiling by white-light interferometry.
further improve the preciseness of the algorithm. For both
error reductions, it is very effective for interferograms to
have small side lobes. The waveform of interferograms
can be controlled by an optical filter installed in the interferometer.
Hence, in this paper, we devise an optimal characteristic
of the optical filter in the sense that the second moment
of the square of the interferogram is minimized with a
fixed band-width. We show that the optical characteristic is given by a sine curve which has a half of the period
as the fixed band-width. We also show that we have a socalled uncertainty principle between the band-width and
the second moment. Simulation results confirm the effectiveness of the optimal characteristic of the optical filter.
2. Surface Profiling by White-Light Interferometry
Figure 1 shows a basic setup of an optical system used
for surface profiling by white-light interferometry. It uses
the Michelson interferometer. A beam from a white-light
source passes through an optical filter. The beam is re-
321
orem for bandpass signals [7]. It is interesting that, since
the squared-envelope function r(z) is the sum of squares
of f (z) and its Hilbert transform, the squared-envelope
function is also reconstructed from samples of f (z), not
those of r(z). Indeed, the following result was established
[9, 5]. The center wavelength and the bandwidth of the
optical filter in Fig. 1 are denoted by λc and 2λb , respectively. Let kl and ku be angular wavenumbers defined by
2
1.5
1
kl =
0.5
5
10
15
20
25
30
z[µm]
Figure 2: An example of a white-light interferogram g(z)
and its sampled values.
flected by the beam splitter A, and divided into two portions by the beam splitter B at the point O. One of the
portions indicated by the dotted line is transmitted to a reference mirror, whose distance from the point O is L. The
other portion indicated by the dashed line is transmitted to
a surface of an object being observed. The height of the
surface from the stage at the point P is denoted by zp . E
is a virtual plane whose distance from the point O is L. z
is the distance of the plane E from the stage.
The two beams reflected by the object surface and the reference mirror are recombined and interfere. As the interferometer scans along the z-axis, the resultant beam intensity varies as is shown in Fig. 2 by the dotted line. It
is called a white-light interferogram or simply an interferogram and denoted by g(z) = f (z) + C, where C is a
constant. Its peak appears in the right side in Fig. 2 if the
height zp is high, while it appears in the left side if zp is
low. Hence, the maximum position of the interferogram
provides the height zp .
The intensity is observed by a charge-coupled device
(CCD) video camera with a shutter speed of 1/1000 second. It has, for example, 512×480 detectors. Each of
them corresponds to a point on the surface to be measured.
Since the CCD camera outputs the intensity, for example,
every 1/60 second, we can utilize only discrete sampled
values of the interferogram shown by ‘•’ in Fig. 2. We
have to estimate the maximum position of the interferogram from these sampled values.
It is known that the envelope function m(z) shown by the
solid line in Fig. 2, or its square r(z), has the same peak
as the interferogram and they are much smoother than the
interferogram. Hence, usually these functions are used for
detection of the peak instead of the interferogram. In this
paper, we use the latter r(z), which we call the squaredenvelope function.
3.
(1)
Two parameters ωl = 2kl and ωu = 2ku are also used.
0
0
2π
2π
, ku =
.
λc + λb
λc − λb
Sampling theorem for squared-envelope
functions
Since the interferogram f (z) is a bandpass signal, it can be
reconstructed from its samples by using the sampling the-
SAMPTA'09
Proposition 1 [5] (Sampling theorem for squaredenvelope functions) Let I be an integer such that
ωl
,
(2)
0≤I≤
ωu − ωl
and ωb be any real number that satisfies
ωu
≤ ωb
(I = 0),
2
ωl
ωu
≤ ωb ≤
(I 6= 0).
2(I + 1)
2I
(3)
Let ωc be a real number defined by
ωc = (2I + 1)ωb .
Let ∆ be a sampling interval given by
π
∆=
,
2ωb
(4)
(5)
and {zn }∞
n=−∞ be sample points defined by
zn = n∆.
(6)
Then, it holds that
1. When z is a sample point zj ,
{ ∞
}2
∑ f (zj+2n+1 )
4
.
r(zj ) = {f (zj )}2 + 2
π
2n + 1
n=−∞
(7)
2. When z is not any sample point,
}2
{ ∞
2 (
∑ f (z2n )
2∆
πz )
r(z) = 2
1 − cos
π
∆
z − z2n
n=−∞
(
+ 1 + cos
πz )
∆
}2
f (z2n+1 )
.
z
− z2n+1
n=−∞
{
∞
∑
(8)
To apply Proposition 1 for surface profiling, we have the
following difficulties. In the proposition, an infinite number of sampled values {f (zn )}∞
n=−∞ of the interferogram
f (z) are used. In practical applications, however, only a
N −1
finite number of sampled values {g(zn )}n=0
of the interferogram g(z) = f (z) + C are available. Hence, we have
to truncate the infinite series in Proposition 1 and approximate the sampled values f (zn ) by g(zn ) − Ĉ, where Ĉ
is an estimate of C. For example, the average of g(zn ) is
used as Ĉ. Now, we are suffered from the truncation error
as well as the estimation error for Ĉ. Both of these errors
severely affect our final goal of precise estimation of zp .
322
Theorem 1 Among second continuously differentiable
functions ψ(k) ∈ C 2 [kl , ku ] satisfying
2
ψ(k) = 0 (k ≤ kl , k ≥ ku ),
ψ(k) ≥ 0 (kl < k < ku ),
∫ ku
{ψ(k)}2 dk = 1,
1.5
(13)
(14)
(15)
kl
1
ψ(k) that minimizes the criterion J[ψ] is given by
π(k − kl )
1
.
ψ(k) = √ sin
2ka
ka
0.5
0
0
5
10
15
20
25
30
z[µm]
Figure 3: A white-light interferogram g(z) when ψ(k) is
rectangular.
4. Optimal characteristics of optical filter
To reduce both of the errors, the following observation is
crucial. As you can see in Fig. 2, only a few number of
samples are located in the main lobe of g(z) while the rest
of them are in side lobes. The latter mostly vanishes once
the constant C is estimated precisely. This implies that,
the smaller the side lobes are, the smaller the truncation
error is. Smaller side lobes also lead us to better estimations of C as shown experimentally in Section 5.
Fortunately, we can control the waveform of the interferogram by the optical filter in the interferometer. Let a(k)
be its characteristic in terms of an angular wavenumber k.
The support of a(k) is the interval kl < k < ku . Averaged attenuation rates of two beams along the dashed
and the dotted lines in Figure 1 are denoted by qo (k) and
qr (k), respectively. Let ψ(k) be
{
2{a(k)}2 qo (k)qr (k)
(k > 0),
ψ(k) =
(9)
0
(k ≤ 0).
It is also supported on the same interval as a(k):
ψ(k) = 0
(k < kl , k > ku ).
(10)
The function ψ(k) is related to the interferogram f (z) as
f (z) =
∫
ku
kl
ψ(k) cos 2k(z − zp ) dk.
(11)
Equation (11) clearly shows that we can control f (z) by
a(k) through ψ(k).
To have smaller side lobes, we can easily arrive at the following idea: we design ψ(k) so that it minimizes the second moment of the square of the interferogram f (z):
∫ ∞
J[ψ] =
(z − zp )2 {f (z)}2 dz.
(12)
−∞
Now, we are at the point to show our main result in this
paper. Let ka be (ku − kl )/2.
SAMPTA'09
The minimum value J0 is given by
( π )3 1
π∆2
J0 =
=
.
2
2
(2ka )
2
(16)
(17)
The following two results are direct consequence of Theorem 1.
Corollary 1 The optimal characteristic a(k) under the
criterion J[ψ] is given by
(
)1/2
sin π(k − kl )/2ka
√
a(k) =
.
(18)
2 ka qo (k)qr (k)
Corollary 2 The optimal waveform of the interferogram
f (z) is given by
where
f (z) = m(z) cos(ku + kl )(z − zp ),
(19)
√
4π ka cos 2ka (z − zp )
.
m(z) =
π 2 − 16ka2 (z − zp )2
(20)
The interferogram shown in Fig. 2 was the optimal one
given by Eqs. (19) and (20) while that shown in Fig. 3 is
generated from a rectangular ψ(k) given by
{ √
1/ ku − kl (kl < k < ku ),
ψ(k) =
0
(otherwise).
Though this ψ(k) is not continuously second differentiable, the conditions (13) ∼ (15) are satisfied. In both
figures, λc = 600[nm] and λb = 30[nm] were used. We
can see that the side lobes in Fig. 2 are much smaller than
those in Fig. 3. The sampling interval used in both figures
is ∆ = 1.425[µm], which is the maximum among those
satisfying Eqs. (2) ∼ (5). We have six samples in the main
lobe in Fig. 2 while only four samples are located there in
Fig. 3 (these samples are displayed by relatively large dots
compared to samples in side lobes). In a nutshell, the optimal characteristic results in fewer samples in the small
side lobes. This results in small errors on the truncation
and the estimation of C, which we demonstrate in the next
section through computer simulations.
Before proceeding simulations, let us make a final remark
in this section.
Corollary 3 Let σ 2 be the value of J[ψ]. Then, the following uncertainty principle holds:
( π )3
,
σ 2 (2ka )2 ≥
2
π
σ2
≥ .
2
∆
2
323
6. Conclusion
1.1
1.05
1
1
0.95
Rectangular
0.9
✲
0.85
0.8
0.75
0.8
0.7
0.65
0.6
10
10.5
11
11.5
12
12.5
13
13.5
0.6
0.4
Optimal
✲
0.2
0
0
5
10
15
20
25
30
z[µm]
Figure 4: Squared-envelope functions (the dashed lines)
and reconstructed functions (the solid lines) from samples
of g(z) for both of the optimal and the rectangular ψ(k).
5.
Simulations
We compare the optimal and the rectangular characteristics ψ(k) by computer simulations. We first sample the interferograms g(z) generated from both ψ(k) with the sampling interval ∆ = 1.425µm. Then, the averages for each
sample values are computed for the estimation of C. Finally, we reconstruct the squared-envelope functions r(z)
by using a finite number of g(zn ) − Ĉ instead of f (zn )
in Proposition 1. The reconstructed functions are shown
in Fig. 4 by the solid lines as well as the original squaredenvelope functions by the dashed lines for both of the optimal and the rectangular ψ(k). The small window in the
top-right side shows the magnified image around the peak.
We can see that the reconstructed function for the optimal ψ(k) provides a much better result than that for the
rectangular ψ(k). We also notice that the latter oscillates
severely.
The normalized truncation errors for the optimal and the
rectangular ψ(k) are 0.45% and 4.68%, respectively. The
former is less than 10% of the latter. When C = 1.10,
its estimation results are 1.10 and 1.06 for the optimal and
the rectangular ψ(k), respectively. Finally, errors for the
estimation of zp are 0.05µm and 0.06µm for the optimal
and the rectangular ψ(k), respectively. Even though the
difference is not so significant, the oscillation of the reconstructed squared-envelope function for the rectangular
ψ(k) may cause difficulties for fast search of the maximum position.
We repeated the same simulations for thirty two values of
zp from 10µm to 20µm. Then, averages of estimation errors were 0.0496µm and 0.0541µm for the optimal and the
rectangular, respectively. They are almost the same value.
However, the averages of truncation errors were 0.35%
and 4.67% for the optimal and the rectangular ψ(k), respectively. The former is less than 7% of the latter. These
results show the effectiveness of the optimal characteristics of the optical filter.
SAMPTA'09
In this paper, we devised an optimal characteristic of the
optical filter that minimizes the second moment of the
square of the interferogram so that both of the truncation
error and the estimation error for the constant in the interferogram are reduced. We showed that the optimal characteristic is given by a sine curve which has a half of the
period as the band-width of the optical filter. Simulation
results showed that the truncation error for the optimal
characteristic is less than 7% of that for the rectangular
one. The estimation error of the constant for the optimal characteristic was also smaller than the rectangular
one. Even though the difference on the estimation error
of the maximum position was not so significant, reconstructed functions for the optimal characteristic was much
smoother than those for the rectangular one. These results
showed the effectiveness of the optimal characteristic. Our
future tasks include to produce a prototype of the optical
filter with the optimal characteristic.
References:
[1] P.J. Caber. Interferometric profiler for rough surfaces. Applied Optics, 32(19):3438–3441, 1993.
[2] S.S.C. Chim and G.S. Kino. Three-dimensional image realization in interference microscopy. Applied
Optics, 31(14):2550–2553, 1992.
[3] P. de Groot and L. Deck. Surface profiling by
analysis of white-light interferograms in the spatial frequency domain. Journal of Modern Optics,
42(2):389–401, 1995.
[4] O.D. Grace and S.P. Pitt. Sampling and interpolation of bandlimited signals by quadrature methods.
The Journal of the Acoustical Society of America,
48(6):1311–1318, 1969.
[5] A. Hirabayashi, H. Ogawa, and K. Kitagawa. Fast
surface profiler by white-light interferometry by use
of a new algorithm based on sampling theory. Applied Optics, 41(23):4876–4883, 2002.
[6] G.S. Kino and S.S.C. Chim. Mirau correlation microscope. Applied Optics, 29(26):3775–3783, 1990.
[7] A. Kohlenberg. Exact interpolation of band-limited
functions. Journal of Applied Physics, 24:1432–
1436, 1953.
[8] K.G. Larkin. Efficient nonlinear algorithm for envelope detection in white light interferometry. Journal
of Optical Society of America, 13(4):832–843, 1996.
[9] H. Ogawa and A. Hirabayashi. Sampling theory in
white-light interferomtery. Sampling Theory in Signal and Image Processing, 1(2):87–116, 2002.
[10] D.W. Rice and K.H. Wu. Quadrature sampling
with high dynamic range. IEEE Transactions on
Aerospace and Electronic Systems, AES-18(4):736–
739, 1982.
[11] W.M. Waters and B.R. Jarrett. Bandpass signal sampling and coherent detection. IEEE Transactions on
Aerospace and Electronic Systems, AES-18(4):731–
736, 1982.
324
SAMPTA'09
Poster Sessions
SAMPTA'09
325
SAMPTA'09
326
Continuous Fast Fourier Sampling
Praveen K. Yenduri(1) and Anna C. Gilbert(2)
(1) University of Michigan, 4438 EECS building, Ann Arbor, MI 48109, USA.
(2) University of Michigan, 2074 East Hall, Ann Arbor, MI 48109, USA.
ypkumar@umich.edu, annacg@umich.edu
Abstract:
Fourier sampling algorithms exploit the spectral sparsity
of a signal to reconstruct it quickly from a small number
of samples. In these algorithms, the sampling rate is subNyquist and the time to reconstruct the dominate frequencies depends on the type of algorithm—some scale with
the number of tones found and others with the length of the
signal. The Ann Arbor Fast Fourier Transform (AAFFT)
scales with the number of desired tones. It approximates
the DFT of a spectrally sparse digital signal on a fixed
block by taking a small number of structured random samples. Unfortunately, to acquire spectral information on a
particular block of interest, the samples acquired must be
appropriately correlated for that block. In other words, the
sampling pattern, though random, depends on the block of
interest. When blocks of interest overlap significantly, the
union of the sampling patterns may not be an optimal one
(it might not be sub-Nyquist anymore). Unlike the much
slower algorithms, the sampling pattern does not accommodate an arbitrary block position. We propose a new
sampling procedure called Continuous Fast Fourier Sampling which allows us to continuously sample the signal
at a sub-Nyquist rate and then apply AAFFT on any arbitrary block. Thus, we have a highly resource-efficient
continuous Fourier sampling algorithm.
1.
Introduction
Let x be a discrete time signal of length n which is sparse
or compressible in the frequency domain but the exact frequency content depends on time. We consider the problem
of computing the frequency content present in different
blocks of the signal in a resource efficient manner. This
problem arises in many applications such as cognitive radio [2] where a wireless node alters its transmission or
reception parameters based on active monitoring of radio
frequency spectrum at various times. Another application
is incoherent demodulation of communication signals [3]
such as FSK, MSK, OOK, etc., where the computed frequency spectrum at different times represents the message
being transmitted itself.
There are several Fourier sampling algorithms [1, 8, 9]
with low sampling costs that reconstruct the entire spectrum of a sampled signal. These algorithms make use of
a uniformly random (not structured) sample set for computations thus allowing us to compute frequencies in any
SAMPTA'09
arbitrary block of interest from the signal. However, the
time to reconstruct the spectrum is superlinear in signal’s
size and hence are slow and inappropriate for the applications involving large signal sizes or bandwidths where
just a few frequencies are of interest. Instead, we consider
a sub-linear time computational method called the AAFFT
(Ann Arbor Fast Fourier Transform) described in [4].
Figure 1: Figure showing the samples acquired in S1 and
the samples required to apply AAFFT on B = [16, 47].
Let y be a fixed block of interest of length N in the discrete time signal x. Since x is sparse in frequency domain, it can be assumed that y has only m dominant
digital frequencies, where m ≪ N . The AAFFT algorithm takes a small number of (correlated) random samples from the block of interest and produces an approximation of its DFT (identifies dominant tones), using time
and storage mpoly(log(N )). If we are interested in a windowed Fourier analysis of x over windows of length N ,
a straightforward approach towards solving our problem
using AAFFT is to divide the signal x into consecutive
non-overlapping blocks of length N , generate appropriately correlated sampling patterns for each block, acquire
the samples and then apply AAFFT on each block. Let
us call this sample set S1. Unfortunately, S1 does not accommodate arbitrary block positions. For example, consider samples acquired in S1 from two consecutive blocks
B1 and B2. Lets say we are now interested in block B
which consists of second half of B1 and first half of B2
(see Figure (1)). However AAFFT cannot be applied on
B since the samples acquired from B will not be appropriately structured for its application. This is illustrated in
Figure (1) for a simple case of N = 32 with a dummy
y-axis and a few samples plotted for clarity.
We propose a new sampling procedure called the Continuous Fast Fourier Sampling that allows us to continu-
327
ously sample the signal (as opposed to division into discrete blocks) at a sub-nyquist rate and then apply AAFFT
on any arbitrary block of interest. The article describes the
algorithm in detail in Section (3.2), proves its correctness
in Section (3.3), followed by a few numerical experiments
and results in Section (3).
2.
The Fourier
(AAFFT)
Sampling
Algorithm
The Fourier Sampling algorithm is predicated upon nonevenly spaced samples unlike many traditional spectral estimation techniques [6, 7] and uses a highly nonlinear reconstruction method that is divided into two stages, frequency identification and coefficient estimation, each of
which includes multiple repetitions of basic subroutines.
A detailed description of the implementation of AAFFT is
available in [5].
Frequency Identification consists of two steps, dominant
frequency isolation and identification. Isolation is carried
out by a two-stage process: (i) pseudo random permutations of the spectrum, followed by (ii) the application of
a filter bank with K = O(m) bands, where m = number
of tones (dominant spikes) in the signal. With high probability, a significant fraction of the dominant tones fall into
individual bands, isolating each tone from the others and
this probability can be increased with additional repetitions. Note that all the above is carried out conceptually
in the frequency domain but instantiated in the time domain. That is, we sample the permuted and filtered signal
in the time domain. To carry out the computations, the
algorithm uses signal samples at time points indexed by
P (t, σ) = {(t + qσ) mod N, q = 0, 1, .., K − 1}, where
(t, σ) is randomly chosen for each repetition. The identification stage performs group testing to determine the dominant frequency value in each of the K outputs of the filterbank. This stage uses the samples indexed at arithmetic
progressions P (tb , σ) formed from each element of the geN
, b = 0, 1, .., log2 (N/2).
ometric progression tb = t+ 2b+1
The estimation stage uses the random sampling similar to
the isolation stage for coefficient estimation of each of the
dominant frequencies identified.
Note that although the (t, σ) pair is chosen randomly in
each repetition, the samples that result from each pair are
highly structured. Let A1 = {(t, σ)} used in the frequency
identification stage and similarly let A2 be defined for the
estimation stage. These two sets define a sampling pattern.
the theorems in Section (3.3) hold. For j = 1, .., J, denote
by Ij the arithmetic progression formed by (t(j), σ),
Ij = {t(j) + qσ, ∀q ≥ 0 : t(j) + qσ ≤ n}
(1)
N
Now, consider the geometric progression tb = t + 2b+1
N
for all b = 0, 1, .., α − 1. For each b, t + 2b+1 , σ is
treated as another (t, σ) pair and the sequence tb (j) and
the corresponding progressions Ijb can be defined.
Do all the above, for each pair (tℓ , σℓ ) in A1 and A2 and
denote the arithmetic progressions produced, by Iℓ,j , for
j = 1, .., Jℓ . DefineSthe union of all such arithmetic
Jℓ
Iℓ,j . Similarly define Iℓb =
progressions as Iℓ = j=0
Sα−1 b
S Jℓ b
B
b=0 Iℓ .
j=0 Iℓ,j for b = 0, .., α − 1. Now define Iℓ =
Finally define
!
!
[
[
B
I(A1 , A2 ) =
(Iℓ ∪ Iℓ ) ∪
Iℓ
(2)
A1
A2
Given a set of indices I, we denote by S x (I) the set of
samples from signal x indexed by I.
Figure 2: Calculation of N -Wraparound t(1) from t.
3.2
The CFFS Algorithm
Preprocessing:
INPUT: N // Block length
(1) Sample-set generation : Choose A1 and A2 as
defined and compute I(A1 , A2 ) (as in Equation (2)).
OUTPUT: I(A1 , A2 ) // Index set
Sample Acquisition
INPUT: I(A1 , A2 ), x
(2) sample signal x at I and obtain samples S x (I).
OUTPUT: S x (I)
Reconstruction
S x (I), (n1 , n2 ) // boundary indices of an
arbitrary block y of length N from signal x
(3) calculate A′1 , A′2 (depend on (n1 , n2 ), defined in
Section (3.3)) and extract S y (I(A′1 , A′2 )) ⊂ S x (I).
(4) apply AAFFT on the sample-set S y (I(A′1 , A′2 ))
OUTPUT:top m frequencies of x in block
y = x[n1 , n2 ]
INPUT:
3.
3.1
Continuous Fast Fourier Sampling
Sample set construction
Let n be the length of signal x which has m dominant
tones that vary over time. Let the block length be N .
Let K = O(m) and α = log2 (N ). Let (t, σ) be a
fixed pair in A1 or A2 . Define a sequence of time points
t(0) = t , t(j) = (t(j − 1) + Q(j − 1)σ)modN for
j = 1, .., J, where Q(j − 1) = smallest integer such that
t(j −1)+Q(j −1)σ ≥ N and J = ⌈ Kσ
N ⌉. We call t(j) the
“N -wraparound” of t(j − 1). Figure (2) illustrates the calculation of a N -wraparound. The choice of J is such that
SAMPTA'09
3.3
Proof of Correctness of CFFS
The arbitrary block y has boundaries (n1 , n2 ). To generate samples from this block, we define new sets A′1
and A′2 as follows. For every (t, σ) in A1 and A2 , let
328
i be the smallest integer such that t + iσ > n1 . Define t′ = (t + iσ)modn1 . Note that t′ is simply the n1 wraparound of t. Put A′1 = {(t′ , σ) : (t, σ) ∈ A1 } and
similarly A′2 . Note that A′1 and A′2 are still random since
A1 and A2 were chosen randomly. To apply AAFFT on
block y we can now use samples of y indexed by the sampling pattern defined (as in Section (2.)) from A′1 and A′2 .
The following theorems together show that the required
samples of y are available in S x (I(A1 , A2 )).
Theorem 1 For sets A′1 and A′2 as defined above,
S y (I(A′1 , A′2 )) ⊂ S x (I(A1 , A2 )).
Theorem 2 AAFFT can be applied on the sample-set
S y (I(A′1 , A′2 )), i.e. the index set I(A′1 , A′2 ) has the required structure explained in Section (2.).
Rather than giving detailed proofs, we prove a proposition
that lies at the heart of the two theorems.
Proposition 3 For every (t′ , σ)
S y (P (t′ , σ)) ⊂ S x (I(A1 , A2 )).
in
A′1
or
A′2 ,
Proof: Let (t, σ) be the pair in A1 or A2 from which
(t′ , σ) was obtained. We will prove that the arithmetic
progressions Ij formed by the sequence of wraparounds
t(j),j = 1, .., J as defined in Section (3.1), induce modN arithmetic in the progression P (t′ , σ) (P as defined in
Section (2.)). Consider the first few terms in P (t′ , σ), till
(t′ + (q0 − 1)σ) mod N where q0 is the smallest integer
such that (t′ + q0 σ) ≥ N . From definition of t′ observe
that t′ = (t+iσ −n1 ). so y(t′ ) = x(n1 +t′ ) = x(t+σ) ∈
S x (I0 ), where I0 is defined in Equation (1). Similarly it
is easy to see that the first q0 terms in S y (P (t′ , σ)) are
′
contained in S x (I0 ). Now call the next term
l (t ′+
m q0 σ)
N −t
′
′
′
− N.
mod N = t (1). Observe that t (1) = t + σ
N −t σ
Similarly observe that t(1) = t + σ σ − N . Now,
′
Substituting t′ = (t + iσ − n1 ) in the
Nexpression
for t (1)
−t+n1 −iσ
′
we get, t (1) = t + iσ − n1 + σ
−N =
σ
t + iσ − n1 + σ Nσ−t + dσ − N = t(1) + (i + d)σ −
n1 , for an appropriately defined d, which can be shown
to be positive. So y((t′ + q0 σ) mod N ) = y(t′ (1)) =
x(t(1) + (i + d)σ) ∈ S x (I1 ), where again I1 is defined
in Equation (1). Let q1 be the smallest integer such that
(t′ (1) + q1 σ) ≥ N . Now it is easy to see that the next
q1 terms in S y (P (t′ , σ)) are contained in S x (I1 ). Repeat
this until all the terms in P (t′ , σ) are covered.
Proposition 4 On average, the storage requirement of
n
CFFS algorithm is O( N
m logO(1) N ), which is of the
same order as a straightforward, fixed boundary sample
set for AAFFT.
4.
Results and Discussion
The Continuous Fast Fourier Sampling algorithm has been
implemented and tested in various settings. In particular,
we performed following three experiments.
First, we consider a model problem for communication devices which use frequency-hopping modulation schemes.
The signal we want to reconstruct has two tones that
SAMPTA'09
Figure 3: The Sparsogram for a synthetic frequencyhopping signal consisting of two tones, as computed by
AAFFT (S1) and by CFFS.
change at regular intervals. We apply both the straightforward AAFFT on S1 and CFFS to identify the location
of the tones. Figure (3) shows the obtained sparsogram
which is a time-frequency plot that displays only the dominant frequencies in the signal. We get the same sparsogram in both cases, as expected. For N = 220 , S1 samples
about 0.94% of the signal whereas CFFS samples about
1.06% of the signal, which is only very slightly larger than
S1. This experiment demonstrates the efficiency and similarity of the two methods and supports the proposition
made in Section (3.3).
Figure 4: Applying CFFS to different blocks of signal x.
While S1-AAFFT cannot be applied to compute the dominant tones in any arbitrary block, the CFFS has no such
limitation. This is demonstrated in the next experiment as
follows. Let y be a signal of length N = 220 , with m = 4
known dominant frequencies. Let x be an arbitrary signal
of length n with N ≪ n. Now let x[n1 , n2 ] be an arbitrary block of interest of length N . Set x(n1 + q) = y(q),
for q = 0, 1, . . . , N − 1. Thus we have placed a copy
of the known signal y in the block of interest. The CFFS
was then applied and the four dominant frequencies in the
block of interest were computed. The obtained values for
frequencies and their coefficients match closely with those
of the signal y and satisfy the error guarantees of AAFFT.
The whole experiment was repeated with different values
for n1 (and corresponding n2 = n1 + N − 1) and the
same results were obtained. Figure (4) shows the sketch
of a signal x, pre-sampled in a predetermined manner (according to CFFS), with copies of y placed at arbitrary positions. Application of AAFFT to any block with copy of y
gives the same results thus demonstrating the correctness
of CFFS.
329
In the final experiment, we consider the frequency hopping signal from the first experiment. Let the block size
be N = 217 with unknown block boundaries. Let f1 and
f2 be the respective frequencies in two adjacent blocks
(f 1 in the left block). We consider the problem of finding
the block boundary using CFFS with an analysis window
of size N. The center of the window can be varied and a
binary search can be performed for the block boundary
in the following manner. If the center is to the left of
the actual boundary, then the coefficient of f 1 produced
by AAFFT will be higher than that of the f 2. This indicates that the center has to be moved to the right from
its current position. Also the search is not strictly binary
since the amount by which f 1 coefficient is higher than
f 2 can be used to shift the center of the window to the
right by an equivalent amount. This step can be iterated a
few times to make the center converge to the actual block
boundary. We express the error as the distance to the true
boundary and determine what percentage of the block this
distance is. Table (1) displays the error and how the error
increases with decreasing SNR. Note that even in the case
SNR(dB)
no noise
10
8
%Error
0.39
0.58
0.70
SNR(dB)
6
4
2
%Error
0.78
0.79
1.56
Table 1: Percentage error in boundary identification.
of no noise there is some inherent ambiguity in the identification of block boundary. This uncertainty is caused
by two factors. First, when the analysis window has portions of both the f 1-block and f 2-block, the net signal is
no longer sparse due to a sudden change in frequency and
has a slowly decaying spectrum. With m = 2 the AFFT
guarantees that the error made in signal approximation is
about as much as the error in optimal 2-term approximation [5]. Hence a slowly decaying spectrum implies more
error in the approximation. A second and more important
factor is the number of samples actually acquired from the
region of uncertainty around the block boundary. From
the entire block, CFFS acquires about 8% samples from
the N = 217 present. Assuming these samples are uniformly distributed (which is not true for CFFS), the number of samples present in the region of uncertainty (0.4%)
is about 40. In practice, CFFS contains even fewer samples in the uncertainty region (about 30 on average). In
terms of samples actually acquired in CFFS, the boundary
estimation is off by only a few samples and hence is negligible, as it does not affect the computations. This will be
true for any sparse sampling method like CFFS. Furthermore, if the uncertainty were to be reduced to 0.3% say,
the boundary identification would improve by only about
6 samples on average, which again is negligible. Hence
the boundary identification through the above method is
accurate enough for all practical purposes.
