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Compressed sensing image reconstruction via adaptive sparse nonlocal regularization

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Abstract

Compressed sensing (CS) has been successfully utilized by many computer vision applications. However,the task of signal reconstruction is still challenging, especially when we only have the CS measurements of an image (CS image reconstruction). Compared with the task of traditional image restoration (e.g., image denosing, debluring and inpainting, etc.), CS image reconstruction has partly structure or local features. It is difficult to build a dictionary for CS image reconstruction from itself. Few studies have shown promising reconstruction performance since most of the existing methods employed a fixed set of bases (e.g., wavelets, DCT, and gradient spaces) as the dictionary, which lack the adaptivity to fit image local structures. In this paper, we propose an adaptive sparse nonlocal regularization (ASNR) approach for CS image reconstruction. In ASNR, an effective self-adaptive learning dictionary is used to greatly reduce artifacts and the loss of fine details. The dictionary is compact and learned from the reconstructed image itself rather than natural image dataset. Furthermore, the image sparse nonlocal (or nonlocal self-similarity) priors are integrated into the regularization term, thus ASNR can effectively enhance the quality of the CS image reconstruction. To improve the computational efficiency of the ASNR, the split Bregman iteration based technique is also developed, which can exhibit better convergence performance than iterative shrinkage/thresholding method. Extensive experimental results demonstrate that the proposed ASNR method can effectively reconstruct fine structures and suppress visual artifacts, outperforming state-of-the-art performance in terms of both the PSNR and visual measurements.

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References

  1. Candès, E.J. et al.: Compressive sampling. In: Proceedings of the International Congress of Mathematicians, vol. 3, pp. 1433–1452. Madrid, Spain (2006)

  2. Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Gehm, M.E., John, R., Brady, D.J., Willett, R.M., Schulz, T.J.: Single-shot compressive spectral imaging with a dual-disperser architecture. Optics Express 15(21), 14013–14027 (2007)

    Article  Google Scholar 

  4. Duarte, M.F., Davenport, M.A., Takhar, D., Laska, J.N., Sun, T., Kelly, K.E., Baraniuk, R.G., et al.: Single-pixel imaging via compressive sampling. IEEE Signal Process. Mag. 25(2), 83 (2008)

  5. Hitomi, Y., Gu, J., Gupta, M., Mitsunaga, T., Nayar, S.K.: Video from a single coded exposure photograph using a learned over-complete dictionary. In: Computer Vision (ICCV), 2011 IEEE International Conference on, pp. 287–294. IEEE (2011)

  6. Candes, E.J., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51(12), 4203–4215 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Figueiredo, Mário A.T., Nowak, Robert D., Wright, Stephen J.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Select. Topics Signal Process. 1(14), 586–597 (2007)

    Article  Google Scholar 

  8. Tropp, J., Gilbert, A.C., et al.: Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. arXiv:math/0307152, (2003)

  10. Tikhonov, A.N., Glasko, V.B.: Use of the regularization method in non-linear problems. USSR Comput. Math. Math. Phys. 5(3), 93–107 (1965)

    Article  MATH  Google Scholar 

  11. Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chan, T., Esedoglu, S., Frederick, P., Yip, A.: Recent developments in total variation image restoration. Math. Models Comput. Vis. 17, 1–18 (2005)

    Google Scholar 

  13. Buades, A., Coll, B., Morel, J.-M.: A non-local algorithm for image denoising. In: Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference, vol. 2, pp. 60–65. IEEE (2005)

  14. Dong, W., Zhang, L., Shi, G., Xiaolin, W.: Image deblurring and super-resolution by adaptive sparse domain selection and adaptive regularization. IEEE Trans. Image Process. 20(7), 1838–1857 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Peyré, G.: Image processing with nonlocal spectral bases. Multiscale Model. Simul. 7(2), 703–730 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, Q., Zhang, C., Guo, Q., Xu, H., Zhou, Y.: Adaptive sparse coding on PCA dictionary for image denoising. Vis. Comput. 32(4), 535–549 (2016)

    Article  Google Scholar 

  17. Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jung, M., Bresson, X., Chan, T.F., Vese, L., et al.: Nonlocal mumford-shah regularizers for color image restoration. IEEE Trans. Image Process. 20(6), 1583–1598 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dong, W., Zhang, L., Shi, G., Li, X.: Nonlocally centralized sparse representation for image restoration. IEEE Trans. Image Process. 22(4), 1620–1630 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Aharon, M., Elad, M., Bruckstein, A.: A k-svd: an algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process. 54(11), 4311–4322 (2006)

    Article  MATH  Google Scholar 

  21. Elad, M., Aharon, M.: Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15(12), 3736–3745 (2006)

    Article  MathSciNet  Google Scholar 

  22. Bhardwaj, A., Raman, S.: Robust PCA-based solution to image composition using augmented Lagrange multiplier (ALM). Vis. Comput. 32(5), 591–600 (2016)

