On a fast bilateral filtering formulation using functional rearrangements

TL;DR

This paper presents an exact reformulation of neighborhood filters, including bilateral filters, using functional rearrangements, enabling efficient computation and convergence analysis across various image dimensions.

Contribution

It introduces a novel rearranged formulation of bilateral filters that is exact, dimension-independent, and extends previous fast filtering methods with proven convergence.

Findings

Reformulation as integral operators in one-dimensional space.

Equivalence between pixel-based and rearranged filter versions.

Numerical results show improved efficiency and quality approximation.

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.