Neighborhood Filters and the Decreasing Rearrangement

Abstract

Nonlocal filters are simple and powerful techniques for image denoising. In this paper, we give new insights into the analysis of one kind of them, the Neighborhood filter, by using a classical although not commonly used transformation: the decreasing rearrangement of a function. Independently of the dimension of the image, we reformulate the Neighborhood filter and its iterative variants as an integral operator defined in a one-dimensional space. The simplicity of this formulation allows to perform a detailed analysis of its properties. Among others, we prove that the filtered image is a contrast change of the original image, an 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. ), among which the pioneering approaches of Perona and Malik [21],Álvarez, Lions and Morel [2] and Rudin, Osher and Fatemi [25] are fundamental. We refer the reader to [9] for a review and comparison of these methods.Among all these filters, the Neighborhood filter is the simplest, but yet useful, method due to its compromise between denoising quality and computational speed. Indeed, although it creates shocks and staircasing effects [8], the computational cost is by far lower than those of other integral kernel filters or PDE's based methods.Since, usually, a single denoising step of the nonlocal filters is not enough, an iteration is performed according to several choices of the iteration actualization, see (3) and (4) for two of such strategies. In this context, the Neighborhood filter and its variants have been analyzed from different points of view. For instance, Barash [3], Elad [13], Barash et al. [4], and Buades et al. [7] investigate the asymptotic relationship between the Yaroslavsky filter and the Perona-Malik PDE. Gilboa et al. [16] study certain applications of nonlocal operators to image processing. In [22], Peyré establishes a relationship between the non-iterative nonlocal filtering schemes and thresholding in adapted orthogonal basis. In a more recent paper, Singer et al. [26] interpret the Neighborhood filter as a stochastic diffusion process, explaining in this way the attenuation of high frequencies in the processed images.In this article, we reformulate the Neighborhood filter in terms of the decreasing rearrangement of the initial image, u, which is defined as the inverse of the distribution function q ∈ R → m u (q) = |{x ∈ Ω : u(x) > q}|, see Section 2 for the precise definition.Realizing that the structure of level sets of u is invariant through the Neighborhood filter operation as well as through the decreasing rearrangement of u allows us to rewrite (1) in terms of a one-dimensional integral expression, see Theorem 1.Although from expression (1) is readily seen that only computation on level lines is needed to perform the filtering, the alternative expression in terms of the decreasing rearrangement offers room for further analysis of the iterative scheme.Perhaps, the most important consequ...

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