Abstract: We present a novel geometric framework called geodesic active fields for general image registration. 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. Here, we propose to embed the deformation field in a weighted minimal surface problem, i.e., we introduce *multiplicative* regularization. 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. Our geometric framework involves two important contributions. Firstly, our general formulation for registration works on any parametrizable, smooth and differentiable surface, including non-flat and multiscale images. Secondly, to the best of our knowledge, this method is the first re-parametrization invariant registration method introduced in the literature. We illustrate our GAF registration framework on different toy examples, including stereo vision disparity map recovery, 2d registration, multiscale registration, sphere-patch and whole sphere registration... As an outlook, we present current research on numerical splitting schemes that solve the GAF energy minimization more efficiently.