Higher Order Tensor Analysis and Higher Order SVD for Diffusion Tensor Analysis
Batchelor P, Atkinson D, G. Hill D, Calamante F, Connelly A, Tournier D
University College London
The standard Singular Value Decomposition (SVD) is one of the main tools to study classical DT-MRI data, when they are represented as symmetric, positive definite tensors of order 2 (written D_ik). Current research in diffusion MR, however, is towards Higher Angular Resolution Diffusion Images (HARDI) data, that can be modeled in different ways as multi-exponential or as Higher Order Tensor (e.g. Ozarsalan). In this context, a generalization of the SVD to such data would appear highly desirable. It turns out that such generalizations, called Higher Order SVD (HOSVD) although not well known, do exist. One possible reason for their reduced use is that the generalization is not unique, and some properties are lost. Nevertheless, it seems a potentially too important tool to be neglected. Thus, our aim here is to introduce and investigate the usefulness of HOSVDs in HARDI, discuss and describe these decompositions, and describe mathematically the transition from 2nd order tensor to higher data.