Retrospective motion correction techniques offer minimal disruptions to sequences and clinical workflows. The computational burden of retrospective techniques can be eased either with alternating minimizations, or true joint estimation but on a reduced model. We provide computational experiments demonstrating the tightly coupled nature of the optimization variable types (motion and voxel values) which hinders the alternating based approaches. The alternating techniques can have an average search direction error of 75%, vs. 22% with reduced modeling. We demonstrate a computational speedup of 17x using our reduced model approach, and present in vivo imaging results comparing TAMER to a state-of-the-art alternating minimization.