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10 March 2006 Apparent diffusion profile estimation from high angular resolution diffusion images
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High angular resolution diffusion imaging (HARDI) has recently been of great interest to characterize non-Gaussian diffusion process. In the white matter of the brain, this occurs when fiber bundles cross, kiss or diverge within the same voxel. One of the important goal is to better describe the apparent diffusion process in these multiple fiber regions, thus overcoming the limitations of classical diffusion tensor imaging (DTI). In this paper, we design the appropriate mathematical tools to describe noisy HARDI data. Using a meaningful modified spherical harmonics basis to capture the physical constraints of the problem, we propose a new regularization algorithm to estimate a smoother and closer diffusivity profile to the true diffusivities without noise. We exploit properties of the spherical harmonics to define a smoothing term based on the Laplace-Beltrami for functions defined on the unit sphere. An additional contribution of the paper is the derivation of the general transformation taking the spherical harmonics coefficients to the high order tensor independent elements. This allows the careful study of the state of the art high order anisotropy measures computed from either spherical harmonics or tensor coefficients. We analyze their ability to characterize the underlying diffusion process. We are able to recover voxels with isotropic, single fiber anisotropic and multiple fiber anisotropic diffusion. We test and validate the approach on diffusion profiles from synthetic data and from a biological rat phantom.
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Maxime Descoteaux, Elaine Angelino, Shaun Fitzgibbons, and Rachid Deriche "Apparent diffusion profile estimation from high angular resolution diffusion images", Proc. SPIE 6144, Medical Imaging 2006: Image Processing, 61441D (10 March 2006);

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