A PDE variational approach to image denoising and restoration. (English) Zbl 1169.35341

Summary: We discuss a general variational model for image restoration based on the minimization of a convex functional of gradient under minimal growth conditions. This approach is related to minimization in bounded variation norm and has a smoothing effect on degraded image while preserving the edge features.


35K55 Nonlinear parabolic equations
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
35A15 Variational methods applied to PDEs
Full Text: DOI


[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0314.46030
[2] Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces (1976), Noordhoff: Noordhoff Leyden · Zbl 0328.47035
[3] Barbu, V., Analysis and Control of Infinite Dimensional Nonlinear Equations (1993), Academic Press: Academic Press Boston, San Diego · Zbl 0776.49005
[4] Catté, F.; Lions, P. L.; Morel, J. M.; Call, T., Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29, 182 (1992) · Zbl 0746.65091
[5] Chen, Y.; Wunderli, W., Adaptive total variation for image restoration in BV spaces, J. Math. Anal. Appl., 272, 117-137 (2002) · Zbl 1020.68104
[6] Kaepfler, G.; Lopez, C.; Morel, J. M., A multiscale algorithm for image segmentation by variational methods, SIAM J. Numer. Anal., 31, 282-299 (1994) · Zbl 0804.68053
[7] Cottet, G. H.; Germain, L., Image processing through reaction combine with nonlinear diffusion, Math. Comput., 61, 659-673 (1993) · Zbl 0799.35117
[8] Diewald, U.; Preusser, T.; Rumpf, M.; Strzodka, R., Diffusion models and their accelerated solutions in image and surface processing, Acta Math. Univ. Comenianae, LXX, 15-21 (2001) · Zbl 0993.65109
[9] Perova, P.; Malik, J., Scale space and edge detection using anizotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12, 629-639 (1990)
[10] Weickert, J.; Haar Romany, B. M.; Viergever, M. A., Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Trans. Image Process., 7, 398-410 (1998)
[11] Rudin, L. I.; Osher, S.; Fatemi, F., Nonlinear total variation based noise removal algorithms, Physica D, 60, 259 (1992) · Zbl 0780.49028
[12] Lim, Jae S., Two-Dimensional Signal and Image Processing (1990), Prentice Hall: Prentice Hall Englewood Cliffs, NJ, pp. 469-476
[13] Lapidus, L.; Pinder, G. F., Numerical solution of partial differential equations in science and engineering, SIAM Rev., 25, 4, 581-582 (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.