  Magnetic Resonance Imaging (MRI) is one of the fields that the compressed
sensing theory is well utilized to reduce the scan time significantly leading
to faster imaging or higher resolution images. It has been shown that a small
fraction of the overall measurements are sufficient to reconstruct images with
the combination of compressed sensing and parallel imaging. Various
reconstruction algorithms has been proposed for compressed sensing, among which
Augmented Lagrangian based methods have been shown to often perform better than
others for many different applications. In this paper, we propose new Augmented
Lagrangian based solutions to the compressed sensing reconstruction problem
with analysis and synthesis prior formulations. We also propose a computational
method which makes use of properties of the sampling pattern to significantly
improve the speed of the reconstruction for the proposed algorithms in
Cartesian sampled MRI. The proposed algorithms are shown to outperform earlier
methods especially for the case of dynamic MRI for which the transfer function
tends to be a very large matrix and significantly ill conditioned. It is also
demonstrated that the proposed algorithm can be accelerated much further than
other methods in case of a parallel implementation with graphics processing
units (GPUs).
