Variational image registration using inhomogeneous regularization

We present a generalization of the convolution basedvariational image registration approach, in which differentregularizers can be implemented by conveniently exchangingthe convolution kernel, even if it is nonseparableor nonstationary. Nonseparable kernels pose a challenge becausethey cannot be efficiently implemented by separate1D convolutions. We propose to use a low-rank tensor decompositionto efficiently approximate nonseparable convolution.Nonstationary kernels pose an even greater challengebecause the convolution kernel depends on, and needs tobe evaluated for, every point in the image. We propose topre-compute the local kernels and efficiently store them inmemory using the Tucker tensor decomposition model. Inour experiments we use the nonseparable exponential kerneland a nonstationary landmark kernel. The exponential kernelreplicates desirable properties of elastic image registration,while the landmark kernel incorporates local prior knowledgeabout corresponding points in the images.We examinethe trade-off between the computational resources neededand the approximation accuracy of the tensor decompositionmethods. Furthermore, we obtain very smooth displacementfields even in the presence of large landmark displacements.