[1] 
U. Clarenz, M. Droske, and M. Rumpf.
Towards fast nonrigid registration.
In Inverse Problems, Image Analysis and Medical Imaging, AMS
Special Session Interaction of Inverse Problems and Image Analysis, volume
313, pages 6784. AMS, 2002. [ bib  .pdf 1 ] A fast multiscale and multigrid method for the matching of images in 2D and 3D is 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} f_{1} o φ f_{2}^{2}, where f_{1}, f_{2} are the intensity maps of the two images to be matched and φ is a deformation. The focus is on a robust and efficient solution strategy. Matching of images, i.e., finding an optimal deformation φ which minimizes E is known to be an illposed problem. Hence, to regularize this problem a regularization of the descent path is considered in a gradient flow method. Thus the initial value problem _{t} φ=  _{g} E[φ] with some regular initial deformation φ(0)=φ_{0} is solved on a suitable space of deformations Ω> Ω. The gradient _{g} is measured w.r.t. a suitable regularizing metric g. Existence and uniqueness of solutions is demonstrated for different types of regularizations. For the implementation a metric based on multigrid cycles on hierarchical grids is proposed, using their superior smoothing properties. This is combined with an effective timestep control in the descent algorithm. Furthermore, to avoid convergence to local minima, multiple scales of the images to be matched are considered. Again, these image scales can be generated applying multigrid operators and we propose to resolve the pyramid of scales on a properly chosen pyramid of hierarchical grids. Examples on 2D and large 3D image matching problems prove the robustness and efficiency of the proposed approach.
