U. Clarenz, S. Henn, and K. Rumpf, M. Witsch.
Relations between optimization and gradient flow methods with
applications to image registration.
In GAMM Seminar on Multigrid and Related Methods for
Optimisation Problems, pages 11-30, 2002.
[ bib | .pdf 1 ]
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[φ] = (1)/(2) Ω |T o φ- R|2, where T, R are the intensity maps of the two images to be matched and φ is a deformation. Matching of images, i.e., finding an optimal deformation φ 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.