# Curve cuspless reconstruction via sub-Riemannian geometry

U. Boscain, R. Duits, F. Rossi, Y. Sachkov

16 Citations (Scopus)

### 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.
Original language English 748-770 ESAIM : Control, Optimisation and Calculus of Variations 20 3 https://doi.org/10.1051/cocv/2013082 Published - 2014

### Keywords

• Mathematics - Optimization and Control
• Mathematics - Differential Geometry