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ISRN Computational Mathematics
Volume 2012 (2012), Article ID 982792, 12 pages
http://dx.doi.org/10.5402/2012/982792
Research Article

Nonconvex Compressed Sampling of Natural Images and Applications to Compressed MR Imaging

1College of Telecommunications and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210046, China
2School of Mathematics and Statistics, Nanjing Audit University, Nanjing 211815, China
3School of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing 210094, China

Received 25 July 2011; Accepted 5 September 2011

Academic Editors: K. T. Miura and E. Weber

Copyright © 2012 Wenze Shao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Skodras, C. Christopoulos, and T. Ebrahimi, “The JPEG 2000 still image compression standard,” IEEE Signal Processing Magazine, vol. 18, no. 5, pp. 36–58, 2001. View at Publisher · View at Google Scholar · View at Scopus
  2. T. Acharya and P. S. Tsai, JPEG 2000 Standard for Image Compression, John Wiley & Sons, Hoboken, NJ, USA, 2005.
  3. A. K. Katsaggelos, R. Molina, and J. Mateos, Super-Resolution of Images and Videos, Morgan and Claypool, 2007.
  4. S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Processing Magazine, vol. 20, no. 3, pp. 21–36, 2003. View at Publisher · View at Google Scholar · View at Scopus
  5. E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4203–4215, 2005. View at Publisher · View at Google Scholar · View at Scopus
  6. E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 489–509, 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Transactions on Information Theory, vol. 52, no. 12, pp. 5406–5425, 2006. View at Publisher · View at Google Scholar · View at Scopus
  8. E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics, vol. 59, no. 8, pp. 1207–1223, 2006. View at Publisher · View at Google Scholar · View at Scopus
  9. E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Problems, vol. 23, no. 3, article no. 008, pp. 969–985, 2007. View at Publisher · View at Google Scholar · View at Scopus
  10. E. Hale, W. Yin, and Y. Zhang, “A fixed-point continuation method for L1-regularized minimization with applications to compressed sensing,” Tech. Rep., Rice University, 2007.
  11. S. Ma, W. Yin, Y. Zhang, and A. Chakraborty, “An efficient algorithm for compressed MR imaging using total variation and wavelets,” in Proceedings of the 26th IEEE Conference on Computer Vision and Pattern Recognition (CVPR '08), pp. 1–8, June 2008. View at Publisher · View at Google Scholar · View at Scopus
  12. K. T. Block, M. Uecker, and J. Frahm, “Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint,” Magnetic Resonance in Medicine, vol. 57, no. 6, pp. 1086–1098, 2007. View at Publisher · View at Google Scholar · View at Scopus
  13. M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magnetic Resonance in Medicine, vol. 58, no. 6, pp. 1182–1195, 2007. View at Publisher · View at Google Scholar · View at Scopus
  14. C. Y. Jong, S. Tak, Y. Han, and W. P. Hyun, “Projection reconstruction MR imaging using FOCUSS,” Magnetic Resonance in Medicine, vol. 57, no. 4, pp. 764–775, 2007. View at Publisher · View at Google Scholar · View at Scopus
  15. H. Jung, J. C. Ye, and E. Kim, “Improved k-t BLAST and k-t SENSE using FOCUSS,” Physics in Medicine and Biology, vol. 52, no. 11, article 018, pp. 3201–3226, 2007. View at Publisher · View at Google Scholar · View at Scopus
  16. M. Seeger and H. Nickisch, “Compressed sensing and Bayesian experimental design,” in Proceedings of the 25th International Conference on Machine Learning, pp. 912–919, July 2008. View at Scopus
  17. J. Yang, Y. Zhang, and W. Yin, “A fast TVL1-L2 minimization algorithm for signal reconstruction from partial Fourier data,” Tech. Rep., Rice University, 2009.
  18. R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Signal Processing Letters, vol. 14, no. 10, pp. 707–710, 2007. View at Publisher · View at Google Scholar · View at Scopus
  19. E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique, vol. 346, no. 9-10, pp. 589–592, 2008. View at Publisher · View at Google Scholar · View at Scopus
  20. S. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “A method for large-scale L1-regularized least squares problems with applications in signal processing and statistics,” Tech. Rep., Stanford University, 2007.
  21. I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Communications on Pure and Applied Mathematics, vol. 57, no. 11, pp. 1413–1457, 2004. View at Publisher · View at Google Scholar · View at Scopus
  22. M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE Journal on Selected Topics in Signal Processing, vol. 1, no. 4, pp. 586–597, 2007. View at Publisher · View at Google Scholar · View at Scopus
  23. I. Daubechies, R. DeVore, M. Fornasier, and S. Güntürk, “Iteratively Re-weighted Least Squares minimization: proof of faster than linear rate for sparse recovery,” In Proceedings of the 42nd Annual Conference on Information Sciences and Systems, pp. 26–29, 2008.
  24. E. J. Candès, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted1 minimization,” Journal of Fourier Analysis and Applications, vol. 14, no. 5-6, pp. 877–905, 2008. View at Publisher · View at Google Scholar · View at Scopus
  25. S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Transactions on Signal Processing, vol. 56, no. 6, pp. 2346–2356, 2008. View at Publisher · View at Google Scholar · View at Scopus
  26. S. D. Babacan, R. Molina, and A. K. Katsaggelos, “Bayesian compressive sensing using laplace priors,” IEEE Transactions on Image Processing, vol. 19, no. 1, Article ID 5256324, pp. 53–63, 2010. View at Publisher · View at Google Scholar · View at Scopus
  27. J. L. Starck, M. Elad, and D. L. Donoho, “Image decomposition via the combination of sparse representations and a variational approach,” IEEE Transactions on Image Processing, vol. 14, no. 10, pp. 1570–1582, 2005. View at Publisher · View at Google Scholar · View at Scopus
  28. Y. Meyer, Oscillating Patterns in IImage Processing and Nonlinear Evolution Equation, vol. 22 of University Lecture Series, American Mathematical Society, 2001.
  29. R. Chartrand and W. Yin, “Iteratively reweighted algorithms for compressive sensing,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 3869–3872, April 2008. View at Publisher · View at Google Scholar · View at Scopus
  30. G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer, New York, NY, USA, 2000.
  31. E. Esser, X. Zhang, and T. Chan, “A general framework for a class of first order primal-dual algorithms for TV minimization,” Tech. Rep., UCLA, 2009.
  32. E. J. Candès and M. B. Wakin, “An introduction to compressive sampling: a sensing/sampling paradigm that goes against the common knowledge in data acquisition,” IEEE Signal Processing Magazine, vol. 25, no. 2, pp. 21–30, 2008. View at Publisher · View at Google Scholar · View at Scopus