5.
Conclusions and Future Work
We described and proved a sub-linear time sparse Fourier
sampling algorithm called the CFFS which along with
AAFFT can be applied to compute the frequency content
SAMPTA'09
of sparse digital signals at any point of time. Once the
block length N is selected, a sub-nyquist sampling pattern can be pre-determined and the samples can be acquired from the signal (during the runtime if required).
The AAFFT can be applied to the samples corresponding to any block of length N of the signal and the dominant frequencies in that block and their coefficients can
be computed in sublinear time. The algorithm requires the
block length N to be fixed beforehand. Designing or extending the algorithm to work for different values of N
can be considered. Adapting the algorithm to further reduce the computational complexity by using known side
information about the signal can also be considered. The
algorithm is also highly parallelizable and can be adapted
for hardware applications. Also, we may be able to extend
this sample set generation to the deterministic sampling
algorithm described in [10].
Acknowledgements
The authors have been partially supported by DARPA
ONR N66001-06-1-2011. ACG is an Alfred P. Sloan Fellow.
References:
[1] E.Candes, J.Romberg and T.Tao, Robust uncertainty principles: Exact signal reconstruction from
highly incomplete frequency information.
IEEE
Trans.Inform.Theory, 52:489–509, 2006.
[2] I.F.Akyildiz, W.Y.Lee, M.C.Vuran, and S.Mohanty,
Next generation dynamic spectrum access cognitive
radio wireless networks: A survey, Computer Networks Journal (Elsevier),50: 2127–2159, Sep 2006.
[3] Simon Haykin, Communication systems. Fourth Edition, John Wiley and Sons, 2005.
[4] A.C.Gilbert, S.Muthukrishnan, and M.J.Strauss, Improved time bounds for near-optimal sparse Fourier
representations. Proc. SPIE Wavelets XI, 59141(A):1–
15, 2005.
[5] A.C.Gilbert, M.J.Strauss, and J.A.Tropp. A Tutorial
on Fast Fourier Sampling IEEE Signal Processing
Magazine, 25(2):57–66, March 2008.
[6] G.K. Smith and D.M. Hawkins, Robust frequency estimation using elemental sets. J.Comput.Graph.Stat,
9(1):196–214, 2000.
[7] G. Harikumar and Y. Bresler, FIR perfect signal reconstruction from multiple convolutions: minimum
deconvolver orders. IEEE Trans.Signal Processing,
46(1):215–218, 1998.
[8] G. Cormode and S. Muthukrishnan. Combinatorial algorithms for compressed sensing. Proc.2006
IEEE Int.Conf.Information Sciences Systems, 230–
294, April 2006.
[9] A.C.Gilbert,
M.J.Strauss,
J.A.Tropp,
and
R.Vershynin, One sketch for all: Fast algorithms for
Compressed Sensing. Proc. 39th ACM Symposium on
Theory of Computing, 237–246, June 2007.
[10] M.A.Iwen, A deterministic sub-linear time sparse
Fourier algorithm via non-adaptive compressed sensing methods. SODA ’08, 20–29, Jan 2008.
330
Double Dirichelet Averages and Complex
B-Splines
Peter Massopust (1,2)
(1) Institute for Biomathematics and Biometry, Helmholtz Zentrum München, Neuberberg, Germany.
(2) Center of Mathematics, Technische Universität München, Garching, Germany.
massopust@ma.tum.de
Abstract:
A relation between double Dirichlet averages and multivariate complex B-splines is presented. Based on this
reationship, a formula for the computation of certain moments of multivariate complex B-splines is derived.
1.
2.
Complex B-Splines
Let n ∈ N and let △n denote the standard n-simplex in
Rn+1 :
(
△n :=
u :=(u0 , . . . , un ) ∈ Rn+1 uj ≥ 0;
j = 0, 1, . . . , n;
Introduction
n
X
uj = 1 .
j=0
Recently, a new class of B-splines with complex order
z, Re z > 1, was introduced in [4]. It was shown that
complex B-splines generate a multiresolution analysis of
L2 (R). Unlike the classical cardinal B-splines, complex
B-splines Bz possess an additional modulation and phase
factor in the frequency domain:
)
The extension of △n to infinite dimensions is done via
projective limits. The resulting infinite-dimensional standard simplex is given by
∞
X
N0
△∞ := u := (uj )j ∈ (R+
uj = 1 ,
0)
j=0
bz (ω) = B
bRe z (ω) ei Im z ln |Ω(ω)| e− Im z arg Ω(ω) ,
B
where Ω(ω) := (1 − e−iω )/(iω). The existence of these
two factors allows the extraction of additional information
from sampled data and the manipulation of images.
In [6] and [9], some further properties of complex Bsplines were investigated. In particular, connections between complex derivatives of Riemann-Liouville or Weyl
type and Dirichlet averages were exhibited. Whereas in
[6] the emphasis was on univariate complex B-splines
and their applications to statistical processes, multivariate complex B-splines were defined in [9] using a wellknown geometric formula for classical multivariate Bsplines [7, 10]. It was also shown that Dirichlet averages are especially well-suited to explore the properties of
multivariate complex B-splines. Using Dirichlet averages,
several classical multivariate B-spline identities were generalized to the complex setting. There also exist interesting relationships between complex B-splines, Dirichlet averages and difference operators, several of which are
highlighted in [5].
This short paper presents a generalization of some results found in [3, 12] to complex B-splines. For this purpose, the concept of double Dirichlet average [1] was introduced and its definition extended via projective limits
to an infinite-dimensional setting suitable for complex Bsplines. Moments of complex B-splines are defined and a
formula for their computation in terms of a special double
Dirichlet average presented.
SAMPTA'09
and endowed with the topology of pointwise convergence,
i.e., the weak∗-topology. We denote by µb = lim µnb
←−
the projective limit of Dirichlet measures µnb on the ndimensional standard simplex △n with density
Γ(b0 ) · · · Γ(bn ) b0 −1 b1 −1
u1
· · · unbn −1 .
u
Γ(b0 + · · · + bn ) 0
(1)
Here, Γ : C\Z−
0 → C denotes the Euler Gamma function.
Let R+ := {x ∈ R | x > 0} and let C+ := {z ∈
C | Re z > 0}.
0
Definition 1 ([6]). Given a weight vector b ∈ CN
+ and
N0
an increasing knot sequence τ := {τk }k ∈ R with the
√
property that limk→∞ k τk ≤ ̺, for some ̺ ∈ [0, e), a
complex B-spline Bz (• | b; τ ) of order z, Re z > 1, with
weight vector b and knot sequence τ is a function satisfying
Z
Z
Bz (t | b; τ )g (z) (t) dt =
g (z) (τ · u) dµb (u) (2)
R
△∞
for all g ∈ S (R).
Here, S (R) denotes
P the space of Schwartz functions on
R, and τ · u = k∈N0 τk uk for u = {uk }k∈N0 ∈ △∞ .
In addition, we used the Weyl or Riemann-Liouville fractional derivative [8, 11, 13] of complex order z, Re z > 0,
W z : S (R) → S (R), defined by
Z ∞
(−1)n dn
z
(W f )(x) :=
(t − x)ν−1 f (t) dt,
Γ(ν) dxn x
331
with n = ⌈Re z⌉, and ν = n − z. Here ⌈ · ⌉ : R → Z,
x 7→ min{n ∈ Z | n ≥ x}, denotes the ceiling function.
To simplify notation, we write f (z) for W z f
It is easy to show that the univariate complex B-spline
Bz (t | b; τ ) is an element of L2 (R) [5].
Remark 2. For finite τ = τ (n) and b = b(n) and z :=
n ∈ N, (2) defines also Dirichlet splines if g is chosen in
C n (R). For, Dirichlet splines Dn ( · | b; τ ) of order n are
defined as those functions for which
Z
Z
g (n) (t)Dn (t| b; τ ) dt =
g (n) (τ · u) dµb (u),
∆n
R
holds true for τ ∈ Rn+1 and for all g ∈ C n (R), and thus
for g ∈ S (R).
To define a multivariate analogue of the univariate complex B-splines, we proceed as follows. Let λ ∈ Rs \ {0}
be a direction, and let g : R → C be a function. The ridge
function corresponding to g is defined as gλ : Rs → C,
gλ (x) = g(hλ, xi) for all x ∈ Rs .
Definition 3 ([9]). Let τ = {τ n }n∈N0 ∈ (Rs )N0 be a
sequence of knots in Rs with the property that
(3)
n→∞
The multivariate complex B-spline B z (• | b, τ ) : Rs → C
0
of order z, Re z > 1, with weight vector b ∈ CN
+ and knot
sequence τ is defined by means of the identity
Z
Z
g(t)Bz (t | b, λτ ) dt,
g(hλ, xi)B z (x | b, τ ) dx =
Rs
R
(4)
where g ∈ S (R), and where λ ∈ Rs \ {0} such that
λτ := {hλ, τ n i}n∈N0 is separated.
As consequence of the fact that Bz (• | b; τ ) ∈ L2 (R),
one obtains from the above definition that B z (• | b, τ ) ∈
L2 (Rs ) [5]. Moreover, it follows from the HermiteGenocchi formula for the univariate complex B-splines
Bz ( • | b, λτ ) and (4), that B z ( x | b, τ ) = 0, when x ∈
/
[τ ], the convex hull of τ .
3.
Dirichlet Averages
Let Ω to be a nonempty open convex set in Cs , s ∈ N, and
0
let b ∈ CN
+ . Let f ∈ S (Ω) := S (Ω, C) be a measurable
function. For τ ∈ ΩN0 ⊂ (Cs )N0 and uP∈ △∞ , define
∞
τ · u to be the bilinear mapping (τ, u) 7→ i=1 ui τ i . The
infinite sum exists if there exists a ̺ ∈ [0, e) so that
p
lim sup n kτ n k ≤ ̺.
(5)
n→∞
Here, k · k now denotes the canonical Euclidean norm on
Cs . (See also [6].)
SAMPTA'09
△∞
where µb = lim µnb is the projective limit of Dirichlet
←−
measures on the n-dimensional standard simplex △n .
We remark that the Dirichlet average is holomorphic in
b ∈ (C+ )N0 when f ∈ C(Ω, C) for every fixed τ ∈ ΩN0 .
(See [2] for the finite-dimensional case and [9] for the
infinite-dimensional setting.)
Definition 5. [1] Let f : Ω ⊂ C → C be continuous.
Let b ∈ Ck+1
and β ∈ Cκ+1
+
+ . Suppose that for fixed
k, κ ∈ N, X ∈ C(k+1)×(κ+1) and that the convex hull
[X] of X is contained in Ω. Then the double Dirichlet
average of f is defined by
Z Z
F (b; X; β) :=
f (u · Xv)dµkb (u)dνβκ (v),
△k
We denote the canonical inner product in Rs by h•, •i and
the norm induced by it by k • k.
p
∃ ̺ ∈ [0, e) : lim sup n kτ n k ≤ ̺.
Definition 4. Let f : Ω ⊂ Cs → C be a measurable
N0
0
→ C over
function. The Dirichlet average F : CN
+ ×Ω
∞
△ is defined by
Z
f (τ · u) dµb (u),
F (b; τ ) :=
where u · Xv :=
Pk
i=0
△κ
Pκ
j=0
ui Xij vj .
Note that F (b; X; β) is holomorphic on Ω in the elements
of b, β, and X.
We again use projective limits to extend the notion of
double Dirichlet average to an infinite-dimenional setting.
To this end, let u, v ∈ △∞ and let µb = lim µnb and
←−
νβ = lim νβn be the projective limits of Dirichlet mea←
n−
n
sures µb and νβ of the form (1) on the n-dimensional
0
standard simplex, where b, β ∈ CN
+ . Now suppose that
N0 ×N0
is a infinite matrix with the property that
X ∈ C
P
∞ P∞
|X
ij | converges. Let
i=0
j=0
u · Xv :=
∞ X
∞
X
ui Xij vj .
i=0 j=0
Suppose that Ω ⊂ C contains the convex hull [X] of X
and that f : Ω → C is continuous. The double Dirichlet
average of f over △∞ is then given by
Z
Z
F (b; X; β) :=
f (u · Xv)dµb (u)dνβ (v). (6)
△∞
△∞
(We use the same symbol for the (double) Dirichlet average over △∞ and its finite-dimensional projections △n .)
It is easy to show that
Z
(7)
F (b; X; β) =
F (β; uX)dµb (u),
△∞
where uX := {hu, Xj i}j∈N0 , with Xj denoting the jcolumn of X.
We note that F (b; X; β) is holomorphic in the elements
of b, β, and X over △∞ .
For z ∈ C+ , we define
Z
Z
F (z) (b; X; β) :=
f (z) (u · Xv)dµb (u)dνβ (v).
△∞
△∞
(See also [9] for the case of a single Dirichlet average.)
332
4.
Double Dirichlet Averages and Complex
B-Splines
Assume now that the matrix X is real-valued and of the
form Xij = 0, for i ≥ s and all j ∈ N0 , some s ∈ N. In
other words, X ∈ Rs×N0 .
Theorem 6. Suppose that β ∈ R∞
+ and that Re z > 1.
Ps−1
Let b := (b0 , b1 , . . . , bs−1 ) ∈ Rs be such that i=0 bi ∈
/
−N0 . Assume that f ∈ S (R+ ). Further assume that uX
is separated for all u ∈ △s−1 . Then
Z
F (z) (b; X; β) =
B z (x | β, X) F (z) (b; x)dx.
Rs
Proof. We prove the formula first for b ∈ Rs+ . To this end,
we identify u = (u0 , u1 , . . . , us−1 , 0, 0, . . .) ∈ △∞ with
(u0 , u1 , . . . , us−1 ) ∈ △s−1 . By the Hermite-Genocchi
formula for complex B-splines (see [6] and to some extend
[9]), we have that
Z
F (z) (β; uX) =
f (z) (u′ · uX) dµβ (u′ )
∞
△
Z
=
f (z) (t)Bz (t | β, uX)dt
R
Substituting this expression into (7) and using (4) gives
F (z) (b;X; β) =
Z
Z
△∞
Rs
f (z) (hu, xi)B z (x | β, uX) dx dµb (u).
Interchanging the order of integration yields the statement
for b ∈ Rs+ . To obtain the general case b ∈ Rs , we
note that by Theorem 6.3-7 in [2], the Dirichlet average
F can beP
holomorphically continued in the b-parameters
s−1
provided i=0 bi ∈
/ −N0 .
Remark 7. Theorem 6 extends Theorem 6.1 in [12] to
complex B-splines and the △∞ -setting.
5.
(See, [2], (6.6-5).)
Now, let p = (p0 , p1 , . . . , ps−1 ) ∈ Rs , s ∈ N, be a multiindex all of whose components satisfy pi < − 21 . The moPs
(z)
ment M|p| (b; X) of order |p| := i=1 pi of the complex
B-spline B z (• | β, X) is defined by
Z
(z)
xp B z (x | β, X) dx.
M|p| (b; X) :=
Rs
Note that since B z (• | β, X) ∈ L2 (Rs ) and B z (• |
β, X) = 0, for x ∈
/ [X], an easy application of the
Cauchy-Schwartz inequality shows that the above integral exists provided the multi-index p satisfies the aforementioned condition on its components.
Using a result from [8], namely Property 2.5 (b), and
requiring that Re z < Re c, we substitute the function
−(c−z)
into (8) to obtain
f := Γ(c−z)
Γ(c) (•)
(z)
R−(c−z) (b; x) = R−c (b; x) =
s−1
Y
xbi i .
i=0
The above considerations together with Theorem 6 immediately yield the next result.
Corollary 8. Suppose that β ∈ R∞
+ and that Re z > 1.
Let b := (b0 , b1 , . . . , bs−1 ) ∈ (−∞, − 12 )s be such that
Ps−1
c :=
/ −N0 . Moreover, suppose that Re z <
i=0 bi ∈
Re c. Then
(z)
(z)
M−c (b; X) = R−(c−z) (b; X; β).
6.
(10)
Acknowledgements
This work was partially supported by the grant MEXTCT-2004-013477, Acronym MAMEBIA, of the European
Commission.
References:
Moments of Complex B-Splines
Following [2], we define the R-hypergeometric function
Ra (b; τ ) : Rs+ × Ωs → C by
Z
(u),
(8)
Ra (b; τ ) :=
(τ · u)a dµs−1
b
△s−1
where Ω := H, H a half-plane in C \ {0}, if a ∈ C \ N,
and Ω := C, if a ∈ N. It can be shown (see [2]) that R−a ,
a ∈ C+ , has a holomorphic continuation in τ to C0 , where
C0 := {ζ ∈ C | − π < arg ζ < π}.
Taking in the definition of the double Dirichlet average
(6)
for f the real-valued function t 7→ t−c , where c :=
Ps−1
i=0 bi , the resulting double Dirichlet average is denoted
by R−c (b; X; β) and generalizes power functions. The
corresponding single Dirichlet average R−c (b; x), where
x = (x0 , . . . , xs−1 ), is given by
R−c (b; x) =
s−1
Y
i=0
SAMPTA'09
i
x−b
,
i
x∈
/ [X].
(9)
[1] B. C. Carlson. Appell functions and multiple averages. SIAM J. Math. Anal., 2(3):420–430, August
1971.
[2] B. C. Carlson. Special Functions of Applied Mathematics. Academic Press, New York, 1977.
[3] B. C. Carlson. B-splines, hypergeometric functions,
and Dirichlet averages. J. Approx. Th., 67:311–325,
1991.
[4] B. Forster, T. Blu, and M. Unser. Complex B-splines.
Appl. Comp. Harmon. Anal., 20:261–282, 2006.
[5] B. Forster and P. Massopust. Multivariate complex
B-splines, Dirichlet averages and difference operators. accepted SAMPTA 2009, 2009.
[6] B. Forster and P. Massopust. Statistical encounters
with complex B-splines. Constr. Approx., 29(3):325–
344, 2009.
[7] S. Karlin, C. A. Micchelli, and Y. Rinott. Multivariate splines: A probabilistic perspective. Journal of
Multivariate Analysis, 20:69–90, 1986.
333
[8] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo.
Theory and Applications of Fractional Differential
Equations. Elsevier B. V., Amsterdam, The Netherlands, 2006.
[9] P. Massopust and B. Forster. Multivariate complex
B-splines and Dirichlet averages. to appear in J. Approx. Th., 2009.
[10] C. A. Micchelli. A constructive approach to Kergin
interpolation in Rk : Multivariate B-splines and Lagrange interpolation. Rocky Mt. J. Math., 10(3):485–
497, 1980.
[11] K. S. Miller and B. Ross. An Introduction to
the Fractional Calculus and Fractional Differential
Equations. Wiley, New York, 1993.
[12] E. Neuman and P. J. Van Fleet. Moments of Dirichlet splines and their applications to hypergeometric
functions. J. Comput. and Appl. Math., 53:225–241,
1994.
[13] S. G. Samko, A. A. Kilbas, and O. I. Marichev. Fractional Integrals and Derivatives. Gordon and Breach
Science Publishers, Minsk, Belarus, 1987.
SAMPTA'09
334
Sampling in cylindrical 2D PET
Yannick Grondin(1,2) , Laurent Desbat(1) and Michel Desvignes(2)
(1) TIMC-IMAG, UMR CNRS 5525, UJF-Grenoble 1 (GU) In3 S, Faculté de Médecine, 38706 La Tronche France
(2) Grenoble-INP/Phelma/ GIPSA-LAB
961 Rue de la houille blanche BP 46 St Martin d’Heres France
Yannick.Grondin@imag.fr, Laurent.Desbat@imag.fr, michel.desvignes@gipsa-lab.inpg.fr
Abstract:
In this paper, we study 2D cylindrical Positron Emission
Tomography (2D PET) sampling. We show that rectangular sampling schemes are more efficient than usual square
schemes.
1. PET and sampling
1.1 PET
The aim of Positron Emission Tomography (PET) is to
map the internal nuclear activity of a patient from exterior measurement. Usually, the patient received some nuclear substance by inhalation or injection. In PET this
substance is tagged with a radioactive isotope, such as
Carbon-11, Fluorine-18, Oxygen-15. This substance has
also chemical and biological properties that enable to visualize metabolism and functions of patient organs (such
as blood flow). This substance, called radiotracer, emits a
positron per decay. The positron annihilates with an electron, which results in the emission of two opposite gamma
rays detected in a PET system. Thanks to detectors surrounding the patient and a powerful electronic processing,
coincident photon pairs can be sorted, meaning that the
emission occurred on the line joining both detectors.
that some activity occurs on the line joining the detectors
(ψ1 , z1 ) and (ψ2 , z2 ). This line is called a LOR (Line Of
Response).
In 2D mode, lead rings called septa, see Fig. 2, are used
to restrict detected LORs to be essentially perpendicular
to the PET cylinder axis. In this case, LORs have only
three parameters (ψ1 , ψ2 , z), see Fig. 3. LORs with a
small oblicity (crossed LORs) are usually approximated to
LORs perpendicular to the axis, between two true detectors rings, creating a virtual detection ring, allowing to improve the sampling rate along the axis direction, see Fig. 2.
Detector rings
Interpolated
LOR
Crossed
LORs
z
Septum
Figure 2: Crossed LORs interpolated to improve axial
sampling .
LOR(ψ1, ψ2, z)
LOR(ψ1, z1, ψ2, z2)
ψ2
r
ψ1
ψ1
ρ
f
support
z
ψ2
r
f
support
z
ρ
z2
Detectors ring
z
z1
Detectors ring
Transverse plane
Figure 1: Parametrization of a LOR with the variables
(ψ1 , z1 , ψ2 , z2 ).
Figure 3: Parametrization of a LOR with the variables
(ψ1 , ψ2 , z).
In a cylindrical PET system of radius r, see Fig. 1, the
unitary detectors are distributed on a cylinder surrounding
the patient (supposed to lie in a cylinder of radius ρ). Each
gamma ray detector localization can be parametrized by
cylindrical coordinates (ψ, z). When the coincidence on
two detectors (ψ1 , z1 ) and (ψ2 , z2 ) is detected, one knows
In 2D PET, after the attenuation correction [5] the measure
can be modeled by g : [0, 2π] × [0, 2π] × R → R, with
Z
f (u (ψ1 , z) + tθ (ψ1 , ψ2 )) dt
g (ψ1 , ψ2 , z) =
SAMPTA'09
R
t
with u (ψ1 , z) = (r cos ψ1 , r sin ψ1 , z) and θ (ψ1 , ψ2 ) =
335
t
1
(cos ψ2 − cos ψ1 , sin ψ2 − sin ψ1 , 0) . Ob)|
viously g satisfies the symmetry relation
2|sin(
ψ1 −ψ2
2
g(ψ1 , ψ2 , z) = g(ψ2 , ψ1 , z).
(1)
1.2 Sampling
2.
3D Sampling in cylindrical PET 2D mode
In [1] we have established the sampling conditions of the
3D Fan-Beam X-ray Transform (3DFBXRT):
Z
f (u)du,
De3 ⊥ f (β, α, z) =
Lβ,α,t
We want to sample a function g being 2π-periodic in its
two first variables and in R in its third variable. This is
a particular case of the general framework of sampling of
function on groups, see for example [2, 3]. In this case, the
Fourier transform of g ∈ C0∞ ([0; 2π[×[0; 2π[×R) can be
defined by:
Z
Z
Z
1
√
ĝ(ξ) =
g(x)e−ix·ξ dx,
(2π)2 2π [0;2π[ [0;2π[ R
where u ∈ R3 , Lβ,α,z is the line in the plane perpendicular to e3 at abscissa z (z ∈ R), joining the source
at r(cos β, sin β, 0)t + ze3 , β ∈ [0, 2π[ and the detector
at angular position α ∈ [−π/2, π/2[, see Fig. 4. This geometry appears in X-ray CT scanner when considering the
reconstruction of many 2D slices. Cylindrical PET in 2D
mode can be linked with the 3DFBXRT in the following
way:
g(x) = D3D f (A(x − eπ ))
(2)
where x = (ψ1 , ψ2 , z)t ∈ [0; 2π[×[0; 2π[×R, ξ =
where x = (ψ1 , ψ2 , z)t , eπ = (0,
(p1 , p2 , ζ)t ∈ Z × Z × R and ξ · x = p1 ψ1 + p2 ψ2 + ζz.
The inverse Fourier transform defined for G a function on
1
0
Z × Z × R is given by
A = − 12 21
Z
0
0
1
G(ξ)eix·ξ
Ǧ(x) = √
2π Z×Z×R
see Fig 4.
Z
1 X X
i(p1 ψ1 +p2 ψ2 +ζz)
=√
G(p1 , p2 , ζ)e
dζ.
p2
2π p ∈Z p ∈Z ζ∈R
1
X
1
(SW g)(x) = √ | det W |
g(y)χ̌K (x − y),
2π
y∈LW
where χK is the indicator function of the set K. The interpolation error is given by
Z
2
||SW g − g||∞ ≤ √
|ĝ(ξ)|dξ.
2π ξ6∈K
R
Thus if K is the essential support of ĝ, i.e., ξ6∈K |ĝ(ξ)|dξ
can be negligible, then the interpolation error is low. The
geometry of the set K can be exploited for the design of
efficient sampling schemes, i.e., the choice of W satisfying the Shannon condition with | det W | maximal in order
to minimize the number of sampling points.
Source
LOR(ψ1 , ψ2 , z)
Detector 1
Ω
2 (r
β
ρ
Detector 2
ψ1
ψ2
ρ
f
Ω
+ ρ)
Ωr
√
2
2 Ωr
v
p1
v
− Ω2 (r + ρ)
−Ωr
Figure 5: Kg : essential support of ĝ for η = ρ/r = 2/3,
slices in the planes (p1 , p2 ) (left) and (v, ζ) (right). The
3D set Kg is just at the intersection of two cylinders of
respective basis the slices in the (p1 , p2 ) and (v, ζ) and
respective axis ζ and the direction perpendicular to (v, ζ)
.
This link allows to easily estimate the essential support of
gb : Z × Z × R → R. Indeed,
Z
Z
Z
g(x)e−ix·ξ dx
gb(ξ) =
[0;2π[ [0;2π[ R
Z
Z
Z
D3D f (A(x − eπ ))e−ix·ξ dx
=
[0;2π[ [0;2π[ R
Z
Z
Z
=
D3D f (Ax)e−ix·ξ+ip2 π dx
=
r
(b)
Figure 4: Fan beam (a) and natural PET (b) parametrization in a transverse plane .
Ω
2 (r
− Ω2 (r − ρ)
=
(a)
− ρ)
[0;2π[
α
Detector
r
ζ
2
Let K ⊂ Z × Z × R, the non-overlapping Shannon condition associated to K for the sampling lattice LW =
W Z3 ∩ ([0; 2π[×[0; 2π[×R) generated by the non singular 3 × 3 matrix W is that the sets K + 2πW −t l, l ∈ Z3
are disjoint sets in Z × Z × R. The Petersen-Middleton
theorem [6, 3] yields the Fourier interpolation formula
f
π, 0)t , and
0
0
1
=
(−1)p2
| det A|
(−1)p2
| det A|
[0;2π[
Z
Z
R
[0;2π[
[0;2π[
Z
Z
Z
Z
[0;2π[
[0;2π[
D3D f (x)e−i(A
−1
x)·ξ
D3D f (x)e−ix·(A
−t
ξ)
R
(−1)p2 \ −t
D3D f (A (ξ))
| det A|
From this link we see that the essential support of gb is
simply a linear transformation of the essential support of
SAMPTA'09
dx
R
336
dx
\
D
3D f . From [1] it can be easily shown that Kg , the essential support of ĝ(p1 , p2 , ζ) when the emission function
f is supposed the be essentially Ω band limited, is given
by
f = χB(c,0.03) + χB(c,0.05) + χB(c,0.07) where χB(c,r) is
the indicator function of the ball of radius r centered on
c = (0.9, 0, 0). The data are simulated for a PET of radius
1.5 with 32 rings and 300 detectors on each ring. (b) and
(d) are based on a Monte Carlo (MC) simulation computed
Kg = {(p1 , p2 , ζ) ∈ Z × Z × R,
with GATE [4]. The phantom f is built with 5 concen2
2 2
2 2
|p1 − p2 | + r ζ < Ω r ; r|p1 + p2 | < ρ|p1 − p2 |} tric weighted ball sources (of radius r expressed in mm):
f = a(χB(c,9) +χB(c,10) +χB(c,11) +χB(c,12) +χB(c,13) ),
where the center c = (130, 0, 0) mm and the activity
see Fig. 5 for a representation.
a = 106 becquerel. The data are simulated for a PET
The angles ψ1 and ψ2 parametrize the same detector ring,
of radius 402 mm with 32 rings and 576 detectors on
thus their sampling must be identical. We consider here
each ring, imitating the ECAT EXACT HR+ scanner of
only standard sampling, i.e. equidistant sampling along
CTI/Siemens. We see that the simulation data are in good
each direction. The most efficient diagonal matrix satisfyagreement with the theoretical results.
ing the non overlapping Shannon conditions, see Fig. 6, is
given by:
r 0 0
1 0 0
2π
0 1 0
2πWS−t = Ω 0 r 0 , WS =
rΩ
0 0 2
0 0 2r
(3)
50
100
150
200
250
300
350
400
450
500
550
100
√
- 2rΩ
300
(a)
(b)
(c)
(d)
400
500
ζ
-
p2
200
√
2
2 rΩ
2Ω
0
√
2
2 rΩ
v
rΩ
rΩ
− Ωr
6
p1
Kg
− 5Ωr
6
copies of Kg
v
Figure 6: Non overlapping conditions for the rectangular
sampling scheme .
