On the Accuracy of Anisotropic Fast Marching

The fast marching algorithm, and its variants, solves numerically the generalized eikonal equation associated to an underlying riemannian metric M. A major challenge for these algorithms is the non-isotropy of the riemannian metric, which magnitude is characterized by the anisotropy ratio κ(M) ∈ [1,∞]. Applications of the eikonal equation to image processing [1, 3] often involve large anisotropy ratios, which motivated the design of new algorithms. A variant of the fast marching algorithm, introduced in [6], addresses the problem of large anisotropies using an algebraic tool named lattice basis reduction. The numerical complexity of this algorithm is insensitive to anisotropy, under extremely weak assumptions. We establish in this paper, in the simplified setting of a constant riemannian metric, that the accuracy of this algorithm is also extremely robust to anisotropy : in an average sense, it does not degrade as κ(M) increases. We also extend this algorithm to higher dimension.