| > In theory, the best results can be obtained with the most basis functions,
| > although the amount of regularisation may need to be tweeked to achieve
this.
|
| In which direction (towards light or towards heavy) we should change
| the regularization if we increase basis functions?
Difficult question. The regularisation can be thought of as being Bayesian,
whereby it attempts to find the solution with the maximum posterior
probability. The posterior probability is proportional to the product
of the prior probability of getting a deformation, and the likelihood (which is
related to the sum of squares difference between the template and the image
you are registering to it). Therefore, the best solution is not simply the
one with the smallest sum of squares difference.
After taking logs, the matching criterion becomes the sum of two terms
- one based on the residual squared difference, and the other being a
term that penalises unlikely deformations. The best solution is obtained
when there is an optimum balance between the terms.
Heavy regularisation basically means that more weight is applied to the
penalty term. Therefore, if the spatial normalisation introduces lots
of excessive warping that is clearly wrong, then more regularisation is
needed. Conversly, if the images do not get warped enough to match the
template, then the amount of regularisation needs to be decreased.
Until recently, I was not aware of proper methods for empirically
determining the best balance. However, more recently I was introduced
to methods of variance component estimation, that I eventually hope to
be able to use in order to get the balance right.
The defaults within SPM99 were those that appeared to give reasonable
solutions for our data. I tried to write the code so that it would
work for a wide variety of different images, but it is very possible
that the defaults may not be optimal for some groups' data.
Basically, I don't know the answer to your question - but hopefully
I will find out in the not too distant future.
All the best,
-John
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