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

TL;DR

This paper develops non-Euclidean statistical methods for covariance matrices, focusing on diffusion tensor imaging, introducing a Procrustes-based mean estimation and a new anisotropy measure.

Contribution

It introduces a novel Procrustes-based approach for estimating mean covariance matrices and a new anisotropy measure tailored for diffusion tensor imaging.

Findings

Procrustes mean estimation outperforms traditional methods.

The new Procrustes Anisotropy provides improved diffusion tensor analysis.

Comparative analysis shows advantages over matrix logarithm and Cholesky methods.

Abstract

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.