An additional step can be introduced to compensate for the effect of
spatial normalisation. When warping a series of images to match a
template, it is inevitable that volumetric differences will be
introduced into the warped images. For example, if one subject's
temporal lobe has half the volume of that of the template, then its
volume will be doubled during spatial normalisation. This will also
result in a doubling of the voxels labeled grey matter. In order to
remove this confound, the spatially normalised grey matter (or other
tissue class) is adjusted by multiplying by its relative volume before
and after warping. If warping results in a region doubling its volume,
then the correction will halve the intensity of the tissue label. This
whole procedure has the effect of preserving the total amount of grey
matter signal in the normalised partitions.
A deformation field is a vector field, where three values are
associated with each location in the field. The field maps from
co-ordinates in the normalised image back to co-ordinates in the
original image. The value of the field at co-ordinate [x y z] in the
normalised space will be the co-ordinate [x' y' z'] in the original
volume. The gradient of the deformation field at a co-ordinate is its
Jacobian matrix, and it consists of a 3x3 matrix:
/ \
| dx'/dx dx'/dy dx'/dz |
| |
| dy'/dx dy'/dy dy'/dz |
| |
| dz'/dx dz'/dy dz'/dz |
\ /
The value of dx'/dy is a measure of how much x' changes if y is changed
by a tiny amount.
The determinant of the Jacobian is the measure of relative volumes of
warped and unwarped structures. The modulation step simply involves
multiplying by the relative volumes.