Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 192895, 11 pages
http://dx.doi.org/10.1155/2013/192895
Research Article

Nonlocal Regularized Algebraic Reconstruction Techniques for MRI: An Experimental Study

Lane Department of Computer Science and Electrical Engineering, West Virginia University, Morgantown, WV 26506-6109, USA

Received 2 April 2013; Revised 20 May 2013; Accepted 26 May 2013

Academic Editor: Ming Li

Copyright © 2013 Xin Li. 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. X. Pan, E. Y. Sidky, and M. Vannier, “Why do commercial CT scanner still employ traditional, filtered back-projection for image reconstruction?” Inverse Problems, vol. 25, no. 12, Article ID 123009, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  2. R. Gordon, R. Bender, and G. T. Herman, “Algebraic Reconstruction Techniques (ART) for three-dimensional electron microscopy and X-ray photography,” Journal of Theoretical Biology, vol. 29, no. 3, pp. 471–481, 1970. View at Google Scholar · View at Scopus
  3. A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the art algorithm,” Ultrasonic Imaging, vol. 6, no. 1, pp. 81–94, 1984. View at Google Scholar · View at Scopus
  4. T. G. Herman and B. L. Meyer, “Algebraic reconstruction techniques can be made computationally efficient,” IEEE Transactions on Medical Imaging, vol. 12, no. 3, pp. 600–609, 1993. View at Publisher · View at Google Scholar · View at Scopus
  5. M. Akay and C. Mello, “Wavelets for biomedical signal processing,” in Proceedings of the 19th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, vol. 6, pp. 2688–2691, November 1997. View at Scopus
  6. 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
  7. J. D. O'Sullivan, “A fast sinc function gridding algorithm for Fourier inversion in computer tomography,” IEEE Transactions on Medical Imaging, vol. 4, no. 4, pp. 200–207, 1985. View at Google Scholar · View at Scopus
  8. Q. Shan, J. Jia, and A. Agarwala, “High-quality motion deblurring from a single image,” ACM SIGGRAPH 2008 Papers, 2008. View at Google Scholar
  9. B. P. Sutton, D. C. Noll, and J. A. Fessler, “Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities,” IEEE Transactions on Medical Imaging, vol. 22, no. 2, pp. 178–188, 2003. View at Publisher · View at Google Scholar · View at Scopus
  10. R. Chartrand, E. Y. Sidky, and X. Pan, “Frequency extrapolation by nonconvex compressive sensing,” in Proceedings of the 8th IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI '11), pp. 1056–1060, April 2011. View at Publisher · View at Google Scholar · View at Scopus
  11. C. E. Shannon, “A mathematical theory of communication,” The Bell System Technical Journal, vol. 27, pp. 379–423; 623–656, 1948. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Transactions on Signal Processing, vol. 51, no. 2, pp. 560–574, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  13. A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM Journal on Scientific Computing, vol. 14, no. 6, pp. 1368–1393, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. N. Nguyen and Q. H. Liu, “The regular Fourier matrices and nonuniform fast Fourier transforms,” SIAM Journal on Scientific Computing, vol. 21, no. 1, pp. 283–293, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S. Kaczmarz, “Angenherte auflsung von systemen linearer gleichungen,” Bulletin International de l'Académie Polonaise des Sciences A, vol. 355, p. 357, 1937. View at Google Scholar
  16. M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” Journal of Research of the National Bureau of Standards, vol. 49, pp. 409–436, 1952. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. H. J. Trussell and M. R. Civanlar, “The landweber iteration and projection onto convex sets,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 33, no. 6, pp. 1632–1634, 1985. View at Google Scholar · View at Scopus
  18. E. Kreyszig, Introductory Functional Analysis with Applications, vol. 130, John Wiley & Sons, New York, NY, USA, 1989. View at MathSciNet
  19. J. Biemond, R. L. Lagendijk, and R. M. Mersereau, “Iterative methods for image deblurring,” Proceedings of the IEEE, vol. 78, no. 5, pp. 856–883, 1990. View at Publisher · View at Google Scholar · View at Scopus
