Biomedical Engineering Reference
In-Depth Information
Definition 2.7
(continued)
S
vol
(
y
)
:
=
Ω
Γ
v
(
det
(
∇
y
(
x
)))
dx
,
(2.20)
·
(
·
)
where
2
is the Frobenius norm (see Eq. (
2.10
)) and Cof
denotes the
cofactor matrix.
Remark 2.3.
A typical choice of
p
and
q
in Definition
2.7
is
p
=
q
=
2[
51
].
balances between
minimizing the data driven energy term (maximizing the image similarity) and
retaining smooth and realistic transformations, which is controlled by the regulariza-
tion energy. As each term of the hyperelastic regularizer has an individual weighting
factor,
In the formulation in Eq. (
2.1
), the positive real number
α
α
can be set to 1 and only
α
l
,
α
a
, and
α
v
need to be determined.
0, i.e.,
y
is a diffeomorphism [
40
]. This allows us to omit the absolute value bars later in
Eq. (
2.45
) of the mass-preserving transformation model. Hyperelastic regularization
has the ability of modeling tissue characteristics like compressibility, improves the
robustness against noise and thus enforces realistic cardiac and respiratory motion
estimates.
For
α
v
>
0, the conditions for
Γ
v
claimed above ensure det
(
∇
y
)
>
2.1.2.4
Relations
To deepen the understanding of the regularization energies described in this section,
we highlight some similarities and differences. This helps us to understand the
relations between the different regularization variants. As we will see in Sect.
2.1.4
,
controlling volume changes plays an important role in connection with the mass-
preserving motion estimation approach V
AMPIRE
. Accordingly, this aspect is
highlighted in particular.
Diffusion
↔
Elastic
2
.
The interpretation of the latter is revealed as a measure of compressibility, i.e., of
volume changes. The first term is the known diffusion regularization term, which is
hence integrated into the elastic energy.
2
2
and
∇
u
(
∇
·
u
)
As we have seen, the elastic energy consists of the two terms
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