Biomedical Engineering Reference
In-Depth Information
Regularization restricts the space of possible transformations to a smaller set of
reasonable functions. Regularization should be chosen depending on the application
and can either be implicit or explicit. If the transformation is regularized by the
properties of the space itself, as in parametric image registration, we speak of
implicit regularization
. For example, for a rigid 3D transformation only a total of
three rotation and three translation parameters need to be estimated instead of a 3D
motion vector for each voxel. In
explicit regularization
, a penalty is added to the
registration functional to avoid non-smooth transformations. The penalty is often
based on a model that is physically sound and fits the processed data.
As the regularization for parametric transformations is implicitly given we will
only analyze penalties for non-linear image registration in this section. In the rest of
this section we will introduce the following regularizers:
Diffusion regularization (Sect.
2.1.2.1
),
Elastic regularization (Sect.
2.1.2.2
), and
Hyperelastic regularization (Sect.
2.1.2.3
).
These regularizers are specifically chosen as diffusion regularization is typically
used for optical flow applications. Further, V
AMPIRE
relies on hyperelastic
regularization, which is a non-linear generalization of linear elastic regularization.
More information about regularization (e.g., curvature or log-elasticity) and
additional constraints, which are beyond the scope of this topic, can be found
in [
3
,
48
,
94
,
99
,
107
].
Hereinafter, we assume a given transformation
y
:
3
3
R
→
R
which is composed
3
3
3
of the spatial position
x
∈
R
and a deformation (or displacement)
u
:
R
→
R
at
position
x
, i.e.,
y
(
x
)=
x
+
u
(
x
)
.
a
b
2
100
50
−
100
−
100
−
100
−
100
50
50
50
50
β
=
.
β
=
.
1
β
=
10
0
1
β
=
10
0
−
2
y
(
x
)
∂
ψ
(
x
)
Fig. 2.1
The Charbonnier penalizing function (
a
) from Eq. (
2.6
) and its derivative (
b
) from
Eq. (
2.8
) are plotted for
β
=
0
.
1and
β
=
10. They both give an approximation to the absolute
value function
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