Biomedical Engineering Reference
In-Depth Information
As shown above in Sect.
2.2.3
, the motion discontinuities can be better modeled
if an L
1
-like penelization term is applied. Using the aforementioned function
ψ
in
Eq. (
2.6
) we get
2
J
(
u
)=
(
(
I,
u
)
)
dx
+
α
(
(
u
))
dx
,
Ω
ψ
D
mp
Ω
ψ
S
(2.73)
mpn
1
2
where
in Eq. (
2.6
). The
corresponding Euler-Lagrange equations can be found in [
37
]. The optical flow
can be estimated by using an iterative scheme similar to that given above in
Algorithm
2.4
for the quadratic case.
ψ
1
and
ψ
2
vary only in the choice of the parameter
β
2.2.5
Multi-level Optimization
As in the case of image registration, the proposed optical flow methods should
be applied in a multi-level framework to avoid local minima during optimization,
cf. Sect.
2.1.5
. The multi-level framework becomes particularly important for
optical flow methods as it allows the treatment of large motion with the Taylor
approximation, which only holds for small displacements. The motion is thus
calculated on a coarse level, where it is still small in terms of voxel size. Starting
at the lowest level we get a rough estimate of the flow vectors. These vectors are
prolonged (interpolated) to the next higher level. On that level the original template
image is interpolated with the current transformation. Again, the flow is estimated,
but on the finer resolution, to figure out the residual motion. This process is repeated
until the maximal resolution is reached.
As the motion vectors are estimated with the mass-preserving motion model,
the transformation of the template image needs to be mass-preserving as well. The
approximation in Eq. (
2.67
)isusedin[
33
] for the mass-preserving transformation
of the template image. For this approximation the time derivative in the equation is
calculated as
. As the optical flow
u
in Eq. (
2.67
) is known after motion
estimation the motion corrected image can be calculated according to
I
t
=
T−R
MC
T
(
x
)=
T
(
x
)+
∇
·
(
T
(
x
)
u
(
x
))
, ∀
x
∈
Ω
.
(2.74)
This equation is accurate for small displacements. However, this assumption
does not hold for potentially large non-linear cardiac displacements. A better,
non-approximated, method is to use the mass-preserving transformation model
in Definition
2.11
based on the exact computation of the Jacobian determinant.
Accordingly, the motion corrected image is calculated as
MC
T
(
x
)=
T
(
x
+
u
(
x
))
·
det
(
∇
(
x
+
u
(
x
)))
, ∀
x
∈
Ω
.
(2.75)
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