Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction
Résumé
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 box 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 establish that the output of the algorithm converges towards the viscosity solution of continuous problem, as the discretization step tends to zero. The algorithm is based on the computation at each grid point of a reduced basis of the unit lattice, with respect to the symmetric positive definite matrix encoding the desired anisotropy at this point.
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