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

The statistical analysis of covariance matrix data is considered, and in particular methodology is discussed which takes into account the non-Euclidean nature of the space of positive semi-definite symmetric matrices. The main motivation for the work is the analysis of diffusion tensors in medical image analysis. The primary focus is on estimation of a mean covariance matrix, and in particular on the use of Procrustes size-and-shape space. Comparisons are made with other estimation techniques, including using the matrix logarithm, matrix square root and Cholesky decomposition. Applications to diffusion tensor imaging are considered, and in particular a new measure of fractional anisotropy called Procrustes Anisotropy is discussed.