Curve cuspless reconstruction via sub-riemannian geometry

Boscain, Ugo; Duits, Remco; Rossi, Francesco; Sachkov, Yuri

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 748-770.

We consider the problem of minimizing ${\int }_0^{\ell }\sqrt{{\xi }^2+K^2\left(s\right)}\mathrm{d}s$ ∫ 0 ℓ ξ 2 + K 2 ( s )   d s 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.

Classification : 94A08, 49J15
Mots clés : curve reconstruction, generalized pontryagin maximum principle

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