Efficient Fast Marching with Finsler metrics.

Jean-Marie Mirebeau (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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 methods. 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.

Mots clés: Hamilton Jacobi; Fast Marching; Anisotropic Finsler metric

Domaines: Analyse numérique [math.NA]

Jean-Marie Mirebeau. Efficient Fast Marching with Finsler metrics.. Numerische Mathematik, 2013, pp.1-43. ⟨10.1007/s00211-013-0571-3⟩. ⟨hal-00736431v2⟩