Thus we see that the most efficient sampling distances are
∆ψ1 = 2π/rΩ(= ∆ψ2 ) and ∆z = π/Ω. lz = ∆z would
thus be the detector axial length. If we approximate the
detector tangential length by lt = r∆ψ1 , we see that the
most efficient relation is lz = lt /2, thus the most efficient
detectors from the sampling point of view are rectangular
detectors. The empirical ring oversampling by rebinning
the crossed LORs as in Fig. 2 yields exactly the factor 2 of
oversampling in the direction z needed for efficient sampling. This is a theoretical justification of this widely used
heuristic rebinning method.
Figure 7: In (a) and (c) the emission function f is the sum
of 3 concentric indicator functions. In (b) and (c) the data
are obtained by a MC simulation of 5 concentric spherical
sources. (a) and (b) slice ζ = 0 of |ĝ(p1 , p2 , ζ)| ; (c) and
(d) 3D visualization of the isosurface at 1% of maximum
of |ĝ(p1 , p2 , ζ)| (|ĝ(p1 , p2 , ζ)| is essentially negligible outside of this surface) .
3.2
Reconstruction resolution
In Fig. 8, Fig. 9 and 10, we present the reconstruction
of the clock phantom, see [8], from simple line integrals.
The simulated cylindrical PET is of radius r = 1.5, the
reconstruction region is of radius ρ = 1. We consider
two sampling schemes with essentially the same number
of data. The square scheme is based on square detectors,
with lt = lz = 0.049. The number of ring is 20. The number of detectors on a ring is 190. The rectangular scheme
is based on rectangular detectors, with lt = 2lz = 0.062.
The number of ring is 32. The number of detectors on a
ring is 150. We see in these numerical experiments that
the rectangular scheme yields better reconstructions than
the square scheme.
3. Numerical experiments
3.1 Essential support
We have computed from numerical phantom the essential
support of |ĝ(p1 , p2 , ζ)| see Fig. 7. In (a) and (c) the simulation is based on simple line integrals of a phantom f
built with 3 concentric weighted ball indicator functions:
SAMPTA'09
4.
Conclusion
We have shown the efficiency of the rectangular sampling
scheme over the square scheme in 2D mode cylindrical
PET. Sampling conditions in fully 3D PET as initiated
in [7] are now being investigated.
337
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Figure 8: A = Original image: transverse view ; B = Image profile ; C = Original image: axial view ; D = Image
profile 1 ; E = Image profile 2 .
0
0
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Figure 10: A = Square scheme image: axial view ; B
= Rectangular scheme image: axial view ; C = Square
scheme image profile 1 ; D = Rectangular scheme image profile 1 ; E = Square scheme image profile 2 ; F =
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References:
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profile .
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[1] L. Desbat, S. Roux, P. Grangeat, and A. Koenig. Sampling conditions of 3D Parallel and Fan-Beam X-ray
CT with application to helical tomography. Phys.
Med. Biol., 49(11):2377–2390, 2004.
[2] A. Faridani. An application of a multidimensional
sampling theorem to computed tomography. In AMSIMS-SIAM Conference on Integral Geometry and Tomography, volume 113, pages 65–80. Contemporary
Mathematics, 1990.
[3] A. Faridani. A generalized sampling theorem for
locally compact abelian groups.
Math. Comp.,
63(207):307–327, 1994.
[4] S. Jan and coll. Gate: a simulation toolkit for pet and
spect. Phys. Med. Biol, 49:4543–4561, 2004.
[5] F. Natterer. The Mathematics of Computerized Tomography. Wiley, 1986.
[6] D.P. Petersen and D. Middleton. Sampling and reconstruction of wavenumber-limited functions in Ndimensional euclidean space. Inf. Control, 5:279–323,
1962.
[7] T. Rodet, J. Nuyts, M. Defrise, and C. Michel. A study
of data sampling in pet using planar detectors. In IEEE
Nuclear Science Symp. Conf. Rec, 2003.
[8] Henrik Turbell. Cone-Beam Reconstruction Using
Filtered Backprojection. PhD thesis, Linkping University, 2001.
338
Significant Reduction of Gibbs’ Overshoot
with Generalized Sampling Method
Yufang Hao(1) , Achim Kempf (1),(2)
(1) Department of Applied Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
(2) Department of Physics, University of Queensland, St Lucia 4072, QLD, Australia
yhao@math.uwaterloo.ca
Abstract:
As is well-known, the use of Shannon sampling to interpolate functions with discontinuous jump points leads to
the Gibbs’ overshoot. In image processing, it can lead to
the problem of artifacts close to edges, known as Gibbs
ringring. Its amplitude cannot be reduced by increasing the sample density. Here we consider a generalized
Shannon sampling method which allows the use of timevarying sample densities so that samples can be taken at a
varying rate adapted to the behavior of the function. Using this generalized sampling method to approximate a
periodic step function, we observe a strong reduction of
Gibbs’ overshoot. In a concrete example, the amplitude of
the Gibbs’ overshoot is reduced by about 70%.
Figure 1: Approximation of the step function by Shannon sam-
1.
Introduction
The Shannon sampling theorem [6] provides the link between continuous and discrete representations of information and has numerous practical uses in communication
engineering and signal processing. For a review on Shannon sampling, see [7, 10, 1]. In addition, the Shannon
sampling theorem has been used to interpolate samples to
approximate a given function.
In the use of Shannon sampling to approximate functions
with discontinuous jump points, the well-known Gibbs’
overshoot [2, 3] has remained a persistent problem, leading to, e.g., Gibbs ringing in image compression [5]. The
clearest example for the Gibb’s phenomenon is the periodic step function H(t), see Figure 1, where H(t) = 1 on
(0, 12 ), H(t) = −1 on ( 12 , 1), H(t) = 0 at t = 0, 12 , 1, and
H(t) has a period T = 1.
In Figure 1, H(t) is approximated using Shannon’s shifted
sinc reconstruction kernel with N = 24 sampling points
on one periodic interval [0, 1). Samples are denoted by
x in the plot, and the solid line at the top indicates the
maximum value of the approximating function, which is
1.0640. Within an error of 0.003, the 6.40% overshoot beyond the maximum amplitude 1 of the step function H(t)
can not be further reduced even if we increase the sampling density.
However, using the generalized sampling method [4, 8, 9],
which allows the reconstruction of a function on a set
of non-equidistant sampling points, chosen adaptively according to the behavior of the function, we show that the
SAMPTA'09
pling.
Gibbs overshoot can be strongly reduced. For an example,
see Figure 2.
In Figure 2, we use the same number of points N = 24
in one period as in the case of Shannon in Figure 1, but
we choose the sampling points to match the behaviour of
the step function. Intuitively, the jump in the step function
contains high frequencies. Thus more samples are taken
near the jump points t = 0, 12 , and 1. In this example, the
maximum value of the approximation is reduced to 1.0074
with an error of 0.0003. This is roughly a 70% reduction of Gibbs’ overshoot without increasing the number of
samples, but only varying the local sample density.
Figure 3 is a zoom-in of Figure 2 near the jump point.
The dashed line on the top indicates the maximum values
of the approximating function using the generalized sampling, while the solid line indicates the overshoot in the
case of Shannon.
2.
Generalized Shannon Sampling Method
The generalized Shannon sampling theory considered here
was not specifically developed for the application of reducing Gibbs’ phenomena. It was originally motivated
by some fundamental physics problem in quantum gravity
[4] and was introduced to engineering for spaces of functions with a new notion of time-varying Nyquist rate [8, 9].
The starting observation is that each set of Nyquist sampling points in Shannon sampling turns to be the eigen-
339
2.1
1
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0
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−0.4
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−0.8
−1
0
0.2
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0.8
1
Figure 2: Approximating the step function by the generalized
sampling method with non-equidistant sampling points..
1.1
1.05
1
0.95
0.9
0.85
0.8
0.4
0.45
0.5
0.55
Figure 3: This is a zoom-in of Figure 2 near the jump point..
values of one of the self-adjoint extensions of a particular simply symmetric multiplication operator T with deficiency indices (1, 1), and the shifted sinc kernels are
the corresponding eigenfunctions. The Shannon sampling
theorem is the special case when the self-adjoint extensions of T have equidistant eigenvalues. By considering a
generic such symmetric operator T , one obtains a generalized sampling method. We can not cover the mathematical
derivations of the new generalized sampling method here,
but we will review the key features of the generalization
along with a comparison to the Shannon sampling theorem.
The Shannon sampling theorem states that if a function
φ(t) is in the space of Ω-bandlimited functions, i.e., φ(t)
has a frequency upper bound Ω, then φ(t) can be perfectly
reconstructed from its sample values {φ(tn )}n taken on a
set of sampling points {tn }n with an equidistant spacing
tn+1 − tn = 1/(2Ω) via:
φ(t) =
∞
X
n=−∞
G t, tn φ(tn )
(1)
The function G(t, tn ) is the so-called reconstruction ker
nel, which is the shifted sinc function sinc 2Ω(t − tn ) .
The frequency upper bound Ω is called the bandwith, and
the sampling rate 1/(2Ω) is the Nyquist sampling rate.
SAMPTA'09
One-Parameter Family of Sampling Lattices
We will call a set of Nyquist sampling points {tn }n a sampling lattice. The Shannon sampling theorem only specifies the constant spacing between adjacent points in one
lattice, but it does not specify an initial sampling point.
Therefore, we can parameterize all possible sampling lattices as:
n+θ
tn (θ) =
, 0≤θ<1
(2)
2Ω
Hence the Shannon sampling method possesses a natural
one-parameter family of sampling lattices, and any function in the function space can be perfectly reconstructed
from its values on any fixed lattice via Eq. (1).
The generalized sampling method also possesses an analogous one-parameter family of sampling lattices, but the
points in each lattice are generally non-equidistant now.
To distinguish from the case of Shannon, we use a different parameter α in {tn (α)}n , 0 ≤ α < 1, and assume that
{tn (α)}n are differentiable with respect to the parameter
α:
dtn (α)
t′n (α) =
dα
Shannon’s family of sampling lattices {tn (θ)}n can be
generated by a single number, namely, the constant bandwidth Ω. It is so simple because the function space in
the case of Shannon has a constant bandwidth Ω. However, in the generalization, since we have a time-varying
‘bandwidth’, in the sense of Nyquist lattices with nonequidistant points, more specification is required. The entire family of sampling lattices is now generated from the
knowledge of a given lattice, say {tn (0)}, and a set of corresponding derivatives {t′n (0)}n by solving for t = tn (α)
in:
X t′ (0)
m
= π cot(πα)
(3)
t
−
tm (0)
m
This equation implies that one sampling lattice and the
correponding derivatives are enough to determine the entire family of sampling lattices, and hence the reconstruction kernel and the function space. This is important for
practical purposes, because one usually takes samples of a
given signal on only one lattice.
The family of sampling lattices {tn (α)}n in the generalization shares many important properties of the uniform
lattices {tn (θ)} of Shannon: as the parameter α (or β in
the case of Shannon) increases from 0 to 1, the sampling
lattices specified by the parameter move to the right on the
real line simultaneously and continuously with the following continuity condition:
tn (1) := lim tn (α) = tn+1 (0), t′n (1) = t′n+1 (0) (4)
α→1−
Hence, together, these sampling points in all lattices again
cover the real line exactly once. Namely, for any t ∈ R,
there exists an unique integer n and an unique α in [0, 1)
such that t = tn (α).
2.2
The Generalized Reconstruction Kernel
From the theory of self-adjoint extensions, if on each fixed
but arbitrary lattice {tn (α)}, α fixed, we let tn = tn (α),
340
2.3
1.2
Interpolation Strategy
To approximate a given function, depending on the behavior of the function, one must select a sampling lattice for
interpolation. Arising from the theory of self-adjoint extensions, the chosen lattice {tn }n must have a minimum
and maximum spacing, namely, there must exist positive
real numbers δmin and ∆max such that:
1
0.8
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0.2
0
0 < δmin ≤ ∆tn = tn+1 − tn ≤ ∆max
for all n (8)
−0.2
−0.4
−0.6
−3
−2
−1
0
1
2
3
Figure 4: An example of generalized sinc function (or reconstruction kernel) on an arbitrary non-equidistant sampling lattice.
The stars on the real line indicate the points in an arbitrary nonequidistant sampling lattice, and the circles denote the same set
of points with an amplitude 1.
t′n = t′n (α), then the reconstruction kernel in the generalized sampling theorem reads:
!−1/2
p
X
t′n
t′m
z(t, tn )
G(t, tn ) = (−1)
| t − tn |
(t − tm )2
m
(5)
where z(t, tn ) is the number of the sampling points
{tm }m between t and tn exclusively.
As functions in t, for each fixed α, the set of functions
o
n
(6)
gn(α) (t) = G(t, tn (α))
n
forms a basis of the function space. Thus, indeed, in the
generalized sampling theorem, every function in the function space specified by the family of sampling lattices can
be expanded using these basis functions.
These continuous functions in Eq. (6) have analogous
properties to the shifted sinc function of Shannon: they
interpolate all the points in the lattice specified by α
gn(α) (tm (α)) = G tm (α), tn (α) = δmn
and their maximum values are all 1 at the sampling points
about which they are ‘centered’. This is important for the
stability of reconstruction. We will refer to these basis
functions as generalized sinc functions. See Figure 4 for
an example.
It
to recall that each set of basis functions
n is important
o
(α)
gn (t) specified by α spans the same function space.
n
This property is remarkable since as in Figure 4, the shape
of the generalized sinc functions is quite non-trivial.
To recover the Shannon sampling theorem as a special
case, we choose any uniform sampling lattice {tn }n with
1
tn+1 − tn = 2Ω
for all n, together with constant deriva′
tives tn = C. Then the reconstruction kernel
in (5) simplifies to the sinc kernel sinc 2Ω(t − tn ) , by using the
following trignometric identity:
+∞
X
1
π2
=
sin2 (πz) k=−∞ (z − k)2
SAMPTA'09
(7)
From Section 2.1 Eq. (3), we know that one also needs a
set of corresponding derivatives to apply the generalized
sampling method. So the question is, for a given lattice
{tn }n , what is a suitable choice of the set of corresponding
derivatives {t′n }n ?
To this end, we notice that the derivative t′n (α) is the velocity with which the sampling points tn (α) are moving to
the right along the real line for increasing α at t = tn (α).
Hence, a good candidate for t′n is the distance travelled in
one period of α, which is the spacing between two adjacent points ∆tn = tn+1 − tn . For symmetry, we set t′n
to be the average distance between tn to its previous and
successive points:
t′n =
1
1
∆tn + ∆tn−1 =
tn+1 − tn−1
2
2
(9)
Here a constant prefactor for the derivatives on a fixed lattice does not matter because the reconstruction kernel is
independent of a scalarp
multiplication of the derivatives:
in (5), the prefactor in t′n -term will cancel out the one
in t′m on the numerator inside the series.
With this set of initial data {tn }n and {t′n }n , we have an
explicit expression of the reconstruction kernel (5). Hence
we can construct theinterpolating
function φ(t) through
all the sample points tn , φ(tn ) n using the reconstruction formula (1).
3.
3.1
Reduction of Gibbs’ Overshoot
Reconstruction of Periodic Functions
The clearest example to demonstrate the reduction of
Gibb’s overshoot using the generalized sampling method
is the periodic step function H(t). One of the reasons for
choosing a periodic function is that the infinite summations in the both reconstruction kernel (5) and the reconstruction formula (1) will simplify to a finite sum. Hence,
we eliminate the truncation error in the summation.
To this end, assume that the function φ(t) has a period of
T , and we take N sampling points on one period [0, T ),
which are denoted by {τ1 , τ2 , . . . , τN } ⊆ [0, T ). Hence,
all the sampling points are
tnN +k = nT + τk ,
1 ≤ k ≤ N, n ∈ N
(10)
and from the periodictiy, we have
t′nN +k = t′k ,
φ(tnN +k ) = φ(tk )
(11)
After a lengthy calculation, the reconstruction kernel (5)
341
amplitude is 1.0193, which is a significant reduction compared to the maximum amplitude 1.0640 in Gibbs’ overshoot (Figure 1).
on this periodic lattice now reads:
p
(−1)z(t,tnN +l ) t′k
|t − tnN +K |
N
−1/2
π X ′ −2 π
tl sin
(t − τl )
T
T
G(t, tnN +K ) =
(12)
l=1
and the reconstruction formula (1) of the T -periodic function φ(t) reads:
φ(t) =
N
X
(−1)z(t,tnN +k )
k=1
N
X
l=1
t′l sin−2
q
π
t′k cot (t − τk )
T
(13)
−1/2
π
φ(tk )
(t − τl )
T
As discussed in Section 2.3, for using the formulae (12)
and (13) to approximate a periodic step function H(t), the
only task now left is to find a sampling lattice adapted to
the behavior of H(t). With a periodic lattice (10), we only
need to pick up a finite number N of them on [0, T ).
3.2
Approximating a Periodic Step Function
Before we discuss how to determine a set of nonequidistant sampling points, let us first consider why the
uniform lattices of Shannon do not work very well. Intuitively, because of the sudden change in the amplitude of
a step function H(t) at its jump points t = 0, 12 and 1,
the function can be considered to suddenly oscillate at an
“infinite” frequency in a sufficiently small neighborhoods
at the jump points, namely to have an ‘infinite’ bandwidth
at t = 0, 12 and 1. Recall that the constant Nyquist spacing
1/(2Ω) in the case of Shannon is inversely proportional to
the bandwidth Ω. A uniform lattice implies uniform bandwidth. Intuitively, the uniform lattice in the case of Shannon is therefore not matched with the increase of bandwidth in the small neighborhoods of jump points.
We therefore choose N sampling points with nonequidistant spacings so that the smallest spacing (the highest bandwidth) occurs near the jump points at t = 0, 12 , 1,
and the spacing gradually increases away from the jump
points (the bandwidth decreases). We used the easiest such
increasing change in spacing, which is linear.
Due to the symmetry of the jump points at t = 0, 12 , 1, we
divide one period [0, 1) into four equal subintervals with
length 14 . On the first subinterval, [0, 14 ), we choose K
points so that their adjacent spacing is linearly increasing.
Let δ be the linear increment in spacing, then
τ1 = 0, τ2 = δ, τ3 = 3δ, . . .
1
τK = K(K − 1)δ
2
(14)
The (K + 1)st point is 14 . The sampling points on ( 14 , 12 ]
are a mirror image of the points on [0, 14 ) with respect to
t = 14 , and the points on [ 12 , 1) repeat the ones on [0, 12 ).
Therefore, we have in total N = 4K points on [0, 1).
The approximation in Figure 2 is obtained in this way with
K = 6. Hence it has the same total number of sampling
points (N = 24) on [0, 1) as in Figure 1. Its maximum
SAMPTA'09
4.
Outlook
The question arises how far one can ultimately reduce the
Gibb’s overshoot? Is the linear change in sampling spacing, as in Eq. (14), the optimal lattice spacing to match
the behavior of a step function? This question will be addressed in a longer following-up paper, in which we will
pursue an analytical optimization of the Gibbs’ overshoot
reduction.
To this end, the fact that the closed form of the reconstruction kernel (12) is available in the case of periodic
functions has an important advantage: it in effect reduces
infinitely many points to a set of finitely many points. We
can then analytically study the behavior of the constructed
approximating functions. Eventually, we hope such an analytical study can lead us to the ultimately goal, which is
to provide solution to design optimally adapted lattices for
arbitrary given functions.
5.
Acknowledgment
This work has been supported by NSERC’s Discovery,
Canada Research Chairs and CGS D2 programs. A.K.
and Y.H. gratefully acknowledge the kind hospitality at
the University of Queensland, where A.K. is currently on
sabbatical.
References:
[1] J.J. Benedetto.
Modern Sampling Theory.
Birkhauser, Boston, 2001.
[2] J.W. Gibbs. Fourier series. Nature, 59:606, 1899.
[3] A.J. Jerri. The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations. Springer,
1998.
[4] A. Kempf. On fields with finite information density.
Phys. Rev. D, 69(124014), 2004.
[5] A. Gelb R. Archilbald. A method to reduce the gibbs
ringing artifact in mri scans while keeping tissue
boundary integrity. IEEE Trans. on Medical Imaging, 21(4):305–319, Apr. 2002.
[6] C.E. Shannon. Communication in the presence of
noise. Proc. IRE, 37:10–21, Jan. 1949.
[7] M. Unser. Sampling - 50 years after shannon. Proc.
IEEE, 88(4):569–587, Apr. 2000.
[8] A. Kempf Y. Hao. On a non-fourier generalization of
shannon sampling theory. Proc. of Canadian Workshop on Information Theory, pages 193–196, 2007.
[9] A. Kempf Y. Hao. On the stability of a generalized
shannon sampling theorem. Proc. of International
Symposium on Information Theorem and its Applications, Dec. 2008.
[10] A.I. Zayed. Advances in Shannon’s Sampling Theory. CRC Press, Boca Baton, 1993.
342
Optimized Sampling Patterns for Practical
Compressed MRI
Muhammad Usman(1) and Philip G. Batchelor(1)
(1) Division of Imaging Sciences, Kings College London, United Kingdom
muhammad.3.usman@kcl.ac.uk, philip.batchelor@kcl.ac.uk
Abstract:
The performance of compressed sensing (CS) algorithms
is dependent on the sparsity level of the underlying
signal, the type of sampling pattern used and the
reconstruction method applied. The higher the
incoherence of the sampling pattern used for undersampling, less aliasing will be noticeable in the aliased
signal space, resulting in better CS reconstruction. In this
work, based on point spread function (PSF) properties,
we compare random, Poisson disc and constrained
random sampling patterns and show their usefulness in
practical compressed sensing applied to dynamic cardiac
magnetic resonance imaging (MRI).
Introduction
One of the main questions that arise in compressed
sensing magnetic resonance imaging (CS-MRI) is: which
type of sampling is optimal? The basic theory of
compressed sensing as proposed by Donoho [1] and
Candes [2] requires acquisition of randomized set of
measurements. For MRI, this corresponds to the random
sampling in Fourier domain (k-space) which results in
incoherent aliasing artefacts in image space. However,
random sampling requires bigger changes in amplitudes
and polarity of MR system gradients, making it
infeasible practically in an MR system.
Figure 1 shows one dimensional gradient variations for
2D random and uniform lattice sampling patterns. From
the figure, it is evident that we have bigger changes in
amplitude and polarity of gradients in case of random
sampling pattern than uniform lattice. The solution to
this problem is to use deterministic sampling patterns.
The uniform lattice pattern is a deterministic pattern but
yields coherent artefacts in its PSF and hence, it does not
satisfy the basic requirements of compressed sensing
theory. Our goal is to find deterministic sampling
patterns that have incoherent artefacts in the PSF and can
yield better CS reconstruction.
of PSF of the ideal sampling pattern: The near zero
region around the main lobe of the PSF should be as
large as possible and outside that region, PSF should
resemble white noise. The samples should be placed
randomly but with a restricted maximum distance
between samples. These two conditions are met by
Poisson disc sampling [4]. Recently, Poisson disc
sampling has been shown to give good results in parallel
MRI due to better reconstruction conditioning [5].
However, it also has impractical gradient requirements.
Gamper [6] defined constrained random pattern with
incoherent artefacts in its PSF. Constrained random
pattern is a normal lattice pattern with samples shifted
along one dimension randomly by -1, 0 and +1. Hence, it
is a normal lattice with constrained randomization added
along one direction and has moderate gradient
requirements. Figure 2 shows the three candidate
sampling patterns (random, Poisson disc and constrained
random) with the corresponding PSFs. Like Poisson disc
sampling, the constrained random sampling has a nearzero region around the main lobe in its PSF and it also
possesses the uniform density of sampling both locally
and globally.
1. Candidate Sampling Patterns in CS
Figure 1: Gradient variation along one dimension in MR
system for (a) random sampling (top) (b) uniform lattice
sampling (bottom)
To have minimum aliasing due to sampling below the
Nyquist rate, Nayak [3] defined the following properties
Additionally, due to added constrained randomization,
the amplitudes of coherent side lobes in the PSF of
constrained random pattern are also suppressed. In CS
SAMPTA'09
343
recovery algorithms like OMP [6] which are based on
picking the most significant component from the aliased
space iteratively, the suppression of coherent artefacts
ensures that only the right candidates are picked up in
successive iterations.
acceleration factors/sampling factors (SF) from 3 to 7.
The x-f space corresponding to each frequency encoding
index was independently reconstructed by our modified
OMP method with adaptive thresholding scheme [9].
The OMP algorithm stops when maximum residual
aliasing intensity in x-f space reaches the intensity level
of noise.
Figure 2: Three candidate sampling patterns and their
corresponding PSFs: top to bottom: random, Poisson disc
and constrained random
2. Experimental Setup
To test the performance of three sampling patterns in
dynamic cardiac MRI, two sets of dynamic cardiac data
of size (nf x n p x nt, nf: number of frequency encoding
indices, np: number of phase encoding indices, nt: number
of time frames) (224x155x50) and (336x178x48) were
acquired with a Philips MRI scanner 1.5 T, SSFP
sequence, FOV 350x350 mm2. We used the jittered grid
approximation of the Poisson disc sampling as proposed
by Cook [7]. For CS based reconstruction, the x-f space
(x: spatial location, f: temporal frequency) is chosen to
be the sparse representation [8]. The x-f space
representation of the dynamic cardiac data is obtained by
taking the Fourier transform of dynamic MR data along
the temporal dimension. Figure 3 shows the dynamic
cardiac MR data and its sparse representation.
For each frequency encoding index, the under-sampled
data was simulated by applying the three sampling
patterns to the fully sampled dynamic cardiac data in kx-t
space (kx: phase encoding index, t: time) with varying
SAMPTA'09
Figure 3: Dynamic cardiac MR data (a) and its x-f space
representation (b), the frequency axis ‘f’ is centered
around dc frequency (f=0)
3. Performance Results
The CS reconstruction results for candidate sampling
patterns are shown in Figure 4 to Figure 9 with different
acceleration factors. For the original cardiac frame
shown in Figure 4 (a), the CS reconstruction results by
OMP method with under-sampling factor (SF) of 3 are
shown in Figure 4 (b), (c) and (d) for random, Poisson
disc and constrained random sampling patterns. The
corresponding temporal profiles are shown in Figure 5.
The CS reconstruction results for acceleration factors of
5 and 7 are shown in Figure 6 to Figure 9. Up to the
acceleration factor of 5, the CS reconstructed data has
same spatial and temporal resolution for all three
sampling patterns with nearly exact signal reconstruction
achieved up to SF=3 (Figure 4 and Figure 5). Below
SF=5, the temporal resolution of CS reconstructed data
344
with constrained random sampling gets worse than that
for the other sampling patterns (Figure 9 d). This is due
to the fact that with very high acceleration factors, many
locations within the constrained random pattern will have
zero probability of being picked up, as the sampling
locations are constrained to only one sample shift from
the uniform lattice grid.
pattern (Poisson disc sampling). Up to the acceleration
factor of 5, the quality of the reconstructed images and
the temporal resolution of the CS reconstructed data are
nearly the same for random, Poisson and constrained
random sampling patterns. Since the constrained random
sampling has moderate gradient requirements when
compared with other optimal sampling schemes, it is an
excellent choice to be used as an optimal sampling
pattern in practical compressed MRI.
References:
Figure 4: CS reconstruction results with SF=3:
(a) original cardiac frame, CS reconstructed data with (b)
random sampling (c) Poisson disc sampling (d)
constrained random sampling
Figure 5:
CS reconstruction
results with SF=3:
(a)original
temporal profile,
CS Reconstructed
temporal profile
with (b) random
sampling
(c) Poisson disc
sampling
(d)
constrained
random sampling
[1] D.L.Donoho, “Compressed Sensing," IEEE
transactions on information theory, vol.52, no. 4, pp.
1289-1306, 2006.
[2] E. Candes,”Compressive sampling," in Proceedings
of the International Congress of Mathematicians, vol. 3,
pp. 1433-1452, Madrid, Spain, 2006
[3] KS Nayak et al, “Randomized trajectories for reduced
aliasing artifact”, in: Proceedings of the ISMRM, p 670.,
Sydney, 1998
[4] J. I. Yellot, “Spectral consequences of photoreceptor
sampling in the rhesus retina.” Science 221, 382–385,
1985
[5] M. Lustig et al., “Autocalibrating Parallel Imaging
Compressed Sensing using L1 SPIR-iT with PoissonDisc Sampling and Joint Sparsity Constraints ISMRM
Workshop on Data Sampling and Image Reconstruction,
Sedona '09
[6] U. Gamper et al,”Compressed sensing in dynamic
MRI,"MRM, vol. 59, no. 2, pp. 365-373, 2008.
[7] R. L. Cook, “ Stochastic sampling in computer
graphics”, ACM Transactions on Graphics (TOG), vol.5,
no. 1, p. 51-72, Jan 1986
[8] S. J. Malik et al, “x-f Choice: reconstruction of
undersampled dynamic MRI by data-driven alias
rejection applied to contrast-enhanced angiography.
