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Computational and Mathematical Methods in Medicine
Volume 2013, Article ID 985819, 16 pages
http://dx.doi.org/10.1155/2013/985819
Research Article

MR Image Reconstruction Based on Iterative Split Bregman Algorithm and Nonlocal Total Variation

1Department of Electronics and Communication Engineering, National Institute of Technology, Tiruchirappalli, Tamil Nadu 620015, India
2Department of Electrical and Computer Engineering, University of Saskatchewan, Saskatoon, SK, Canada
3Department of Medical Imaging, Royal University Hospital, University of Saskatchewan, Saskatoon, SK, Canada

Received 26 April 2013; Revised 26 June 2013; Accepted 11 July 2013

Academic Editor: Dimitri Van De Ville

Copyright © 2013 Varun P. Gopi 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. 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
  2. D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289–1306, 2006. View at Publisher · View at Google Scholar · View at Scopus
  3. L. He, T. C. Chang, S. Osher, T. Fang, and P. Speier, “MR image reconstruction by using the iterative refinement method and nonlinear inverse scale space methods,” UCLA CAM Report, 2006. View at Google Scholar
  4. 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
  5. 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 IEEE Conference on Computer Vision and Pattern Recognition (CVPR '08), pp. 1–8, Anchorage, Alaska, USA, June 2008. View at Publisher · View at Google Scholar · View at Scopus
  6. J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE Journal on Selected Topics in Signal Processing, vol. 4, no. 2, pp. 288–297, 2010. View at Publisher · View at Google Scholar · View at Scopus
  7. J. Huang, S. Zhang, and D. Metaxas, “Efficient MR image reconstruction for compressed MR imaging,” Medical Image Analysis, vol. 15, no. 5, pp. 670–679, 2011. View at Publisher · View at Google Scholar · View at Scopus
  8. T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM Journal on Imaging Sciences, vol. 2, no. 2, pp. 323–343, 2009. View at Google Scholar
  9. D. C. Dobson and F. Santosa, “Recovery of blocky images from noisy and blurred data,” SIAM Journal on Applied Mathematics, vol. 56, no. 4, pp. 1181–1198, 1996. View at Google Scholar · View at Scopus
  10. A. Chambolle and P. L. Lions, “Image recovery via total variation minimization and related problems,” Numerische Mathematik, vol. 76, no. 2, pp. 167–188, 1997. View at Google Scholar · View at Scopus
  11. Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures, vol. 22 of University Lecture Series, American Mathematical Society, Boston, Mass, USA, 2001.
  12. S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Modeling and Simulation, vol. 4, no. 2, pp. 460–489, 2005. View at Publisher · View at Google Scholar · View at Scopus
  13. Z. Tian, X. Jia, K. Yuan, T. Pan, and S. B. Jiang, “Low-dose CT reconstruction via edge-preserving total variation regularization,” Physics in Medicine and Biology, vol. 56, no. 18, pp. 5949–5967, 2011. View at Publisher · View at Google Scholar · View at Scopus
  14. G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Modeling and Simulation, vol. 7, no. 3, pp. 1005–1028, 2008. View at Publisher · View at Google Scholar · View at Scopus
  15. X. Zhang, M. Burger, X. Bresson, and S. Osher, “Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,” SIAM Journal on Imaging Sciences, vol. 3, no. 3, pp. 253–276, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. G. Peyre, S. Bougleux, and L. Cohen, “Non-local regularization of inverse problems,” in Computer Vision—ECCV 2008, vol. 5304 of Lecture Notes in Computer Science, pp. 57–68, 2008. View at Google Scholar
  17. Y. Lou, X. Zhang, S. Osher, and A. Bertozzi, “Image recovery via nonlocal operators,” Journal of Scientific Computing, vol. 42, no. 2, pp. 185–197, 2010. View at Publisher · View at Google Scholar · View at Scopus
  18. X. Zhang, M. Burger, X. Bresson, and S. Osher, “Bregmanized nonlocal regularization for deconvolution and sparse reconstruction,” SIAM Journal on Imaging Sciences, vol. 3, no. 3, pp. 253–276, 2010. View at Publisher · View at Google Scholar · View at Scopus
  19. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004.
