There are several neurological diseases, including Alzheimer's disease and schizophrenia, where the connections between different brain structures are damaged. A detailed knowledge of the human brain circuitry is vital to understand and treat these diseases. High Angular Resolution Diffusion Imaging (HARDI), a non-invasive imaging technique, allows us to trace brain connectivity pathways. HARDI produces in-vivo images of biological tissues by exploiting the constrained diffusion properties of water molecules. More precisely, the fiber directions can be correlated with the directions of maximum diffusion. Thus, HARDI can be used to infer the organization of several structures in biological tissues.

Many challenging questions need to be addressed for HARDI to be beneficial in diagnosis and clinical applications. First of all, given a set of diffusion weighted images, how do we efficiently compute the probability density function (PDF) of the water diffusion? Second, given the diffusion PDFs, how do we perform various key preprocessing operations such as averaging, interpolation, filtering and convolution, and statistical analysis operations such as principal components analysis? While these operations are well understood when the data are Euclidean, they are less well understood when the data are no longer Euclidean. For example, how do we average a given set of diffusion PDFs to reduce the effects of noise in the data? Likewise, given the PDFs of two populations, how do we perform multi-variate hypothesis testing?

We first present an estimation framework that constructs the diffusion orientation distribution function (ODF), i.e., the angular profile of the diffusion PDF, directly from HARDI signals. We constrain the estimated ODF to be non-negative and sum up to 1. We incorporate spatial regularization into the estimation framework. We then present a novel mathematical framework capable of dealing with the rich information present in HARDI data This framework is the first of its kind that does not require any prior assumptions about the specific properties of water diffusion. Unlike existing frameworks, our framework has the ability of taking full advantage of the given diffusivity information. Experimental results show that this framework offers valuable clues about the structural differences of the human brain in population studies.