Multigrid based total variation image registration

We consider the image registration problem to find a reasonable displacement field, such that a transformed template image becomes similar to a so-called reference image. The minimization of the similarity measure (exemplarily based on the gray-value difference) yields a nonlinear ill-posed inverse problem. The necessary regularization is done by replacing the ill-conditioned Hessian by a multidimensional total variation $$(\mathcal{TV})$$ norm. This allows steep gradients and discontinuities in the displacement field in contrast to the common approach by elastic regularization which leads to globally smooth displacement fields. We propose and investigate a multigrid algorithm as inner iteration for $$\mathcal{TV}$$ registration. As we use Neumann boundary conditions which lead to singular systems, a special treatment before and during the FAS multigrid algorithm is required, e.g. the introduction and solution of an augmented system. We describe the necessary modifications for the multigrid algorithm and present convergence results as well as first registration experiments demonstrating the capabilities of the proposed approach.