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Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction

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.