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Image Recovery based on Local and Nonlocal Regularizations

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Abstract

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.

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References

  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):1–39

    Article  MathSciNet  MATH  Google Scholar 

  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):2992–3004

    Article  MathSciNet  Google Scholar 

  3. Buades A, Coll B, Morel JM (2005) A review of image denoising algorithms, with a new one. Multiscale Modeling & Simulation 4(2):490–530

    Article  MathSciNet  MATH  Google Scholar 

  4. Cai JF, Candès EJ, Shen Z (2010) A singular value thresholding algorithm for matrix completion. SIAM J Optim 20(4):1956–1982

    Article  MathSciNet  MATH  Google Scholar 

  5. Candes EJ, Wakin MB, Boyd SP (2008) Enhancing sparsity by reweighted l1 minimization. J Fourier Anal Appl 14(5–6):877–905

    Article  MathSciNet  MATH  Google Scholar 

  6. Dabov K, Foi A, Katkovnik V (2007) Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans Image Process 16(8):2080–2095

    Article  MathSciNet  Google Scholar 

  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):1109–1122

    Article  Google Scholar 

  8. Dong W, Shi G, Li X, Zhang L (2014) Compressive Sensing via Nonlocal Low-Rank Regularization. IEEE Trans Image Process 23(8):3618–3632

    Article  MathSciNet  MATH  Google Scholar 

  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 1:549–552

    Google Scholar 

  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 2156–2162

    Google Scholar 

  11. Feng L, Sun H, Sun Q, Xia G (2016) Compressive sensing via nonlocal low-rank tensor regularization. Neurocomputing 216(C):45–60

    Article  Google Scholar 

  12. Huang J, Zhang T, Metaxas D (2011) Learning with structured sparsity. The Journal of Machine Learning Research 12:3371–3412

    MathSciNet  MATH  Google Scholar 

  13. Li W, Wang P, Qiao H (2010) Double Least Squares Pursuit for Sparse Decomposition. International Conference on Intelligent Information Processing, Berlin, Heidelberg, pp 357–363

  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 16:18–27

    Article  Google Scholar 

  15. Mohan K, Fazel M (2010) Reweighted nuclear norm minimization with application to system identification. IEEE American Control Conference, Baltimore, pp 2953–2959

    Google Scholar 

  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 655–661

    Google Scholar 

  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):843–856

    Article  Google Scholar 

  18. Shu X, Yang J, Ahuja N (2014) Non-local compressive sampling recovery. IEEE International Conference on Computational Photography (ICCP), Santa Clara, pp 1–8

    Google Scholar 

  19. Shu X, Ahuja N (2010) Hybrid Compressive Sampling via a New Total Variation TVL1. European Conference on Computer Vision, Santa Clara, pp 393–404

  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):3135–3144

    Article  MathSciNet  Google Scholar 

  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):4642–4659

  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):4842–4857

  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):1081–1091

    Article  Google Scholar 

  24. Zhang J, Zhao D, Zhao C (2012) Compressed sensing recovery via collaborative sparsity. IEEE Data Compression Conference, Snowbird, pp 287–296

  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 2836–2839

  26. Zhang Y, Peterson BS, Ji G, Dong Z (2014) Energy preserved sampling for compressed sensing MRI. Computational & Mathematical Methods in Medicine 2014(5):546814

    MATH  Google Scholar 

  27. Zhang J, Zhao D, Gao W (2014) Group-based sparse representation for image restoration. IEEE Trans Image Process 23(8):3336–3351

    Article  MathSciNet  MATH  Google Scholar 

  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):115–132

    Article  Google Scholar 

  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 161–170

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Acknowledgments

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).

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Authors and Affiliations

  1. School of Computer Engineering, Jinling Institute of Technology, Nanjing, 211169, China

    Jun Zhu

  2. College of Computer and Information Engineering, Nanjing Xiaozhuang University, Nanjing, 210017, China

    Changwei Chen

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  1. Jun Zhu
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  2. Changwei Chen
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Correspondence to Changwei Chen.

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Zhu, J., Chen, C. Image Recovery based on Local and Nonlocal Regularizations. Multimed Tools Appl 77, 22841–22855 (2018). https://doi.org/10.1007/s11042-018-5935-3

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  • DOI: https://doi.org/10.1007/s11042-018-5935-3

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