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IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. ??, NO. ?, APR 2011
1
Blind Image Watermarking Using a Sample
Projection Approach
Mohammad Ali Akhaee*, Member, IEEE, Sayed Mohammad Ebrahim Sahraeian, Student Member, IEEE, and
Craig Jin, Member, IEEE
Abstract—This paper presents a robust image watermarking
scheme based on a sample projection approach. While we consider the human visual system in our watermarking algorithm,
we use the low frequency components of image blocks for data
hiding to obtain high robustness against attacks. We use four
samples of the approximation coefficients of the image blocks
to construct a line segment in the 2-D space. The slope of this
line segment, which is invariant to the gain factor, is employed
for watermarking purpose. We embed the watermarking code
by projecting the line segment on some specific lines according
to message bits. To design a maximum likelihood decoder,
we compute the distribution of the slope of the embedding
line segment for Gaussian samples. The performance of the
proposed technique is analytically investigated and verified via
several simulations. Experimental results confirm the validity
of our model and its high robustness against common attacks
in comparison with similar watermarking techniques that are
invariant to the gain attack.
Index Terms—Image watermarking, Maximum Likelihood detector, sample projection, gain attack.
I. I NTRODUCTION
Digital watermarking embeds information within a digital
work as a part of the media. Watermarking techniques falls into
three categories of robust, semi fragile, and fragile methods
according to their specific applications [1]–[3]. Robust watermarking mainly serves for identification purposes while the
fragile and semi fragile watermarking are usually employed
in authentication applications. Since a good watermarking
scheme should always be able to deal with some kinds of
attacks, studies in the watermarking research area mostly target
robust watermarking problems.
Several robust watermarking techniques have been proposed
so far. Cox et al. [4] have proposed an additive watermarking
approach based on spread spectrum concept which remains
highly robust against noise and cropping attacks. Several other
studies have improved this approach further [5]–[10]. Based
on this observation that boosting the watermarking power
increases the barrier against attacks, most of the effective
watremarking schemes try to match the characteristics of
the watermark to those of the image asset. Multiplicative
Copyright (c) 2010 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to [email protected].
M. A. Akhaee and Craig Jin are with the Computing and Audio Research Laboratory (CARLAB), Department of Electrical and Information
Engineering, University of Sydney, NSW, Australia, 2006 (e-mail: akhaee,
[email protected]).
S.M.E. Sahraeian is with the Genomic Signal Processing (GSP) Lab,
Department of Electrical and Computer Engineering, Texas A&M University,
College Station, TX 77843, USA (e-mail: [email protected]).
watermarking, as an example, has been introduced in [11] and
has been widely studied later on using local optimum decoders
in multiresolutional transform domains such as wavelet and
contourlet domains [12]–[16]. Besides, a universal optimal
detector for scaling based watermarking schemes is presented
in [17]. These schemes are highly robust against noise and
compression attacks. To satisfy robustness against geometric
attacks and reduce the watermark synchronization problem,
Tang and Hang [18] proposed a watermarking scheme which
employs a feature extraction and image normalization approach. Based on log polar mapping (LPM) and phase correlation, Zheng et al. [19] proposed an image watermarking
technique which embeds the watermark into the LPMs of the
image Fourier magnitude spectrum. This scheme is invariant
to rotation and remains comparatively robust to scaling attack.
One of the common but effective attacks in watermarking
systems is volumetric distortions ( i.e., any kind of amplitude scaling or gamma compensation). For image signals, it
may happen due to the scanning process where light is not
distributed uniformly over the paper. This simple attack is the
main drawback of most recent studies like [12], [14]–[16] and
even the quantization index modulation (QIM) algorithm [20]
which has attained great popularity due to its lossless performance with lattice-based codebooks. Three types of solutions
can tackle this problem, especially for the QIM method:
1) adopting auxiliary pilots through the watermarked signal
known at both the encoder and decoder [21]; 2) using spherical
codewords [22] with correlator decoders [23], or using angle
QIM (AQIM) [24], [25]; iii) introducing a domain in which the
embedding process is invariant to the gain attack [26]. Some
improvement/generalization on this scheme which is referred
to as rational dither modulation (RDM) are proposed in [27]–
[29].
The first solution against the gain attack decreases the
security of the algorithm, since the malicious attacker can
change either the watermark or the pilot signals. Besides, pilots
are deterministic objects in the main signal and can be easily
detected. Although the second and third approach keep the
security of the QIM algorithm, they cause high computational
cost. Besides, the low robustness of AQIM against additive
white Gaussian noise (AWGN) and the high peak to average
power ratio (PAPR) of RDM algorithm which happens due
to its momentarily large quantization step size are the main
drawbacks that should be addressed. Thus, no approach has
been presented so far that both proposes an optimal decoder
and remains invariant to the gain attack.
