Strong solutions for the generalized Perona-Malik equation for image restoration. (English) Zbl 1194.35503

Summary: We establish the existence and uniqueness of strong solutions for the generalized Perona-Malik equation of the fourth order for image restoration in dimension one.


35R30 Inverse problems for PDEs
35K35 Initial-boundary value problems for higher-order parabolic equations
35K55 Nonlinear parabolic equations
68U10 Computing methodologies for image processing
Full Text: DOI


[1] Perona, P.; Malik, J., Scale-space and edge detection using anisotropic diffusion, IEEE trans. pattern anal. Mach. intell., 12, 629-639, (1990)
[2] Aubert, G.; Kornprobst, P., ()
[3] Catte, F.; Lions, P.L.; Morel, J.M.; Coll, T., Image selective smoothing and edge detection by nonlinear diffusion, SIAM. J. numer. anal., 29, 1, 182-193, (1992) · Zbl 0746.65091
[4] Bertozzi, A.L.; Greer, J.B., Low curvature image simplifiers: global regularity of smooth solutions and Laplacian limiting schemes, Comm. pure appl. math., 57, 6, 764-790, (2004) · Zbl 1058.35083
[5] Chambolle, A.; Lions, P., Image recovery via total variation minimization and related problems, Numer. math., 76, 2, 167-188, (1997) · Zbl 0874.68299
[6] Chan, T.; Marquina, A.; Mulet, P., High-order total variation-based image restoration, SIAM J. sci. comput., 22, 2, 503-516, (2000) · Zbl 0968.68175
[7] Lysaker, M.; Lundervold, A.; Tai, X.C., Noise removal using fourth order partial differential equations with applications to medical magnetic resonance images in space and time, IEEE trans. image process., 12, 1579-1590, (2003) · Zbl 1286.94020
[8] J. Tumblin, G. Turk, LCIS: a boundary hierarchy for detail-preserving contrast reduction, in: Proceedings of the SIGGRAPH 1999 Annual Conference on Computer Graphics, August 8-13, 1999, Los Angeles, CA USA, 1999, pp. 83-90.
[9] Wei, G.W., Generalized perona – malik equation for image processing, IEEE signal process. lett., 6, 7, 165-167, (1999)
[10] You, Y.-L.; Kaveh, M., Fourth-order partial differential equations for noise removal, IEEE trans. image process., 9, 10, 1723-1730, (2000) · Zbl 0962.94011
[11] Greer, J.B.; Bertozzi, A.L., \(H^1\) solution of a class of fourth order nonlinear equations for image processing, Discrete contin. dyn. syst., 10, 1, 2, 349-366, (2004) · Zbl 1159.68619
[12] Greer, J.B.; Bertozzi, A.L., Travelling wave solutions of fourth order nonlinear equations for image processing, SIAM J. math. anal., 36, 1, 38-68, (2004) · Zbl 1082.35080
[13] Novick-Cohen, A., On the cahn – hilliard type equations, Nonlinear anal., 15, 797-814, (1990) · Zbl 0731.35057
[14] Lions, J.L., Quelques méthodes de résolution des problèmes aux limites non linéaries, (1969), Dunod · Zbl 0189.40603
[15] Evans, L.C., Partial differental equations, (1998), American Mathematical Society
[16] L. Nirenberg, Topics in Nonlinear Functional Analysis, Lecuture Notes, New York University, 1974. · Zbl 0286.47037
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.