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.

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.

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.

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.

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.

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.

Z. Opal, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bull. Amer. Math. Soc., Vol. 73, 1967, pp. 591-597.

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.

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.

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.

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.

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.

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.