The new fast full-search motion estimation algorithm for optimal motion estimation is proposed in this paper. This algorithm presents fast computing method that calculates the tighter boundaries faster by exploiting the computational redundancy. The proposed algorithm, first, determines the possible motion vectors (PMVs) that are not rejected by the first two tighter boundaries to facilitate the prediction of the best initial motion vector (IMV) for the follow-up search. Thereafter, an optimal motion vector (OMV) will be traced out in the PMV set. The IMV greatly helps in the early rejection of impossible candidate blocks while tracing the OMV. Experimental results show that the proposed algorithm outperforms the other previous optimal motion estimation algorithms in reducing the number of computations on several video sequences. The proposed algorithm achieves about 156–456 speed-up gain over full search on several video sequences. But, the state-of-the-art algorithms such as adaptive MSEA and Winner Update algorithm with Integral image algorithms can achieve only about 72–382 speed-up gain over full search on the same video sequences. Finally, the proposed new fast full-search motion estimation algorithm is modified to suboptimal motion estimation algorithm, resulting only in a trivial average peak signal-to-noise ratio drop of about 0.2 dB, but it achieves a very fast computational speed.