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
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.
L. Edelstein-KESHET, Mathematical Models in Biology. McGraw-Hill, New York 1988.
A. Gierer, H. Meinhardt, A theory of biological pattern formation. Kybernetik 12 (1972), 30–39.
R.C. Gonzalez, R.E. Woods, Digital Image Processing. Addison-Wesley, Reading, 1992.
B.M. Ter HAAR ROMENY (Ed.), Geometry—Driven Diffusion in Computer Vision. Kluwer, Dordrecht 1994.
R. Kapral, Pattern formation in chemical systems. Physica D86 (1995), 149–157.
T. Lindeberg, Scale-space: A framework for handling image structures at multiple scales. Report, Stockholm, 1996.
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.
D. Marr, E. Hildreth, Theory of edge detection. Proc. Roy. Soc. London Ser. B207 (1980), 187–217.
H. Meinhardt, Models of Biological Pattern Formation. Academic Press, London, 1982.
D. Mumford, J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42 (1989), 577–685.
J.D. Murray, Mathematical Biology. Springer, Berlin 1989.
H. Neumann,Mechanisms of neural architecture for visual contrast and brightness perception. Neural Networks 9 (1996), 921–936.
S. Osher, J. Sethian, Fronts Propagating with Curvature Dependent Speed Algorithms Based on the Hamilton-Jacobi Formulation. J. Comp. Physics 79 (1988), 12–49.
T. Poggio, V. Torre, C. Koch, Computational vision and regularization theory. Nature 317 (1985), 314–319.
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.
F.W. Schneider, A.F. MüNSTER, Nichtlineare Dynamik in der Chemie. Spektrum Akademischer Verlag, Heidelberg, 1996.
C. Schnörr, R. Sprengel, A nonlinear regularization approach to early vision. Biol. Cybern. 72 (1994), 141–149.
R. Seydel, Practical Bifurcation and Stability Analysis. From Equilibrium to Chaos. Second Edition. Springer Interdisciplinary Applied Mathematics, New York, 1994.
R. Seydel,Nonlinear Computation. Int. J. of Bifurcation and Chaos 7 (1997), 2105–2126.
J. Stoer, R. Bulirsch, Introduction to Numerical Analysis. Springer, New York, 1980.
A.M. Turing, The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Lond. B237 (1952), 37–72.
J. Weickert, Anisotropic Diffusion in Image Processing. Teubner, Stuttgart, 1998.
A. Witkin, M. Kass,Reaction—Diffusion textures. Computer Graphics 25 (1991), 299–308.
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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
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