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Existence of generalized minimizers and dual solutions for a class of variational problems with linear growth related to image recovery. (English. Russian original) Zbl 1331.49014
J. Math. Sci., New York 210, No. 4, 458-475 (2015); translation from Probl. Mat. Anal. 81, 107-120 (2015).
Summary: We continue the analysis of modifications of the total variation image inpainting method formulated on the space $$BV(\Omega)^M$$ and treat the case of vector-valued images where we do not impose any structure condition on the density $$F$$ and the dimension of the domain $$\Omega$$ is arbitrary. We discuss the existence of generalized solutions of the corresponding variational problem and show the unique solvability of the associated dual variational problem. We establish the uniqueness of the absolutely continuous part $$\nabla^au$$ of the gradient of $$BV$$-solutions $$u$$ on the domain $$\Omega$$ and get the uniqueness of $$BV$$-solutions outside the damaged region $$D$$. We also prove new density results for functions of bounded variation and for Sobolev functions.
Reviewer: Reviewer (Berlin)

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 49N15 Duality theory (optimization) 26B30 Absolutely continuous real functions of several variables, functions of bounded variation 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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##### References:
 [1] Bildhauer, M; Fuchs, M, “on some perturbations of the total variation image inpainting method. part 2: relaxation and dual variational formulation”, J. Math. Sci., New York, 205, 121-140, (2015) · Zbl 1321.49054 [2] Burger, M; He, L; Schönlieb, C-B, Cahn-Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imaging Sci., 2, 1129-1167, (2009) · Zbl 1180.49007 [3] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer, New York (2002). · Zbl 1109.35002 [4] T. Lehmann, W. Oberschelp, E. Pelikan, and R. Repges, Bildverarbeitung fur die Medizin: Grundlagen, Modelle, Methoden, Anwendungen, Springer, Berlin (1997). · Zbl 0948.68527 [5] Lehmann, T; Gönner, C; Spitzer, K, Survey: interpolation methods in medical imaging processing, IEEE Trans. Medical Imaging, 18, 1049-1075, (1999) [6] Blomgren, P; Chan, TF, Color TV: total variation methods for restoration of vector-valued images, IEEE Trans. Medical Imaging, 7, 304-309, (1998) [7] K. Papafitsoros, B. Sengul, and C.-B. Schnlieb, Combined First and Second Order Total Variation Inpainting Using Split Bregman, IPOL Preprint (2012). [8] J. Shen, Inpainting and the fundamental problem of image processing, SIAM News36, No. 5, 1-4 (2003). [9] P. Arias, V. Caselles, G. Facciolo, V. Lazcano, and R. Sadek, “Nonlocal variational models for inpainting and interpolation,” Math. Models Methods Appl. Sci.22, No. Suppl. 2 (2012). · Zbl 1267.68278 [10] 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, ACM Press/Addison-Wesley Publishing Co. (2000). [11] Chan, TF; Kang, SH; Shen, J, Euler’s elastica and curvature based inpaintings, SIAM J. Appl. Math., 63, 564-592, (2002) · Zbl 1028.68185 [12] Chan, TF; Shen, J, Nontexture inpainting by curvature-driven diffusions, J. Visual Commun. Image Represen., 12, 436-449, (2001) [13] T. F. Chan and J. Shen, “Mathematical models for local nontexture inpaintings,” SIAM J. Appl. Math.62, No. 3, 1019-1043 (2001/02). · Zbl 1050.68157 [14] 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 [15] Bildhauer, M; Fuchs, M, “on some perturbations of the total variation image inpainting method. part I: regularity theory”, J. Math. Sci., New York, 202, 154-169, (2014) · Zbl 1321.49060 [16] Bildhauer, M; Fuchs, M, “on some perturbations of the total variation image inpainting method. part III: minimization among sets with finite perimeter”, J. Math. Sci., New York, 207, 142-146, (2015) · Zbl 1335.49024 [17] Bildhauer, M; Fuchs, M, “image inpainting with energies of linear growth. A collection of proposals”, J. Math. Sci., New York, 196, 490-497, (2014) · Zbl 1302.49004 [18] M. Bildhauer, M. Fuchs, and C. Tietz, “On a class of variational problems with linear growth related to image inpainting,” Algebra Anal. [To appear] · Zbl 1335.49058 [19] P. Arias, V. Casseles, and G. Sapiro, “A variational framework for nonlocal image inpainting,” In: Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 345-358, Springer (2009). · Zbl 0658.49005 [20] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford (2000). · Zbl 0957.49001 [21] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Basel (1984). · Zbl 0545.49018 [22] R. A. Adams, Sobolev Spaces, Academic Press, New York etc. (1975). · Zbl 0314.46030 [23] Anzellotti, G; Giaquinta, M, Convex functionals and partial regularity, Arch. Rat. Mech. Anal., 102, 243-272, (1988) · Zbl 0658.49005 [24] Demengel, F; Temam, R, Convex functions of a measure and applications, Indiana Univ. Math. J., 33, 673-709, (1984) · Zbl 0581.46036 [25] Giaquinta, M; Modica, G; Souček, J, Functionals with linear growth in the calculus of variations. I, Commentat. Math. Univ. Carol., 20, 143-156, (1979) · Zbl 0409.49006 [26] 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 [27] M. Fuchs and G. Seregin, Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids, Springer, Berlin etc. (2000). · Zbl 0964.76003 [28] M. Bildhauer, Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions, Lect. Notes Math. 1818, Springer, Berlin etc. (2003). · Zbl 1033.49001 [29] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North Holland, Amsterdam (1976). · Zbl 0322.90046 [30] C. Tietz, $$C$$1,α-Interior Regularity for Minimizers of a Class of Variational Problems with Linear Growth Related to Image Inpainting in Higher Dimensions, Preprint No. 356, Saarland University (2015). [31] M. Giaquinta, G. Modica, and J. Souček, Cartesian Currents in the Calculus of Variations II Springer, Berlin etc. (1998). [32] J. Malý and W. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Am. Math. Soc., Providence, RI (1997). · Zbl 0882.35001 [33] H. W. Alt, Lineare Funktionalanalysis, Springer, Berlin etc. (1985). · Zbl 0577.46001 [34] M. Meier, Reguläre und singuläre Lösungen quasilinearer elliptischer Gleichungen und Systeme I,II, Preprint Bonn University/ SFB 72 No. 245, 246, (1979). · Zbl 0405.35034 [35] Bildhauer, M; Fuchs, M, “partial regularity for a class of anisotropic variational integrals with convex hull property,”, Asymp. Anal., 32, 293-315, (2002) · Zbl 1076.49018 [36] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1989). · Zbl 0691.35001
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