# Curve cuspless reconstruction via sub-riemannian geometry

Ugo Boscain; Remco Duits; Francesco Rossi; Yuri Sachkov

ESAIM: Control, Optimisation and Calculus of Variations (2014)

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

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topBoscain, 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 > 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 > 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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