Biomedical Engineering Reference
In-Depth Information
Other investigators have proposed methods for enforcing pairwise consis-
tent transformations. For example, Woods
et al
. [24] computes all pairwise reg-
istrations of a population of image volumes using a linear transformation model,
i.e., a 3
×
3 matrix transformation. They then average the transformation from
T
to
S
with all the transformations from
T
to
X
to
S
. The original transformation
from
T
to
S
is replaced with average transformation. The procedure is repeated
for all the image pairs until convergence. This technique is limited by the fact
that it can not be applied to two data sets. Also, there is no guarantee that the
generated set of consistent transformations are valid. For example, a poorly reg-
istered pair of images can adversely affect all of the pairwise transformations.
The method described in this chapter is most similar to the approach de-
scribed by Thirion [6]. Thirion's idea was to iteratively estimate the forward
h
,
reverse
g
, and residual
r
=
h
◦
g
transformations in order to register the images
T
and
S
. At each iteration, half of the residual
r
is added to
h
and half of the
residual
r
is mapped through
h
and added to
g
. After performing this opera-
tion,
h
◦
g
is close to the identity transformation. The advantage of Thirion's
method is that it enforces the inverse consistency constraint without having
to explicitly compute the inverse transformations as in Eq. (6.2). The resid-
ual method is an approximation to the inverse consistency method in that the
residual method approximates the correspondences between the forward and
reverse transformations while the inverse consistency method computes those
correspondences. Thus, the residual approach only works under a small defor-
mation assumption since the residual is computed between points that do not
correspond to one another. This drawback limits the residual approach to small
deformations and it therefore can not be extended to nonlinear transformation
models. On the other hand, the approach presented in this paper can be ex-
tended to the nonlinear case by modifying the procedure used to calculate the
inverse transformation to include nonlinear transformations.
6.2
Inverse Consitent Image Registration
6.2.1
Problem Statement
Assume that
T
and
S
correspond to two continuous images defined on the co-
ordinate system
=
[0
,
1)
3
. Traditionally, the image registration problem has
