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On a Fast Bilateral Filtering Formulation Using Functional Rearrangements

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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.

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Notes

  1. C++  code downloaded from http://www.cs.cityu.edu.hk/~qiyang

  2. C++  code downloaded from http://graphics.stanford.edu/papers/permutohedral

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Acknowledgments

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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  1. Department of Mathematics, Universidad de Oviedo, c/ Calvo Sotelo, 33007, Oviedo, Spain

    Gonzalo Galiano & Julián Velasco

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  1. Gonzalo Galiano
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  2. Julián Velasco
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Correspondence to Gonzalo Galiano.

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Galiano, G., Velasco, J. On a Fast Bilateral Filtering Formulation Using Functional Rearrangements. J Math Imaging Vis 53, 346–363 (2015). https://doi.org/10.1007/s10851-015-0583-y

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