Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a fast bilateral filtering formulation using functional rearrangements

Gonzalo Galiano, Julián Velasco

TL;DR

The paper addresses the computational burden of bilateral and related neighborhood filters and presents an exact reformulation using functional rearrangements. It introduces the decreasing rearrangement $u_*$ and the relative rearrangement $v_{*u}$ to express the filter as a one-dimensional integral over level-set measures, resulting in a pixel-independent range kernel. The authors prove the equivalence with the standard pixel-based formulation, establish convergence of discrete constant-wise approximations to the continuous solution, and demonstrate the approach with experiments comparing quality and speed to state-of-the-art methods. This work enables dimension-free computation and provides a rigorous framework for fast bilateral filtering across different spatial kernels, with practical acceleration and a solid theoretical foundation.

Abstract

We introduce an exact reformulation of a broad class of neighborhood filters, among which the bilateral filters, in terms of two functional rearrangements: the decreasing and the relative rearrangements. Independently of the image spatial dimension (one-dimensional signal, image, volume of images, etc.), we reformulate these filters as integral operators defined in a one-dimensional space corresponding to the level sets measures. We prove the equivalence between the usual pixel-based version and the rearranged version of the filter. When restricted to the discrete setting, our reformulation of bilateral filters extends previous results for the so-called fast bilateral filtering. We, in addition, prove that the solution of the discrete setting, understood as constant-wise interpolators, converges to the solution of the continuous setting. Finally, we numerically illustrate computational aspects concerning quality approximation and execution time provided by the rearranged formulation.

On a fast bilateral filtering formulation using functional rearrangements

TL;DR

The paper addresses the computational burden of bilateral and related neighborhood filters and presents an exact reformulation using functional rearrangements. It introduces the decreasing rearrangement and the relative rearrangement to express the filter as a one-dimensional integral over level-set measures, resulting in a pixel-independent range kernel. The authors prove the equivalence with the standard pixel-based formulation, establish convergence of discrete constant-wise approximations to the continuous solution, and demonstrate the approach with experiments comparing quality and speed to state-of-the-art methods. This work enables dimension-free computation and provides a rigorous framework for fast bilateral filtering across different spatial kernels, with practical acceleration and a solid theoretical foundation.

Abstract

We introduce an exact reformulation of a broad class of neighborhood filters, among which the bilateral filters, in terms of two functional rearrangements: the decreasing and the relative rearrangements. Independently of the image spatial dimension (one-dimensional signal, image, volume of images, etc.), we reformulate these filters as integral operators defined in a one-dimensional space corresponding to the level sets measures. We prove the equivalence between the usual pixel-based version and the rearranged version of the filter. When restricted to the discrete setting, our reformulation of bilateral filters extends previous results for the so-called fast bilateral filtering. We, in addition, prove that the solution of the discrete setting, understood as constant-wise interpolators, converges to the solution of the continuous setting. Finally, we numerically illustrate computational aspects concerning quality approximation and execution time provided by the rearranged formulation.

Paper Structure

This paper contains 20 sections, 3 theorems, 60 equations, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Prior work
  3. Preceding applications of functional rearrangement
  4. Objective of the article
  5. Outline
  6. Neighborhood filters in terms of functional rearrangements
  7. The decreasing rearrangement
  8. Motivation
  9. The relative rearrangement
  10. Example: Rearrangement of constant-wise functions
  11. Main results
  12. Examples
  13. Proofs
  14. Experiments
  15. Yaroslavsky filter
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\Omega\subset\mathbb{R}^d$ be an open and bounded set, $d\geq 1$, $\mathcal{K}\in L^\infty(\mathbb{R},\mathbb{R}_+)$ and $w_\rho\in L^\infty(\mathbb{R}_+,\mathbb{R}_+)$. Assume that $u\in L^\infty(\Omega)$ is, without loss of generality, non-negative. Consider $\mathop{\mathrm{F}}\nolimits u(\m Then $\mathop{\mathrm{F}}\nolimits_* u(\mathbf{x}) = \mathop{\mathrm{F}}\nolimits u(\mathbf{x})$ fo

Figures (4)

  • Figure 1: Example of construction of the relative rearrangement. (c) shows $u_*$ and $v$ transported as by the displacement of the level sets of $u$ in the construction of $u_*$. (d) shows the decreasing rearrangement of $v$ restricted to the level sets of $u$, that is $v_{*u}$.
  • Figure 2: Image Clock, $h=8$. The columns are, from left to right, the denoised image, a detail of the image, the contour plot of the detail, and the histogram of the denoised image. The rows give, from above to below, the noisy image, the results of BPB, BRR20, Y8, YRR and P, and in the last row, the ground truth clean image.
  • Figure 3: Image Boat, $h=16$. The columns are, from left to right, the denoised image, a detail of the image, the contour plot of the detail, and the histogram of the denoised image. The rows give, from above to below, the noisy image, the results of BPB, BRR20, Y8, YRR and P, and in the last row, the ground truth clean image.
  • Figure 4: Image Still life, $h=32$. The columns are, from left to right, the denoised image, a detail of the image, the contour plot of the detail, and the histogram of the denoised image. The rows give, from above to below, the noisy image, the results of BPB, BRR20, Y8, YRR and P, and in the last row, the ground truth clean image.

Theorems & Definitions (5)

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