Variational Image Registration Allowing for Discontinuities in the Displacement Field
S. Kabus and A. Franz and B. Fischer
    363--377  (2007)

Registration of medical images is an active field of current research. The problem is to find a transformation which aligns two given images. The resulting displacement field may be described as a linear combination of pre-selected basis functions (parametric approach), or, as in our case, it may be computed as a minimizer of a functional (non-parametric or variational approach). This functional combines 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 the displacement field. The minimizing task is tackled by computing the Gateaux derivative of the functional resulting in a set of nonlinear partial differential equations for the displacement field. These equations are linearized by means of a fixed--point iteration scheme and discretized by a standard finite difference approach. A conventional variational method results in a globally smooth displacement field. However, a variety of clinical applications involve topology changes between the two images as for instance brain shift or tumor appearance or resection. For such applications a generalization of the standard method is needed which allows for localized discontinuities in the displacement field. The variational image registration approach presented here assumes a segmentation of the images into corresponding subdomains. At the interfaces between neighbouring subdomains the influence of the smoothness term can be suppressed by introducing a spatially dependent weighting function. By choosing it appropriately this allows for opening or closing of a gap between image regions. We demonstrate the capability of this new registration method by means of a one-dimensional synthetic example and a two-dimensional MR head image. However, our method can be applied to images of arbitrary dimensionality.