Stable discretization of scalar and constrained vectorial Perona–Malik equation

Abstract

We survey recent results on analysis and numerics of the scalar Perona–Malik equation. A vectorial Perona–Malik equation is introduced to evolve unit vector fields for directional diffusion. For both cases, scalar and vectorial, fully discrete schemes are proposed which fulfill a discrete energy law, and satisfy a discrete sphere constraint in the vectorial case. Computational experiments are provided to illustrate quantitative behaviors, and compare with scalar total variation flow and heat flow of $p$-harmonic maps.