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

TL;DR

This paper introduces a novel formulation of nonlocal image filters using functional rearrangements, enabling simplified analysis and discretization, and reveals their behavior as contrast-changing shock filters with staircasing effects.

Contribution

It reformulates nonlocal filters via decreasing and relative rearrangements, providing a unified integral operator framework and detailed analysis of filtering properties.

Findings

Filtered images are contrast changes of original images.

Filtering acts asymptotically as a shock filter with border diffusion.

Discretization converges to the continuous filter solution.

Abstract

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.