Anisotropic fast-marching on cartesian grids using Voronoi ’ s first reduction of quadratic forms

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, 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. 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. The proposed numerical scheme shows good behavior with Riemannian metrics of condition numbers of 10 and more, can be enhanced by the use of second order finite differences, and is easier to implement, generalize, and up to 5 times faster than the related semi-Lagrangian discretization [18].