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Geodesics on Shape Spaces with Bounded Variation and Sobolev Metrics

dc.contributor.authorNardi, Giacomo
dc.contributor.authorPeyré, Gabriel
HAL ID: 1211
dc.contributor.authorVialard, François-Xavier
dc.date.accessioned2014-04-03T11:24:40Z
dc.date.available2014-04-03T11:24:40Z
dc.date.issued2016
dc.identifier.issn1936-4954
dc.identifier.urihttps://basepub.dauphine.fr/handle/123456789/13050
dc.language.isoenen
dc.subjectBV 2-curves
dc.subjectMartingale
dc.subjectGeodesics
dc.subjectshape registration
dc.subject.ddc519en
dc.titleGeodesics on Shape Spaces with Bounded Variation and Sobolev Metrics
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenThis paper studies the space of $BV^2$ planar curves endowed with the $BV^2$ Finsler metric over its tangent space of displacement vector fields. Such a space is of interest for applications in image processing and computer vision because it enables piecewise regular curves that undergo piecewise regular deformations, such as articulations. The main contribution of this paper is the proof of the existence of a shortest path between any two $BV^2$ curves for this Finsler metric. % The method of proof relies on the construction of a martingale on a space satisfying the Radon-Nikodym property and on the invariance under reparametrization of the Finsler metric. This method applies more generally to similar cases such as the space of curves with $H^k$ metrics for $k\geq 2$ integer. When $k \geq 2$ is integer, this space has a strong Riemannian structure and is geodesically complete. Thus, our result shows that the exponential map is surjective, which is complementary to geodesic completeness in infinite dimensions. We propose a finite element discretization of the minimal geodesic problem, and use a gradient descent method to compute a stationary point of a regularized energy. Numerical illustrations shows the qualitative difference between $BV^2$ and $H^2$ geodesics.
dc.publisher.cityParisen
dc.relation.isversionofjnlnameSIAM Journal on Imaging Sciences
dc.relation.isversionofjnlvol9
dc.relation.isversionofjnlissue1
dc.relation.isversionofjnldate2016
dc.relation.isversionofjnlpages238-274
dc.relation.isversionofdoi10.1137/15100518X
dc.identifier.urlsitehttps://arxiv.org/abs/1402.6504v8
dc.relation.isversionofjnlpublisherSociety for Industrial and Applied Mathematics
dc.subject.ddclabelProbabilités et mathématiques appliquéesen
dc.description.submittednonen
dc.description.ssrncandidatenon
dc.description.halcandidateoui
dc.description.readershiprecherche
dc.description.audienceInternational
dc.relation.Isversionofjnlpeerreviewedoui
dc.date.updated2016-10-10T08:01:27Z


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