Biomedical Engineering Reference
In-Depth Information
Definition 2.12 (D
mp
- Data term for mass-preserving optical flow).
For
and a deformation (or velocity) field
u
the
data term for mass-preserving optical flow
is defined as
x
x
two images
I
(
,
t
)
and
I
(
,
t
+
1
)
u
u
D
mp
(
I,
)=
∇
·
(
I
)
=
I
x
u
+
I
y
v
+
I
z
w
+
I
(
u
x
+
v
y
+
w
z
)+
I
t
.
(2.69)
As usual a regularization term is added to the functional, e.g., the diffusion
regularization term of the Horn-Shunck method in Eq. (
2.62
). The mass-preserving
optical flow functional to be minimized is therefore
2
dx
u
u
u
dx
J
(
)=
D
mp
(
I,
)
+
α
S
(
)
,
(2.70)
mp
Ω
Ω
where
is the regularization parameter. The corresponding Euler-Lagrange equa-
tions are given by
α
0
=
D
x
I
+
αΔ
u
,
0
=
D
y
I
+
αΔ
v
,
(2.71)
0
=
D
z
I
+
αΔ
w
,
where
D
x
,
D
z
are the derivatives of
D
mp
in the corresponding directions. Solving
this system leads to the following iterative scheme
D
y
,
u
k
+
1
u
k
=
+
D
x
I
+
αΔ
u
,
v
k
+
1
v
k
=
+
D
y
I
+
αΔ
v
,
(2.72)
w
k
+
1
w
k
=
+
D
z
I
+
αΔ
w
.
Algorithm 2.4
Mass-preserving optical flow method
Input:
Images
I
(
x
,
t
)
and
I
(
x
,
t
+
1
)
Output:
Motion estimate
u
(for all voxels)
initialize
u
for all voxels, e.g., null vector
while
change in
u
>
threshold
do
for
each voxel
x
do
compute
u
(
x
)
as given in Eqs. (
2.72
)
end for
end while
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