In this paper, we propose a novel gain invariant watermark-
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IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. ??, NO. ?, APR 2011
ing scheme based on a sample projection scheme. Embedding
the watermark bits into the approximation coefficients of
the image blocks makes the algorithm highly robust against
noise and compression attacks. Any possible selection of four
approximation coefficients, that may be selected using a secret
key, constructs a line segment whose slope is considered for
data hiding. We embed the watermark bits by projecting the
line segment on some specific coding lines while preserving
its center of mass. In this way, the slope of the line segment
carries the watermark information while the distortion imposed
to its constructive samples is minimal. Since our embedding
process is linear, it can be denoted by multiplication of specific
embedding matrices. To implement the maximum likelihood
(ML) detector for data extraction, we should calculate the
distribution of the slope of the line segment. To this aim,
we consider the fact that the approximation coefficients of
most of the image blocks can be well-modeled by Gaussian distribution [15]. The performance of our watermarking
method is analytically investigated and verified via simulation
on artificial signal. Several experiments on sample images
confirm the effectiveness of the proposed scheme in resisting
against common noise attacks.
The rest of the paper is organized as follows. In Section
II, we describe the model of the system. The watermark
embedding and decoding process are introduced in Section
III. Section IV analyzes and evaluates the performance of the
proposed scheme. Experimental results are demonstrated in
Section V, and Section VI concludes the paper.
II. S YSTEM M ODELING
In this section, we first introduce the model considered
for our watermarking algorithm. To this aim, we calculate
the distribution of the watermarking variable. We assume
to have four samples of an independently and identically
distributed (i.i.d) Gaussian random variable as the host signal.
We show this signal as u = [u1 , u2 , u3 , u4 ] with the Gaussian
distribution of N (0, σu2 ). These four samples form two points
p = [u1 , u2 ] and q = [u3 , u4 ] in the 2-D space. We employ
2
c = uu43 −u
−u1 , the slope of the line going through these two points
as our watermarking variable. We can see that the numerator
and the denominator of c are distributed as N (0, 2σu2 ).
For independent Gaussian variables a ∼ N (0, σa2 ) and b ∼
N (0, σb2 ), their ratio c = ab which is the ratio of two zero-mean
independent Gaussian variables is Cauchy as:
σ
fC (c) =
a
1
σb
,
π c2 + ( σσab )2
(1)
As shown in Appendix A, for the case that the two Gaussian
variables a and b are correlated with the correlation coefficient
r=
E[(a − µa )(b − µb )]
,
σa σb
the probability density function (PDF) of the variable c is given
as:
√
σa σb 1 − r 2
1
,
(2)
fC (c) =
a 2
2
2
π σb2 (c − rσ
σb ) + σa (1 − r )
and its cumulative distribution function (CDF), FC (c), can be
computed as:
FC (c) =
1
1
σb c − rσa
+ tan−1 √
.
2 π
σa 1 − r 2
(3)
As we will see in the next sections, we use the probabilistic
characteristics of the variable c, the slope of the mentioned
line segment, to design an optimal decoder and later analyze
its performance.
III. P ROPOSED M ETHOD
In this section, we introduce our blind watermarking
scheme. As discussed in the previous section, we assume the
host signal as a four-sample i.i.d. Gaussian random signal.
In practical applications, these four samples can come from
approximation coefficients of the image blocks which satisfy
our i.i.d. Gaussian assumption according to KolmogorovSmirnov test results.
A. Watermark embedding
Let us represent the four samples of the host signal as
u = [u1 , u2 , u3 , u4 ]. We model these four samples as two
points p = [u1 , u2 ] and q = [u3 , u4 ] in the 2-D space. Fig.
1(a) illustrates these two points as well as the line segment
connecting them. We denote the slope of this line segment as
3 u2 +u4
θ. The center of this (p, q) line is located at [ u1 +u
, 2 ].
2
By translating the center of this line to the origin, we reach
to the points pc and qc , where
( ′ ) ( u1 −u3 )
( ′ ) ( u3 −u1 )
u1
u3
2
2
pc =
= u2 −u4 , qc =
= u4 −u
. (4)
2
u′2
u′4
2
2
Now, to embed the M -ary watermark code, we project this
line segment to one of the M coding lines (L1 to LM if θ > 0
or their counterparts L′1 to L′M if θ < 0), shown in Fig. 1
(b), depending on the watermark code. We use projection and
keep the center of the line segment to impose less distortion
and thereby cause more invisibility of the watermarked signal.
Fig. 1 (a) illustrates the projection steps in details. We call the
resulted points in the mapped line as p⊥ and q⊥ . The slope of
the ith coding line is αi (for Li ) if θ, the slope of the primary
line connecting (p, q), is positive and −αi (for L′i ) otherwise,
where αi is given by
π M − 2i + 1
−
β).
(5)
4
2
Here, β is the angle between two consecutive coding lines.
The position of p⊥ for the case that the slope of the mapped
line is k, can be computed using the intersection of two
following lines:
{
y = kx
(6)
y − u′2 = − k1 (x − u′1 ).
