Compact finite-difference approximations for anisotropic image smoothing and painting

Finite-difference approximations are succinctly represented by their stencils, a set of weights that when applied to adjacent samples of a function approximate some differential operator. In image processing the samples are pixels or voxels, and the differential operators must be inverted for smoothing or painting applications. For efficiency in such applications requiring inverses, the finitedifference stencils should be compact, using only a small 3× 3 set of nine pixels or a 3× 3× 3 set of twenty-seven voxels. From 2 × 2 and 2 × 2 × 2 approximations to gradient operators I obtain 3 × 3 (9-point) and 3 × 3 × 3 (27-point) stencils that approximate Laplacian operators. The latter may include tensor coefficients that make them anisotropic. By deriving finite-difference stencils for anisotropic Laplacians in this way, that is, from approximations to gradients, we guarantee that our approximations to anisotropic Laplacians are symmetric and positive-semidefinite. And by choosing the gradient approximations carefully, discretization errors can be made isotropic to leading order.