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Therefore, we have
ξ
ξ
2
+θ
ξ
ξ
2
X
/
L
i
=1
∑
F
[
i
](
x
)
F
−
1
ξ
B
[
i
]
ζ
γ
F
0
c
h
(
ξ
)
2
x
∈
F
0
c
h
2
+
2
ξ
)
2
as
where
γ
is 1 or -1. By defining
γ
∑
ξ
ξ
θ
∑
ξ
ξ
∑
ξ
(
κ
2
,wegetthe
following approximation:
X
/
L
i
=1
∑
κ
2
F
−
1
B
[
i
]
ζ
[
i
](
x
)
(9)
x
∈
By assuming that the displacement estimation error
[
i
](
x
) is statistically indepen-
dent of the image intensity derivatives and inserting the above equations into equa-
tion (1), we get the following approximation of prediction error per pixel:
ζ
X
/
L
i
=1
σ[
i
]
2
1
X
1
F
−
2
+
2
F
−
1
+
α
α
α
(10)
3
1
,
2
,
where
α
α
α
3
are as follows:
d
dx
f
t
−
1
(
x
)
2
X
/
L
i
=1
∑
α
1
=
κ
1
X
x
∈
B
[
i
]
dx
f
t
−
1
(
x
)
X
/
L
i
=1
∑
2
κ
2
X
(
x
)
d
α
2
=
φ
x
∈
B
[
i
]
X
/
L
i
=1
∑
n
(
x
)
2
+
(
x
)
2
1
X
α
3
=
φ
x
∈
B
[
i
]
2.2
Mathematical Model of Temporal Down-Sampling by Mean
Filter
In this subsection, we establish a mathematical model of the relationship between
frame-rate and bit-rate for temporally down-sampled sequences with due consid-
eration of the effect of the integral phenomenon associated with the open interval
of the shutter. Let
f
t
(
x
,
) denote a one-dimensional signal at position
x
in the
t
-
th frame which was taken with the shutter open in the time interval between
t
and
t
+
δ
. Pixel values in each frame are quantized with 8 [bits] at any interval of shutter
open. When the shutter open interval is increased to
m
δ
δ
(
m
is a natural number), the
corresponding signal
f
mt
(
x
,
m
δ
) is given by the following equation:
m
(
t
+1)
−
1
)=
1
m
f
mt
(
x
,
m
∑
τ=
mt
f
τ
(
x
,
δ
δ
)
(11)
