article

Constructing an atlas for the diffeomorphism group of a compact manifold with boundary, with application to the analysis of image registrations

Authors:

Carole J. Twining,

Stephen MarslandAuthors Info & Claims

Journal of Computational and Applied Mathematics, Volume 222, Issue 2

Pages 411 - 428

https://doi.org/10.1016/j.cam.2007.11.008

Published: 20 December 2008 Publication History

Abstract

This paper considers the problem of defining a parameterization (chart) on the group of diffeomorphisms with compact support, motivated primarily by a problem in image registration, where diffeomorphic warps are used to align images. Constructing a chart on the diffeomorphism group will enable the quantitative analysis of these warps to discover the normal and abnormal variation of structures in a population. We construct a chart for particular choices of boundary conditions on the space on which the diffeomorphism acts, and for a particular class of metrics on the diffeomorphism group, which define a class of diffeomorphic interpolating splines. The geodesic equation is computed for this class of metrics, and we show how it can be solved in the spline representation. Furthermore, we demonstrate that the spline representation generates submanifolds of the diffeomorphism group, and we study this mapping. Explicit computational examples are included, showing how this chart can be constructed in practice, and that the use of the geodesic distance allows better classification of variation than those obtained using just a Euclidean metric on the space of warps.

References

[1]

Adams, M., Ratiu, T. and Schmid, R., The Lie group structure of diffeomorphism groups and invertible Fourier integral operators with applications. In: Kac, V. (Ed.), Mathematical Sciences Research Institute Publications, vol. 4. Springer-Verlag.

[2]

Arnold, V.I., Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications í l'hydrodynamique des fluides parfaits. Annales de L'Institut Fourier (Grenoble). v16 i1. 319-361.

[3]

Arsigny, V., Commowick, O., Pennec, X. and Ayache, N., A log-euclidean framework for statistics on diffeomorphisms. In: Larsen, R., Nielsen, M., Sporring, J. (Eds.), Lecture Notes in Computer Science, vol. 4190. Springer-Verlag. pp. 924-931.

[4]

Boggio, T., Sulle funzioni di Green d'ordine m. Rendiconti - Circolo Matematico di Palermo. v20. 97-135.

[5]

Bookstein, F.L., Morphometric Tools for Landmark Data: Geometry and Biology. 1997. Cambridge University Press, Cambridge, UK.

[6]

Camion, V. and Younes, L., Geodesic interpolating splines. In: Figueiredo, M., Zerubia, J., Jain, A.K. (Eds.), Lecture Notes in Computer Science, vol. 2134. Springer-Verlag. pp. 513-527.

Digital Library

[7]

Cootes, T.F., Taylor, C.J., Cooper, D.H. and Graham, J., Active shape models-their training and application. Computer Vision and Image Understanding. v61 i1. 38-59.

Digital Library

[8]

Duchon, J., Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. Revue Française d'Automatique, Informatique, Recherche Opérationelle (RAIRO) Analyse Numerique. v10. 5-12.

[9]

Dupuis, P., Grenander, U. and Miller, M.I., Variational problems on flows of diffeomorphisms for image matching. Quarterly of Applied Mathematics. v56 i3. 587-600.

Digital Library

[10]

Ebin, D.G. and Marsden, J.E., Groups of diffeomorphisms and the motion of an incompressible fluid. Annals of Mathematics, 2nd Series. v92 i1. 102-163.

[11]

Holm, D.D., Ratnanather, J.T., Trouvé, A. and Younes, L., Soliton dynamics in computational anatomy. NeuroImage. v23 iSuppl. 1. S170-S178.

[12]

Holm, D.D., Marsden, J.E. and Ratiu, T.S., The Euler-Poincaré equations and semidirect products, with applications to continuum theories. Advances in Mathematics. v137 i1. 1-81.

[13]

Holm, D.D. and Marsden, J.E., Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff Equation. In: Progressive Mathematics, vol. 232. Birkhäuser, Boston. pp. 203-235.

[14]

Joshi, S.C. and Miller, M.M., Landmark matching via large deformation diffeomorphisms. IEEE Transactions on Image Processing. v9 i8. 1357-1370.

Digital Library

[15]

Kopell, N., Commuting Diffeomorphisms. In: Proceedings of the Symposium of Pure Mathematics, vol. 14. American Mathematical Society. pp. 165-184.

[16]

Kouranbaeva, S., The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. Journal of Mathematical Physics. v40 i2. 857-868.

[17]

Kriegl, A. and Michor, P.W., . In: Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence, RI.

[18]

Marsden, J.E. and Ratiu, T.S., Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. 1999. 2nd edition. Springer, Berlin.

[19]

Marsland, S. and Twining, C.J., Constructing diffeomorphic representations for the groupwise analysis of the nonrigid registrations of medical images. IEEE Transactions on Medical Imaging. v23 i8. 1006-1020.

[20]

Michor, P.W., Some geometric evolution equations arising as geodesic equations on groups of diffeomorphism, including the Hamiltonian approach. In: Bove, A., Colombini, F., Del Santo, D. (Eds.), Progress in Non Linear Differential Equations and Their Applications, vol. 69. Birkhauser Verlag. pp. 133-215.

