Ideally the cost function should be continuous with respect to the transformation. However, by working with discrete data (to reduce computational load) some discontinuities are usually introduced, as can be seen in figure 4a. The exact nature of the discontinuities depend on both the data and the interpolation method. Such discontinuities can be reduced by methods such as Partial Volume Interpolation [Maes et al., 1997], but these are not well adapted to certain transformations such as scaling changes.
Although it should be possible to place bounds on the difference between discrete cost evaluations for small changes in transformation, it is a difficult analysis and is left as a topic for further investigation. Moreover, in practice, the optimisation methods usually converge satisfactorily and simple re-samplings can avoid problems associated with grid alignment [Pluim et al., 2000].