Abstract
The similar image patches should have similar underlying structures. Thus the matrix constructed from stacking the similar patches together has low rank. Based on this fact, the nuclear norm minimization, which is the convex relaxation of low rank minimization, leads to good denoising results. Recently, the weighted nuclear norm minimization has been applied to image denoising. This approach presents state-of-the-art result for image denoising. In this paper, we further study the weighted nuclear norm minimization problem for general image recovery task. For the weights being in arbitrary order, we prove that such minimization problem has a unique global optimal solution in the closed form. Incorporating this idea with the celebrated total variation regularization, we then investigate the image deblurring problem. Numerical experimental results illustratively clearly that the proposed algorithms achieve competitive performance.
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Aharon, M., Elad, M., Bruckstein, A.M.: The K-SVD: an algorithm for designing of overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process. 54, 4311–4322 (2006)
Breiman, L.: Better subset regression using the nonnegative garrote. Technometrics 37, 373–384 (1995)
Buades, A., Coll, B., Morel, J.M.: A review of image denoising algorithms, with a new one. Multiscale Model. Simul. 4, 490–530 (2005)
Cai, J., Candès, E., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20, 1956–1982 (2010)
Cai, J., Chan, R., Nikolova, M.: Fast two-phase image deblurring under impulse noise. J. Math. Imaging Vis. 36, 46–53 (2010)
Cai, J., Huang, S., Ji, H., Shen, Z., Ye, G.: Data-driven tight frame construction and image denoising. Appl. Comput. Harmon. Anal. 37, 89–105 (2014)
Cai, J., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8, 337–369 (2010)
Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009)
Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004)
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2004)
Chantas, G., Galatsanos, N.P., Molina, R., Katsaggelos, A.K.: Variational Bayesian image restoration with a product of spatially weighted total variation image priors. IEEE Trans. Image Process. 19, 351–362 (2010)
Chan, T., Zhou, H.M.: Total variation wavelet thresholding. J. Sci. Comput. 32, 315–341 (2007)
Chen, K., Dong, H., Chan, K.S.: Reduced rank regression via adaptive nuclear norm penalization. Biometrika 100, 901–920 (2013)
Cho, T.S., Joshi, N., Zitnick, C.L., Kang, S.B., Szeliski, R., Freeman, W.T.: A content-aware image prior. In: Proceedings of CVPR, pp. 169–176 (2010)
Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3D transform-domain collaborative filtering. IEEE Trans. Image Process. 16, 2080–2095 (2007)
Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image restoration by sparse 3D transform domain collaborative filtering. SPIE Electronic Imaging (2008)
Danielyan, A., Katkovnik, V., Egiazarian, K.: BM3D frames and variational image deblurring. IEEE Trans. Image Process. 21, 1715–1728 (2012)
Daubechies, I., Teschkeb, G.: Variational image restoration by means of wavelets: simultaneous decomposition, deblurring, and denoising. Appl. Comput. Harmon. Anal. 19, 1–16 (2005)
Dong, Y., Hintermüler, M., Neri, M.: An efficient primal-dual method for L1-TV image restoration. SIAM J. Imaging Sci. 2, 1168–1189 (2009)
Effros, A., Leung, T.: Texture synthesis by non-parametric sampling. In: Proceedings of 7th IEEE ICCV, pp. 1033–1038 (1999)
Elad, M., Aharon, M.: Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15, 3736–3745 (2006)
Elad, M., Figueiredo, M., Ma, Y.: On the role of sparse and redundant representations in image processing. In: Proceedings of the IEEE Special Issue on Applications of Sparse Representation and Compressive Sensing, vol. 98, pp. 972–982 (2010)
Fazel, M.: Matrix rank minimization with applications. Ph.D. thesis, Stanford University (2002)
Gao, H.: Wavelet shrinkage denoising using the non-negative garrote. J. Comput. Graph. Stat. 7, 469–488 (1998)
Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)
Goldstein, T., Osher, S.: The split Bregman method for L1 regularized problems. SIAM J. Imaging Sci. 2, 323–343 (2009)
Gu, S., Zhang, L., Zuo, W., Feng, X.: Weighted nuclear norm minimization with application to image denoising. In: Proceedings of IEEE CVPR, pp. 2862–2869 (2014)
Hu, Z., Huang, J., Yang, M.: Single image deblurring with adaptive dictionary learning. In: Proceedings of IEEE ICIP, pp. 1169–1172 (2010)
Ji, H., Liu, C., Shen, Z., Xu, Y.: Robust video denoising using low rank matrix completion. In: Proceedings of IEEE CVPR, pp. 1791–1798 (2010)
Liu, G., Lin, Z., Yu, Y.: Robust subspace segmentation by low-rank representation. In: Proceedings of ICML, pp. 663–670 (2010)
Lou, Y., Bertozzi, A., Soatto, S.: Direct sparse deblurring. J. Math. Imaging Vis. 39, 1–12 (2011)
Lou, Y., Zhang, X., Osher, S., Bertozzi, A.: Image recovery via nonlocal operators. J. Sci. Comput. 42, 185–197 (2010)
Lu, C., Tang, J., Yan, S., Lin, Z.: Generalized nonconvex nonsmooth low-rank minimization. In: Proceedings of IEEE CVPR, pp. 4130–4137 (2014)
Lu, C., Zhu, C., Xu, C., Yan, S., Lin, Z.: Generalized singular value thresholding. In: Proceedings of AAAI (2015)
Ma, L., Ng, M., Yu, J., Zeng, T.: Efficient box-constrained TV-type-\(l^1\) algorithms for restoring images with impulse noise. J. Comput. Math. 31, 249–270 (2013)
Ma, L., Moisan, L., Yu, J., Zeng, T.: A dictionary learning approach for Poisson image deblurring. IEEE Trans. Med. Imaging 32, 1277–1289 (2013)
Ma, L., Yu, J., Zeng, T.: Sparse representation prior and total variation-based image deblurring under impulse noise. SIAM J. Imaging Sci. 6, 2258–2284 (2013)
Maleki, A., Narayan, M., Baraniuk, R.G.: Anisotropic nonlocal means denoising. Appl. Comput. Harmon. Anal. 35, 452–482 (2013)
Neelamani, R., Choi, H., Baraniuk, R.: ForWaRD: Fourier wavelet regularized deconvolution for ill-conditioned systems. IEEE Trans. Image Process. 52, 418–433 (2004)
Olshausen, B., Field, D.: Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381, 607–609 (1996)
Quan, Y., Ji, H., Shen, Z.: Data-driven multi-scale non-local wavelet frame construction and image recovery. J. Sci. Comput. (2014). doi:10.1007/s10915-014-9893-2
Ron, A., Shen, Z.: Affine system in L2(Rd): the analysis of the analysis operator. J. Funct. Anal. 148, 408–447 (1997)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1992)
Wen, Y., Ng, M., Huang, Y.: Efficient total variation minimization methods for color image restoration. IEEE Trans. Image Process. 17, 2081–2088 (2008)
Wu, C., Tai, X.-C.: Augmented Lagrangian method, dual Methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imaging Sci. 3, 300–339 (2010)
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)
Yan, M., Yang, Y., Osher, S.: Exact low-rank matrix completion from sparsely corrupted entries via adaptive outlier pursuit. J. Sci. Comput. 56, 433–449 (2013)
Zeng, T., Malgouyres, F.: Matching pursuit shrinkage in Hilbert spaces. Sig. Process. 91, 2754–2766 (2011)
Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM J. Imaging Sci. 3, 253–276 (2010)
Zhang, J.P., Chen, K., Yu, B.: An iterative Lagrange multiplier method for constrained total variation-based image denoising. SIAM J. Numer. Anal. 50, 983–1003 (2012)
Zoran, D., Weiss, Y.: From learning models of natural image patches to whole image restoration. In: Proceedings of ICCV, pp. 479–486 (2011)
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The research is supported by the National Natural Science Foundation of China (Nos. 61402462, 11271049, 11201455, 11671383, 61503202), RGC 211911, 12302714, and RFGs of HKBU.
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Ma, L., Xu, L. & Zeng, T. Low Rank Prior and Total Variation Regularization for Image Deblurring. J Sci Comput 70, 1336–1357 (2017). https://doi.org/10.1007/s10915-016-0282-x
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DOI: https://doi.org/10.1007/s10915-016-0282-x