Summary

No guarantee of finding the global minimum is available using this method. However, this is true for most global optimisation methods which at most provide only statistical guarantees for convergence which can never be met in practice (as they require infinite time in theory). The timing for this method, on the other hand, is more or less guaranteed to be less than an hour for our implementation.

Some assumptions in the standard multi-resolution approach have been relaxed here, while others are retained. In particular, the assumptions that are being used are:

• The global minimum for the 1mm resolution represents the desired solution.
• Significant change in the position of corresponding minima only occurs between the 8mm and 4mm resolutions.
• Anisotropic scaling and skew are minimal and can be ignored for the 8mm and 4mm resolutions.
• The search grid will evaluate one point in the global minimum's basin of attraction for the 8mm resolution and that this will be a local minima in the fine grid.
• Initial estimates of the translation and scale, as found from the optimised transformations at each point in the coarse grid, are sufficiently accurate for fine grid cost evaluations.

Of these, some assumptions can be further relaxed, but at the expense of computation time which can easily become excessive.

Much is based on empirical observations; however, the performance will not be worse than using the standard (no search) multi-resolution approach as this is equivalent to having M=N=1 and using no perturbations.