Eigenvalues of Random Matrices with Isotropic Gaussian Noise and the Design of Diffusion Tensor Imaging Experiments

Area of Science:

• Multivariate statistics
• Material sciences
• Medical imaging

Background:

• Tensor-valued and matrix-valued measurements are common in material sciences and medical imaging.
• Eigenvalues and eigenvectors of this data offer unique insights but require complex statistical analysis.
• This work focuses on symmetric random matrices with isotropic matrix-variate Gaussian noise.

Purpose of the Study:

• Derive distributions of eigenvalues and eigenvectors for symmetric random matrices.
• Analyze how these distributions depend on the symmetries of the mean tensor.
• Apply these derivations to diffusion tensor imaging (DTI) for symmetry detection and experimental design.

Main Methods:

• Derivation of eigenvalue and eigenvector distributions for m x m symmetric random matrices.
• Analysis of distribution properties based on mean tensor symmetries.
• Application to diffusion tensor imaging (DTI) with m=3.
• Utilizing quadrature rules for experimental design in DTI.

Main Results:

• Eigenvalue distributions depend strongly on mean tensor symmetries.
• Non-Gaussian asymptotic distributions and eigenvalue repulsion observed when the mean tensor has repeated eigenvalues.
• A simple asymptotic distribution for eigenvalue central moments in 3D DTI with isotropic Gaussian noise allows for isotropy testing.
• Designed DTI experiments where tensor isotropy implies Fisher information isotropy.

Conclusions:

• The derived distributions and methods offer a framework for analyzing tensor-valued data in DTI.
• The study provides a method for detecting symmetries in diffusion tensors.
• The proposed experimental design in DTI can improve the detection of tensor isotropy.