We consider the problem of minimizing {formula omitted} 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 ¿>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.

title = "Curve cuspless reconstruction via sub-Riemannian geometry",

abstract = "We consider the problem of minimizing {formula omitted} 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 ¿>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.",

author = "U. Boscain and R. Duits and F. Rossi and Y. Sachkov",

s.n., 2012. 28 p. (arXiv.org; Vol. 1203.3089 [math.OC]).

Research output: Book/Report › Report › Academic

TY - BOOK

T1 - Curve cuspless reconstruction via sub-Riemannian geometry

AU - Boscain, U.

AU - Duits, R.

AU - Rossi, F.

AU - Sachkov, Y.

PY - 2012

Y1 - 2012

N2 - We consider the problem of minimizing {formula omitted} 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 ¿>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.

AB - We consider the problem of minimizing {formula omitted} 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 ¿>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.

M3 - Report

T3 - arXiv.org

BT - Curve cuspless reconstruction via sub-Riemannian geometry