Sparse Non-negative Stencils for Anisotropic Diffusion

Authors:
Jérôme Fehrenbach;Jean-Marie Mirebeau
Affiliations:
Institut de Mathématiques de Toulouse, Université Paul Sabatier, Toulouse Cedex 9, France 31062;CNRS, Laboratory CEREMADE, UMR 7534, University Paris Dauphine, Paris Cedex 16, France 75775
Venue:
Journal of Mathematical Imaging and Vision
Year:
2014

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Abstract

We introduce a new discretization scheme for Anisotropic Diffusion, AD-LBR, on two and three dimensional Cartesian grids. The main features of this scheme is that it is non-negative and has sparse stencils, of cardinality bounded by 6 in 2D, by 12 in 3D, despite allowing diffusion tensors of arbitrary anisotropy. The radius of these stencils is not a-priori bounded however, and can be quite large for pronounced anisotropies. Our scheme also has good spectral properties, which permits larger time steps and avoids e.g. chessboard artifacts.AD-LBR relies on Lattice Basis Reduction, a tool from discrete mathematics which has recently shown its relevance for the discretization on grids of strongly anisotropic Partial Differential Equations (Mirebeau in Preprint, 2012). We prove that AD-LBR is in 2D asymptotically equivalent to a finite element discretization on an anisotropic Delaunay triangulation, a procedure more involved and computationally expensive. Our scheme thus benefits from the theoretical guarantees of this procedure, for a fraction of its cost. Numerical experiments in 2D and 3D illustrate our results.