Riemannian fast-marching on cartesian grids using Voronoi's first reduction of quadratic forms

Jean-Marie Mirebeau (1)

1 CNRS - Centre National de la Recherche Scientifique

We address the numerical computation of distance maps with respect to Riemannian metrics of strong anisotropy. For that purpose we solve generalized eikonal equations, discretized using adaptive upwind finite differences on a Cartesian grid, in a single pass over the domain using a variant of the fast marching algorithm. The key ingredient of our PDE numerical scheme is Voronoi's first reduction, a tool from discrete geometry which characterizes the interaction of a quadratic form with an additive lattice. This technique, never used in this context, which is simple and cheap to implement, allows us to efficiently handle Riemannian metrics of eigenvalue ratio $10^2$ and more. Two variants of the introduced scheme are also presented, adapted to sub-Riemannian and to Rander metrics, which can be regarded as degenerate Riemannian metrics and as Riemannian metrics perturbed with a drift term respectively. We establish the convergence of the proposed scheme and of its variants, with convergence rates. Numerical experiments illustrate the effectiveness of our approach in various contexts, in dimension up to five, including an original sub-Riemannian model related to the penalization of path torsion.

Mots clés: Riemannian metric; Sub-Riemannian metric; Rander metric; Eikonal equation; Viscosity solution; Fast Marching Method; Voronoi Reduction

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Jean-Marie Mirebeau. Riemannian fast-marching on cartesian grids using Voronoi's first reduction of quadratic forms. 2017. ⟨hal-01507334v4⟩