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Fourth-order partial differential equations for image enhancement. (English) Zbl 1124.65096
The authors propose a method for noise removal which based on a combination of a model by F. Catté, P.-L. Lions, J.-M. Morel and T. Coll [SIAM J. Numer. Anal. 29, No. 1, 182–193 (1992; Zbl 0746.65091)] with fourth-order terms. They prove the correctness of the proposed model and show that it is numerically superior to that of P. Perona and J. Malik [Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)] Catté et al. [loc. cit.], and T. Chan, A. Marquina and P. Mulet [Second order differential functionals in total variation-based image restoration. (to appear)].

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K30 Initial value problems for higher-order parabolic equations
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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[1] Acar, R.; Vogel, C.R., Analysis of bounded variation penalty methods for ill-posed problems, Inverse probl., 10, 1217-1229, (1994) · Zbl 0809.35151
[2] Brezis, H., Analyse fonctionnelle, théorie et applications, (1987), Masson
[3] Catte’, F.; Lions, P.L.; Morel, J.M.; Coll, T., Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. numer. anal., 129, 182-193, (1992) · Zbl 0746.65091
[4] Chan, T.F.; Golub, G.H.; Mulet, P., A nonlinear primal-dual method for total variation-based image restoration, SIAM J. sci. comput., 20, 1964-1977, (1996) · Zbl 0929.68118
[5] Evans, L.C., Partial differential equations, Grad. stud. math., 19, (1998), AMS
[6] Gonzalez, R.C.; Woods, R.E., Digital image processing, (1992), Addison-Wesley
[7] Hutson, V.; Pym, J.S., Applications of functional analysis and operator theory, (1980), Academic Press · Zbl 0426.46009
[8] Jain, A.K., Fundamentals of digital image processing, (2003), Prentice Hall
[9] Perona, P.; Malik, J., Scale space and edge detection using anisotropic diffusion, IEEE trans. pattern anal. Mach. intell., 12, 629-639, (1990)
[10] Rudin, L.I.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 28, 288-305, (1993)
[11] T. Chan, A. Marquina, P. Mulet, Second order differential functionals in total variation-based image restoration. Available from: <http://www.math.ucla.edu/chan>. · Zbl 0968.68175
[12] Vogel, C.R.; Oman, M.E., Iterative methods for total variation denoising, SIAM J. sci. comput., 17, 227-238, (1996) · Zbl 0847.65083
[13] You, Y.-L.; Xu, W.; Tannenbaum, A.; Kaveh, M., Behavioral analysis of anisotropic diffusion in image processing, IEEE trans. image process., 5, 1539-1553, (1996)
[14] You, Y.-L.; Kaveh, M., Fourth-order partial differential equations for noise removal, IEEE trans. image process., 9, 1723-1730, (2000) · Zbl 0962.94011
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