Contributors: Computer and biological vision (ODYSSEE); Département d'informatique - ENS Paris (DI-ENS); École normale supérieure - Paris (ENS-PSL); Université Paris Sciences et Lettres (PSL)-Université Paris Sciences et Lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS-PSL); Université Paris Sciences et Lettres (PSL)-Université Paris Sciences et Lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Inria Sophia Antipolis - Méditerranée (CRISAM); Institut National de Recherche en Informatique et en Automatique (Inria)-Inria Paris-Rocquencourt; Institut National de Recherche en Informatique et en Automatique (Inria)-École des Ponts ParisTech (ENPC); Geometry, algebra, algorithms (GALAAD); Inria Sophia Antipolis - Méditerranée (CRISAM); Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Nice Sophia Antipolis (1965 - 2019) (UNS)-Centre National de la Recherche Scientifique (CNRS); Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe SIGNAL; Signal, Images et Systèmes (Laboratoire I3S - SIS); Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S); Université Nice Sophia Antipolis (1965 - 2019) (UNS)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UniCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UniCA)-Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S); Université Nice Sophia Antipolis (1965 - 2019) (UNS)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UniCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UniCA); Alexander; Gee; Whitaker; ANR-06-BLAN-0074,DECOTES,Decompositions Tensorielles et applications(2006)
نبذة مختصرة : International audience ; 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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