The classical TV (Total Variation) model has been applied to gray texture image denoising and inpainting previously based on the non local operators, but such model can not be directly used to color texture image inpainting due to coupling of different image layers in color images. In order to solve the inpainting problem for color texture images effectively, we propose a non local CTV (Color Total Variation) model. Technically, the proposed model is an extension of local TV model for gray images but we take account of the coupling of different layers in color images and make use of concepts of the non-local operators. As the coupling of different layers for color images in the proposed model will in-crease computational complexity, we also design a fast Split Bregman algorithm. Finally, some numerical experiments are conducted to validate the performance of the proposed model and its algorithm.
G. Aubert. and P. Kornprobst, “Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations 2nd ed. Berlin, Germany: Springer-Verlag, 2006.
M. Bertalmío, G. Sapiro, V. Caselles and C. Ballester, “Image Inpainting,” In Proceedings of SIGGRAPH, 2000, pp. 417-424. doi:10.1145/344779.344972
M. Bertalmio, A. Bertozzi and G. Sapiro, “Navier-Stokes, Fluid-Dynamics, and Image and Video Inpainting,” In Proc. of IEEE-CVPR, 2001, pp. 355-362.
T. Chan and J. Shen, “Mathematical Models of Local Non-texture Inpaintings,” SIAM Journal of Applied Mathematics, Vol. 62, No. 3, 2002, pp. 1019-1043. doi:10.1137/S0036139900368844
L. Rudin, S. Osher and E. Fatemi, “Nonlinear Total Variation Based Noise Removal Algorithms,” Physica D:Nonlinear Phenomena, Vol. 60, No. 1/4, 1992, pp. 259-268. doi:10.1016/0167-2789(92)90242-F
A. Bertozzi, S. Esedoglu and A. Gillette, “Inpainting of Binary Images Using the Cahn-Hilliard Equation,” IEEE Trans. Image Processing, Vol. 16, No. 1, 2007, pp. 285-291. doi:10.1109/TIP.2006.887728
M. Nitzberg, D. Mumford and T. Shiota, Filering, segmentation, and depth, LNCS, Vol. 662, Berlin, Germany: Springer-Verlag, 1993. http://dx.doi.org/10.1007/3-540-56484-5
T. Chan and J. Shen, “Non-texture Inpainting by Curvature-driven Diffusions (CDD),” Journal of Visual Communication and Image Representation., Vol. 12, No. 4, 2001, pp. 436-449. doi:10.1006/jvci.2001.0487
T. Chan, S. Kang and J. Shen, “Euler's Elastica and Curvature-based Image Inpainting,” SIAM J. Appl. Math., Vol. 63, No. 2, 2002, pp. 564-592.
S. Esedoglu and J. Shen, “Digital Inpainting Based on the Mumford-Shah-Euler Image Model,” European Journal of Applied Mathematics, Vol. 13, No. 4, 2002, pp. 353-370. doi:10.1017/S0956792502004904
X. Tai, J. Hahn and G. Chung, “A Fast Algorithm for Euler’s Elastica Model Using Augmented Lagrangian Method,” SIAM J. Appl. Math., Vol. 4, No. 1, 2011, pp. 313-344.
A. Efros and T. Leung, “Texture Synthesis by NonparaMetric Sampling,” In Proc. of the IEEE ICCV, 1999, pp. 1033-1038.
A. Criminisi, P. Pérez, and K. Toyama, “Region Filling and Object Removal by Exemplar-based Inpainting,” IEEE Transactions on Image Processing, Vol. 13, No. 9, 2004, pp. 1200-1212. doi:10.1109/TIP.2004.833105
Z. Xu and S. Jian, “Image Inpainting by Patch Propagation Using Patch Sparsity,” IEEE Transactions on Image Processing, Vol. 19, No. 3, 2010, pp. 1153-1165.
