> Does increasing the number of basis actually increase the accuracy of
> image registration in SPM and AIR ?
Sometimes. Nonlinear registration is (approximately) formulated as a MAP
problem, which maximises:
P(warp , data) = P(data | warp) * P(warp)
this is equivalent to minimising:
-log(P(data | warp)) - log(P(warp))
The regularisation term encodes P(warp) as a multi-variate Gaussian, whereby
(in SPM2)
-log(P(warp)) = the "bending energy" of the deformation
If the form and magnitude of the regularisation are an accurate estimate of
the prior probability distribution, then having more basis functions should
improve the result. If they are not, then increasing the number of basis
functions can decrease the accuracy.
> I am assuming that most image registration techniques (based on basis
> function expansion) estimate the lower order coefficients first and
> estimate the higher order coefficients later.
In SPM, they are all estimated together. Higher frequencies tend to be
regularised more heavily than lower frequencies though, so they tend to be
fitted last. In SPM, there is also a rough coarse-to-fine strategy, whereby
the earlier iterations use more regularisation.
The coarse-to-fine strategy is a bit of a mess (although it works not too
badly), and should probably be re-written to use Laplace approximations to an
ML-II (REML) scheme, in order to estimate the best trade-off between
regularisation and likelihood terms. The non-IID nature of the residuals
makes this slightly difficult though (see my very old attempts in 1D in
http://www.fil.ion.ucl.ac.uk/~john/misc/nonlin_reml.m and
http://www.fil.ion.ucl.ac.uk/~john/misc/nonlin_reml_smo.m ).
> In this sense, variance of
> the low order coefficients is larger than the variance of the higher order
> coefficients. Is my logic correct?
It's fairly correct. Look around line 256 of spm_normalise.m. You will see
an inverse covariance matrix, which is used to generate the penalties. Lower
frequencies are penalised much less than higher frequencies.
All the best,
-John
