Magnetic Resonance Imaging (MRI) is unique in its ability to image human tissue with varying contrasts and to probe the physical properties of the tissue. In addition, MRI signal is often redundant (or sparse) in nature, especially over time with short update intervals. The goal of this thesis was to better understand and utilize T1relaxation time mapping schemes in the heart to ultimately image infiltrative disease. A second aim was to improve upon state of the art methods for high frame rate liver imaging by utilizing local (patch) based constraints. iii