@INPROCEEDINGS{ClHeRu02, author = {Clarenz, U. and Henn, S. and Rumpf, M. and Witsch, K.}, title = {Relations between optimization and gradient flow methods with applications to image registration}, booktitle = {GAMM Seminar on Multigrid and Related Methods for Optimisation Problems}, year = {2002}, pages = {11--30}, abstract = {Fast multiscale and multigrid methods for the matching of images in 2D and 3D are presented. Especially in medical imaging this problem - denoted as the registration problem - is of fundamental importance in the handling of images from multiple image modalities or of image time series. The paper restricts to the simplest matching energy to be minimized, i.e., $E[\phi] = \frac{1}{2} \int_\Omega |T \circ \phi - R|^2$, where $T$, $R$ are the intensity maps of the two images to be matched and $\phi$ is a deformation. Matching of images, i.e., finding an optimal deformation $\phi$ which minimizes $E$ is known to be an ill-posed problem. Here, the focus is on regularization methods. We compare different iterative methods where the necessary regularization is incorporated via an additional convex energy functional. Furthermore we give a new interpretation of these methods in the framework of gradient flows. Hence, a regularization is interpreted as a regular metric used to measure length on the descent path in the gradient flow method. Typically regularizing energies appear together with a small coefficient. We obtain an interpretation of this coefficient as a time-step parameter. Examples on 2D and large 3D image matching problems prove the robustness and efficiency of the proposed approach.}, pdf = {http://numod.ins.uni-bonn.de/research/papers/public/ClHeRu02.pdf} }