# Curve cuspless reconstruction via sub-riemannian geometry

• Volume: 20, Issue: 3, page 748-770
• ISSN: 1292-8119

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## Abstract

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We consider the problem of minimizing ${\int }_{0}^{\ell }\sqrt{{\xi }^{2}+{K}^{2}\left(s\right)}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}s$ ∫ 0 ℓ ξ 2 + K 2 ( s )   d s for a planar curve having fixed initial and final positions and directions. The total lengthℓ is free. Here s is the arclength parameter, K(s) is the curvature of the curve and ξ &gt; 0 is a fixed constant. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. We finally give properties of the set of boundary conditions for which there exists a solution to the problem.

## How to cite

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Boscain, Ugo, et al. "Curve cuspless reconstruction via sub-riemannian geometry." ESAIM: Control, Optimisation and Calculus of Variations 20.3 (2014): 748-770. <http://eudml.org/doc/272891>.

@article{Boscain2014,
abstract = {We consider the problem of minimizing $\int _\{0\}^\ell \sqrt\{\xi ^2 +K^2(s)\}\, \{\rm d\}s$ ∫ 0 ℓ ξ 2 + K 2 ( s )   d s for a planar curve having fixed initial and final positions and directions. The total lengthℓ is free. Here s is the arclength parameter, K(s) is the curvature of the curve and ξ &gt; 0 is a fixed constant. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. We finally give properties of the set of boundary conditions for which there exists a solution to the problem.},
author = {Boscain, Ugo, Duits, Remco, Rossi, Francesco, Sachkov, Yuri},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {curve reconstruction; generalized pontryagin maximum principle},
language = {eng},
number = {3},
pages = {748-770},
publisher = {EDP-Sciences},
title = {Curve cuspless reconstruction via sub-riemannian geometry},
url = {http://eudml.org/doc/272891},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Boscain, Ugo
AU - Duits, Remco
AU - Rossi, Francesco
AU - Sachkov, Yuri
TI - Curve cuspless reconstruction via sub-riemannian geometry
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 3
SP - 748
EP - 770
AB - We consider the problem of minimizing $\int _{0}^\ell \sqrt{\xi ^2 +K^2(s)}\, {\rm d}s$ ∫ 0 ℓ ξ 2 + K 2 ( s )   d s for a planar curve having fixed initial and final positions and directions. The total lengthℓ is free. Here s is the arclength parameter, K(s) is the curvature of the curve and ξ &gt; 0 is a fixed constant. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. We finally give properties of the set of boundary conditions for which there exists a solution to the problem.
LA - eng
KW - curve reconstruction; generalized pontryagin maximum principle
UR - http://eudml.org/doc/272891
ER -

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