Stochastic sampling in computer graphics”, MRM,
vol.56, p. 811-823, 2006
[9] M. Usman et al, “Adaptive thresholding scheme for
OMP method applied to dynamic MRI”, Proc.
ESMRMB, Valencia, vol. 25, no. 766, pp. 389, Oct 2008
4. Conclusion
We showed that the PSF properties of constrained
random sampling are similar to the optimal sampling
SAMPTA'09
345
Figure 6: CS reconstruction results with SF=5:
(a) original cardiac frame, CS reconstructed data with (b)
random sampling (c) Poisson disc sampling (d)
constrained random sampling
Figure 7:
CS reconstruction
results with SF=5:
(a)original
temporal profile,
CS Reconstructed
temporal profile
with (b) random
sampling
(c) Poisson disc
sampling
(d)
constrained
random sampling
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Figure 8: CS reconstruction results with SF=7:
(a) original cardiac frame, CS reconstructed data with (b)
random sampling (c) Poisson disc sampling (d)
constrained random sampling
Figure 9:
CS reconstruction
results with SF=7:
(a)original
temporal profile,
CS Reconstructed
temporal profile
with (b) random
sampling
(c) Poisson disc
sampling (d)
constrained
random sampling
346
A Study on Sparse Signal Reconstruction from
Interlaced Samples by l1-Norm Minimization
Akira Hirabayashi (1)
(1) Yamaguchi University, 2-16-1, Tokiwadai, Ube City, Yamaguchi 755-8611, Japan.
a-hira@yamaguchi-u.ac.jp
Abstract:
We propose a sparse signal reconstruction algorithm from
interlaced samples with unknown offset parameters based
on the l1 -norm minimization principle. A typical application of the problem is superresolution from multiple lowresolution images. The algorithm first minimizes the l1 norm of a vector that satisfies data constraint with the
offset parameters fixed. Second, the minimum value is
further minimized with respect to the parameters. Even
though this is a heuristic approach, the computer simulations show that the proposed algorithm perfectly reconstructs sparse signals without failure when the reconstruction functions are polynomials and with more than 99%
probability for large dimensional signals when the reconstruction functions are Fourier cosine basis functions.
1. Introduction
Sampling theory is at the interface of analog/digital conversion, and sampling theorems provide bridges between
the continuous and the discrete-time worlds. A fundamental framework of the sampling theorems consists of data
acquisition (sampling) process of a target signal and reconstruction process from the data. Classical studies assumed that both processes are fixed and known. Then,
sampling theorems yield in linear formulations [9].
On the other hand, recent studies assume that sampling or
reconstruction processes contain unknown factors. Then,
sampling theorems become nonlinear. For example, Vetterli et al. discussed problems in which locations of reconstruction functions are unknown [11], [5]. They introduced the notion of rate of innovation, and provided perfect reconstruction procedures for signals with finite rate
of innovation. The recent hot topic, compressive sampling, assumes that signals are sparse in the sense that
signals are expressed by a small subset of reconstruction
functions, but the subset is unknown [3], [1], [4]. It is interesting that the solution is obtained by the l1 -norm minimization.
In contrast to the above studies, problems with unknown
factors in the sampling process have also been discussed.
For example, sampling locations are assumed to be unknown and completely arbitrary in [8] and [2]. A more
restricted sampling process is interlaced sampling [7], in
which a signal is sampled by a sampling device several
times with slightly shifted locations. If the offset parame-
SAMPTA'09
ters are unknown, the sampling theorem becomes nonlinear. A typical application is superresolution from a set of
multiple low-resolution images. A replacement of a single
high-rate A/D converter by multiple lower rate converters
also yields within this formulation.
To this problem, Vandewalle et al. proposed perfect reconstruction algorithms under a condition that the total
number of unknown parameters is less than or equal to the
number of samples [10]. We can find, however, practical
situations in which the condition is not true. The method
proposed in [2] can be applied to such situations. However, it hardly provides a high quality stable result. In
order to solve these difficulties, the present author proposed an algorithm that reconstructs the closest function
to a mean signal under data constraint assuming that signals are generated from a probability distribution [6]. The
mean signal is, however, not always available.
Hence, in this paper we propose a signal reconstruction
algorithm from interlaced samples with unknown offsets
using a relatively weak a priori knowledge, sparsity. The
algorithm first minimizes the l1 -norm of a vector that satisfies data constraint with the offset parameters fixed. Then,
the minimum value is further minimized with respect to
the parameters. Even though this is a heuristic approach,
the computer simulations show that the proposed algorithm perfectly reconstructs sparse signals without failure when the reconstruction functions are polynomials and
with more than 99% probability for large dimensional signals when the reconstruction functions are Fourier cosine
basis functions.
This paper is organized as follows. Section 2 formulates
the fundamental framework and defines the notion of sparsity. Section 3 introduces interlaced sampling and summarizes the conventional studies. In Section 4, we propose the l1 -norm minimization algorithm. Section 5 evaluates the algorithm through simulations, and shows that the
algorithm perfectly reconstruct sparse signals with high
probability. Section 6 concludes the paper.
2. Sparse Signals
A signal f to be reconstructed is defined on a continuous
domain D. We assume that f belongs to a Hilbert space
H = H(D) of a finite dimension K. The inner product
for f and g in H √
is denoted by hf, gi, and the norm is
induced as kf k = hf, f i. By using an arbitrarily fixed
347
basis {ϕk }K−1
k=0 , any f in H is expressed as
f=
K−1
∑
H
ak ϕk .
(1)
k=0
f=
A K-dimensional vector with k-th element ak is denoted
by a.
Definition 1 A signal f is J-sparse if at most J coefficients of {ak }K−1
k=0 in Eq. (1) are non-zero and the rest are
zero.
It should be noted that unknown factors in J-sparse signals
are not only values of non-zero coefficients but also their
locations. Hence, there are 2J unknown factors in a Jsparse signal. If 2J ≥ K, then the number of unknown
factors is more than K, which is the number of the original
unknown coefficients {ak }K−1
k=0 without sparsity. Hence,
in order for sparsity to be meaningful, we assume that
J < K/2.
In real applications, J is supposed to be much smaller than
K/2.
3.
Interlaced Sampling
Interlaced sampling means that a signal f is sampled
M times by an identical observation device with offsets
−1
(0)
{δ (m) }M
= 0. An M -dimensional vector
m=0 , where δ
(m)
with m-th element δ
is denoted by δ. The observation
−1
device is characterized by sampling functions {ψn }N
n=0 ,
which are given a priori. Then, the sampling function for
the n-th sample in the m-th sequence is given by
and the sample is expressed as
dn(m) = hf, ψn(m) i.
(2)
(m)
Let d be an M N -dimensional vector in which dn is the
n+mN -th element. An M N ×K matrix with the n+mN ,
(m)
k-th element hϕk , ψn i is denoted by Bδ . Substituting
Eq. (1) into Eq. (2) yields
(3)
For simplicity, we assume that the column vectors of Bδ
are linearly independent. Figure 1 illustrates the formulation of interlaced sampling.
In order to reconstruct the signal f from interlaced samples with unknown offsets, we have to determine both
(m) M −1
{ak }K−1
}m=1 . To this problem, Vandek=0 and {δ
walle et al. proposed perfect reconstruction algorithms
under a condition that the number of unknown parameters is less than or equal to the number of samples
(m) N −1 M −1
{{dn }n=0
}m=0 , or
K + M − 1 ≤ M N.
(4)
We can find, however, practical situations in which the
condition is not true. The method in [2] can be applied
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ak ϕk
k=0
Reconstruction
CK
a0
a = ...
aK−1
Sampling
Bδ
d=
CM N
d0
..
.
(M −1)
dN −1
Figure 1: Formulation of sampling and reconstruction.
The vector a is to be estimated from the vector d. Note
that there are unknown offset parameters δ in Bδ .
to the situation without Eq. (4). However, the results obtained by the method tend to be unstable. The present author proposed an algorithm which uses a mean signal as a
prior [6]. However, the mean signal is not always available. Hence, in this paper, we propose perfect reconstruction algorithms using a relatively weak prior, sparsity.
4. l1 -Norm Minimization Algorithm
The problem which we are going to solve in this paper is
stated as follows.
Problem 1 Determine J-sparse vector a and δ which satisfy Eq. (3) under the condition that the column vectors of
Bδ are linearly independent.
ψn(m) (x) = ψn (x − δ (m) ),
Bδ a = d.
K−1
∑
Because of the linear independentness, a vector a that satisfies Bδ a = d is uniquely determined as
a = Bδ† d,
where Bδ† is the Moore-Penrose generalized inverse of Bδ .
Let us define a matrix Bε by setting an arbitrarily fixed
parameter ε instead of δ. By using this matrix, a vector cε
is defined as
cε = Bε† d.
(5)
Then, our problem becomes a problem of finding a parameter ε such that the vector cε is J-sparse.
It is well-known that l1 -norm minimization is effective to
promote sparsity as is used in the compressed sensing [3],
[1], [4]. Hence, we also employ this principle to find Jsparse vector cε . Now, our problem becomes the following problem.
Problem 2 Determine ε that makes column vectors of the
matrix Bε linearly independent, and minimizes l1 -norm of
cε in Eq. (5):
ε̂ = argminε kcε kl1 = argminε kBε† dkl1 .
(6)
348
Table 1: Parameters K, J, N and M used in simulations.
K
J
N
M
4
1
2
2
6
2
3
2
8
3
4
2
10
4
5
2
12
5
6
2
✁
✆
✁
☎
✁
✄
✁
✄
✞
The solution to Problem 2 is different from that to Problem 1 in general. Similar to the compressed sensing, the
former agrees with the latter in some cases. Theoretical
analyses for the agreement are still under consideration.
Instead, we show simulation results in this paper.
✁
☎
✁
✆
✁
✝
✁
✂
✞
✞
✞
✞
✁
✂
✄
✄
✁
✂
☎
☎
✁
✂
✆
✆
✁
✂
✝
5. Simulations
(a) Reconstruction result
We show computer simulations which demonstrate that
the proposed algorithm perfectly reconstructs sparse signals under certain conditions. We consider two reconstruction functions, polynomial and Fourier cosine basis.
✁
✁
✄
✞
✁
✄
✁
✁
☎
✞
✁
☎
✁
✁
✆
✞
✁
✆
✁
✁
5.1 Polynomial reconstruction
✞
Let H be a space spanned by functions
ϕk (x) = xk (0 ≤ k < K)
for [0, l] where l is a positive real number. The inner prod∫l
uct is defined by hf, gi = 1l 0 f (x)g(x)dx. Sampling
(m)
is assumed to be ideal, i.e., dn = f (xn + δ (m) ). The
sample point xn is given by
✝
✞
✁
✝
✁
✁
✞
✁
✁
(2n + 1)l
xn =
(n = 0, 1, . . . , N − 1),
2N
which we call the base sequence. Let l = N so that the
sampling interval becomes one.
Figure 2 (a) shows a simulation result, in which the dimension of H is K = 8, sparsity parameter is J = 3, the
number of samples in each sequence is N = 4, and the
sequence was used M = 2 times. The offset parameter
is δ (1) = −0.4. The black line shows the target signal
f , and ‘o’ and ‘x’ respectively show the base and the first
sequences. The red line shows the reconstructed signal,
from which we can see the target signal is perfectly recovered. Figure 2 (b) shows that the l1 -norm of cε is indeed
minimized at ε = −0.4. We repeated the simulation for
one thousand target signals with the values shown in Table
1. Then, all of the signals are perfectly recovered as well
as the offset parameters.
5.2 Fourier cosine basis reconstruction
We used the same setup except that the reconstruction
functions are
{
1
(k = 0),
ϕk (x) = √
kπx
2 cos
(0 < k < K).
l
Under the above defined inner product, {ϕk }K−1
k=0 is an
orthonormal basis.
SAMPTA'09
✁
✂
✄
✁
✂
☎
✁
✂
✆
✁
✂
✝
✁
✁
✂
✝
✁
✂
✆
✁
✂
☎
✁
✂
✄
(b) l1 -norm of cε
Figure 2: Simulation result. The black line shows the target signal f , and ‘o’, ‘x’, and ‘+’ respectively show the
base, the first, and the second sequences. The red line
shows the reconstructed signal which perfectly matches to
the target signal.
Figure 3 (a) shows a simulation result, in which the dimension of H is K = 60, sparsity parameter is J = 15,
the number of samples in each sequence is N = 20, and
the sequence was used M = 3 times. The offset parameters are δ (1) = −0.2 and δ (2) = 0.3. The black line shows
the target signal f , and ‘o’, ‘x’, and ‘+’ respectively show
the base, the first, and the second sequences. The red line
shows the reconstructed signal, from which we can see the
target signal is perfectly recovered.
Unfortunately, perfect reconstruction is not always
achieved. Figure 4 shows failure rates [%] of perfect
reconstruction with respect to K. The dotted red and
the solid blue lines show the rates when J = K/4 and
J = K/6, respectively. The failure rate for J = K/4 arrives at less than or equal to 1% when K > 32, while that
for J = K/6 does so when K > 30.
Even though these results are only verified through simu-
349
tions are Fourier cosine basis functions. Because of the
computational efficiency, the proposed algorithm is very
attractive. Theoretical analyses of these results are our
most important future task.
6
4
2
Acknowledgment
0
This work was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for
Young Scientists (B), 20700164, 2008.
-2
-4
References:
-6
0
2
4
6
8
10
12
14
16
18
20
Figure 3: Simulation result for Fourier cosine basis functions. The black line shows the target signal f , and ‘o’,
‘x’, and ‘+’ respectively show the base, the first, and the
second sequences. The red line shows the reconstructed
signal which perfectly matches to the target signal.
lations, the proposed approach is attractive because of its
computational efficiency. It takes less than 0.4 second to
find the solution for the case of K = 60, N = 20, and
M = 3.
[%]
30
25
J = K/4
20
15
10
J = K/6
5
0
0
10
20
30
40
50
60
K
Figure 4: Failure rates of signal recovery when reconstruction functions are Fourier cosine basis functions.
6.
Conclusion
We proposed a sparse signal reconstruction algorithm
from interlaced samples with unknown offset parameters.
The algorithm is based on the l1 -norm minimization principle: First, it minimizes the l1 -norm with the offset parameters fixed. Second, the minimum value is further minimized with respect to the parameters. Even though this
is a heuristic approach, the computer simulations showed
that the proposed algorithm perfectly reconstructs sparse
signals without failure when the reconstruction functions
are polynomials and with more than 99% probability for
large dimensional signals when the reconstruction func-
SAMPTA'09
[1] R.G. Baraniuk. Compressive sensing [lecture notes].
IEEE Signal Processing Magazine, 24(4):118–121,
July 2007.
[2] J. Browning. Approximating signals from nonuniform continuous time samples at unknown locations. IEEE Transactions on Signal Processing,
55(4):1549–1554, April 2007.
[3] E.J. Candes, J. Romberg, and T. Tao. Robust uncertainty principles: Exact signal reconstruction from
highly incomplete frequency information. IEEE
Transactions on Information Thoery, 52(2):489–
509, February 2006.
[4] E.J. Candes and M.B. Wakin. An introduction to
compressive sampling [a sensing/sampling paradigm
that goes against the common knowledge in data
acquisition]. IEEE Signal Processing Magazine,
25(2):21–30, March 2008.
[5] P. L. Dragotti, M. Vetterli, and T. Blu. Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets Strang-Fix. IEEE Transactions on Signal Processing, 55(5):1741–1757, May
2007.
[6] Akira Hirabayashi and Laurent Condat. A study on
interlaced sampling with unknown offsets. In Proceedings of European Signal Processing Conference
2008 (EUSIPCO2008), volume CD-ROM, 2008.
[7] R.J. Marks II. Introduction to Shannon Sampling and
Interpolation Theory. Springer-Verlag, New York,
1991.
[8] P. Marziliano and M. Vetterli. Reconstruction of irregularly sampled discrete-time bandlimited signals
with unknown sampling locations. IEEE Transactions on Signal Processing, 48(12):3462–3471, December 2000.
[9] M. Unser. Sampling—50 Years after Shannon. Proceedings of the IEEE, 88(4):569–587, April 2000.
[10] P. Vandewalle, L. Sbaiz, J. Vandewalle, and M. Vetterli. Super-resolution from unregistered and totally
aliased signals using subspace methods. IEEE Transactions on Signal Processing, 55(7):3687–3703, July
2007.
[11] M. Vetterli, P. Marziliano, and T. Blu. Sampling signals with finite rate of innovation. IEEE Transactions on Signal Processing, 50(6):1417–1428, June
2002.
350
Multiresolution analysis on multidimensional
dyadic grids
Douglas A. Castro(1) , Sônia M. Gomes(1) , Anamaria Gomide(2) , Andrielber S. Oliveira (1), Jorge Stolfi(2)
(1) IMECC-Unicamp, Caixa Postal 6065, CEP 13083-859 Campinas-SP, Brazil.
(2) IC-Unicamp, Caixa Postal 6176, CEP 13081-970 Campinas-SP, Brazil.
{douglas,andriel,soniag}@ime.unicamp.br {anamaria,stolfi}@ic.unicamp.br
Abstract:
We propose a modified adaptive multiresolution scheme
for representing d-dimensional signals which is based on
cell-average discretization in dyadic grids. A dyadic grid
is an hierarchy of meshes where a cell at a certain level is
partitioned into two equal children at the next refined level
by hyperplanes perpendicular to one of the coordinate axes
which varies cyclically from level to level. Adaptivity is
obtained by interrupting the refinement at the locations
where appropriate scale (wavelet) coefficients are sufficiently small. One important aspect of such multiresolution representation is that we can use a binary tree data
structure in all dimensions, that helps to compress data
while still being able to navigate through it. Dyadic grids
provide a more gradual refinement as compared with traditional multiresolution analyses that use, for instance, different quad-trees or oct-trees in 2D or 3D multiresolution
applications. The cells may have different scales in different directions, this property can be explored to improve
data compression of signals having anisotropic aspects.
1.
dyadic grids to multiresolution analysis, using cell averaging as the discretization method.
The paper is organized as follows. In Section 2 we define
dyadic grids and related concepts. In Section 3 we present
a general overview of multiresolution analysis. Section 4
contains numerical results on sample problems to show
the efficiency of the proposed scheme.
2. Dyadic grids
Let the coordinates of Rd be indexed from 0 to d − 1. An
infinite dyadic grid is a hierarchy of meshes that begins
with a d-cube at level k = 0, and, for each higher level
k > 0, is the result of dividing each cell of level k into
two equal children by a hyperplane perpendicular to the
coordinate axis (k mod d) [2]. Figure 1 illustrates five
steps of the refinement process for d = 3.
Introduction
In recent years, many multiscale techniques have been
used to provide more efficient algorithms than those that
use just one level of resolution. In such frameworks, the
differences between the information at consecutive levels
of refinement are computed, and only the significant coefficients are stored. These are the principles of wavelet
compression which have being successfully applied in
many different contexts [3]. For example, multiresolution
finite volume schemes of Müller [6] and Domingues et
al. [4] use adaptive grids that are dynamically obtained by
taking local regularity information indicated by wavelet
coefficients in the context of multiresolution analysis for
cell averages of signals. Such adaptive discretizations allow the efficient solution of problems with vastly different
scales of detail in different parts of the domain.
For computational efficiency, one important aspect of such
multiresolution methods is the topology of the mesh and
data structure used to represent it. Often quad-grids and
oct-grids are used for 2D and 3D domains, respectively,
represented by quad-tree and oct-tree data structures [1].
We describe here a type of mesh, the dyadic grid, that can
be efficiently represented by a binary tree, in domains of
arbitrary dimension. For illustration, we apply adaptive
SAMPTA'09
Figure 1: 3D dyadic grids.
In practice, one uses only finite segments of this grid,
where the subdivision stops at a maximum level. In a regular dyadic grid, the refinement stops at the same level everywhere. In an irregular grid, the maximum level varies
from place to place.
The topology of a dyadic grid can be represented by a 02 binary tree. This is a data structure consisting of a set
351
R
Detail coefficients. In the structure we do not store the
averages (f¯ck or fˆck ), but only the details or wavelet coeficients. Each detail dkc is the difference between the exact
average in the cell c and the value predicted for it by formulas (3) and (4) from the cell’s parent and its neighbors:
d−1
c
cr
c
c
dk+1
= f¯ck+1 − fˆck+1 .
c
i
Note that the detail of the root cell is not defined.
Figure 2: Definition of c− , c+ , c, cr .
of elements named nodes, among which there is a special
node r, the root; every node has either zero or two children
nodes; and every node, except the root, has exactly one
parent node. A node that has no children is called a leaf
node. The children of a non-leaf node t are called the left
child tℓ and the right child tr .
Each node of this tree represents a cell that appeared at
some level of the subdivision; the leaf nodes represent the
cells that weren’t divided. By convention, the left child
tℓ of a non-leaf node t in level k represents the “lower”
half cℓ of the cell c represented by t; that is, the half
whose projection on the axis i = k mod d has smallest
i-coordinates.
3.
Multiresolution analysis
In mutiresolution analysis, signals can be represented in
two ways, as ordinary samples at each scale, or as differences between two consecutive scales. Connecting these
two views are the prediction and the restriction operators.
The prediction operator Pkk+1 takes information from a
coarse level k and gives an estimate for the information at
the next finer level k + 1. Conversely, the restriction opk
erator Pk+1
takes information from a fine level k + 1 and
gives an estimate of the information at a coarser level k.
In this paper, the samples of a d−dimensional signal f
are averages computed over the cells of a d-dimensional
dyadic grid. That is, the sample associated with a cell c in
level k of the grid is
Z
1
f (x)dx,
(1)
f¯ck =
|c| c
where |c| is the volume of c. The restriction operation is
therefore (trivially and exactly) the sum of the averages in
the children cells,
i
1h
f¯ck = f¯ck+1
+ f¯ck+1
.
(2)
r
ℓ
2
In the other direction, we predict the cell average of a child
cell cr or cℓ by the formulas
f¯ck+1
≈ fˆck+1
r
r
f¯ck+1
≈ fˆck+1
ℓ
ℓ
i
1h
= f¯ck + f¯ck+ − f¯ck−
8
i
h
1
k
= f¯c − f¯ck+ − f¯ck−
8
(3)
(4)
where c− and c+ are the two closest neighbor cells of c at
level k in the direction of refinement. See Figure 2. These
estimators are exact for quadratic polynomials.
SAMPTA'09
(5)
Analysis and synthesis. The analysis algorithm computes the details of every cell, given the average values f¯ck
for every cell c. It scans the tree bottom-up, level by level.
For each non-leaf cell c in level k, it executes
h
i
¯k+1 ;
δ̄ ← 12 f¯ck+1
−
f
cℓ
h r
i
δ̂ ← 18 f¯ck+ − f¯ck− ;
(6)
δ ← δ̄ − δ̂
dkcr ← +δ;
dkcℓ ← −δ.
Once the detail dkc of a cell has been computed, its average
f¯ck is no longer needed, so we may store the detail in its
place. In the root node r, however, we must still keep the
average f¯r0 of the function over the whole domain.
The inverse of the analysis algorithm is the syntesis algorithm, which recomputes the averages f¯ck from the details.
It scans the tree top down, level by level. At each cell c in
level k, it executes
h
i
δ̂ ← 18 f¯ck+ − f¯ck− ;
δ̄ ← δ̂ − dk+1
cr ;
¯k + δ̄
f¯ck+1
=
f
c
r
f¯ck+1
= f¯ck − δ̄.
ℓ
(7)
After this step, the details dk+1
and dk+1
of the children
cr
cℓ
are no longer needed, and can be overwritten with the reconstructed averages f¯ck+1
and f¯ck+1
.
r
ℓ
Compact representation. These algorithms show that
knowledge of the cell averages for all leaves is equivalent
to knowledge of the average value f¯r0 for the root cell together with the detail of every right child cell. To save
space, we could store the detail of the right child in its
parent’s node (and keep the domain average f¯r0 in variable
external to the tree). Then the leaf nodes would carry no
information, and could be omitted from the structure. We
will refer to this variant (which is an ordinary binary tree)
as the compact tree representation.
Adaptive resolution grid. As in any wavelet representation, we can save space and processing time by pruning
all sub-trees which do not contribute significantly to the
reconstructed signal. If we start with a tree of sufficient
depth, we can eliminate all sibling leaf nodes cℓ and cr
such that dkc falls below a prescribed tolerance ǫk . This
will
condition implies that the predictions fˆck+1
and fˆck+1
r
ℓ
k+1
¯
be very close to the actual averages fcℓ and f¯ck+1
.
Here
r
we use Harten’s thresholds [5],
ǫk = (1 − q)q (L−k) ǫ,
(8)
352
where q and ǫ are specified by the user, with ǫ > 0 and
0 < q < 1, and L is the maximum level of the initial tree.
4.
Numerical results
In order to compare the efficiencies of dyadic grids and
quad-grids, we performed the multiresolution analyses
of two different examples in 2D, using cell-average discretization. √In all tests, the root cell was the rectangle
[0, 1] × [0, 22 ], and the starting tree was a uniform grid
with 210 × 210 = 220 = 1, 048, 576 leaf cells. This
corresponds to tree structures with L = 20 and L = 10
levels for dyadic grid and quad-grid frameworks, respectively. The cell averages f¯cL were computed for every
leaf cell c by Gaussian quadrature with 5 × 5 sampling
points. The trees were pruned as described in the previous section, with threshold parameters ǫ = 0.1 and the
q = 0.5.PThe number of non-leaf nodes in the initial
19
tree was i=0 2i = 1, 048, 575 for the dyadic grid, and
P9
i
i=0 4 = 349, 525 for the quad-grid.
In the first test, we used the signals
Figure 3: Pruned dyadic tree (top) and dyadic grid
(bottom) for the first signal at t = 0.1.
f (x, y) = 1−tanh(100(x−0.2−t)+0.001(y −1)), (9)
for t varying from 0 to 0.6 in steps of 0.1. Equation (9) describes a 2D smooth step function with an almost vertical
straight front, moving form left to right. Figure 3 shows
the dyadic grid and the corresponding tree at t = 0.1, after
pruning cells with small details. Figure 4 shows the corresponding quad-grid and quad-tree. Figure 5 shows the
number of leaf cells in both grids for each time step, as a
percentage of the number of leaves in the uniform grid.
For the second test, we used the signals
f (x, y) =
p
1 if (x − 0.5)2 + (y − 0.35)2 < t,
0 otherwise.
(10)
for t varying between 0.05 and 0.35 in steps of 0.05. Equation (10) describes a step function with a sharp circular
front, expanding from the center of the domain. Figure 6
shows the dyadic grid and its tree at t = 0.2, and Figure 7
shows the corresponding quad-grid and its quad-tree. The
number of leaves is plotted in Figure 8.
Space efficiency. If leaves are explicitly represented in
the tree structure, and all nodes have the same fields, then
the space E used by the structure is E = (pA + B)n,
where n is the number of nodes, p is the number of pointers in each node, A is the size of a pointer in bytes, and
B is the size of any additional information stored in each
node (such as the detail coefficients dkc ). In all these trees
we have n = (pm − 1)/(p − 1) ≈ mp/(p − 1), where m
is the number of leaf nodes.
From the plots in Figure 5, we see that, in the first test, the
quad-grid (p = 4) had about 8 times as many leaf cells
as the dyadic grid (p = 2), and therefore about 5 times as
many tree nodes, for the same accuracy. Assuming A = 4
and B = 8 bytes, we conclude that the quad-tree used
5(24/16) ≈ 7.5 times as much storage as the dyadic grid.
SAMPTA'09
Figure 4: Pruned quad-tree (top) and quad-grid
(bottom) for the first signal at t = 0.1..
5
quad−grid
dyadic grid
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Figure 5: Leaf count in the pruned trees for the first
test..
353
In the second test, the quad-grid had about 1.32 times as
many leaf cells as the dyadic grid, and therefore about 0.88
as many tree nodes. With the same A and B, the quad grid
still used 0.88(24/16) ≈ 1.32 times as much space as the
dyadic grid.
Had we used the compact representation of the tree, with
omitted leaves, the storage cost would be E = (pA +
(p − 1)B)(n − m). The quad-tree would use 7.5 times as
much storage as the dyadic tree in the first example, and
1.1 times as much in the second example.
5. Conclusions
Our tests show that adaptive dyadic grids are substantially
more efficient than quad-grids for the same level of accuracy, both in terms of space needed to store the topology
(tree structure) of the grid, and in the number of leaf cells
retained — which determines the time cost of most adaptive numeric algorithms.
6. Acknowledgments
Figure 6: Pruned dyadic tree (top) and dyadic grid
(bottom) for the second signal at t = 0.2..
The authors thank CNPq (grants 06631/07-5, 472402/072, and 142191/06-0) and FAPESP (07/52015-0) for financial support.
References:
[1] B. L. Bihari and A. Harten. Multiresolution schemes
for the numerical solution of 2-d conservation laws.
SIAM J. Sci. Comput., 18(2):315–354, 1997.
[2] C. G. S. Cardoso, M. C. Cunha, A. Gomide, D. J.
Schiozer, and J. Stolfi. Finite elements on dyadic
grids with applications. Mathematics and Computers
in Simulation, 73:87–104, 2006.
[3] A. Cohen. Wavelet Methods in Numerical Analysis.
Handbook of Numerical Analysis. in: Ph. Ciarlet and
J. L. Lions (Eds.), Handbook of Numerical Analysis,
Vol VII, Elsevier, Amsterdam, 2000.
