Global motion generally describes the motion of the camera, although it may comprise large object motion. The region of support for global motion representation consists of the entire image frame. Therefore, estimating global motion parameters tends to be computationally costly due to the involvement of all the pixels in the calculation. Efficient global motion estimation (GME) techniques are sought after in many applications such as video coding, image stabilization and super-resolution. In this paper, we propose to select only a small subset of the pixels in estimating the global motion parameters, based on a combination of fixed and random subsampling patterns. Simulation results demonstrate that the proposed method was able to speed up the conventional all-pixel GME approach by up to 7 times, without significant loss in the estimation accuracy. The combined subsampling patterns were also found to provide better motion estimation accuracy/complexity tradeoffs than those achievable by using either fixed or random patterns alone.