Abstract
Nonlocal self-similarity shows great potential in image denoising. Therefore, the denoising performance can be attained by accurately exploiting the nonlocal prior. In this paper, we model nonlocal similar patches through the multi-linear approach and then propose two tensor-based methods for image denoising. Our methods are based on the study of low-rank tensor estimation (LRTE). By exploiting low-rank prior in the tensor presentation of similar patches, we devise two new adaptive tensor nuclear norms (i.e., ATNN-1 and ATNN-2) for the LRTE problem. Among them, ATNN-1 relaxes the general tensor N-rank in a weighting scheme, while ATNN-2 is defined based on a novel tensor singular-value decomposition (t-SVD). Both ATNN-1 and ATNN-2 construct the stronger spatial relationship between patches than the matrix nuclear norm. Regularized by ATNN-1 and ATNN-2 respectively, the derived two LRTE algorithms are implemented through the adaptive singular-value thresholding with global optimal guarantee. Then, we embed the two algorithms into a residual-based iterative framework to perform nonlocal image denoising. Experiments validate the rationality of our tensor low-rank assumption, and the denoising results demonstrate that our proposed two methods are exceeding the state-of-the-art methods, both visually and quantitatively.
Similar content being viewed by others
References
Milanfar P (2013) A tour of modern image filtering: new insights and methods, both practical and theoretical. IEEE Signal Process Mag 30(1):106–128
Budak C, Tiirk M, Toprak A (2015) Reduction in impulse noise in digital images through a new adaptive artificial neural network model. Neural Comput Appl 26:835–843
Chatterjee P, Milanfar P (2010) Is denoising dead? IEEE Trans Image Process 19(4):895–911
Buades A, Coll B, Morel J (2005) A non-local algorithm for image denoising. CVPR 2:60–65
Shao L, Yan R, Li X, Liu Y (2014) From heuristic optimization to dictionary learning: a review and comprehensive comparison of image denoising algorithms. IEEE Trans Cybern 44(7):1001–1013
Dabov K, Foi A, Katkovnik V, Egiazarian K (2007) Image denoising by sparse 3D transform-domain collaborative filtering. IEEE Trans Image Process 16(8):2080–2095
Mairal J, Bach F, Ponce J et al (2009) Non-local sparse models for image restoration. In: 2009 IEEE 12th international conference on computer vision, Kyoto, 29 September–2 October 2009, pp 2272–2279. doi:10.1109/ICCV.2009.5459452
Dong W, Zhang L, Shi G, Li X (2013) Nonlocally centralized sparse representation for image restoration. IEEE Trans Image Process 22(4):1620–1630
Wang S, Zhang L, Liang Y (2012) Nonlocal spectral prior model for low-level vision. In: Lee KM, Matsushita Y, Rehg JM, Hu Z (eds) Computer vision – ACCV 2012, vol 7726, pp 231–244
Dong W, Shi G, Li X (2013) Nonlocal image restoration with bilateral variance estimation: a low-rank approach. IEEE Trans Image Process 22(2):700–711
Andrews H, Patterson C (1976) Singular value decompositions and digital image processing. IEEE Trans Acoust Speech Signal Process 24(1):26–53
Chang S, Yu B, Vetterli M (2000) Adaptive wavelet thresholding for image denoising and compression. IEEE Trans Image Process 9(9):1532–1546
Candes E, Recht B (2009) Exact matrix completion via convex optimization. Found Comput Math 9(6):717–772
Cai J, Candes E, Shen Z (2010) A singular value thresholding algorithm for matrix completion. SIAM J Optim 20(4):1956–1982
Chen K, Dong H, Chan K (2013) Reduced rank regression via adaptive nuclear norm penalization. Biometrika 100(4):901–920
Gu S, Zhang L, Zuo W, Feng X (2014) Weighted nuclear norm minimization with application to image denoising. In: 2014 IEEE conference on computer vision and pattern recognition (CVPR), Columbus, 23–28 June 2014, pp 2862–2869. doi:10.1109/CVPR.2014.366
Nguyen T, Park J, Kim S, Lee G (2011) Automatically improving image quality using tensor voting. Neural Comput Appl 20:1017–1026
Zhang M, Ding C (2013) Robust tucker tensor decomposition for effective image representation. In: 2013 IEEE international conference on computer vision, Sydney, 1–8 December 2013, pp 2448–2455. doi:10.1109/ICCV.2013.304
Rajwade A, Rangarajan A, Banerjee A (2013) Image denoising using the higher order singular value decomposition. IEEE Trans Pattern Anal Mach Intell 35(4):849–862
Lathauwer L, Moor B, Vandewalle J (2000) A multilinear singular value decomposition. SIAM J Matrix Anal Appl 21(4):1253–1278
