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A Discrete Theory and Efficient Algorithms for Forward-and-Backward Diffusion Filtering

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Abstract

Image enhancement with forward-and-backward (FAB) diffusion lacks a sound theory and is numerically very challenging due to its diffusivities that are negative within a certain gradient range. In our paper, we address both problems. First we establish a comprehensive theory for space-discrete and time-continuous FAB diffusion processes. It requires approximating the gradient magnitude with a non-standard discretisation. Then, we show that this theory carries over to the fully discrete case, when an explicit time discretisation with a fairly restrictive step-size limit is applied. To come up with more efficient algorithms, we propose three accelerated schemes: (i) an explicit scheme with global time step size adaptation that is also well suited for parallel implementations on GPUs, (ii) a randomised two-pixel scheme that offers optimal adaptivity of the time step size, (iii) a deterministic two-pixel scheme which benefits from less restrictive consistency bounds. Our experiments demonstrate that these algorithms allow speed-ups by up to three orders of magnitude without compromising stability or introducing visual artefacts.

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Acknowledgements

J.W. and G.G. would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Variational Methods and Effective Algorithms for Imaging and Vision, when final work on this paper was undertaken. This work was supported by EPSRC Grant Number EP/K032208/1 and by a Rothschild Distinguished Visiting Fellowship for J.W.

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Authors and Affiliations

  1. Private University for Health Sciences, Medical Informatics and Technology (UMIT), Eduard-Wallnöfer-Zentrum 1, 6060, Hall/Tyrol, Austria

    Martin Welk

  2. Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Campus E1.7, Saarland University, 66041, Saarbrücken, Germany

    Joachim Weickert

  3. Electrical Engineering Department, Technion – IIT, Technion City, 32000, Haifa, Israel

    Guy Gilboa

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  1. Martin Welk
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Correspondence to Martin Welk.

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Welk, M., Weickert, J. & Gilboa, G. A Discrete Theory and Efficient Algorithms for Forward-and-Backward Diffusion Filtering. J Math Imaging Vis 60, 1399–1426 (2018). https://doi.org/10.1007/s10851-018-0847-4

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