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A Short Introduction to Diffusion-Like Methods

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Mathematical Methods for Signal and Image Analysis and Representation

Part of the book series: Computational Imaging and Vision ((CIVI,volume 41))

Abstract

This contribution aims to give a basic introduction to diffusion-like methods. There are many different methods commonly used for regularization tasks. Some of them will be briefly introduced and their connection to diffusion shown. In addition to this we will go into some detail for diffusion-like methods in a narrower sense, i.e. methods based on PDEs similar to diffusion PDEs known from physics. Main issues highlighted here are which PDE to use, how diffusivities in such a PDE are constructed, and which discretization is suitable for a given task.

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Notes

  1. 1.

    Considering e.g. Poisson or shot noise and low intensities, this is not a good approximation.

  2. 2.

    Diffusivities calculated by e.g. Tuckey or Cup functions may become 0 and thus a diffusion tensor based on them is not guaranteed to be positive definite, but positive semi-definite.

  3. 3.

    In the case of an image, where r j are spatially distributed on the pixel grid this is of course a smoothness assumption on the underlying signal s.

  4. 4.

    Optical flow is derived as ratio of the first and third, and second and third components of the eigenvector e 3 to the smallest eigenvalue μ 3 of a structure tensor calculated from a spatio-temporal gradient.

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Authors and Affiliations

  1. Institute for Chemistry and Dynamics of the Geosphere, ICG-3, Forschungszentrum Jülich GmbH, 52425, Jülich, Germany

    Hanno Scharr & Kai Krajsek

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  1. Hanno Scharr
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  2. Kai Krajsek
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Correspondence to Hanno Scharr .

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Editors and Affiliations

  1. Dept. Mathematics & Computer Science, Eindhoven University of Technology, Eindhoven, 5600 MB, The Netherlands

    Luc Florack

  2. Dept. Mathematics & Computer Science, Eindhoven University of Technology, P.O. Box 513, Eindhoven, 5600 MB, The Netherlands

    Remco Duits

  3. Dept. Applied Mathematics, Delft University of Technology, Delft, 2628 CD, The Netherlands

    Geurt Jongbloed

  4. Probability & Stochastic Networks (PNA), Centrum Wiskunde & Informatica, Amsterdam, 1098 XG, The Netherlands

    Marie-Colette van Lieshout

  5. Fakultät für Mathematik, Universität Duisburg-Essen, Essen, 45141, Germany

    Laurie Davies

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Scharr, H., Krajsek, K. (2012). A Short Introduction to Diffusion-Like Methods. In: Florack, L., Duits, R., Jongbloed, G., van Lieshout, MC., Davies, L. (eds) Mathematical Methods for Signal and Image Analysis and Representation. Computational Imaging and Vision, vol 41. Springer, London. https://doi.org/10.1007/978-1-4471-2353-8_1

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