Efficient fast marching with Finsler metrics

Abstract

We study the discretization of the escape time problem: find the length of the shortest path joining an arbitrary point \(z\) of a domain \(\Omega \), to the boundary \(\partial \Omega \). Path length is measured locally via a Finsler metric, potentially asymmetric and strongly anisotropic. This optimal control problem can be reformulated as a static Hamilton–Jacobi partial differential equation, or as a front propagation model. It has numerous applications, ranging from motion planning to image segmentation. We introduce a new algorithm, fast marching using anisotropic stencil refinement (FM-ASR), which addresses this problem on a two dimensional domain discretized on a cartesian grid. The local stencils used in our discretization are produced by arithmetic means, like in the FM-LBR (Mirebeau in Anisotropic fast Marching on Cartesian grids, using lattice basis reduction, preprint 2012), a method previously introduced by the author in the special case of Riemannian metrics. The complexity of the FM-ASR, in an average sense over all grid orientations, only depends (poly-)logarithmically on the anisotropy ratio of the metric, while most alternative approaches have a polynomial dependence. Numerical experiments show, in several occasions, that the accuracy/complexity compromise is improved by an order of magnitude or more.