  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.
