Recursivity and PDEs in image processing

Recursive filtering structures reduce drastically the computational effort required for different tasks in image processing. These operations are done with a fixed number of operations per output point independently of the size of the neighbourhood considered. In this paper we show that implicit numerical implementations of some partial differential equations (PDEs) provide algorithms that can be interpreted in terms of recursive filters. We show, in particular, that the classical second order recursive filter introduced by Deriche (1987, 1990) is in fact a particular implementation of the heat equation. Using the well-known Neumann boundary condition for the heat equation, we propose some new implementation of the filter. We extend this linear filter to a nonlinear recursive smoothing filter, following the general idea of anisotropic diffusion. We present some comparison results with the classical Perona-Malik model