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Discussion

In this report the problem of global optimisation for brain image registration was examined. Only affine registration was examined as although this is a much easier problem than general, non-linear transformations, finding the global minimum is still difficult. Furthermore, many non-linear methods rely on an initial affine registration to find a good starting position, and so having a good method of affine registration is important.

The standard mathematical formulation of the registration problem was detailed and its assumptions examined. This included an analysis of the asymptotic behaviour of several multi-modal cost functions and showed that only the Correlation Ratio, Mutual Information and Normalised Mutual Information correctly yielded high costs for these poor registrations. Consequently, only these functions (out of those considered) are suitable for the global optimisation formulation. In addition, the distribution of local minima was sampled empirically for large sub-sampling (8mm cubed voxels) which demonstrated that there were still many local minima, so that a local optimisation method would not be sufficient to reliably find the global minimum.

A global optimisation method, which was tailored to this registration problem, was then proposed together with a complete discussion of the implementation details required to turn the method into a practical registration tool. This method uses the Correlation Ratio (although any suitable cost function could be used) and combines a common local optimisation method (Powell's method -- adapted for the voxel size to improve the efficiency) with several search strategies. The main search is a global search through the transformation space using the most sub-sampled volumes, with smaller neighbourhood-based searches taking place as the sub-sampling decreases. In such methods, the major constraint is the amount of time that is considered reasonable. In the design of this method the search time was limited to 20 minutes using our implementation. Even with this constraint it was still possible to perform a full search and identify the global minimum more reliably.

Evaluation of this registration method was done in two ways. Firstly, the software has been used routinely in the FMRIB Centre as an experimental trial, allowing many people to use and comment on the performance. The qualitative feedback was positive in its own right and in comparison with other available methods. This method has now been used to satisfactorily solve thousands of registration problems.

Secondly, quantitative results were found for a consistency test. This test is designed to examine the robustness of a registration method. Results showed that the method was highly consistent on a set of difficult images. Furthermore, several other available packages were tested on the same set of images and did not achieve the same level of consistency, sometimes demonstrating substantial inconsistencies.

The global optimisation method proposed here does not, however, guarantee finding the global minimum. This is typical though, as even methods such as Simulated Annealing only provide a statistical guarantee which cannot be met in practice. The results, though, are encouraging, and by using finer search grids, the likelihood of finding the global minimum can be increased. This requires that there be sufficient time at hand, or a sufficiently fast computer. However, even with modest resources this method can find the global minimum (solving the registration problem) within one hour, more reliably than the other methods tested.

Optimisation is only one aspect of the registration problem, although it is practically a very important one. Other aspects such as interpolation, alternative cost functions and understanding the properties of existing cost functions remain important areas for further work. In addition, for higher dimensional transformations, the optimisation problem becomes even more difficult, and so this too is an important area for future research.

Finally, as stated before, it is important to be precise about the implementational details of such methods. This allows (1) the methods to be more easily re-implemented by others, (2) the various methods to be compared fairly (by using the author's parameters) and (3) the results to be repeated. The alternative is finding the best value for various implementation parameters by trial and error which is extremely tedious and error prone. In addition to including the implementation details in this report the source code for the FLIRT package is available for downloading from www.fmrib.ox.ac.uk. This should avoid others needing to re-implement the method and facilitate the evaluation of the method, hopefully leading to further improvements.


next up previous
Next: Acknowledgements Up: No Title Previous: Consistency Test
Mark Jenkinson
2000-05-10