Kolda T, Bader B (2009) Tensor decompositions and applications. SIAM Rev 51(3):455–500
Gandy S, Recht B, Yamada I (2011) Tensor completion and low-n-rank tensor recovery via convex optimization. Inverse Probl 27(2):025010
Chen Y, Hsu C, Liao H (2014) Simultaneous tensor decomposition and completion using factor priors. IEEE Trans Pattern Anal Mach Intell 36(3):577–591
Chen J, Saad Y (2009) On the tensor SVD and the optimal low rank orthogonal approximation of tensors. SIAM J Matrix Anal Appl 30(4):1709–1734
Acar E, Dunlavy D, Kolda T, Morup M (2011) Scalable tensor factorization for incomplete data. Chemometrics Intell Lab Syst 106(1):41–56
Liu J, Musialski P, Wonka P, Ye J (2013) Tensor completion for estimating missing values in visual data. IEEE Trans Pattern Anal Mach Intell 35(1):208–220
Tomioka R, Suzuki T (2013) Convex tensor decomposition via structured schatten norm regularization. In: Advances in neural information processing systems 26, pp 1331–1339
Signoretto M, Dinh Q, Lathauwer L, Suykens J (2014) Learning with tensors: a framework based on convex optimization and spectral regularization. Mach Learn 94:303–351
Fazel M (2002) Matrix rank minimization with applications. Ph.D. dissertation, Stanford University
Semerci O, Hao Ning, Kilmer M, Milller E (2014) Tensor-based factorization and nuclear norm regularization for multienergy computed tomography. IEEE Trans Image Process 23(4):1678–1693
Kilmer M, Braman K, Hao N, Hoover R (2013) Third-order tensors as operators on matrices: a theoretical and computational framework with applications in imaging. SIAM J Matrix Anal Appl 34(1):148–172
Zhang Z, Ely G, Aeron S et al (2014) Novel methods for multilinear data completion and de-noising based on tensor-SVD. In: 2014 IEEE conference on computer vision and pattern recognition (CVPR), pp 3842–3849. doi:10.1109/CVPR.2014.485
Georges F, Nadakuditi R (2012) The singular values and vectors of low rank perturbations of large rectangular random matrices. J Multivar Anal 111:120–135
Stewart G (1990) Perturbation theory for the singular value decomposition. Technical Report UMIACS-TR-90-124
Golub G, Hoffman A, Stewart G (1987) A generalization of the Eckart-Young-Mirsky matrix approximation theorem. Linear Algebra Appl 88:317–327
Nadakuditi R (2014) OptShrink: an algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage. IEEE Trans Inf Theory 60(5):3002–3018
Boyd S, Parikh N, Chu E et al (2010) Distributed optimization and statistical learning via the alternating direction method and multipliers. Found Trends Mach Learn 3(1):1–122
Lin Z, Chen M, Ma Y (2009) The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. Technical report UILU-ENG-09-2215, UIUC (arXiv:1009.5055)
Brand M (2006) Fast low-rank modifications of the thin singular value decomposition. Linear Algebra Appl 415(1):20–30
Osher S, Burger M, Goldfarb D et al (2005) An iterative regularization method for total variation-based image restoration. Multiscale Model Simul 4(2):460–489
Charest M, Milanfar P (2008) On iterative regularization and its application. IEEE Trans Circuits Syst Video Technol 18(3):406–411
Milanfar P (2014) Symmetrizing smoothing filters. SIAM J Imaging Sci 6(1):263–284
Wang Z, Bovik A, Sheikh H, Simoncelli E (2004) Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process 13(4):600–612
Donoho D, Johnstone I (1993) Ideal spatial adaptation by wavelet shrinkage. Biometrika 81(3):425–455
Recht B (2011) A simpler approach to matrix completion. J Mach Learn Res 12:3413–3430
Nie F, Huang H, Ding C (2012) Low-rank matrix recovery via efficient Schatten p-norm minimization. In: Proceedings of the twenty-sixth (AAAI) conference on artificial intelligence, Toronto, 22–26 July 2012, pp 655–661
Hu Y, Zhang D, Ye J et al (2013) Fast and accurate matrix completion via truncated nuclear norm regularization. IEEE Trans Pattern Anal Mach Intell 35(9):2117–2130
Lu C, Tang J, Yan S, Lin Z (2014) Generalized nonconvex nonsmooth low-rank minimization. In: 2014 IEEE conference on computer vision and pattern recognition (CVPR), Columbus, 23–28 June 2014, pp 4130–4137. doi:10.1109/CVPR.2014.526
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, C., Hu, W., Jin, T. et al. Nonlocal image denoising via adaptive tensor nuclear norm minimization. Neural Comput & Applic 29, 3–19 (2018). https://doi.org/10.1007/s00521-015-2050-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00521-015-2050-5