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Stabilised Nonlinear Inverse Diffusion for Approximating Hyperbolic PDEs

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Scale Space and PDE Methods in Computer Vision (Scale-Space 2005)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 3459))

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Abstract

Stabilised backward diffusion processes have shown their use for a number of image enhancement tasks. The goal of this paper is to show that they are also highly useful for designing shock capturing numerical schemes for hyperbolic conservation laws. We propose and investigate a novel flux corrected transport (FCT) type algorithm. It is composed of an advection step capturing the flow dynamics, and a stabilised nonlinear backward diffusion step in order to improve the resolution properties of the scheme. In contrast to classical FCT procedures, we base our method on an analysis of the discrete viscosity form. This analysis shows that nonlinear backward diffusion is necessary. We employ a slope limiting type approach where the antidiffusive flux determined by the viscosity form is controlled by a limiter that prohibits oscillations. Numerical experiments confirm the high accuracy and shock capturing properties of the resulting scheme. This shows the fruitful interaction of PDE-based image processing ideas and numerical analysis.

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Author information

Authors and Affiliations

  1. Computational Mathematics, Technical University Brunswick, Pockelsstraße 14, 38106, Brunswick, Germany

    Michael Breuß & Thomas Sonar

  2. Mathematical Image Analysis Group, Faculty of Math. and Computer Science, Saarland University, Building 27, 66041, Saarbrücken, Germany

    Thomas Brox & Joachim Weickert

Authors
  1. Michael Breuß
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  2. Thomas Brox
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  3. Thomas Sonar
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  4. Joachim Weickert
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Editor information

Editors and Affiliations

  1. Dept. of Computer Science, Technion – Israel Institute of Technology, 32000, Haifa, Israel

    Ron Kimmel

  2. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Nir A. Sochen

  3. Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, Building E1.1, 66041, Saarbrücken, Germany

    Joachim Weickert

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© 2005 Springer-Verlag Berlin Heidelberg

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Breuß, M., Brox, T., Sonar, T., Weickert, J. (2005). Stabilised Nonlinear Inverse Diffusion for Approximating Hyperbolic PDEs. In: Kimmel, R., Sochen, N.A., Weickert, J. (eds) Scale Space and PDE Methods in Computer Vision. Scale-Space 2005. Lecture Notes in Computer Science, vol 3459. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11408031_46

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  • DOI: https://doi.org/10.1007/11408031_46

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25547-5

  • Online ISBN: 978-3-540-32012-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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