[4] M. O. Domingues, S. M. Gomes, O. Roussel, and
K. Schneider. An adaptive multiresolution scheme
with local time-stepping for evolutionary pdes. Journal of Computational Physics, 227:3758–3780, 2008.
[5] A. Harten. Multiresolution representation of cellaveraged data. Technical Report CAM/Report/94-21,
UCLA, Los Angeles, US, July 1994.
[6] S. Muller. Adaptive Multiscale Schemes for Conservation Laws. Vol. 27 of Lecture Notes in Computational
Science and Engineering, Springer, Heidelberg, 2003.
Figure 7: Pruned quad-tree (top) and quad-grid
(bottom) for the second signal at t = 0.2..
2.5
quad−grid
dyadic grid
2
1.5
1
0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Figure 8: Leaf count in the pruned trees for the second
test..
SAMPTA'09
354
Adaptive and Ultra-Wideband Sampling via
Signal Segmentation and Projection
Stephen D. Casey (1) , Brian M. Sadler(2)
(1) Department of Mathematics and Statistics, American University, Washington, DC, USA .
(2) Army Research Laboratory, Adelphi, MD, USA.
scasey@american.edu, bsadler@arl.army.mil
Abstract:
Adaptive frequency band (AFB) and ultra-wide-band
(UWB) systems require either rapidly changing or very
high sampling rates. Conventional analog-to-digital devices have non-adaptive and limited dynamic range. We
investigate AFB and UWB sampling via a basis projection
method. The method decomposes the signal into a basis
over time segments via a continuous-time inner product
operation and then samples the basis coefficients in parallel. The signal may then be reconstructed from the basis coefficients to recover the signal in the time domain.
We develop the procedure of this method, analyze various methods for signal segmentation and close by creating
systems designed for binary signals.
1.
Introduction
Adaptive frequency band (AFB) and ultra-wide-band
(UWB) systems, requiring either rapidly changing or
very high sampling rates, stress classical sampling approaches. At UWB rates, conventional analog-to-digital
devices have limited dynamic range and exhibit undesired nonlinear effects such as timing jitter. Increased
sampling speed leads to less accurate devices that have
lower precision in numerical representation. This motivates alternative sampling schemes that use mixed-signal
approaches, coupling analog processing with parallel sampling, to provide improved sampling accuracy and parallel data streams amenable to lower speed (parallel) digital
computation.
We investigate AFB and UWB sampling via a basis projection method. The method was introduced as a means
of UWB parallel sampling by Hoyos et. al. [7] and applied to UWB communications systems [8, 9, 10]. The
method first decomposes the signal into a basis over time
segments via a continuous-time inner product operation
and then samples the basis coefficients in parallel. The signal may then be reconstructed from the basis coefficients
to recover time domain samples, or further processing may
be carried out in the new domain [7].
We address several issues associated with the basisexpansion and sampling procedure, including choice of
basis, truncation error, rate of convergence and segmentation of the signal. We develop a mathematical model
of the procedure, using both standard (sine, cosine) basis
elements and general basis elements, and give this rep-
SAMPTA'09
resentation in both the time and frequency domains. We
compute exact truncation error bounds, and compare the
method with traditional sampling. We close by developing the method for binary signals.
2.
Sampling via Projection
Let f be a signal of finite energy whose Fourier transform
fb has compact support, i.e., f, fb ∈ L2 , with supp(fb) ⊂
[−Ω, Ω]. The signal is in the Paley-Wiener class P W (Ω).
For a block of time Tc , let
X
f (t) =
f (t)χ[(k)Tc ,(k+1)Tc ] (t) .
k∈Z
For this original development, we keep Tc fixed. We
later let Tc be adaptive and will denote the adaptive
time segmentation as τc (t). A given block fk (t) =
f (t)χ[(k)Tc ,(k+1)Tc ] (t) can be Tc − periodically continued,
getting
(fk )◦ (t) = (f (t)χ[(k)T ,(k+1)T ] (t))◦ .
c
c
◦
Expanding (fk ) (t) in a Fourier series, we get
X
(2πint/Tc )
◦
[
, where
(fk )◦ (t) =
(f
k ) [n]e
n∈Z
1
◦
[
(f
k ) [n] =
Tc
Z
(k+1)Tc
f (t)e(−2πint/Tc ) dt .
(k)Tc
Given that the original function f is Ω band-limited, we
can estimate the value of n for which fk [n] is non-zero.
At minimum, fk [n] is non-zero if
n
≤ Ω , or equivalently, n ≤ Tc · Ω .
Tc
Let
N = ⌈Tc · Ω⌉ .
(Note that the truncated block functionsfk are not bandlimited. We discuss this in section 3.) For this choice of
N , we compute
f (t) =
X
f (t)χ[(k)Tc ,(k+1)Tc ] (t)
k∈Z
X
◦
(fk ) (t) χ[(k)Tc ,(k+1)Tc ] (t)
=
k∈Z
≈
X n=N
X
k∈Z n=−N
(2πint/Tc ) χ
◦
[
(fk ) [n]e
[(k)Tc ,(k+1)Tc ] (t) .
355
Given this choice of the standard (sines, cosines) basis, we
can, for a fixed value of N , adjust to a large bandwidth Ω
by choosing small time blocks Tc . Also, after a given set
of time blocks, we can deal with a increase or decrease
in bandwidth Ω by again adjusting the time blocks, e.g.,
given an increase in Ω, decrease the time blocks adaptively
to τc (t), and vice versa. There is, of course, a price to be
paid. The quality of the signal, as expressed in the accuracy the representation of f , depends on N , Ω and Tc .
Theorem : [The Projection Formula] Let f , fb ∈ L2 (R)
and f ∈ P WΩ , i.e. supp(fb) ⊂ [−Ω, Ω]. Let Tc be a fixed
block of time. Then, for N = ⌈Tc · Ω⌉, f (t) ≈ fP (t),
where
fP (t) =
N
X X
(2πint/Tc )
fk [n]e
k∈Z n=−N
χ[kT
c ,(k+1)Tc ]
(t).
the other two to fluctuate. The easiest and most practical
parameter from the design factor to fix is N . For situations in which the bandwidth does not need flexibility, it
is possible to fix Ω and Tc by the equation N = ⌈Tc · Ω⌉.
However, if greater bandwidth Ω is need, choose shorter
time blocks Tc .
The Projection Method adapts to general orthonormal systems, much as Kramer-Weiss extends sampling to general orthonormal bases. Given a function f such that
f ∈ P WΩ , let Tc be a fixed time block. Define f (t),
fk (t) and fk ◦ (t) as in the beginning of the computation
above. Now, let {ϕn } be a general orthonormal system
for L2 [0, Tc ]. Then,
∞
X
fk ◦ (t) =
fk [n]ϕn (t), where fk [n] = hfk ◦ , ϕn i.
n=−∞
(1)
The Projection Method can adapt to changes in the signal.
Suppose that the signal f (t) has a band-limit Ω(t) which
changes with time. For example, let f be a signal from a
cell phone which changes from voice to a highly detailed
musical piece. This change effects the time blocking τc (t)
and the number of basis elements N (t). This, of course,
makes the analysis more complicated, but is at the heart
of the advantage the Projection Method has over conventional methods.
During a given τc (t), let Ω(t) = sup {Ω(t) : t ∈ τc (t)}.
For a signal f that is Ω(t) band-limited, we can estimate
the value of n for which fk [n] is non-zero. At minimum,
fk [n] is non-zero if
n
≤ Ω(t) , or equivalently, n ≤ τc (t) · Ω(t) .
τc (t)
Since f ∈ P WΩ , there exists N = N (Tc , Ω) such that
fk [n] = 0 for all n > N . Therefore, f (t) ≈ fP (t), where
fP (t) =
∞ X
N
X
k=−∞ n=−N
fk [n]ϕn (t) χ[kTc ,(k+1)Tc ] (t).
(3)
Given characteristics of the input class signals, the choice
of basis functions used in the the Projection Method can be
tailored to optimal representation of the signal or a desired
characteristic in the signal. We develop a Walsh system for
binary signals in section 4.
We close this section with a different system of segmentation for the time domain. This was created because it is
relatively easy to implement, cuts down on frequency error and has no loss of data in time. It was developed by
studying the de la Vallée-Poussin kernel used in Fourier
series. Let 0 < r < Tc /2 and let
Let
TriL (t) = max{[((Tc /(4r)) + r) − |t|/(2r)], 0} ,
N (t) = ⌈τc (t) · Ω(t)⌉ .
For this choice of N (t), we have the following.
Theorem : [The Adaptive Projection Formula] Let f ,
fb ∈ L2 (R) and f have a variable but bounded bandlimit Ω(t). Let τc (t) be an adaptive block of time and
given τc (t), let Ω(t) = sup {Ω(t) : t ∈ τc (t)}. Then, for
N (t) = ⌈τc (t) · Ω(t)⌉ , f (t) ≈ fP (t), where
fP (t) =
(t)
X N
X
k∈Z n=−N (t)
fk [n]e(2πint/τc ) χ[kτc ,(k+1)τc ] (t).
(2)
In comparison, Shannon Sampling examines the function
at specific points, then uses those individual points to
recreate the signal. The Projection Method breaks the
signal into segments in the time domain and then approximates their respective periodic expansions with a
Fourier series. This process allows the system to individually evaluate each piece and base its calculation on the
needed bandwidth. The individual Fourier series are then
summed, recreating a close approximation of the original
signal. It is important to note that instead of fixing Tc , the
method allows us to fix any of the three while allowing
SAMPTA'09
TriS (t) = max{[((Tc /(4r)) + r − 1) − |t|/(2r)], 0}
and
Trap(t) = TriL (t) − TriS (t) .
The Trap function has perfect overlay in the time domain
and 1/ω 2 decay in frequency space. When one time block
is ramping down, the adjacent block is ramping up at exactly the same rate. This leads to the Projection formula
N
X X
k∈Z n=−N
3.
((f ·Trap)k [n]e(2πint/(Tc +r)) Trap(t−k(Tc /2)) .
Error Analysis
To compute truncation error, we first calculate the Fourier
transform of both sides of the equation. Let f ∈ P W (Ω),
so f ∈ L2 and Ω band-limited. For N = ⌈Tc · Ω⌉,
fP (t) =
N
X X
k∈Z n=−N
(2πint/Tc )
fk [n]e
χ[kT
c ,(k+1)Tc ]
(t)
356
Taking the transform of both sides and evoking the relationship between the transform and convolution gives
N
X X
(2πint/Tc ) b
c
fP (ω) =
(ω) ∗
fk [n] e
k∈Z n=−N
χ[kT
(t) b(ω)
c ,(k+1)Tc ]
Performing the indicated transforms using the definition
results in
N
X X
n
c
fk [n] δ(ω − ) ∗
fP (ω) =
Tc
k∈Z n=−N
1
sin(πTc ω)
e(2πi(k− 2 )Tc ω)
πω
It is important to note that f · χ[kTc ,(k+1)Tc ] is no longer
band-limited, but it does decay at a rate less than or equal
to ω1 in frequency. Using the relationship between translation and modulation, we get the following.
Theorem : [The Fourier Transform of the Projection Formula] Let f , fb ∈ L2 (R) and f ∈ P WΩ , i.e.
supp(fb) ⊂ [−Ω, Ω]. Let Tc be a fixed block of time.
Then, for N = ⌈Tc · Ω⌉,
∞ X
N
X
1
n
fc
(ω)
=
fk [n]e(2πi(k− 2 )Tc (ω− Tc )
P
k=−∞ n=−N
n
c
sin(π( ωT
2 − 2 ))
π(ω − Tnc )
!
This replaces the sinc term in the equation above. The
Fourier coefficients are also different, and are computed in
the same method as the de la Vallée-Poussin kernel used
in Fourier series.
In the formula for the Projection Method, there is a reliance on a number N , representative of the number of
Fourier series components. In order to ensure maximum
utility from the formula, the difference between the infinitely summed series and the truncated must be made
a minimum. To do this, the mean square error must be
calculated. We compute this as a truncation error on the
number of Fourier coefficients used to represent a given
block fk . For a fixed N , the mean square error is
Computing and then simplifying gives
Z (k+1)Tc
X
1
|fk ◦ (t) −
fk [n]e(2πitn/Tc ) |2 dt
e2N =
Tc kTC
|n|≤N
=
1
Tc
Z
(k+1)Tc
kTC
SAMPTA'09
|
X
|n|>N
fk [n]e(2πitn/Tc ) |2 dt .
|n|>N
≤
X
|n|>N
1
|fk [n]| ·
Tc
2
Z
(k+1)Tc
kTc
12 dt =
X
|fk [n]|2
|n|>N
This demonstrates that the value of N has to be chosen
carefully. This truncation error perpetuates over all the
blocks.
The Projection Method experiences error due to truncation
in two separate categories: time and frequency. The error
in frequency is a function of the errors on each block due
to the choice of N . By duality, this gives us errors in time.
We can also get an error in time by loss of a given block
or blocks of information. This is easier to compute. Given
any lost or partially transmitted block fk,L , error is simply
kfk − fk,L k2 .
Error over the entire signal is computed by simply adding
up the blocks. Cell phone users are used to lost information blocks, which gives rise to the following frequently
used phrase – “Can you hear me now?”
4.
Binary Signals
(4)
The system using overlapping Trap functions has the
advantage
of 1/ω 2 decay in frequency. Let βL =
p
pTc /(4r) + r, αL = Tc /(4r) + r/2, βS =
Tc /(4r) + r − 1, αS = Tc /(4r) − r/2. The Fourier
transform of Trap is
2
2
sin(2παL (ω)
sin(2παS (ω)
(βL )
− (βS )
.
π(ω)
π(ω)
2
e2N = kfk − fk,N k22 = kfbk − fd
k,N k2 .
Applying the triangle inequality to the right side and then
exploiting the fact that e(2πitn/Tc ) is an orthonormal system, thus |e(2πitn/Tc ) | = 1, we arrive at the following:
Z (k+1)Tc X
1
|
fk [n]e(2πitn/Tc ) |2 dt (5)
e2N =
Tc kTC
The Walsh functions {ωn } form an orthonormal basis for
L2 [0, 1]. The basis functions have the range {1, −1}, with
values determined by a dyadic decomposition of the interval. The Walsh functions are of modulus 1 everywhere.
The functions are give by the rows of the unnormalized
Hadamard matrices, which are generated recursively by
1 1
H(2) =
1 −1
H(2k ) H(2k )
(k+1)
k
.
) = H(2) ⊗ H(2 ) =
H(2
H(2k ) −H(2k )
We point out that although the rows of the Hadamard matrices give the Walsh functions, the elements have to be
reordered into sequency order. Walsh arranged the components in ascending order of zero crossings (see [1]). The
Walsh functions can also be interpreted as the characters
of the group G of sequences over Z2 , i.e., G = (Z2 )N .
The Walsh basis is a well-developed system for the study
of a wide variety of signals, including binary. The Projection Method works with the Walsh system to create a
wavelet-like system to do signal analysis.
First assume that the time domain is covered by a uniform block tiling χ[kTc ,(k+1)Tc ] (t). Translate and scale
the function on this kth interval back to [0, 1] by a linear
mapping. Denote the resultant mapping as fk , which is an
element of L2 [0, 1]. Given that f ∈ P W (Ω), there exists an N > 0 (N = N (Ω)) such that hfk , ωn i = 0 for
all n > N . The decomposition of fk into Walsh basis
elements is
N
X
hfk , ωn i ωn .
n=0
357
Translating and summing up gives the Projection representation fP
fP (t) =
N
X X
k∈Z n=0
hfk , ωn i ωn χ[kTc ,(k+1)Tc ] (t).
fP (t) =
(6)
Next assume that the time domain is covered by a uniform
overlapping trapezoidal tiling Trap(t − k(Tc /2)). Note
that the construction of the trapezoidal system results in
the loss of no signal data, for just as a given block is ramping down, the subsequent block is ramping up at exactly
the same rate. Again translate and scale the function on
this kth interval back to [0, 1] by a linear mapping. Denote
the resultant mapping as fkT . The resultant function is an
element of L2 [0, 1]. Given that f ∈ P W (Ω), there exists
an M > 0 (M = M (Ω)) such that hfkT , ωn i = 0 for
all n > M . The decomposition of fkT into Walsh basis
elements is
M
X
hfk , ωn i ωn .
n=0
Translating and summing up gives the Projection representation fPT
fPT (t) =
N
X X
k∈Z n=0
5.
hfkT , ωn i ωn Trap(t−k(Tc /2)). (7)
Conclusions
The Projection Method gives a method for analog-todigital encoding which is an alternative to Shannon Sampling. Projection gives a procedure for the sampling of
a signal of variable or ultra-wide bandwidth Ω by varying the time blocks Tc . If f is Ω band-limited, we can
estimate the value of n for which the Fourier coefficients
fk [n] of a given time block are non-zero. At minimum,
fk [n] is non-zero if Tnc ≤ Ω, or equivalently, n ≤ Tc · Ω.
If N = ⌈Tc · Ω⌉, then, f (t) ≈ fP (t), where
fP (t) =
N
X X
k∈Z n=−N
(2πint/Tc )
fk [n]e
χ[kT
c ,(k+1)Tc ]
(t).
For fixed N , if greater bandwidth Ω is need, choose
shorter time blocks Tc . The price paid for this flexibility is in signal error, which has been computed above. The
Projection Method can also adapt to changes in the signal, e.g., f (t) has a band-limit Ω(t) which changes with
time. This change effects the time blocking τc (t) and the
number of basis elements N (t). During a given τc (t), let
Ω(t) = sup {Ω(t) : t ∈ τc (t)}. For a signal f that is Ω(t)
band-limited, we can estimate the value of n for which
fk [n] is non-zero. At minimum, fk [n] is non-zero if
n
≤ Ω(t) , or equivalently, n ≤ τc (t) · Ω(t) .
τc (t)
We let
N (t) = ⌈τc (t) · Ω(t)⌉ ,
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and have
(t)
X
X N
(2πint/τc )
fk [n]e
k∈Z n=−N (t)
χ[kτ
c ,(k+1)τc ]
(t).
This adaptable time segmentation makes the analysis more
complicated, but is at the heart of the advantage the Projection Method has over conventional methods. Subsequent work on this method will focus on minimizing error,
creating systems based on the Projection Method tailored
to different types of signals and optimizing signal reconstruction in a noisy environment.
References:
[1] Beauchamp, K. G., Applications of Walsh and Related Functions, Academic Press, London, 1984.
[2] Benedetto, J. J., Harmonic Analysis and Applications, CRC Press, Boca Raton, FL, 1997.
[3] Casey, S. D., “Sampling and reconstruction on
unions of non-commensurate lattices via complex
interpolation theory,” 1999 International Workshop on Sampling Theory and Applications, 48–
53, 1999.
[4] Casey, S. D., and Sadler, B. M., “New directions in
sampling and multi-rate A-D conversion via number theoretic methods,” IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2000), 3, 1417–1420, 2000.
[5] Casey, S. D., and Walnut, D. F., “Residue and
sampling techniques in deconvolution,” Chapter 9
in Modern Sampling Theory: Mathematics and
Applications, Birkhauser Research Monographs,
ed. by P. Ferreira and J. Benedetto, 193–217,
Birkhauser, Boston, 2001.
[6] Casey, S. D., “Two Problems from Industry and
Their Solutions via Harmonic and Complex Analysis, to appear in The Journal of Applied Functional Analysis, 31 pp., 2009.
[7] Hoyos, S., and Sadler, B. M. “Ultra wideband
analog-to-digital conversion via signal expansion,”
IEEE Transactions on Vehicular Technology, Invited Special Section on UWB Wireless Communications, vol. 54, no. 5, pp. 1609–1622, September 2005.
[8] Hoyos, S., Sadler, B. .M., and Arce, G., “Broadband multicarrier communication receiver based
on analog to digital conversion in the frequency
domain,” IEEE Transactions on Wireless Communications, vol. 5, no. 3, pp. 652–661, March 2006.
[9] Hoyos, S., and Sadler, B. .M. “Frequency domain
implementation of the transmitted-reference ultrawideband receiver,” IEEE Transactions on Microwave Theory and Techniques, Special Issue on
Ultra-Wideband, vol. 54, no. 4, Part II, pp. 1745–
1753, April 2006.
[10] Hoyos, S., and Sadler, B. M. “UWB mixed-signal
transform-domain direct-sequence receiver,” IEEE
Transactions on Wireless Communications, vol. 6,
no. 8, pp. 3038-3046, August 2007.
358
Non-Uniform Sampling Methods for MRI
Steven Troxler
(1) Arizona State University, Tempe AZ 85287-1804 USA.
Steven.Troxler@asu.edu
1.
Introduction
Simple Cartesian scans, which collect Fourier transform
data on a uniformly-spaced grid in the frequency domain,
are by far the most common in MRI. But non-Cartesian
trajectories such as spirals and radial scans have become
popular for their speed and for other benefits, like making motion-correction easier [12]. A major problem in
such scans, however, is reconstructing from nonuniform
data, which cannot be performed by a standard fast Fourier
transform (FFT) as in the Cartesian case.
Here, we briefly describe the most common reconstruction methods and the non-uniform fast Fourier transform
(NFFT) needed to complete the computations quickly.
We then give an overview of several current methods for
choosing a density compensation function (DCF) and suggest some possible improvements.
2.
Reconstruction Methods
The most common method for nonuniform reconstruction
in MRI is the Riemann approach, which approximates the
integral difining the inverse (continuous) Fourier transform using a Riemann sum
fw (x) =
J
X
wj fˆ(ξ j )e2πiξj ·x ,
(1)
j=1
where x ∈ ZdN are the pixel locations and ξ j , j =
1, ..., J, are the frequency locations at which we measure
the Fourier transform (we assume J ≥ N d ). As the subscript w suggests, this approach requires finding appropriate weights wj for each sample point in the reconstruction,
a major theoretical problem. An alternative method, called
implicit discretization (ID), assumes that the image itself
is a sum of evenly spaced delta impulses at the pixel points
of the final image, so that its Fourier transform is a finitedimensional, harmonic trigonometric polynomial. We can
then find a least-squares solution to the resulting system
of equations
X
(2)
f (x)e−2πix·ξj
fˆ(ξ j ) =
x∈Zd
N
This model, which is known to have negligible error (the
model error is the Gibb’s error that would appear in a
SAMPTA'09
Cartesian reconstruction), has the important advantage of
not depending on our arbitrary choice of weights.
These two approaches can be described in terms of matrix
algebra as follows: Let G be a J × N d matrix given by
Gj,x = e−2πiξj ·x .
Then we see immediately that
f w = G∗ W f˜,
(3)
where f w is the N d × 1 vector, indexed by ZdN , whose
xth entry is fw (x), f˜ is the J × 1 vector of measurements
whose jth entry is fˆ(ξ j ), and W is the N d × N d diagonal
matrix with diagonal equal to w. Once we have w, whose
determination is the main problem of interest, the remaining issue is one of computational complexity, since G∗ is
a very large unstructured matrix.
Fortunately, there is a fast method for computing products called the nonuniform fast Fourier transform (NFFT),
based on the approximate factorization
G ≈ C φ F Dφ ,
(4)
where C φ is a sparse, banded N d × J matrix of convolution interpolation coefficients which depends on our
choice of convolution kernel φ, F is the uniform M d ×M d
DFT matrix for some M > N, products of which are
rapidly computed via the FFT, and D φ is an M d × N d
modified diagonal deconvolution matrix, also depending
on φ, whose extra rows are zero. Since it is easy to compute products with all three factors, this algorithm can
be used to quickly approximate matrix products involving either G or G∗ . The theory of the NFFT, as applied
to MRI, was first laid out in [11] and [8]. Later, [4] found
bounds on the errors for Gaussian interpolation, and [23]
and [5] gave general estimates and gave sharper bounds
for Gaussian kernels. The most complete discussion of
NFFT theory is given in [16], while [15] presents many
of the proofs. Practical considerations like computational
load and numerical stability were addressed in [3] and
[17], while [1] and [6] presented two methods of efficient
interpolation using Kaiser-Bessel and Gaussian kernels.
In matrix form, the ID problem attempts to find a leastsquares solution to the problem
f˜ = Gf .
359
The ordinary least squares solution f OLS satisfies the normal equation
(5)
G∗ Gf OLS = G∗ fˆ.
Although the matrix G∗ G is far too large to invert, it is
symmetric, so we may use iterative methods like conjugate gradients to find the solution. The resulting solution
typically has excellent quality, but convergence is often
slow, making ordinary least squares expensive.
Conjugate gradients converges fastest when G∗ G is close
to the identity, which is unfortunately rarely the case unless the sampling density is reasonably close to unity. In
order to improve the convergence of conjugate gradients,
we introduce the weighted least squares problem, which
finds the least squares solution to
W 1/2 f˜ = W 1/2 Gf
by solving the normal equations
G∗ W bvecGf W LS = G∗ W fˆ,
where W is the modified diagonal density compensation
matrix used for the Riemann method. We expect an improvement in convergence because we know that the Riemann method gives much better results with W than without, which means G∗ W G approximate the identity much
better than G∗ G. From a signal processing perspective,
this has the additional benefit that we weight errors heavier at highly isolated observations of the Fourier transform, which heuristically contain more information about
the objective function than less isolated observations.
For either method, then, determining an appropriate value
of w is important. It is more essential in the Riemann approach, where a poor choice of w will lead to useless results. The ID method is known to converge quite well after
only a few iterations, even when a very rough approximation to w is used, but the better w, the fewer iterations are
required. It is worth noting that the first iteration, which
always moves in the direction of the residual, is actually
just a rescaling of the Riemann solution.
3.
Determination of an optimal DCF
3.1
Algebraic and Analytic Approaches
Since the equation
The weighted conjugate gradient method described at the
end of the previous section, whose first iteration performs
best when the matrix is as close to the identity as possible,
leads to a similar but slightly simpler condition, that
G∗ W G ≈ I,
in the sense that the eigenvalues of G∗ W G be as closely
clustered as possible. Several techniques have been proposed to use these conditions to find an algebraically ideal
DCF via use of a singular value decomposition or some
similar approach [22], [20]. These methods, however, tend
to have high computational complexity. This is a problem
if the same trajectory is not always used, as is the case
in many MRI applications in which iterative reconstruction is used to compensate for field inhomogeneities and
other measurement imperfections. Moreover, although
such algebraic methods generally give workable results,
other methods which take analytic considerations into account often perform better empirically. Possible reasons
why the theoretically optimal algebraic solutions fail to
give the best results include numerical instability and illconditioning. In some cases, the algebraic approaches
even result in DCF’s with negative weights at some points.
This contradicts our intuition, and empirical studies indicate that such DCF’s tend to perform relatively poorly.
The simplest analytic approaches to determining w are
based on the fact that the goal of the Riemann method
is to approximate a Riemann sum. For radial and analytic spiral trajectories, which may be smoothly parameterized, methods have been proposed which use the Jacobian of a change-of-coordinates [10], [7]. These techniques give very good results for certain spirals, although
for radial trajectories they tend to underweight points near
the center. An alternative analytic method, which works
for arbitrary nonuniform sampling schemes, is to construct
a Voronoi diagram, which partitions the sampled part of
frequency space into polygons about each sample point,
and weight the samples according to the area or volume
of those polygons [19]. This typically results in a good
image for radial trajectories. With other trajectories, the
results are generally inferior to alternative point-spreadfunction methods, although it was demonstrated in [9] that
performing a few iterations of the weighted conjugate gradient method using Voronoi weights produces an excellent
image.
f˜ = Gf ,
used directly in the CG reconstruction, provides an accurate mathematical model for the measurements which does
not depend on the choice of a sampling density w, the
clearest method of evaluating a DCF w is to require that
f˜ ≈ Gf w ,
where
f w = G∗ W f˜.
This is the same as requiring that
∗
G W
approximate the pseudoinverse (G∗ G)−1 G∗ of G.
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3.2
The Point Spread Function
Most of the best-performing methods for determining the
DCF when the trajectory is anything other than an analytic
spiral are based on analysis of the point-spread-function
(PSF). The PSF is defined as the inverse Fourier transform
w̌ of the DCF, where we
P view the DCF as a distribution on
Rd defined by w := j wj δξj . The PSF w̌ is then given
by
J
X
w̌(x) =
(6)
wj e2πix·ξj .
j=1
This is what the algorithm would produce if the true object
were a delta impulse located at zero. The observed data
would be a vector of all ones, so the reconstruction would
360
be the result of applying G∗ to w itself, i.e., the function
defined by (6).
If f is a more general object, it follows from the convolution theorem (for distributions) that the reconstructed
function fw will be equal to the convolution f ∗ w̌ of the
actual object f with the PSF. The more closely the PSF
resembles a delta impulse, the better the reconstruction.
It is important to note that, since the PSF is a (nonharmonic) trigonometric polynomial, it will not decay at infinity. Clearly, then, the best that we can hope for is that
w̌ will resemble a delta impulse in some compact neighborhood of the origin. Recall that, by accepting the ID
model as having negligible error, we are assuming that f
is a finite-dimensional vector defined on ZdN which we associate with a distribution supported on ZdN for notational
convenience when dealing with convolutions. Since the
terms w̌(z)f (x − z) defining
X
fw (x) = w̌ ∗ f (x) =
w̌(z)f (x − z)
(7)
z∈Zd
ZdN ,
are nonzero only if (x − z) ∈
and we only want
to find the reconstruction fw (x) for x ∈ ZdN , we conclude that the only values of z for which w̌(z) matter
are z ∈ Zd2N . It is also worth noting that not all points
z ∈ Zd2N appear equally often in the convolution defining f W . The origin will appear in one term of every sum,
whereas values of z near the edge of Zd2N will appear only
occasionally.
