Natural gradients for deformable registration

We apply the concept of natural gradients to deformable registration. The motivation stems from the lack of physical interpretation for gradients of image-based difference measures. The main idea is to endow the space of deformations with a distance metric which reflects the variation of the difference measure between two deformations. This is in contrast to standard approaches which assume the Euclidean frame. The modification of the distance metric is realized by treating the deformations as a Riemannian manifold. In our case, the manifold is induced by the Riemannian metric tensor based on the approximation of the Fisher Information matrix, which takes into account the information about the chosen difference measure and the input images. Thus, the resulting natural gradient defined on this manifold inherently takes into account this information. The practical advantages of the proposed approach are the improvement of registration error and faster convergence for low-gradient regions. The proposed scheme is applicable to arbitrary difference measures and can be readily integrated into standard variational deformable registration methods with practically no computational overhead.