A Riemannian approach for estimating orientation distribution function (ODF) images from high-angular resolution diffusion imaging (HARDI)

High-angular resolution diffusion imaging (HARDI) is a magnetic resonance technique estimating the direction of self-diffusion of water molecules in biological tissue. HARDI encodes at each pixel (voxel) the orientation distribution function (ODF) of water diffusion molecules, i.e. the probability distribution function of finding a water molecule which moved in a certain direction during the observation time. As a consequence ODF images differ from usual gray scale images with respect to their underlying geometry as well as with respect to their error distribution. We present a Bayesian estimator for ODF images considering these differences. To this end, we derive a likelihood function based on the Rician distribution of the NMR signals and propose prior distributions considering ODFs as Riemanian manifolds. Utilizing properties of spherical harmonics and the square root representation of ODFs allows us to effectively reconstruct and regularize ODF images in one step within this Riemannian framework. Experiments demonstrate the merits of our approach on synthetic as well as on real data.