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Optimisation

Once a cost function has been chosen it is necessary to search for the transformation which will yield the minimum cost value. To do this, an optimisation method is used which searches through the parameter space of allowable transformations. Note that rigid-body transformations are specified by 6 parameters (3 rotations and 3 translations) while affine transformations are specified by 12 parameters. Consequently, even for linear transformation, the optimisation takes place in a high dimensional space; $ {\cal R}^{n}$, where $ 6 \le n \le 12$. While the problem specified in equation 1 is a global optimisation, quite often local optimisation methods are employed as they are simpler and faster.1 However, this can result in the method returning a transformation that corresponds to a local minimum of the cost function, rather than the desired global minimum. Such cases often appear as mis-registrations, of varying severity, and are a major cause of registration failure. Unfortunately, there are very few global optimisation methods that are suitable for a 3D brain image registration problem. This is because, in terms of operations, the cost function is expensive to evaluate and most global optimisation methods require a great many evaluations leading to unacceptable execution times (e.g. days).

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next up previous
Next: Multi-Resolution Techniques Up: Materials Previous: Interpolation
Peter Bannister 2002-05-03