Biomedical Engineering Reference
In-Depth Information
The image registration problem can be stated mathematically as finding the
transformations
h
and
g
that minimize the cost function
2
2
dx
C
(
µ,η
)
=
|
T
(
x
+
u
(
x
))
−
S
(
x
)
|
+|
S
(
x
+
w
(
x
))
−
T
(
x
)
|
||
h
(
x
)
−
g
−
1
(
x
)
||
2
+||
g
(
x
)
−
h
−
1
(
x
)
||
2
dx
+
χ
2
dx
+||
L
w
(
x
)
||
2
dx
+
ρ
||
Lu
(
x
)
||
The constants
σ
,
χ
and
ρ
are used to enforce/balance the constraints (see [26]
for complete details on how to minimize this cost function).
6.2.2
Symmetric Similarity Cost Function
A problem with many image registration techniques is that the image similarity
function does not uniquely determine the correspondence between two image
volumes. In general, similarity cost functions have many local minima due to
the complexity of the images being matched and the dimensionality of the trans-
formation. It is these local minima (ambiguities) that cause the estimated trans-
formation from image
T
to
S
to be different from the inverse of the estimated
transformation from
S
to
T
. In general, this becomes more of a problem as the
dimensionality of the transformation increases.
To overcome correspondence ambiguities, the transformations from image
T
to
S
and from
S
to
T
are jointly estimated. This is accomplished by defining a cost
function to measure the shape differences between the deformed image
T
◦
h
and image
S
and the differences between the deformed image
S
◦
g
and image
T
. Ideally, the transformations
h
and
g
should be inverses of one another, i.e.,
h
=
g
−
1
. In this work, the transformations
h
and
g
are estimated by minimizing
a cost function
2
dx
C
SIM
(
T
◦
h
,
S
)
+
C
SIM
(
S
◦
g
,
T
)
=
|
T
(
h
(
x
))
−
S
(
x
)
|
2
dx
|
S
(
g
(
x
))
−
T
(
x
)
|
(6.1)
+
where the intensities of
T
and
S
are scaled between 0 and 1. To use this similarity
function, the images
T
and
S
must correspond to the same imaging modality
and they may require preprocessing to equalize the intensities of the image. In
practice, MRI images require intensity equalization while CT images do not. A
