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predicted from the previous frame by using estimated displacement (
d
[
i
]), the pre-
diction error is given as follows:
d
[
i
])
[
i
]
2
=
∑
2
σ
B
[
i
]
{
f
t
(
x
)
−
f
t
−
1
(
x
+
}
x
∈
d
[
i
]) +
n
(
x
)
∑
2
=
B
[
i
]
{
f
t
−
1
(
x
+
d
[
i
](
x
))
−
f
t
−
1
(
x
+
}
x
∈
d
dx
f
t
−
1
(
x
)
(
x
)+
n
(
x
)
2
∑
=
ζ
[
i
](
x
)+
φ
x
∈
B
[
i
]
[
i
](
x
) is displacement estimation error between estimated displacement
d
[
i
]
and true displacement
d
[
i
](
x
) at position
x
as follows:
Where,
ζ
d
[
i
]
ζ
[
i
](
x
)=
d
[
i
](
x
)
−
φ
(
x
) is the second order remainder term of the Taylor expansion, and
n
(
x
) is the
noise element.
Let us consider the summation of
[
i
]
2
over all segments (
B
[
i
]
i
(= 1
,
2
,
,
X
/
L
)).
By using the first order approximation of Taylor expansion and the assumption that
the noise element is zero-mean white noise and is statistically independent of the
video signal, we obtain:
σ
···
d
dx
f
t
−
1
(
x
)
2
X
/
L
i
=1
σ[
i
]
2
X
/
L
i
=1
∑
[
i
](
x
)
2
ζ
x
∈
B
[
i
]
(
x
)
d
dx
f
t
−
1
(
x
)
X
/
L
i
=1
∑
+ 2
B
[
i
]
φ
ζ
[
i
](
x
)
x
∈
X
/
L
i
=1
∑
n
(
x
)
2
+
(
x
)
2
+
φ
(1)
x
∈
B
[
i
]
In the following, we describe the relationship between displacement and frame-
rate. Based on modeling the non-uniform motion of pixels within a block, we have
the following approximations about displacement estimation error, as a function of
frame-rate
F
:Let
d
F
[
i
] and
d
F
[
i
](
x
) be the estimated displacement of segment
B
[
i
]
and the true displacement, respectively, at position
x
at frame-rate
F
.
According to the study by Zhen
et al.
[22], statistically, block matching based
on the sum of squared differences (SSD) criterion will result in displacement that
is most likely to be the displacement of block centers. Let
x
c
[
i
] be the position of
the center of block
B
[
i
]. Therefore, we have the following approximation about es-
timated displacement at frame-rate
F
=
F
0
:
d
F
0
[
i
](
x
c
[
i
]) (2)
Additionally, [22] says that the difference in displacement at position
x
from that
at
d
F
0
[
i
]
x
can be modeled as a zero-mean Gaussian distribution whose variance is
