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where
y
k−
1
k−L
denotes the set of watermarked samples (
y
k−L
...y
k−
1
) and
L
is the num-
ber of previous watermarked samples used to calculate the function
g
(), the function
Q
m
k
(
.
) is the standard quantization operation, so that the quantized samples belong to
the shifted lattices:
Q
m
k
(
.
)=
2
ʔ
Z
if
m
k
=0
(3)
2
ʔ
Z
+
ʔ
if
m
k
=1
where
ʔ
is the fixed quantization step size.
At the decoding side, suppose
z
k
is a possibly distorted sample. The hidden bit is
recovered by applying standard quantization decoding procedure to the ratio between
z
k
and its previous samples
z
k−
1
k−L
:
z
k
g
(
z
k−
1
z
k
g
(
z
k−
1
m
k
=
arg min
m
k
||
k−L
)
−
Q
m
k
(
k−L
)
)
||
,m
k
∈{
0
,
1
}
(4)
As to the choice of
g
(), a very large possible functions can be chosen, including the
l
p
−
norms, given by:
i
=1
|
L
k−L
)=(
1
g
(
y
k−
1
p
)
1
/p
y
k−i
|
(5)
L
In this paper, the
l
1
norm is considered, as in [11] and [12].
2.2
Improved RDM Watermarking
A weakness of the basic RDM algorithm is that attacking noise has big influence on the
decoding quantization step size, though the influence can be decreased by increasing
L
. Hence, we modify the basic RDM algorithm to increase the quantization step size,
then better robustness can be obtained. In the basic RDM algorithm, a ratio is computed
using a un-watermarked sample and several past watermarked samples, thus only the
un-watermarked sample can be modified. Different from the basic RDM method, we
compute a ratio of two un-watermarked samples, thus two samples can be modified
simultaneously to embed watermark, which increases the quantization step size.
Let
x
i
∈
R,i
=1
,
2 be two samples, the ratio of them
r
x
is computed as:
r
x
=
min(
x
1
,x
2
)
max(
x
1
,x
2
)
(6)
Obviously,
r
x
is in the range of 0 and 1. The watermarked ratio
r
y
is quantized as
follow:
r
y
=
Q
m
k
(
r
x
)
,m
k
∈{
0
,
1
}
(7)
where
Q
m
k
(
.
) is the quantization function, which is defined as:
⊧
⊨
ʔ
[
r
Δ
] if
mod
([
r
Δ
]
,
2) =
m
k
ʔ
[
r
Δ
]+
ʔ
if
mod
([
r
Δ
]
,
2)
=
m
k
and
Q
m
k
(
r
x
)=
ʔ
[
r
Δ
]
(8)
≥
r
x
or
r
x
=0
⊩
ʔ
if
mod
([
r
Δ
]
,
2)
=
m
k
and
ʔ
[
r
Δ
]
<r
x
or
r
x
=1
ʔ
[
r
Δ
]
−
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