Undersampled CS image reconstruction using nonconvex nonsmooth mixed constraints

Compressed sensing magnetic resonance imaging (CS-MRI) has attracted considerable attention due to its great potential in reducing scanning time and guaranteeing high-quality reconstruction. In conventional CS-MRI framework, the total variation (TV) penalty and L1-norm constraint on wavelet coefficients are commonly combined to reduce the reconstruction error. However, TV sometimes tends to cause staircase-like artifacts due to its nature in favoring piecewise constant solution. To overcome the model-dependent deficiency, a hybrid TV (TV^1,2) regularizer is introduced in this paper by combining TV with its second-order version (TV²). It is well known that the wavelet coefficients of MR images are not only approximately sparse, but also have the property of tree-structured hierarchical sparsity. Therefore, a L0-regularized tree-structured sparsity constraint is proposed to better represent the measure of sparseness in wavelet domain. In what follows, we present our new CS-MRI framework by combining the TV^1,2 regularizer and L0-regularized tree-structured sparsity constraint. However, the combination makes CS-MRI problem difficult to handle due to the nonconvex and nonsmooth natures of mixed constraints. To achieve solution stability, the resulting composite minimization problem is decomposed into several simpler subproblems. Each of these subproblems has a closed-form solution or could be efficiently solved using existing numerical method. The results from simulation and in vivo experiments have demonstrated the good performance of our proposed method compared with several conventional MRI reconstruction methods.