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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 119))

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Abstract

Images can be processed by integrating reaction-diffusion equations. Patterns in an image may be stabilized depending on the parameters of the differential equations. This paper shows a method for designing parameters of dynamical systems of reaction-diffusion type such that specific modes of the images become stable.

Dedicated to Prof. Dr. R. Bulirsch on the occasion of his 65th birthday.

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References

  1. L. Alvarez, F. Guichard, P.-L. Lions, J.-M. Morel, Axioms and fundamental equations of image processing. Arch. Rat. Mech. Anal. 123 (1993), 199–257.

    MathSciNet  MATH  Google Scholar 

  2. L. Edelstein-KESHET, Mathematical Models in Biology. McGraw-Hill, New York 1988.

    Google Scholar 

  3. A. Gierer, H. Meinhardt, A theory of biological pattern formation. Kybernetik 12 (1972), 30–39.

    Google Scholar 

  4. R.C. Gonzalez, R.E. Woods, Digital Image Processing. Addison-Wesley, Reading, 1992.

    Google Scholar 

  5. B.M. Ter HAAR ROMENY (Ed.), Geometry—Driven Diffusion in Computer Vision. Kluwer, Dordrecht 1994.

    Google Scholar 

  6. R. Kapral, Pattern formation in chemical systems. Physica D86 (1995), 149–157.

    MathSciNet  Google Scholar 

  7. T. Lindeberg, Scale-space: A framework for handling image structures at multiple scales. Report, Stockholm, 1996.

    Google Scholar 

  8. J. Malik, P. Perona, Scale-space and edge detection using anisotropic diffusion. Report No. UCB/CSD 88/483, Computer Science Division EECS, University of California, Berkeley, 1988.

    Google Scholar 

  9. D. Marr, E. Hildreth, Theory of edge detection. Proc. Roy. Soc. London Ser. B207 (1980), 187–217.

    Google Scholar 

  10. H. Meinhardt, Models of Biological Pattern Formation. Academic Press, London, 1982.

    Google Scholar 

  11. D. Mumford, J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42 (1989), 577–685.

    Article  MathSciNet  MATH  Google Scholar 

  12. J.D. Murray, Mathematical Biology. Springer, Berlin 1989.

    Google Scholar 

  13. H. Neumann,Mechanisms of neural architecture for visual contrast and brightness perception. Neural Networks 9 (1996), 921–936.

    Google Scholar 

  14. S. Osher, J. Sethian, Fronts Propagating with Curvature Dependent Speed Algorithms Based on the Hamilton-Jacobi Formulation. J. Comp. Physics 79 (1988), 12–49.

    Article  MathSciNet  MATH  Google Scholar 

  15. T. Poggio, V. Torre, C. Koch, Computational vision and regularization theory. Nature 317 (1985), 314–319.

    Google Scholar 

  16. C.B. Price, P. Wambacq, A. Oosterlinck, Applications of reaction-diffusion equations to image processing. Third Int. Conf. on Image Processing and its Applications, 1989.

    Google Scholar 

  17. F.W. Schneider, A.F. MüNSTER, Nichtlineare Dynamik in der Chemie. Spektrum Akademischer Verlag, Heidelberg, 1996.

    Google Scholar 

  18. C. Schnörr, R. Sprengel, A nonlinear regularization approach to early vision. Biol. Cybern. 72 (1994), 141–149.

    MATH  Google Scholar 

  19. R. Seydel, Practical Bifurcation and Stability Analysis. From Equilibrium to Chaos. Second Edition. Springer Interdisciplinary Applied Mathematics, New York, 1994.

    Google Scholar 

  20. R. Seydel,Nonlinear Computation. Int. J. of Bifurcation and Chaos 7 (1997), 2105–2126.

    Google Scholar 

  21. J. Stoer, R. Bulirsch, Introduction to Numerical Analysis. Springer, New York, 1980.

    Google Scholar 

  22. A.M. Turing, The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Lond. B237 (1952), 37–72.

    Google Scholar 

  23. J. Weickert, Anisotropic Diffusion in Image Processing. Teubner, Stuttgart, 1998.

    Google Scholar 

  24. A. Witkin, M. Kass,Reaction—Diffusion textures. Computer Graphics 25 (1991), 299–308.

    Article  Google Scholar 

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Seydel, R. (2000). A Design Problem for Image Processing. In: Doedel, E., Tuckerman, L.S. (eds) Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol 119. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1208-9_19

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  • DOI: https://doi.org/10.1007/978-1-4612-1208-9_19

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7044-7

  • Online ISBN: 978-1-4612-1208-9

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