TV Sparsifying MR Image Reconstruction in Compressive Sensing


In this paper, we apply alternating minimization method to sparse image reconstruction in compressed sensing. This approach can exactly reconstruct the MR image from under-sampled k-space data, i.e., the partial Fourier data. The convergence analysis of the fast method is also given. Some MR images are employed to test in the numerical experi-ments, and the results demonstrate that our method is very efficient in MRI reconstruction.

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Zhu, Y. and Yang, X. (2011) TV Sparsifying MR Image Reconstruction in Compressive Sensing. Journal of Signal and Information Processing, 2, 44-51. doi: 10.4236/jsip.2011.21007.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, Vol. 52, 2006, pp. 489-509.
[2] D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, Vol. 52, 2006, pp. 1289-1306.
[3] E. Candes and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Problems, Vol. 23, 2007, pp. 969-985.
[4] B. K. Natarajan, “Sparse approximate solutions to linear systems,” SIAM J. Comput., Vol. 24, 1995, pp.227-234.
[5] D. Donoho and X. Huo, “Uncertainty principles and ideal atomic decompositions,” IEEE Trans. Inf.Theory, Vol. 47, 2001, pp. 2845-2862.
[6] J. J. Fuchs, “On sparse representations in arbitrary redundant bases,” IEEE Trans. Inf. Theory, Vol. 50, 2004, pp. 1341-1344.
[7] L. Rudin, S. Osher and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D, Vol. 60, 1992, pp. 259-268.
[8] 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, 2006,06-35.
[9] M. Lustig, D. Donoho and J. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magnetic Resonance in Medicine, Vol. 58,2007,pp. 1182-1195.
[10] Y. Wang, J. Yang, W. Yin and Y. Zhang, “A alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imag. Sci., Vol. 1, 2008,pp. 248-272.
[11] R. Courant, “Variational methods for the solution of problems with equilibrium and vibration,” Bull.Amer. Math. Soc., Vol. 49, 1943, pp. 1-23.
[12] M. K. Ng, R. H. Chan and W. C. Tang, “A fast algorithm for deblurring models with Neumann boundary conditions,” SIAM J. Sci. Comput., Vol. 21, pp. 851-866, 1999.
[13] Z. Opal, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bull. Amer. Math. Soc., Vol. 73, 1967, pp. 591-597.
[14] J. Yang, W. Yin and Y.Wang, “A Fast Algorithm for Edge-Preserving Variational Multichannel Image Restoration,” SIAM J. Sci. Comput., Vol. 2, 2009, pp. 569-592.
[15] J. Yang, Y. Zhang and W. Yin, “A Fast Alternating Direction Method for TVL1-L2 Signal Reconstruction From Partial Fourier Data,” IEEE Journal of Selected Topics in Signal Processing, Vol. 4, 2010, pp.288-297.
[16] E. T. Hale, W. Yin and Y. Zhang, “A Fixed-Point Continuation for l1?regularization with Application to Compressed Sensing,” Rice University CAAM Technical Report TR07-07, 2007, pp. 1-45.
[17] E. T. Hale, W. Yin and Y. Zhang, “Fixed-Point Continuation for l1?Minimization: Methodology and Convergence,” SIAM J. Sci. Comput., Vol. 2, 2009, pp. 569-592.
[18] J. Bioucas-Dias and M. Figueiredo, “A new TwIST: Two-step iterative thresholding algorithm for image restoration,” IEEE Trans. Imag. Process., Vol. 16, 2007, pp. 2992-3004.
[19] I. Daubechies, M. Defriese and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math., Vol. 57, 2004, pp. 1413-1457.
[20] A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision, Vol. 20, 2004, pp. 89-97.

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