Biomedical Engineering Reference
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y
(
Ω
)
T
(
x
)
dx
=
Ω
T
(
y
(
x
))
|
(
∇
y
(
x
))
|
dx
.
det
(2.44)
With respect to PET, Eq. (
2.44
) guarantees already the same total amount of
radioactivity before and after applying the transformation
y
to
T
. The Jacobian
determinant
accounts for the volume change induced by the transfor-
mation
y
, representing the mass-preserving component. As
y
should reflect cardiac
and respiratory motion, transformations that are not bijective are anatomically
not meaningful and have therefore to be excluded. For example, the hyperelastic
regularization functional in Sect.
2.1.2.3
guarantees
y
to be diffeomorphic and
orientation preserving, which allows us to drop the absolute value bars
|
det
(
∇
y
(
x
))
|
y
(
Ω
)
T
(
x
)
dx
=
Ω
T
(
y
(
x
))
det
(
∇
y
(
x
))
dx
.
(2.45)
We derived the V
AMPIRE
(Variational Algorithm for Mass-Preserving Image
REgistration) based on the above considerations in our previous work [
51
]. The
source code of V
AMPIRE
can be downloaded at [
52
].
MP
Definition 2.11 (
M
- Mass-preserving transformation model).
For an
3
3
3
image
T
:
Ω
→
R
on the domain
Ω
⊂
R
and a transformation
y
:
R
→
R
the
mass-preserving transformation model
is defined as
MP
M
(
T ,
y
)
:
=(
T◦
y
)
·
det
(
∇
y
)=
T
(
y
)
·
det
(
∇
y
)
.
(2.46)
In the mass-preserving transformation model of V
AMPIRE
the template image
is transformed by interpolation on the deformed grid
y
with an additional
multiplication by the volume change. The multiplication by the Jacobian is a phys-
iological and realistic modeling for density-based images [
128
,
138
]. It guarantees
similar intensities at corresponding points after transformation with a simultaneous
preservation of the total amount of radioactivity.
A simple 2D example for the mass-preserving transformation model is shown in
Fig.
2.3
for illustration. We will use the same notation as in the 3D case for
T
T
,
R
,
,
y
, and
x
. Thus, we redefine
2
,
y
:
2
2
, and
x
T
2
Ω
Ω
⊂
R
R
→
R
=(
x
,
y
)
∈
R
for the
2D example.
T
:
Ω
→
R
and
R
:
Ω
→
R
are defined in the same way but for the
new domain
.
The images of the example are constructed to be similar to SA views of the heart
to a certain degree, cf. Fig.
1.14
b. Further, they have the same total mass (by con-
struction) and thus exactly fulfill the conditions for mass-preservation. The template
and reference image represent smooth signals based on Gaussian distributions. The
template image is created by subtracting two Gaussian distributions with different
standard deviation
Ω
σ
1
,
σ
2
∈
R
>
0
and
σ
1
>
σ
2
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