  20. G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins, Baltimore, Md, USA, 2nd edition, 1989. View at MathSciNet
  21. A. Jain, Fundamentals of Digital Image Processing, Prentice Hall, Upper Saddle River, NJ, USA, 1989.
  22. Z.-P. Liang and P. C. Lauterbur, Principles of Magnetic Resonance Imaging: A Signal Processing Perspective, IEEE Press, Piscataway, NJ, USA, 1999.
  23. F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  24. X. Li, “Fine-granularity and spatially-adaptive regularization for projection-based image deblurring,” IEEE Transactions on Image Processing, vol. 20, no. 4, pp. 971–983, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  25. N. P. Galatsanos and A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Transactions on Image Processing, vol. 1, no. 3, pp. 322–336, 1992. View at Publisher · View at Google Scholar · View at Scopus
  26. D. F. Yu and J. A. Fessler, “Edge-preserving tomographic reconstruction with nonlocal regularization,” IEEE Transactions on Medical Imaging, vol. 21, no. 2, pp. 159–173, 2002. View at Publisher · View at Google Scholar · View at Scopus
  27. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  28. X. Li, “The magic of nonlocal Perona-Malik diffusion,” IEEE Signal Processing Letters, vol. 18, no. 9, pp. 533–534, 2011. View at Publisher · View at Google Scholar · View at Scopus
  29. P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629–639, 1990. View at Publisher · View at Google Scholar · View at Scopus
  30. A. Buades, B. Coll, and J.-M. Morel, “A non-local algorithm for image denoising,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR '05), vol. 2, pp. 60–65, June 2005. View at Publisher · View at Google Scholar · View at Scopus
  31. M. Elad and M. Aharon, “Image denoising via sparse and redundant representations over learned dictionaries,” IEEE Transactions on Image Processing, vol. 15, no. 12, pp. 3736–3745, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  32. K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Transactions on Image Processing, vol. 16, no. 8, pp. 2080–2095, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  33. 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 Zentralblatt MATH · View at MathSciNet
  34. J. Mairal, F. Bach, J. Ponce, G. Sapiro, and A. Zisserman, “Non-local sparse models for image restoration,” in Proceedings of the 12th International Conference on Computer Vision (ICCV '09), pp. 2272–2279, October 2009. View at Publisher · View at Google Scholar · View at Scopus
  35. W. Dong, X. Li, L. Zhang, and G. Shi, “Sparsity-based image denoising via dictionary learning and structural clustering,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR '11), pp. 457–464, June 2011. View at Publisher · View at Google Scholar · View at Scopus
  36. K. Egiazarian, A. Foi, and V. Katkovnik, “Compressed sensing image reconstruction via recursive spatially adaptive filtering,” in Proceedings of the 14th IEEE International Conference on Image Processing (ICIP '07), vol. 1, pp. I549–I552, San Antonio, Tex, USA, September 2007. View at Publisher · View at Google Scholar · View at Scopus
  37. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Transactions on Circuits and Systems, vol. 25, no. 9, pp. 694–702, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. J. Portilla and E. P. Simoncelli, “Parametric texture model based on joint statistics of complex wavelet coefficients,” International Journal of Computer Vision, vol. 40, no. 1, pp. 49–71, 2000. View at Publisher · View at Google Scholar · View at Scopus
  39. S. Ramani and J. A. Fessler, “Parallel MR image reconstruction using augmented lagrangian methods,” IEEE Transactions on Medical Imaging, vol. 30, no. 3, pp. 694–706, 2011. View at Publisher · View at Google Scholar · View at Scopus
  40. A. Blake and A. Zisserman, Visual Reconstruction, MIT Press, Cambridge, Mass, USA, 1987. View at MathSciNet
