Information Technology Reference
In-Depth Information
f
mt
(
x
,
m
When segment
B
[
i
] in
) is predicted from the previous frame by using
estimated displacement (
d
m
[
i
]), the prediction error is given as follows:
δ
X
/
L
i
=1
σ
X
/
L
i
=1
∑
d
m
,
m
δ)
|
m
[
i
]=
f
mt
(
x
,
m
δ)
−
f
m
(
t
−
1)
(
x
+
2
B
[
i
]
|
x
∈
f
m
(
t
−
1)
(
x
+
d
m
(
x
)
,
m
δ)
−
f
m
(
t
−
1)
(
x
+
d
m
,
m
δ)+
n
(
m
)
2
X
/
L
i
=1
∑
=
x
∈
B
[
i
]
1
m
j
=0
{
f
m
(
t
−
1)+
j
(
x
+
d
m
(
x
)
,
δ)
−
f
m
(
t
−
1)+
j
(
x
+
d
m
,
δ)
}
+
n
(
m
)
2
X
/
L
i
=1
∑
m
−
1
=
x
∈
B
[
i
]
m
−
1
j
=0
dx
f
m
(
t
−
1)+
j
(
x
,
δ)
m
⎨
⎩
⎬
⎭
2
d
X
/
L
i
=1
∑
=
ζ
m
[
i
](
x
)+φ(
x
)+
n
(
m
)
(12)
x
∈
B
[
i
]
where
(
x
) is the second order remainder term of the Taylor expansion, and
n
(
m
)
is the noise element.
φ
ζ
m
[
i
](
x
) is displacement estimation error between estimated
d
m
[
i
] and the true displacement
d
m
[
i
](
x
) at position
x
as follows:
displacement
d
m
[
i
]
ζ
m
[
i
](
x
)=
d
m
[
i
](
x
)
−
Henceforth, we substitute
f
t
(
x
) for
f
t
(
x
,
) for simplicity, unless otherwise stated.
By inserting the above equation into equation (7) (9) and using the first order
approximation of the Taylor expansion and the assumption that the noise element is
statistically independent of the video signal, we obtain:
δ
X
/
L
i
=1
σ
m
[
i
]
2
β
1
(
m
)
F
−
2
+
β
2
(
m
)
F
−
1
+
β
3
(
m
)
(13)
where
β
1
(
m
),
β
2
(
m
),
β
3
(
m
) are as follows:
1
m
2
X
/
L
i
=1
∑
m
1
j
=0
−
d
dx
f
m
(
t
−
1)+
j
(
x
)
β
1
(
m
)=
κ
1
x
∈
B
[
i
]
1
m
dx
f
m
(
t
−
1)+
j
(
x
)
X
/
L
i
=1
∑
m
1
j
=0
−
d
β
2
(
m
)=2
κ
2
φ
(
x
)
x
∈
B
[
i
]
X
/
L
i
=1
∑
(
x
)
2
+
n
(
m
)
2
β
3
(
m
)=
B
[
i
]
{
φ
}
x
∈
Henceforth, we set
m
1
j
=0
−
μ
mt
(
x
)=
1
m
f
m
(
t
−
1)+
j
(
x
)
