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Weighted nuclear norm minimization-based regularization method for image restoration. (English) Zbl 1499.68375

Summary: Regularization methods have been substantially applied in image restoration due to the ill-posedness of the image restoration problem. Different assumptions or priors on images are applied in the construction of image regularization methods. In recent years, matrix low-rank approximation has been successfully introduced in the image denoising problem and significant denoising effects have been achieved. Low-rank matrix minimization is an NP-hard problem and it is often replaced with the matrix’s weighted nuclear norm minimization (WNNM). The assumption that an image contains an extensive amount of self-similarity is the basis for the construction of the matrix low-rank approximation-based image denoising method. In this paper, we develop a model for image restoration using the sum of block matching matrices’ weighted nuclear norm to be the regularization term in the cost function. An alternating iterative algorithm is designed to solve the proposed model and the convergence analyses of the algorithm are also presented. Numerical experiments show that the proposed method can recover the images much better than the existing regularization methods in terms of both recovered quantities and visual qualities.

MSC:

68U10 Computing methodologies for image processing

Software:

RASL; RecPF; FTVd
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Full Text: DOI

References:

[1] Afonso, MV; Bioucas-Dias, JM; Figueiredo, MAT, Fast image recovery using variable splitting and constrained optimization, IEEE Trans. Image Process., 19, 9, 2345-2356 (2010) · Zbl 1371.94018 · doi:10.1109/TIP.2010.2047910
[2] Beck, A.; Teboulle, M., Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Trans. Image Process., 18, 11, 2419-2434 (2009) · Zbl 1371.94049 · doi:10.1109/TIP.2009.2028250
[3] Buades, A., Coll, B., Morel, J.M.: A non-local algorithm for image denoising. In: IEEE International Conference on Computer Vision and Pattern Recognition, pp. 60-65 (2005) · Zbl 1108.94004
[4] Buades, A.; Coll, B.; Morel, JM, A review of image denoising algorithms, with a new one, SIAM Multiscale Model. Simul., 4, 2, 490-530 (2005) · Zbl 1108.94004 · doi:10.1137/040616024
[5] Buades, A., Coll, B., Morel, J.M.: Image denoising by non-local averaging. In: IEEE International Conference on Acoustics, Speech, and Signal Processing 2, pp. 25-28 (2005)
[6] Chambolle, A.; Pock, T., A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40, 1, 120-145 (2011) · Zbl 1255.68217 · doi:10.1007/s10851-010-0251-1
[7] Chambolle, A., An algorithm for total variation minimization and application, J. Math. Imaging Vision, 20, 1-2, 89-97 (2004) · Zbl 1366.94048 · doi:10.1023/B:JMIV.0000011320.81911.38
[8] Chambolle, A.; Lions, P., Image recovery via total variation minimization and related problems, J. Numer. Math., 76, 2, 167-188 (1997) · Zbl 0874.68299 · doi:10.1007/s002110050258
[9] Cai, J.; Osher, S.; Shen, Z., Split Bregman methods and frame based image restoration, SIAM Multiscale Model. Simul., 8, 2, 337-369 (2009) · Zbl 1189.94014 · doi:10.1137/090753504
[10] Combettes, P., Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53, 5-6, 475-504 (2004) · Zbl 1153.47305 · doi:10.1080/02331930412331327157
