On a new formulation of nonlocal image filters involving the relative rearrangement
This work provides a theoretical analysis for image denoising methods, but it is incremental as it builds on existing nonlocal filter frameworks.
The authors tackled the problem of analyzing nonlocal image filters by reformulating them using functional rearrangements, which allowed them to prove equivalency and discretization convergence, and revealed that the filtered image is a contrast change with asymptotic shock filter behavior and border diffusion.
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.