Biomedical Engineering Reference
In-Depth Information
Another reason why it is difficult to produce inverse consistent transforma-
tions is that numerical optimization techniques used to find the optimal image
transformation often take a long time to converge or get stuck in local minima.
The large number of parameters estimated and the nonlinearity introduced by
the images being mapped makes it very difficult to find the optimal transforma-
tion. Placing a limit on the acceptable inverse consistency error may be one way
of specifying the stopping criteria for a particular optimization technique.
Formally, transformations
h
AB
:
→
and
h
BA
:
→
are said to
be inverses of one another if the transformation
h
BA
exists and satisfies
h
AB
(
h
BA
(
x
))
=
x
and
h
BA
(
h
AB
(
x
))
=
x
for all
x
∈
. A set of linear transfor-
mations
H
is said to have the invertibility property if
h
AB
(
h
BA
(
x
))
=
x
for all
A
,
B
∈
Q
and
x
∈
.
The average Inverse Consistency (IC) error within a region of interest (ROI)
M is defined as
1
M
E
AIC
(
h
AB
,
h
BA
,
M
)
=
M
||
h
AB
(
h
BA
(
x
))
−
x
||
dx
(6.6)
and the maximum IC error is defined as
E
MIC
(
h
AB
,
h
BA
,
M
)
=
max
x
∈
M
||
h
AB
(
h
BA
(
x
))
−
x
||
.
(6.7)
Eqs. (6.6 and 6.7) are discretized for implementation.
It is important to define a ROI because the amount of padding applied to
the image data can have a significant effect on the average error calculation.
The ROI restricts the error measurements to areas of interest preventing the
situation where the largest error occurs in the background of the image.
6.3.2
Transitivity Property
Image registration algorithms that have a difficult time producing inverse consis-
tent transformations have an even harder time producing transformations that
satisfy the transitivity property. In the paper we investigate how an algorithm
that reduces the inverse consistency error compared to another also reduces
the transitivity error.
A set of image transformations
H
is said to have the transitivity property if
h
CB
(
h
BA
(
x
))
=
h
CA
(
x
) or equivalently if
h
AC
(
h
CB
(
h
BA
(
x
)))
=
x
for all
A
,
B
,
C
∈
Q
and
x
∈
.