For notational convenience, let A be the field of view
[−N/2, N/2]d and let B be the region of optimization
[−N, N ]d . PSF optimization techniques find some computational way of minimizing the error
E = w̌ − δ
over this region of optimization B.
By carefully looking at (7), we see that the frequency with
which a PSF error at x actually occurs in the final image is
proportional to p = χA∗χA. Since errors are unavoidable
and we would like to minimize the important errors, we
introduce the weighted error, given by
E = pw̌ − δ,
(8)
where p is called the error profile. This error can be expressed in the Fourier domain as
Ê = p̂ ∗ w − 1 = χ̂2A ∗ w − 1.
(9)
Our goal is to minimize these errors, thereby minimizing
the error in the final reconstruction fw = w̌ ∗ f.
Although this optimal kernel p was suggest only recently
in [13], convolution techniques for minimizing the Fourier
domain PSF error Ê have been used for some time. In
one of the early gridding papers, Jackson et. al. proposed
taking w to be equal to
w1 =
w0
,
φ ∗ w0
(10)
where w0 is a DCF of unity (in distributional form) and
φ is the gridding kernel [8]. This method predates PSF
SAMPTA'09
techniques, and was instead motivated by the intuitive idea
that φ ∗ w0 would give a reasonable, estimate of the sampling density. Later researchers noted, however, that if we
φ with p̂, we would expect this ratio correction to make
w1 ∗ p̂ closer to unity than w0 ∗ p̂ regardless of the initial density w0 [14]. An iterative technique, based on this
observation, starts with a constant DCF w0 and takes
wi+1 =
wi
.
p̂ ∗ wi
(11)
Since p̂ can be effectivly truncated, each iteration can be
computed quickly, particularly if an efficient sorting algorithm is used to avoid time-consuming searches for the
nonzero terms p̂(ξk − ξ j )w(ξ j ) in the convolution [13].
Another iterative algorithm, aimed at the same goal of
achieving Ê = 1, uses an additive correction instead of
a ratio-based correction, taking
wi = wi−1 + σ(1 − p̂ ∗ wi−1 ),
where σ ∈ (0, 1) is a parameter controlling convergence
[18]. Taking σ close to 1 may result in the fastest convergence, but could also lead to instability and a failure to
converge.
The advantage of these iterative techniques is that they
are conceptually simple, computationally fast, and empirically give results as good as any current methods when
the correct error profile p is used and the number of iterations is determined experimentally. A disadvantage is that,
although they work conceptually and empirically, there is
no theoretical basis for claiming that they converge to the
optimal solution, and, in fact, experimental evidence indicates that the mean square error in fw can actually rises if
the algorithm is allowed to run too long. This may be due
to numerical instability, or to a failure of the mathematical
algorithm itself to technically converge.
An algebraic method of optimizing the PSF, which has
more theoretical grounding than convolution-based methods, attempts to directly solve the inverse problem
GG∗ w = u,
where u is a vector of all ones. The direct solution to
this problem via conjugate gradients using the NFFT was
proposed in [21], but as with the algebraic solutions for
w based on the least-squares method, this can result in a
w with wide variations and sometimes even negative entries, which does not match our expectation for a density
and empirically gives inferior results. A regularization of
this method was proposed in [2] which instead attempts to
solve
(GG∗ + σ 2 I)w = u + σ 2 w1 ,
where w1 is an initial nonnegative and smoothly varying estimate of the density, say, Jackson’s weight (10), or,
more optimally, the result of one or two iterations of (11).
This second approach ensures that the solution behaves as
we would expect a DCF to behave, and empirically gives
better results than the unregularized method. The algorithm given in [2] also incorporates Jacobi preconditioning
to speed convergence of the conjugate gradient iterations.
Knowing that Pipe and Johnson’s error profile p provides
an optimal weight on errors in the point-spread function,
361
it might be preferable to modify the approach in [2] in
two ways. The first is that, since we need to minimize
PSF errors over twice the support of f, we replace the
NDFT matrix G with G1 , where the uniform grid has
twice the radius of that used by G. This avoids the risk
that we might ignore PSF errors which, according to the
convolution defining f w , appear in the Riemann reconstruction. The second is replacing G1 G∗1 , which treats
all PSF errors as equally important, with G1 P G∗1 , where
P contains the values of the optimal error profile p. To
the author’s knowledge, this has never been tried, but in
light of experiments reported by [13] indicating that the
approaches taken in [2] and [13] both yield the some of
the best results of methods proposed to date for arbitrary
trajectories, combining their methods might produce the
best results seen yet.
4.
Acknowledgments
This work was part of an undergraduate research project
under the supervision of Dr. Svetlana Roudenko. The author would also like to thank Ken Johnson and Dr. Jim
Pipe for providing graphical illustrations and for helpful discussions of this content, as well as Dr. Doug
Cochran for his assistance and input. The project was partially supported by NSF-DUE # 0633033 and NSF-DMS
# 0652853.
References:
[1] Philip J. Beatty, Dwight G. Nishimura, and John M.
Pauly. Rapid gridding reconstruction with a minimal oversampling ratio. IEEE Trans. Med. Imag.,
24(6):799–808, June 2005.
[2] Mark Bydder, Alexey A. Samsonov, and Jiang Du.
Evaluation of optimal density weighting for regridding. Magnetic Resonance Imaging, 25:695–702,
2007.
[3] S. Dunis and D. Potts. Time and memory requirements of the nonequispaced fft. Sampling Theory in
Signal and Image Processing, 7:77–100, 2008.
[4] A. Dutt and V. Rokhlin. Fast fourier transforms
for nonequispaced data. SIAM Journal of Scientific
Computing, 14(6):1368–1393, 1993.
[5] B. Elbel and G. Steidl. Fast fourier transform for
nonequispaced data. In C. K. Chui and L. L. Schumaker, editors, Approximation Theory IX. Vanderbilt
University Press, Nashville, 1998.
[6] Leslie Greengard and June-Yub Lee. Accelerating
the nonuniform fast fourier transform. SIAM Review,
46(3):443–454, 2004.
[7] R.D Hodge, R.K.S. Kwan, and G.B. Pike. Density
compensation functions for spiral MRI. Magnetic
Resonance in Medicine, 38:117–128, 1997.
[8] J. Jackson, C. Meyer, D. Nishimura, and A. Macovski. Selection of a convolution function for fourier
enversion using gridding. IEEE Trans. Med. Imag.,
10(3):473–478, Sep. 1991.
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[9] Tobias Knopp, Stefan Kunis, and Daniel Potts. A
note on the iterative mri reconstruction from nonuniform k-space data. International Journal of Biomedical Imaging, 2007.
[10] C. Meyer, B. S. Hu, D. Nishimura, and A. Macovski. Fast spiral coronary artery imaging. Magnetic
Resonance in Medicine, 28:202–213, 1992.
[11] J. O’sullivan. A fast sinc function gridding algorithm
for fourier inversion in computerized tomography.
IEEE Transactions on Medical Imaging, MI-4:200–
207, 1985.
[12] James G. Pipe.
[13] J.G. Pipe and Kenneth Johnson. Convolution kernel
design and efficient algorithm for sampling density
correction. Preprint, 2008.
[14] J.G. Pipe and P. Menon. Sampling density compensation in MRI: Rationale and an iterative numerical
solution. Magnetic Resonance in Medicine, 41:799–
808, June 2005.
[15] D. Potts. Schnelle Fourier-Transformationen für
nichtäquidistante Daten und Anwendungen. Habilitation, Universität zu Lübeck, 2003.
[16] D. Potts, G. Steidl, and M. Tasche. Fast fourier transforms for nonequispaced data: A tutorial. In J. J.
Benedetto and P. J. S. G. Ferreira, editors, Modern
Sampling Theory: Mathematics and Applications,
pages 247–270. Birkhäuser, Boston, 2001.
[17] D. Potts and M Tasche. Numerical stability of nonequispaced fast fourier transforms. Journal of Computational Applied Mathematics, 222:655–674, 2008.
[18] Y. Qian, J. Lin, and D. Jin. Reconstruction of mr
images from data acquired on an arbitrary k-space
trajectory using the same-image weight. Magnetic
Resonance in Medicine, 48:306–311, 2002.
[19] V. Rasche, R. Proksa, R. Sinkus, P. Bornert, and
H. Eggers. Resampling of data between arbitrary
grids using convolution interpolation. IEEE Trans.
Med. Imag., 18:427–434, 1999.
[20] D. Rosenfeld. An optimal and efficient new gridding
algorithm using singular value decomposition. Magnetic Resonance in Medicine, 40:14–23, 1998.
[21] A.A. Samsonov, E.G. Kholmovski, and C.R. Johnson. Determination of the sampling density compensation function using a point spread function modeling approach and gridding approximation. volume 11, 2003.
[22] Hossein Sedarat and Dwight G. Nishimura. On the
optimality of the gridding reconstruction algorithm.
IEEE Transactions on Medical Imaging, 19(4):306–
317, 2000.
[23] G. Steidl. A note on fast fourier transforms for
nonequispaced grids.
Advanced Computational
Mathematics, 9:337–353, 1998.
362
On approximation properties of sampling
operators defined by dilated kernels
Andi Kivinukk (1) and Gert Tamberg (2)
(1) Dept. of Mathematics, Tallinn University, Narva Road 25, 10120 Tallinn, Estonia.
(2) Dept. of Mathematics, Tallinn University of Technology, Ehitajate tee 5 19086 Tallinn, Estonia.
andik@tlu.ee, gert.tamberg@mail.ee
Abstract:
In this paper we consider some generalized Shannon sampling operators, which are defined by band-limited kernels. In particular, we use dilated versions of some previously known kernels. We give also some examples of
using sampling operators with dilated kernels in imaging
applications.
1.
Introduction
For the uniformly continuous and bounded functions f ∈
C(R) the generalized sampling series with a kernel function s ∈ L1 (R) are given by (t ∈ R; W > 0)
∞
X
(SW f )(t) :=
k=−∞
where
∞
X
k
f ( )s(W t − k)
W
s(u − k) = 1,
[16]), in Signal Analysis in particular. Many kernels can
be defined by (3), e.g.
1) λ(u) = 1 defines the sinc function;
2) λ(u) = 1 − u defines the Fejér kernel (cf. [15])
sF (t) =
3) λH (u) := cos2
kernel (see [7])
sH (t) :=
|s(u − k)| < ∞
sC,b (t) :=
(2)
(4)
m
1X
bj sinc(t − j) + sinc(t + j)
2 j=0
(5)
provided
m
⌊X
2 ⌋
(u ∈ R).
λ(u) cos(πtu) du =
r
π ∧
λ (πt).
2
0
(3)
These types of kernels arise in conjunction with window
functions widely used in applications (e.g. [1], [2], [11],
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bj cos jπu
defines the Blackman-Harris kernel (see [9])
If the kernel function is s(t) = sinc (t) := sinπtπt , we
get the classical (Whittaker-Kotel’nikov-)Shannon operasinc
tor SW
. The idea to replace the sinc kernel (sinc (·) 6∈
1
L (R)) by another kernel function s ∈ L1 (R) appeared
first in [15], where the case s(t) = (sinc (t))2 was considered. A systematic study of sampling operators (1) for arbitrary kernel functions s was initiated at RWTH Aachen
by P. L. Butzer and his students since 1977 (see [3], [4],
[14] and references cited there).
In this paper we consider the generalized sampling series
with even band-limited kernels s, defined as the Fourier
transform of an even window function λ ∈ C[−1,1] ,
λ(0) = 1, λ(u) = 0 (|u| > 1) by the equality
s(t) := s(λ; t) :=
m
X
j=0
(1)
k=−∞
Z1
1 sinc t
= O(|t|−3 );
2 1 − t2
4) the general cosine window
and their operator norms are
kSW k =
= 12 (1 + cos πu) defines the Hann
λC,b (u) :=
k=−∞
∞
X
πu
2
1
t
sinc 2 = O(|t|−2 );
2
2
b2j =
j=0
⌊ m+1
2 ⌋
X
b2j−1 =
j=1
1
.
2
(6)
From approximation theory point of view at least two
problems for the generalized sampling operators SW :
C(R) → C(R) have some interest:
1) to calculate the operator norms
kSW k = sup
u∈R
∞
X
|s(u − k)|;
(7)
k=−∞
2) to estimate the order of approximation
kf − SW f kC 6 M ωk (f,
1
)
W
(8)
in terms of the k-th modulus of smoothness ωk (f, δ).
2.
Interpolating generalized sampling operators with dilated kernels
Let us consider the dilated kernel sα (t) = αs(αt). The
Shannon operators with sinc kernel satisfy the interpolatory conditions
sinc
(SW
)(
k
k
) = f ( ) (k ∈ Z).
W
W
(9)
363
When we replace the sinc kernel with a band-limited one
(3), we may lose the interpolatory property (9), but using
the dilated kernel s̃(t) = 2s(2t), we can recover the interpolatory property. If s ∈ Bπ1 , then sα ∈ Bα1 π , and the
condition (2) is valid for 0 < α 6 2, therefore we get
∞
the sampling operator SW,α : C(R) → BαπW
⊂ C(R).
p
Here Bσ stands for the Bernstein class consisting of those
bounded functions f ∈ Lp (R) (1 6 p 6 ∞) which can be
extended to an entire function f (z) (z ∈ C) of exponential
type σ.
Using the Nikolskii inequality [13], we get the bounds for
the operator norm.
∞
Theorem 1. Let the operators SW : C(R) → BW
π ⊂
∞
C(R), SW,α : C(R) → BαW π ⊂ C(R) are defined by
(1) with kernels s and sα , respectively. Then
ksk1 6 kSW,α k 6 (1 + απ)kSW k (0 < α 6 2).
The order of approximation by operators SW,α we can estimate via modulus of smoothness ωk (f, σ). Next theorem
generalizes slightly the result in [10] (Th. 1.3).
Theorem 2. Let SW : C(R) → C(R), SW,α : C(R) →
∞
BαW
π ⊂ C(R) be sampling operators defined by (1) with
kernel functions s ∈ Bπ1 , sα ∈ Bα1 π , respectively.
1) If 0 < α 6 1, then there exist positive constants C1,α
and C2,α such that
C1,α kSαW f −f kC 6 kSW,α f −f kC 6 C2,α kSαW f −f kC .
2) Moreover, if 0 < α < 2, then
kSW f − f kC 6 Mk ωk (f,
1
),
W
(10)
implies
Example. The Blackman-Harris sampling operator CW,b
is defined by the window function
bj cos(πju).
j=0
In [9] we proved that for some values of the parameters
b = (b0 , b1 , . . . , bm ) ∈ Rm+1 we can estimate the order
∞
of approximation by operators CW,b : C(R) → BW
π ⊂
1
C(R) via the modulus of continuity ω2ℓ (f, W ) (ℓ 6 m).
More precisely (see [9], Th. 3), let ℓ, 1 6 ℓ 6 m, be fixed.
If for every k = 0, . . . , ℓ − 1
m
X
j 2k bj = 0 (00 = 1),
(11)
j=0
then
1
).
(12)
W
Now by Theorem 2 we obtain for the corresponding di∞
lated sampling operator CW,b;α : C(R) → BαW
π ⊂
C(R) with 0 < α < 2 the estimate
kf − CW,b f kC 6 Mb,ℓ ω2ℓ (f,
kf − CW,b;α f kC 6 Mb,ℓ,α ω2ℓ (f,
SAMPTA'09
1
).
W
∞
then S̃W : C(R) → B2W
π ⊂ C(R) is an interpolating
sampling operator.
Examples. For the Hann window function λH (u) the condition (15) holds and we get the interpolating Hann sam∞
pling operator H̃W : C(R) → B2W
π ⊂ C(R). Taking
b0 = 1/2, b2j = 0(j ∈ N) in (11) gives us the BlackmanHarris window function for which the condition (15) is
fullfilled (see [10]).
1
In the case when s ∈ Bβπ
, 0 < β < 1 and (15) holds
for the corresponding window function we can prove the
following theorem.
Theorem 4. Let the sampling operator S̃W be defined by
(1) using the kernel s̃(t) := 2s(2t), where the kernel s ∈
1
⊂ L1 (R), 0 < β < 1, is generated by (3) with a
Bβπ
window function λ. If (15) is valid, then for every k ∈ N
there exist a constant Mk such that
1
).
W
Example. So-called Lanczos n-kernels
for some constant Mk,α > 0.
λC,b :=
Theorem 3. Let the sampling operator S̃W be defined by
(1) using the kernel s̃(t) := 2s(2t), where the kernel s ∈
Bπ1 ⊂ L1 (R) is generated by (3) with a window function
λ. If
λ(u) + λ(1 − u) = 1 (u ∈ [0, 1])
(15)
kS̃W f − f kC 6 Mk ωk (f,
1
kSW,α f − f kC 6 Mk,α ωk (f, )
W
m
X
The case m = ℓ = 1 gives the Hann sampling operator HW : C(R) → C(R), which often has been
used in practise. For the corresponding dilated operator
∞
HW,α : C(R) → BαW
π ⊂ C(R) for 0 < α < 2 we
obtain
1
kf − HW,α f kC 6 Mα ω2 (f, ).
(14)
W
See Figure 2 for corresponding kernels.
The next theorem gives hints how to construct the interpolating sampling series.
(13)
s̃L,n (t) := sinc
t
sinc t,
n
which has been often used in image processing. The Lanczos 3-kernel is especially popular in imaging ((see [16]
and references cited there). They are defined by De la
Vallée Poussin window function
1,
0 6 u 6 n−1
2n ,
1
n−1
(1 + n(1 − 2u)), 2n < u < n+1
λL,n (u) :=
2
2n ,
0,
u > n+1
2n .
If n > 1, then the De la Vallée Poussin window function
λL,n satisfies the conditions (15) and s̃L,n ∈ B(1n+1 )π ,
2n
hence Theorem 4 is applicable. If n = 1, then we get the
Fejér sampling operator (cf. [15]), for which we do not
have even an estimate via the modulus of continuity ω1 .
3.
Applications in 2D imaging
A natural application of sampling operators with dilated
kernels is imaging. We can represent an discrete 2D image
f as a continuous function using sampling series
X
f (j, k)s1 (x − j)s2 (y − k). (16)
(Sf )(x, y) :=
j,k
364
Figure 1: Original image, derivatives with Hann kernel s̃H (t) = 2sH (2t) and sH,1/4 (t) = 12 sH ( 14 t) (ϕ =
Many image resizing (resampling) algorithms use such
type of representation (see [16], [12], [6]). If the image
data is exact, then we can take interpolating kernels s1
and s2 , like interpolating Hann, Blackman-Harris or Lanczos, and enlarge (up-sample) image, having (Sf )(j, k) =
f (j, k). If we want to reduce the image size (downsample) (magnification γ < 1) then, for eliminating artifacts, we can choose a dilated kernel sα with in some sense
optimal value of α = 2γ (see Figure 2). The artifacts in
down-sampled images appear, because details that are resized to smaller than one pixel will be misrepresented by
larger aliases (see [5], [6]). Depending on the choice of
∞
the parameter value α we have SW,α : C(R) → BαW
π
i.e. a function belonging to a class for bandlimited functions, for which the Fourier’ transform vanishes outside
of the interval [−αW π, αW π]. This approach eliminates
higher spatial frequencies, being equivalent to the use of
low-pass filter. Also in the case, when the resolution of
the optical system is less than the resolution of the sensor, we can choose the value of the dilation parameter α
accordingly.
Using the representation (16) we can apply different imaging technics. For image enhancement we can use the unsharp masking (see [5], [6]), i.e. to subtract a blurred
version of an image from the image itself. For the representation of original image f (x, y) we can choose in (16)
the interpolating kernels (dilation by α = 2), but to get
blurred version fb (x, y), we choose in (16) the dilated kernels with small parameter α, like sH,1/2 in Figure 2. We
can control the amount of unsharp masking choosing the
parameter a < 0:
fusm (x, y) = (1 − a)f (x, y) + afb (x, y).
Another well-known image enhancement method uses the
derivatives of image. First derivatives in image processing
are implemented using the magnitude of the gradient. The
representation (16) gives us a natural way to implement
derivatives. Indeed
X
f (j, k)s′1 (x − j)s2 (y − k),
fx (x, y) :=
j,k
fy (x, y) :=
X
j,k
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f (j, k)s1 (x − j)s′2 (y − k).
2π
3 ).
Surprisingly, if we choose Hann kernel s1 = s2 = sH and
x, y ∈ Z, then the discrete convolution
fx (p, q) ≈
p+1
X
q+1
X
f (j, k)s′H (p−j)sH (q −k) (17)
j=p−1 k=q−1
gives us the well-known Sobel filter (see [5], [6])
1 0 −1
1
2 1 0 −1 = 2 0 −2 .
1 0 −1
1
Indeed, sH (k) = 0 (k ∈ Z) if |k| > 1 (see Figure 1) and
we get 41 (1, 2, 1). For s′H we use the first 3 values only,
i.e. 38 (1, 0, −1).
We can easily compute a directional derivative
X
fϕ (x, y) :=
f (j, k)s′1 (x − j) cos ϕ − (y − k) sin ϕ ×
j,k
×s2 (y − k) cos ϕ + (x − j) sin ϕ ,
To get the edges with different spatial frequency, we
choose the dilation parameter (see Figure 1).
Second derivatives in image processing are implemented
using the Laplacian. Using the representation (16) we get
△ f (x, y) := fxx (x, y) + fyy (x, y) =
X
f (j, k) s′′ (x − j)s(y − k) + s(x − j)s′′ (y − k) .
j,k
In image processing we use the derivatives for edge detection. Changing the dilation parameter α for the kernel
sα (t) = αs(αt) we can detect edges with different spatial
frequencies.
In calculations we must use the truncated sampling series
(p, q ∈ Z)
(Sf )mn (p, q) :=
p+m
X
q+n
X
f (j, k)s1 (p − j)s2 (q − k)
j=p−m k=q−n
and have the truncation error. We can use kernels with
finite support like the combinations of B-splines, considered in [4], to get rid of the truncation error, but in some
365
grants 6943, 7033, and by the Estonian Min. of Educ. and
Research, projects SF0132723s06, SF0140011s09. The
second author wants to thank Archimedes Foundation for
support.
References:
Figure 2: Unsharp mask with Hann kernel sH,1/2 =
1
1
2 sH ( 2 t), a = −1.7.
1.0
0.8
0.6
0.4
0.2
-4
2
-2
4
-0.2
Figure 3: Hann kernel s̃H (t) = O(|t|−3 ), Lanczos kernel
sL,3 (t) = O(|t|−2 ) and sinc(t) = O(|t|−1 ) .
cases other types of kernels are more suitable. For minimizing the truncation error the kernel s(t) must decrease
rapidly when |t| → ∞. The sinc function does not belong even to L1 . Therefore using the kernels in form
s(t) = θ(t) sinc t, where θ(t) is some window function
(see [11]), is well-known. In most cases of we lose the
important property (2) and do not get a generalized sampling series anymore. The kernels in our approach, i.e.
kernels defined via Fourier transform of window functions, allow us to get good approximation properties and
are rapidly decreasing. In Figure 3 we take the Hann kernel s̃H (t) = O(|t|−3 ) and compare it with the Lanczos
kernel sL,3 (t) = O(|t|−2 ), which is one of the most used
kernels in imaging (see [16]). In the case of BlackmanHarris kernels (5), considered more precisely in [9], we
have sC,b = (|t|−2ℓ−1 ) if for every k = 0, . . . , ℓ − 1
m
X
j 2k bj = 0.
j=0
We defined many rapidly decreasing kernels also in [8],
[7], [10].
4.
Acknowledgments
This research was supported by European Social Fund
Funds Doctoral Studies and Internationalisation Programme DoRa, by the Estonian Science Foundation,
SAMPTA'09
[1] H. H. Albrecht. A family of cosine-sum windows
for high resolution measurements. In IEEE International Conference on Acoustics, Speech and Signal
Processing, Salt Lake City, Mai 2001, pages 3081–
3084. Salt Lake City, 2001.
[2] R. B. Blackman and J. W. Tukey. The measurement
of power spectra. Wiley-VCH, New York, 1958.
[3] P. L. Butzer, G. Schmeisser, and R. L. Stens. An
introduction to sampling analysis. In F Marvasti,
editor, Nonuniform Sampling, Theory and Practice,
pages 17–121. Kluwer, New York, 2001.
[4] P. L. Butzer, W. Splettster, and R. L. Stens. The sampling theorems and linear prediction in signal analysis. Jahresber. Deutsch. Math-Verein, 90:1–70, 1988.
[5] R. C. Gonzalez and R. E. Woods. Digital Image Processing. Second Edition. Prentice-Hall, 2002.
[6] B. Jähne. Digital Image Processing: Concepts, Algorithms, and Scientific Applications. Springer Verlag, Basel, Stuttgart, 1997.
[7] A. Kivinukk and G. Tamberg. On sampling operators defined by the Hann window and some of their
extensions. Sampling Theory in Signal and Image
Processing, 2:235–258, 2003.
[8] A. Kivinukk and G. Tamberg. Blackman-type windows for sampling series. J. of Comp. Analysis and
Applications, 7:361–372, 2005.
[9] A. Kivinukk and G. Tamberg. On Blackman-Harris
windows for Shannon sampling series. Sampling
Theory in Signal and Image Processing, 6:87–108,
2007.
[10] A. Kivinukk and G. Tamberg. Interpolating generalized Shannon sampling operators, their norms and
approximation properties. Sampling Theory in Signal and Image Processing, 8:77–95, 2009.
[11] R. J. Marks. Fourier Analysis and Its Applications.
Oxford University Press, New York, 2009.
[12] E. Meijering and et al. Quantitative comparison of
sinc-approximating kernels for medical image interpolation. In C. Taylor and A. Colchester, editors,
Medical Image Computing and Computer-Assisted
Intervention, pages 210–217. Springer, Berlin, 1999.
[13] S. M. Nikolskii. Approximation of Functions of Several Variables and Imbedding Theorems. Springer,
Berlin, 1975. (Orig. Russian ed. Moscow, 1969).
[14] R. L. Stens. Sampling with generalized kernels. In
J. R. Higgins and R. L. Stens, editors, Sampling Theory in Fourier and Signal Analysis: Advanced Topics. Clarendon Press, Oxford, 1999.
[15] M. Theis. Über eine interpolationsformel von de la
Vallee-Poussin. Math. Z., 3:93–113, 1919.
[16] K. Turkowski. Filters for common resampling tasks.
In A. S. Glassner, editor, Graphics Gems I, pages
147–165. Academic Press, 1990.
366
Reconstruction of signals in a shift-invariant space from
nonuniform samples
Junxi Zhao
College of Mathematics and Physics,
Nanjing university of posts and telecommunications,
Nanjing,21003, P.R. China
junxi_zhao@163.com
practical algorithm for implementing the optimal signal
reconstruction. In Section , some numerical examples
of reconstructing signals demonstrate the effectiveness of
the proposed method.
Abstract- In this paper, we consider the method for
reconstructing a signal from a finite number of samples. In
shift-invariant space framework, we derive an
aproximately min-max optimal interpolator to to
reconstruct a signal on
an interval. An effective
non-iterative algorithm for signal reconstruction is given
also. Numerical examples show the effectiveness of the
proposed method.
Index Terms–sampling, signal reconstruction, scaling
function, shift-invariant space
.THE PROBLEM FORMULATION
Throughout the paper, we focus on one-dimensional
signals and denote the space of signals of finite energy on
R by L2 (R) . Let || f || 2 = ∫ | f (t ) |2 dt be the energy
of
a
signal
Given
K
scaling
functions ϕ1 (t ), ϕ2 (t ),A , ϕ K (t ) ∈ L (R ) , the shiftinvariant space V (ϕ1 , ϕ2 ,A , ϕ K ) is a Hilbert space
defined as
V (ϕ1 ,ϕ2 ,A,ϕK ) = close f (t ) ∈ L2 (R) :
. INTRODUCTION
The problem of signal reconstruction is pervasive in
many areas of signal processing, such as in designs of
nonuniform antenna arrays, sparse array beamforming, the
restoration of signals with missing samples, image
acquisition, etc [1-3]. The classical Shannon’s sampling
theorem was extended theoretically to general
shift-invariant subspaces, and various generalized
sampling theorems concerning band-limited and
nonband-limited signals have been proposed [4-9].
However, the problem of reconstructing a continuous-time
signal from its finite number of nonuniform samples is
often encountered in practical applications, and truncating
infinite reconstruction leads to errors.
Bandlimited and non bandlimited signals are often
modeled by shift-invariant spaces. Some authors have
proposed interpolation methods and iterative methods for
reconstructing signals in shift-invariant spaces in the
signal processing literatures[10-14]. A non-iterative
reconstruction method is effective to reconstruct
continuous-time signals from a finite number of samples
by using a suitable interpolator. The Yen interpolator is
well known to reconstruct band-limited signals in both
minimal energy and least squares senses [13]. Some
interpolation methods in shift-invariant spaces were
given[9-10,12-14].
In this paper, we are interested in optimally constructing
signals in a shift-invariant space from a finite number of
nonuniform samples, and develop a new method for
reconstructing continuous-time signal on a interval. The
upper bound of reconstruction error is derived. We also
propose a practical reconstruction algorithm. The method
of signal reconstruction can be regarded as a
generalization of Yen’s in general shift-invariant spaces.
The paper is organized as follows: in Section
we
formulate the optimal reconstruction problem in
shift-invariant spaces, and Section
derives a new
method to reconstruct a signal from a finite number of
arbitrarily distributed samples. Section
propose a
SAMPTA'09
f (t ) ∈ L (R ) .
