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In High Angular Resolution Diffusion Imaging (HARDI) processing, Orientation Distribution Function (ODF) in every voxel is the probability density function that is used to describe the probability of the fiber direction. How to find an appropriate way to process this kind of manifold-valued data is one of the most important issues in HARDI field.

Here we proposed a natural, intrinsic Riemannian framework for ODF computing. It is based on the information geometry theory and sparse representation of orthonormal basis. In this Riemannian framework, the exponential map, logarithmic map and geodesic have closed forms. And the weighted Frechet mean and median exist uniquely on this manifold. We also propose a novel scalar measurement, named Geometric Anisotropy (GA), which is the Riemannian geodesic distance between the ODF and the isotropic ODF. The Renyi entropy of the ODF can be computed from the GA. Moreover, we present an Affine-Euclidean framework and a Log-Euclidean framework so that we can work in an Euclidean space.

Lagrange interpolation on ODF field is proposed based on weighted Frechet mean. Compared with existing Riemannian frameworks on ODF, our framework is model-free. The estimation of the parameters, i.e. Riemannian coordinates, is robust and fast. Moreover it should be noted that our theoretical results can be used for any probability density function (PDF) under an orthonormal basis representation.

Publications on this topic include:

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