Novel Schemes for Hyperbolic PDEs Using Osmosis Filters from Visual Computing

  • Kai Hagenburg
  • Michael Breuß
  • Joachim Weickert
  • Oliver Vogel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6667)

Abstract

Recently a new class of generalised diffusion filters called osmosis filters has been proposed. Osmosis models are useful for a variety of tasks in visual computing. In this paper, we show that these filters are also beneficial outside image processing and computer graphics: We exploit their use for the construction of better numerical schemes for hyperbolic partial differential equations that model physical transport phenomena.

Our novel osmosis-based algorithm is constructed as a two-step, predictor-corrector method. The predictor scheme is given by a Markov chain model of osmosis that captures the hyperbolic transport in its advection term. By design, it also incorporates a discrete diffusion process. The corresponding terms can easily be identified within the osmosis model. In the corrector step, we subtract a stabilised version of this discrete diffusion. We show that the resulting osmosis-based method gives correct, highly accurate resolutions of shock wave fronts in both linear and nonlinear test cases. Our work is an example for the usefulness of visual computing ideas in numerical analysis.

Keywords

diffusion filtering osmosis diffusion-advection drift-diffusion hyperbolic conservation laws finite difference methods predictor-corrector schemes stabilised inverse diffusion 

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References

  1. 1.
    Rudin, L.I.: Images, Numerical Analysis of Singularities and Shock Filters. PhD thesis, California Institute of Technology, Pasadena, CA (1987)Google Scholar
  2. 2.
    Osher, S., Rudin, L.I.: Feature-oriented image enhancement using shock filters. SIAM Journal on Numerical Analysis 27, 919–940 (1990)CrossRefMATHGoogle Scholar
  3. 3.
    Alvarez, L., Mazorra, L.: Signal and image restoration using shock filters and anisotropic diffusion. SIAM Journal on Numerical Analysis 31, 590–605 (1994)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Kornprobst, P., Deriche, R., Aubert, G.: Image coupling, restoration and enhancement via PDEs. In: Proc.1997 IEEE International Conference on Image Processing, Washington, DC, vol. 4, pp. 458–461 (October 1997)Google Scholar
  5. 5.
    Osher, S., Rudin, L.: Shocks and other nonlinear filtering applied to image processing. In: Tescher, A.G. (ed.) Applications of Digital Image Processing XIV. Proceedings of SPIE, vol. 1567, pp. 414–431. SPIE Press, Bellingham (1991)Google Scholar
  6. 6.
    Breuß, M., Welk, M.: Analysis of staircasing in semidiscrete stabilised inverse linear diffusion algorithms. Journal of Computational and Applied Mathematics 206, 520–533 (2007)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Pollak, I., Willsky, A.S., Krim, H.: Image segmentation and edge enhancement with stabilized inverse diffusion equations. IEEE Transactions on Image Processing 9(2), 256–266 (2000)CrossRefMATHGoogle Scholar
  8. 8.
    Gilboa, G., Sochen, N.A., Zeevi, Y.Y.: Forward-and-backward diffusion processes for adaptive image enhancement and denoising. IEEE Transactions on Image Processing 11(7), 689–703 (2002)CrossRefGoogle Scholar
  9. 9.
    Welk, M., Gilboa, G., Weickert, J.: Theoretical foundations for discrete forward-and-backward diffusion filtering. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 527–538. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Breuß, M., Zimmer, H., Weickert, J.: Can variational models for correspondence problems benefit from upwind discretisations? Journal of Mathematical Imaging and Vision 39(5), 230–244 (2011)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Grahs, T., Meister, A., Sonar, T.: Image processing for numerical approximations of conservation laws: nonlinear anisotropic artificial dissipation. SIAM Journal on Scientific Computing 23(5), 1439–1455 (2002)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Grahs, T., Sonar, T.: Entropy-controlled artificial anisotropic diffusion for the numerical solution of conservation laws based on algorithms from image processing. Journal of Visual Communication and Image Representation 13(1/2), 176–194 (2002)CrossRefGoogle Scholar
  13. 13.
    Wei, G.: Shock capturing by anisotropic diffusion oscillation reduction. Computer Physics Communications 144, 317–342 (2002)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Boris, J.P., Book, D.L.: Flux corrected transport. I. SHASTA, a fluid transport algorithm that works. Journal of Computational Physics 11(1), 38–69 (1973)CrossRefMATHGoogle Scholar
  15. 15.
    Burgeth, B., Pizarro, L., Breuß, M., Weickert, J.: Adaptive continuous-scale morphology for matrix fields. International Journal of Computer Vision 92(2), 146–161 (2011)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Breuß, M., Weickert, J.: A shock-capturing algorithm for the differential equations of dilation and erosion. Journal of Mathematical Imaging and Vision 25, 187–201 (2006)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Breuß, M., Weickert, J.: Highly accurate schemes for PDE-based morphology with general structuring elements. International Journal of Computer Vision 92(2), 132–145 (2011)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Breuß, M., Brox, T., Sonar, T., Weickert, J.: Stabilised nonlinear inverse diffusion for approximating hyperbolic PDEs. In: Kimmel, R., Sochen, N., Weickert, J. (eds.) Scale-Space 2005. LNCS, vol. 3459, pp. 536–547. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Weickert, J., Hagenburg, K., Vogel, O., Breuß, M., Ochs, P.: Osmosis models for visual computing. Technical report, Department of Mathematics, Saarland University, Saarbrücken, Germany (2011)Google Scholar
  20. 20.
    LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser, Basel (1992)CrossRefMATHGoogle Scholar
  21. 21.
    van Leer, B.: Towards the ultimate conservative difference scheme, V. A second order sequel to Godunov’s method. Journal of Computational Physics 32(1), 101–136 (1979)Google Scholar
  22. 22.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics - A Practical Introduction, 2nd edn. Springer, Berlin (1999)MATHGoogle Scholar
  23. 23.
    Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)MATHGoogle Scholar
  24. 24.
    Seneta, E.: Non-negative Matrices and Markov Chains. Series in Statistics. Springer, Berlin (1980)Google Scholar
  25. 25.
    LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)CrossRefMATHGoogle Scholar
  26. 26.
    Breuß, M.: The correct use of the Lax-Friedrichs method. ESAIM: Mathematical Modeling and Numerical Analysis 38(3), 519–540 (2004)CrossRefMATHGoogle Scholar
  27. 27.
    Breuß, M.: An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws. ESAIM: Mathematical Modeling and Numerical Analysis 39(5), 965–993 (2005)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Kai Hagenburg
    • 1
  • Michael Breuß
    • 1
  • Joachim Weickert
    • 1
  • Oliver Vogel
    • 1
  1. 1.Mathematical Image Analysis Group, Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany

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