Curve cuspless reconstruction via sub-Riemannian geometry

2012/03/14 by Ugo Boscain, Remco Duits, Boscain, Ugo +6 · 3 citations
Computer Science · Engineering · Mathematics · #3D Shape Modeling and Analysis #Computer Graphics and Visualization Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Geology #Geometry #Mathematics #Optimization and Control (math.OC) #Riemannian geometry #Topological and Geometric Data Analysis #math.DG #math.OC

paper · pdf · doi:10.48550/arxiv.1203.3089

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2012/03/14 · arxiv created 2013/04/25 · arxiv updated 2013/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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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• A variational method for curve extraction with curvature-dependent energies

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