Numerical optimization for constrained image registration

Image registration or image matching is a technique to establish meaningful correspondences between points in different scenes. It is a mandatory tool for various applications in medicine, geoscience, and other disciplines. However, obtaining plausible deformations is a complex task. For example, many applications require the transformations to be locally invertible, or even harder, keep volume changes within a reasonable bandwidth. In this work, solutions to the registration problem are obtained by direct imposition of a volume constraint on each voxel in a discretized domain. In contrast to previous work, the focus here is on development of an efficient and robust numerical algorithm and in particular, the study of an augmented Lagrangian method with a multigrid solver. The paper demonstrates that this combination yields an almost optimal solver (i.e. linear time) for the problem.