Biomedical Engineering Reference
In-Depth Information
Fig. 2.8
SAD V
AMPIRE
:
T
in (
a
)isregisteredto
R
in (
b
) leading to the overlaid transformation
grid in (
a
). The transformed template image is shown in (
c
). A magnification with a comparison
of the estimated vectors (
red
) and the ground-truth vectors (
blue
)isshownin(
d
). (Colored figures
are only available in the online version.)
Inserting Eq. (
2.54
)into(
2.53
) leads to
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
0
=
x
I
(
x
,
y
,
z
,
t
)
·
u
y
I
(
x
,
y
,
z
,
t
)
·
v
z
I
(
x
,
y
,
z
,
t
)
·
w
t
I
(
x
,
y
,
z
,
t
)
=
I
x
u
+
I
y
v
+
I
z
w
+
I
t
u
T
=
∇
I .
(2.55)
For the sake of notation simplicity we will use the shortterm
I
for
I
(
x
,
t
)
in the
following. The terms
I
x
,
I
y
, and
I
z
above represent the derivatives of
I
. Similarly,
u
is shortened to
u
. Equation (
2.55
) immediately gives us the data term for
standard optical flow algorithms.
(
x
)
2
:
u
T
2
2
D
(
I,
u
)
=(
∇
I
)
=(
I
x
u
+
I
y
v
+
I
z
w
+
I
t
)
.
(2.56)
As we now have an equation in three variables (the motion along the three
directions) additional conditions have to be formulated to make the problem
solvable. Different conditions have been proposed to this end.
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