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Denoising and inpainting of images using TV-type energies: theoretical and computational aspects. (English. Russian original) Zbl 1381.94011
J. Math. Sci., New York 219, No. 6, 899-910 (2016); translation from Probl. Mat. Anal. 87, 69-78 (2016).
Summary: We discuss variational approaches towards the denoising of images and towards the image inpainting problem combined with simultaneous denoising. Our techniques are based on variants of the TV-model, but in contrast to this case a complete analytical theory is available in our setting. At the same time, numerical experiments illustrate the advantages of our models in comparison with some established techniques.

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
49J10 Existence theories for free problems in two or more independent variables
68U10 Computing methodologies for image processing
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[1] R. Acar and C. R., Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Probl.10, No. 6, 1217-1229 (1994). · Zbl 0809.35151
[2] Aubert, G; Vese, L, A variational method in image recovery, SIAM J. Numer. Anal., 34, 1948-1979, (1997) · Zbl 0890.35033
[3] P. Blomgren, T. F. Chan, P. Mulet, L. Vese, and W. L. Wan, “Variational PDE models and methods for image processing,” In: Numerical Analysis 1999, pp. 43-67, Chapman and Hall/CRC, Boca Raton, FL (2000). · Zbl 0953.68621
[4] Caselles, V; Chambolle, A; Novaga, M, Regularity for solutions of the total variation denoising problem, Rev. Mat. Iberoam., 27, 233-252, (2011) · Zbl 1228.94005
[5] Chambolle, A; Lions, P-L, Image recovery via total variation minimization and related problems, Numer. Math., 76, 167-188, (1997) · Zbl 0874.68299
[6] Chan, TF; Esedoglu, S, Aspects of total variation regularized \(L\)1 function approximation, SIAM J. Appl. Math., 65, 1817-1837, (2005) · Zbl 1096.94004
[7] Chan, T; Shen, J; Vese, L, Variational PDE models in image processing, Notices Am. Math. Soc., 50, 14-26, (2003) · Zbl 1168.94315
[8] Chen, Y; Levine, S; Rao, M, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66, 1383-1406, (2006) · Zbl 1102.49010
[9] Kawohl, B, Variational versus PDE-based approaches in mathematical image processing, CRM Proc. Lect. Notes, 44, 113-126, (2008) · Zbl 1144.35401
[10] Rudin, L; Osher, S; Fatemi, E, Nonlinear total variation based noise removal algorithms, Physica D, 60, 259-268, (1992) · Zbl 0780.49028
[11] Vese, L, A study in the BV space of a denoising-deblurring variational problem, Appl. Math. Optimization, 44, 131-161, (2001) · Zbl 1003.35009
[12] Bildhauer, M; Fuchs, M, A variational approach to the denoising of images based on different variants of the TV-regularization, Appl. Math. Optim., 66, 331-361, (2012) · Zbl 1260.49074
[13] P. Arias, V. Caselles, G. Facciolo, V. Lazcano, and R. Sadek, “Nonlocal variational models for inpainting and interpolation,” Math. Models Methods Appl. Sci.22, Suppl.2, 1230003 (2012). · Zbl 1267.68278
[14] P. Arias, V. Caselles, and G. Sapiro, “A variational framework for non-local image inpainting,” In: Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 345-358, Springer, Berlin (2009). · Zbl 0581.46036
[15] Arias, P; Facciolo, G; Caselles, V; Sapiro, G, A variational framework for exemplarbased image inpainting, Int. J. Comput. Vis., 93, 319-347, (2011) · Zbl 1235.94015
[16] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer, New York (2002). · Zbl 1109.35002
[17] M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester, “Image inpainting,” In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, pp. 417-424,
[18] Burger, M; He, L; Schӧnlieb, C-B, Cahn-Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imaging Sci., 2, 1129-1167, (2009) · Zbl 1180.49007
[19] Chan, TF; Kang, SH; Shen, J, Euler’s elastica and curvature based inpaintings, SIAM J. Appl. Math., 63, 564-592, (2002) · Zbl 1028.68185
[20] Chan, TF; Shen, J, Nontexture inpainting by curvature-driven diffusions, J. Visual Commun. Image Represen., 12, 436-449, (2001)
[21] Chan, TF; Shen, J, Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math., 62, 1019-1043, (2001) · Zbl 1050.68157
[22] Esedoglu, S; Shen, J, Digital inpainting based on the Mumford-Shah-Euler image model, European J. Appl. Math., 13, 353-370, (2002) · Zbl 1017.94505
[23] K. Papafitsoros, B. Sengul, and C.-B. Schӧnlieb, Combined First and Second Order Total Variation Impainting Using Split Bregman, IPOL Preprint (2012).
[24] J. Shen, “Inpainting and the fundamental problem of image processing,” SIAM News36, No. 5, 1-4 (2003).
