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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;
, where
.
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).
Subsections
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Peter Bannister
2002-05-03