Physical space is defined relative to some arbitrary set of 3-D
reference coordinates in the real world. The head of the patient in
the scanner and in the operating room can be modeled by two volumes in
physical space that are related by a non-linear transformation 
. This non-linearity arises from the fact that these volumes
represent the anatomy of the patient at two different times (the time
of image acquisition and the time of surgery), and some anatomical
changes might occur between the two situations making the volumes
slightly different. However this non-linear part of the transformation
between the volumes is expected to be small in any realistic situation.
Therefore, we consider the two volumes to be related by a rigid body
transformation since in general the rectangular coordinate systems
associated with the scanner and surgical spaces are different, i.e.,
rotated and translated, with respect to each other.