    Article  Google Scholar 

  23. Ignácio, U., Jung, C.R.: Block-based image inpainting in the wavelet domain. Vis. Comput. 23(9–11), 733–741 (2007)

    Article  Google Scholar 

  24. Elad, M., Yavneh, I.: A plurality of sparse representations is better than the sparsest one alone. IEEE Trans. Inf. Theory 55(10), 4701–4714 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Protter, M., Yavneh, I., Elad, M.: Closed-form mmse estimation for signal denoising under sparse representation modeling over a unitary dictionary. IEEE Trans. Signal Process. 58(7), 3471–3484 (2010)

    Article  MathSciNet  Google Scholar 

  26. Wu, X., Zhang, X., Wang, J.: Model-guided adaptive recovery of compressive sensing. In: Data Compression Conference, 2009. DCC’09., pp. 123–132. IEEE (2009)

  27. Ravishankar, S., Bresler, Y.: Mr image reconstruction from highly undersampled k-space data by dictionary learning. IEEE Trans. Med. Imaging 30(5), 1028–1041 (2011)

    Article  Google Scholar 

  28. He, L., Carin, L.: Exploiting structure in wavelet-based bayesian compressive sensing. IEEE Trans. Signal Process. 57(9), 3488–3497 (2009)

    Article  MathSciNet  Google Scholar 

  29. He, L., Chen, H., Carin, L.: Tree-structured compressive sensing with variational bayesian analysis. IEEE Signal Process. Lett. 17(3), 233–236 (2010)

    Article  Google Scholar 

  30. Yang, J., Liao, X., Yuan, X., Llull, P., Brady, D.J., Sapiro, G., Carin, L.: Compressive sensing by learning a gaussian mixture model from measurements. IEEE Trans. Image Process. 24(1), 106–119 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mun, S., Fowler, J.E.: Block compressed sensing of images using directional transforms. In: Image Processing (ICIP), 2009 16th IEEE International Conference on, pp. 3021–3024. IEEE (2009)

  32. Candès, E.J., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. Goldstein, T., Osher, S.: The split bregman method for l1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. SIAM J. Imaging Sci. 3(3), 253–276 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lin, Z., Chen, M., Ma, Y.: The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. (2010). arXiv:1009.5055

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Xu, L., Lu, C., Xu, Y., Jia, J.: Image smoothing via l 0 gradient minimization. In: ACM Transactions on Graphics (TOG), vol. 30, pp. 174. ACM (2011)

  38. Sharifi, K., Leon-Garcia, A.: Estimation of shape parameter for generalized gaussian distributions in subband decompositions of video. IEEE Trans. Circuits Syst. Video Technol. 5(1), 52–56 (1995)

    Article  Google Scholar 

  39. Dong, W., Shi, G., Li, X., Zhang, L., Xiaolin, W.: Image reconstruction with locally adaptive sparsity and nonlocal robust regularization. Signal Process. Image Commun. 27(10), 1109–1122 (2012)

    Article  Google Scholar 

  40. Mairal, J., Bach, F., Ponce, J., Sapiro, G.: Online learning for matrix factorization and sparse coding. J. Mach. Learn. Res. 11, 19–60 (2010)

    MathSciNet  MATH  Google Scholar 

  41. Wang, S., Zhang, L., Liang, Y., Pan, Q.: Semi-coupled dictionary learning with applications to image super-resolution and photo-sketch synthesis. In: Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on, pp. 2216–2223. IEEE (2012)

  42. Huang, R., Ye, M., Pei, X., Li, T., Dou, Y.: Learning to pool high-level features for face representation. Vis. Comput. 31(12), 1683–1695 (2015)

    Article  Google Scholar 

  43. Zhan, J., Zhuo, S., Hefeng, W., Luo, X.: Robust tracking via discriminative sparse feature selection. Vis. Comput. 31(5), 575–588 (2015)

    Article  Google Scholar 

  44. Mun, S., Fowler, J.E.: Residual reconstruction for block-based compressed sensing of video. In: Data Compression Conference (DCC), 2011, pp. 183–192. IEEE (2011)

  45. Becker, S., Bobin, J., Candès, E.J.: Nesta: a fast and accurate first-order method for sparse recovery. SIAM J. Imaging Sci. 4(1), 1–39 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhang, J., Zhao, D., Zhao, C., Xiong, R., Ma, S., Gao, W.: Image compressive sensing recovery via collaborative sparsity. IEEE J. Emerg. Select. Topics Circuits Syst. 2(3), 380–391 (2012)

    Article  Google Scholar 

  47. Egiazarian, K., Foi, A., Katkovnik, V.: Compressed sensing image reconstruction via recursive spatially adaptive filtering. In: Image Processing, 2007. ICIP 2007. IEEE International Conference. vol. 1, pp. I–549. IEEE (2007)

  48. Zhang, J., Zhao, C., Zhao, D., Gao, W.: Image compressive sensing recovery using adaptively learned sparsifying basis via l0 minimization. Signal Process. 103, 114–126 (2014)

    Article  Google Scholar 

  49. Canh, T.N, Dinh, K.Q., Jeon, B.: Multi-scale/multi-resolution kronecker compressive imaging. In Image Processing (ICIP), 2015 IEEE International Conference on, pp. 2700–2704. IEEE (2015)

  50. Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18(11), 2419–2434 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work was supported by the Natural Science Foundation of China (61462052, 61571220) and the open research fund of National Mobile Commune. Research Lab., Southeast University (No.2015D08).

