Benefits and shortcomings of partial volume interpolation for MI based image registration

The alignment of images is a well known clinical procedure to detect and monitor the evolution of medical conditions. Mutual Information (MI) is a frequently used alignment criterion. Although MI based alignment criteria have reached maturity, some issues regarding the occurrence of artifacts and smoothness, that can hamper or slow down the alignment process, still remain a research topic. Interpolation in combination with grid alignment often introduces spurious local maxima in the calculated MI functional. If interpolation is followed by binning the functional will even be piecewise constant. Classical partial volume MI estimation is an approach that avoids - in a similar way as Parzen windowing - the interpolation/binning succession, and guarantees continuity and differentiability of the MI estimate almost everywhere, while being computationally very cheap. Despite this, the Partial Volume (PV) approach cannot avoid the occurrence of artificial local maxima with a turning point type anomaly. In this paper, MI based on PV interpolation will be reformulated by means of contribution functions of pixel pairs. Starting from this reformulation, partial volume interpolation will be adapted to guarantee differentiability while maintaining the idea of PV interpolation at a low computational cost. It is shown that, if closed form expressions exist for the derivatives of the transformation parameters, a closed form expression for the derivatives of the adapted partial volume based mutual information functional is available.