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.
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Rudin, L.I.: Images, Numerical Analysis of Singularities and Shock Filters. PhD thesis, California Institute of Technology, Pasadena, CA (1987)
Osher, S., Rudin, L.I.: Feature-oriented image enhancement using shock filters. SIAM Journal on Numerical Analysis 27, 919–940 (1990)
Alvarez, L., Mazorra, L.: Signal and image restoration using shock filters and anisotropic diffusion. SIAM Journal on Numerical Analysis 31, 590–605 (1994)
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)
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)
Breuß, M., Welk, M.: Analysis of staircasing in semidiscrete stabilised inverse linear diffusion algorithms. Journal of Computational and Applied Mathematics 206, 520–533 (2007)
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)
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)
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)
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)
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)
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)
Wei, G.: Shock capturing by anisotropic diffusion oscillation reduction. Computer Physics Communications 144, 317–342 (2002)
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)
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)
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)
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)
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)
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)
LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser, Basel (1992)
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)
Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics - A Practical Introduction, 2nd edn. Springer, Berlin (1999)
Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)
Seneta, E.: Non-negative Matrices and Markov Chains. Series in Statistics. Springer, Berlin (1980)
LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)
Breuß, M.: The correct use of the Lax-Friedrichs method. ESAIM: Mathematical Modeling and Numerical Analysis 38(3), 519–540 (2004)
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)
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Hagenburg, K., Breuß, M., Weickert, J., Vogel, O. (2012). Novel Schemes for Hyperbolic PDEs Using Osmosis Filters from Visual Computing. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2011. Lecture Notes in Computer Science, vol 6667. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24785-9_45
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DOI: https://doi.org/10.1007/978-3-642-24785-9_45
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