Eigenvalues of random matrices with isotropic Gaussian noise and the design of Diffusion Tensor Imaging experiments
| Autor: | Dario Gasbarra, Peter J. Basser, Sinisa Pajevic |
|---|---|
| Přispěvatelé: | Department of Mathematics and Statistics |
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
FOS: Computer and information sciences
General Mathematics Gaussian IMAGES STATISTICAL-ANALYSIS singular hypothesis testing 01 natural sciences Article Gaussian orthogonal ensemble 030218 nuclear medicine & medical imaging Methodology (stat.ME) LATENT ROOTS 010104 statistics & probability 03 medical and health sciences Matrix (mathematics) symbols.namesake 0302 clinical medicine spherical t-design 111 Mathematics Symmetric matrix Tensor SYMMETRIC-MATRICES 0101 mathematics Eigenvalues and eigenvectors Statistics - Methodology Physics 62E20 62K 62P10 92C55 94A08 sphericity test Applied Mathematics Isotropy Mathematical analysis EIGENVECTORS EXPANSION 113 Computer and information sciences eigenvalue and eigenvector distribution asymptotics Gaussian noise DTI TESTS symbols Random matrix MRI |
| Popis: | 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. 40+4 pages, 14 figures |
| Databáze: | OpenAIRE |
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