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Computational and Mathematical Methods in Medicine
Volume 2014, Article ID 134058, 11 pages
http://dx.doi.org/10.1155/2014/134058
Research Article

Sparse-Representation-Based Direct Minimum -Norm Algorithm for MRI Phase Unwrapping

1Department of Biomedical Engineering, Zhejiang University, Hangzhou 310027, China
2School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia

Received 6 December 2013; Revised 8 February 2014; Accepted 12 February 2014; Published 26 March 2014

Academic Editor: Xiaobo Qu

Copyright © 2014 Wei He 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. J. Szumowski, W. R. Coshow, F. Li, and S. F. Quinn, “Phase unwrapping in the three-point Dixon method for fat suppression MR imaging,” Radiology, vol. 192, no. 2, pp. 555–561, 1994. View at Google Scholar · View at Scopus
  2. A. Rauscher, M. Barth, J. R. Reichenbach, R. Stollberger, and E. Moser, “Automated unwrapping of MR phase images applied to BOLD MR-venography at 3 tesla,” Journal of Magnetic Resonance Imaging, vol. 18, no. 2, pp. 175–180, 2003. View at Publisher · View at Google Scholar · View at Scopus
  3. H. Bagher-Ebadian, Q. Jiang, and J. R. Ewing, “A modified fourier-based phase unwrapping algorithm with an application to MRI venography,” Journal of Magnetic Resonance Imaging, vol. 27, no. 3, pp. 649–652, 2008. View at Publisher · View at Google Scholar · View at Scopus
  4. P. Chiang, Y. Cai, K. H. Mak, and J. Zheng, “A B-spline approach to phase unwrapping in tagged cardiac MRI for motion tracking,” Magnetic Resonance in Medicine, vol. 69, pp. 1297–1309, 2013. View at Publisher · View at Google Scholar
  5. P. Munger, G. R. Greller, T. M. Peters, and G. B. Pike, “An inverse problem approach to the correction of distortion in EPI images,” IEEE Transactions on Medical Imaging, vol. 19, no. 7, pp. 681–689, 2000. View at Google Scholar · View at Scopus
  6. X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Optics and Lasers in Engineering, vol. 42, no. 3, pp. 245–261, 2004. View at Publisher · View at Google Scholar · View at Scopus
  7. M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Applied Optics, vol. 41, no. 35, pp. 7437–7444, 2002. View at Google Scholar · View at Scopus
  8. S. Fang, L. Meng, L. Wang, P. Yang, and M. Komori, “Quality-guided phase unwrapping algorithm based on reliability evaluation,” Applied Optics, vol. 50, no. 28, pp. 5446–5452, 2011. View at Publisher · View at Google Scholar · View at Scopus
  9. K. Zhou, M. Zaitsev, and S. Bao, “Reliable two-dimensional phase unwrapping method using region growing and local linear estimation,” Magnetic Resonance in Medicine, vol. 62, no. 4, pp. 1085–1090, 2009. View at Publisher · View at Google Scholar · View at Scopus
  10. S. Witoszynskyj, A. Rauscher, J. R. Reichenbach, and M. Barth, “Phase unwrapping of MR images using ΦUN—a fast and robust region growing algorithm,” Medical Image Analysis, vol. 13, no. 2, pp. 257–268, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Science, vol. 23, no. 4, pp. 713–720, 1988. View at Google Scholar · View at Scopus
  12. D. C. Ghiglia and L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” Journal of the Optical Society of America A, vol. 13, no. 10, pp. 1999–2013, 1996. View at Google Scholar · View at Scopus
  13. T. Mario Costantini, “A novel phase unwrapping method based on network programming,” IEEE Transactions on Geoscience and Remote Sensing, vol. 36, no. 3, pp. 813–821, 1998. View at Publisher · View at Google Scholar · View at Scopus
  14. C. W. Chen and H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: Intractability and two new algorithms,” Journal of the Optical Society of America A, vol. 17, no. 3, pp. 401–414, 2000. View at Google Scholar · View at Scopus
  15. J. M. B. Dias and J. M. N. Leitäo, “The πM algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Transactions on Image Processing, vol. 11, no. 4, pp. 408–422, 2002. View at Publisher · View at Google Scholar · View at Scopus
  16. J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Transactions on Image Processing, vol. 16, no. 3, pp. 698–709, 2007. View at Publisher · View at Google Scholar · View at Scopus
  17. K. Zhang, L. Ge, Z. Hu, A. H.-M. Ng, X. Li, and C. Rizos, “Phase unwrapping for very large interferometric data sets,” IEEE Transactions on Geoscience and Remote Sensing, vol. 49, no. 10, pp. 4048–4061, 2011. View at Publisher · View at Google Scholar · View at Scopus
