Dear Torben,
> Wouldn't it be a good idea to move the smoothing procedure
> from the "spatial preprocessing" to a step just before the result
> section. The idea is, that since smoothing (convolution with a
> kernel) and beta estimation (projection) are both linear operations
> they commute, and thus the order in which they are applied does not
> matter. So why not smooth a few beta images and the ResMS.img instead
> of hundreds raw-images.
>
> I forgot to say that switching the order of smoothing and estimation is
> a problem in first level analyses because the RPV.img is calculated in
> the "estimate" section and not in the "results" section. So how about
> moving the two steps: smoothing and resel per voxel estimation, to the
> results section?
You are absolutely right about the commutative nature of the estimation
and convolution operators
y*K = X*B*K + e*K ....smooth before
B*K = pinv(X)*y*K
y = X*B + e
B = pinv(X)*y
B*K = pinv(X)*y*K ....smooth after
However nonlinearities enter with error vaiance estimation. One would
have to smooth the residual images and then compute the sum of
squares. This part is not commutative and the number of residual
images equals the number of raw images.
i.e. diag(K'*e'*e*K) ~= K'*diag(e'*e)
Taking the sum of squared smoothed residuals is not the same as
smoothing the sum of sqaured residuals.
The reason one can smooth after a 1st level analysis is because the sum
of squared residuals do not enter into second level.
With best wishes - Karl
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