  41. L. Mancera and J. Portilla, “Non-convex sparse optimization through deterministic annealing and applications,” in Proceedings of the IEEE International Conference on Image Processing (ICIP '08), pp. 917–920, October 2008. View at Publisher · View at Google Scholar · View at Scopus
  42. R. T. Rockafellar, “Augmented Lagrange multiplier functions and duality in nonconvex programming,” SIAM Journal on Control and Optimization, vol. 12, pp. 268–285, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. J. I. Jackson, C. H. Meyer, D. G. Nishimura, and A. Macovski, “Selection of a convolution function for Fourier inversion using gridding,” IEEE Transactions on Medical Imaging, vol. 10, no. 3, pp. 473–478, 1991. View at Publisher · View at Google Scholar · View at Scopus
  44. H. Schomberg and J. Timmer, “Gridding method for image reconstruction by Fourier transformation,” IEEE Transactions on Medical Imaging, vol. 14, no. 3, pp. 596–607, 1995. View at Publisher · View at Google Scholar · View at Scopus
  45. P. J. Beatty, D. G. Nishimura, and J. M. Pauly, “Rapid gridding reconstruction with a minimal oversampling ratio,” IEEE Transactions on Medical Imaging, vol. 24, no. 6, pp. 799–808, 2005. View at Publisher · View at Google Scholar · View at Scopus
  46. D. Rosenfeld, “An optimal and efficient new gridding algorithm using singular value decomposition,” Magnetic Resonance in Medicine, vol. 40, no. 1, pp. 14–23, 1998. View at Publisher · View at Google Scholar · View at Scopus
  47. K. P. Pruessmann, M. Weiger, P. Börnert, and P. Boesiger, “Advances in sensitivity encoding with arbitrary k-space trajectories,” Magnetic Resonance in Medicine, vol. 46, no. 4, pp. 638–651, 2001. View at Publisher · View at Google Scholar · View at Scopus
  48. E. N. Yeh, C. A. McKenzie, M. A. Ohliger, and D. K. Sodickson, “Parallel magnetic resonance imaging with adaptive radius in k-space (PARS): constrained image reconstruction using k-space locality in radiofrequency coil encoded data,” Magnetic Resonance in Medicine, vol. 53, no. 6, pp. 1383–1392, 2005. View at Publisher · View at Google Scholar · View at Scopus
  49. N. Seiberlich, F. A. Breuer, M. Blaimer, K. Barkauskas, P. M. Jakob, and M. A. Griswold, “Non-Cartesian data reconstruction using GRAPPA operator gridding (GROG),” Magnetic Resonance in Medicine, vol. 58, no. 6, pp. 1257–1265, 2007. View at Publisher · View at Google Scholar · View at Scopus
  50. A. A. Samsonov, “On optimality of parallel MRI reconstruction in k-space,” Magnetic Resonance in Medicine, vol. 59, no. 1, pp. 156–164, 2008. View at Publisher · View at Google Scholar · View at Scopus
  51. 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
  52. X. Ye, Y. Chen, W. Lin, and F. Huang, “Fast MR image reconstruction for partially parallel imaging with arbitrary k-space trajectories,” IEEE Transactions on Medical Imaging, vol. 30, no. 3, pp. 575–585, 2011. View at Publisher · View at Google Scholar · View at Scopus
  53. D. K. Sodickson and W. J. Manning, “Simultaneous acquisition of spatial harmonics (SMASH): fast imaging with radiofrequency coil arrays,” Magnetic Resonance in Medicine, vol. 38, no. 4, pp. 591–603, 1997. View at Publisher · View at Google Scholar · View at Scopus
  54. K. P. Pruessmann, M. Weiger, M. B. Scheidegger, and P. Boesiger, “SENSE: sensitivity encoding for fast MRI,” Magnetic Resonance in Medicine, vol. 42, pp. 952–962, 1999. View at Google Scholar
  55. S. Ahn and R. M. Leahy, “Analysis of resolution and noise properties of nonquadratically regularized image reconstruction methods for PET,” IEEE Transactions on Medical Imaging, vol. 27, no. 3, pp. 413–424, 2008. View at Publisher · View at Google Scholar · View at Scopus
  56. 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 MathSciNet
  57. S. Ravishankar and Y. Bresler, “MR image reconstruction from highly undersampled k-space data by dictionary learning,” IEEE Transactions on Medical Imaging, vol. 30, no. 5, pp. 1028–1041, 2011. View at Publisher · View at Google Scholar · View at Scopus
  58. P. Vandewalle, J. Kovacević, and M. Vetterli, “Reproducible research in signal processing: what, why, and how,” IEEE Signal Processing Magazine, vol. 26, no. 3, pp. 37–47, 2009. View at Publisher · View at Google Scholar · View at Scopus