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

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http://hdl.handle.net/10138/228570

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Gasbarra , D , Pajevic , S & Basser , P J 2017 , ' Eigenvalues of Random Matrices with Isotropic Gaussian Noise and the Design of Diffusion Tensor Imaging Experiments ' , SIAM journal on imaging sciences , vol. 10 , no. 3 , pp. 1511-1548 . https://doi.org/10.1137/16M1098693

Title: Eigenvalues of Random Matrices with Isotropic Gaussian Noise and the Design of Diffusion Tensor Imaging Experiments
Author: Gasbarra, Dario; Pajevic, Sinisa; Basser, Peter J.
Contributor: University of Helsinki, Department of Mathematics and Statistics
Date: 2017
Language: eng
Number of pages: 38
Belongs to series: SIAM journal on imaging sciences
ISSN: 1936-4954
URI: http://hdl.handle.net/10138/228570
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 x 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, (D) over bar. When (D) over bar has repeated eigenvalues, the eigenvalues of D are not asymptotically Gaussian, and repulsion is observed between the eigenvalues corresponding to the same (D) over bar 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 >= 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.
Subject: eigenvalue and eigenvector distribution
asymptotics
sphericity test
singular hypothesis testing
DTI
spherical t-design
Gaussian orthogonal ensemble
STATISTICAL-ANALYSIS
SYMMETRIC-MATRICES
LATENT ROOTS
MRI
EIGENVECTORS
EXPANSION
IMAGES
TESTS
111 Mathematics
113 Computer and information sciences
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