The parameters in the area of no interest can initially be divided into
those in the null space of (associated with zero columns) and the remainder.
All null parameters can be treated in the same way as null parameters formed
from shapes of interest, and therefore need not be considered separately.
The remaining parameters all contribute towards the term.
For small subshapes, each parameter contributes a small partial volume
amount to a single image voxel. Consider the image voxel
in the
area of no interest, which is modelled by
subshapes of equal
size. Therefore each subshape contributes a partial volume amount of
and is associated with a column in
of the form:
. Such a set of subshapes contains 1
effective parameter of no interest and
effective degenerate
parameters.
To see the degeneracy more clearly it is necessary to transform this
set of parameters. Let this set of subshape parameters be
with
associated columns from
. Furthermore,
define a transformation
where
is an
by
matrix such that
where
.
These matrices are
It is clear from this that the first parameter in the transformed set,
, associated with the first column of
, is the
effective parameter of no interest and that all the others are null
parameters. Consequently, the integration over this set of subshape
parameters can be achieved using this reparameterisation. Note that
this integration can be done separately for each measured voxel in the
area of no interest, since the contributions to separate voxels are
orthogonal in both the prior and posterior, which is the case for flat
priors.
Applying this to the previously derived posterior gives:
The expression for is derived using that fact that the columns
of
and
are all mutually orthogonal - that is,
. This is true since
only have non-zero entries
in voxels inside the area of no interest.
This expression for the posterior is exactly the same form as
previously derived (equation 7) without having an area of no
interest. The difference between the two forms is that is
replaced by
and the residual operator removes
signals related to both
and
, rather than just
. As the signals spanned by
are voxels in the area
of no interest, of which there are
, it is completely
equivalent to ignoring these area completely - that is, by
effectively removing them from the measured image and doing the
analysis as if only
voxels had been measured.
Finally, the total number of parameters used to model the
area of no interest does not appear in the final posterior. Only the
number of uninteresting parameters (equal to the number of measured
voxels totally in the area of no interest) is important. Consequently,
there is no limit on the number of subshapes which may be used to
model area of interest, and the subshapes can become infinitesimal,
justifying the implicit assumption that it is possible to treat subshapes
as only ever contributing to one image voxel at a time, regardless of the
spatial transformation, .
Note that a similar analysis applies to voxels which only partially
cover the area of no interest. In this case, only
subshapes will contribute towards the voxel intensity, such that
and the non-null parameter becomes a pure partial volume
parameter, with
being the partial volume fraction.
These will be treated in the next section.