αi = tan(
With a similar scheme, we can find q⊥ . The obtained points
are as follows:
( u′ +ku′ )
( u′ +ku′ )
1
p⊥ =
2
k2 +1
ku′1 +k2 u′2
k2 +1
3
,
q⊥ =
4
k2 +1
ku′3 +k2 u′4
k2 +1
.
(7)
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IMAGE WATERMARKING
3


 
u′′1
u1
u′′2 
 u2 
 ′′  = T(k)   ,
 u3 
 u3 
′′
u4
u4
as:
(9)
using the transfer matrix T(k), given as:
 2
k +2
 k
1

T(k) = 2
2k + 2  k 2
−k

k
k2
−k
2k 2 + 1
−k
1 
 . (10)
−k,
k2 + 2
k 
1
k
2k 2 + 1
Therefore, the watermarking embedding process can be
figured as implementation of (9) where k is defined based
on the watermark bit and θ, the slope of the primary (p, q)
line, as:
{
αi
when θ ≥ 0
k=
(11)
−αi when θ < 0.
(a)
In Fig. 1(b), we can see that for each coding word we have
two lines, a solid one (Li ) for positive θ and a dashed one
(L′i ) for negative θ.
In this way, we obtain the watermarked signal u′′ =
′′
[u1 , u′′2 , u′′3 , u′′4 ].
B. Watermark decoding
(b)
Fig. 1.
(a) Steps of the proposed watermarking embedding scheme:
translation to the origin, projection, and translation back. (b) Coding space.
Solid lines: coding lines with positive slopes (L1 · · · LM ); dashed lines:
the counterpart coding lines with negative slopes (L′1 · · · L′M ); dotted lines:
Decision boundaries.
As the final step, we just need to translate back the mapped
line to its original center to obtain points pw = [u′′1 , u′′2 ] and
qw = [u′′3 , u′′4 ] as:
)
( ′′ ) ( u′1 +ku′2
u1 +u3
u1
k2 +1 +
2
pw =
= ku′ +k2 u′
1
2
4
u′′2
+ u2 +u
k2 +1
2
)
(
( ′′ )
u′3 +ku′4
u1 +u3
u3
k2 +1 +
2
.
qw =
= ku′ +k2 u′
3
4
4
u′′4
+ u2 +u
k2 +1
2
To extract the hidden bits, an optimum decoder is implemented using M-Hypothesis test as follows. We denote the
received signal as y = u′′ + n, where y = [y1 , y2 , y3 , y4 ]
represents the watermarked signal contaminated with zero
mean AWGN n ∼ N (0, σn2 ).
For the i.i.d. Gaussian host signal u, the watermarked signal
u′′ is also Gaussian as it is obtained by a linear transformation
through the matrix T(k). Thus, the received samples y are
Gaussian with the variance of σy2 = σu′′2 + σn2 . Now, we can
use the model given in Section II. We depict the four samples
in y as two points pr = [y1 , y2 ] and qr = [y3 , y4 ] in the 2-D
space and calculate the slope of the line segment connecting
them as:
y4 − y2
u′′ − u′′2 + n4 − n2
c=
= 4′′
.
(12)
y3 − y1
u3 − u′′1 + n3 − n1
As we fixed the slope of the line connecting pw = [u′′1 , u′′2 ]
and qw = [u′′3 , u′′4 ] to k, defined in (11), we have:
k=
u′′4 − u′′2
⇒ u′′4 − u′′2 = k(u′′3 − u′′1 ).
u′′3 − u′′1
(13)
Therefore, by defining v = u′′3 − u′′1 , we can rewrite (12) as:
c=
kv + n4 − n2
.
v + n3 − n1
(14)
Using (9) and (10), we have
1
(−u1 − ku2 + u3 + ku4 ).
+1
Thus, v is a Gaussian random variable with mean µv = 0
2
2
and variance σv2 = 1+k
2 σu . Consequently, if we show the
numerator and the denominator of c in (14) respectively as a
and b, we will have a ∼ N (0, k 2 σv2 +2σn2 ) and b ∼ N (0, σv2 +
v = u′′3 − u′′1 =
(8)
By inserting (4) in (8) we can summarize the whole procedure
k2
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IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. ??, NO. ?, APR 2011
2σn2 ). The correlation coefficient between a and b can also be
computed as:
r=√
kσv2
(k 2 σv2 + 2σn2 )(σv2 + 2σn2 )
.
can be written as:
P + (e|i) =P (tan (−ϕi +
β
β
) < c|i) + P (c < tan (−ϕi − )|i)
2
2
(
(
β )
β )
=FC|i tan (ϕi − ) − FC|i tan (−ϕi + )
2
2
(
(
β )
β )
+1 − FC|i tan (ϕi + ) + FC|i tan (−ϕi − ) ,
2
2
(21)
(15)
+P (tan (ϕi +
Now we have c = ab , where a and b are two correlated
zero mean Gaussian random variables. Thus, the distribution
function of c can be given as in (2).
