Discussion

This paper has examined the problem of optimisation for fully automatic registration of brain images. In particular, the problem of avoiding local minima is addressed in two ways. Firstly, a general apodization (smoothing) method was formulated for cost functions in order to eliminate small discontinuities formed by discontinuous changes of the number of voxels in the overlapping field of view with changing transformation parameters. Secondly, a novel hybrid global-local optimisation method was proposed which uses prior knowledge about the problem (size of the brain, expected changes in scalings, etc.) while combining the speed of local optimisation with an initial search phase and multi-start optimisation component. The latter two components are similar to those used in Simulated Annealing and Genetic Algorithms -- two popular, but slow, global optimisation techniques. Only affine (linear) registration was examined in this paper as, although this is a much easier problem than general non-linear registration, 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 global optimisation method proposed here does not, however, guarantee finding the global minimum. This is typical though, as even methods such as Simulated Annealing and Genetic Algorithms 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 and solve the registration problem within one hour (and often much less) 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, a theoretical justification for the current method and finding a method suitable for higher dimensional transformations are important areas for future research. The implementations of the registration and motion correction methods (FLIRT and MCFLIRT) were tested using experiments designed to demonstrate the improved robustness and accuracy. These issues are important and each is examined separately.