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arXiv · 1203.3089

Curve cuspless reconstruction via sub-Riemannian geometry

Ugo Boscain·Remco Duits·Francesco Rossi·Yuri Sachkov
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Abstract

We consider the problem of minimizing $\int_{0}^L \sqrt{\xi^2 +K^2(s)}\, ds $ for a planar curve having fixed initial and final positions and directions. The total length $L$ is free. Here $s$ is the variable of arclength parametrization, $K(s)$ is the curvature of the curve and $\xi>0$ a parameter. 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.

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Ugo Boscain, Remco Duits, Francesco Rossi, Yuri Sachkov. 2012-03-14. Curve cuspless reconstruction via sub-Riemannian geometry. https://arxiv.org/abs/1203.3089

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