Having the distribution of the slope function, we can simply
use the Maximum Likelihood detector as:
i∗ = arg max fC (c|i),
(16)
i∈1,··· ,M
where fC (c|i) represents the distribution function for the ith
coding line. Here, without loss of generality, we suppose that
the received c is positive. The same argument also holds for
the negative case. Thus, the conditional distribution function
is
√
σ
σ
1 − r|i2
a
b
|i
|i
1
fC (c|i) =
,
(17)
π σ 2 (c − r|i σa|i )2 + σa2 (1 − r2 )
b|i
σb|i
|i
where, c is the slope of the detected line, and FC|i (.) is its
CDF if the embedded line is Li . Substituting FC (c) from (3)
and defining
β
π M − 2i
= −
β,
2
4
2
β
π M − 2(i − 1)
= ϕi − = −
β,
2
4
2
ϕ′i
= ϕi +
ϕ′i−1
and di =
σb|i
σa|i
P + (e|i)
=
di tan ϕ′i−1 − r|i
1
√
tan−1
π
1 − r2
|i
where,
−
2
2αi2 2
σ + 2σn2 , σb2|i =
σ 2 + 2σn2 , (18)
1 + αi2 u
1 + αi2 u
2αi
σ2
αi σu2
1+α2i u
=√ 2
r|i =
.
σa|i σb|i
(αi σu2 + (1 + αi2 )σn2 )(σu2 + (1 + αi2 )σn2 )
(19)
σa2|i =
′
−1 di tan(−ϕi−1 )
1
tan
π
+ 1−
+
√
1 − r|i2
Pe+
M
1 ∑ +
=
P (e|i),
M i=1
− r|i
di tan(ϕ′i ) − r|i
1
√
tan−1
π
1 − r|i2
di tan(−ϕ′i ) − r|i
1
√
,
tan−1
π
1 − r2
(23)
|i
where σa|i , σb|i , and r|i are defined as in (18) and (19).
Moreover, by defining
Hi (ϕ) =
di tan ϕ − r|i
di tan(ϕ) + r|i
1
1
√
+ tan−1
,
tan−1 √
2
π
π
1 − r|i
1 − r|i2
we can simplify (23) as:
P + (e|i) = 1 + Hi (ϕ′i−1 ) − Hi (ϕ′i ).
(24)
For i = 1 and i = M , we can see that:
β
β
P + (e|1) = 1 − P (tan (−ϕ1 − ) < c < tan (ϕ1 + )|1)
2
2
= 1 − H1 (ϕ′1 ),
(25)
IV. P ERFORMANCE E VALUATION
Here, we analytically study the error probability of the
proposed watermarking scheme in the presence of AWGN.
First, we compute Pe+ , the error probability when the slope
of embedding line (p, q) is positive, given as:
(22)
, we have:
|i
These parameters are computed by substituting k as defined
in (11) in definitions of variances of a, b and their correlation
coefficient r.
As shown in Appendix B, the decision boundary between
each two coding line is their bisector. Considering this ML
decision for all pairs of coding lines, we deduce that fC (c|i)
is maximum if the angle of the received line with slope c
falls between the bisectors of ith coding line and its two
neighboring coding lines. Therefore, we obtain the decision
boundaries as shown in Fig. 1(b) with dotted lines. It is
notable that the deduced decoder is independent of β, the angle
between two consecutive coding lines.
β
β
) < c < tan (ϕi − )|i)
2
2
P + (e|M )
=
1 − P (c > tan (ϕM −
− P (c < tan (−ϕM +
β
)|M )
2
β
)|M ) = HM (ϕ′M −1 ),
2
(26)
Substituting (24),(25), and (26) in (20),we have:
=
M
M −1
1 ∑
M −1
1 ∑
+
Hi (ϕ′i−1 ) −
Hi (ϕ′i )
M
M i=2
M i=1
=
M −1
1 ∑
M −1
+
[Hi+1 (ϕ′i ) − Hi (ϕ′i )],
M
M i=1
(20)
where P + (e|i) represents the error probability when the
embedding line is Li , the ith positive slope coding line.
According to (16), for i = 2 · · · M − 1, the error probability
Pe+
(27)
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IMAGE WATERMARKING
5
(a)
(b)
(c)
Fig. 2. Comparison between the theoretical error probability and experimental one for a Gaussian random variable with σu = 40. (a) one bit embedding
(M = 2), (b) two bits embedding (M = 4), and (c) three bits embedding (M = 8).
Fig. 3. Comparing the probability of error with DWR of 19 dB for different
bit rates.
With a similar argument we can see that the error probability
for the negative slope embedding line is the same as Pe+ ; that
is Pe− = Pe+ . Besides, it is easy to show that the positive
and the negative embedding are equally probable in general;
therefore, we can write the error probability of the decoder as
Pe =
1 + 1 −
P + Pe = Pe+ .
