Mathematical Models for Magnetic Resonance Imaging Reconstruction: An Overview of the Approaches, Problems, and Future Research Areas

Since its inception in the early 1970s [1], magnetic resonance imaging (MRI) has revolutionized radiology and medicine. Apart from high-quality data acquisition, image reconstruction is an important step to guarantee high image quality in MRI. Although the very first MR images were obtained from data resembling radial projections of the imaged object by applying an iterative reconstruction algorithm [1], non-Cartesian acquisition and iterative reconstruction techniques were not adopted in clinical MRI for many years, and, even today, their use is very limited. The reason for this is twofold. First, the underlying assumption that the measured data are radial projections of the imaged object fails in the presence of B0 field inhomogeneity and/or gradient waveform imperfections. Second, the long reconstruction times associated with iterative reconstruction algorithms limit their practical application.