In this paper we present a novel framework called geodesic active fields for general image registration on Riemannian manifolds. In image registration, one looks for the underlying deformation field that best maps one image onto another. This is a classic ill-posed inverse problem, which is usually solved by adding a regularization term. In this paper, we define a new geometric approach to image registration. More specifically, we propose to embed the deformation field in a higher dimensional manifold. Then, the deformation field is driven by a minimization flow towards a harmonic map corresponding to the solution of the registration problem, much like geodesic active contours in image segmentation. The energy of the deformation field is measured with the Polyakov energy weighted by a suitable image distance, borrowed from standard registration models. We investigate three different weighting functions, the squared error and the approximated absolute error for monomodal images, and the local joint entropy for multimodal images. Further, the concept of exponential maps of vector fields is integrated into the framework, to allow for smoothly invertible diffeomorphic deformation fields. Finally, we illustrate the validity of our framework on a set of different image registration tasks, including non-flat and multiscale images.