R
2
2
{
f (t ) = ∑∑ ck (n)ϕk (t − n)
K
, ( ck (n) ) ∈ l 2 (Z), k = 1, 2,A, K}
k =1 n
We assume that
{ϕk (t − n) |1 ≤ k ≤ K , n ∈ Z }
forms a
frame of V (ϕ1 , ϕ2 ,A , ϕ K ) , i.e., there exist two constants
A > 0 and B < +∞ such that
A || f ||2 ≤ ∑ ∑ | ( f , ϕ k (t − n)) |2 ≤ B || f ||2
K
k =1
(1)
for any f ∈ V (ϕ1 , ϕ 2 ,A , ϕ K ) .
To make the sampling of functions in
V (ϕ1 , ϕ2 ,A , ϕ K ) well-defined, we additionally assume that
there exists a constant C such that
n
∑ ∑|ϕ
K
k =1
t ∈ [0, 1]
k
(t − n) |< C
(2)
n
f (t ) = ∑ ∑ ck (n)ϕk (t − n) with ( ck (n) ) ∈ l 2 (Z) .
for
any
.
To
see
this,
let
K
k =1
From
n
(2) and (3) it follows that
⎛ K
⎞ 2⎛ K
⎞ 2
| f (t ) |≤ ⎜ ∑ ∑ | ϕk (t − n) |2 ⎟ ⎜ ∑ ∑|ck (n) |2 ⎟
⎝ k =1 n
⎠ ⎝ k =1 n
⎠ (3)
≤ C ' || f ||, t ∈ R
where C ' is a constant.
It is known from [8] that the assumption (3) implies that
V (ϕ1 , ϕ 2 ,A , ϕ K ) is a reproducing kernel Hilbert space. For
a function f (t ) in V (ϕ1 , ϕ2 ,A , ϕ K ) , we adopt the
1
1
1
367
E (τ ) = ∑ || ek ,τ −∑ hm (τ )ek , m ||2 =
f# (t ) = ∑ hm (t ) f (tm )
following interpolator
K
M
k =1
∑ || e τ || −2∑ e
m =1
K
to reconstruct f (t ) on the interval containing sampling
where
Ak =
| f (t ) − f# (t ) |2
inf sup
,
h
|| f ||2
f
interpolator, we discuss the optimization
D. DERIVATION OF SIGNAL RECONSTRUCTION
t1 , t2 ,A , tM ∈ [a, b]
,
let
a
function
nonuniform
τ ∈ [a, b] and
f (τ ) is determined by appropriate coefficients hm (τ ) ’s.
So ,we study the following optimization
Letting
| f (τ ) − f# (τ ) |2
|| f ||2
f ∈V (ϕ1 ,A,ϕ K )
sup
k
k
{ϕ#k (t − n) |1 ≤ k ≤ K , n ∈ Z } is the
{ϕk (t − n) |1 ≤ k ≤ K , n ∈ Z } . We have
where,
and
≤ ∑∑| ck,n (n) |
n k =1
n k =1
m=1
m
k
where
eTk ,τ = (ϕk (τ − n))n
m =1
,
k = 1, 2,A , K . So, it follows that
− n) |
2
m
be
K
M
k =1
m =1
approximate f (t ) at τ .
seen
that
T
k
e k ,τ .
(9)
aij( k ) = ϕ k (t j − i )
⎞
⎟
−1
⎟
⎠
H(τ ) = ( h1 (τ ) , h2 (τ ) ,A, hM (τ ) ) = X∑ ATk ek ,τ =
∑ X(∑ϕ (t −n)ϕ (τ −n),A,∑ϕ (t
k =1
k
1
k
k
k =1
M
−n)ϕk (τ −n))T
= (∑∑∑ x1lϕk ( tl − n ) ϕk (τ − n),A,
n
K
n
(11)
M
k =1 l =1
n
∑∑∑ x
K
M
k =1 l =1
Ml
n
ϕk ( tl − n ) ϕk (τ − n))T .
So, the optimal reconstruction of f (t ) can be expressed
as
,
f# (t ) = ∑ f (tm )hm (t )
M
m =1
(12)
⎡K M
⎤
= ∑ f (tm ) ⎢ ∑∑∑ xmlϕ k ( tl − n ) ϕ k (t − n) ⎥.
m =1
⎣ k =1 l =1 n
⎦
Let us take a look at the case K = 1 in detail. Given
M samples of x(t ) ∈ V (ϕ ) at instants t1 , t2 ,A , t M for a
M
minimizing
make
K
T
f# (τ )
proper scaling function ϕ (t ) ∈ L2 (R ) , we can express the
To minimize E (τ ) , we further express E (τ ) as
SAMPTA'09
with
K
eTk ,m = (ϕk (tm − n)) n
E (τ ) = ∑ || ek ,τ − ∑ hm (τ )ek , m || 2 can
can
∑A
⎞ −1 K
⎟
⎟
⎟
k =1
⎠
interpolating vector as
(6)
K
M
| f (τ ) − f# (τ ) |2
−1
e
sup
||
≤
−
A
hm (τ )e k ,m || 2
∑
∑
k
,
τ
2
|| f ||
f
k =1
m =1
.
It
Ak
A k = (aij( k ) )
⎜
⎝ k =1
| f (τ ) − f# (τ ) |2 ≤ A−1 || f ||2 ∑ || ek ,τ − ∑ hm (τ )ek , m || 2 , (7)
k =1
T
k
the others for each f ∈ V (ϕ1 ,A , ϕ M ) .
The above can furthermore be expressed explicitly in
vector form as
M
∑A
⎛ K
⎜
⎜
⎜
⎝ k =1
M
m=1
K
= 0 yields that
exist one sample in { f (tm )}m =1 which can be expressed by
,
M
n k =1
(8)
samples { f (tm )}m =1 is indepentant, that is, there doesn’t
(5)
≤ A−1 || f || 2 ∑∑| ϕk (τ − n) − ∑ hm (τ )ϕk (tm − n) |2 .
K
∂E (τ )
∂H (τ )
⎛ K
⎜
M
k
k = 1, 2,A, K . It is
for
and X = ( xij ) = ⎜ ∑ ATk A k ⎟ , we then rewrite the optimal
∑∑| ϕ (τ − n) − ∑ h (τ )ϕ (t
2
and
1
Letting
m=1
K
⎞
k , M ⎟⎠
M
M
K
H (τ ) = ( h1 (τ ), h2 (τ ), A , hM (τ ))T
K
dual frame of
=| ∑∑ ck (n)[ϕk (τ − n) − ∑hm (τ ) ϕk (tm − n)] |2
n k =1
⎟
⎠
Note that the matrix ∑ ATk A k is invertible when the
| f (τ ) − f# (τ ) |2
K
⎜
⎝ k =1
k =1
ck (n ) = ( f , ϕ#k (t − n )) , k = 1, 2, A , K , n ∈ Z
k =1
⎞
⎟
K
K
⎡K
⎤ 2 (10)
A || f || ⎢ ∑ || e k ,τ −A k (∑ ATl A l )−1 ∑ ATm em,τ ||2 ⎥ .
l =1
m =1
⎣ k =1
⎦
K
n
⎛ K
⎜
−1
(4)
∑ ∑ c (n )ϕ (t − n ),
f (t ) =
Ak H(τ ) + HT (τ ) ⎜ ∑ATk Ak ⎟ H(τ ).
k ,τ
and the minimal error between f (τ ) and f# (τ ) is given
by
| f (τ ) − f# (τ ) |≤ r (τ ) =
m =1
h (τ )
k =1
e , e k , 2 ,A , e
⎛
⎜ k ,1
⎝
m =1
T
H (τ ) =
M
inf
2
Therefore, solving
f# (τ ) = ∑ hm (τ ) f (tm ) . The optimal estimation f# (τ ) of
instants
K
followed that
K
K
∂E (τ )
= −2∑ ATk ek ,τ + 2∑ ATk A k H (τ ) .
∂H (τ )
k =1
k =1
which yields a min-max type interpolator.
of
at
k,
k =1
instants t1 ,A , t M . In order to obtain an optimal
M
samples
Given
f (t ) ∈ V (ϕ1 , ϕ2 ,A, ϕ K )
M
optimal interpolating vector as H (τ ) = ( AT A)−1 AT eτ ,
2
368
where
A = (e1 , e2 ,A, e M )
and
eτ = (ϕ (τ − n))Tn
reconstruction error is strongly related to the sampling
pattern. As pointed in [18], the reconstruction errors are
smaller in the neighborhood of the sampling instants. So,
the quality of reconstruction should be evaluated in a
pointwise manner. From (10) we know that the min-max
reconstruction error is pointwise upper-bounded by
,
em = (ϕ (tm − n))Tn for m = 1, 2,A, M . It is easy to see
that the optimal interpolating vector H (τ ) is exactly the
orthogonal projection of vector eτ onto the subspace
e1 , e2 ,A, e M
by
and
hence
from
#
#
(12) x(t ) ∈ V (ϕ ) with x(tm ) = x(tm ) for m = 1, 2,A, M .
This implies that the reconstructed signal best fits the
sampling
data.
Especially,
for
σ >0
and
#
ϕ (t ) = sinc(σ t ) , the optimal reconstruction x(t ) of
K
K
⎡K
⎤2
r(τ ) ≤ A−1/ 2|| f || ⎢∑ || ek ,τ −Ak (∑ ATl Al )−1∑ ATmem,τ ||2 ⎥ (14)
l =1
m=1
⎣ k =1
⎦
spanned
x(t ) ∈ V (ϕ ) from samples x(t1 ), x(t2 ),A , x(tM ) is also
x(tm ) = x# (tm )
for
band-limited
to
σ
with
m = 1, 2,A, M . It is easy to show that the interpolator
obtained here is just Yen’s for band-limited signals.
1
and it can estimate the quality of reconstruction when the
sampling instants are known.
F. DEMONSTRATIVE EXAMPLES
Some numerical examples are given to demonstrate the
performance of the proposed method, where signals are
selected randomly in shift-invariant subspaces, and the
sampling instants are generated by adding random
perturbation, distributed uniformly in the interval [−u , u ]
for u > 0 , to each equally-spaced sampling instant, i.e.,
the sampling instants are mT + um , where um randomly
distributes in [−u , u ] for m = 1, 2,A , M .
Example 1 For the first example, we choose arbitrarily a
signal of band [−π , π ] . We reconstruct it on [0, 40] from
42 samples. The average sampling period is T = 0.995 s
and u = 0.7T . It is clear that the average sampling period
is almost critical. The reconstructed signal, its
reconstruction from its nonuniform samples and the errors
between the original signal and its reconstructions were
plotted in Fig.1. From this experiment, it can be seen that
under such a relaxed condition, the reconstruction of a
signal is quite satisfying,
Example 2 For non band-limited signals, we choose the
cubic spline [19] as a scaling function, and randomly
choose a signal in the shift-invariant space. The average
sampling period is T = 1.0s and the maximum of
irregular perturbation is u = 0.5T . The signal to be
reconstructed, its reconstruction and the reconstruction
error (in dB) were plotted in Fig.2, respectively. Note that,
although the sampling density is much lower than that
estimated in [5](see the examples therein for details), the
quality of signal reconstruction is still considerably high.
In addition, although the cubic spline is supported
compactly, the method given in [10] could not be used in
this case because the maximal gap between adjacent
sampling instants is too large. In fact, the local
reconstruction methods in [10] required the condition that
the maximal gap of the sampling instants must be less 1
and the number of the samples must be bigger than the
length of the reconstruction interval, but our method
doesn’t rely on any sampling condition. In contrast to the
results in [10], we also give another example to show the
performance of the proposed method in Fig.3. In this
example, we chose the same scaling function, a Gaussian
function, and the similar sampling condition as in [10].
E. ALGORITHM AND DISCUSSION
In the previous section we have derived an interpolator
for signal reconstruction. However, because computing
the min-max interpolator requires calculating the inverse
of a matrix with possibly larger dimension, the
reconstruction formula (12) wound be unfeasible when the
number of samples is much large. To circumvent this
problem, we reshape (12) as
⎛M K M
⎞
f# (τ ) = ∑ ⎜ ∑∑∑ f (tm ) xmlϕk ( tl − n ) ⎟ϕk (t − n) . (13)
n ⎝ l =1 k =1 m =1
⎠
From this, a non-iterative reconstruction algorithm can be
given as follows.
Algorithm:
(1) Let f = ( f (t1 ), f (t2 ),A , f (tM ))T ,
Ek (t ) = (A , ϕ k (t − n), ϕ k (t − n + 1),A)T ,
(k )
(k )
A k = (bmn
) with bmn
= ϕk (tm − n) for
i = 1, 2,A , M and k = 1, 2,A , K ;
(2) Compute T = ∑ A k A Tk
K
k =1
(3) Solve Th = f ;
(4) f# (t ) = h T ∑ A k Ek (t ) .
K
k =1
The most crucial step is solving the equation system of
equations Th = f . This can be done effectively by
computing the Cholesky factorization of the
matrix T when T is invertible. In fact, the cholesky
factorization of T gives a upper triangular matrix S
such that T = ST S . Then the solution of the
system Th = f can be obtained by sequentially solving the
systems ST b = f and Sh = b by Gaussian elimination.
This procedure is faster and more robust than directly
computing the inverse of T . When T is not invertible, the
equation can be solved effectively by the least squares
method.
Note that although the proposed method has no
restriction on sampling locations, the obtained
SAMPTA'09
3
369
Fig. 1 Top : original signal and sampling points marked by dots; middle:
reconstructed signal obtained by the proposed method; bottom: normalized
errors between the original signal the its reconstruction obtained by the
proposed method and Yen interpolator.
Fig. 2 original signal with sampling points marked by stars, reconstructed signal obtained by the
proposed method, normalized error(in dB) between the original signal the its reconstruction.
SAMPTA'09
4
370
Fig. 3 original signal with sampling points marked by stars, reconstructed signal obtained by the
proposed method, normalized error(in dB) between the original signal the its reconstruction with
scaling function ϕ (t ) = e −t
2
/ 2σ 2
, σ = 0.81 , and sampling density 0.85.
Fig. 4 original signal with sampling points marked by stars, reconstructed signal obtained by the
proposed method, normalized error(in dB) between the original signal the its reconstruction.
ϕ1 (t ) = a1e
Example
3.
−t 2 / 4
and ϕ 2 (t ) = a2 (t + t )e
Finally,
we
select
3
two
−t 2 / 4
functions
as scaling
functions, where a1 and a2 are normalized constants.
Here the average sampling period is T = 0.8 s and the
maximum of irregular perturbation u = 0.6T . The
SAMPTA'09
simulation results were showed in Fig.4. This example also
indicates the feasibility of the proposed method for signal
reconstruction in a shift-invariant spaced with several
scaling functions
. CONCLUSION
5
371
The proposed method of reconstructing a signal from
its finite nonuniform samples has the following
advantages: (a) the method doesn’t require the usual
hypotheses on the maximal gap between adjacent
sampling instants and the compactness of the scaling
functions of the shift-invariant space as in the literature,
and therefore can be applied in various shift-invariant
spaces with sampling locations distributed arbitrarily. (b)
The reconstruction error function as sensitivity functions
[17] can measure the quality of the reconstruction prior
to the practical implementation. (c) the method can be
used effectively in a multi-wavelet space and can be
extended straightforward to two-dimensional spaces.
However, our method does not incorporate the case when
samples are noisy, which we will investigate in future.
[9] C. Ford and D.M. Etter, “Wavelet basis
reconstruction of nonuniformly sampled data”, IEEE
Trans on Circuits and Systems II, vol.45,
pp.1165–1168, 1998.
REFERENCES
[12] P. J. S. G. Ferreira, “Noniterative and faster
iterative methods for interpolation and extrapolation, ”
IEEE Trans. on Signal Processing, vol. 42(11), pp.
3278-3282, 1994.
[10] K. Grochenig and H. Schwab, “Fast local
reconstruction methods for nonuniform sampling in
shift-invariant spaces,” SIAM Journal of Matrix
Analysis and Applications, vol.24 , no.4, pp.899-913,
2003.
[11] A. Aldroubi and H. Feichtinger, “Complete iterative
reconstruction algorithms for irregular sampled data in
spline-like spaces”, in IEEE Acoustics, Speech and
Signal Process International Conf.(ICASSP-97), vol.3,
pp.1857-1860, 1997.
[1] S. D. Berger, “Nonuniform sampling reconstruction
Applied to sparse array beamforming,” Proc. IEEE
Radar Conf. 2002, pp.98–103, 2002,
[13] Hyeokho Choi and D. C. Munson, “Analysis and
Design of minimax-optimal interpolators”, IEEE Trans.
on Signal Processing, vol.46, pp.1571–1579, 1998.
[2] D. S. Early and D.G. Long, “Image reconstruction
and enhanced resolution imaging from irregular
samples,” IEEE Trans. on Geoscience and Remote
Sensing, vol.39, pp.291–302, 2001.
[14] Y. Rolain, J. Schoukens and G. Vandersreen, “Signal
reconstruction for non-equidistant finite length sample
sets: a KIS approach,” IEEE Trans. on Instrument
and Measurement, vol.47, no.5, pp.1046-1052,1998.
[3] R. Stasinski and J. Konrad, “POCS-based image
reconstrunction from nonuniform samples,” http://
iss.bu.edu/jkonrad/Publications/local/cpapers/Stas00ici
p.pdf.
[15]I. A. Aldroubi and M.
Unser, “Sampling
procedures in function spaces and asymptotic
equivalence with Shannon sampling,” Numer. Funct.
Anal. Optimiz., vol.15, pp. 1-21,1994.
[4] G. G. Water, “A sampling theorem for wavelet
subspaces,” IEEE Trans. on Inform. Theory, vol. 38,
pp.881-884, 1992.
[16] P.P. Vaidyanathan, “Generalizations of the sampling
theorem: Seven decades after Nyquist”, IEEE, Trans.
on Circuits and Systems I: Fundamental Theory and
Applications, vol.48, pp.1094–1109, 2001.
[5] Wen Chen, S. Itoh and J. Shiki, “On Sampling in
Shift Invariant Spaces,” IEEE Trans. on Information
Theory, vol.48, pp.2802–28010, 2002.
[6] I. Djokovic and P.P. Vaidyanathan, “Generalized
sampling theorem in multiresolution subspaces,” IEEE,
Trans. on Signal Process, vol.45, pp.583–599, 1997.
[17] R. G. Shenoy and T.W. Parks, “An optimal recovery
approach to interpolation”, IEEE, Trans. on Signal
Processing, Vol.40, pp.1987-1996, 1992.
[7] I. W. Selesnick, “Interpolating multiwavelet bases
and sampling theory,” IEEE Trans. on Signal Process,
vol.47, pp.1615–11621, 1999.
[18]A. Tarczynski, “Sensitivity of signal reconstruction”,
IEEE Signal Processing Letter, vol.4, pp.192–194,
1997.
[8] A. Aldroubi and K. Grochenig, “Nonuniform
sampling and reconstruction in shift-invariant spaces”,
SIAM Rev., 2001, no.4, pp.585-620.
SAMPTA'09
[19]C. K. Chui, An Introduction to wavelets, Academic
Press, Inc. 1992.
6
372
Spline Interpolation in Piecewise
Constant Tension
Masaru Kamada(1) and Rentsen Enkhbat(2)
(1) Ibaraki University, Hitachi, Ibaraki 316-8511, Japan.
(2) National University of Mongolia, P. O. Box 46/635, Ulaanbaatar, Mongolia.
kamada@mx.ibaraki.ac.jp, renkhbat46@ses.edu.mn
Abstract:
Locally supported splines in tension are constructed where
the tension, which has ever been constant over the entire
domain, is allowed to change at sampling points.
∫
∞
(f (2) (t))2 + p(t)2 (f (1) (t))2 dt
of its squared second derivative f (2) and squared first
derivative f (1) subject to the constraints
f (tk ) = fk , k = 0, ±1, ±2, · · · .
1. Introduction
A cubic spline gives the interpolation of data that minimizes the square integral of its second derivative [3, 5, 9]
and is crowned as the smoothest interpolation in this sense.
A linear spline gives the piecewise linear interpolation
that is most straight but nonsmooth. The linear spline is
characterized as minimizing the square integral of its first
derivative [3, 9]. A spline in tension [1, 10] was devised as
a generalization of those two splines. It minimizes the integral of a weighted sum of the squared second derivative
and the squared first derivative. By increasing the weight
called tension, we can make a spline in tension approach
the most straight linear spline while retaining smoothness
similar to that of the cubic spline.
The spline in tension has been known for more than 40
years. It has been extended even to the multidimensional
cases [2, 7] and is now supported by a standard software
library [8]. But the tension has ever been a single constant
over the entire domain.
In this paper, we look at the splines as the output of a linear
dynamical system with a series of delta functions input.
That is the same way as how the exponential splines and
their locally supported basis were successfully constructed
in [12, 13]. In addition, attending to that the linear dynamical system theory [6] allows for time-varying dynamical parameters, we shall place different tension in each
sampling interval. Then we will obtain locally supported
splines in piecewise constant tension that can change the
interpolation characteristics from a sampling interval to
another.
(1)
−∞
(2)
In the case p = 0, f is identical with the cubic spline
[3, 5, 9]. By increasing p, f approaches the most straight
linear spline as if the curve were pulled from both ends.
That is why p is called tension [1, 10].
The tension p has originally been a single constant over
the entire domain [10]. We shall now relax the tension to
be a non-negative constant in each sampling interval, i.e.,
p(t) = pk ≥ 0, for t ∈ [tk , tk+1 ),
(3)
which can change at the sampling points.
By the calculus of variation, the minimization problem
is reduced to solution of the Euler-Lagrange differential
equation
f (4) (t) − 2p(t)p(1) (t)f (1) (t) − p(t)2 f (2) (t) = w(t), (4)
where w is a series of the Dirac delta functions
w(t) =
∞
∑
wn δ(t − tn )
n=−∞
to be determined so that (2) holds good. We do not have,
however, a practical means to decide the coefficients {wn }
for given {(tk , fk )}.
In practice, it is convenient to have locally supported functions {yn } satisfying
yn (t) = 0, for t ∈
/ [tn , tn+4 ]
(5)
of which linear combination
f (t) =
∞
∑
cn yn (t)
(6)
n=−∞
2.
Preliminaries
A spline f in tension interpolating the data
{(tk , fk )}∞
k=−∞ given at strictly increasing sampling
points (· · · < t−2 < t−1 < t0 < t1 < t2 < · · · ) on the
real line is defined as the twice-differentiable function
that minimizes the integral of a weighted sum
SAMPTA'09
represents any possible f . This yn can be constructed by
yn(4) (t) − 2p(t)p(1) (t)yn(1) (t) − p(t)2 yn(2) (t) = un (t) (7)
for some appropriately chosen
un (t) =
4
∑
ul,n δ(t − tn+l )
(8)
l=0
373
as long as the sampled data system (7) with the impulse
input (8) is completely controllable [4]. Once we obtain
yn (t), we have only to determine the coefficients {cn } by
the linear equations
fk =
∞
∑
cn yn (tk ), k = 0, ±1, ±2, · · ·
n=−∞
from {(tk , fk )}. Although infinitely many coefficients and
data are involved in the equations, we can solve the linear equations for finitely many {cn } from finitely many
{(tk , fk )} in practice because {yn } are locally supported.
3. Construction of locally supported splines
in piecewise constant tension
A state-space representation of the differential equation
(7) is
x(1)
n (t) = F (t)xn (t) + gun (t), yn (t) = hxn (t),
(9)
where
0
1
0
0
F (t) =
0
0
0 2p(t)p(1) (t)
yn
y (1)
n
xn (t) = (2) , g =
yn
(3)
yn
0
1
0
p(t)2
0
0
,
1
0
0
0
, h = [1 0 0 0] . (10)
0
1
The state xn can be expressed by
∫ t
xn (t) = Φ(t, v)xn (v) +
Φ(t, τ )gun (τ ) dτ,
(11)
v
(i) In the open interval (tn+l , tn+l+1 ), (11) with v =
tn+l +0 is reduced to
xn (t) = Φ(t, tn+l +0)xn (tn+l +0), l = 0, 1, 2, 3 (14)
because un (t) = 0 for t ∈
in (10) is a constant matrix
0
0
F (t) =
0
0
(tn+l , tn+l+1 ). Besides, F (t)
1
0
0
0
0
1
0
p2n+l
0
0
1
0
(15)
because of (3) so that we can calculate the state-transition
matrix by the matrix exponential function [11] as follows:
Φ(t, tn+l +0)
Rt
F (υ) dυ
= etn+l +0
(t−tn+l )2
(t−tn+l )3
1
t
−
t
n+l
2
6
(t−tn+l )2
0
1
t
−
t
n+l
2
0
0
1
t − tn+l
0
0
0
1
if pn+l = 0
cosh(pn+l (t−tn+l ))−2
1 t − tn+l
p2n+l
sinh(pn+l (t−tn+l ))
0
1
pn+l
=
0
(t − tn+l ))
0
cosh(p
n+l
sinh(p
(t − tn+l ))
0
0
p
n+l
n+l
sinh(pn+l (t−tn+l ))−2pn+l (t−tn+l )
p3n+l
cosh(pn+l (t−tn+l ))−2
2
p
n+l
sinh(pn+l (t−tn+l ))
pn+l
cosh(pn+l (t − tn+l ))
if pn+l > 0.
(16)
for any real numbers t and v, in terms of the statetransition matrix function Φ and the input un [11].
In the special case that t = tn+l+1 -0, we have the state
transition from xn (tn+l +0) to xn (tn+l+1 -0) as follows:
Since un (t) = 0 for t ∈
/ {tn , tn+1 , tn+2 , tn+3 , tn+4 }, it
follows from (11) that
xn (tn+l+1 -0) = Φ(tn+l+1 -0, tn+l +0)xn (tn+l +0),
xn (t)
0, t < tn
Φ(t, tn+l +0)xn (tn+l +0),
=
(12)
tn+l < t < tn+l+1 , (l = 0, 1, 2, 3)
Φ(t, tn+4 +0)xn (tn+4 +0), tn+4 < t.
Because of the top and bottom lines of (12), yn = hxn
is locally supported as (5) if xn (tn+4 +0) = 0. In order
to avoid the trivial case u0,n = u1,n = u2,n = u3,n =
u4,n = 0 that would result in un ≡ yn ≡ 0, let us fix one
of them as u0,n = 1. Then the problem of constructing a
locally supported yn becomes a dead-beat control problem
of finding u1,n , u2,n , u3,n , and u4,n that make the terminal
state dead as
xn (tn+4 +0) = 0.
(13)
Once the terminal state is controlled to 0, it will stay at 0
forever for t > tn+4 without any beats.
We shall consider two types of state transitions: (i) Those
within each sampling interval (tn+l , tn+l+1 ), and (ii) one
across each sampling point tn+l .
SAMPTA'09
l = 0, 1, 2, 3.
(17)
The matrix Φ(tn+l+1 -0, tn+l +0) can be evaluated by the
right hand side of (16) with t replaced by tn+l+1 .
(ii) The state transition from xn (tn+l -0) to xn (tn+l +0)
across the sampling point tn+l , (l = 0, 1, 2, 3, 4) finds a
trouble that F (t) in (10) contains a derivative of the function p being discontinuous at tn+l as defined by (3). We
had better consider this transition by way of the original
differential equation (7). An equivalent form of (7) is
)
d (
p(t)2 yn(1) (t) = un (t)
(18)
yn(4) (t) −
dt
and its integration gives
∫ t
un (τ )dτ + c,
(19)
yn(3) (t) = p(t)2 yn(1) (t) +
tn −0
where c is an integral constant. Substituting tn+l +0 and
tn+l -0 for t of (19), we have
yn(3) (tn+l +0) = p(tn+l +0)2 yn(1) (tn+l +0)
+u0,n + · · · + un+l,n + c
(20)
374
and
yn(3) (tn+l -0) = p(tn+l -0)2 yn(1) (tn+l -0)
+u0,n + · · · + un+l−1,n + c, (21)
respectively. Recall that the spline in tension is sought
among the twice-differentiable functions and attend to the
definition (3) of p. Then we can reduce (20) and (21) to
yn(3) (tn+l +0) = p2n+l yn(1) (tn+l )
+u0,n + · · · + un+l,n + c
and
(22)
yn(3) (tn+l -0) = p2n+l−1 yn(1) (tn+l )
+u0,n + · · · + un+l−1,n + c, (23)
respectively. Subtracting (23) from (22), we have
y (3) (tn+l +0) − y (3) (tn+l -0)
=
(p2n+l − p2n+l−1 )y (1) (tn+l ) + ul,n ,
(24)
which tells how to update the state variable y (3) at tn+l
and implies that the other state variables y (2) , y (1) , and y
are continuous at tn+l . So we can write the state transition
across the sampling point tn+l as follows:
xn (tn+l +0) = Φ(tn+l +0, tn+l -0)xn (tn+l -0) + gul,n ,
l = 0, 1, 2, 3, 4, (25)
where
1
0
Φ(tn+l +0, tn+l -0) =
0
0
0
1
0
p2n+l − p2n+l−1
0
0
1
0
0
0
. (26)
0
1
The two types of state transitions (17) and (25) can be
combined into the recurrence formulae
xn (tn +0) = gu0,n = g,
xn (tn+l +0) = Ψn+l xn (tn+l−1 +0) + gul,n ,
l = 1, 2, 3, 4, (27)
where we have set
Ψn+l = Φ(tn+l +0, tn+l -0)Φ(tn+l -0, tn+l−1 +0),
l = 1, 2, 3, 4,
(28)
and used our choice u0,n = 1 and the initial state
x(tn -0) = 0. By these recurrence formulae, we can write
the terminal state as follows:
xn (tn+4 +0) = Ψn+4 Ψn+3 Ψn+2 Ψn+1 g
+Ψn+4 Ψn+3 Ψn+2 gu1,n
+Ψn+4 Ψn+3 gu2,n
+Ψn+4 gu3,n
+gu4,n .
