Biomedical Engineering Reference
In-Depth Information
Definition 2.10
(continued)
The concrete numbers for the matrix
T
d
(including boundary treatment)
are
⎡
⎣
⎤
⎦
15
.
1
−
5
.
0
−
7
.
2
−
0
.
3
0
−
5
.
0
23
.
9
−
4
.
5
−
7
.
2
−
0
.
3
−
7
.
2
−
4
.
5
24
.
0
−
4
.
5
−
7
.
2
−
0
.
3
−
.
−
.
−
.
.
−
.
−
.
−
.
0
3
7
2
4
5
24
0
4
5
7
2
0
3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
n
d
,
n
d
T
d
=
∈
R
−
0
.
3
−
7
.
2
−
4
.
5
24
.
0
−
4
.
5
−
7
.
2
−
0
.
3
−
0
.
3
−
7
.
2
−
4
.
5
24
.
0
−
4
.
5
−
7
.
2
−
0
.
3
−
7
.
2
−
4
.
5
23
.
9
−
5
.
0
0
−
0
.
3
−
7
.
2
−
5
.
0
15
.
1
(
2.41)
. The entries of the matrix
T
d
for
d
∈{
1
,
2
,
3
}
are obtained by calculating
n
d
n
d
theintegralinEq.(
2.40
) for each index
i
∈{
1
,...,
}
and
j
∈{
1
,...,
}
,
d
∈{
1
,
2
,
3
}
.
2.1.4
Mass-Preserving Image Registration
The standard transformation model for image registration in Eq. (
2.2
) does not
guarantee the preservation of mass, i.e., in general
Ω
T
(
x
)
dx
=
Ω
T
(
y
(
x
))
dx
.
(2.42)
Further, the distance functional
typically entails the assumption of similar
intensities at corresponding points. This assumption is not valid for the standard
transformation model
D
std
in case of dual gated PET as discussed in connection
with Fig.
1.11
. Consequently, the standard transformation model needs some modi-
fication to express this feature.
The requirement for the desired transformation model
M
MP
M
is the preservation
of mass
MP
Ω
T
(
x
)
dx
=
Ω
M
(
T ,
y
(
x
))
dx
.
(2.43)
From the integration by substitution theorem for multiple variables we know that
the following equation holds
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