Regularisation, interpolation and visualisation of diffusion tensor images using non-Euclidean statistics

Practical statistical analysis of diffusion tensor images is considered, and we focus primarily on methods that use metrics based on Euclidean distances between powers of diffusion tensors. First, we describe a family of anisotropy measures based on a scale invariant power-Euclidean metric, which are useful for visualisation. Some properties of the measures are derived and practical considerations are discussed, with some examples. Second, we discuss weighted Procrustes methods for diffusion tensor imaging interpolation and smoothing, and we compare methods based on different metrics on a set of examples as well as analytically. We establish a key relationship between the principal-square-root-Euclidean metric and the size-and-shape Procrustes metric on the space of symmetric positive semi-definite tensors. We explain, both analytically and by experiments, why the size-and-shape Procrustes metric may be preferred in practical tasks of interpolation, extrapolation and smoothing, especially when observed tensors are degenerate or when a moderate degree of tensor swelling is desirable. Third, we introduce regularisation methodology, which is demonstrated to be useful for highlighting features of prior interest and potentially for segmentation. Finally, we compare several metrics in a data set of human brain diffusion-weighted magnetic resonance imaging, and point out similarities between several of the non-Euclidean metrics but important differences with the commonly used Euclidean metric.