  The development of compressed sensing methods for magnetic resonance (MR)
image reconstruction led to an explosion of research on models and optimization
algorithms for MR imaging (MRI). Roughly 10 years after such methods first
appeared in the MRI literature, the U.S. Food and Drug Administration (FDA)
approved certain compressed sensing methods for commercial use, making
compressed sensing a clinical success story for MRI. This review paper
summarizes several key models and optimization algorithms for MR image
reconstruction, including both the type of methods that have FDA approval for
clinical use, as well as more recent methods being considered in the research
community that use data-adaptive regularizers. Many algorithms have been
devised that exploit the structure of the system model and regularizers used in
MRI; this paper strives to collect such algorithms in a single survey. Many of
the ideas used in optimization methods for MRI are also useful for solving
other inverse problems.
