This posterior form above can be integrated in the case where
all the parameters, , are associated with a column in
that is linearly independent and has a norm greater than one.
However, this is not commonly the case, and the set of parameters must
be split into subsets. These subsets (see figure 2)
are: interesting (int), for shapes inside the valid field of
view (i.e. contained in the observed image); null (null), for
shapes totally outside the field of view; uninteresting (un),
for voxels inside the field of view but not within a modelled shape
(i.e. in `areas of no interest'); and partial volume (pv),
for shapes that overlap the field of view by less than one voxel.
That is:
|
The dimensionality of the subsets will be denoted as:
such that
. Note that the composition and dimension of these subsets
depends heavily on the spatial transformation,
. Hence, like
,
they implicitly encode dependence on
, but as
is not being
marginalised, this dependence will be left implicit. It is important
to remember, however, that these are not constants.
Also, many of these subsets may be empty (zero dimensional) for
certain spatial transformations, , or always empty in certain
applications (e.g. where there are no areas of no interest). The way
that these subsets are defined and how they are treated in the
integrations will be the topic of the next sections.