A model based approach called k-t BLAST/SENSE, has drawn significant attentions from MR imaging community due to its improved spatio-temporal resolution. Recently, we showed that k-t BLAST/SENSE corresponds to the special case of a new dynamic MRI algorithm called k-t FOCUSS that is asymptotically optimal from compressed sensing perspective. The k-t FOCUSS exploits the sparsity of x-f support of dynamic scene and converts imaging problem into an L1 minimization problem that can be solved using FOCal Underdetermined System Solver (FOCUSS). In this paper, we extend the idea of k-t FOCUSS and introduce motion estimation and compensation (ME/MC) based prediction step and residual encoding step. The ME/MC based prediction step exploits the temporal redundancies using the motion field estimation and provides much sparser residual signals. The sparse residual signal can then be effectively encoded using much smaller number of k-t samples. Simulation results demonstrate that high resolution dynamic MR images can be accurately obtained even from very limited data samples.