A PDE approach to image restoration problem with observation on a meager domain. (English) Zbl 1239.94005

Summary: We present here a new nonlinear PDE approach to image restoration (the inpainting problem) using a meager blurred image or a finite number of observation points. To this end, one uses a least square approach with the \(H^{-1}\) distributional metric. Some important theoretical and numerical results are provided.


94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
35Q94 PDEs in connection with information and communication
Full Text: DOI


[1] Catte, F.; Lions, P.; Morel, M.; Coll, T., Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. numer. anal., 29, 182-193, (1992) · Zbl 0746.65091
[2] T. Chan, S. Osher, J. Shen, The digital TV filter and nonlinear denoising, Department of Mathematics, University of California, Los Angeles, CA 90095-1555, Technical Report #99-34, October 1999.
[3] Chan, T.; Shen, J., Image processing and analysis, (2005), SIAM Philadelphia
[4] Lysaker, M.; Lundervold, A.; Tai, X.-C., Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE trans. image process., 12, 12, 1579-1590, (2003) · Zbl 1286.94020
[5] Marquina, A.; Osher, S., Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal, SIAM J. sci. comput., 22, 387-405, (2000) · Zbl 0969.65081
[6] S. Osher, M. Burger, D. Goldfarb, J. Xu, W. Yin, Using geometry and iterated refinement for inverse problems (1): total variation based image restoration, Department of Mathematics, UCLA, LA, CA 90095, CAM Report #04-13, 2004.
[7] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, (2003), Springer Verlag New York · Zbl 1026.76001
[8] Osher, S.; Solé, A.; Vese, L., Image decomposition and restoration using total variation minimization and the \(H^{- 1}\) norm, SIAM J. multiscale model. simul., 1, 2, 349-370, (2003) · Zbl 1051.49026
[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.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 60, 259-268, (1992) · Zbl 0780.49028
[11] Barbu, T.; Barbu, V.; Biga, V.; Coca, D., A PDE variational approach to image denoising and restorations, Nonlinear anal. RWA, 10, 1351-1361, (2009) · Zbl 1169.35341
[12] Kim, S., PDE-based image restoration: a hybrid model and color image denoising, IEEE trans. image processing, 15, 5, 1163-1170, (2006)
[13] V.B. Surya Prasath, A. Singh, Well-posed inhomogeneous nonlinear diffusion scheme for digital image denoising, J. Appl. Math. doi:10.1155/2010/763847. · Zbl 1189.94024
[14] Jin, Z.; Yang, X., Analysis of a new variational model for multiplicative noise removal, J. math. anal. appl., 362, 415-426, (2010) · Zbl 1191.68788
[15] Kim, S.; Lim, H., A non-convex diffusion model for simultaneous image denoising and edge enhancement, Electron. J. differential equations, 175-192, (2007), Conference 15 · Zbl 1112.35096
[16] Brezis, H., Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, () · Zbl 0278.47033
[17] Barbu, V., Nonlinear differential equations of monotone type in Banach spaces, (2010), Springer Verlag New York
[18] Brezis, H.; Strauss, W., Semilinear elliptic equations in \(L^1\), J. math. soc. Japan, 25, 565-590, (1975) · Zbl 0278.35041
[19] A. Kalaitzis, Image Inpainting with Gaussian Processes, Master of Science Thesis, School of Informatics, University of Edinburgh, 2009.
[20] Chan, T.; Shen, J., Mathematical models for local nontexture inpaintings, SIAM J. appl. math., 62, 3, 1019-1043, (2001) · Zbl 1050.68157
[21] M. Bertalmio, G. Sapiro, V. Caselles, C. Ballester, Image Inpainting. SIGGRAPH 2000, 417-424.
[22] Cao, Y.; Yin, J.; Liu, Q.; Li, M., A class of nonlinear parabolic-hyperbolic equations applied to image restoration, Nonlinear anal. RWA, 11, 253-261, (2010) · Zbl 1180.35378
[23] Prasath, V.B.S.; Singh, A., A hybrid convex variational model for image restoration, Appl. math. comput., 215, 3655-3664, (2010) · Zbl 1185.94016
[24] Jin, Z.; Yang, X., Strong solutions for the generalized perona – malik equation for image restoration, Nonlinear anal., 73, 1077-1084, (2010) · Zbl 1194.35503
[25] Janev, M., Fully fractional anisotropic diffusion for image denoising, Math. comput. model., 54, 729-741, (2011) · Zbl 1225.94003
[26] Liu, X.; Huang, L., Split Bregman iteration algorithm for total bounded variation regularization based image deblurring, J. math. anal. appl., 372, 486-495, (2010) · Zbl 1202.94062
[27] Guo, Z.; Liu, Q.; Sun, J.; Wu, B., Reaction – diffusion systems with-growth for image denoising, Nonlinear anal. RWA, 12, 2904-2918, (2011) · Zbl 1219.35340
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