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中文摘要：
提出小波稀疏的MR图像重构的交替最小化方法，分析证明了这一方法的收敛性。利用半二次罚函数方法将小波稀疏的MR图像重构最优化问题分裂成两个子最优化问题：X-子问题和Y-子问题，通过对两个子问题的交替最小化得到原问题的最优解。利用1维软阈值收缩方法求解Y-子问题，利用Fourier变换的方法求解X-子问题解，进而给出原问题求解的分裂算法。利用Phantom图像和一些实际的MR图像与最新的算子分裂算法进行数值实验比较，其结果是交替最小化方法重构的图像的信噪比比算子分裂算法的高，而相对误差和CPU时间较低，从而表明交替最小化方法是稀疏MR图像重构的一种快速算法。
英文摘要：
An alternating minimization method for the reconstruction of MR images with wavelet sparsity, including the convergence analysis of this algorithm, is presented in this paper. Our algorithm is base on a half-quadratic penalty method. The optimization problem with wavelet sparsity is splited into two sub-problems： the X-subproblem and the Y-subproblem. The solution for the original problem can be obtained by alternately solving the two subproblems, in which the Y-subproblem is solved via a 1 D soft-thresholding or shrinkage and the X-subproblem is solved by a Fourier transform approach. Thus, the splitting algorithm for solving the original problem is generated. The phantom image and some real MR images are employed to test our approach in the numerical experiments. The alternating minimization algorithm is also compared to the state-of-the-art algorithm, operator splitting algorithm. The experimental results demonstrate that the alternating minimization algorithm has not only a greater signal-to-noise ratio（SNR） ,but has also less relative errors and is faster than the operator splitting algorithm. Therefore the alternating minimization method is a fast reconstruction method for sparse MR images.
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