Variational Image Registration Allowing for Discontinuities in the Displacement Field

Summary:
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.