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Smoothness a la SPM
The current implementations of SPM (SPM99) use the smoothness
estimation as described in Kiebel et al [5].
For a Gaussian random field the smoothness is defined as
where is the dimensionality
of the field and
the covariance matrix of it's
first partial derivatives. An unbiased estimator for the covariance of
the partial derivatives is given by

(10) 
where is the number of degrees of freedom, the number of
voxel positions () and the number of time points () in a FMRI time series. In fact we don't have to evaluate this equation for the
whole covariance matrix. The offdiagonal elements will be zero so we
only need to calculate the diagonal elements of

(11) 
The partial derivative is evaluated via the gradient operator

(12) 
for nonedge voxels, where and are s
neighbouring voxels in the dimension along which the derivative is
currently being evaluated.
Now we can calculate the FWHM of the theoretical Gaussian responsible
for the observed smoothness

(13) 
where is the variance, , of the Gaussian
pointspreadfunction computed from the covariance measure for the
th dimension

(14) 
A combined value for the FWHM is given by the geometric mean of the
various FWHM

(15) 
Next: A more robust smoothness
Up: Smoothness estimation
Previous: Smoothness estimation
David Flitney
20011129