P. Arias, V. Caselles, G. Facciolo and G. Sapiro, “A Variational Framework for Exemplar-based Image Inpainting,” International Journal of Computer Vision, Vol. 93, No. 3, 2011, pp. 1-29. doi:10.1007/s11263-010-0418-7
P. Arias, V. Caselles and G. Facciolo, “Analysis of a Variational Framework for Exemplar-Based Image Inpainting,” SIAM Multiscale Modeing Simulation, Vol. 10, No. 2, 2012, pp. 473-514. doi:10.1137/110848281
M. Bertalmio, L. Vese, G. Sapiro and S. Osher, “Simultaneous Structure and Texture Image Inpainting,” IEEE Transactions Image Processing, Vol. 12, No. 8, 2003, pp. 882-889. doi:10.1109/TIP.2003.815261
J. Cai, R. Chan and Z. Shen, “A Framelet-based Image Inpainting Algorithm,” Applied and Computational Harmonic Analysis, Vol. 24, No. 2, 2008, pp. 131-149. doi:10.1016/j.acha.2007.10.002
J. Cai, R. Chan and Z. Shen, “Simultaneous Cartoon and Texture Inpainting,” Inverse Problems and Imaging, Vol. 4, No. 3, 2010, pp. 379-395. doi:10.3934/ipi.2010.4.379
A. Buades, B. Coll and J. Morel, “A Review of Image Denoising Algorithms, with a New One,” SIAM Multiscale Modeling Simulation, Vol. 4, No. 2, 2005, pp. 490-530. doi:10.1137/040616024
G. Gilboa and S. Osher, “Nonlocal Operators with Applications to Image Processing,” SIAM Multiscale Modeling Simulation, Vol. 7, No. 3, 2008, pp. 1005-1028. doi:10.1137/070698592
G. Peyré, S. Bougleux and L. Cohen, “Non-local Regularization of Inverse Problems,” LNCS, Vol. 5304, 2008, pp. 57-68.
Y. Lou, X. Zhang, S. Osher and A. Bertozzi, “Image recovery via non local operators,” Journal of Science Computer, Vol. 42, No. 2, 2010, pp. 185-197.
X. Zhang, M. Burger, X. Bresson and S. Osher, “Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction,” SIAM Journal Imag. Science, Vol. 3, No. 3, 2010, pp. 253-276.
W. Ma and S. Osher, “A TV Bregman iterative model of Retinex theory,” Univ. California, Los Angeles, UCLA CAM Rep. 10-13, 2010.
D. Zosso, G. Tran and S. Osher, “A Unifying Retinex Model Based on Non-local Differential Operators,” Univ. California, Los Angeles, UCLA CAM Rep. 13-03, 2013.
M. Jung, X. Bresson, T. F. Chan and L. A. Vese. “Nonlocal Mumford-Shah Regularizers for Color Image Restoration,” IEEE Transactions on Image Processing, Vol. 20, No. 6, 2011, pp. 1583-1598. doi:10.1109/TIP.2010.2092433
D. Mumford and J. Shah, “Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems,” CPAM, Vol. XLII, 1989, pp. 577-685.
P. Blomgren and T. Chan, “Color TV: Total Variation Methods for Restoration of Vector-Valued Images,” IEEE Transactions on Image Processing, Vol. 7, No. 3, 1998, pp. 304-309. doi:10.1109/83.661180
J. Yang, W. Yin, Y. Zhang and Y. Wang, “A Fast Algorithm for Edge-Preserving Variational Multichannel Image Restoration,” SIAM Journal on Imaging Sciences, Vol. 2, No. 2, 2009, pp. 569-592. doi:10.1137/080730421
Y. Yu, Z. Pan, W. Wei and J. Jiang, “Edge Preserving of Some Variational Models for Vectorial Image Denoising,” Jounal of Graphics and Images, Vol. 16, No. 12, 2011, pp. 2223-2230.
T. Goldstein and S. Osher, “The Split Bregman Algorithm for L1 Regularized Problems,” SIAM Journal on Imaging Sciences, Vol. 2, No. 2, 2009, pp. 323-343. doi:10.1137/080725891
Q. Wang, Z. Pan, W. Wei, Z. Zhang, and C. Wang , “The Generalized MTV-L1 and its Split Bregman Algorithm for Noise Removal of Color Images with Textures,” in The Chn. Conf. Pattern Recognition (CCPR). 2010, pp. 1-6.