Processing and segmentation of high angular resolution diffusion images described by orientation distribution functions

Quantitative characterization of the brain circuitry is an important problem in neuroradiology because a damage to this circuitry is often indicative of a neurological disease. Diffusion magnetic resonance imaging (DMRI) is presently the only available non-invasive technique to investigate the neural architecture of the brain in vivo. DMRI produces images of biological tissues by measuring the constrained diffusion properties of water molecules. More precisely, the fiber directions can be correlated with the directions of maximum diffusion. This unique property has generated enthusiasm for developing techniques such as high angular resolution diffusion imaging (HARDI). HARDI enables the reconstruction of the orientation distribution function (ODF), which describes diffusion using non-parametric statistics. This offers improved accuracy in resolving intra-voxel complexities over the tensor model, currently the de facto standard for neuroimaging.;Many challenging issues need to be addressed for HARDI to be studied in the correct mathematical setting and to be beneficial in clinical research. For instance, one needs to accurately estimate the ODF from diffusion weighted images and then focus on the problem of processing ODF images. Once these fundamental problems are addressed, higher-level problems such as segmentation, i.e., separating an ODF image into regions with distinct diffusion properties, can be studied.;In this thesis, we present a Riemannian framework to perform essential processing operations such as averaging, interpolation, and filtering of ODF images. By treating the information on the shape and orientation of an ODF as entities on separate manifolds, we show how to perform these operations on the joint space of these manifolds. The proposed framework resolves issues with prior Riemannian frameworks by eliminating physically and anatomically meaningless configurations. We also present a method to segment ODF images into multiple regions. Our approach integrates tools from sparse representation theory and Riemannian geometry into a graph theoretic segmentation framework. This method can incorporate weak supervision in the form of user interaction to differentiate between anatomically distinct regions with similar ODFs and group different ODFs in the same fiber tract. The proposed method successfully segments important white matter tracts.