2 e
2
(28)
Here, we obtained a closed form solution for the error
probability of the decoder. In Fig. 2 we compared this theoretical error probability with the experimental case of Gaussian
random variable with σu = 40 for different capacities of
one bit (M = 2), two bits (M = 4), and three bits
(M = 8). In this figure, the probability of error is shown
versus various noise strengths, measured by the watermark-tonoise ratio (WNR), which is the ratio between the embedding
distortion and the attacking noise distortion. As, we can see
the theoretical and experimental results match perfectly.
Fig. 4. Comparing the probability of error for the proposed method and the
AQIM method [25] with DWRs of 19 dB and 22 dB.
V. E XPERIMENTAL R ESULTS
We performed several experiments to test the proposed algorithm and evaluate its performance against different attacks.
A. AWGN attack on the artificial signal
In the first experiment, to validate our method, we present
the performance results for an artificial Gaussian signal under
AWGN attack. We conducted the simulations for various bit
rates and noise levels. The strength of the watermarking in
terms of the document-to-watermark ratio (DWR) is fixed to
19dB and bit rates are set to one bit (M = 2), two bits (M =
4), and three bits (M = 8) per each four samples. The results
versus various WNRs are given in Fig. 3. As expected, the
performance degrades as M increases. However, we observe
that the proposed scheme is practically applicable for M = 2,
M = 4, and M = 8. Nevertheless, for further experiments we
just use M = 2 and M = 4.
In Fig. 4, we also compare the robustness for M = 2 with
an AQIM based method [25] in similar situations for DWRs of
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Fig. 5. Comparing the probability of error for the proposed method and the
hyperbolic RDM scheme [28] under power-law attack.
19 dB and 22 dB. As we can see, the proposed method which
employs an optimal decoder has lower error probability values
even for very low WNRs.
B. Power-law attack on the artificial signal
Here, we show the robustness of the proposed watermarking
algorithm against the power-law attack which can model the
gamma correction attack (as a non-linear gain attack) that is
a common process on image and video signals [28]. If we
represent the watermarked signal with u′′ , the AWGN signal
with n, and the attacked signal with z, the power-law attack
is defined as z = λ(u′′ + n)γ , where λ is the gain factor and
γ is the exponential parameter.
Fig. 5 shows the performance of our sample projection
approach against this attack for λ = 0.7 and γ = 1.2 along
with the results obtained by [28] with different value of L
in similar conditions. Here, L determines the memory of the
system. As seen, we have high robustness against the powerlaw attack which is an important factor of our algorithm caused
form the invariance of the proposed sample projection scheme
to the gain attack.
C. Tests on the image signal
Then, we conduct several experiments to verify the performance of the proposed technique in the real application
of image watermarking. Throughout our experiments, we use
the Daubechies length-8 Symslet filters with two levels of
decomposition to compute the 2-D DWT. The watermark data
is embedded in the second level approximation coefficients
of each block. The results are obtained by averaging over 100
runs with 100 different pseudorandom binary sequences as the
watermark bits.
For this study, we use four natural images of size 512×512:
Plane, Pirate, Boat, and Bridge. The original test images and
their watermarked versions using the proposed method with
16 × 16 block size and 128 bits message length are shown in
Fig. 6. Original (top) and watermarked (bottom) test images; Left-right:
Plane, Pirate, Boat, and Bridge.
Fig. 6. Besides, β, the angle between two consecutive coding
lines used in (11) is set to β = 40o for M = 2 and β = 20o
for M = 4. These values are hand-optimized to achieve the
highest robustness while keep the watermark almost imperceptible. As we can see, the watermark invisibility is perfectly
satisfied. The mean peak-signal-to-noise-ratio (PSNR) of the
watermarked images are 40.39dB, 40.44dB, 39.89, and 40.85,
respectively. To quantitatively evaluate the imperceptibility of
the watermarked images, we use the mean structural similarity
index (MSSIM) [30] which is highly matched with the human
visual system and effectively captures local errors in the spatial
domain. The respective MSSIM values for four test images
are 0.9983, 0.9984, 0.9982, and 0.9984 which confirms the
imperceptibility of the proposed algorithm.
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IMAGE WATERMARKING
7
TABLE III
BER(%) R ESULTS OF EXTRACTED WATERMARK
ATTACK .
(a)
0
0.05
0.1
w
0.15
4.00
8.00
16.00
32.00
0.00
0.00
0.00
0.00
0.83
2.24
0.00
0.00
2.17
5.62
0.00
0.00
3.93
8.02
0.00
0.00
0.2
0.25
5.88
10.71
0.00
0.00
7.54
13.43
0.00
0.00
TABLE IV
E FFECT OF BLOCK SIZE AND MESSAGE LENGTH ON THE WATERMARK
ROBUSTNESS (BER (%)).
(b)
Fig. 7. AWGN attack for various noise variances, (a) for M = 2 (binary
case) and (b) for M = 4 (2-bits).
(a)
Segment
Size
UNDER VARIABLE GAIN
Blk.