(29)
Then we can determine u1 , u2 , u3 , and u4 that makes the
terminal state xn (tn+4 +0) be zero by
u1,n
u2,n
−1
u3,n= − [Ψn+4 Ψn+3 Ψn+2 g Ψn+4 Ψn+3 g Ψn+4 g g]
Ψn+4 Ψn+3 Ψn+2 Ψn+1 g.
u4,n
(30)
Existence of the inverse matrix is equivalent to the complete controllability of the sampled-data system with the
SAMPTA'09
impulse control un input. We do not have the condition
in a simpler form due to the complication caused by timevarying dynamics and non-uniform sampling. Even the
uniform sampling case is yet to be investigated.
For the numerical evaluation of yn , we first compute the
states {xn (tn+l +0)}3l=0 by (27) from {ul,n }4l=0 . Then we
can evaluate yn by
0, t ≤ tn
hΦ(t, tn+l +0)xn (tn+l +0),
yn (t) =
(31)
tn+l < t ≤ tn+l+1 , (l = 0, 1, 2, 3)
0, tn+4 ≤ t
and
hΦ(t, tn+l +0)
]
[
(t−tn+l )2 (t−tn+l )3
if pn+l = 0
1
t
−
t
n+l
2
6
[
n+l ))−2
(32)
=
1 t − tn+l cosh(pn+lp(t−t
2
n+l
]
sinh(pn+l (t−tn+l ))−2pn+l (t−tn+l )
if pn+l > 0
p3
n+l
which follow from (12), (16), and the continuity of yn over
the entire domain.
4. Numerical examples
Test data were prepared by concatenating a sampled
smooth curve and a sampled polygonal line. Their interpolation was computed as a linear combination of the locally
supported splines in tension.
The cubic spline interpolation (equivalent to the case
p(t) ≡ 0) is shown in Fig. 1. The cursive part is reproduced in a good shape but the polygonal part suffers
from inter-sample vibration. The linear spline interpolation (equivalent to the case p(t) → ∞) in Fig. 2 behaves
in the opposite way. Reproduction of the polygonal part
is perfect but there is no smootheness. Interpolation by a
spline in constant tension (p(t) ≡ 10) in Fig. 3 provides
a good compromise between the cubic and linear spline
interpolation. It is fairly smooth and has less vibration.
Some may say that the cursive part is not smooth enough
and rather polygonal in Fig. 3. In this case, we can obtain
a better interpolation by varying the tension in time. Figure 4 is an interpolation by a spline in piecewise constant
tension. Higher tensions are imposed on the polygonal
part to suppress the vibration. The interpolation is kept
smooth elsewhere. The locally supported splines used to
construct this curve are plotted in Fig. 5 where the plots
are vertically scaled to have a common peak value.
5. Conclusions
Locally supported splines in tension were constructed
where the tension is constant within each sampling interval and variable at the sampling points. They will hopefully contribute to the variety of curve drawing modules
in the graphical design tools. Another application may be
image enlargement tools which allow users to put higher
tension manually at the portions where they want to suppress ringing effects.
375
f(t)
t0
f(t)
t1
t2
t3
t4
t5
t6
t7 t8 t9 t10
t
Figure 1: Interpolation by a cubic spline (p(t) ≡ 0).
t0
t1
t2
t3
t4
t5
t6
t7 t8 t9 t10
t
Figure 2: Interpolation by a linear spline (p(t) → ∞).
References:
f(t)
[1] J. H. Ahlberg, E. N. Nilson and J. L. Walsh. The
Theory of Splines and Their Applications. Academic
Press, London, 1967.
[2] M. N. Benbourhim and A. Bouhamidi. Approximation of vector fields by thin plate splines with tension.
J. Approx. Theory, 136:198–229, 2005.
[3] C. de Boor. Best approximation properties of spline
functions of odd degree. J. Math. Mech., 12:747–
750, 1963.
[4] Y. C. Ho, R. E. Kalman and K. S. Narendra. Controllability of linear dynamical systems. Contrib. Diff.
Eqs., 1:189–213, 1963.
[5] J. C. Holladay. Smoothest curve approximation.
Math. Tables and Aids to Comput., 11:223–243,
1957.
[6] R. E. Kalman. A new approach to linear filtering
and prediction problems. Trans. ASME, 82(Series
D):35–45, 1960.
[7] H. Mitasova and L. Mitas. Interpolation by regularized spline with tension: I. theory and implementation. Mathematical Geology, 25:641–655, 1993.
[8] A. Polyakov and V. Brusentsev. Graphics Programming with GDI+ & DirectX. A-List Publishing,
Wayne, PA, 2005.
[9] I. J. Schoenberg. On interpolation by spline functions and its minimal properties. In P. L. Butzer
and J. Korevaar, editors, On Approximation Theory,
pages 109–129, June 1964.
[10] D. G. Schweikert. An interpolation curve using a
spline in tension. J. Math. Phys., 45:312–317, 1966.
[11] E. D. Sontag.
Mathematical Control Theory.
Springer, New York, 1990.
[12] M. Unser and T. Blu. Cardinal exponential splines:
Part I—Theory and filtering algorithms. IEEE Transactions on Signal Processing, 53(4):1425–1438,
April 2005.
[13] M. Unser. Cardinal exponential splines: Part II—
Think analog, act digital. IEEE Transactions on Signal Processing, 53(4):1439–1449, April 2005.
t0
t1
t2
t3
t4
t5
t6
t7 t8 t9 t10
t
Figure 3: Interpolation by a spline in constant tension
(p(t) ≡ 10).
f(t)
t0
t1
t2
t3
t4
t5
t6
t7 t8 t9 t10
t
Figure 4: Interpolation by a spline in piecewise constant
tension (p(t) = 0 for the cursive part (t < t4 ), p(t) = 10
for the straight parts (t4 ≤ t < t6 and t7 ≤ t), and p(t) =
30 for the breaking part (t6 ≤ t < t7 ) ).
yn(t)
t0
t1
t2
t3
t4
t5
t6
t7 t8 t9 t10
t
Figure 5: Locally supported splines used to construct the
curve in Fig. 4.
SAMPTA'09
376
The Effect of Sampling Frequency on a FFT
Based Spectral Estimator
Saeed Ayat
Payame Noor University, Najafabad, Iran.
dr.ayat@pnu.ac.ir
Abstract:
This paper reviews the effect of sampling frequency on a
FFT-based spectral estimator. In signal processing
applications usually a fix window size is used for
obtaining the current frame spectral.
For an application like speech enhancement this accuracy
of this estimation has a great influence in the quality of
the system, because listener feeling is very important in
this subject.
In our proposed method we divided the well-known
spectral subtraction method in two phases. Then by using
different frame sizes that we used in these two phases the
overall quality of the system has increased in different
sampling frequencies.
Now with subtracting this estimation of noise spectrum
from the spectrum of each noisy speech frames we can
achieve enhanced speech signal.
The paper is organized as follows: In section 2 we have a
review on spectral subtraction method. In section 3 we
proposed our method and in section 4 we present the
simulation results.
2. Spectral Subtraction
There are many different versions for spectral
subtraction. In a generalized spectral subtraction [4] we
have:
1
)
⎧
⎫
α
α
S (w) = max⎨( S (w) − β N (w) ) α , γ N (w) ⎬
⎩
⎭
)
S (w)
(1)
1. Introduction
Where S (w) , N (w)
One of the first methods introduced for speech
enhancement is spectral subtraction. Till now, different
versions of spectral subtraction have been proposed to
increase the performance of this method, for example [1,
2, 3].
Despite of its high noise removal, it can cause an
annoying noise called musical noise and hence it can
reduce overall quality. Musical noise is produced
because, we don’t have the needed spectra exactly, so we
have to use their estimations.
In signal processing applications usually a fix window
size is used for obtaining the current frame spectral. As
we know if the frame length is L the frequency
resolution in Fourier spectral analysis is Fs/L. For
example if Fs=11025Hz and L=256 then Fs/L is 43Hz
and this resolution may not be enough for speech signal.
As we know a clean speech signal consists of some
sections that have speech and some others that have no
speech and we call them silences.
In a noisy speech signal these silence sections have only
noise and other sections have noisy speech signals. If the
noise is stationary we can estimate its spectrum in the
noise sections.
In spectral subtraction method, after framing the noisy
speech signal we use a silence detector or a voice activity
detector for separating noisy speech frames and noise
frames.
After that with applying, FFT we have the spectrum of
each frame. By calculating the average of the noise
frames spectra we have estimation for noise spectra.
spectrum of noisy speech, estimation of noise and
enhanced speech. β is the oversubtraction factor and γ
SAMPTA'09
and
are magnitude
is spectral floor. Both β and γ are adjusted to improve
the quality of enhanced speech.
By the assuming that the noise is stationary, a good
estimation can be resulted by computing the average of
the noise in silence frames spectra. We called such
average W (w) .
In presence of nonstationary noises, an adaptation
technique can be used. Given an initial value W 0 ( w) , if
the current frame is silence,
Wm (w) is updated using
this equation:
W m ( w ) = (1 − f ) W m −1 ( w ) + f Ym ( w )
(2)
In this formula Ym (w) is the spectrum of current
silence frame and f is a coefficient called forgetting
factor. This factor is changed depending on the noise
changing rate.
The main problem of spectral subtraction method is the
production of musical noise. Musical noise is produced
because we don’t have the exact spectrum of the noise
signal.
377
3. Proposed Method
SNRout = 10 log10
In our method that estimates the spectrum better than the
basic averaging method, after separating speech and
silence frames in the noisy signal with a basic analysis
frame, we can increase the analysis frame length until it
covers all the current silence frames. As in periodogram
estimator technique the accuracy improves by increasing
the number of signal samples, by using this adaptive
analysis frame length we can have a better spectral
estimation for noise and noisy signal and so the system
can produce a better enhanced signal with less musical
noise.
As we know if the frame length is L the frequency
resolution in Fourier spectral analysis is Fs/L. For
example if Fs=11025Hz and L=256 then Fs/L is 43Hz
and this resolution may not be enough for speech signal.
In our method we first apply a SAD algorithm with
L=256 and L/2=128 points overlap to detect the silence
frames. Now we can increase the analysis frame length
until it covers all the current silence frames. By this
method we have larger window length and hence better
frequency resolution. If we have several silence areas
with the new frame length, the average of them is the
overall noise spectrum.
By applying such method we have better noise spectrum
estimation with less musical noises.
In section 4 we give experimental results that confirm
this improvement clearly.
(3)
( n)
(6)
2
in the enhanced signal. At this point, β and SNR
improvement is recorded. This is done for SNR equal to
5dB and different frame lengths with 256, 512, 1024 and
2048 samples.
This test was evaluated for different sampling
frequencies equal to 8000Hz, 11025 Hz and
16000Hz. α is fixed to 1.0 and γ to 0.0. Note that the
frame length is 256 in silence detection step.
Tables 1 to 3 show the results for β and SNR
improvement at the appearance of musical noise in the
enhanced signal for tested SNRs.
L
256
512
1024
2048
SNRimp
0.8
1.0
1.4
1.44
β
0.1
0.15
0.25
0.45
Table 1:
β
and SNR improvement at the start of
musical noise (Fs=8000Hz)
L
y (n) = s(n) + w(n)
∑ (sˆ(n) − s(n))
2
In this experiment a listener listens to the enhanced
signal and increases β until the musical noise appeares
256
512
1024
2048
SNRimp
1.0
1.8
2.3
3.1
β
0.15
0.3
0.5
0.9
4. Simulation Results
In this section we explain our simulation. The speech
signal that used for these tests was chosen from TIMIT
data base and was pronounced with a female speaker.
Then this sentence converted to different sampling
frequencies by cool edit software. All these sentences
degraded by additive Gaussian white noise, so we can
have the noisy signal in required SNR, here 5dB.
For evaluating our method we calculate SNR
improvement as below.
If s (n) is the clean speech, y(n) the noisy, sˆ( n) the
enhanced signal and w(n) the noise then we have:
∑s
Table 2:
β
and SNR improvement at the start of
musical noise (Fs=11025Hz)
L
256
512
1024
2048
SNRimp
1.2
2.3
3.1
3.8
β
0.2
0.5
0.7
1.0
Table 3:
β
and SNR improvement at the start of
musical noise (Fs=16000Hz)
and the SNR improvement is computed as follows[5]:
SNRimp = SNR out − SNRin
(4)
In which SNRin and SNRout are the SNRs for noisy and
enhanced:
SNRin = 10 log10
SAMPTA'09
∑s
∑ ( y(n) − s(n))
2
( n)
2
(5)
As we can see the SNR improvement is better for longer
frame lengths in all different sampling frequency rates.
This show that the musical noise arises from inaccurate
noise estimation and reduces as the frame length
increases, and this result is true for different sampling
frequencies.
So with a greater frame length, we can choose a
greater β without production of musical noise and by
increasing it we can have less noise in the enhanced
signal and then achieve more SNR improvement, too.
378
5. Conclusions
In this paper we studied the effect of sampling frequency
on a FFT-based spectral estimator. We also proposed an
improved spectral subtraction method by increasing the
accuracy of spectral estimator.
This adaptive estimator can give better spectral
estimation by increasing the analysis frame length that
achieves in silence regions.
In this method for separating silence frames we use a
basic analyzing frame and for estimation the spectrum
we use an adaptive frame length that can increase until it
covers all current silence region. By this method we
could have a better spectral estimation for noise and
noisy signal and so the system can produce a better
enhanced signal with less musical noise.
SAMPTA'09
References:
[1] S. F. Boll, "Suppression of Acoustic Noise in Speech
using Spectral Subtraction,” IEEE Trans. Acoustics,
Speech and Signal processing, vol. ASSP-27, No. 2,
pp. 113-120, 1977.
[2] H. Hu, F. Kuo, H. Wang, "Supplementary Schemes
to Spectral Subtraction for Speech Enhancement,"
Speech Communication, 2002.
[3] H. Gustafsson, S. Nordholm, “Speech Subtraction
using Reduced Delay Convolution and Adaptive
Averaging”, IEEE Trans. Speech and Audio
Processing, vol. 9, No. 8, pp. 799-807, 2001.
[4] J. S. Lim, A. V. Oppenheim, "Enhancement and
Bandwidth Compression of Noisy Speech",
Proceedings of the IEEE, vol. 67, 1972.
[5] S. Ayat, “Enhanced Human-Computer Speech
Interface Using Wavelet Computing", IEEE
International Conference on Virtual Environments,
Human-Computer Interfaces and Measurement
Systems, 2008.
379
SAMPTA'09
380
Nonlinear Locally Adaptive Wavelet Filter
Banks
Gerlind Plonka (1) and Stefanie Tenorth (1)
(1) Department of Mathematics, University of Duisburg-Essen, 47048 Duisburg, Germany.
gerlind.plonka@uni-due.de, stefanie.tenorth@uni-due.de
Abstract:
In this paper we introduce a new construction of nonlinear
locally adaptive wavelet filter banks by connecting the lifting scheme with the idea of image smoothing by nonlinear
diffusion methods.
In this paper we wish to construct a new invertible nonlinear wavelet filter bank by connecting the two concepts
of the lifting scheme and the discrete nonlinear diffusion.
The main goal is to adapt the local geometry of images
suitably, in order to obtain highly efficient sparse image
representations.
1. Introduction
2.
A crucial problem in data analysis is to construct efficient low-level representations, thereby providing a precise characterization of features which compose it, such as
edges and texture components. Fortunately, in many relevant applications, the components of given multidimensional data are not independent, and the strong correlation
between neighboring data points can be suitably exploited.
In the two-dimensional case, tensor-product wavelets are
not optimal for representing geometric structures because
their support is not adapted to directional geometric properties.
Instead of choosing a priori a basis or a frame to approximate the image, one can try to adopt the approximation
scheme to the image geometry. Within the last years, different approaches have been developed in this direction,
see e.g. [1, 4, 5, 7, 10, 12, 13]. In particular, the construction of non-linear filter banks by the lifting scheme has
been proposed already in [4, 8]. Since that time, there have
been different attempts to construct adaptive and directional lifting based, invertible transforms for sparse image
representation, see [2, 5, 6, 9, 12]. The lifting scheme for
representation of wavelet filter banks has originally been
suggested and analyzed by Sweldens [16]. It provides a
flexible tool for the construction of new nonlinear wavelet
filter banks. The main feature of lifting is that it provides
an entirely spatial-domain interpretation of the transform.
Besides wavelet shrinkage, other approaches like regularization techniques and PDE-based methods (as nonlinear
diffusion) have been shown to be powerful tools in signal and image restoration in image processing, e.g., for
denoising purposes. In particular, the choice of nonlinear diffusion filters leads to impressive results by removing insignificant, small-scale variations while preserving
important features such as discontinuities [3, 11, 17, 18].
In [15], certain connections between explicit discrete onedimensional schemes for non-linear diffusion and shiftinvariant Haar wavelet shrinkage have been established.
SAMPTA'09
2.1
Lifting and Nonlinear Diffusion
The Lifting Scheme
The typical lifting scheme consists of three steps: Split,
Predict and Update.
1. Split. Usually, in this step, the given data is split into
even and odd components. Let N ∈ N be of the form
N = 2l r with l, r ∈ N. For a given digital image of the
N ×N
−1
, we split the data into
form a = (a(i, j))N
i,j=0 ∈ R
the following two sets of equal size,
ae
−1
:= (ai,j )N
i,j=0,i+j even ,
ao
−1
:= (ai,j )N
i,j=0,i+j odd ,
and we denote the components of ae and ao by aei,j and
aoi,j , respectively. The data sets ae and ao split the image
a like a checkerboard.
2. Predict. The goal of the prediction step is to find a
good approximation ão of the data ao of the form
ão = P1 (ao ) + P2 (ae ).
Here P1 and P2 can be nonlinear operators. Afterwards,
we consider the residual
do := ao − ão = ao − (P1 (ao ) + P2 (ae )).
We have to assume that the mapping (ae , ao ) 7→ (ae , do )
is invertible, i.e., the operator I − P1 needs to be invertible
for arbitrary data ao . The operators P1 and P2 need to be
chosen such that the residual do is very small.
3. Update. In the third step, we aim to find a smoothed
approximation of the data ae that can be regarded as a lowpass filtered and subsampled version of the original image
a. The general update has the form
ãe := U1 (do ) + U2 (ae )
with (possibly nonlinear) operators U1 and U2 , where we
again want to assume the invertibility of the mapping
381
(ae , do ) 7→ (ãe , do ), i.e., U2 is assumed to be invertible
such that
ae = U2−1 (ãe − U1 (do )).
We use a discretization of the form
− ukij
uk+1
ij
= g(|uki+1,j − uki,j |)(uki+1,j − uki,j )
τ
−g(|uki,j − uki−1,j |)(uki,j − uki−1,j )
The complete scheme is illustrated in Figure 1.
+g(|uki,j+1 − uki,j |)(uki,j+1 − uki,j )
ae
U2
a
+
I − P1
−g(|uki,j − uki,j−1 |)(uki,j − uki,j−1 ),
U1
−P2
ao
ãe
do
+
Figure 1: Illustration of the nonlinear filter bank using the
lifting scheme.
(2)
where u0i,j := aij for i, j = 0, . . . , N − 1. Here, k denotes
the iteration step and τ is the step size of time discretization. In our numerical examples we will use the step size
τ = 1/4.
3.
The Nonlinear Diffusion Filter Bank
2.2 Nonlinear Diffusion
The nonlinear diffusion has been shown to be a very successful model for image denoising. For Ω = (0, N1 ) ×
(0, N1 ) we consider the diffusion equation
³
´
∂u
= div g(|∇u|) ∇u
∂t
on Ω × (0, ∞)
(1)
with a given noisy image a as initial state
u(x, 0) = a(x),
x∈Ω
∂u
and with Neumann boundary conditions ∂n
= 0 on ∂Ω.
T
,
∂u/∂x
Here, ∇u = (ux1 , ux2 )T = (∂u/∂x
2 ) denotes
p 1
the gradient of u, and |∇u| := u2x1 + u2x2 .
The time t in (1) is a scale parameter. Increasing t corresponds to stronger filtering. The diffusivity function
g(|∇u|) is a non-negative function that determines the
amount of diffusion. It is decreasing in |∇u| in order to
ensure that strong edges are hardly blurred by the diffusion filter while small variations (noise) are smoothed
much stronger. Frequently used bounded diffusivities are
the Perona-Malik diffusivity
g(x) :=
Choice of the Prediction Operator
Using equation (2) with the notations aoi,j := u0i,j , ãoi,j :=
u1i,j for i + j odd, and aei,j = u0i,j for i + j even, we obtain
£
ãoi,j = aoi,j + τ g(|aei+1,j − aoi,j |)(aei+1,j − aoi,j )
+ g(|aei−1,j − aoi,j |)(aei−1,j − aoi,j )
¤
+ g(|aei,j−1 − aoi,j |)(aei,j−1 − aoi,j ) .
A prediction could now be of the form
x = 0,
x > 0,
see [14, 17]. One may also take a “robust” diffusivity of
the form
½
1 0 ≤ x < θ,
g(x) :=
0 |x| ≥ θ,
as it has been used in [14] with a suitably chosen threshold
θ.
Replacing g(|∇u|) by g(|∇uσ |), where uσ denotes the
slightly smoothed image by convolution with the Gaussian kernel, uσ := Kσ ⋆ u, existence and uniqueness of a
solution of (1) have been shown in [3].
For application of the diffusion approach to digital images
we follow [11] and replace (1) by the following slightly
modified equation
∂u
= ∂x1 (g(|∂x1 u|) ∂x1 u) + ∂x2 (g(|∂x2 u|) ∂x2 u).
∂t
SAMPTA'09
3.1
+ g(|aei,j+1 − aoi,j |)(aei,j+1 − aoi,j )
1
,
1 + x2 /λ2
or the Weickert diffusivity
(
1
³
´
g(x) :=
1 − exp −3.315
(x/λ)4
Now we want to apply the nonlinear diffusion filter for
the construction of prediction and update operators in the
lifting scheme, in order to obtain a new sparse representation of images. The nonlinear filter bank should satisfy
the following demands.
1. For linear (bivariate) polynomials, the residual do found
in the prediction step should vanish. This condition is
equivalent with two vanishing moments of the high-pass
filter in a wavelet filter bank.
2. Near discontinuities (edges) of u, the residual do should
remain small.
3. The data ãe should be a suitable (downsampled) approximation of the image a with good low-pass filter properties in smooth areas of a and without blurring of edges.
doi,j = aoi,j − ãoi,j
1
h X
i
g(|aei+µ,j+ν − aoi,j |)(aei+µ,j+ν − aoi,j ) .
= −τ
µ,ν=−1
|µ|+|ν|=1
Unfortunately, with this coice of prediction the desired invertibility of the mapping (ae , ao ) 7→ (ae , do ) is not guaranteed since the nonlinear diffusion g depends on the data
aoi,j . Therefore, we replace the values aoi,j that are used for
the computation of the function values of g by the median
of its four direct neighbors,
aoi,j ≈ median {aei,j+1 , aei,j−1 , aei+1,j , aei−1,j }:= med aoi,j .
A normalization with
gij :=
1
X
µ,ν=−1
|µ|+|ν|=1
g(|aei+µ,j+ν − med aoi,j |)
382
now yields the prediction
doi,j :=
−τ
gij
1
P
g(|aei+µ,j+ν− med aoi,j |)(aei+µ,j+ν− aoi,j )
µ,ν=−1
|µ|+|ν|=1
= τ aoi,j −
τ
gij
1
P
µ,ν=−1
|µ|+|ν|=1
g(|aei+µ,j+ν− med aoi,j |) aei+µ,j+ν .
Now,the invertibility of the prediction is ensured for τ > 0
and we have
aoi,j =
do
i,j
τ
+ g1ij
1
P
µ,ν=−1
|µ|+|ν|=1
g(|aei+µ,j+ν −med aoi,j |)aei+µ,j+ν .
Observe that the term gij is positive for all i, j if we take
Perona-Malik diffusivity or Weickert diffusivity. At the
boundary of the image, where not all four neighbors of a
data point are given, we slightly change the operator and
use only the three available neighbors in the sum (or even
only two neighbors at a vertex). Because of the normalization with the (correspondingly defined constants gij )
the properties of the prediction operator will not change.
application of the diffusion filter bank. Secondly, we apply
the shrinkage function
½
x |x| ≥ θ,
Sθ (x) :=
0 |x| < θ,
to the residual coefficients. In our numerical experiments
we will take a level-independent threshold θ. Finally, we
reconstruct the image with the modified residual coefficients.
4.
Properties of the Diffusion Filter Bank
We can show the following
Theorem 1.
Let g be a diffusivity function satisfying 0 < g(|x|) ≤ 1
for x ∈ R. The diffusion filter bank determined in Section
3 reproduces linear polynomials.
Proof. We consider a linear polynomial of the form
a(x1 , x2 ) = a0 + b0 x1 + c0 x2 ,
a0 , b0 , c0 ∈ R.
Let the digital image now be given by
3.2 Choice of the Update Operator
As update operator we simply apply a linear operator of
the form
√
ãei,j = 2aei,j + 14 (doi+1,j + doi−1,j + doi,j+1 + doi,j−1 ).
ai,j = a(ih, jh) = a0 + b0 ih + c0 jh.
Then we obtain for data that are not at the boundary
med aoi,j = median {a0 + b0 (i − 1)h + c0 jh, a0 +
b0 (i + 1)h + c0 jh, a0 + b0 ih + c0 (j − 1)h,
Invertibility is obviously satisfied and we find
¡
¢
aei,j = √12 ãei,j − 41 (doi+1,j + doi−1,j + doi,j+1 + doi,j−1 ) .
At the boundary, where aei,j has only three neighbors, we
slightly change the operator. For example, for 0 < i <
N − 1 and j = 0, we take
√
ãei,0 := 2aei,0 + 31 (doi+1,0 + doi−1,0 + doi,1 ),
etc.. Analogously, at vertices, only two neighbors are
taken into account.
Observe that the low-pass filtered values ãei,j are amplified
√
by 2 here (as it is usual also for orthogonal wavelet filter
banks).
3.3 Iterative Application of the Filter Bank
In order to obtain a suitable sparse representation of the
digital image a, we now iteratively apply the nonlinear
filter bank described above, and we use a hard threshold
procedure to suppress small residual values doi,j .
After the first application of the filter bank, the (small)
residual data doi,j , i, j = 0, . . . , N − 1, i + j odd, are
stored and we consider only the N 2 /2 values ãei,j , i, j =
0, . . . , N − 1, i + j even. For a second application of
(1)
the filter bank to ãei,j , we rename these data by ak,l :=
e
ãk−l,k+l , where k = 0, . . . , N − 1 and l = − min{k, N −
1 − k}, . . . , min{k, N − 1 − k}, and apply the filters now
to this data set, etc..
As usual, the complete procedure involves the following
three steps. First, we decompose the image by an iterative
SAMPTA'09
a0 + b0 ih + c0 (j + 1)h}
= a0 + b0 ih + c0 jh +
median {−b0 h, b0 h, −c0 h, c0 h}
= a0 + b0 ih + c0 jh = aoi,j
and
doi,j
=
−τ
gij
1
P
g(|aei+µ,j+ν− aoi,j |)(aei+µ,j+ν− aoi,j )
µ,ν=−1
|µ|+|ν|=1
=
−τ
gij
= 0.
£
g(b0 h)(aei+1,j + aei−1,j − 2aoi,j )
+g(c0 h)(aei,j+1 + aei,j−1 − 2aoi,j )
¤
Hence the prediction operator leads to doi,j = 0 and the
√
update yields ãei,j = 2aei,j for all i, j with i + j even.
¤
Further, one can show in case studies, that the proposed filter bank behaves well at vertical, horizontal and diagonal
edges, i.e., the obtained residual values using the nonlinear
prediction operator remain to be small.
5.
Numerical Results
We apply the above described nonlinear diffusion filter
bank in order to achieve sparse image representations.
In the experiment, we consider the monarch image. We
use the Perona-Malik diffusivity with λ = 28 and with
τ = 0.25. We apply 8 levels of the nonlinear filter bank,
383
PSNR= 26.41 dB
PSNR= 24.73 dB
10
10
10
20
20
20
30
30
30
40
40
40
50
50
50
60
60
10
20
30
40
50
60
60
10
20
30
40
50
60
10
20
30
40
50
60
Figure 2: Original image Monarch (left), sparse image representation with 449 coefficients using the proposed nonlinear
diffusion filter bank (middle) and the biorthogonal filter bank with 7-9 filter (right).
i.e., there will remain 16 low-pass coefficients. For thresholding we use the hard shrinkage function with θ = 13.
In Figure 2(left), we present the original image. Figure
2(middle) shows the obtained compressed image with 449
remaining coefficients using the new diffusion filter bank.
For comparison, we apply 8 decomposition levels of the
two-dimensional biorthogonal wavelet shrinkage with the
7−9 filter with the same number of 449 remaining nonzero
coefficients, see Figure 2(right). As we can see, the nonlinear filter bank not only gives an optically better result
but also achieves a better PSNR value (26.41 dB) while
the biorthogonal filter bank achieves a PSNR of 24.73 dB.
We remark that our method is especially designed for constructing efficient low-level representations and does not
work well for image denoising.
6. Acknowledgement
The research in this paper is supported by the project PL
170/13-1 of the German Research Foundation (DFG). This
is gratefully acknowledged.
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