Biomedical Engineering Reference
In-Depth Information
new shape will be
similar
to the original shape but at a different scale. It should
be noticed that we will use the term similarity transformation to represent a
rotation, translation and scaling only, no shear or torsion or other deformable
transformations are included.
The minimization of Eq. (1.3) is very difficult because
d
(
Ry
i
+
t
,
S
) is highly
nonlinear since the correspondence between
y
i
and
S
is not known beforehand.
To understand the challenges involved in solving the registration problem
one needs to understand the following:
1. For two data sets, if the transformation from one to the other is pre-
cisely known, then the registration process would be trivial. But when
the transformation is only approximately known, the problem becomes
more difficult. It is here that most researchers have addressed this prob-
lem. However, few researchers have attempted to solve the problem when
the transformation is totally unknown.
2. The search for an unknown, optimal transformation invariably assumes an
initial transformation which is iteratively refined through the minimization
of some evaluation function. Such search may lead to a local minimum and,
unless a global optimizer is used [3], it is difficult to reach a global solution.
3. For many applications (e.g., intraoperative registration), the registration
time can be very critical and near real-time registration process is still
needed.
The registration process must compensate for three very important problems,
which are translational offset, rotational misalignment, and partial data sets.
Error due to translational offset occurs when the coordinate origins of the data
sets are not the same point in
N
-dimensional space. This can be demonstrated by
calculating the point by point error of two identical surfaces located at different
locations in the
N
-dimensional space. Even though the data sets are identical,
the average experimental data error will be equal to the distance of the offset
between the two sets.
Registration must also correct for error due to rotational misalignment be-
tween the data sets. This can be visualized by viewing a non-symmetrical surface
from two different angles. The two different views may appear very different
even though they come from the same surface. Once the views are rotated into
correct alignment, an accurate value for the error measure can be obtained.
