Discrimination of Volumetric Shapes Using Orthogonal Tensor Decomposition

Hayato Itoh¹⁸ &
Atsushi Imiya¹⁹

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11167))

Included in the following conference series:

International Workshop on Shape in Medical Imaging

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Abstract

Organs, cells and microstructures in cells dealt with in medical image analysis are volumetric data. Sampled values of volumetric data are expressed as three-way array data. For the quantitative discrimination of multiway forms from the viewpoint of principal component analysis (PCA)-based pattern recognition, distance metrics for subspaces of multiway data arrays are desired. The paper aims to extend pattern recognition methodologies based on PCA for vector spaces to those for multilinear data. First, we extend the canonical angle between linear subspaces for vector-based pattern recognition to the canonical angle between multilinear subspaces for tensor-based pattern recognition. Furthermore, using transportation between the Stiefel manifolds, we introduce a new metric for a collection of linear subspaces. Then, we extend the transportation of between Stiefel manifolds in vector space to the transportation of the Stiefel manifolds in multilinear spaces for the discrimination analysis of multiway array data.

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Mathematical Aspects of Tensor Subspace Method

Relaxed Optimisation for Tensor Principal Component Analysis and Applications to Recognition, Compression and Retrieval of Volumetric Shapes

Approximation of N-Way Principal Component Analysis for Organ Data

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Authors and Affiliations

Graduate School of Informatics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
Hayato Itoh
Institute of Management and Information Technologies, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba, 263-8522, Japan
Atsushi Imiya

Authors

Hayato Itoh
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Atsushi Imiya
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Corresponding author

Correspondence to Atsushi Imiya .

Editor information

Editors and Affiliations

German Center for Neurodegenerative Diseases, Bonn, Germany
Martin Reuter
Ludwig Maximilian University of Munich, Munich, Germany
Christian Wachinger
École de Technologie Supérieure, Montreal, QC, Canada
Hervé Lombaert
Kitware Inc., Carrboro, NC, USA
Beatriz Paniagua
University of Basel, Basel, Switzerland
Marcel Lüthi
Massachusetts Institute of Technology, Cambridge, MA, USA
Bernhard Egger

Appendix

Image SVD (imageSVD) [8, 9] for the image array $\varvec{X}\in \mathbf{R}^{m\times n}$ establishes the decomposition

$$\begin{aligned} \varvec{X}=\varvec{U}\varvec{X}\varvec{V}^\top +\varvec{E}, \end{aligned}$$

where $\varvec{U}\varvec{X}\varvec{V}^\top $ has low-rank and $\varvec{E}$ is the residual error. This decomposition is performed by minimising $|\varvec{E}|_{\mathrm {F}}^2$ with respect to the conditions

$$\begin{aligned} \varvec{U}^\top \varvec{U}= \left( \begin{array}{cc} \varvec{I}_{k},&{} \varvec{O}\\ \varvec{O},&{} \varvec{O} \end{array}\right) , \, \, \, \varvec{V}^\top \varvec{V}= \left( \begin{array}{cc} \varvec{I}_{l},&{} \varvec{O}\\ \varvec{O},&{} \varvec{O} \end{array}\right) \end{aligned}$$

for $k\le m$ and $l\le n$. Eigenmatrices of $\varvec{X}\varvec{X}^\top $ and $\varvec{X}^\top \varvec{X}$ derive matrices $\varvec{U}$ and $\varvec{V}$, respectively.

For a collection of $m\times n$ matrices $\{\varvec{X}_i\}_{i=1}^N$, where $N\gg \max (m, n)$, we assume that

$$\begin{aligned} \frac{1}{N}\sum _{i=N}\varvec{X}_i=\varvec{O}. \end{aligned}$$

The matrix PCA derives a pair of matrices $\varvec{U}$ and $\varvec{V}$ by minimising the criterion

$$\begin{aligned} J(\varvec{U}, \varvec{V})=\frac{1}{N}\sum _{i=1}^N|\varvec{X}_i-\varvec{U}\varvec{X}\varvec{V}^\top |_{\mathrm {F}}^2 \end{aligned}$$

with the constraints $\varvec{U}^\top \varvec{U}=\varvec{I}_{m}$ and $\varvec{V}^\top \varvec{V}=\varvec{I}_{n}$. A pair of orthogonal matrices $\varvec{U}$ and $\varvec{V}$ are eigenmatrices of

$$\begin{aligned} \varvec{M}=\sum _{i=1}^N\varvec{X}_i\varvec{X}_i^\top , \, \, \varvec{N}=\sum _{i=1}^N\varvec{X}_i^\top \varvec{X}_i. \end{aligned}$$

For a pair of $k\times m \times n$ three-ways $\varvec{F}=((f_{\alpha \beta \gamma }))$ and $\varvec{G}=(( g_{\alpha \beta \gamma }))$, Euclidean distance $d_E$ and the transportation $d_T$ of intensities are

$$\begin{aligned} d_E(\varvec{F},\varvec{G})^2= & {} \sum _{\alpha =1}^k\sum _{\beta =1}^m\sum _{\gamma =1}^n |f_{\alpha \beta \gamma }- g_{\alpha \beta \gamma }|^2\\ d_T(\varvec{F},\varvec{G})^2= & {} \min _{ c_{\alpha \alpha '\beta \beta '\gamma \gamma '} } \sum _{\alpha \, \alpha '=1}^k\sum _{\beta \, \beta '=1}^m \sum _{\gamma \, \gamma '=1}^n k_{\alpha \beta \gamma }^{\alpha '\beta '\gamma '} c_{\alpha \alpha '\beta \beta '\gamma \gamma '} \end{aligned}$$

with respect to

$$\begin{aligned} g_{\alpha '\beta '\gamma '}\ge \sum _{\alpha =1}^k\sum _{\beta =1}^m \sum _{\gamma =1}^n c_{\alpha \alpha '\beta \beta '\gamma \gamma '}, \, \, \, f_{\alpha \beta \gamma }\ge \sum _{\alpha '=1}^k\sum _{\beta '=1}^m \sum _{\gamma '=1}^n c_{\alpha \alpha '\beta \beta '\gamma \gamma '}, \, \, \, c_{\alpha \alpha '\beta \beta '\gamma \gamma '}\ge 0 \end{aligned}$$

for $ k_{\alpha \beta \gamma }^{\alpha '\beta '\gamma '} = |f_{\alpha \beta \gamma }- g_{\alpha '\beta '\gamma '}|^2$.

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Itoh, H., Imiya, A. (2018). Discrimination of Volumetric Shapes Using Orthogonal Tensor Decomposition. In: Reuter, M., Wachinger, C., Lombaert, H., Paniagua, B., Lüthi, M., Egger, B. (eds) Shape in Medical Imaging. ShapeMI 2018. Lecture Notes in Computer Science(), vol 11167. Springer, Cham. https://doi.org/10.1007/978-3-030-04747-4_26

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DOI: https://doi.org/10.1007/978-3-030-04747-4_26
Published: 23 November 2018
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-04746-7
Online ISBN: 978-3-030-04747-4
eBook Packages: Computer ScienceComputer Science (R0)

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