Biomedical Engineering Reference
In-Depth Information
be shown that the versions of normalized mutual information in Equations
3.28 and 3.30 are closely related.
( )
HA
,
B
IA
,
B
1
˜
1
A
,
B
HA
()
HA
,
B
HB
()
˜
3
A
,
B
(3.31)
------------------------------------
---------------------
--------------------------------
(
)
1
(
)
(
)
(
)
2
3.4.9
Optimization and Capture Ranges
With the exception of registration using the Procrustes technique described
in section 3.4.1, and, under certain circumstances, the registration algorithm
in the SPM software,
33
all the registration algorithms reviewed in this chapter
require an iterative approach in which an initial estimate of the transforma-
tion is gradually refined by trial and error. In each iteration, the current esti-
mate of the transformation is used to calculate a similarity measure. The
optimization algorithm then makes another (better, we hope) estimate of the
transformation, evaluates the similarity measure again, and continues until
the algorithm converges, at which point no transformation can be found that
results in a better value of the similarity measure, to within a preset tolerance.
A review of optimization algorithms can be found in Press et al.
18
One of the difficulties with optimization algorithms is that they can con-
verge to an incorrect solution called a “local optimum.” It is sometimes useful
to consider the parameter space of values of the similarity measure. For rigid-
body registration, there are six degrees of freedom, giving a six-dimensional
parameter space, and for an affine transformation with twelve degrees of
freedom, the parameter space has twelve dimensions. Each point in the
parameter space corresponds to a different estimate of the transformation.
Nonaffine registration algorithms have more degrees of freedom (often many
hundreds or thousands), in which case the parameter space has correspond-
ingly more dimensions. The parameter space can be thought of as a high
dimensionality image in which the intensity at each location corresponds to
the value of the similarity measure for that transformation estimate. If we
consider dark intensities as good values of similarity and high intensities as
poor ones, an ideal parameter space image would contain a sharp low inten-
sity optimum with monotonically increasing intensity with distance away
from the optimum position. The job of the optimization algorithm would
then be to find the optimum location given any possible starting estimate.
Unfortunately, parameter spaces for image registration are frequently not
this simple. There are often multiple optima within the parameter space, and
registration can fail if the optimization algorithm converges to the wrong
optimum. Some of these optima may be very small, caused either by interpo-
lation artifacts (discussed further in Section 3.5), or a local good match
between features or intensities. These small optima can often be removed
from the parameter space by blurring the images prior to registration. In
fact, a hierarchical approach to registration is common: the images are first
registered at low resolution, then the transformation solution obtained at
