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Defining the Centre of the Volume

In the above formula it is assumed that the volume of integration is centred about . If this is not the case (when the origin of the world coordinates is not at the centre of the brain) then the matrix can be simply modified to include a general centre.

That is, let be the desired centre, and define a new coordinate

 (33)

This new coordinate is now zero when , so that the above derivation is valid for . Furthermore, this translation can be expressed in matrix form (using homogeneous coordinates) as where

Therefore, , giving . This implies that and so that

 (34)

Mark Jenkinson 2003-02-11