A global CAT approach for graylevel diffusion

This paper proposes an alternative to partial differential equations (PDEs) for the solution of diffusion (Perona and Malik scheme), using the heat transfer problem. Traditionally, the method for solving such physics-based problems is to discretize and solve a PDE by a mathematical process. We propose to use the global heat equation and decompose it into simpler laws. Some of these laws admit an exact global version since they arise from conservation principles while the assumptions on the others can be made wisely, taking into account knowledge about the problem. A computational algebraic topology-based image model allows us to write directly discrete equations. The numerical scheme is derived in a straightforward way from the problem modeled. It thus provides a physical explanation of each solving step in the solution. Finally, we present results for nonlinear diffusion.