A geometric analysis of ODFs as oriented surfaces for interpolation, averaging and denoising in HARDI data

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 pseudo-geodesics 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 comparison and averaging of a collection of PDFs. The latter tools, in turn, are useful for interpolation, denoising, and improved tractography in HARDI data. We demonstrate these ideas using both synthetic and real HARDI data.