Optimization methods for MR image reconstruction

The development of compressed sensing methods for MR image reconstruction led to an explosion of research on models and optimization algorithms for MRI. Roughly 10 years after such methods first appeared in the MRI literature, the US FDA approved the commercial use of certain compressed sensing methods, 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. One impetus for this paper is that "off the shelf" optimization methods have rarely been the best choice for solving optimization problems in MR image reconstruction, due to the large volume of MRI data collected by clinical systems and practical time constraints on processing time. Instead, special purpose 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.