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attack [6,7]. Moreover, some amplitude-scale invariant features were used to embed
watermark. In the angle QIM (AQIM) [8], the angle of a vector of image samples
was quantized. Zhu introduced a normalized dither modulation (NDM) [9]. The main
idea of NDM was to construct a gain-invariant vector with zero mean for quantization.
Nezhadarya proposed the gradient direction watermarking (GDWM) [10], where the
direction of gradient vectors was uniformly quantized.
Fernando proposed an alternative QIM method [11], called rational dither modula-
tion (RDM), where a gain-invariant adaptive quantization step size at both embedder
and decoder was used to against gain attacks. Inspired by their previous work, we pro-
pose an improved version of the basic RDM in DWT domain to obtain better robustness,
because robustness is a basic requirement for watermarking used for copyright protec-
tion. To this aim, we improve the basic RMD watermarking algorithm mainly in the
following three aspects: First, two coefficients instead of only one coefficient in the
basic RDM are modified to embed watermark, then the allowed quantization step size
can be increased. Second, several modification rules are defined to reduce embedding
distortion and to improve robustness. In addition, significant coefficients in DWT do-
main are selected to embed watermark, because they are more robust to resist various
kinds of attacks. A wide range of attacks are tested to evaluate the performance of our
method, such as amplitude scaling, image filtering, JPEG compression, noise addition,
rotation and resizing. We can see that our method is not only robust to amplitude scal-
ing attack but also robust to common signal processing attacks. We compare our method
with the basic RDM watermarking method [11] and two state-of-the-art watermarking
methods proposed in [10] and [13]. Experimental results demonstrate that our method
outperforms the three compared watermarking methods.
The rest of this paper is structured as follows. Section 2 introduces the details of
RDM watermarking method. Section 3 presents the proposed watermarking method.
Then, experimental results are shown in Section 4. Conclusions are given in Section 5.
2
Improved RDM Watermarking
2.1
Basic RDM Watermarking
RDM watermarking as a QIM approach was first proposed by Gonzalez [11] to against
amplitude scaling attack. The quantization step size of RDM can be seen as a variable
step quantizer, whose size is a function of several past watermarked samples. In this
paper, samples denote as coefficients in the lowest sub-band in DWT domain. Then a
gain invariant adaptive quantization step size is obtained at both embedder and decoder.
In the basic RDM, the set of rational functions
g
:
L
R
ₒ
R
,L
≥
1 are used, which
have the property that:
L
.
g
(
ˁy
)=
ˁg
(
y
)
,forallˁ>
0
,y
∈
R
(1)
Given a host signal vector,
x
=(
x
1
...x
N
) and a watermarked signal vector,
y
=
(
y
1
...y
M
), then the
k
th bit
m
k
∈{
0
,
1
}
of a watermark message is embedded in the
L
th-order RDM as:
x
k
g
(
y
k−
1
y
k
=
g
(
y
k−
1
k−L
)
Q
m
k
(
k−L
)
)
(2)
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