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Image restoration with mixed or unknown noises. (English) Zbl 1380.94021

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
90C25 Convex programming
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[1] F. J. Anscombe, The transformation of Poisson, binomial and negative-binomial data, Biometrika, 35 (1948), pp. 246–254. · Zbl 0032.03702
[2] G. Aubert and J.-F. Aujol, A variational approach to removing multiplicative noise, SIAM J. Appl. Math., 68 (2008), pp. 925–946. · Zbl 1151.68713
[3] L. Bar, N. Kiryati, and N. Sochen, Image deblurring in the presence of impulse noise, Internat. J. Comput. Vis., 70 (2006), pp. 279–298. · Zbl 1119.68520
[4] L. Bar, N. Sochen, and N. Kiryati, Image deblurring in the presence of salt-and-pepper noise, in Scale Space and PDE Methods in Computer Vision, Lecture Notes in Comput. Sci. 3459, Springer-Verlag, Berlin, 2005, pp. 107–118. · Zbl 1119.68520
[5] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), pp. 183–202. · Zbl 1175.94009
[6] J.-F. Cai, B. Dong, S. Osher, and Z. Shen, Image restoration: Total variation, wavelet frames, and beyond, J. Amer. Math. Soc., 25 (2012), pp. 1033–1089. · Zbl 1277.35019
[7] J.-F. Cai, S. Osher, and Z. Shen, Linearized Bregman iterations for frame-based image deblurring, SIAM J. Imaging Sci., 2 (2009), pp. 226–252. · Zbl 1175.94010
[8] J.-F. Cai, S. Osher, and Z. Shen, Split Bregman methods and frame based image restoration, Multiscale Model. Simul., 8 (2009), pp. 337–369. · Zbl 1189.94014
[9] T. Chan, Y. Wang, and H. Zhou, Denoising natural color photos in digital photography, Preprint, 2007.
[10] T. Chen and H. R. Wu, Adaptive impulse detection using center-weighted median filters, IEEE Trans. Signal Process. Lett., 8 (2001), pp. 1–3.
[11] I. Daubechies, B. Han, A. Ron, and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal., 14 (2003), pp. 1–46. · Zbl 1035.42031
[12] B. Dong, H. Ji, J. Li, Z. Shen, and Y. Xu, Wavelet frame based blind image inpainting, Appl. Comput. Harmon. Anal., 32 (2012), pp. 268–279. · Zbl 1261.94006
[13] B. Dong and Z. Shen, MRA-based wavelet frames and applications, in The Summer Program on “The Mathematics of Image Processing,” IAS Lecture Notes Ser. 19, Park City Mathematics Institute, Salt Lake City, UT, 2010.
[14] Y. Dong, R. F. Chan, and S. Xu, A detection statistic for random-valued impulse noise, IEEE Trans. Image Process., 16 (2007), pp. 1112–1120.
[15] D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation, Comput. Math. Appl., 2 (1976), pp. 17–40. · Zbl 0352.65034
[16] R. Glowinski and A. Marrocco, Sur l’approximation par éléments finis d’ordre un, et la resolution par pénalisation-dualité d’une classe de problèmes de Dirichlet non linéaires, Revue Française Automat. Informat. Recherche Opérationnelle RAIRO Analyse Numér., 9 (1975), pp. 41–76.
[17] T. Goldstein and S. Osher, The split Bregman algorithm for L1-regularized problems, SIAM J. Imaging Sci., 2 (2009), pp. 323–343. · Zbl 1177.65088
[18] H. Ji, C. Liu, Z. Shen, and Y. Xu, Robust video denoising using low rank matrix completion, in IEEE Conference on Computer Vision and Pattern Recognition (CVPR), San Francisco, CA, 2010, pp. 1791–1798.
[19] H. Ji, Z. Shen, and Y. Xu, Wavelet frame based image restoration with missing/damaged pixels, East Asia J. Appl. Math., 1 (2011), pp. 108–131. · Zbl 1286.94017
[20] T. Le, R. Chartrand, and T. J. Asaki, A variational approach to reconstructing images corrupted by poisson noise, J. Math. Imaging Vis., 27 (2007), pp. 257–263.
[21] F. Luisier, Image denoising in mixed Poisson-Gaussian noise, IEEE Trans. Image Process., 20 (2011), pp. 696–707. · Zbl 1372.94168
[22] F. Murtagh, J.-L. Starck, and A. Bijaoui, Image restoration with noise suppression using a multiresolution support, Astron. Astrophys. Supplement Series, 112 (1995), pp. 179–189.
[23] A. Nemirovski and D. Yudin, Problem Complexity and Method Efficiency in Optimization, Wiley, New York, 1983.
[24] Y. Nesterov, A method of solving a convex programming problem with convergence rate \({O}(1/k^2)\), Soviet Math. Dokl., 27 (1983), pp. 372–376. · Zbl 0535.90071
[25] Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program., 103 (2005), pp. 127–152. · Zbl 1079.90102
[26] J. Portilla, V. Strela, M. J. Wainwright, and E. P. Simoncelli, Image denoising using scale mixtures of Gaussians in the wavelet domain, IEEE Trans. Image Process., 12 (2003), pp. 1338–1351. · Zbl 1279.94028
[27] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. · Zbl 0193.18401
[28] R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), pp. 97–116. · Zbl 0402.90076
[29] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877–898. · Zbl 0358.90053
[30] A. Ron and Z. Shen, Affine systems in \(l_2(\mathbb{R}^d)\): The analysis of the analysis operator, J. Funct. Anal., 148 (1997), pp. 408–447. · Zbl 0891.42018
[31] S. Setzer, G. Steidl, and T. Teuber, Deblurring Poissonian images by split Bregman techniques, J. Vis. Commun. Image Rep., 21 (2010), pp. 193–199.
[32] Z. Shen, Wavelet frames and image restorations, in Proceedings of the International Congress of Mathematicians, Vol. IV, Rajendra Bhatia, ed., Hindustan Book Agency, Hyderabad, India, 2010, pp. 2834–2863. · Zbl 1228.42036
[33] Z. Shen, K.-C. Toh, and S. Yun, An accelerated proximal gradient algorithm for frame-based image restoration via the balanced approach, SIAM J. Imaging Sci., 4 (2011), pp. 573–596. · Zbl 1219.94012
[34] J. Shi and S. Osher, A nonlinear inverse scale space method for a convex multiplicative noise model, SIAM J. Imaging Sci., 1 (2008), pp. 294–321. · Zbl 1185.94018
[35] G. Steidl and T. Teuber, Removing multiplicative noise by Douglas-Rachford splitting, J. Math. Imaging Vis., 36 (2010), pp. 168–184. · Zbl 1287.94016
[36] S. Yun and H. Woo, A new multiplicative denoising variational model based on m-th root transformation. IEEE Trans. Image Process., 21 (2012), pp. 2523–2533. · Zbl 1373.94476
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