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A Riemannian Framework for Orientation Distribution Function Computing

Jian Cheng 1, 2, * Aurobrata Ghosh 2 Tianzi Jiang 1 Rachid Deriche 2
* Corresponding author
2 ODYSSEE - Computer and biological vision
DI-ENS - Département d'informatique de l'École normale supérieure, CRISAM - Inria Sophia Antipolis - Méditerranée , ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
Abstract : Compared with Diffusion Tensor Imaging (DTI), High Angular Resolution Imaging (HARDI) can better explore the complex microstructure of white matter. Orientation Distribution Function (ODF) is used to describe the probability of the fiber direction. Fisher information metric has been constructed for probability density family in Information Geometry theory and it has been successfully applied for tensor computing in DTI. In this paper, we present a state of the art Riemannian framework for ODF computing based on Information Geometry and sparse representation of orthonormal bases. In this Riemannian framework, the exponential map, logarithmic map and geodesic have closed forms. And the weighted Frechet mean exists 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 H_{1/2} 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. As an application, Lagrange interpolation on ODF field is proposed based on weighted Frechet mean. We validate our methods on synthetic and real data experiments. Compared with existing Riemannian frameworks on ODF, our framework is model-free. The estimation of the parameters, i.e. Riemannian coordinates, is robust and linear. Moreover it should be noted that our theoretical results can be used for any probability density function (PDF) under an orthonormal basis representation.
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https://hal.inria.fr/inria-00424764
Contributor : Jian Cheng <>
Submitted on : Sunday, October 18, 2009 - 12:18:58 AM
Last modification on : Thursday, July 1, 2021 - 5:58:02 PM
Long-term archiving on: : Tuesday, June 15, 2010 - 9:26:51 PM

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Jian Cheng, Aurobrata Ghosh, Tianzi Jiang, Rachid Deriche. A Riemannian Framework for Orientation Distribution Function Computing. Medical Image Computing and Computer-Assisted Intervention – MICCAI 2009, Sep 2009, London, United Kingdom. ⟨10.1007/978-3-642-04268-3_112⟩. ⟨inria-00424764⟩

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