Size
Bits
MSSIM
JPEG
20%
AWGN
15
Gaus.
3×3
Crop
5%
Scale
0.75
Rot.
-5
8
8
8
8
16
16
16
32
32
128
256
512
1024
128
256
512
128
256
0.9995
0.9989
0.9980
0.9961
0.9983
0.9965
0.9932
0.9906
0.9850
10.92
11.38
13.60
16.93
1.92
2.58
4.96
1.09
6.27
9.14
9.89
12.22
14.82
4.27
4.81
7.99
1.70
7.13
10.61
11.25
12.67
12.71
1.18
2.31
2.33
0.12
0.48
4.14
4.38
4.87
5.24
4.29
4.71
5.89
9.49
11.35
3.81
3.57
4.36
4.09
0.19
0.12
0.07
0.00
0.00
7.95
8.07
8.54
9.09
2.60
2.75
3.56
4.90
5.44
(b)
Fig. 8. JPEG compression attack for various quality factors, (a) for M = 2
(binary case) and (b) for M = 4 (2-bits).
Image
Median
Filtering
3×3
M =2
Plane
Pirate
Boat
Bridge
1.50
4.05
3.34
3.80
0.38
1.58
1.74
1.00
1.15
2.38
2.43
2.66
1.15
2.82
2.84
2.86
M =4
TABLE I
BER(%) R ESULTS OF EXTRACTED WATERMARK UNDER MEDIAN AND
G AUSSIAN FILTERING ATTACKS .
Gaussian
Filtering
3×3 5×5 7×7
Plane
Pirate
Boat
Bridge
6.57
7.38
7.02
6.93
4.61
3.95
6.05
4.59
5.52
5.66
9.22
6.87
5.79
5.70
9.38
7.16
1) AWGN attack: As the first attack, we investigate the
effect of AWGN to the proposed watermarking scheme. Fig.
7 shows the bit error rate (BER) of the proposed method for
various images versus different noise powers. As we see, the
method has a great resistance against noise attack for both
M = 2 and M = 4.
2) JPEG attack: Secondly, the proposed technique is tested
against JPEG compression with different quality factors. As
demonstrated in Fig 8, since low frequency components of the
image blocks are used for watermarking, the proposed method
is highly robust against JPEG with different quality factor up
to 10%.
3) Filtering attack: Table I shows the BER results for
median filtering, and Gaussian low-pass filtering attacks for
different test images. It can be seen that the proposed scheme
is highly robust against these attacks.
4) Editing attack: In Table II, we also report the performance of our algorithm under several image editing attacks
such as cropping, scaling, and rotation. As we can see in this
table, the proposed scheme remains highly robust against these
attacks as well.
5) Gain attack: To show the robustness of the proposed
algorithm to constant and variable gain attack, we conduct
an experiment on the four test images. To model the variable
gain attack, we segment each image to small non-overlapping
blocks and multiply all the pixels in each block by the gain
factor of 1 + wδ where δ is a random variable uniformly
distributed between 0 and 1, and w is a constant determining
the strength of the attack . Thus, over each block, the gain
factor remains constant, while for each block we use different
gain factors. This modeling can realize the variable gain attack
and fading attack happens through scanning process. Results
for different segmenting block sizes and strength factors ws
are shown in Table III ( Results are averaged over the four test
images for M = 2). As we see, for block sizes of 16 and 32
our algorithm is invariant to the gain attack. This is because
our watermarking approach embeds information in blocks of
size 16; hence, the gain will be constant in each embedding
block, and thud in the approximation band. That is, duo to the
invariance of the sample projection scheme to the constant
gain factor, we will have 0% BER for segmentation of 16
and 32. For segments of size 8 and 4 we will have variable
gain factors in each embedding block. However, we can see
that even for such variable gains, our algorithm shows high
robustness.
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8
IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. ??, NO. ?, APR 2011
TABLE II
BER(%) R ESULTS OF EXTRACTED WATERMARK UNDER CROPPING , SCALING , AND ROTATION
M =2
Image
Plane
Pirate
Boat
Bridge
Cropping
5%
10%
3.95
4.42
3.57
5.23
5.71
5.71
6.20
11.34
ATTACKS .
M =4
0.5
Scaling
0.75
1.5
-10
0.38
0.70
0.72
1.42
0.00
0.38
0.00
0.37
0.00
0.00
0.00
0.00
2.23
5.06
2.72
6.67
Rotation
-5
5
1.45
3.77
1.54
3.64
0.69
2.73
0.35
4.75
10
1.09
3.11
1.12
5.87
Cropping
5%
10%
4.00
5.57
4.68
4.80
5.97
5.72
6.84
10.94
0.5
Scaling
0.75
1.5
-10
2.54
4.08
4.25
4.74
0.00
0.60
0.48
0.76
0.00
0.81
0.00
0.24
5.16
6.30
7.53
5.50
Rotation
-5
5
3.83
5.31
4.53
4.70
10
2.37
4.23
2.58
3.37
2.26
4.45
3.20
5.62
TABLE V
C OMPARISON BETWEEN OUR WATERMARKING METHOD AND [24] : BER
(%) UNDER AWGN ATTACK .
