Jan Modersitzki

Institute of Mathematics,

Medical University of Lubeck, Germany

Image registration is one of the fundamental tasks in image processing. Registration is needed whenever images are spatially distorted. Typically, one has to deal with two images of an object, where the images are taken at different times, from different perspectives, or with different imaging devices. Another important source for registration are images stemming from different but similar objects. Given the two images, the goal of registration is to find a transformation, such that the deformed image matches the other image subject to a suitable distance measure.

In this talk, different medical applications are presented, each demanding for its own particular registration technique. A unified mathematical formulation for image registration based on a variational approach is given. In this formulation the desired transformation is characterized as a minimizer of a certain functional, which does combine a similarity and a smoothness measure.

A variety of similarity and smoothness measures is discussed.
In particular, the * diffusion *- and the
* elastic* registration
are considered in detail. For the * diffusion*-registration an
O(n) algorithm based on an additive operator splitting
scheme is derived whereas for the * elastic*-registration an
O(n\log n) algorithm based on fast Fourier transformation type
techniques is presented. Here, n denotes the number of voxels.

Finally, the performance of the schemes is demonstrated for typical medical applications.