Curve cuspless reconstruction via sub-riemannian geometry
ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, p. 748-770

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 length 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.

DOI : https://doi.org/10.1051/cocv/2013082
Classification:  94A08,  49J15
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 via sub-riemannian geometry},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {20},
number = {3},
year = {2014},
pages = {748-770},
doi = {10.1051/cocv/2013082},
language = {en},
url = {http://www.numdam.org/item/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, Volume 20 (2014) no. 3, pp. 748-770. doi : 10.1051/cocv/2013082. http://www.numdam.org/item/COCV_2014__20_3_748_0/

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