Multimedia Tools and Applications

, Volume 77, Issue 17, pp 22841–22855 | Cite as

Image Recovery based on Local and Nonlocal Regularizations

  • Jun Zhu
  • Changwei ChenEmail author


Recently, a nonlocal low-rank regularization based compressive sensing approach (NLR) which exploits structured sparsity of similar patches has shown the state-of-the-art performance in image recovery. However, NLR cannot efficiently preserve local structures because it ignores the relationship between pixels. In addition, the surrogate logdet function used in NLR cannot well approximate the rank. In this paper, a novel approach based on local and nonlocal regularizations toward exploiting the sparse-gradient property and nonlocal low-rank property (SGLR) has been proposed. Weighted schatten-p norm and lq norm have been used as better non-convex surrogate functions for the rank and l0 norm. In addition, an efficient iterative algorithm is developed to solve the resulting recovery problem. The experimental results have demonstrated that SGLR outperforms existing state-of-the-art CS algorithms.


Compressive sensing Nonlocal low-rank regularization Total variation Weighted schatten-p norm Alternating direction methods of multipliers 



This work was supported by the Natural Science Fund for Colleges and Universities in Jiangsu Province (Grant No. 16KJB520014), the Doctor Initial Captional of Jinling Institute of Technology Nanjing (No. jit-b-201508), the Scientific Research Starting Foundation of Jinling Institute of Technology for Introducing Talents (No. jit-rcyj-201505), and sponsored by the Jiangsu Key Laboratory of Image and Video Understanding for Social Safety (Nanjing University of Science and Technology) (Grant No. 30916014107).


