Geodesic
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.

Voir la notice de l'article provenant de la source Numdam

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.

DOI : 10.1016/j.anihpc.2010.06.004
Classification : 49J99, 26B30, 68U10
Keywords: 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://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2010.06.004/}
}
TY  - JOUR
AU  - Fonseca, I.
AU  - Leoni, G.
AU  - Maggi, F.
AU  - Morini, M.
TI  - Exact reconstruction of damaged color images using a total variation model
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 1291
EP  - 1331
VL  - 27
IS  - 5
PB  - Elsevier
UR  - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2010.06.004/
DO  - 10.1016/j.anihpc.2010.06.004
LA  - en
ID  - AIHPC_2010__27_5_1291_0
ER  - 
%0 Journal Article
%A Fonseca, I.
%A Leoni, G.
%A Maggi, F.
%A Morini, M.
%T Exact reconstruction of damaged color images using a total variation model
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 1291-1331
%V 27
%N 5
%I Elsevier
%U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2010.06.004/
%R 10.1016/j.anihpc.2010.06.004
%G en
%F AIHPC_2010__27_5_1291_0
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://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2010.06.004/

[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 | MR | JFM

Cité par Sources :