The Convergence of Two Algorithms for Compressed Sensing Based Tomography


The constrained total variation minimization has been developed successfully for image reconstruction in computed tomography. In this paper, the block component averaging and diagonally-relaxed orthogonal projection methods are proposed to incorporate with the total variation minimization in the compressed sensing framework. The convergence of the algorithms under a certain condition is derived. Examples are given to illustrate their convergence behavior and noise performance.

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Li, X. and Zhu, J. (2012) The Convergence of Two Algorithms for Compressed Sensing Based Tomography. Advances in Computed Tomography, 1, 30-36. doi: 10.4236/act.2012.13007.

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The authors declare no conflicts of interest.


[1] A. C. Kak and M. Slaney, “Principles of Computerized Tomographic Imaging,” Society of Industrial and Applied Mathematics, Philadelphia, 2001. doi:10.1137/1.9780898719277
[2] P. B. Eggermont, G. T. Herman and A. Lent, “Iterative Algorithm for Larger Partitioned Linear Systems, with Applications to Image Reconstruction,” Linear Algebra and Its Applications, Vol. 40, 1981, pp. 37-67. doi:10.1016/0024-3795(81)90139-7
[3] G. Cimmino, “Calcolo Approssimato Per le Soluzioni dei Sistemi di Equazioni Lineari,” La Ricerca Scientifica, Series II, Vol. 9, 1938, pp. 326-333.
[4] Y. Censor, D. Gordan and R. Gordan, “Component Aveging: An Efficient Iterative Parallel Algorithm for Large and Sparse Unstructured Problems,” Parallel Computing, Vol. 27, No. 6, 2001, pp. 777-808. doi:10.1016/S0167-8191(00)00100-9
[5] Y. Censor, T. Elfving, G. T. Herman and T. Nikazad. “On Diagonally-Relaxed Orthogonal Projection Methods,” SIAM Journal on Scientific Computing, Vol. 30, No. 1, 2008, pp. 473-504. doi:10.1137/050639399
[6] R. Aharoni and Y. Censor, “Block-Iterative Projection Methods for Parallel Computation of Solutions to Convex Feasibility Problems,” Linear Algebra and Its Applications, Vol. 120, 1989, pp. 65-175. doi:10.1016/0024-3795(89)90375-3
[7] Y. Censor and Z. Stavors, “Parallel Optimization: Theory, Algorithms, and Applications,” Oxford University Press, Oxford, 1997.
[8] E. Candes, 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, 2006, pp. 489-509. doi:10.1109/TIT.2005.862083
[9] E. Candes and M. Wakin, “An Introduction to Compressive Sampling,” IEEE Signal Processing Magazine, Vol. 25, No. 2, 2008, pp. 21-30. doi:10.1109/MSP.2007.914731
[10] D. Donoho, “Compressed Sensing,” IEEE Transactions on Information Theory, Vol. 52, No. 4, 2006, pp. 1289-1306. doi:10.1109/TIT.2006.871582
[11] C. E. Shannon, “Communication in the Presence of Noise,” Proceedings of the IEEE, Vol. 86, No. 2, 1998, pp. 447-457. doi:10.1109/JPROC.1998.659497
[12] E. Candes and M. Wakin, “Enhancing Sparsity by Reweighted L1 Minimization,” Journal of Fourier Analysis and Applications, Vol. 14, No. 5-6, 2008, pp. 877-905. doi:10.1007/s00041-008-9045-x
[13] H. Yu and G. Wang, “Compressed Sensing Based Interior Tomography,” Physics in Medicine and Biology, Vol. 54, No. 9, 2009, pp. 2791-2805. doi:10.1088/0031-9155/54/9/014
[14] X. Li and J. Zhu, “Convergence of Block Cyclic Projection and Cimmino Algorithms for Compressed Sensing Based Tomography,” Journal of X-Ray Science and Technology, Vol. 18, No. 4, 2010, pp. 1-11.
[15] D. Butnariu, R. Davidi, G. T. Herman and I. G. Kazantsev, “Stable Convergence Behavior under Summable Perturbation of a Class Projection Methods for Convex Feasibility and Optimization Problems,” IEEE Journal of Selected Topics in Signal Processing, Vo. 1, No. 4, 2007, pp. 540547.
[16] J. Zhu, X. Li, Y. Ye and G. Wang, “Analysis on the StripBased Projection for Discrete Tomography,” Discrete Applied Mathematics, Vol. 156, No. 12, 2008, pp. 2359-2367. doi:10.1016/j.dam.2007.10.011
[17] X. Li and J. Zhu, “A Note of Reconstruction Algorithm of the Strip-Based Projection Model for Discrete Tomography,” Journal of X-Ray Science and Technology, Vol. 16, No. 4, 2008, pp. 253-260.

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