Efficient Fast Marching with Finsler metrics

TL;DR

This paper introduces FM-ASR, a new fast marching algorithm for solving anisotropic Eikonal equations with Finsler metrics, offering improved efficiency and accuracy over existing methods, especially in highly anisotropic cases.

Contribution

The paper presents a novel Fast Marching algorithm with anisotropic stencil refinement that reduces complexity dependence on anisotropy ratio, enhancing performance in anisotropic path problems.

Findings

Complexity depends logarithmically on anisotropy ratio.

Numerical experiments show significant accuracy and efficiency improvements.

Method outperforms most existing approaches in highly anisotropic scenarios.

Abstract

We study the discretization of the Escape Time problem: find the length of the shortest path joining an arbitrary point of a domain, to the domain's boundary. 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, or Anisotropic Eikonal, Partial Differential Equation, as well 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. 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.