Biomedical Engineering Reference
In-Depth Information
correspondence between any two coordinate systems is determined explicitly
by one of the displayed transformations or indirectly by concatenating two of
the transformations. For example, a point
x
in coordinate system B is mapped
to
y
in coordinate system C by the mapping
y
=
h
BA
(
h
AC
(
x
)), etc.
Figure 6.3 demonstrates that it is advantageous to design pairwise registra-
tion algorithms rather than
N
-wise registration algorithms that satisfy the tran-
sitivity property. The first advantage is that a pairwise algorithm only needs to
compute
N
−
1 pairwise transformations as opposed to (
N
−
1)! pairwise trans-
formations. This reduces computation time and computer storage requirements
by a factor of (
N
−
2) factorial. Another advantage is that a pairwise algorithm
only requires one additional set of pairwise transformations to be computed to
add a new data set to the population. An
N
-wise registration algorithm requires
that all of the transformations to be recomputed to produce a set of transforma-
tions with the transitivity property.
In general, pairwise image registration algorithms do not produce trans-
formations that have the transitivity property. The degree of transitivity can
be evaluated by measuring the difference between the identity mapping
and the composition the transformations from image
A
-to-
B
,
B
-to-
C
, and
C
-
to-
A
.
6.4
Inverse Consistent Registration
Algorithms
6.4.1
Intensity-Based Small Deformation Inverse
Consistent Linear Elastic (I-SICLE) Image
Registration
The intensity-based small deformation inverse consistent linear elastic (I-SICLE)
image registration algorithm [26,33-35] jointly estimates the forward and reverse
transformations
h
and
g
by minimizing Eq. (6.1). The image intensity alone
is used to register the images with this algorithm. A small deformation linear
elastic continuum mechanical model is used to regularize the transformations.
A generalization of this technic allows for multiple template image modalities
or subvolumes to be registered to corresponding target modalities. The cost
