Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T20:28:35.422Z Has data issue: false hasContentIssue false
  • English
  • Français

Time-delay regularization of anisotropic diffusion and image processing

Published online by Cambridge University Press:  15 April 2005

Abdelmounim Belahmidi  and
Antonin Chambolle
Abdelmounim Belahmidi
Affiliation:
CEREMADE (CNRS UMR 7534), Université de Paris Dauphine, 75775 Paris Cedex 16, France. [email protected]
Antonin Chambolle
Affiliation:
CMAP (CNRS UMR 7641), École Polytechnique, 91128 Palaiseau Cedex, France. [email protected]
Get access

Abstract

We study a time-delay regularization of the anisotropic diffusion model for image denoising of Perona and Malik [IEEE Trans. Pattern Anal. Mach. Intell12 (1990) 629–639], which has been proposed by Nitzberg and Shiota [IEEE Trans. Pattern Anal. Mach. Intell14 (1998) 826–835]. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images.

Keywords

Image restorationedge detectionPerona-Malik equationtime-delay regularization.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 39 , Issue 2 , March 2005 , pp. 231 - 251
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alvarez, L., Guichard, F., Lions, P.-L. and Morel, J.-M., Axioms and fundamental equations of image processing. Arch. Rational Mech. Anal. 123 (1993) 199–257. CrossRef
A. Belahmidi, Équations aux dérivées partielles appliquées à la restauration et à l'agrandissement des images. Ph.D. thesis, CEREMADE, Université de Paris-Dauphine, Paris (2003). Available at http://tel.ccsd.cnrs.fr.
Canny, J., A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8 (1986) 679–698. CrossRef
Catté, F., Lions, P.-L., Morel, J.-M. and Coll, T., Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29 (1992) 182–193. CrossRef
P.G. Ciarlet, Introduction à l'analyse numérique matricielle et à l'optimisation. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris (1982).
Cottet, G.H. and El-Ayyadi, M., A volterra type model for image processing. IEEE Trans. Image Process. 7 (1998) 292–303. CrossRef
Esedoḡlu, S., An analysis of the Perona-Malik scheme. Comm. Pure Appl. Math. 54 (2001) 1442–1487. CrossRef
Giga, Y. and Goto, S., Motion of hypersurfaces and geometric equations. J. Math. Soc. Japan 44 (1992) 99–111. CrossRef
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition.
Heinz, E., An elementary analytic theory of the degree of mapping in n-dimensional space. J. Math. Mech. 8 (1959) 231–247.
Höllig, K. and Nohel, J.A., A diffusion equation with a nonmonotone constitutive function, in Systems of nonlinear partial differential equations (Oxford, 1982), Reidel, Dordrecht. NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 111 (1983) 409–422.
O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, volume 13 of Mathématiques & Applications (Berlin). Springer-Verlag, Paris (1993).
Kawohl, B. and Kutev, N., Maximum and comparison principle for one-dimensional anisotropic diffusion. Math. Ann. 311 (1998) 107–123. CrossRef
Kichenassamy, S., The Perona-Malik paradox. SIAM J. Appl. Math. 57 (1997) 1328–1342. CrossRef
Marr, D. and Hildreth, E., Theory of edge detection. Proc. Roy. Soc. London B. 207 (1980) 187–217. CrossRef
Meyers, N.G., An Lpe-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17 (1963) 189–206.
Nitzberg, M. and Shiota, T., Nonlinear image filtering with edge and corner enhancement. IEEE Trans. Pattern Anal. Mach. Intell. 14 (1992) 826–833. CrossRef
Perona, P. and Malik, J., Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990) 629–639. CrossRef
Whitaker, R.T. and Pizer, S.M., A multi-scale approach to nonuniform diffusion. CVGIP: Image Underst. 57 (1993) 99–110. CrossRef
You, Y., Xu, W., Tannenbaum, A. and Kaveh, M., Behavioral analysis of anisotropic diffusion in image processing. IEEE Trans. Image Process. 5 (1996) 1539–1553.