Biomedical Engineering Reference
In-Depth Information
A least squares registration between the points and is then car-
ried out using the method* described in Section 3.4.1. The set of data points
is then transformed to using the calculated rigid-body transforma-
tion, and then the closest points once again identified. The algorithm ter-
minates when the change in mean square error between iterations falls
below a defined threshold.
The optimization can be accelerated by keeping track of the solutions at
each iteration. If there is good alignment between the solutions (to within
some tolerance), then both a parabola and straight line are fitted through
the solutions, and the registration estimate is updated using one of these
estimates based on a slightly ad-hoc method to be “on the safe side.”
As the algorithm iterates to the local minimum closest to the starting
position, it may not find the correct match. The solution proposed by Besl
and McKay
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is to start the algorithm multiple times, each with a different
estimate of the rotation alignment, and choose the minimum of the minima
obtained.
{}
q
i
{}
p
i
{}
p
i
p
i
3.4.3
Voxel Similarity Measure
Registration using voxel similarity measure involves calculating the registra-
tion transformation
by optimizing some measure calculated directly
from the voxel values in the images rather than from geometrical structures
such as points or surfaces derived from the images. As stated in Section 3.2,
with voxel similarity measures we are iteratively determining
T
, whereas in
the case of point registration or surface matching we first identify corre-
sponding features, determine
T
directly or iteratively from these, and finally
infer
T
.
In Sections 3.4.1 and 3.4.2, we did not distinguish between registration
where images
A
and
B
are of the same modality and registration of
A
and
B
when they are of different modalities. For registration using voxel similar-
ity measures this is an important distinction, as seen from the following
example. A common reason for carrying out same modality, or intramodal-
ity, registration is to compare images from a subject taken at slightly differ-
ent times in order to ascertain whether there have been any subtle changes
in anatomy or pathology. If there has been no change in the subject, we
might expect that after registration and subtraction there will be no struc-
ture in the difference image, just noise. Where there is a small amount of
change in the structure, we would expect to see noise in most places in the
images, with a few regions visible in which there has been some change. If
there were a registration error, we would expect to see artifactual structure
in the difference image resulting from the poor alignment. In this applica-
tion, various voxel similarity measures suggest themselves. We could, for
example, iteratively calculate
T
while minimizing the structure in the dif-
ference image on the grounds that at correct registration there will be either
T
* In fact, the authors used the equivalent quaternion method.
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