Curve cuspless reconstruction via sub-Riemannian geometry∗∗∗. (25th June 2014) - Boscain, Ugo; Duits, Remco; Rossi, Francesco; Sachkov, Yuri
<x content-type="archive" xml:space="preserve">Abstract</x> We consider the problem of minimizing <alternatives> <![CDATA[\hbox{$\int_{0}^\ell \sqrt{\xi^2 +K^2(s)}\, {\rm d}s $}]]> <inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2jt01x6r" xmlns:xlink="http://www.w3.org/1999/xlink" />∫0ℓξ2+K2(s) ds</alternatives> 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.