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Abstract:
In this paper we extract the geometric characteristics from an antipodally symmetric spherical function (ASSF), which can be described equivalently in the spherical harmonic (SH) basis, in the symmetric 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), represents 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.
BibTeX:
@inproceedings{gtdcmd-miccai-2008,
TITLE = {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},
X-INTERNATIONAL-AUDIENCE ={yes},
AUTHOR = {Ghosh, Aurorata and Tsigaridas, Elias and
Descoteaux, Maxime and Comon, Pierre and Mourrain,
Bernard and Deriche, Rachid},
BOOKTITLE = {11th Int. Conf. on Medical Image Computing and
Computer Assisted Intervention (MICCAI), Workshop on
Computational Diffusion MRI},
PAGES = {237--248 },
ADDRESS = {New York, USA },
EDITOR = {Alexander and Gee and Whitaker },
YEAR = 2008,
URL = {http://hal.archives-ouvertes.fr/hal-00340600/en/},
X-PROCEEDINGS ={yes},
abstract = "In this paper we extract the geometric
characteristics from an antipodally symmetric
spherical function (ASSF), which can be described
equivalently in the spherical harmonic (SH) basis,
in the symmetric 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), represents 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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