Geodesic Active Fields - A Geometric Framework for Image Registration

In this paper 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. The energy of the deformation field is measured with the Polyakov energy weighted by a suitable image distance, borrowed from standard registration models. Minimizing this weighted Polyakov energy drives the deformation field toward a minimal surface, while being attracted by the solution of the registration problem. 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.