Convergence of an Iterative Nonlinear Scheme for Denoising of Piecewise Constant Images

In this paper we present a new efficient iterative nonlinear scheme for recovering of a piecewise constant image from an observed image containing additive noise. We apply an adaptive neighborhood filter which comes from robust statistics and completely rejects outliers being greater than a certain constant. We prove that the iterated application of the scheme leads to a piecewise constant image. This observation generalizes the known results on convergence of nonlinear diffusion schemes to a constant steady-state. Moreover, we show that the partition of the image determining the piecewise constant steady-state after an infinite iteration process can already be found after a finite number of iteration steps. This result can be used for a fast approximation of the piecewise constant image by a mean value procedure. We examine the relations of our scheme to average and bilateral filtering, diffusion filtering and wavelet shrinkage. Numerical experiments illustrate the performance of the algorithm.