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Curve cuspless reconstruction via sub-Riemannian geometry∗∗∗

Published online by Cambridge University Press:  27 May 2014

Ugo Boscain,
Remco Duits,
Francesco Rossi and
Yuri Sachkov
Ugo Boscain
Affiliation:
Centre National de Recherche Scientifique (CNRS), CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France, and Team GECO, INRIA-Centre de Recherche Saclay. ugo.boscain@polytecnique.edu
Remco Duits
Affiliation:
Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands. Department of Mathematics and Computer Science; R.Duits@tue.nl
Francesco Rossi
Affiliation:
Aix-Marseille Univ, LSIS, 13013, Marseille, France; francesco.rossi@lsis.org
Yuri Sachkov
Affiliation:
Program Systems Institute Pereslavl-Zalessky, Russia

Abstract

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We consider the problem of minimizing \hbox{$\int_{0}^\ell \sqrt{\xi^2 +K^2(s)}\, {\rm d}s $} for a planar curve having fixedinitial and final positions and directions. The total length ℓ is free. Heres is thearclength parameter, K(s) is the curvature of the curveand ξ > 0 is a fixed constant. This problem comes from a model of geometry ofvision due to Petitot, Citti and Sarti. We study existence of local and global minimizersfor this problem. We prove that if for a certain choice of boundary conditions there is noglobal minimizer, then there is neither a local minimizer nor a geodesic. We finally giveproperties of the set of boundary conditions for which there exists a solution to theproblem.

Keywords

Curve reconstructiongeneralized pontryagin maximum principle

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 3 , July 2014, pp. 748 - 770
Copyright
© EDP Sciences, SMAI, 2014

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