[11] Dabov, K.; Foi, A.; Katkovnik, V.; Egiazarian, K., Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Trans. Image Process., 16, 8, 2080-2095 (2007) · doi:10.1109/TIP.2007.901238
[12] Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image restoration by sparse 3-D transform-domain collaborative filtering. In: Proc. SPIE Electron. Imaging 6812 (2008)
[13] Dong, W., Zhang, L., Shi, G.: Centralized sparse representation for image restoration. In: IEEE International Conference on Computer Vision, pp. 1259-1266 (2011)
[14] Dong, W., Li, X., Zhang, D., Shi, G.: Sparsity-based image denoising via dictionary learning and structural clustering. In: IEEE International Conference on Computer Vision and Pattern Recognition, pp. 457-464 (2011)
[15] Dong, W.; Shi, G.; Li, X.; Ma, Y.; Huang, F., Compressive sensing via nonlocal low-rank regularization, IEEE Trans. Image Process., 23, 8, 3618-3632 (2014) · Zbl 1374.94085 · doi:10.1109/TIP.2014.2329449
[16] Dong, W.; Zhang, L.; Lukac, R.; Shi, G., Nonlocal centralized sparse representation for image restoration, IEEE Trans. Image Process., 22, 4, 1620-1630 (2013) · Zbl 1373.94104 · doi:10.1109/TIP.2012.2235847
[17] Dong, W., Shi, G., Li, X.: Image deblurring with low-rank approximation structured sparse representation. In: IEEE APSIPA ASC (2012)
[18] Elad, M.; Aharon, M., Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15, 12, 3736-3745 (2006) · doi:10.1109/TIP.2006.881969
[19] Galatsanos, N.; Katsaggelos, A., Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation, IEEE Trans. Image Process., 1, 3, 322-336 (1992) · doi:10.1109/83.148606
[20] Ge, Q.; Jing, X.; Wu, F.; Wei, Z.; Xiao, L.; Shao, W.; Yue, D.; Li, H., Structure-based low-rank model with graph nuclear norm regularization for noise removal, IEEE Trans. Image Process., 26, 7, 3098-3112 (2017) · Zbl 1409.94178 · doi:10.1109/TIP.2016.2639781
[21] Gu, S., Zhang, L., Zuo, W., Feng, X.: Weighted nuclear norm minimization with application to image denoising. In: IEEE International Conference on Computer Vision and Pattern Recognition, pp. 2862-2867 (2014)
[22] Gu, S.; Xie, Q.; Meng, D.; Zuo, W.; Feng, X.; Zhang, L., Weighted nuclear norm minimization and its applications to low level vision, International Journal of Computer Vision, 121, 2, 183-208 (2017) · Zbl 1458.68231 · doi:10.1007/s11263-016-0930-5
[23] Ji, H., Liu, C., Shen, Z., Xu, Y.: Robust video denoising using low rank matrix completion. In: IEEE International Conference on Computer Vision and Pattern Recognition, pp. 1791-1798 (2010)
[24] Liu, X.; Tanaka, M.; Okutomi, M., Single-image noise level estimation for blind denoising, IEEE Trans. Image Process., 22, 12, 5226-5237 (2013) · doi:10.1109/TIP.2013.2283400
[25] Ma, L.; Yu, J.; Zeng, T., A dictionary learning approach for Poisson image deblurring, IEEE Trans. Medical Imaging, 32, 7, 1277-1289 (2013) · doi:10.1109/TMI.2013.2255883
[26] Ma, L.; Xu, L.; Zeng, T., Low rank prior and total variation regularization for image deblurring, Journal of Scientific Computing, 70, 3, 1336-1357 (2017) · Zbl 1366.65041 · doi:10.1007/s10915-016-0282-x
[27] Ma, S.; Goldfarb, D.; Chen, L., Fixed point and Bregman iterative methods for matrix rank minimization, Math. Program. A, 128, 1, 321-353 (2011) · Zbl 1221.65146 · doi:10.1007/s10107-009-0306-5
[28] Mairal, J., Bach, F., Ponce, J., Sapiro, G., Zisserman, A.: Non-local sparse models for image restoration. In: IEEE International Conference on Computer Vision, pp. 2272-2279 (2009)
