Changes in image topology occur in medical images due to normal variation in anatomy, image artifacts, and the presence of pathology. Non-rigid registration of images undergoing topological change for the purpose of atlas-based segmentation or deformation analysis is challenging since non-smooth geometric transformations must be introduced. As most registration methods impose a smoothness constraint on the allowable transformations they either do not model such changes or perform poorly in their presence. In this paper we describe an approach to non-rigid registration treating the images as embedded maps that deform in a Riemannian space. We show that smooth transformations representing topological changes in the original images can be obtained and describe the evolution in terms of a partial differential equation. Two-dimensional examples from brain morphometry are used to illustrate the method