New Detector for Spread-Spectrum Based ImageWatermarking using Underdetermined ICA

Hafiz Malik, Ashfaq Khokhar, and Rashid Ansari

Department of Electrical and Computer Engineering, University of Illinois At Chicago, USAEmail: {hmalik, ashfaq, ansari}@ece.uic.edu

ABSTRACT

This paper presents a novel scheme for detection of watermarks embedded in multimedia signals using spreadspectrum (SS) techniques. The detection method is centered on using the model that the embedded watermarkand the host signal are mutually independent. The proposed detector assumes that the host signal and thewatermark obey non-Gaussian distributions. The proposed blind watermark detector employs underdeterminedblind source separation (BSS) based on independent component analysis (ICA) for watermark estimation from thewatermarked image. The mean-field theory based undetermined BSS scheme is used for watermark estimation.Analytical results are presented showing that the proposed detector performs significantly better than the existingcorrelation based blind detectors traditionally used for SS-based image watermarking.

Keywords: Watermarking, Spread Spectrum, Mean-Field, Independent Component Analysis, Correlation

1. INTRODUCTION

Digital watermarking refers to the process of imperceptible embedding information (watermark) into a digitaldocument (host data) in order to provide content protection and/or content authentication. Watermark embed-ding schemes can be classified into two major categories: (1) blind embedding, in which the watermark embedderdoes not exploit host signal information during the watermark embedding process. Watermarking schemes basedon spread-spectrum [1, 2, 4] fall into this category. (2) informed embedding, in which the watermark embedderexploits properties of the host signal during the watermark embedding process. Watermarking schemes basedon quantization index modulation [1, 2, 5, 6] belong to this category. Existing watermark detectors may be clas-sified into two categories: (a) informed detectors, which assume that the host signal is available at the detectorduring watermark detection process, and (b) blind detectors, which assume that the host signal is not availableat the detector for watermark detection. Although the performance expected from a given watermarking sys-tem depends on the target application area [1] robust embedding schemes and efficient detection procedures areinherently desired.

The watermark detection schemes for SS-based watermarking employ statistical characterization of the hostsignal to develop an optimal or near-optimal watermark detector (in the maximum likelihood (ML) sense)[6, 7]. The salient features of such schemes include increased robustness against interference (i.e. antijammingcapability), simplicity, and low computational complexity. Existing blind detectors for SS-based watermarkingperform poorly, particularly in terms of decoding error probability due to the presence of host-signal interferenceat the decoder. The nonzero decoding error probability at the watermark decoder even in the absence of attack-channel distortion is one of the limitations of existing blind watermark detectors. The main motivation ofthis paper is to design a blind detector for SS-based watermarking schemes capable of canceling host signalinterference at the detector, hence improving decoding as well as detection performance. Towards this end,we use the framework of ICA by posing the problem of watermark detection as that of BSS. We have shownin [8] that the watermark estimation problem in SS-based watermarking systems is equivalent to BSS. Therefore,ICA framework could be used to estimate the watermark in SS-based watermarked signals. The proposed ICA-based detector first estimates hidden independent components (i.e., the watermark and the host signal) fromthe watermarked signal using underdetermined ICA framework i.e. more sources than sensors. The estimatedcomponents are used to detect the presence of the embedded watermark. Theoretical analysis shows thatthe proposed ICA-based detector performs significantly better than existing blind detectors operating withoutreducing host signal interference at the detector [6, 7]. Simulation results also show that the proposed detector

1

outperforms the correlation based detector against a variety of signal manipulations and degradations appliedto the watermarked data. The proposed blind watermark detector is applicable to SS-based watermarking ofall media types, i.e. audio, video and images. However, in this paper digital images are used as host media forwatermark embedding, detection, and performance analysis.

