Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging

The statistical analysis of covariance matrices occurs in m any important applications, e.g. in diffusion tensor imaging or longitudinal data analysis. We consider the situation where it is of interest to estimate an average covariance matrix, describe its anisotropy and to carry out principal geodesic analysis of covariance matrices. In medical image analysis a particular type of covariance matrix arises in diffusion weighted imaging called a diffusion tensor. The diffusion tensor is a 3 × 3 covariance matrix which is estimated at each voxel in the brain, and is obtained by fittin g a physically-motivated model on measurements from the Fourier transform of the molecule displacement density (Basser et al., 1994). A strongly anisotropic diffusion tensor indicates a strong direction of white matter fibre tracts, and plots of measures of anisotropy are very useful t o neurologists. A measure that is very commonly used in diffusion tensor imaging is Fractional Anisotropy

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