Lars Hoemke

Institute of Medicine

Research Center Juelich

Germany

A Multigrid Method for Anisotrophic PDE’s in Elastic Image Registration

The goal of digital image registration is to compute a spatial

transformation that minimizes the difference between two images. This

problem can be defined as the minimization of a non-linear

least-square functional which measures the image difference. Generally

this is an ill-posed problem. Hence, a regularization term that is

borrowed from the theory of linear elasticity is added to the

functional.

We study inexact Newton methods for solving this problem, i.e. we

linearize the functional around a current approximation and replace

the Hessian by a suitable operator, in order to obtain well-posed

subproblems in each step of the iteration.

These subproblems are solved using a multigrid solver. The underlying

equations exhibit anisotropies at object boundaries (edges). The

magnitude of these"jumps" depends on the degree of regularization. Due

to the anisotropies in the coefficients, standard multigrid solvers

suffer from poor convergence rates. We discuss modifications to the

multigrid components, specifically to the smoothing procedure, the

prolongation and the coarse grid correction. Numerical results that

demonstrate the improvements obtained with these new components are

given.