σn
4
Method
[24]
Proposed
1
2
3
1.00
0.00
2.00
0.00
3.00
0.12
4.00
0.12
5
6
7
15.00
0.43
25.00
0.85
44.00
1.05
TABLE VI
C OMPARISON BETWEEN OUR WATERMARKING METHOD AND WANG ’ S
METHOD [31] : BER (%) UNDER M EDIAN F ILTERING ATTACK WITH
WINDOW SIZE 3 × 3.
Method
Barbara
Wang [31]
Proposed
Fig. 9.
24.95
7.00
Image
Baboon Peppers
31.65
9.88
29.35
3.25
Goldhill
25.60
1.65
Capacity of the proposed method for BER less than 5%.
6) Block size and bit rate effect: In this experiment, we
study the effect of changing the block size and message
length on the performance of the proposed algorithm. To this
aim, we vary the block size from 8 to 32, and the number
of blocks engaged, that is the message length, from 128 to
1024. The robustness for M = 2, averaged over the four test
images, against set of attacks as well as the MSSIM value,
measure of imperceptibility, is demonstrated in Table IV. Our
experiments on various block sizes and watermark message
lengths indicated that while other block size and bit rates also
can be picked based on the application, the case of 16 × 16
block size and 128 bit message length is the best compromise
between imperceptibility, capacity, and robustness.
7) Capacity: To show the capacity of the proposed scheme,
a graph of capacity versus noise standard deviation is shown
in Fig. 9. In this graph, we have considered a fixed BER of
5% as a typical sufficiently small error rate and investigated
the maximum message length in bits for 16 × 16 block size
and M = 2 which obtains this error rate. The average message
length over four test images as well as the average MSSIM
value for each case is shown in Fig. 9. As shown in this figure,
even for heavy noise powers of σn = 20 , the capacity of the
proposed method in obtaining the BER of less than 5% is near
140 bits.
8) Comparison With Other Watermarking Schemes: Finally, we compare our watermarking algorithm for M = 2
with two of the recent blind watermarking techniques, [24] and
[31], in the same situation, for AWGN and median filtering
attacks. The simulation results are shown in Tables V, VI.
As shown in these tables, the robustness of our method is
considerably better than these two techniques.
VI. C ONCLUSION
In this article, we presented a novel blind watermarking
approach with the optimal decoder. Watermark embedding is
performed by multiplication of M specific matrices to the vector of samples of size four. These matrices translate and project
the vector of samples on the predefined coding lines depending
on the message symbol. Assuming the host samples to be i.i.d
Gaussian, which is often valid for approximation coefficients
of image blocks [15], we obtained a closed form PDF of
noisy watermarked samples. Having this distribution function,
we designed an optimum ML decoder. We analytically studied
and verified the error probability of the proposed decoder in
a noisy environment. The proposed algorithm is applied to
image signals by using four approximation coefficients of the
image blocks which may be selected according to a secret
key. In addition to be invariant to the volumetric distortions,
several simulations showed that the proposed algorithm is
highly robust against common watermarking attacks such as
AWGN, compression, and filtering. However, the algorithm
is sensitive to the collusion attack. The future work may be
proposing a solution to this problem in order to make the
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IMAGE WATERMARKING
9
algorithm robust to the this attack as well.
where,
A PPENDIX A
C OMPUTING THE P ROBABILISTIC C HARACTERISTICS OF
THE D ECISION VARIABLE
To compute the PDF and the CDF of the variable c = ab , for
correlated Gaussian variables a ∼ N (0, σa2 ) and b ∼ N (0, σb2 )
with correlation coefficient r, consider their joint distribution,
given as:
fab (a, b) =
1
√
2
2
1
a
2rab
b
− 2(1−r
2 ) ( σ 2 − σa σ + 2 )
2πσa σb 1 − r2
e
a
b
σ
b
. (29)
a
b
Using this distribution function, the CDF of c =
can be
computed as:
}
{a
≤c
FC (c) = P
{b
}
{
}
= P a ≤ bc, b > 0 + P a ≥ bc, b < 0
∫ ∞ ∫ bc
=
fab (a, b) da db
b=0 a=−∞
∫ 0
∫ ∞
+
fab (a, b) da db,
(30)
b=−∞
a=bc
and its probability density function can be defined as:
∫ ∞
′
fC (c) = FC (c) =
|b|fab (bc, b) db.
(31)
−∞
Substituting (29) in (31) and considering the fact that fab (a, b)
is an even function with respect to a and b, we have:
∫ ∞
2
− b2
1
σ02
√
√
fC (c) =
,
be 2σ0 db =
2πσa σb 1 − r2 0
πσa σb 1 − r2
(32)
where
1 − r2
σ02 = c 2
.
