Recursive Green's Function Registration

Non-parametric image registration is still among the most challenging problems in both computer vision and medical imaging. Here, one tries to minimize a joint functional that is comprised of a similarity measure and a regularizer in order to obtain a reasonable displacement field that transforms one image to the other. A common way to solve this problem is to formulate a necessary condition for an optimizer, which in turn leads to a system of partial differential equations (PDEs). In general, the most time consuming part of the registration task is to find a numerical solution for such a system. In this paper, we present a generalized and efficient numerical scheme for solving such PDEs simply by applying 1-dimensional recursive filtering to the right hand side of the system based on the Green's function of the differential operator that corresponds to the chosen regularizer. So in the end we come up with a general linear algorithm. We present the associated Green's function for the diffusive and curvature regularizers and show how one may efficiently implement the whole process by using recursive filter approximation. Finally, we demonstrate the capability of the proposed method on realistic examples.