On a new formulation of nonlocal image filters involving the relative rearrangement

Nonlocal filters are simple and powerful techniques for image denoising. In this paper we study the reformulation of a broad class of nonlocal filters in terms of two functional rearrangements: the decreasing and the relative rearrangements. Independently of the dimension of the image, we reformulate these filters as integral operators defined in a one-dimensional space corresponding to the level sets measures. We prove the equivalency between the original and the rearranged versions of the filters and propose a discretization in terms of constant-wise interpolators, which we prove to be convergent to the solution of the continuous setting. For some particular cases, this new formulation allows us to perform a detailed analysis of the filtering properties. Among others, we prove that the filtered image is a contrast change of the original image, and that the filtering procedure behaves asymptotically as a shock filter combined with a border diffusive term, responsible for the staircaising effect and the loss of contrast.