  20. J. Nocedal and S. Wright, Numerical Optimization, Springer, New York, NY, USA, 2nd edition, 2006.
  21. 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. 3, pp. 200–217, 1967. View at Google Scholar · View at Scopus
  22. A. Osher, Y. Mao, B. Dong, and W. Yin, “Fast linearized Bregman iterations for compressed sensing and sparse denoising,” UCLA CAM Report, University of California at Los Angeles, Los Angeles, Calif, USA, 2008. View at Google Scholar
  23. L. He, T. C. Chang, and S. Osher, “MR image reconstruction from sparse radial samples by using iterative refinement procedures,” in Proceedings of the 13th Annual Meeting of ISMRM, p. 696, 2005.
  24. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, vol. 60, no. 1–4, pp. 259–268, 1992. View at Google Scholar · View at Scopus
  25. A. Chambolle, “An algorithm for total variation minimization and applications,” Journal of Mathematical Imaging and Vision, vol. 20, no. 1-2, pp. 89–97, 2004. View at Publisher · View at Google Scholar · View at Scopus
  26. M. Mignotte, “A non-local regularization strategy for image deconvolution,” Pattern Recognition Letters, vol. 29, no. 16, pp. 2206–2212, 2008. View at Publisher · View at Google Scholar · View at Scopus
  27. A. Elmoataz, O. Lezoray, and S. Bougleux, “Nonlocal discrete regularization on weighted graphs: a framework for image and manifold processing,” IEEE Transactions on Image Processing, vol. 17, no. 7, pp. 1047–1060, 2008. View at Publisher · View at Google Scholar · View at Scopus
  28. X. Jia, Y. Lou, B. Dong, Z. Tian, and S. Jiang, “4D computed tomography reconstruction from few-projection data via temporal non-local regularization,” in Medical Image Computing and Computer-Assisted Intervention—MICCAI 2010, vol. 6361 of Lecture Notes in Computer Science, pp. 143–150, 2010. View at Publisher · View at Google Scholar · View at Scopus
  29. A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Modeling and Simulation, vol. 4, no. 2, pp. 490–530, 2005. View at Publisher · View at Google Scholar · View at Scopus
  30. A. Spira, R. Kimmel, and N. Sochen, “A short-time Beltrami kernel for smoothing images and manifolds,” IEEE Transactions on Image Processing, vol. 16, no. 6, pp. 1628–1636, 2007. View at Publisher · View at Google Scholar · View at Scopus
  31. C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” in Proceedings of the 6th IEEE International Conference on Computer Vision, pp. 839–846, Bombay, India, January 1998. View at Publisher · View at Google Scholar · View at Scopus
  32. L. P. Yaroslavsky, Digital Picture Processing—An Introduction, Springer, Berlin, Germany, 1985.
  33. R. R. Coifman, S. Lafon, A. B. Lee et al., “Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps,” Proceedings of the National Academy of Sciences of the United States of America, vol. 102, no. 21, pp. 7426–7431, 2005. View at Publisher · View at Google Scholar · View at Scopus
  34. A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Modeling and Simulation, vol. 4, no. 2, pp. 490–530, 2005. View at Publisher · View at Google Scholar · View at Scopus
  35. A. Szlam, M. Maggioni, and R. R. Coifman, “A general framework for adaptive regularization based on diffusion processes on graphs,” Journal of Machine Learning Research, vol. 9, pp. 1711–1739, 2008. View at Google Scholar
  36. W. S. Geisler and M. S. Banks, “Visual performance,” in Handbook of Optics, vol. 1, McGraw-Hill, New York, NY, USA, 1995. View at Google Scholar
  37. A. B. Watson and L. B. Kreslake, “Measurement of visual impairment scales for digital video,” in 6th Human Vision and Electronic Imaging, vol. 4299 of Proceedings of SPIE, pp. 79–89, San Jose, Calif, USA, January 2001. View at Publisher · View at Google Scholar · View at Scopus
  38. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Transactions on Image Processing, vol. 13, no. 4, pp. 600–612, 2004. View at Publisher · View at Google Scholar · View at Scopus
  39. Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Processing Letters, vol. 9, no. 3, pp. 81–84, 2002. View at Publisher · View at Google Scholar · View at Scopus
  40. L. Zhang, L. Zhang, X. Mou, and D. Zhang, “FSIM: a feature similarity index for image quality assessment,” IEEE Transactions on Image Processing, vol. 20, no. 8, pp. 2378–2386, 2011. View at Publisher · View at Google Scholar · View at Scopus