We consider the problem of minimizing ∫ 0 ℓ ξ 2 + K 2 ( s ) d s for a planar curve having fixed initial and final positions and directions. The total length ℓ 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.
Keywords: curve reconstruction, generalized pontryagin maximum principle
@article{COCV_2014__20_3_748_0,
author = {Boscain, Ugo and Duits, Remco and Rossi, Francesco and Sachkov, Yuri},
title = {Curve cuspless reconstruction \protect\emph{via }sub-riemannian geometry},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {748--770},
year = {2014},
publisher = {EDP Sciences},
volume = {20},
number = {3},
doi = {10.1051/cocv/2013082},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2013082/}
}
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 SP - 748 EP - 770 VL - 20 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2013082/ DO - 10.1051/cocv/2013082 LA - en ID - COCV_2014__20_3_748_0 ER -
%0 Journal Article %A Boscain, Ugo %A Duits, Remco %A Rossi, Francesco %A Sachkov, Yuri %T Curve cuspless reconstruction via sub-riemannian geometry %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 748-770 %V 20 %N 3 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2013082/ %R 10.1051/cocv/2013082 %G en %F COCV_2014__20_3_748_0
Boscain, Ugo; Duits, Remco; Rossi, Francesco; Sachkov, Yuri. Curve cuspless reconstruction via sub-riemannian geometry. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 748-770. doi: 10.1051/cocv/2013082
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