( σa ) − σ2rc
+ σ12
a σb
b
As a consequence,
fC (c) =
√
1
σa σb 1 − r 2
.
a 2
2
2
π σb2 (c − rσ
σb ) + σa (1 − r )
In addition, FC (c) can be computed as:
∫ c
1
1
σb c − rσa
FC (c) =
fC (t) dt = + tan−1 √
.
2
π
σa 1 − r 2
−∞
(33)
(34)
A PPENDIX B
C OMPUTING THE D ECISION B OUNDARIES OF THE ML
D ECODER
For each two coding lines Li and Lj we have:
fC (c|i) ≷ij fC (c|j).
(35)
Substituting (17) in (35) and after some simplifications, we
obtain:
m2 c2 + m1 c + m0 ≷ij 0,
(36)
m2
=
m1
=
m0
=
σb|j √
σb|i √
2
1 − r|i2 −
1 − r|j
σa|j
σa|i
√
√
2 − 2r
2r|i 1 − r|j
1 − r|i2
|j
σa|j √
σa|i √
2.
1 − r|i2 −
1 − r|j
σb|j
σb|i
(37)
To find the roots for this quadratic equation, we show these
coefficients using ϕi = arctan(αi ), the angle of the coding
σ2
line Li . Thus, by defining γ 2 = σu2 , we can write the
n
parameters in (18) and (19) as:
σa2|i = 2σn2 (γ 2 sin2 (ϕi ) + 1),
r|i =
σb2|i = 2σn2 (γ 2 cos2 (ϕi ) + 1)
sin(ϕi ) cos(ϕi )γ 2
sin(2ϕi )σu2
=√
.
σa|i σb|i
(γ 2 sin2 (ϕi ) + 1)(γ 2 cos2 (ϕi ) + 1)
(38)
Then, it is easy to see that:
√
√
2σn2 γ 2 + 1
2
1 − r|i =
.
σa|i σb|i
(39)
Using these equations, we can rewrite m2 in (37) as:
√
2σn2 γ 2 + 1
m2 = (σb2|j − σb2|i )
σa|i σb|i σa|i σb|j
[
]
2
= R cos (ϕj ) − cos2 (ϕi ) ,
= R sin(ϕi + ϕj ) sin(ϕi − ϕj ),
(40)
√ 2
4 2
4σ γ
γ +1
where R = σa nσb σa σb . Similarly m0 can be computed as
|i
|i
|i
|j
m0 = −m2 , and m1 can be rewritten as:
[
]
m1 = R sin(2ϕi ) − sin(2ϕj )
= 2R cos(ϕi + ϕj ) sin(ϕi − ϕj ).
(41)
Therefore, the roots of (36) can be found as:
c = tan(
ϕi + ϕj
) or
2
c=−
1
ϕ +ϕ
tan( i 2 j )
,
(42)
which means the decision boundary between each two coding
lines Li and Lj . is either their bisector or its perpendicular
line. Since we first realize the sign of c and then detect its
corresponding coded word, the only valid solution will be the
bisector line.
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Mohammad Ali Akhaee (S’07–M’10) was born in
Tehran, Iran, in 1982. He received the B.S. degree
in both electronic and communication engineering
from Amirkabir University of Technology and the
M.S. degree from Sharif University of Technology,
Tehran, Iran, in 2003 and 2005, respectively. He
received his PhD degree from Sharif University of
Technology in 2009. He has awarded the Endeavour
research fellowship from Australia in 2010. He is
now a post-doc researcher at the Computing and
Audio Research Laboratory at the University of
Sydney.
His research interest includes multimedia security, statistical signal processing, acoustic, and image processing.
Sayed Mohammad Ebrahim Sahraeian (S’08)
was born in Shiraz, Iran, in 1983. He received
the B.S. and M.S. degrees in electrical engineering
from Sharif University of Technology, Tehran, Iran,
in 2005 and 2008, respectively. He is currently
pursuing the Ph.D. degree in electrical and computer
engineering at Texas A&M University, College Station.
His research interests include image and multidimensional signal processing, genomic signal processing, bioinformatics, and computational biology.
Craig Jin (M’92) received the M.S. degree in
applied physics from the California Institute of Technology, Pasadena, in 1991 and the Ph.D. degree in
electrical engineering from the University of Sydney,
Sydney, Australia, in 2001. He is a Senior Lecturer
in the School of Electrical and Information Engineering, University of Sydney and also a Queen Elizabeth II Fellow. He is a Director of the Computing
and Audio Research Laboratory at the University
of Sydney and also a co-founder of three start-up
companies. His research focuses on spatial audio and
neuromorphic engineering and he is the author or co-author of more than 50
journal or conference papers in these areas and holds six patents. Dr. Jin has
received national recognition in Australia for his invention of a spatial hearing
aid.
Copyright (c) 2011 IEEE. Personal use is permitted. For any other purposes, Permission must be obtained from the IEEE by emailing [email protected].