Biomedical Engineering Reference
In-Depth Information
accurately as well. Fiducial registration error is typically reported as a mean,
commonly in the root-mean-square (RMS) sense of the distance between cor-
responding fiducial points after a point-based registration has been effected.
Thus,
N
FRE
2
2
Tp
()
q
i
,
(6.2)
i
where
N
equals the number of fiducials used in the registration process and
in the two image spaces.
TRE is more meaningful as a measure of registration success than FRE for
two reasons. First, as pointed out above, TRE can be measured at clinically
relevant points, whereas FRE is by definition limited to fiducial features
whose positions are clinically relevant only by coincidence. Second, FRE may
overestimate or underestimate registration error. The clinical relevance of
TRE makes it dominant over FRE as a general measure of meaningful error,
but if FRE were itself a good estimate of TRE, then it could serve as an easily
measurable surrogate.
Unfortunately, it is not. The first problem is underestimation. Some compo-
nents of registration error will not be reflected in FRE because the registration
system uses these same fiducial point locations in its determination of the
registering transformation. These hidden errors come about because the sys-
tem does its best, within the limits of the set of transformations at its disposal,
to align fiducial points pairs identified as corresponding, regardless of
whether they do indeed correspond. For example, if the transformations
being considered by the registration system are limited to rigid-body motion,
then fiducial localization errors of the same magnitude and direction for
every fiducial in a given image space will make no contribution to FRE. As a
specific simple case, suppose FLE is exactly 3 mm in the
p
and
q
are positions of fiducial
i
i
i
x
direction for all
N
fiducial points in the first image space and 4 mm in the
direction for all fidu-
cials in the second. Because there exists a rigid transformation that will bring
each of these purportedly corresponding pairs into perfect alignment, namely
a translational motion of the first image of
z
, relative
to the second, the resulting FRE for this case will be zero, while TRE will be
displacement whose magnitude is 5 mm at all points. In general, for all
rigid-body registration systems, to the extent that a given set of fiducial
localization errors can be duplicated by means of a rigid-body motion, the
registration error will be underestimated by FRE.
For nonrigid motion the situation may be worse. Any set of point pairs,
however badly localized in either or both spaces, can be perfectly aligned,
given a sufficiently versatile set of geometrical transformations. The FLE at
each of the fiducial points will be interpolated exactly or approximately at
all other points. Thus, by overfitting the fiducials in one space to their posi-
tions in the other space, the transformation may produce an FRE smaller
than the true TRE. This problem is less insidious if the set of transformations
3 mm in
x
and 4 mm in
z