  18. D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” Journal of the Optical Society of America A, vol. 11, no. 1, pp. 107–117, 1994. View at Google Scholar · View at Scopus
  19. G. Nico, G. Palubinskas, and M. Datcu, “Bayesian approaches to phase unwrapping: theoretical study,” IEEE Transactions on Signal Processing, vol. 48, no. 9, pp. 2545–2556, 2000. View at Google Scholar · View at Scopus
  20. L. Ying, Z.-P. Liang, D. C. Munson Jr., R. Koetter, and B. J. Frey, “Unwrapping of MR phase images using a Markov random field model,” IEEE Transactions on Medical Imaging, vol. 25, no. 1, pp. 128–136, 2006. View at Publisher · View at Google Scholar · View at Scopus
  21. Z.-P. Liang, “A model-based method for phase unwrapping,” IEEE Transactions on Medical Imaging, vol. 15, no. 6, pp. 893–897, 1996. View at Publisher · View at Google Scholar · View at Scopus
  22. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software, John Wiley & Sons, New York, NY, USA, 1998.
  23. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” Journal of the Optical Society of America A, vol. 14, no. 10, pp. 2692–2701, 1997. View at Google Scholar · View at Scopus
  24. G. Dardyk and I. Yavneh, “A multigrid approach to two-dimensional phase unwrapping,” Numerical Linear Algebra with Applications, vol. 11, no. 2-3, pp. 241–259, 2004. View at Publisher · View at Google Scholar · View at Scopus
  25. I. Shalem and I. Yavneh, “A multilevel graph algorithm for two dimensional phase unwrapping,” Computing and Visualization in Science, vol. 11, no. 2, pp. 89–100, 2008. View at Publisher · View at Google Scholar · View at Scopus
  26. P. Mistry, S. Braganza, D. Kaeli, and M. Leeser, “Accelerating phase unwrapping and affine transformations for optical quadrature microscopy using CUDA,” in Proceedings of the 2nd Workshop on General Purpose Processing on Graphics Processing Units (GPGPU '09), pp. 28–37, Washington, DC, USA, March 2009. View at Publisher · View at Google Scholar · View at Scopus
  27. R. P. Tewarson, Sparse Matrices, Academic Press, New York, NY, USA, 1973.
  28. U. W. Pooch and A. Nieder, “A survey of indexing techniques for sparse matrices,” ACM Computing Surveys, vol. 5, pp. 109–133, 1973. View at Publisher · View at Google Scholar
  29. J. A. Scott and Y. Hu, “Experiences of sparse direct symmetric solvers,” ACM Transactions on Mathematical Software, vol. 33, no. 3, Article ID 1268772, p. 18, 2007. View at Publisher · View at Google Scholar · View at Scopus
  30. D. S. Watkins, Fundamentals of Matrix Computations, vol. 64, John Wiley & Sons, New York, NY, USA, 2nd edition, 2002.
  31. A. Krishnamoorthy and D. Menon, “Matrix inversion using Cholesky decomposition,” http://arxiv.org/abs/1111.4144.
  32. I. S. Duff, A. M. Erisman, and J. K. Reid, Direct Methods for Sparse Matrices, Oxford University Press, New York, NY, USA, 1986.
  33. N. I. M. Gould, J. A. Scott, and Y. Hu, “A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations,” ACM Transactions on Mathematical Software, vol. 33, no. 2, Article ID 1236465, p. 10, 2007. View at Publisher · View at Google Scholar · View at Scopus
  34. Y. Chen, T. A. Davis, W. W. Hager, and S. Rajamanickam, “Algorithm 887: CHOLMOD, supernodal sparse cholesky factorization and update/downdate,” ACM Transactions on Mathematical Software, vol. 35, no. 3, article 22, pp. 1–14, 2008. View at Publisher · View at Google Scholar · View at Scopus
  35. I. S. Duff, “MA57—a code for the solution of sparse symmetric definite and indefinite systems,” ACM Transactions on Mathematical Software, vol. 30, no. 2, pp. 118–144, 2004. View at Publisher · View at Google Scholar · View at Scopus
  36. S. Chavez, Q.-S. Xiang, and L. An, “Understanding phase maps in MRI: a new cutline phase unwrapping method,” IEEE Transactions on Medical Imaging, vol. 21, no. 8, pp. 966–977, 2002. View at Publisher · View at Google Scholar · View at Scopus
  37. K. Chen, J. Xi, Y. Yu, and J. F. Chicharo, “Fast quality-guided flood-fill phase unwrapping algorithm for three-dimensional fringe pattern profilometry,” in Optical Metrology and Inspection for Industrial Applications, vol. 78550X of Proceedings of the SPIE, Beijing, China, October 2010. View at Publisher · View at Google Scholar · View at Scopus
  38. W. He, Y. Cheng, L. Xia, and F. Liu, “A new particle swarm optimization-based method for phase unwrapping of MRI data,” Computational and Mathematical Methods in Medicine, vol. 2012, Article ID 475745, 9 pages, 2012. View at Publisher · View at Google Scholar
  39. B. Spottiswoode, “2D phase unwrapping algorithms,” http://www.mathworks.com/matlabcentral/fileexchange/22504.