In this paper, we attempt to place segmentation schemes utilizing the pyramid architecture on a firm footing. We show that there are some images which cannot be segmented in principle. An efficient segmentation scheme is also developed using pyramid relinking. This scheme will normally have a time complexity which is a sublinear function of the image diameter, which compares favorably to other schemes. The efficacy of our approach to segmentation using pyramid schemes is demonstrated in the context of region matching. The global features we use are compared to those used in previous approaches and this comparison will indicate that our approach is more robust than the standard moment-based techniques.