Information Technology Reference
In-Depth Information
proportional to the square of the distance between these positions. Here, position
x
and
x
are local coordinates in segment
B
[
i
] (
i
= 0
,...,
X
/
L
). According, we have
the following statistical model:
E
[
(
d
F
0
[
i
](
x
)
d
F
0
[
i
](
x
)
2
]
c
h
(
x
x
)
2
−
−
(3)
where
E
[
] is expectation operator and
c
h
is constant parameter that depends on
the original video signal. This model gives a good approximation of the ensemble
mean of the difference in displacement (
d
F
0
[
i
](
x
)
·
d
F
0
[
i
](
x
)) for every segment
−
B
[
i
](
i
= 0
,...,
X
/
L
).
From the approximation (2) (3), we create the following approximation :
X
/
L
i
=1
∑
d
F
0
[
i
])
2
(
d
F
0
[
i
](
x
)
−
x
∈
B
[
i
]
X
/
L
i
=1
∑
(
d
F
0
[
i
](
x
)
d
F
0
[
i
](
x
c
[
i
]))
2
−
x
∈
B
[
i
]
c
h
ξ
ξ
2
(4)
where
c
h
=
L
c
h
and
ξ
is
x
−
x
c
[
i
].
ξ
is a relative coordinate (its origin lies at the center
of segment
B
[
i
]). In other words,
is the distance from the center of segment
B
[
i
].
We consider the relationship between displacements at different frame-rates. We
have the assumption that a moving object exhibits uniform motion across successive
frames. This is a highly plausible assumption for high frame rate video signals. In
this case, object displacement is proportional to the frame interval. In other words,
the displacement is inversely proportional to the frame-rate. It leads to the following
equation
ξ
F
−
1
F
−
1
0
d
F
[
i
](
x
)
d
F
0
[
i
](
x
)
(5)
From the approximation (2) (4) (5), we create the following approximation at frame-
rate
F
:
X
/
L
i
=1
∑
d
F
[
i
](
x
)
d
F
[
i
]
2
−
x
∈
B
[
i
]
F
−
1
F
−
1
0
2
X
/
L
i
=1
∑
d
F
0
[
i
](
x
)
d
F
0
[
i
]
−
x
∈
B
[
i
]
=
F
−
1
F
−
1
0
2
X
/
L
i
=1
∑
d
F
0
[
i
](
x
)
d
F
0
[
i
]
2
−
x
∈
B
[
i
]
F
−
2
F
0
c
h
ξ
ξ
2
(6)
