The Florida State University College of Arts and Sciences a Novel Riemannian Metric for Analyzing Spherical Functions with Applications to Hardi Data

We propose a novel Riemannian framework for analyzing orientation distribution functions (ODFs), or their probability density functions (PDFs), in HARDI data sets for use in comparing, interpolating, averaging, and denoising PDFs. This is accomplished by separating shape and orientation features of PDFs, and then analyzing them separately under their own Riemannian metrics. We formulate the action of the rotation group on the space of PDFs, and define the shape space as the quotient space of PDFs modulo the rotations. In other words, any two PDFs are compared in: (1) shape by rotationally aligning one PDF to another, using the Fisher-Rao distance on the aligned PDFs, and (2) orientation by comparing their rotation matrices. This idea improves upon the results from using the Fisher-Rao metric in analyzing PDFs directly, a technique that is being used increasingly, and leads to interpolation paths that are biologically feasible. This framework leads to definitions and efficient computations for the Karcher mean that provide tools for improved interpolation and denoising. We demonstrate these ideas, using an experimental setup involving several PDFs.