From Adaptive Averaging to Accelerated Nonlinear Diffusion Filtering

Weighted averaging filters and nonlinear partial differential equations (PDEs) are two popular concepts for discontinuity-preserving denoising. In this paper we investigate novel relations between these filter classes: We deduce new PDEs as the scaling limit of the spatial step size of discrete weighted averaging methods. In the one-dimensional setting, a simple weighted averaging of both neighbouring pixels leads to a modified Perona-Malik-type PDE with an additional acceleration factor that provides sharper edges. A similar approach in the two-dimensional setting yields PDEs that lack rotation invariance. This explains a typical shortcoming of many averaging filters in 2-D. We propose a modification leading to a novel, anisotropic PDE that is invariant under rotations. By means of the example of the bilateral filter, we show that involving a larger number of neighbouring pixels improves rotational invariance in a natural way and leads to the same PDE formulation. Numerical examples are presented that illustrate the usefulness of these processes.