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Diffusion MRI is a non-invasive imaging technique that allows the measurement of water molecule diffusion through tissue in vivo. The directional features of water diffusion allow one to infer the connectivity patterns prevalent in tissue and possibly track changes in this connectivity over time for various clinical applications. In this paper, we present a novel statistical model for diffusion-weighted MR signal attenuation which postulates that the water molecule diffusion can be characterized by a continuous mixture of diffusion tensors. An interesting observation is that this continuous mixture and the MR signal attenuation are related through the Laplace transform of a probability distribution over symmetric positive definite matrices. We then show that when the mixing distribution is a Wishart distribution, the resulting closed form of the Laplace transform leads to a Rigaut-type asymptotic fractal expression, which has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until now. Our model not only includes the traditional diffusion tensor model as a special instance in the limiting case, but also can be adjusted to describe complex tissue structure involving multiple fiber populations. Using this new model in conjunction with a spherical deconvolution approach, we present an efficient scheme for estimating the water molecule displacement probability functions on a voxel-by-voxel basis. Experimental results on both simulations and real data are presented to demonstrate the robustness and accuracy of the proposed algorithms.

Diffusion magnetic resonance imaging (MRI) is a relatively new imaging modality which is capable of measuring the diffusion of water molecules in biological systems noninvasively. The measurements from diffusion MRI provide unique clues for extracting orientation information of brain white matter fibers and can be potentially used to infer the brain connectivity in vivo using tractography techniques. Diffusion tensor imaging (DTI), currently the most widely used technique, fails to extract multiple fiber orientations in regions with complex microstructure. In order to overcome this limitation of DTI, a variety of reconstruction algorithms have been introduced in the recent past. One of the key ingredients in several model-based approaches is deconvolution operation which is presented in a unified deconvolution framework in this paper. Additionally, some important computational issues in solving the deconvolution problem that are not addressed adequately in previous studies are described in detail here. Further, we investigate several deconvolution schemes towards achieving stable, sparse, and accurate solutions. Experimental results on both simulations and real data are presented. The comparisons empirically suggest that nonnegative least squares method is the technique of choice for the multifiber reconstruction problem in the presence of intravoxel orientational heterogeneity.