  1. 1.
    Becker S, Bobin J, Candes EJ (2011) NESTA: a fast and accurate first-order method for sparse recovery. SIAM Journal on Imaging Sciences 4(1):1–39MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bioucas-Dias JM, Figueiredo MAT (2007) A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration. IEEE Trans Image Process 16(12):2992–3004MathSciNetCrossRefGoogle Scholar
  3. 3.
    Buades A, Coll B, Morel JM (2005) A review of image denoising algorithms, with a new one. Multiscale Modeling & Simulation 4(2):490–530MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cai JF, Candès EJ, Shen Z (2010) A singular value thresholding algorithm for matrix completion. SIAM J Optim 20(4):1956–1982MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Candes EJ, Wakin MB, Boyd SP (2008) Enhancing sparsity by reweighted l1 minimization. J Fourier Anal Appl 14(5–6):877–905MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dabov K, Foi A, Katkovnik V (2007) Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans Image Process 16(8):2080–2095MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dong W, Shi G, Li X, Zhang L (2012) Image reconstruction with locally adaptive sparsity and nonlocal robust regularization. Signal Process Image Commun 27(10):1109–1122CrossRefGoogle Scholar
  8. 8.
    Dong W, Shi G, Li X, Zhang L (2014) Compressive Sensing via Nonlocal Low-Rank Regularization. IEEE Trans Image Process 23(8):3618–3632MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Egiazarian K, Foi A, Katkovnik V (2007) Compressed sensing image reconstruction via recursive spatially adaptive filtering. IEEE International Conference on Image Processing, San Antonio, TX, USA 1:549–552Google Scholar
  10. 10.
    Fazel M, Hindi H, Boyd SP (2003) Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices, vol 3. IEEE American Control Conference, Denver, pp 2156–2162Google Scholar
  11. 11.
    Feng L, Sun H, Sun Q, Xia G (2016) Compressive sensing via nonlocal low-rank tensor regularization. Neurocomputing 216(C):45–60CrossRefGoogle Scholar
  12. 12.
    Huang J, Zhang T, Metaxas D (2011) Learning with structured sparsity. The Journal of Machine Learning Research 12:3371–3412MathSciNetzbMATHGoogle Scholar
  13. 13.
    Li W, Wang P, Qiao H (2010) Double Least Squares Pursuit for Sparse Decomposition. International Conference on Intelligent Information Processing, Berlin, Heidelberg, pp 357–363Google Scholar
  14. 14.
    Manjón JV, Coupé P, Buades A, Louis Collins DL, Robles M (2012) New methods for MRI denoising based on sparseness and self-similarity. Med Image Anal 16:18–27CrossRefGoogle Scholar
  15. 15.
    Mohan K, Fazel M (2010) Reweighted nuclear norm minimization with application to system identification. IEEE American Control Conference, Baltimore, pp 2953–2959Google Scholar
  16. 16.
    Nie F, Huang H, Ding C (2012) Low-rank matrix recovery via efficient schatten p-norm minimization. Twenty-Sixth AAAI Conference on Artificial Intelligence, Toronto, pp 655–661Google Scholar
  17. 17.
    Qu X, Hou Y, Lam F, Guo D, Zhong J, Chen Z (2014) Magnetic resonance image reconstruction from undersampled measurements using a patch-based nonlocal operator. Med Image Anal 18(6):843–856CrossRefGoogle Scholar
  18. 18.
    Shu X, Yang J, Ahuja N (2014) Non-local compressive sampling recovery. IEEE International Conference on Computational Photography (ICCP), Santa Clara, pp 1–8Google Scholar
  19. 19.
    Shu X, Ahuja N (2010) Hybrid Compressive Sampling via a New Total Variation TVL1. European Conference on Computer Vision, Santa Clara, pp 393–404Google Scholar
  20. 20.
    Wang J, Wang M, Hu X (2015) Visual data denoising with a unified Schatten-p norm and lq norm regularized principal component pursuit. Pattern Recogn 48(10):3135–3144MathSciNetCrossRefGoogle Scholar
  21. 21.
    Xie Y, Qu Y, Tao D, Wu W, Yuan Q, Zhang W (2016) Hyperspectral Image Restoration via Iteratively Regularized Weighted Schatten p-Norm Minimization. IEEE Trans Geosci Remote Sens 54(8):4642–4659Google Scholar
  22. 22.
    Xie Y, Gu S, Liu Y, Zuo W, Zhang W, Zhang L (2015) Weighted schatten, p-norm minimization for image denoising and background subtraction. IEEE Trans Image Process 25(10):4842–4857Google Scholar
  23. 23.
    Zhang Y, Wu L, Peterson B, Dong Z (2011) A Two-Level Iterative Reconstruction Method for Compressed Sensing MRI. Journal of Electromagnetic Waves and Applications 25(8–9):1081–1091CrossRefGoogle Scholar
  24. 24.
    Zhang J, Zhao D, Zhao C (2012) Compressed sensing recovery via collaborative sparsity. IEEE Data Compression Conference, Snowbird, pp 287–296Google Scholar
  25. 25.
    Zhang J, Liu S, Xiong R, Ma S, Zhao D (2013) Improved total variation based image compressive sensing recovery by nonlocal regularization. IEEE International Symposium on Circuits and Systems, Beijing, pp 2836–2839Google Scholar
  26. 26.
    Zhang Y, Peterson BS, Ji G, Dong Z (2014) Energy preserved sampling for compressed sensing MRI. Computational & Mathematical Methods in Medicine 2014(5):546814zbMATHGoogle Scholar
  27. 27.
    Zhang J, Zhao D, Gao W (2014) Group-based sparse representation for image restoration. IEEE Trans Image Process 23(8):3336–3351MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zhang Y, Wang S, Ji G, Dong Z (2015) Exponential wavelet iterative shrinkage thresholding algorithm with random shift for compressed sensing magnetic resonance imaging. Inf Sci 322(1):115–132CrossRefGoogle Scholar
  29. 29.
    Zhao C, Zhang J, Ma S, Gao W (2016) Nonconvex Lp Nuclear Norm based ADMM Framework for Compressed Sensing. IEEE Data Compression Conference (DCC2016), Snowbird, pp 161–170Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Computer EngineeringJinling Institute of TechnologyNanjingChina
  2. 2.College of Computer and Information EngineeringNanjing Xiaozhuang UniversityNanjingChina

Personalised recommendations