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Multimedia Tools and Applications

, Volume 77, Issue 10, pp 12853–12869 | Cite as

Image compressed sensing based on non-convex low-rank approximation

  • Yan Zhang
  • Jichang GuoEmail author
  • Chongyi Li
Article

Abstract

Nonlocal sparsity and structured sparsity have been evidenced to improve the reconstruction of image details in various compressed sensing (CS) studies. The nonlocal processing is achieved by grouping similar patches of the image into the groups. To exploit these nonlocal self-similarities in natural images, a non-convex low-rank approximation is proposed to regularize the CS recovery in this paper. The nuclear norm minimization, as a convex relaxation of rank function minimization, ignores the prior knowledge of the matrix singular values. This greatly restricts its capability and flexibility in dealing with many practical problems. In order to make a better approximation of the rank function, the non-convex low-rank regularization namely weighted Schatten p-norm minimization (WSNM) is proposed. In this way, both the local sparsity and nonlocal sparsity are integrated into a recovery framework. The experimental results show that our method outperforms the state-of-the-art CS recovery algorithms not only in PSNR index, but also in local structure preservation.

Keywords

Image compressed sensing Low-rank approximation Weighted Schatten p-norm Non-convex optimization 

Notes

Acknowledgements

The authors would like to give thanks to the anonymous reviewers for their valuable comments that were useful to improve the quality of the paper. This work was supported by Natural Science Foundation of Tianjin (Grant No. 15JCYBJC15500).