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Authors and Affiliations

  1. School of Electronic Science and Engineering, Nanjing University, Nanjing, 210023, China

    Zhiyuan Zha, Xinggan Zhang, Yang Chen, Yechao Bai & Qiong Wang

  2. The Center for Machine Vision and Signal Analysis, University of Oulu, Finland and School of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an, Shanxi, China

    Xin Liu

  3. Department of Electronic Science and Engineering, Nanjing University, and National Mobile Commun. Research Lab., Southeast University, Nanjing, 210023, China

    Lan Tang

  4. School of Information Engineering and Automation, Kunming University of Science and Technology, Kunming, 650500, China

    Zhenhong Shang

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  1. Zhiyuan Zha
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  2. Xin Liu
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  4. Yang Chen
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  5. Lan Tang
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Correspondence to Zhiyuan Zha or Xinggan Zhang.

Appendix: Proof of Theorem 1

Appendix: Proof of Theorem 1

Owing to the assumption that \({{{\varvec{e}}}}{({{\varvec{j}}})}\) follows an independent zero mean distribution with variance \(\sigma ^2\), namely, \({{\varvec{E}}}[{{{\varvec{e}}}}{({{\varvec{j}}})}]\) and \({{\varvec{Var}}}[{{{\varvec{e}}}}{({{\varvec{j}}})}]=\sigma ^2\). Thus, it can be deduced that each \({{{\varvec{e}}}}{({{\varvec{j}}})}^2\) is also independent, and the meaning of each \({{{\varvec{e}}}}{({{\varvec{j}}})}^2\) is:

$$\begin{aligned} {{\varvec{E}}}[{{{\varvec{e}}}}{({{\varvec{j}}})}^2]={{\varvec{Var}}}[{{{\varvec{e}}}}{({{\varvec{j}}})}] +[{{\varvec{E}}}[{{{\varvec{e}}}}{({{\varvec{j}}})}]]^2=\sigma ^2, \quad j =1,2,\ldots ,N \end{aligned}$$
(29)

By invoking the law of Large numbers in probability theory, for any \(\epsilon >0\), it leads to \(\lim \limits _{N \rightarrow \infty }{{{\varvec{P}}}}\{|\frac{1}{N }\Sigma _{j =1}^{N }{{{\varvec{e}}}}{({{\varvec{j}}})}^2-\sigma ^2|<\frac{\epsilon }{2}\}=1\), namely,

$$\begin{aligned} \lim \limits _{N \rightarrow \infty }{{{\varvec{P}}}}\left\{ \left| \frac{1}{N } ||{{\varvec{x}}}-{{\varvec{r}}}||_2^2-\sigma ^2 \right| <\frac{\epsilon }{2} \right\} =1 \end{aligned}$$
(30)

Next, we denote the concatenation of all the patches \({{{\varvec{x}}}}_i\) and \({{{\varvec{r}}}}_i,\ i =1,2,\ldots ,n \), by \({{{\varvec{x}}}}_l\) and \({{{\varvec{r}}}}_l\) , respectively. Meanwhile, we denote the error of each element of \({{{\varvec{x}}}}_l-{{{\varvec{r}}}}_l\) by \({{{\varvec{e}}}}_l{(k) },\ k =1,2,\ldots ,K \). We have also denote \({{{\varvec{e}}}}_l{(k) }\) following an independent zero mean distribution with variance \(\sigma ^2\).

Therefore, the same process applied to \({{{\varvec{e}}}}_l{(k) }^2\) yields \(\lim \limits _{N \rightarrow \infty }{{{\varvec{P}}}}\{|\frac{1}{N }\Sigma _{k =1}^{N }{{{\varvec{e}}}}_l{({k })}^2-\sigma ^2|<\frac{\epsilon }{2}\}=1\), i.e.,

$$\begin{aligned} \lim \limits _{N \rightarrow \infty }{{{\varvec{P}}}} \left\{ \left| \frac{1}{N }\Sigma _{i =1}^{n }||{{{\varvec{x}}}}_l-{{{\varvec{r}}}}_l||_2^2-\sigma ^2 \right| <\frac{\epsilon }{2} \right\} =1 \end{aligned}$$
(31)

Obviously, considering Eqs. (30) and (31) together, we can prove Eq. (19).

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Zha, Z., Liu, X., Zhang, X. et al. Compressed sensing image reconstruction via adaptive sparse nonlocal regularization. Vis Comput 34, 117–137 (2018). https://doi.org/10.1007/s00371-016-1318-9

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