volume | PIER Journals

Abstract

Compressed sensing (CS) relies on the sparse priorin posed on the signal to solve the ill-posed recovery problem in an under-determined linear system (ULS). Motivated by the theory, this paper proposes a new algorithm called regularized re-weighted inverse trigonometric smoothed function approximating L₀-norm minimization (RRITSL0) algorithm, where the inverse trigonometric (IT) function, iteratively re-weighted scheme and regularization mechanism constitute the core of the proposed RRITSL0 algorithm. Compared with other state-of-the-art functions, our proposed IT function cluster can better approximate the L₀-norm, thus improving the reconstruction accuracy. And the new re-weighted scheme we adopted can promote sparsity and speed up convergence. Moreover, the regularization mechanism makes the RRITSL0 algorithm more robust against noise. The performance of the proposed algorithm is verified via numerical experiments with additive noise. Furthermore, the experiments prove the superiority of the RRITSL0 algorithm in magnetic resonance (MR) image recovery.

1. Chen, Z., Y. Fu, Y. Xiang, and R. Rong, "A novel iterative shrinkage algorithm for CS-MRI via adaptive regularization," IEEE Signal Processing Letters, Vol. 99, 1-1, 2017.
doi:10.1109/LSP.2017.2647810

2. Yazdanpanah, A. P. and E. E. Regentova, "Compressed sensing MRI using curvelet sparsity and nonlocal total variation: CS-NLTV," Electronic Imaging, Vol. 2017, No. 13, 5-9, 2017.
doi:10.2352/ISSN.2470-1173.2017.13.IPAS-197

3. Candes, E. J., "Compressive sampling," IEEE Proceedings of the International Congress of Mathematicians, 2006.

4. Candes, E. J. and M. B. Wakin, "An introduction to compressive sampling," IEEE Signal Processing Magazine, Vol. 25, No. 2, 21-30, 2008.
doi:10.1109/MSP.2007.914731

5. Li, S., H. Yin, and L. Fang, "Remote sensing image fusion via sparse representations over learned dictionaries," IEEE Transactions on Geoscience & Remote Sensing, Vol. 51, No. 9, 4779-4789, 2013.
doi:10.1109/TGRS.2012.2230332

6. Zhang, J., D. Zhao, F. Jiang, and W. Gao, "Structural group sparse representation for image compressive sensing recovery," Data Compression Conference, Vol. 6, No. 3, 331-340, 2013.

7. Josa, M., D. Bioucas, and M. A. T. Figueiredo, "A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration," IEEE Transactions on Image Processing, Vol. 16, No. 12, 2992-3004, 2007.
doi:10.1109/TIP.2007.909319

8. Beck, A. and M. Teboulle, "A fast iterative shrinkage-thresholding algorithm for linear inverse problems," Siam Journal on Imaging Sciences, Vol. 2, No. 1, 183-202, 2009.
doi:10.1137/080716542

9. Ghadimi, E., A. Teixeira, and I. Shames, "Optimal parameter selection for the alternating direction method of multipliers (ADMM): quadratic problems," IEEE Transactions on Automatic Control, Vol. 60, No. 3, 644-658, 2015.
doi:10.1109/TAC.2014.2354892

10. Ramdas, A. and R. J. Tibshirani, "Fast and flexible ADMM algorithms for trend filtering," Journal of Computational and Graphical Statistics, Vol. 25, No. 3, 839-858, 2014.
doi:10.1080/10618600.2015.1054033

11. Wright, S. J., R. D. Nowak, and M. A. T. Figueiredo, "Sparse reconstruction by separable approximation," IEEE International Conference on Acoustics, Speech and Signal Processing, 2479-2493, 2008.

12. Ye, X., W. Zhu, A. Zhang, and Q. Meng, "Sparse channel estimation in MIMO-OFDM systems based on an improved sparse reconstruction by separable approximation algorithm," Journal of Information & Computational Science, Vol. 10, No. 2, 609-619, 2013.

13. Figueiredo, M. A. T., R. D. Nowak, and S. J.Wright, "Gradient projection for sparse reconstruction: Application To Compressed Sensing And Other Inverse problems," IEEE Journal of Selected Topics in Signal Processing, Vol. 1, No. 4, 586-597, 2008.
doi:10.1109/JSTSP.2007.910281

14. Zibulevsky, M. and M. Elad, "L1-L2 optimization in signal and image processing," IEEE Signal Processing Magazine, Vol. 27, No. 3, 76-88, 2010.
doi:10.1109/MSP.2010.936023

15. Pant, J. K., W. S. Lu, and A. Antoniou, "New improved algorithms for compressive sensing based on Lp NORM," IEEE Transactions on Circuits & Systems II Express Briefs, Vol. 61, No. 3, 198-202, 2014.
doi:10.1109/TCSII.2013.2296133

16. Ye, X., W. P. Zhu, A. Zhang, and J. Yan, "Sparse channel estimation of MIMO-OFDM systems with unconstrained smoothed L0 -norm-regularized least squares compressed sensing," Eurasip Journal on Wireless Communications & Networking, Vol. 2013, No. 1, 282, 2013.
doi:10.1186/1687-1499-2013-282

17. Zhang, Y., B. S. Peterson, G. Ji, and Z. Dong, "Energy preserved sampling for compressed sensing MRI," Computational and Mathematical Methods in Medicine, Vol. 2014, No. 5, 546814, 2014.

18. Zhang, Y., S. Wang, G. Ji, and Z. Dong, "Exponential wavelet iterative shrinkage thresholding algorithm with random shift for compressed sensing magnetic resonance imaging," Information Sciences, Vol. 10, No. 1, 116-117, 2015.

19. Mohimani, H., M. Babaie-Zadeh, and C. Jutten, "A fast approach for overcomplete sparse decomposition based on smoothed L0 norm," IEEE Transactions on Signal Processing, Vol. 57, No. 1, 289-301, 2009.
doi:10.1109/TSP.2008.2007606

20. Zhao, R., W. Lin, H. Li, and S. Hu, "Reconstruction algorithm for compressive sensing based on smoothed L0 norm and revised newton method," Journal of Computer-Aided Design & Computer Graphics, Vol. 24, No. 4, 478-484, 2012.

21. Candes, E. J., M. B. Wakin, and S. P. Boyd, "Enhancing sparsity by re-weighted L1 minimization," Journal of Fourier Analysis and Applications, Vol. 14, No. 5, 877-905, 2008.
doi:10.1007/s00041-008-9045-x

22. Pant, J. K., W. S. Lu, and A. Antoniou, "Reconstruction of sparse signals by minimizing a re-weighted approximate L0-norm in the null space of the measurement matrix," IEEE International Midwest Symposium on Circuits and Systems, 430-433, 2010.

23. Zibetti, M. V. W., C. Lin, and G. T. Herman, "Total variation superiorized conjugate gradient method for image reconstruction," Inverse Problems, Vol. 34, No. 3, 2017.

24. Wen, F., Y. Yang, P. Liu, R. Ying, and Y. Liu, "Efficient l_q minimization algorithms for compressive sensing based on proximity operator," Mathematics, 2016.

25. Ye, X. and W. P. Zhu, "Sparse channel estimation of pulse-shaping multiple-input-multiple-output orthogonal frequency division multiplexing systems with an approximate gradient L₂SL₀ reconstruction algorithm," Iet Communications, Vol. 8, No. 7, 1124-1131, 2014.
doi:10.1049/iet-com.2013.0571

Dr. Xiangjun Yin

Dr. Linyu Wang

Dr. Huihui Yue

Dr. Jianhong Xiang