Biomedical Engineering Reference
In-Depth Information
the fiducial markers in a given space relative to the object to be registered, the
use of Equation 6.3 will underestimate FLE, and hence TRE.* Such consistency
will be negligible for displacements caused by the effects of noise or voxeliza-
tion on the marker localization algorithm. There may be a consistent effect
from spatially nonuniform brightness, such as that caused in MR by radiofre-
quency (RF) inhomogeneity, but such an effect will be negligible because of the
small extent of the typical fiducial feature. It may well be appreciable, however,
for geometrical distortion resulting from static field inhomogeneity in MR
scanners. Such distortion patterns may, for example, produce a consistent
translation of a marker's image relative to the head image when it is located
just external to the head. Scaling and skewing errors in the image, which,
for example, may be caused by errors in MR gradient strength or CT gantry
angle, will invalidate the relationship in Equation 6.3, with both over- and
underestimation of FLE possible. Image distortion is discussed at greater
length in Chapter 5.
6.2.1.3
Other Error Measures
Other measures of alignment error may be used as well. Distances between
lines and, more commonly, between surfaces may be clinically relevant in
some cases, but they reveal only part of the displacement error. Distances
between lines are insensitive to displacements in either space parallel to the
line in the respective space; distances between surfaces are similarly insensi-
tive to displacements parallel to the surfaces. Distances between surfaces,
used in some systems as cues for registration, are known to be poorly related
to TRE for such systems
7
but are still used for validation when no other
8,9
Combinations may be used also with distances mea-
sured between points and lines, points and surfaces, or lines and surfaces.
For rigid-body registration, error reports may include angular displace-
ment. Angular displacements specified relative to the directions of the coor-
dinate axes are independent of the positions of those axes and the origin of
the coordinate system. (The translational components of the error do, how-
ever, depend on the origin when there are nonzero rotational components.) It
is also possible to identify for any rigid motion a single axis of rotation such
that rigid motion is completely specified by a single angular displacement
about that axis along with a single translational component in the direction of
that axis. Since the inverse of a rigid transformation is rigid and the composition
of any two rigid transformations is likewise rigid, any error in the determina-
tion of a rigid transformation has itself the form of a rigid transformation.
Thus, an angular displacement would appear to be a simple, convenient
method for specifying alignment error. Unfortunately, its meaning is severely
limited by two factors: it is a global measure that cannot be focused on a clin-
ically important region, and the magnitude of local displacement that results
from rotation depends on the distance from an axis of rotation that has no
means is available.
* To be precise, the problem only occurs if the rigid displacement relative to the object is different
in the two spaces to be registered.