2. BASICS OF SS-BASED WATERMARKING

The SS based watermarking system can be modeled using a classical secure communication model [1], as shownin Figure 1. In Figure 1, s ∈ RN is a vector containing coefficients of an appropriate transform of the host

Figure 1. Perception Based Data Hiding System with Blind Receiver modeled as a standard Secure CommunicationModel

signal. It is assumed that the coefficients, si : i = 0, 1, · · · , N − 1, are independent and identically distributed(i.i.d.) random variables (r.v.) with mean zero and variance σ2

si. A watermark, W, is generated using (i) a binary

information vector B = {b1, b2, · · · , bL} that encodes input message M , (ii) pseudo-random sequence w ∈ {±1}N

generated using a secret key K, and (iii) perceptual mask, α, estimated based on the human auditory system(HAS) and the host signal s. We assume that the watermark Wi and the host signal coefficient si are mutuallyindependent. The watermarked signal x is obtained by adding an amplitude-modulated watermark W to thehost signal s. In order to meet the fidelity requirement of the watermarked media x, i.e, to ensure that theembedded watermark is imperceptible, the amplitude-modulated watermark is spectrally shaped according toestimated perceptual mask α.

The watermarked signal x can be expressed as

x = s + α¯wb (1)

where ¯ denotes element-wise product of the two vectors.

The embedding error can be expressed as:de = x− s (2)

The mean-squared error embedding distortion is expressed as:

De = E{‖de‖2} = E{‖x− s‖2} = E{‖α ·w‖2} =1N

N−1∑

i=0

α2i = σ2

W (3)

where ‖ · ‖ represents the Euclidian norm, E{·} denotes expected value of a r.v., and σ2W is the variance of

watermark W.

The distortion introduced due to an active adversary attack can be modeled as an additive channel noise, n,as shown in Figure 1. The resulting watermarked signal, x, is processed for watermark detection and is expressedas:

x = x + n (4)

2

The SS-based watermarking schemes use probabilistic characterization of the host data to develop an optimalor near-optimal watermark detector (in ML sense). The statistical characterizations of real-world host data areavailable in spatial domain as well as in the transform domain. For example, discrete cosine transform (DCT)and discrete wavelet transform (DWT) coefficients of natural images can be approximated by the generalizedGaussian distribution [3], i.e.,

fs(τ) = Ae−|βτ |c (5)

where A and β can be expressed as

β =1σs

(Γ( 3

c )Γ( 1

c )

) 12

, A =cβ

2Γ( 1c )

Here Eq. (5) shows that the Gaussian as well as Laplacian distributions are the special cases of the generalizedGaussian distribution, e.g., c = 2 gives Gaussian probability density function (pdf) and c = 1 represents Laplacianpdf. This has been shown in [3] that coefficients of natural-images in low-frequency range can be modeledby either the generalized Gaussian distribution with c = 1/2 or by the Laplacian distribution, whereas high-frequency range coefficients can be modeled by either the Laplacian distribution or by the Gaussian distribution.Therefore, it is reasonable to model mid-range frequency coefficients (generally used for watermark embedding)by the Laplacian distribution which is given as,

fsi(τ) =βi

2e−βi|τ |, | τ |< ∞ (6)

where βi =√

2σsi

The error probability, Pe, of a decoded bit under zero-channel distortion, i.e. ni = 0, can be calculatedunder the follow assumptions (a) the watermarked image coefficient xi is obtained by adding binary amplitude-modulated watermark Wi, i.e. αiwib and (b) the detection is performed using Perez-Gonzalez et al’s optimaldecoder (in ML sense) [6,7], operating without canceling the host-signal interference. In addition, for performanceanalysis we will consider two information embedding scenarios: 1) one bit, b ∈ {±1}, of information is embeddedin each coefficient of the host media, si, and 2) one bit of information is embedding in in |ζ| coefficients of thehost image s.

