Curve cuspless reconstruction via sub-Riemannian geometry
نویسنده
چکیده
We consider the problem of minimizing ∫ ` 0 √ ξ2 +K2(s) ds for a planar curve having fixed initial and final positions and directions. The total length ` 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.
منابع مشابه
Cuspless Sub-Riemannian Geodesics within the Euclidean Motion Group SE(d)
We consider the problem Pcurve of minimizing ∫ ` 0 √ β 2 + |κ(s)|2ds for a planar curve having fixed initial and final positions and directions. Here κ is the curvature of the curve with free total length `. This problem comes from a 2D model of geometry of vision due to Petitot, Citti and Sarti. Here we will provide a general theory on cuspless sub-Riemannian geodesics within a sub-Riemannian ...
متن کاملIdentification of Riemannian foliations on the tangent bundle via SODE structure
The geometry of a system of second order differential equations is the geometry of a semispray, which is a globally defined vector field on TM. The metrizability of a given semispray is of special importance. In this paper, the metric associated with the semispray S is applied in order to study some types of foliations on the tangent bundle which are compatible with SODE structure. Indeed, suff...
متن کاملOn Some Sub-riemannian Objects in Hypersurfaces of Sub-riemannian Manifolds
We study some sub-Riemannian objects (such as horizontal connectivity, horizontal connection, horizontal tangent plane, horizontal mean curvature) in hypersurfaces of sub-Riemannian manifolds. We prove that if a connected hypersurface in a contact manifold of dimension more than three is noncharacteristic or with isolated characteristic points, then given two points, there exists at least one p...
متن کاملIntrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling
We relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the ma...
متن کاملSub-Lorentzian Geometry on Anti-de Sitter Space
Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. Sub-Riemannian Geometry enjoys major differences from the Riemannian being a generalization of the latter at the same time, e.g., geodesics are not unique and may be singular, the Hausdorff dimension is larger than the manifold topological dimension. There exists a la...
متن کامل