Time-delay regularization of anisotropic diffusion and image processing
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 2, p. 231-251

We study a time-delay regularization of the anisotropic diffusion model for image denoising of Perona and Malik [IEEE Trans. Pattern Anal. Mach. Intell 12 (1990) 629-639], which has been proposed by Nitzberg and Shiota [IEEE Trans. Pattern Anal. Mach. Intell 14 (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.

DOI : https://doi.org/10.1051/m2an:2005010
Classification:  68U10,  35K55,  35M10
Keywords: image restoration, edge detection, Perona-Malik equation, time-delay regularization
@article{M2AN_2005__39_2_231_0,
     author = {Belahmidi, Abdelmounim and Chambolle, Antonin},
     title = {Time-delay regularization of anisotropic diffusion and image processing},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {2},
     year = {2005},
     pages = {231-251},
     doi = {10.1051/m2an:2005010},
     zbl = {1101.68102},
     mrnumber = {2143948},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2005__39_2_231_0}
}
Belahmidi, Abdelmounim; Chambolle, Antonin. Time-delay regularization of anisotropic diffusion and image processing. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 2, pp. 231-251. doi : 10.1051/m2an:2005010. http://www.numdam.org/item/M2AN_2005__39_2_231_0/

[1] L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing. Arch. Rational Mech. Anal. 123 (1993) 199-257. | Zbl 0788.68153

[2] 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.

[3] J. Canny, A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8 (1986) 679-698.

[4] F. Catté, P.-L. Lions, J.-M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29 (1992) 182-193. | Zbl 0746.65091

[5] P.G. Ciarlet, Introduction à l'analyse numérique matricielle et à l'optimisation. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris (1982). | Zbl 0488.65001

[6] G.H. Cottet and M. El-Ayyadi, A volterra type model for image processing. IEEE Trans. Image Process. 7 (1998) 292-303.

[7] S. Esedoḡlu, An analysis of the Perona-Malik scheme. Comm. Pure Appl. Math. 54 (2001) 1442-1487. | Zbl 1031.68133

[8] Y. Giga and S. Goto, Motion of hypersurfaces and geometric equations. J. Math. Soc. Japan 44 (1992) 99-111. | Zbl 0739.53005

[9] 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. | MR 1814364 | Zbl 1042.35002

[10] E. Heinz, An elementary analytic theory of the degree of mapping in n-dimensional space. J. Math. Mech. 8 (1959) 231-247. | Zbl 0085.17105

[11] K. Höllig and J.A. Nohel, 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. | Zbl 0531.35045

[12] O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques, volume 13 of Mathématiques & Applications (Berlin). Springer-Verlag, Paris (1993). | MR 1276944 | Zbl 0797.58005

[13] B. Kawohl and N. Kutev, Maximum and comparison principle for one-dimensional anisotropic diffusion. Math. Ann. 311 (1998) 107-123. | Zbl 0909.35025

[14] S. Kichenassamy, The Perona-Malik paradox. SIAM J. Appl. Math. 57 (1997) 1328-1342. | Zbl 0887.35071

[15] D. Marr and E. Hildreth, Theory of edge detection. Proc. Roy. Soc. London B. 207 (1980) 187-217.

[16] N.G. Meyers, An L p e-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17 (1963) 189-206. | Numdam | Zbl 0127.31904

[17] M. Nitzberg and T. Shiota, Nonlinear image filtering with edge and corner enhancement. IEEE Trans. Pattern Anal. Mach. Intell. 14 (1992) 826-833.

[18] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990) 629-639.

[19] R.T. Whitaker and S.M. Pizer, A multi-scale approach to nonuniform diffusion. CVGIP: Image Underst. 57 (1993) 99-110.

[20] Y. You, W. Xu, A. Tannenbaum and M. Kaveh, Behavioral analysis of anisotropic diffusion in image processing. IEEE Trans. Image Process. 5 (1996) 1539-1553.