Biomedical Engineering Reference
In-Depth Information
simple but effective method for intensity equalizing MRI data is to compute the
histograms of the two images, scale the axis of one histogram so that the gray-
and white-matter maximums match, and then apply the intensity scaling to the
image.
This joint estimation approach applies to both linear and non-linear trans-
formations. In general, the squared-error similarity functions in Eq. (6.1) can
be replaced by any suitable similarity function—mutual information [27, 28],
demons [6], an intensity variance cost function [24], etc.—where the choice is
dependent on the particular registration application.
6.2.3
Inverse Consistency Constraint
Minimizing a symmetric cost function like Eq. (6.1) is not sufficient to guarantee
that
h
and
g
are inverses of each other because the contributions of
h
and
g
to
the cost function are independent. In order to couple the estimation of
h
with
that of
g
, an inverse consistency constraint is imposed that is minimized when
h
=
g
−
1
. The inverse consistency constraint is given by
||
h
(
x
)
−
g
−
1
(
x
)
||
2
dx
+
||
g
(
x
)
−
h
−
1
(
x
)
||
2
dx
C
ICC
(
u
, w
)
+
C
ICC
(
w,
u
)
=
2
dx
+
2
dx
.
(6.2)
=
||
u
(
x
)
−
w
(
x
)
||
||
w
(
x
)
−
u
(
x
)
||
Notice that the inverse consistency constraint is written in a symmetric form
like the symmetric cost function for similar reasons.
6.2.4
Computation of the Inverse Transformation
The procedure used to compute the inverse transformation of a transformation
with minimum Jacobian greater than zero is as follows. Assume that
h
(
x
)is
a continuously differentiable transformation that maps
onto
and has a
positive Jacobian for all
x
∈
. The fact that the Jacobian is positive at a point
x
∈
implies that it is locally one-to-one and therefore has a local inverse. It
is therefore possible to select a point
y
∈
and iteratively search for a point
x
∈
such that
||
y
−
h
(
x
)
||
is less than some threshold provided that the initial
guess of
x
is close to the final value of
x
.
The inverse transformation is computed in the following way [26]. First,
note that all images including transformed images are discrete. Therefore, it is
