Eigenvalues of random matrices with isotropic Gaussian noise and the design of Diffusion Tensor Imaging experiments

TL;DR

This paper derives the distributions of eigenvalues and eigenvectors for symmetric random matrices with Gaussian noise, and applies these results to optimize diffusion tensor imaging experiments for detecting tensor symmetries.

Contribution

It provides new statistical distributions for eigenvalues/eigenvectors under Gaussian noise and proposes experimental design methods for DTI to test tensor isotropy.

Findings

Eigenvalue distributions depend on tensor symmetries.

Repulsion occurs between eigenvalues with the same eigenspaces.

Designs using quadrature rules can test for isotropy in DTI.

Abstract

Tensor-valued and matrix-valued measurements of different physical properties are increasingly available in material sciences and medical imaging applications. The eigenvalues and eigenvectors of such multivariate data provide novel and unique information, but at the cost of requiring a more complex statistical analysis. In this work we derive the distributions of eigenvalues and eigenvectors in the special but important case of $m \times m$ symmetric random matrices, $D$, observed with isotropic matrix-variate Gaussian noise. The properties of these distributions depend strongly on the symmetries of the mean tensor/matrix, $\bar D$. When $\bar D$ has repeated eigenvalues, the eigenvalues of $D$ are not asymptotically Gaussian, and repulsion is observed between the eigenvalues corresponding to the same $\bar D$ eigenspaces. We apply these results to diffusion tensor imaging (DTI), with $m=3$, addressing an important problem of detecting the symmetries of the diffusion tensor, and seeking an experimental design that could potentially yield an isotropic Gaussian distribution. In the 3-dimensional case, when the mean tensor is spherically symmetric and the noise is Gaussian and isotropic, the asymptotic distribution of the first three eigenvalue central moment statistics is simple and can be used to test for isotropy. In order to apply such tests, we use quadrature rules of order $t \ge 4$ with constant weights on the unit sphere to design a DTI-experiment with the property that isotropy of the underlying true tensor implies isotropy of the Fisher information. We also explain the potential implications of the methods using simulated DTI data with a Rician noise model.