Abstract
Although much effort has been spent in the recent decade to establish a theoretical foundation of certain partial differential equations (PDEs) as scale-spaces, it is almost never taken into account that, in practice, images are sampled on a fixed pixel grid1. For nonlinear PDE-based filters, usually straightforward finite difference discretizations are applied in the hope that they reflect the nice properties of the continuous equations. Since scale-spaces cannot perform better than their numerical discretizations, however, it would be desirable to have a genuinely discrete nonlinear framework which reflects the discrete nature of digital images. In this paper we discuss a semidiscrete scale-space framework for nonlinear diffusion filtering. It keeps the scale-space idea of having a continuous time parameter, while taking into account the spatial discretization on a fixed pixel grid. It leads to nonlinear systems of coupled ordinary differential equations. Conditions are established under which one can prove existence of a stable unique solution which preserves the average grey level. An interpretation as a smoothing scale-space transformation is introduced which is based on an extremum principle and the existence of a large class of Lyapunov functionals comprising for instance p-norms, even central moments and the entropy. They guarantee that the process is not only simplifying and information-reducing, but also converges to a constant image as the scale parameter t tends to infinity.
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References
Acton, S.T.: Edge enhancement of infrared imagery by way of the anisotropic diffusion pyramid. Proc. IEEE Int. Conf. Image Processing (ICIP-96, Lausanne, Sept. 16–19, 1996), Vol. 1, 865–868 (1996).
Barenblatt, G.I., Bertsch, M., Dal Passo, R, Ughi, R.: A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow. SIAM J. Math. Anal. 24, 1414–1439 (1993).
Benhamouda, B.: Parameter adaptation for nonlinear diffusion in image processing. M.Sc. thesis, Dept. of Mathematics, University of Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany (1994).
Catté, F., Lions, P.-L., Morel, J.-M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29, 182–193 (1992).
Cong, G., Ma, S.D.: Nonlinear diffusion for early vision. Proc. 13th Int. Conf. Pattern Recognition (ICPR 13, Vienna, Aug. 25–30, 1996), Vol. A, 403–406 (1996).
Dzhu Magazieva, S.K.: Numerical study of a partial differential equation. U.S.S.R. Comput. Maths. Math. Phys. 23, No. 4, 45–49 (1983).
Fröhlich, J., Weickert, J.: Image processing using a wavelet algorithm for nonlinear diffusion. Report No. 104, Laboratory of Technomathematics, University of Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany (1994).
De Giorgi, E.: Congetture riguardanti alcuni problemi di evoluzione - a paper in honor of John Nash. Preprint CV-GMT-96040102, Math. Dept., Scuola Normale Superiore, Piazza dei Cavalieri, 56126 Pisa, Italy, 1995 (in English).
De Giorgi, E.: Su alcuni problemi instabili legati alla teoria della visione. Paper in honour of C. Ciliberto, Math. Dept., Scuola Normale Superiore, Piazza dei Cavalieri, 56126 Pisa, Italy (in Italian).
Höllig, K.: Existence of infinitely many solutions for a forward-backward heat equation. Trans. Amer. Math. Soc. 278, 299–316 (1983).
Hummel, R.A.: Representations based on zero-crossings in scale space. Proc. IEEE Comp. Soc. Conf. Computer Vision and Pattern Recognition (CVPR ‘86, Miami Beach, June 22–26, 1986), IEEE Computer Society Press, Washington, 204–209 (1986).
Kichenassamy, S.: Nonlinear diffusions and hyperbolic smoothing for edge enhancement. In Berger, M.-O., Deriche, R., Herlin, I, Jaffré, J., Morel, J.-M. (eds.): ICAOS ‘86: Images, wavelets and PDEs. London: Springer 1996 (Lecture Notes in Control and Information Sciences, vol. 219, pp. 119–124 ).
Kichenassamy, S.: The Perona-Malik paradox. SIAM J. Appl. Math., to appear.
Lindeberg, T.: Scale-space theory in computer vision. Boston: Kluwer 1994.
Nitzberg, M., Shiota, T.: Nonlinear image filtering with edge and corner enhancement. IEEE Trans. Pattern Anal. Mach. Intell. 14, 826–833 (1992).
Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990).
Perona, P., Shiota, T., Malik, J.: Anisotropic diffusion. In ter Haar Romeny, B.M. (ed.): Geometry-driven diffusion in computer vision. Dordrecht: Kluwer 1994 (pp. 72–92 ).
Posmentier, E.S.: The generation of salinity finestructure by vertical diffusion. J. Phys. Oceanogr. 7, 298–300 (1977).
Weickert, J.: Theoretical foundations of anisotropic diffusion in image processing. Computing, Suppl. 11, 221–236 (1996).
Weickert, J.: Anisotropic diffusion in image processing. Ph.D. thesis, Dept. of Mathematics, University of Kaiserslautern, Germany, 1996. Revised version to be published by Teubner Verlag, Stuttgart.
Weickert, J.: Nonlinear diffusion scale-spaces: From the continuous to the discrete setting. In Berger, M.-0., Deriche, R., Herlin, I, Jaffré, J., Morel, J.-M. (eds.): ICAOS ‘86: Images, wavelets and PDEs. London: Springer 1996 (Lecture Notes in Control and Information Sciences, vol. 219, pp. 111–118 ).
Weickert, J.: Recursive separable schemes for nonlinear diffusion filters. In ter Haar Romeny, B., Florack, L., Koenderink, J., Viergever, M. (eds.): Scale-space theory in computer vision. Berlin: Springer 1997 (Lecture Notes in Computer Science, vol. 1252, pp. 260–271 ).
Whitaker, R.T., Pizer, S.M.: A multi-scale approach to nonuniform diffusion. CVGIP: Image Understanding 57, 99–110 (1993).
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Weickert, J., Benhamouda, B. (1997). A semidiscrete nonlinear scale-space theory and its relation to the Perona—Malik paradox. In: Solina, F., Kropatsch, W.G., Klette, R., Bajcsy, R. (eds) Advances in Computer Vision. Advances in Computing Science. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6867-7_1
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DOI: https://doi.org/10.1007/978-3-7091-6867-7_1
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