Variational methods are presented which allow to correlate pairs of implicit shapes in 2D and 3D images, to morph pairs of explicit surfaces, or to analyse motion pattern in movies. A particular focus is on joint methods. Indeed, fundamental tasks in image processing are highly interdependent: Registration of image morphology significantly benefits from previous denoising and structure segmentation. On the other hand, combined information of different image modalities makes shape segmentation significantly more robust. Furthermore, robustness in motion extraction of shapes can be significantly enhanced via a coupling with the detection of edge surfaces in space time and a corresponding feature sensitive space time smoothing. The methods are based on a splitting of image morphology into a singular part consisting of the edge geometry and a regular part represented by the field of normals on the ensemble of level sets. Mumford-Shah type free discontinuity problems are applied to treat the singular morphology both in image matching and in motion extraction. For the discretization a multi scale finite element approach is considered. It is based on a phase field approximation of the free discontinuity problems and leads to effective and efficient algorithms. Numerical experiments underline the robustness of the presented approaches.