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

Abstract

The mathematical and computational backgrounds of pattern recognition are the geometries in Hilbert space used for functional analysis and the applied linear algebra used for numerical analysis, respectively. Organs, cells and microstructures in cells dealt with in biomedical image analysis are volumetric data. We are required to process and analyse these data as volumetric data without embedding into higher-dimensional vector spaces from the viewpoint of object-oriented data analysis . Therefore, sampled values of volumetric data are expressed as three-way array data. The aim of the paper is to develop relaxed closed forms for tensor principal component analysis (PCA) for the recognition , classification , compression and retrieval of volumetric data. Tensor PCA derives the tensor Karhunen-Loève transform, which compresses volumetric data, such as organs, cells in organs and microstructures in cells, preserving both the geometric and statistical properties of objects and spatial textures in the space.

Appendix

The linear reduction operation R and its dual E are defined as

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} g(\boldsymbol{x})= Rf(\boldsymbol{y}) &\displaystyle =&\displaystyle \int_{{\mathbf{R}}^3}w_3(\boldsymbol{u})f(2\boldsymbol{x}-\boldsymbol{u})d\boldsymbol{u}, \end{array} \end{aligned} $$

(8.60)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} Eg(\boldsymbol{x})&\displaystyle =&\displaystyle 2^3 \int_{\mathbb{R}^3} w_3(\boldsymbol{u})g \left(\frac{\boldsymbol{x}-\boldsymbol{u}}{2}\right)d\boldsymbol{u}. \end{array} \end{aligned} $$

(8.61)

where w ₃(x) = w(x)w(y)w(z) for x = (x, y, z)^⊤ and

$$\displaystyle \begin{aligned} \begin{array}{rcl} w(s)=\left\{\begin{array}{ll} \frac{1}{2}(1-\frac{|s|}{2}), & |s|\leq 2\\ 0, & |s|> 2 \end{array}\right. \end{array} \end{aligned} $$

(8.62)

These processes are achieved by computing a weighted average of the image values in a finite small region, which is called the window for the operation.

Setting $w_{\pm 1}=\frac {1}{4}$ and $w_0=\frac {1}{2}$, for the two-dimensional sampled function f _ijk = f(Δi, Δj, Δk), the transforms of Eqs. (8.60) and (8.61) are respectively described as

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} R f_{kmn}&\displaystyle =&\displaystyle \sum_{p,q,r=-1}^{1}w_{p}w_{q}w_{r} f_{2k-p\, \, 2m-q\,\,2n-p}, \end{array} \end{aligned} $$

(8.63)

$$\displaystyle \begin{aligned} \begin{array}{rcl} E f_{kmn}&\displaystyle =&\displaystyle \frac{1}{2^3} \sum_{p,q,r=-2}^{2} w _{p}w_{q}w_{r} f_{\frac{k-p}{2}\, \, \frac{m-q}{2}\,\,\frac{n-r}{2}}, \end{array} \end{aligned} $$

(8.64)

where the summation is achieved for $\frac {(k-p)}{2}$, $\frac {(m-q)}{2}$ and $\frac {(n-r)}{2}$ being integers. These procedures are called the pyramid transform and extension, respectively. These two operations involve the reduction and expansion of the image sizes. Therefore, image features are extracted in the higher-layer images of the pyramid transform.

Setting

$$\displaystyle \begin{aligned} \boldsymbol{R}=\frac{1}{4}(\boldsymbol{I}\otimes(0,1)^\top)(\boldsymbol{D}+4\boldsymbol{I})\end{aligned} $$

(8.65)

for the second-order differential matrix D with the Neumann condition such that

$$\displaystyle \begin{aligned} \boldsymbol{D}=\left(\begin{array}{ccccccc} -1& 1&0 &0& \cdots & 0& 0\\ 1&-2&1 &0& \cdots & 0& 0\\ 0& 1&-2&1& \cdots & 0& 0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0& 0& 0& \cdots & 0& 1& -1 \end{array}\right),\end{aligned} $$

(8.66)

the three-dimensional pyramid transform

$$\displaystyle \begin{aligned} g_{pqr}=\sum_{i,j=-1}^{1}w_{i}w_{j}w_{k} f_{2p-i\,\,2q-j\,\,2r-k, },\end{aligned} $$

(8.67)

is redescribed

$$\displaystyle \begin{aligned} \mathcal{Y}=\mathcal{X} \times_1 \boldsymbol{R}\times_2 \boldsymbol{R}\times_3 \boldsymbol{R} \end{aligned} $$

(8.68)

using the Tucker-3 decomposition of $\mathcal {X}$.

Since the eigenmatrix of D is the DCT-II matrix the three-dimensional pyramid transform processes the following property.

Property 8.2

Setting $L^N={\mathsf {L}}( \{\boldsymbol {\varphi }_{k} \}_{k=0}^{2^N-1} )$, for three-dimensional images, the pyramid transform is a linear transform from L ^N × L ^N × L ^N to $L^{\frac {N}{2}}\times L^{\frac {N}{2}} \times L^{\frac {N}{2}}$.

Equation (8.68) directly derives outlines of volumetric shapes by enforcing and inhibiting low- and high-frequency parts, respectively, on the DCT of the volumetric shape. Therefore, the dominant operation in the pyramid transform is the relaxed Karhunen-Loève transform using the DCT.