Abstract: This paper presents a novel framework for image and shape deformation, and applies the presented method to 2D shape deformation and image/video magnifying. Our proposed framework maintains two distinct advantages. The first is to enforce the topology preservation constraints on a given displacement field. This allows us to develop a foldover-free constrained deformation approach. The second is the ability to easily incorporate the constraint of As-Rigid-As Possible deformation. It guarantees that the resulting distortions are as small as possible. Experiments are carried out on both 2D shape deformations and images/video magnifying.