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\(C^{1,\alpha}\)-interior regularity for minimizers of a class of variational problems with linear growth related to image inpainting. (English) Zbl 1335.49058
St. Petersbg. Math. J. 27, No. 3, 381-392 (2016) and Algebra Anal. 27, No. 3, 51-65 (2015).
Summary: A modification of the total variation image inpainting method is investigated. By using DeGiorgi type arguments, the partial regularity results established previously are improved to the \(C^{1,\alpha}\) interior differentiability of solutions of this new variational problem.
Reviewer: Reviewer (Berlin)

MSC:
49N60 Regularity of solutions in optimal control
49Q20 Variational problems in a geometric measure-theoretic setting
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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[1] [Ad] R. A. Adams, Sobolev spaces, Pure Appl. Math., vol. 65, Acad. Press, New York–London, 1975.
[2] [ACFLS] P. Arias, V. Caselles, G. Facciolo, V. Lazcano, and R. Sadek, Nonlocal variational models for inpainting and interpolation, Math. Models Methods Appl. Sci. 22 (2012), suppl. 2, 1230003. · Zbl 1267.68278
[3] [ACS] P. Arias, V. Casseles, and G. Sapiro, A variational framework for non-local image inpainting, IMA Preprint Sers., no. 2265, 2009.
[4] [afcs] P. Arias, G. Facciolo, V. Casseles, and G. Sapiro, A variational framework for exemplar-based image inpainting, Int. J. Comput. Vis. 93 (2011), no. 3, 319–347. · Zbl 1235.94015
[5] [AK] G. Aubert and P. Kornprobst, Mathematical problems in image processing, Appl. Math. Sci., vol. 147, Springer-Verlag, New York, 2002. · Zbl 1109.35002
[6] [BBCS] M. Bertalmio, C. Ballester, G. Sapiro, and V. Caselles, Image inpainting, Proc. 27th. Conf. Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley Publ. Co., 2000, pp. 417–424.
[7] [Bi1] M. Bildhauer, Convex variational problems: linear, nearly linear and anisotropic growth conditions, Lecture Notes in Math., vol. 1818, Springer-Verlag, Berlin, 2003. · Zbl 1033.49001
[8] [BF1] M. Bildhauer and M. Fuchs, A variational approach to the denoising of images based on different variants of the TV-regularization, Appl. Math. Optim. 66 (2012), no. 3, 331–361. · Zbl 1260.49074
[9] [BF2] \bysame, On some perturbations of the total variation image inpainting method. Part 1: Regularity theory, J. Math. Sci. 202 (2014), no. 2, 154–169. · Zbl 1321.49060
[10] [BF3] \bysame, On some perturbations of the total variation image inpainting method. Part 2: Relaxation and dual variational formulation, J. Math. Sci. 205 (2015), no. 2, 121–140. · Zbl 1321.49054
[11] [BFW] M. Bildhauer, M. Fuchs, and J. Weickert, Denoising and inpainting of images using TV-type energies: computational and theoretical aspects. (to appear) · Zbl 1381.94011
[12] [BHS] M. Burger, L. He, and C.-B. Sch\"onlieb, Cahn–Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imaging Sci. 2 (2009), no. 4, 1129–1167. · Zbl 1180.49007
[13] [CKS] T. E. Chan, S. H. Kang, and J. Shen, Euler’s elastica and curvature based inpaintings, SIAM J. Appl. Math. 63 (2002), no. 2, 564–592. · Zbl 1028.68185
[14] [CS1] T. E. Chan and J. Shen, Nontexture inpainting by curvature-driven diffusions, J. Vis. Comm. Image Represent. 12 (2001), no. 4, 436–449.
[15] [CS2] \bysame, Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math. 62 (2001/02), no. 3, 1019–1043. · Zbl 1050.68157
[16] [ES] S. Esedoglu and J. Shen, Digital inpainting based on the Mumford–Shah–Euler image model, European J. Appl. Math. 13 (2002), no. 4, 353–370. · Zbl 1017.94505
[17] [FS] J. Frehse and G. Seregin, Regularity for solutions of variational problems in the deformation theory of plasticity with logarithmic hardening, Proc. St. Petersburg Math. Soc., vol. 5 (1998), 127–152; English transl., Amer. Math. Soc. Transl. Ser. 2, vol. 193, Amer. Math. Soc., Providence, RI, 1999, pp. 127–152.
[18] [Giu] E. Giusti, Minimal surfaces and functions of bounded variation, Monogr. Math., vol. 80, Birkh\"auser Verlag, Basel, 1984.
[19] [GT] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren Math. Wiss., Bd. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[20] [KS] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Pure Appl. Math., vol. 88, Acad. Press, New York, 1980. · Zbl 0457.35001
[21] [PSS] K. Papafitsoros, B. Sengul, and C.-B. Sch\"onlieb, Combined first and second order total variation impainting using split bregman, IPOL Preprint, 2012.
[22] [Sh] J. Shen, Inpainting and the fundamental problem of image processing, SIAM News 36 (2003), no. 5, 1–4.
[23] [St] G. Stampacchia, Le probl\`eme de Dirichlet pour les \'equations elliptiques du second ordre \`a coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. 1, 189–258. · Zbl 0151.15401
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