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

The statistical analysis of covariance matrix data is consideredand, in particular, methodology is discussed which takes into accountthe non-Euclidean nature of the space of positive semi-definite sym-metric matrices. The main motivation for the work is the analysis ofdiffusion tensors in medical image analysis. The primary focus is onestimation of a mean covariance matrix and, in particular, on the useof Procrustes size-and-shape space. Comparisons are made with otherestimation techniques, including using the matrix logarithm, matrixsquare root and Cholesky decomposition. Applications to diffusiontensor imaging are considered and, in particular, a new measure offractional anisotropy called Procrustes Anisotropy is discussed