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A polynomial based approach to extract the maxima of an antipodally symmetric spherical function and its application to extract fiber directions from the Orientation Distribution Function in Diffusion MRI

Aurorata Ghosh 1, * Elias P. Tsigaridas 2 Maxime Descoteaux 1 Pierre Comon 3 Bernard Mourrain 2 Rachid Deriche 1
* Corresponding author
1 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
2 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis (... - 2019), CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : In this paper we extract the geometric characteristics from an antipodally symmetric spherical function (ASSF), which can be de- scribed equivalently in the spherical harmonic (SH) basis, in the symmet- ric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. All three bases span the same vector space and are bijective when the rank of the SH series equals the order of the ST and equals the degree of the HP. We show, therefore, how it is possible to extract the maxima and minima of an ASSF by computing the stationary points of a constrained HP. In Diffusion MRI, the Orientation Distribution Function (ODF), repre- sents a state of the art reconstruction method whose maxima are aligned with the dominant fiber bundles. It is, therefore, important to be able to correctly estimate these maxima to detect the fiber directions. The ODF is an ASSF. To illustrate the potential of our method, we take up the example of the ODF, and extract its maxima to detect the fiber directions. Thanks to our method we are able to extract the maxima without limiting our search to a discrete set of values on the sphere, but by searching the maxima of a continuous function. Our method is also general, not dependent on the ODF, and the framework we present can be applied to any ASSF described in one of the three bases.
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  • HAL Id : hal-00340600, version 1

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Aurorata Ghosh, Elias P. Tsigaridas, Maxime Descoteaux, Pierre Comon, Bernard Mourrain, et al.. A polynomial based approach to extract the maxima of an antipodally symmetric spherical function and its application to extract fiber directions from the Orientation Distribution Function in Diffusion MRI. 11th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI), Sep 2008, New York, United States. p. 237-248. ⟨hal-00340600⟩

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