Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction

TL;DR

This paper presents an improved Fast Marching Algorithm that efficiently solves the generalized eikonal equation on Cartesian grids with extreme anisotropy by using lattice basis reduction, maintaining low computational complexity.

Contribution

The authors introduce a novel modification of the Fast Marching Algorithm utilizing lattice basis reduction to handle high anisotropy with reduced computational cost.

Findings

Algorithm achieves logarithmic complexity relative to anisotropy ratio.

Proven consistency of the modified algorithm.

Numerical experiments demonstrate efficiency and accuracy.

Abstract

We introduce a modification of the Fast Marching Algorithm, which solves the generalized eikonal equation associated to an arbitrary continuous riemannian metric, on a two or three dimensional domain. The algorithm has a logarithmic complexity in the maximum anisotropy ratio of the riemannian metric, which allows to handle extreme anisotropies for a reduced numerical cost. We prove the consistence of the algorithm, and illustrate its efficiency by numerical experiments. The algorithm relies on the computation at each grid point of a special system of coordinates: a reduced basis of the cartesian grid, with respect to the symmetric positive definite matrix encoding the desired anisotropy at this point.