Block-based motion estimation is widely used for exploiting temporal redundancies in arbitrarily shaped video objects, which is computationally the most demanding part within the MPEG-4 standard. One of the main differences between MPEG-4 video and the previously standardized video coding schemes is the support of arbitrarily shaped video objects, for which most of the existing fast motion estimation algorithms are not suitable. In MPEG-4, motion estimation is performed on two kinds of macroblocks (MBs): the boundary MB and the opaque MB. Since the motion activities in opaque MBs are highly correlated with the neighbouring boundary MBs, a new priority search algorithm (PSA) for motion estimation is proposed in this paper, which performs motion estimation on all boundary MBs first in contrast to the conventional raster-scanning approach. This search strategy works well if the motion vectors in the boundary MBs truly represent the moving video object. Consequently, the full search algorithm is applied to the boundary MB in order to ensure its accuracy. However, the computational burden of motion estimation of the boundary MBs must be reduced. Fast search algorithms offered in the past tend to reduce the amount of computation by limiting the number of locations to be searched. Nearly all of these algorithms assume that the distortion function increases monotonically as the search location moves away from the global minimum. Unfortunately, this is usually not true in boundary MBs. We can reasonably assume that it is monotonic in a small neighbourhood around the global minimum. Consequently, one simple strategy, but perhaps the most efficient and reliable one, is to place the checking point as close as possible to the global minimum. In this paper, we also propose a fast search algorithm, which incorporates the binary alpha-plane to predict accurately the motion vectors of boundary MBs such that these motion vectors can be used in the PSA. Experimental results show that, when compared to conventional methods, our approach requires low computational complexity and provides significant improvement in terms of accuracy in motion-compensated video object planes.