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$ C^{1,\alpha}$-interior regularity for minimizers of a class of variational problems with linear growth related to image inpainting

Authors: M. Bildhauer, M. Fuchs and C. Tietz
Original publication: Algebra i Analiz, tom 27 (2015), nomer 3.
Journal: St. Petersburg Math. J. 27 (2016), 381-392
MSC (2010): Primary 49N60, 49Q20
Published electronically: March 30, 2016
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Abstract | References | Similar Articles | Additional Information

Abstract: 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.

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  • 1. Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • 2. Pablo Arias, Vicent Caselles, Gabriele Facciolo, Vanel Lazcano, and Rida Sadek, Nonlocal variational models for inpainting and interpolation, Math. Models Methods Appl. Sci. 22 (2012), no. suppl. 2, 1230003, 65. MR 2974178, 10.1142/S0218202512300037
  • 3. P. Arias, V. Casseles, and G. Sapiro, A variational framework for non-local image inpainting, IMA Preprint Sers., no. 2265, 2009.
  • 4. Pablo Arias, Gabriele Facciolo, Vicent Caselles, and Guillermo Sapiro, A variational framework for exemplar-based image inpainting, Int. J. Comput. Vis. 93 (2011), no. 3, 319–347. MR 2787013, 10.1007/s11263-010-0418-7
  • 5. Gilles Aubert and Pierre Kornprobst, Mathematical problems in image processing, Applied Mathematical Sciences, vol. 147, Springer-Verlag, New York, 2002. Partial differential equations and the calculus of variations; With a foreword by Olivier Faugeras. MR 1865346
  • 6. 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. Michael Bildhauer, Convex variational problems, Lecture Notes in Mathematics, vol. 1818, Springer-Verlag, Berlin, 2003. Linear, nearly linear and anisotropic growth conditions. MR 1998189
  • 8. Michael Bildhauer and Martin 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. MR 2996430, 10.1007/s00245-012-9174-0
  • 9. M. Bildhauer and M. Fuchs, On some perturbations of the total variation image inpainting method. Part I: Regularity theory, J. Math. Sci. (N.Y.) 202 (2014), no. 2, Problems in mathematical analysis. No. 76 (Russian), 154–169. MR 3391330, 10.1007/s10958-014-2039-0
  • 10. 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. (N.Y.) 205 (2015), no. 2, Problems in mathematical analysis. No. 77 (Russian), 121–140. MR 3391337, 10.1007/s10958-015-2237-4
  • 11. M. Bildhauer, M. Fuchs, and J. Weickert, Denoising and inpainting of images using TV-type energies: computational and theoretical aspects. (to appear)
  • 12. Martin Burger, Lin He, and Carola-Bibiane Schönlieb, Cahn-Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imaging Sci. 2 (2009), no. 4, 1129–1167. MR 2559162, 10.1137/080728548
  • 13. Tony F. Chan, Sung Ha Kang, and Jianhong Shen, Euler’s elastica and curvature-based inpainting, SIAM J. Appl. Math. 63 (2002), no. 2, 564–592. MR 1951951, 10.1137/S0036139901390088
  • 14. T. E. Chan and J. Shen, Nontexture inpainting by curvature-driven diffusions, J. Vis. Comm. Image Represent. 12 (2001), no. 4, 436-449.
  • 15. Tony F. Chan and Jianhong Shen, Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math. 62 (2001/02), no. 3, 1019–1043. MR 1897733, 10.1137/S0036139900368844
  • 16. Selim Esedoglu and Jianhong Shen, Digital inpainting based on the Mumford-Shah-Euler image model, European J. Appl. Math. 13 (2002), no. 4, 353–370. MR 1925256, 10.1017/S0956792502004904
  • 17. G. A. Seregin and J. Frehse, Regularity of solutions to variational problems of the deformation theory of plasticity with logarithmic hardening, Proceedings of the St. Petersburg Mathematical Society, Vol. V, Amer. Math. Soc. Transl. Ser. 2, vol. 193, Amer. Math. Soc., Providence, RI, 1999, pp. 127–152. MR 1736908
  • 18. Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682
  • 19. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • 20. David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
  • 21. K. Papafitsoros, B. Sengul, and C.-B. Schönlieb, Combined first and second order total variation impainting using split bregman, IPOL Preprint, 2012.
  • 22. J. Shen, Inpainting and the fundamental problem of image processing, SIAM News 36 (2003), no. 5, 1-4.
  • 23. Guido Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 189–258 (French). MR 0192177

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Additional Information

M. Bildhauer
Affiliation: Department of Mathematics, Saarland University, P.O. Box 151150, 66041 Saarbrücken, Germany

M. Fuchs
Affiliation: Department of Mathematics, Saarland University, P.O. Box 151150, 66041 Saarbrücken, Germany

C. Tietz
Affiliation: Department of Mathematics, Saarland University, P.O. Box 151150, 66041 Saarbrücken, Germany

Keywords: Image inpainting, variational method, TV-regularization
Received by editor(s): November 20, 2014
Published electronically: March 30, 2016
Dedicated: Dedicated to Professor N. N. Ural’tseva on her jubilee
Article copyright: © Copyright 2016 American Mathematical Society