Exact reconstruction of damaged color images using a total variation model

Fonseca, I.; Leoni, G.; Maggi, F.; Morini, M.

doi:10.1016/j.anihpc.2010.06.004

Fonseca, I. ; Leoni, G. ; Maggi, F. ; Morini, M.

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1291-1331.

Résumé

In this paper the reconstruction of damaged piecewice constant color images is studied using an RGB total variation based model for colorization/inpainting. In particular, it is shown that when color is known in a uniformly distributed region, then reconstruction is possible with maximal fidelity.

MR Zbl

DOI : 10.1016/j.anihpc.2010.06.004

Classification : 49J99, 26B30, 68U10
Mots clés : Energy minimization, Calibrations, RGB total variation models, Colorization, Inpainting, Image restoration

@article{AIHPC_2010__27_5_1291_0,
     author = {Fonseca, I. and Leoni, G. and Maggi, F. and Morini, M.},
     title = {Exact reconstruction of damaged color images using a total variation model},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1291--1331},
     publisher = {Elsevier},
     volume = {27},
     number = {5},
     year = {2010},
     doi = {10.1016/j.anihpc.2010.06.004},
     mrnumber = {2683761},
     zbl = {1198.49045},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.06.004/}
}

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Fonseca, I.; Leoni, G.; Maggi, F.; Morini, M. Exact reconstruction of damaged color images using a total variation model. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 5, pp. 1291-1331. doi : 10.1016/j.anihpc.2010.06.004. http://www.numdam.org/articles/10.1016/j.anihpc.2010.06.004/

Bibliographie
Cité par

[1] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York (2000) | MR | Zbl

[2] G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. 135 (1983), 293-318 | MR | Zbl

[3] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat. 6 (1967), 551-556 | MR | Zbl

[4] G. Aronsson, M.G. Crandall, P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. 41 (2004), 439-505 | MR | Zbl

[5] R.G. Bartle, The Elements of Real Analysis, John Wiley & Sons, New York, London, Sydney (1976) | MR | Zbl

[6] G. Bellettini, V. Caselles, M. Novaga, The total variation flow in $ℝ^{N}$ , J. Differ. Equations 184 (2002), 475-525 | MR | Zbl

[7] J. Buriánek, D. Sýkora, J. Žára, Unsupervised colorization of black-and-white cartoons, in: Proc. 3rd Int. Symp. Non-Photorealistic Animation and Rendering, 2004, pp. 121–127.

[8] T.F. Chan, J. Shen, Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math. 62 (2001/2002), 1019-1043 | MR | Zbl

[9] T.F. Chan, J. Shen, Variational image inpainting, Comm. Pure Appl. Math. 58 (2005), 579-619 | MR | Zbl

[10] D. Cohen-Or, R. Irony, D. Lischinski, Colorization by example, in: Proc. Eurograph. Symp. Rendering, 2005, pp. 201–210.

[11] F. Demengel, Functions locally almost 1-harmonic, J. Appl. Anal. 83 (2004), 865-896 | MR | Zbl

[12] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics vol. 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999) | MR | Zbl

[13] I. Fonseca, G. Leoni, Modern Methods in the Calculus of Variations: $L^{p}$ Spaces, Springer Monographs in Mathematics, Springer, New York (2007) | MR | Zbl

[14] M. Fornasier, Nonlinear projection recovery in digital inpainting for color image restoration, J. Math. Imaging Vision 24 (2006), 359-373 | MR

[15] M. Fornasier, Faithful recovery of vector valued functions from incomplete data, Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science, Springer, Berlin, Heidelberg (2009), 116-127

[16] M. Fornasier, R. March, Restoration of color images by vector valued BV functions and variational calculus, SIAM J. Appl. Math. 68 (2007), 437-460 | MR | Zbl

[17] S.H. Kang, R. March, Variational models for image colorization via chromaticity and brightness decomposition, IEEE Trans. Image Process. 16 (2007), 2251-2261 | MR

[18] B. Kawohl, F. Schuricht, Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem, Commun. Contemp. Math. 9 (2007), 515-543 | MR | Zbl

[19] M.D. Kirszbraun, Ber die zusammenziehende und Lipschitzsche Transformationen, Fund. Math. 22 (1934), 77-108 | EuDML | JFM

[20] A. Levin, D. Lischinsk, Y. Weiss, Colorization using optimization, Proc. SIGGRAPH Conf. vol. 23 (2004), 689-694

[21] E.J. Mcshane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842 | MR | Zbl

[22] L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992), 259-268 | MR | Zbl

[23] G. Sapiro, Inpainting the colors, Proc. IEEE Int. Conf. Image Processing vol. 2 (2005), 698-701

[24] G. Sapiro, L. Yatziv, Fast image and video colorization using chrominance blending, IEEE Trans. Image Process. 15 (2006), 1120-1129

[25] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89 | JFM | MR

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