[25] M. Bildhauer and M. Fuchs, “Image inpainting with energies of linear growth. A collection of proposals,” J. Math. Sci., New York196, No. 4, 490-497 (2014). · Zbl 1302.49004
[26] M. Bildhauer and M. Fuchs, “On some perturbations of the total variation image inpainting method. Part I: Regularity theory,” J. Math. Sci., New York202, No. 2, 154-169 (2014). · Zbl 1321.49060
[27] M. Bildhauer and M. Fuchs, “On some perturbations of the total variation image inpainting method. Part II: Relaxation and dual variational formulation,” J. Math. Sci., New York202, No. 2, 121-140 (2015). · Zbl 1321.49054
[28] M. Bildhauer and M. Fuchs, “On some perturbations of the total variation image inpainting method. Part III: Minimization among sets with finite perimeter,” J. Math. Sci., New York207, No. 2, 142-146 (2015). · Zbl 1335.49024
[29] M. Bildhauer, M. Fuchs, and C. Tietz, “\(C\)\^{}{1}-interior regularity for minimizers of a class of variational problems with linear growth related to image inpainting,” St. Petersb. Math. J.27, No. 3, 381-392 (2016). · Zbl 1335.49058
[30] R. A. Adams, Sobolev Spaces, Academic Press, New York (1975), · Zbl 0314.46030
[31] E. Giusti Minimal Surfaces and Functions of Bounded Variation, Birkhӓuser, Basel etc. (1984). · Zbl 0545.49018
[32] Demengel, F; Temam, R, Convex functions of a measure and applications, Ind. Univ. Math. J., 33, 673-709, (1984) · Zbl 0581.46036
[33] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations [in Russian], Nauka, Moscow (1964); English transl.: Academic Press, New York (1968). · Zbl 1235.94015
[34] M. Fuchs and C. Tietz, “Existence of generalized minimizers and of dual solutions for a class of variational problems with linear growth related to image recovery,” J. Math. Sci., New York210, No. 4, 458-475 (2015). · Zbl 1331.49014
[35] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford (2000). · Zbl 0957.49001
[36] Anzellotti, G; Giaquinta, M, Convex functionals and partial regularity, Arch. Rat. Mech. Anal., 102, 243-272, (1988) · Zbl 0658.49005
[37] Yu. G. Reshetnyak, “Weak convergence of completely additive vector functions on a set” [in Russian], Sib. Mat. Zh.9, No. 6, 1386-1394 (1968); English transl.: Sib. Math. J.9, No. 6, 1039-1045 (1968).
[38] Spector, D, Simple proofs of some results of reshetnyak, Proc. Am. Math. Soc., 139, 1681-1690, (2011) · Zbl 1215.49025
[39] Beck, L; Schmidt, T, On the Dirichlet problem for variational integrals in BV, J. Reine Angew. Math., 674, 113-194, (2013) · Zbl 1260.49079
[40] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, Philadelphia, PA (2007). · Zbl 1127.65080
[41] K. W. Morton and L. M. Mayers, Numerical Solution of Partial Differential Equations. An Introduction, Cambridge University Press, Cambridge (1994). · Zbl 0811.65063
[42] Fučik, S; Kratochvil, A; Nečas, J, Kačanov-Galerkin method, Commentat. Math. Univ. Carol., 14, 651-659, (1973) · Zbl 0268.49035
[43] J. Weickert, J. Heers, C. Schnӧrr, K. J. Zuiderveld, O. Scherzer, and H. S. Stiehl, “Fast parallel algorithms for a broad class of nonlinear variational diffusion approaches,” Real-Time Imaging7, No. 1, 31-45 (2001). · Zbl 1010.68934
[44] Geman, D; Yang, C, Nonlinear image recovery with half-quadratic regularization, IEEE Trans. Image Process., 4, 932-945, (1995)
[45] E. Zeidler, Nonlinear Functional Analysis and its Applications II/A: Linear Monotone Operators, Springer, New York etc. (1990). · Zbl 0684.47028
[46] Gerschgorin, S, Fehlerabschӓtzung für das differenzenverfahren zur Lӧsung partieller differentialgleichungen, Z. Angew. Math., 10, 373-382, (1930) · JFM 56.0467.03
[47] Whittaker, ET, A new method of graduation, Proc. Edinburgh Math. Soc., 41, 65-75, (1923)
[48] A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method” [in Russian], Dokl. Akad. Nauk SSSR, 151, 501-504 (1963); English transl.: Sov. Math., Dokl.5, 1035-1038 (1963). · Zbl 1228.94005
[49] Bertero, M; Poggio, TA; Torre, V, Ill-posed problems in early vision, Proc. IEEE, 76, 869-889, (1988)
[50] Beck, A; Teboulle, M, Fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2, 183-202, (2009) · Zbl 1175.94009
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