Other studies of existing registration methods have specified the difference
between transformations as the basis for error measures
[HDaYC+93][BDC+93][ABKC90][FL93]. The results here are different from,
and may not be directly compared to, previous studies of the ILM
technique because of general case specificity [NGD+94], the
measure of error used and the dependence of the errors on the
configuration of the points sets has been explicitly isolated. Hill et
al [HHH+92] only studied the error's dependence on configuration
by contrasting the resultant mean distance measurements at specific
points from points generated dispersely on a grid and bound on a
single spherical surface. The use of the mean distance to evaluate
error may sufficiently specify the error at specific points, but it
inadequately provides academic insight into the factors which confound
the translation and rotation separately. As opposed to what had been
done here, a difference in transformations was not calculated. The
mean distance measure between specific points calculated there did
not allow the description of the errors in all of the image space.
This may only be done if the knowledge of the translation errors in
each dimension and the rotation errors in each plane about a specific
center of rotation are specified. Additionally, their use of a simple
phantom which attempts to model the grid simulation inadequately
provided for which is present in dealing with complex
structures. Neelin et al [NCH+92] provided much of the insight
in this study by specifying the standard deviation as the measure of
the distributions of the translation and rotation errors, but their
simulations involved variable configurations from sample to sample and
therefore produce rotation errors which are approximately equal in all
planes because they have been averaged. This is contrary to what has
been discovered in the present work because this study has allowed the
evaluation of the radial dependence of the rotation error by using the
same dimensions for the configurations. It has been shown in the last
chapter that the rotation errors decrease as the radial dispersion of
the points from the centroid increase. In Neelin's study, the
translation errors which are constant across dimensions may not be
expected in SPECT/MR registrations because of the anisotropy of the
resolution of SPECT.
Similar to what has been done here, Turkington et al [TJP+93] used
external fiducials on the Hoffman brain phantom as a basis for error
measurements from exact registrations, but they measured error for a
single registration as the distance between the fiducial locations
averaged over the number of fiducials used. Their measure was an
average over space where the error is generally radially dependent and
not over N registrations. In general, a repetition of any
registration technique will effect a distribution of translation and
rotation errors because of finite spatial resolution.
and
from the finite
may manifest itself from the variability of surface or volume mask
generation with voxel threshold or the existence of multiple solutions
in global search methods.
In Chen's investigation [Che93] of a landmark based registration technique involving outlier stripping and parameter accumulation algorithms, he defined an equivalent error angle, f, in terms of the Frobenius norm of the difference between real and estimated rotation matrices. This is essentially a measure of the mean error angle in 3-space and therefore does not describe the distribution of the dimensionally dependent rotation errors. It is sufficient though in the context of that study where f was evaluated in terms of its variation with a signal to noise ratio measurement in simulations on a unit sphere.
From the non-homogeneous results of the error distributions which were found in the previous chapter, it is recommended that, regardless of the registration technique, certain factors be provided when specifying registration errors. Briefly:
The last point is also necessary because it allows the
determination of the approximate errors in all space if the rotation errors
are given. Recall though that the external fiduciaries were provided on the
brain phantom to explicitly provide test points for exact registration,
minute imperfect marker location gave non-zero mean translations and
rotations (see table 5.2). It might be incorrectly thought that
since and
are specified, any set of test points may in fact be used
to provide the basis for true registrations - as long as the exact same
test points are used for N registrations of dimensionally similar
configurations. This is not necessarily true because anatomical homology is
necessary as a basis for the error measurement. At the least, real scan
results must be compared to point simulation results. That the real scan
results here were within about 0.3 mm of the point simulation results is
witness to the degree of error that may be expected in error measurements
based on supposed true registrations with non-zero anatomical homology
(compare table 5.2 with figure 5.6 ).
There is another constraint which validates the use of the fiducials
in this study. The centroid of the test points should be approximately
coincident with the centroid of the clinical registration points. This
allows: (a) the measure of
from the same centroid,
and (b) direct calibration with the point simulation studies.