author = {Clarenz, U. and Droske, M. and Rumpf, M.},
  title = {Towards fast non--rigid registration},
  booktitle = {Inverse Problems, Image Analysis and Medical Imaging, AMS Special
	Session Interaction of Inverse Problems and Image Analysis},
  year = {2002},
  volume = {313},
  pages = {67--84},
  publisher = {AMS},
  abstract = {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., $E[\phi] = \frac{1}{2} \int_\Omega |f_1 \circ
	\phi - f_2|^2$, where $f_1$, $f_2$ are the intensity maps of the
	two images to be matched and $\phi$ is a deformation. The focus is
	on a robust and efficient solution strategy. Matching of images,
	i.e., finding an optimal deformation $\phi$ which minimizes $E$ is
	known to be an ill-posed problem. Hence, to regularize this problem
	a regularization of the descent path is considered in a gradient
	flow method. Thus the initial value problem $\partial_t \phi = -
	\grad_g E[\phi ]\,$ with some regular initial deformation $\phi(0)=\phi_0$
	is solved on a suitable space of deformations $\Omega \rightarrow
	\Omega$. The gradient $\grad_g$ 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 time-step 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.},
  pdf = { 1},
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