Discrimination of Volumetric Shapes Using Orthogonal Tensor Decomposition

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.

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

Fig. 3.

Wasserstein distances between the first and jth frames in sequence of volumetric images. We use the sum of the singular values, the sum of the eigenvalues and the energy for constraints. The left, middle and right columns show the distances for the case of using sum of the singular values, sum of the eigenvalues and the energy, respectively, for constraints. The top, middle and bottom rows show the distances for $p=1, 2, \infty$, respectively. The vertical axis represents the distance between frames. The horizontal axis represents the number j of the target frame.