Higher Order Tensor Analysis and Higher Order SVD for Diffusion Tensor Analysis

P. G. Batchelor, F. Calamante, D. Atkinson, D. L. G. Hill, D. Tournier, A. Connelly University College London, London, United Kingdom, Brain Research Institute, Melbourne, Australia Introduction. 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. Much current research in diffusion MR, however, is directed towards Higher Angular Resolution Diffusion Images (HARDI) data, that can be modeled in a number of different ways, e.g. as multi-exponential [1] or a natural generalization as Higher Order Tensor (e.g. [2]) to take account of multiple fibre directions within individual voxels. In this context, a generalization of SVD to such data would appear highly desirable, and such generalizations, called Higher Order SVD (HOSVD) although not commonly used, do exist [3]. Thus, our aim here is to introduce and investigate the usefulness of HOSVDs in HARDI, and describe mathematically the transition from 2 order tensor to higher data. Theory-Methods. Multiexponentials: Physically, a voxel can contain multiple contributions to diffusion due to partial volume effect. The most straightforward description of the measured signal for a given diffusion gradient orientation g is to assume that it is a linear combination of the signals from individual Gaussian diffusion [1]. This leads to a multi-exponential description of the signal: s(g)/s0 =f exp(-bgD1g) + (1-f)exp(-bg D2g). The factors f and 1-f measure the size of each contribution; b-factors encode the amount of diffusion weighting. As a first theoretical result, this signal is asymptotically weighted average of the tensors (this can be seen from a Taylor expansion in b): log(s(g)/s0) ≈ -b g Dg ≈ -bg (fD1 + (1-f)D2)g + (higher order terms), (1) The fibre direction is extracted as the principal eigendirection of the tensor D. When b becomes larger, the expansion to second order becomes inaccurate, and higher order (even) terms, of at least 4 order have to be considered (expressions of the type Dijkl). This turns out to have a surprisingly simple expression for two compartments: the coefficient of b in the Taylor expansion, i.e., the 4 order tensorial term in (1) ≈ f(1-f)(g(D1 – D2)g) 2 . (2) 4 Order Stejskal-Tanner: An alternative view is the one proposed in [2], where instead of treating s(g) as a multi-exponential, it is viewed directly as an exponential of higher order tensors, log(s(g)/s0) ~ -b Dijkl gi gj gk gl (summation over repeated coordinate indices, this is eq. (11) in [2]). Thus, in both cases tensors of at least 4 order are needed. HOSVD: Standard SVDs can be described as a decomposition of the tensor in a weighted sum of rank 1 terms (dyadic products). Two generalizations of this interpretation of the SVD to expressions of the form Dijkl are available: the PARAFAC and Tucker-HOSVD (see [3] for details). As a simulation, we construct a pure ‘crossing’ situation of two tensors D1 and D2 with orthogonal principal directions forming an angle of 90° (arbitrary non-principal directions, both with FA= 0.86, f=0.75).