Variational Image Registration Allowing for Discontinuities in the Displacement Field

Sven Kabus, Bernd Fischer, Astrid Franz


Registration of medical images is an active field of current research.The problem is to find a transformation which aligns two given images. The re-sulting displacement field may be described as a linear combination of pre-selectedbasis functions (parametric approach), or, as in our case, it may be computed as aminimizer of a functional (non-parametric or variational approach). This functionalcombines a similarity measure and a smoothness term. The first one puts the com-parison of the images into quantifiable terms whereas the latter one regularizes thedisplacement field. The minimizing task is tackled by computing the Gâteaux deriv-ative of the functional resulting in a set of nonlinear partial differential equationsfor the displacement field. These equations are linearized by means of a fixed–pointiteration scheme and discretized by a standard finite difference approach.A conventional variational method results in a globally smooth displacementfield. However, a variety of clinical applications involve topology changes betweenthe two images as for instance brain shift or tumor appearance or resection. Forsuch applications a generalization of the standard method is needed which allowsfor localized discontinuities in the displacement field.The variational image registration approach presented here assumes a segmenta-tion of the images into corresponding subdomains. At the interfaces between neigh-bouring subdomains the influence of the smoothness term can be suppressed byintroducing a spatially dependent weighting function. By choosing it appropriatelythis allows for opening or closing of a gap between image regions.We demonstrate the capability of this new registration method by means of aone-dimensional synthetic example and a two-dimensional MR head image. However,our method can be applied to images of arbitrary dimensionality.
Original languageEnglish
Title of host publicationImage Processing Based on Partial Differential Equations
EditorsXue-Cheng Tai, Knut-Andreas Lie, Tony F. Chan, Stanley Osher
Number of pages15
Place of PublicationBerlin, Heidelberg
PublisherSpringer Berlin Heidelberg
Publication date2007
ISBN (Print)978-3-540-33266-4
ISBN (Electronic)978-3-540-33267-1
Publication statusPublished - 2007
EventInternational Conference on PDE-Based Image Processing and Related Inverse Problems 2005 - CMA, Norway
Duration: 08.08.200512.08.2005


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