References

  1. 1.
    Baraniuk RG, Cevher V, Duarte MF, Hegde C (2010) Model-based compressive sensing. IEEE Trans Inf Theory 56(4):1982–2001MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Cai JF, Candès EJ, Shen Z (2010) A singular value thresholding algorithm for matrix completion. SIAM J Optim 20(4):1956–1982MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Candès EJ, Recht B (2009) Exact matrix completion via convex optimization. Found Comput Math 9(6):717–772MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Candès EJ, Romberg J (2005) Practical signal recovery from random projections. Proc SPIE Comput Imag 5674:76–86CrossRefGoogle Scholar
  6. 6.
    Candès EJ, Romberg J, Tao T (2006) Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory 52(2):489–509MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Candès EJ, Wakin MB, Boyd SP (2008) Enhancing sparsity by reweighted ℓ1 minimization. J Fourier Anal Appl 14(5–6):877–905MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Candès EJ, Li X, Ma Y, Wright J (2011) Robust principal component analysis? J ACM 58(3):11MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen SS, Donoho DL, Saunders MA (2001) Atomic decomposition by basis pursuit. SIAM Rev 43(1):129–159MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dabov K, Foi A, Katkovnik V, Egiazarian K (2007) Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans Image Process 16(8):2080–2095MathSciNetCrossRefGoogle Scholar
  11. 11.
    Daubechies I, Defrise M, De Mol C (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun Pure Appl Math 57(11):1413–1457MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dong W, Zhang L, Shi G (2011) Centralized sparse representation for image restoration. In 2011 International Conference on Computer Vision, pp 1259–1266Google Scholar
  13. 13.
    Dong W, Shi G, Li X, Zhang L, Wu X (2012) Image reconstruction with locally adaptive sparsity and nonlocal robust regularization. Signal Process Image Commun 27(10):1109–1122CrossRefGoogle Scholar
  14. 14.
    Dong W, Shi G, Li X, Ma Y, Huang F (2014) Compressive sensing via nonlocal low-rank regularization. IEEE Trans Image Process 23(8):3618–3632MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Donoho DL (2006) Compressed sensing. IEEE Trans Inf Theory 52(4):1289–1306MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Egiazarian K, Foi A, Katkovnik V (2007) Compressed sensing image reconstruction via recursive spatially adaptive filtering. In 2007 I.E. International Conference on Image Processing, 1, pp I-549–I-552Google Scholar
  17. 17.
    Foucart S, Lai MJ (2009) Sparsest solutions of underdetermined linear systems via ℓq-minimization for 0<q⩽1. Appl Comput Harmon Anal 26(3):395–407MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gu S, Zhang L, Zuo W, Feng X (2014) Weighted nuclear norm minimization with application to image denoising. In 2014 I.E. Conference on Computer Vision and Pattern Recognition, pp 2862–2869Google Scholar
  19. 19.
    Huang J, Zhang T, Metaxas D (2011) Learning with structured sparsity. J Mach Learn Res 12:3371–3412MathSciNetzbMATHGoogle Scholar
  20. 20.
    Jiang J, Ma X, Chen C, Lu T, Wang Z, Ma J (2017) Single image super-resolution via locally regularized anchored neighborhood regression and nonlocal means. IEEE Trans Multimedia 19(1):15–26CrossRefGoogle Scholar
  21. 21.
    Lin Z, Chen M, Wu L, Ma Y (2009) The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices. Dept. Electr. Comput. Eng., Univ. Illinois Urbana-Champaign, Urbana, IL, USA, Tech. Rep. UILU-ENG-09-2215Google Scholar
  22. 22.
    Liu R, Lin Z, De la Torre F, Su Z (2012, June) Fixed-rank representation for unsupervised visual learning. In 2012 I.E. Conference on Computer Vision and Pattern Recognition, pp 598–605Google Scholar
  23. 23.
    Liu L, Huang W, Chen DR (2014) Exact minimum rank approximation via Schatten p-norm minimization. J Comput Appl Math 267:218–227MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lu YM, Do MN (2008) Sampling signals from a union of subspaces. IEEE Signal Process Mag 25(2):41–47CrossRefGoogle Scholar
  25. 25.
    Lu T, Xiong Z, Wan Y, Yang W (2016) Face hallucination via locality-constrained low-rank representation. In 2016 I.E. international conference on acoustics, speech and signal processing, pp. 1746-1750Google Scholar
  26. 26.
    Mairal J, Bach F, Ponce J, Sapiro G, Zisserman A (2009) Non-local sparse models for image restoration. In 2009 I.E. 12th International Conference on Computer Vision, pp 2272–2279Google Scholar
  27. 27.
    Massa A, Rocca P, Oliveri G (2015) Compressive sensing in electromagnetics-a review. IEEE Antennas and Propagation Magazine 57(1):224–238CrossRefGoogle Scholar
  28. 28.
    Mirsky L (1975) A trace inequality of John von Neumann. Monatshefte für Mathematik 79(4):303–306MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Natarajan BK (1995) Sparse approximate solutions to linear systems. SIAM J Comput 24(2):227–234MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nie F, Wang H, Cai X, Huang H, Ding C (2012, December) Robust matrix completion via joint schatten p-norm and lp-norm minimization. In 2012 I.E. 12th International Conference on Data Mining, pp 566–574Google Scholar
  31. 31.
    Pan JS, Li W, Yang CS, Yan LJ (2015) Image steganography based on subsampling and compressive sensing. Multimedia Tools and Applications 74(21):9191–9205CrossRefGoogle Scholar
  32. 32.
    Sendur L, Selesnick IW (2002) Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency. IEEE Trans Signal Process 50(11):2744–2756CrossRefGoogle Scholar
  33. 33.
    Software l1magic [Online] (2006) Available: http://www.acm.caltech.edu/l1magic
  34. 34.
    Tropp J, Gilbert AC (2005) Signal recovery from partial information via orthogonal matching pursuitGoogle Scholar
  35. 35.
    Wang Y, Wang J, Xu Z (2014) Restricted p-isometry properties of nonconvex block-sparse compressed sensing. Signal Process 104:188–196CrossRefGoogle Scholar
  36. 36.
    Wang B, Lu T, Xiong Z (2016) Adaptive boosting for image denoising: Beyond low-rank representation and sparse coding. In 2016 International Conference on Pattern RecognitionGoogle Scholar
  37. 37.
    Wipf DP, Rao BD (2004) Sparse Bayesian learning for basis selection. IEEE Trans Signal Process 52(8):2153–2164MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wu X, Dong W, Zhang X, Shi G (2012) Model-assisted adaptive recovery of compressed sensing with imaging applications. IEEE Trans Image Process 21(2):451–458MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Xie Y, Gu S, Liu Y, Zuo W, Zhang W, Zhang L (2015) Weighted Schatten p-norm minimization for image Denoising and background subtraction. arXiv preprint arXiv:1512.01003Google Scholar
  40. 40.
    Xu BH, Cen YG, Wei Z, Cen Y, Zhao RZ, Miao ZJ (2016) Video restoration based on PatchMatch and reweighted low-rank matrix recovery. Multimedia Tools and Applications 75(5):2681–2696CrossRefGoogle Scholar
  41. 41.
    Zhang D, Hu Y, Ye J, Li X, He X (2012) Matrix completion by truncated nuclear norm regularization. In 2012 I.E. Conference on Computer Vision and Pattern Recognition, pp 2192–2199Google Scholar
  42. 42.
    Zhang J, Xiang Q, Yin Y, Chen C, Luo X (2016) Adaptive compressed sensing for wireless image sensor networks. Multimedia Tools and Applications: 1–16Google Scholar
  43. 43.
    Zuo W, Meng D, Zhang L, Feng X, Zhang D (2013) A generalized iterated shrinkage algorithm for non-convex sparse coding. In Proceedings of the IEEE international conference on computer vision, pp 217–224Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Electronic Information EngineeringTianjin UniversityTianjinChina
  2. 2.School of Computer and Information EngineeringTianjin Chengjian UniversityTianjinChina

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