[21]

Michor, P.W. and Ratiu, T.S., On the geometry of the Virasoro-Bott group. Journal of Lie Theory. v8 i2. 293-309.

[22]

Miller, M., Trouvé, A. and Younes, L., Geodesic shooting for computational anatomy. Journal of Mathematical Imaging and Vision. v24 i2. 209-228.

Digital Library

[23]

Miller, M.I., Trouvé, A. and Younes, L., On the metrics and Euler-Lagrange equations of computational anatomy. Annual Review of Biomedical Engineering. v4. 375-405.

[24]

Newcomb, W.A., Lagrangian and Hamiltonian methods in magnetohydrodynamics. Nuclear Fusion: Suppl. Part 2. 451-463.

[25]

Omori, H., On the group of diffeomorphisms on a compact manifold. In: Proceedings of the Symposium of Pure Mathematics, vol 15. American Mathematical Society. pp. 167-184.

[26]

Schmid, R., Infinite dimensional Lie groups with applications to mathematical physics. Journal of Geometry and Symmetry in Physics. v1. 54-120.

[27]

Shkoller, S., Geometry and curvature of diffeomorphism groups with H1 metric and mean hydrodynamics. Journal of Functional Analysis. v160. 337-365.

[28]

Toga, A.W., Brain Warping. 1999. Academic Press, San Diego.

[29]

Trouvé, A., Diffemorphism groups and pattern matching in image analysis. International Journal of Computer Vision. v28 i3. 213-221.

Digital Library

[30]

Vaillant, M., Miller, M., Younes, L. and Trouvé, A., Statistics on diffeomorphisms via tangent space representations. NeuroImage. v23 iSuppl. 1. S161-S169.

[31]

Zitová, B. and Flusser, J., Image registration methods: A survey. Image and Vision Computing. v21. 977-1000.

Cited By

Abbad ZMaliani AAlaoui SHassouni MAbbassi M(2022)The Geodesic Distance on the Generalized Gamma Manifold for Texture Image RetrievalJournal of Mathematical Imaging and Vision10.1007/s10851-021-01063-x64:3(243-260)Online publication date: 1-Mar-2022
https://dl.acm.org/doi/10.1007/s10851-021-01063-x
Verdoolaege GScheunders P(2012)On the Geometry of Multivariate Generalized Gaussian ModelsJournal of Mathematical Imaging and Vision10.1007/s10851-011-0297-843:3(180-193)Online publication date: 1-Jul-2012
https://dl.acm.org/doi/10.1007/s10851-011-0297-8
De Craene MSukno FTobon-Gomez CButakoff CVentura RHoogendoorn CPiella GDuchateau NMuñoz-Moreno ESebastian RCamara OFrangi A(2010)Atlas construction and image analysis using statistical cardiac modelsProceedings of the First international conference on Statistical atlases and computational models of the heart, and international conference on Cardiac electrophysiological simulation challenge10.5555/1889198.1889200(1-13)Online publication date: 20-Sep-2010
https://dl.acm.org/doi/10.5555/1889198.1889200

Index Terms

Constructing an atlas for the diffeomorphism group of a compact manifold with boundary, with application to the analysis of image registrations
1. Mathematics of computing
  1. Discrete mathematics

Recommendations

Deep neural networks on diffeomorphism groups for optimal shape reparametrization
Abstract
One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, ...
Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics ...
Synthesis of an Individualized Cranial Atlas with Dysmorphic Shape
MMBIA '96: Proceedings of the 1996 Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA '96)

Abstract: A new method for non-rigid registration of a normal infant CT head atlas with CT data of infants with abnormal skull shape is presented. An individualized atlas is synthesized computing a volume transformation from the normal atlas to the ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Journal of Computational and Applied Mathematics

Journal of Computational and Applied Mathematics Volume 222, Issue 2

December, 2008

507 pages

ISSN:0377-0427

Issue’s Table of Contents

Copyright © Elsevier B.V. © 2007.

Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 20 December 2008

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

3
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 21 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Abbad ZMaliani AAlaoui SHassouni MAbbassi M(2022)The Geodesic Distance on the Generalized Gamma Manifold for Texture Image RetrievalJournal of Mathematical Imaging and Vision10.1007/s10851-021-01063-x64:3(243-260)Online publication date: 1-Mar-2022
https://dl.acm.org/doi/10.1007/s10851-021-01063-x
Verdoolaege GScheunders P(2012)On the Geometry of Multivariate Generalized Gaussian ModelsJournal of Mathematical Imaging and Vision10.1007/s10851-011-0297-843:3(180-193)Online publication date: 1-Jul-2012
https://dl.acm.org/doi/10.1007/s10851-011-0297-8
De Craene MSukno FTobon-Gomez CButakoff CVentura RHoogendoorn CPiella GDuchateau NMuñoz-Moreno ESebastian RCamara OFrangi A(2010)Atlas construction and image analysis using statistical cardiac modelsProceedings of the First international conference on Statistical atlases and computational models of the heart, and international conference on Cardiac electrophysiological simulation challenge10.5555/1889198.1889200(1-13)Online publication date: 20-Sep-2010
https://dl.acm.org/doi/10.5555/1889198.1889200

View Options

View options

Media

Figures

Other

Tables

View Issue’s Table of Contents