Curve cuspless reconstruction via sub-Riemannian geometry

2012/03/14 by Boscain, Ugo, Duits, Remco, Rossi, Francesco +1 · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1203.3089

Abstract

We consider the problem of minimizing ∫₀^L √(ξ² +K²(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 ξ>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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