The diffusion tensor MRI (DTMRI) data has been used for studying brain structures, their connectivities, and functionalities consisting of fields of diffusion tensor matrices in the imaged volume. The problems involving denoising, interpolation, tractography, and related analyses for such data require a framework for comparing, averaging, and manipulating these tensor structures. In recent years, the increased strengths of MRI scanners have led to the possibility of measuring diffusion orientations beyond the three canonical directions, and have led to HARDI (High Angular Resolution Diffusion Imaging) technology that measures fluid flow at each voxel in numerous directions [1]. The basic unit of HARDI data at each anatomical location is an orientation diffusion function (ODF) that captures the diffusivity as a function of the direction. Using smooth interpolations (e.g. using spherical harmonics) one can obtain an ODF on the full sphere. From those limited observations, an ODF can be viewed as a nonnegative function on a sphere. The space of ODFs is, thus, the set of all such functions on a sphere.
Many researchers tend to focus on the shape and orientations of the ODFs by removing the scale variability . They do so by forming a corresponding space of probability density functions (PDFs) by normalizing the space of ODFs. The space of all PDFs forms a Banach manifold, and it can become a Hilbert manifold by choosing a Riemannian structure. The FisherRao Metric [2] has become increasingly popular in recent years. In HARDI data analysis, this metric was first proposed in Chang et al. [3], and later described in several other papers, including [4, 5]. In this paper, we highlight a major disadvantage of using the FisherRao metric directly. The main problem is that in a direct formulation, the shape and the orientation features are intermingled and, as a result, the geodesic paths between PDFs go through PDFs that are not amenable to biological interpretations.
In our work, we propose a novel Riemannian framework for analyzing PDFs in HARDI data sets for use in comparing, interpolating, averaging, and denoising PDFs. This is accomplished by separating the 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 FisherRao distance on the aligned PDFs, and (2) orientation by comparing their rotation matrices. By separating the shapes and the orientations of the PDFs, This allows for a formal Riemannian framework for analyzing PDFs, while generating geodesic paths that are biologically feasible. This framework leads to definitions and efficient computations for the Karcher mean and covariance that provide tools for improved interpolation and denoising.

