  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.
