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Neighborhood filters and the decreasing rearrangement

Gonzalo Galiano, Julián Velasco

TL;DR

This work reformulates the Neighborhood Filter (NF) in terms of the decreasing rearrangement to obtain a one-dimensional integral representation that preserves level-set structure and enables rigorous analysis of the nonlinear iterative scheme. It proves that NF iterations maintain monotonicity in the rearranged space and reveals the asymptotic behavior as a shock-filter with border diffusion, explaining staircasing and contrast loss. The authors demonstrate computational advantages by reducing complexity to depend on the number of intensity levels, and they connect NF to histogram-based segmentation, supported by numerical experiments in denoising and MRI brain segmentation. Overall, the rearranged NF provides a fast, analyzable framework with practical segmentation applications, though its denoising quality can be inferior for images with flat histograms.

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 very common transformation: the decreasing rearrangement of a function (the image). 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 filter behaves asymptotically as a shock filter combined with a border diffusive term, responsible for the staircaising effect and the loss of contrast.

Neighborhood filters and the decreasing rearrangement

TL;DR

This work reformulates the Neighborhood Filter (NF) in terms of the decreasing rearrangement to obtain a one-dimensional integral representation that preserves level-set structure and enables rigorous analysis of the nonlinear iterative scheme. It proves that NF iterations maintain monotonicity in the rearranged space and reveals the asymptotic behavior as a shock-filter with border diffusion, explaining staircasing and contrast loss. The authors demonstrate computational advantages by reducing complexity to depend on the number of intensity levels, and they connect NF to histogram-based segmentation, supported by numerical experiments in denoising and MRI brain segmentation. Overall, the rearranged NF provides a fast, analyzable framework with practical segmentation applications, though its denoising quality can be inferior for images with flat histograms.

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 very common transformation: the decreasing rearrangement of a function (the image). 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 filter behaves asymptotically as a shock filter combined with a border diffusive term, responsible for the staircaising effect and the loss of contrast.

Paper Structure

This paper contains 9 sections, 5 theorems, 54 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Neighborhood filters in terms of the decreasing rearrangement
  3. Examples
  4. Properties of the nonlinear varying-kernel iterative scheme
  5. Discretization and numerical examples
  6. Numerical examples for denoising
  7. Numerical examples for segmentation
  8. Summary
  9. Acknowledgments

Key Result

Theorem 1

Let $\Omega\subset\mathbb{R}^d$ be an open and bounded set, $d\geq 1$, and $\mathcal{K}:\mathbb{R}\to\mathbb{R}_+$ be a Borel function. Let $u^{(0)}\in L^\infty(\Omega)$ be, without loss of generality, non-negative and set $v_0 =u^{(0)}_*$. Then, for $m=0$ (resp. $m=n$), the iterative scheme (def.GF and $c_m(t)=\int_0^{| \Omega |} \mathcal{K}_h(v_{m}(t)-v_{m}(s))ds$. In addition, for $n\in \mathbb

Figures (8)

  • Figure 1: Test images Boat, Texture and Squares. Second and third columns show the decreasing rearrangement and the histogram, respectively, of the original images (clean) and their counterparts obtained after the addition of a Gaussian noise with SNR$=10$.
  • Figure 2: Graphic of function $f(x)=x\sin(10x)$ (top panel) and that of its decreasing rearrangement, $f_*$ (bottom panel). Although $f\in C^\infty (-\pi,\pi)$, $f_*$ is not even once continuously differentiable in $(0,2\pi)$.
  • Figure 3: Denoising experiment. Columns: noisy image, Neighborhood, NLM, and Bilateral filters. Rows: noisy image, detail of the image, histograms, decreasing rearrangements, and level curves of image details shown in row 2.
  • Figure 4: Denoising experiment. Columns: noisy image, Neighborhood, NLM, and Bilateral filters. Rows: noisy image, detail of the image, histograms, decreasing rearrangements, and level curves of image details shown in row 2.
  • Figure 5: Denoising experiment. Columns: noisy image, Neighborhood, NLM, and Bilateral filters. Rows: noisy image, detail of the image, histograms, decreasing rearrangements, and level curves of image details shown in row 2.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Corollary 1
  • Corollary 2
  • Theorem 3
  • Remark 2