Fast Marching methods for Curvature Penalized Shortest Paths
Abstract
We introduce numerical schemes for computing distances and shortest paths with respect to several planar paths models, featuring curvature penalization and data-driven velocity: the Dubins car, the Euler/Mumford elastica, and a two variants of the Reeds-Shepp car. For that purpose, we design monotone and causal discretizations of the associated Hamilton-Jacobi-Bellman PDEs, posed on the three dimensional domain R2 × S1. Our discretizations involves sparse, adaptive and anisotropic stencils on a cartesian grid, built using techniques from lattice geometry. A convergence proof is provided, in the setting of discontinuous viscosity solutions. The discretized problems are solvable in a single pass using a variant of the Fast-Marching algorithm. Numerical experiments illustrate the applications of our schemes in motion planning and image segmentation.
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