Resolution Conversion of Volumetric Array Data for Multimodal Medical Image Analysis

Abstract

This paper aims to clarify statistical and geometric properties of linear resolution conversion for registration between different resolutions observed using the same modality. The pyramid transform for higher-dimensional array with rational-order is formulated by means of tensor decomposition. For fast processing of volumetric data, compression of data is an essential task. Three-dimensional extension of the pyramid transform reduces the sizes of the volumetric data by factor 2. Extension of matrix expression of the pyramid transform to the operation of tensors using the mode product of a tensor and matrix derives the pyramid transform for volumetric data of the rational orders. The pyramid transform is achieved by downsampling after linear smoothing. The dual operation of the pyramid transform is achieved by linear interpolation after upsampling. The rational-order pyramid transform is decomposed into upsampling by linear interpolation and the traditional pyramid transform with the integer order. By controlling ratio between upsampling for linear interpolation and downsampling in the pyramid transform, the rational-order pyramid transform is computed. The tensor expression of the volumetric pyramid transform clarifies that the transform yields the orthogonal base systems for any ratios of the rational pyramid transform.

Appendix: Discrete Heat Equation

For the heat equation $\frac{\partial f}{\partial \tau }=\frac{1}{2}\cdot \frac{\partial ^2 f}{\partial x^2}$ in $\mathbf{R}^2\times \mathbf{R}_+$, the semi-implicit discretisation with the Neumann boundary condition

$$ \frac{\varvec{f}^{(k+1)}-\varvec{f}^{(k)}}{\tau }=\frac{1}{2}\varvec{D}\varvec{f}^{(k+1)}, $$

and the eigenvalue decomposition of the matrix $\varvec{D}$ yield the iteration form [20, 21]

$$\begin{aligned} \varvec{f}^{(k+1)}&=\varvec{\varPhi }\left( \varvec{I}-\frac{\tau }{2}\varvec{\varLambda }\right) ^{-k}\varvec{\varPhi }^\top \varvec{f},&\varvec{\varLambda }&=((\lambda _i\delta _{ij})), \end{aligned}$$

where $\lambda _0>\lambda _2>\cdots >\lambda _{n-1}$, for $\varvec{f}=(f_0,f_1,\cdots ,f_{n-1})^\top$. This iteration form implies that the discrete scale transform is a linear transform from $\mathsf {L}\{\varvec{\varphi }_i\}_{i=0}^{n-1}$ to $\mathsf {L}\{\varvec{\varphi }_i\}_{i=0}^{n-1}$.