We propose a Riemannian framework for analyzing orientation distribution
functions (ODFs), or corresponding probability density functions
(PDFs), in HARDI for use in comparing, interpolating, averaging, and denoising.
Recent approaches based on the Fisher-Rao Riemannian metric
result in geodesic paths that have limited biological interpretations. As
an alternative, we develop a framework where we separate the shape and
orientation features of PDFs, compute geodesics under their respective
Riemannian metrics and then combine them to form a pseudo-geodesic
on the product space. These pseudo-geodesic paths have better biological
interpretation (in terms of interpolating points between given PDFs
by preserving shape diffusivity and anisotropy) and provide tools for pairwise
comparing and averaging a collection of PDFs. The latter tools, in
turn, are useful for interpolation, denoising, and improved tractography
HARDI. We demonstrate these ideas using both synthetic and real HARDI
data.