Curve cuspless reconstruction via sub-Riemannian geometry

Abstract

We consider the problem of minimizing $+Ks $ 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. © EDP Sciences, SMAI, 2014.