KSII Transactions on Internet and Information Systems (TIIS)

Korean Society for Internet Information (한국인터넷정보학회)
권호 표지
DOI QR코드

DOI QR Code

Fast Linearized Bregman Method for Compressed Sensing

  • Yang, Zhenzhen (College of Communication & Information Engineering, Nanjing University of Posts and Telecommunications) ;
  • Yang, Zhen (Key Lab of "Broadband Wireless Communication and Sensor Network Technology" (Nanjing University of Posts and Telecommunications), Ministry of Education)
  • Received : 2013.06.24
  • Accepted : 2013.09.21
  • Published : 2013.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, a fast and efficient signal reconstruction algorithm for solving the basis pursuit (BP) problem in compressed sensing (CS) is proposed. This fast linearized Bregman method (FLBM), which is inspired by the fast method of Beck et al., is based on the fact that the linearized Bregman method (LBM) is equivalent to a gradient descent method when applied to a certain formulation. The LBM requires $O(1/{\varepsilon})$ iterations to obtain an ${\varepsilon}$-optimal solution while the FLBM reduces this iteration complexity to $O(1/\sqrt{\varepsilon})$ and requiring almost the same computational effort on each iteration. Our experimental results show that the FLBM can be faster than some other existing signal reconstruction methods.

Keywords

References

  1. D. L. Donoho, "Compressed sensing," IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289-1306, 2006. https://doi.org/10.1109/TIT.2006.871582
  2. D. L. Donoho and Y. Tsaig, "Extensions of compressed sensing," Signal Processing, vol. 86, no.3, pp. 533-548, 2006. http://www.sciencedirect.com/science/article/pii/S0165168405002215 https://doi.org/10.1016/j.sigpro.2005.05.028
  3. E. J. Candes and M. B. Wakin, "An introduction to compressive sampling," IEEE Signal Processing Magazine, vol. 25, no. 2, pp. 21-30, 2008. https://doi.org/10.1109/MSP.2007.914731
  4. M. A. Davenport, M. F. Duarte, Y. C. Eldar and G. Kutyniok, "Compressed Sensing: Theory and Applications," Cambridge University Press, 2012. http://www.math.tu-berlin.de/fileadmin/i26fg-kutyniok/Kutyniok/Papers/SurveyCS.pdf
  5. J. Romberg, "Imaging via compressive sampling," IEEE Signal Processing Magazine, vol. 25, no. 2, pp. 14-20, 2008.
  6. S. S. Chen, D. L. Donoho and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM Review, vol. 43, no. 1, pp. 129-159, 2001. https://doi.org/10.1137/S003614450037906X
  7. S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, "An iteration regularization method for total variation-based image restoration," SIAM Journal on Multiscale Modeling and Simulation, vol. 4, pp. 460-489, 2005. https://doi.org/10.1137/040605412
  8. W.Yin, M. Burger, D. Goldfarb and J. Darbon, "Bregman iteration algorithm for ${\iota}_1$ -minimization with applications to compressed sensing," SIAM Journal on Imaging Sciences, vol. 1, no.1, pp. 143-168, 2008. https://doi.org/10.1137/070703983
  9. J. F. Cai, S. Osher and Z. W. Shen, "Linearized Bregman iterations for compressed sensing," Mathematics of Computation, vol. 78, no. 267, pp. 1515-1536, 2009. https://doi.org/10.1090/S0025-5718-08-02189-3
  10. W. Yin, "Analysis and generalizations of the linearized Bregman method," SIAM Journal on Imaging Sciences, vol. 3, no.1 pp. 856-877, 2010.
  11. S. Osher, Y. Mao, B. Dong and W. Yin, "Fast linearized Bregman iterations for compressed sensing and sparse denoising," Communications in Mathematical Sciences, vol. 8, pp. 93-111, 2010. https://doi.org/10.4310/CMS.2010.v8.n1.a6
  12. B. Huang, S. Q. Ma and D. Goldfarb. "Accelerated linearized Bregman method," Journal of Scientific Computing, vol. 54, no.2-3, pp. 428-453, 2013. https://doi.org/10.1007/s10915-012-9592-9
  13. A. Beck and M. Teboulle, "A fast iterative shrinkage/thresholding algorithm for linear inverse problems," SIAM Journal on Imaging Sciences, vol. 2, no.1, pp. 183-202, 2009. https://doi.org/10.1137/080716542
  14. L. M. Bregman, "The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming," USSR Computational Mathematics and Mathematical Physics, vol. 7, no.4, pp. 200-217, 1967.
  15. R.O. Preda and D.N. Vizireanu, "A robust digital watermarking scheme for video copyright protection in the wavelet domain," Measurement, vol. 43, no.10, pp. 1720-1726, 2010. https://doi.org/10.1016/j.measurement.2010.07.009
  16. R.O. Preda and D.N. Vizireanu, "Quantisation-based video watermarking in the wavelet domain with spatial and temporal redundancy," International Journal of Electronics, vol. 98, no.3, pp. 393-405, 2011. https://doi.org/10.1080/00207217.2010.547810
  17. Z. Z. Yang and Z. Yang, " ${\iota}_0$-regularisation signal reconstruction based on fast alternating direction method of multipliers for compressed sensing," Journal of Electronics & Information Technology, vol. 35, no.4, pp. 826-831, 2013. http://jeit.ie.ac.cn/CN/abstract/abstract16337.shtml
  18. J. Tropp and A. Gilbert, "Signal recovery from random measurements via orthogonal matching pursuit," IEEE Transactions on Information Theory, vol. 53, no.12, pp. 4655-4666, 2007. https://doi.org/10.1109/TIT.2007.909108