In medical images, intensity changes caused by certain pathology can change the topology of image level-sets and are thus commonly referred to as topological changes. Topological changes cause false deformation in existing deformable registration algorithms, which in turn leads to unreliable observations in the clinical study that relies on the deformation fields, such as deformation based morphometry (DBM). In this work, we develop a new deformable registration algorithm for images with topological changes. In our proposed algorithm, 3D images are embedded as 4D surfaces in a Riemannian space. The registration is therefore conducted as a surface evolution, which is modeled by a diffusion process. Our algorithm differs from existing methods in the sense that it takes an a-priori estimation of areas with topological change as an additional input and generates dense deformation vector fields which are free of false deformation. In particular, the output of our algorithm is composed of a diffeomorphic deformation field and an intensity displacement which corrects intensity difference caused by topological changes. By conducting multiple sets of experiments, we demonstrate that our proposed algorithm is capable of accurately registering images with considerable topological changes. More importantly, the resulting deformation field is not impacted by topological changes, i.e., there is no false deformation. / Ph. D.