[29] Mairal, J.; Elad, M.; Sapiro, G., Sparse representation for color image restoration, IEEE Trans. Image Process., 17, 1, 53-69 (2008) · Zbl 1194.49041 · doi:10.1109/TIP.2007.911828
[30] Ng, MK; Chan, RH; Tang, WC, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21, 3, 851-866 (1999) · Zbl 0951.65038 · doi:10.1137/S1064827598341384
[31] Ng, MK; Weiss, P.; Yuan, X., Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods, SIAM J. Sci. Comput., 32, 5, 2710-2736 (2010) · Zbl 1217.65071 · doi:10.1137/090774823
[32] Paragios, N.; Chen, C.; Faugeras, O., Handbook of Mathematical Models in Computer Vision (2006), New York: Springer, New York · Zbl 1083.68500 · doi:10.1007/0-387-28831-7
[33] Peng, Y.; Ganesh, A.; Wright, J.; Xu, W.; Ma, Y., RASL: Robust alignment by sparse and low-rank decomposition for linearly correlated images, IEEE Trans. Pattern Anal. Mach. Intell., 34, 11, 2233-2246 (2012) · doi:10.1109/TPAMI.2011.282
[34] Rajwade, A.; Rangarajan, A.; Banerjee, A., Image denoising using the higher order singular value decomposition, IEEE Trans. Pattern Anal. Mach. Intell., 35, 4, 849-862 (2013) · doi:10.1109/TPAMI.2012.140
[35] Rudin, L.; Osher, S., Total variation based image restoration with free local constraints, Proc. IEEE ICIP, 1, 31-35 (1994)
[36] Rudin, L.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D, 60, 1-2, 259-268 (1992) · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[37] Teuber, T.; Lang, A., Nonlocal filters for removing multiplicative noise, Scale Space and Variational Methods in Computer Vision, 6667, 50-61 (2012) · doi:10.1007/978-3-642-24785-9_5
[38] Teuber, T.; Lang, A., A new similarity measure for nonlocal filtering in the presence of multiplicative noise, Comput. Statist. Data Anal., 56, 12, 3821-3842 (2012) · Zbl 1254.94010 · doi:10.1016/j.csda.2012.05.009
[39] Tikhonov, A., Solution of incorrectly formulated problems and the regularization method, Soviet Math. Dokl., 4, 1035-1038 (1963) · Zbl 0141.11001
[40] Tikhonov, A.; Goncharsky, A., Ill-Posed Problem in Natural Sciences (1987), Moscow: “Mir Publishers”, Moscow
[41] Wang, Y.; Yang, J.; Yin, W.; Zhang, Y., A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imaging Sci., 1, 3, 248-272 (2008) · Zbl 1187.68665 · doi:10.1137/080724265
[42] Wang, Z.; Bovik, AC; Sheikh, HR; Simoncelli, EP, Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process., 13, 4, 600-612 (2004) · doi:10.1109/TIP.2003.819861
[43] Xie, Y.; Gu, S.; Liu, Y.; Zuo, W.; Zhang, W.; Zhang, L., Weighted schatten \(p\)-norm minimization for image denoising and background subtraction, IEEE Trans. Image Process., 25, 10, 4842-4857 (2016) · Zbl 1408.94731
[44] Xie, Q., Meng, D., Gu, S., Zhang, L., Zuo, W., Feng, X., Xu, Z.: On the optimal solution of weighted nuclear norm minimization. Technical Report (2014)
[45] Yang, J.; Yin, W.; Zhang, Y.; Wang, Y., A fast algorithm for edge-preserving variational multichannel image restoration, SIAM J. Imaging Sci., 2, 2, 569-592 (2009) · Zbl 1181.68304 · doi:10.1137/080730421
[46] Yang, J.; Zhang, Y.; Yin, W., An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31, 4, 2842-2865 (2009) · Zbl 1195.68110 · doi:10.1137/080732894
[47] Zhang, C.; Hu, W.; Jin, T.; Mei, Z., Nonlocal image denoising via adaptive tensor nuclear norm minimization, Neural Comput Appl., 29, 1, 3-19 (2018) · doi:10.1007/s00521-015-2050-5
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