The determination of the image space position corresponding to given
physical space location is what is all about. This can be
achieved, once the transformation 
has been estimated (in
other words, once 
has been found), by using the inverse
transformation 
. But, in general, an arbitrary point
in physical space will map into a point for which the intensity in
image space have not been measured (recall that measurement in image
space is discrete). Moreover, the surgeon may want to look at the
image data through slices in arbitrary orientations (which is in
general different from the planes originally acquired). Then, the
interpolation of the image data is required, which is performed by
reformatting.
Reformatting can be defined in this particular case as the
mapping of image intensities of the model space (
) onto the points of an arbitrary plane taken through the model
space [40].
The orientation of the reformatted plane is fully determined if the
spatial coordinate of three of its points are known. If 
,
, 
are the 3-D position vectors of these points in
scanner space and 
, 
, 
, their 2-D
positions in the reformatted plane, vectors in scanner space and in
the reformatted plane are related by the following relations:
where
Note that 
and 
are 
and 
column vectors respectively. Once the transformation 
has
been defined, the values of intensity in the reformat plane can be
retrieved from the values in the scanner space.
where 
is given by s2r. The fact that the
intensity is not known at every arbitrary point implies that the value
at 
must be interpolated from the known values in scanner
space.
