# zbMATH — the first resource for mathematics

On some perturbations of the total variation image inpainting method. II: Relaxation and dual variational formulation. (English. Russian original) Zbl 1321.49054
J. Math. Sci., New York 205, No. 2, 121-140 (2015); translation from Probl. Mat. Anal. 77, 3-18 (2014).
Summary: We continue the analysis of some strongly elliptic modifications of the total variation image inpainting model formulated in the space $$\mathrm{BV}(\Omega)$$ and investigate the corresponding dual variational problems. Remarkable features are the uniqueness of the dual solution and the uniqueness of the absolutely continuous part $$\nabla^a u$$ of the gradient of $$\mathrm{BV}$$-solutions $$u$$ on the whole domain. Additionally, any $$\mathrm{BV}$$-minimizer $$u$$ automatically satisfies the inequality $$0 \leqslant u \leqslant 1$$, which means that $$u$$ measures the intensity of the grey level. Outside of the damaged region we even have the uniqueness of $$\mathrm{BV}$$-solutions, whereas on the damaged domain the $$L^2$$-deviation $$\| (u-v) \|_{L^2}$$ of different solutions is governed by the the total variation of the singular part $$\nabla^s (u-v)$$ of the vector measure $$\nabla(u-v)$$. Moreover, the dual solution is related to the $$\mathrm{BV}$$-solutions through an equation of stress-strain type.
Editorial remark: for part I, see [Zbl 1321.49060].

##### MSC:
 49N15 Duality theory (optimization) 49J45 Methods involving semicontinuity and convergence; relaxation 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 68U10 Computing methodologies for image processing
Full Text:
##### References:
 [1] 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 [2] P. Arias, V. Casseles, and G. Sapiro, A Variational Framework for Non-Local Image Inpainting, IMA Preprint Series No. 2265 (2009). [3] 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 [4] M. Bertalmio, V. Caselles, S. Masnou, and G. Sapiro, Inpainting. www.math.univlyon1.fr/masnou/fichiers/publications/survey.pdf · Zbl 1260.49079 [5] Chan, TF; Kang, SH; Shen, J, Euler’s elastica and curvature based inpaintings, SIAM J. Appl. Math., 63, 564-592, (2002) · Zbl 1028.68185 [6] 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 [7] K. Papafitsoros, B. Sengul, and C.-B. Sch¨onlieb, Combined First and Second Order Total Variation Impainting Using Split Bregman, IPOL Preprint (2012). [8] Shen, J, Inpainting and the fundamental problem of image processing, SIAM News, 36, 1-4, (2003) [9] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Basel (1984). · Zbl 0545.49018 [10] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford (2000). · Zbl 0957.49001 [11] M. Bildhauer, Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions, Springer, Berlin etc. (2003). · Zbl 1033.49001 [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] R. A. Adams, Sobolev Spaces, Academic Press, New York etc. (1975). · Zbl 0314.46030 [14] Anzellotti, G; Giaquinta, M, Convex functionals and partial regularity, Arch. Rat. Mech. Anal., 102, 243-272, (1988) · Zbl 0658.49005 [15] Demengel, F; Temam, R, Convex functions of a measure and applications, Ind. Univ. Math. J., 33, 673-709, (1984) · Zbl 0581.46036 [16] 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 [17] Giaquinta, M; Modica, G; Souček, J, Functionals with linear growth in the calculus of variations. II, Commentat. Math. Univ. Carol., 20, 157-172, (1979) · Zbl 0409.49007 [18] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1989). · Zbl 0691.35001 [19] M. Fuchs and G. Seregin, Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids, Springer, Berlin etc. (2000). · Zbl 0964.76003 [20] Goffman, C; Serrin, J, Sublinear functions of measures and variational integrals, Duke Math. J, 31, 159-178, (1964) · Zbl 0123.09804 [21] Beck, L; Schmidt, T, On the Dirichlet problem for variational integrals in BV, J. Reine Angew. Math., 674, 113-194, (2013) · Zbl 1260.49079 [22] Bildhauer, M; Fuchs, M, A geometric maximum principle for variational problems in spaces of vector valued functions of bounded variation, Zap. Nauchn. Semin. POMI, 385, 5-17, (2010) [23] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North Holland, Amsterdam (1976). · Zbl 0322.90046
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.