  Magnetic Resonance Imaging (MRI) is a kind of medical imaging technology used
for diagnostic imaging of diseases, but its image quality may be suffered by
the long acquisition time. The compressive sensing (CS) based strategy may
decrease the reconstruction time greatly, but it needs efficient reconstruction
algorithms to produce high-quality and reliable images. This paper focuses on
the algorithmic improvement for the sparse reconstruction of CS-MRI, especially
considering a non-smooth convex minimization problem which is composed of the
sum of a total variation regularization term and a $\ell_1$-norm term of the
wavelet transformation. The partly motivation of targeting the dual problem is
that the dual variables are involved in relatively low-dimensional subspace.
Instead of solving the primal model as usual, we turn our attention to its
associated dual model composed of three variable blocks and two separable
non-smooth function blocks. However, the directly extended alternating
direction method of multipliers (ADMM) must be avoided because it may be
divergent, although it usually performs well numerically. In order to solve the
problem, we employ a symmetric Gauss-Seidel (sGS) technique based ADMM.
Compared with the directly extended ADMM, this method only needs one additional
iteration, but its convergence can be guaranteed theoretically. Besides, we
also propose a generalized variant of ADMM because this method has been
illustrated to be efficient for solving semidefinite programming in the past
few years. Finally, we do extensive experiments on MRI reconstruction using
some simulated and real MRI images under different sampling patterns and
ratios. The numerical results demonstrate that the proposed algorithms
significantly achieve high reconstruction accuracies with fast computational
speed.

Optimization and Control