Biomedical Engineering Reference
In-Depth Information
Definition 2.2
(continued)
where
is the continuously differentiable Charbonnier penalizing
function [
28
] with a small positive constant
ψ
:
R
→
R
β
∈
R
>
0
x
2
2
ψ
(
x
)=
+
β
.
(2.6)
Remark 2.1.
Different choices for the function
ψ
in Definition
2.2
are possible,
cf. [
19
].
Remark 2.2.
The function
ψ
in Eq. (
2.6
) is always greater than zero. Consequently,
SAD
D
(
T ,T
)
=
0. This issue could be fixed by subtracting
β
x
2
ˆ
2
ψ
(
x
)
:
=
+
β
−
β
.
(2.7)
For minimization of the registration functional we need to compute the deriva-
tives of the functional and hence the first
x
x
2
∂ψ
(
)=
x
(2.8)
2
+
β
and second derivative of
ψ
2
β
2
∂
ψ
(
x
)=
(2.9)
(
x
2
+
β
2
)
3
/
2
are given here for completeness.
The behavior of the penalizing function
ψ
for different values of
β
is shown in
Fig.
2.1
. It can be observed that
ψ
becomes more quadratic around zero for higher
β
values. This might prevent abrupt jumps in the first derivative between
−
1 and 1
during optimization, thus stabilizing the optimization process.
2.1.2
Regularization
According to Hadamard [
61
] a problem is called
well-posed
if there
1.
Exists
a solution that is
2.
Unique
and when
3. Small changes in data lead only to small changes in the result.
In image registration, small changes in the data can lead to large changes in the
results. Further, as the uniqueness is generally not given, image registration is
ill-posed
[
48
]. This makes regularization inevitable and essential to find feasible
transformations.
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