Journal of Mathematical Imaging and Vision

, Volume 53, Issue 3, pp 346–363 | Cite as

On a Fast Bilateral Filtering Formulation Using Functional Rearrangements

  • Gonzalo Galiano
  • Julián Velasco


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.


Neighborhood filters Bilateral filter Decreasing rearrangement Relative rearrangement Denoising 

Mathematics Subject Classification




The authors are partially supported by the Spanish DGI Project MTM2013-43671-P. The authors thank to the anonymous reviewers for their interesting comments and suggestions, that highly contributed to the improvement of our work.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversidad de OviedoOviedoSpain

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