Consider one bit per coefficient embedding scenario first, given b = 1 was embedded, in [7] it is shown thatthe optimal ML decoder estimates b = 1 if xiwi > 0 and an error will occur when xiwi < 0. Assuming that wi

takes values ±1 with probability 12 and L = 1, the average error probability Pe is given by

Pe = Pr{xiwi < 0|b = 1} =∫ 0

−∞fsi(τ − α) dτ (7)

For Laplacian pdf, fsi(τ), this can be shown

Pe =12e−√

2/λi (8)

where λi = σsi

σWiwhich is generally referred as signal-to-watermark ratio (SWR), when expressed in dB i.e.

SWR = 20 log10 λ.

From Eq. (8) it is clear that even in the absence of attack-channel distortion, zero decoding error probabilityis not achievable and Pe is a function of parameter λ. The value of the parameter λ offers a tradeoff betweenfidelity of the embedded watermark and decoding error probability, with a larger value of λ providing betterfidelity but poor Pe performance. Perez-Gonzalese et al in [6] have also shown that for second embeddingscenario the decoding error probability Pe in the absence of attack-channel distortion is also non-zero, and thePe is a function of parameter λ. In addition, Perez-Gonzalese et al in [6] have shown that the detection/decodingperformance of SS-based watermarking schemes is inherently bounded by the host signal interference at thedetector. The main objective of this paper is to design a watermark detector for SS-based watermarking schemeswith improved watermark detection, decoding, and maximum watermarking-rate performances by rejecting thehost-signal interference at the detector and BSS based on underdetermined ICA is used to achieve this goal.

3

3. INDEPENDENT COMPONENT ANALYSIS

Independent Component Analysis (ICA) is a statistical framework for estimating underlying hidden factors orcomponents of multivariate statistical data. In the ICA model, the data variables are assumed to be linear ornonlinear mixtures of some unknown latent variables, and the mixing system is also unknown [9]. Moreover, thesehidden variables are assumed to be non-Gaussian and mutually independent. The ICA model can be consideredas an extension of the principal component analysis (PCA) and factor analysis [9]. In fact, ICA can be treated asnon-Gaussian factor analysis, since data is modeled as a linear mixture of underlying non-Gaussian factors. Inthe following we will review only the linear ICA framework since that is applicable to the SS-based watermarkingmodel. In general, the linear ICA model can be defined for noise-free and noisy scenarios as follows.

Noise-free ICA model: ICA of a random vector x consists of estimating the following generative model ofthe data:

x = As (9)

where x represents the observed m-dimensional random vector, s is a vector of hidden variables and the matrixA is a m × n1 mixing matrix. The hidden variables (components) s(j) in the vector s = [s(1), · · · , s(n1)]T areassumed statistically independent.

Noisy ICA model: ICA of a random vector x consists of estimating the following generative model of thedata:

x = As + n (10)

where n is m-dimensional random noise vector , while x, s, and A are the same as in the noise-free model.

BSS is one of the most widely explored applications of the ICA [9]. The BSS using ICA framework triesis to find a linear representation in which the underlying components are statistically independent. For ex-ample, consider observation x corresponding to K realizations of an m-dimensional random vector xj(t) : j =1, 2, · · · ,m, t = 1, 2, · · · , K which follow the linear statistical model

x(t) = As(t) + n(t) (11)

where A ∈ Rm×n1 is unknown mixing matrix, s(t) are realizations of n1-dimensional underlying independentcomponents or sources, and n(t) is a noise which is assumed to be independent of s(t). The goal is to recoverthe underlying independent sources s, from the observation vector x. The cocktail party problem is a classicalexample of BSS e.g. separating voices of different speakers (sources) from the microphone-recordings of severalpeople simultaneously speaking in the room (observations/sensors). In other words objective is to estimate thedemixing matrix B from the observed data x. The estimated demixing matrix B is the inverse (or generalizedinverse in case of degenerate ICA) of mixing matrix A, i.e., B = A† = (AHA)−1AH . This is worth mentioningthat blind watermark estimation problem requires BSS based on underdetermined ICA framework. In this paper,we extended the noisy ICA generative model to design an ICA-based watermark detector for SS-based water-marking schemes. The proposed ICA-based watermark detector attempts to estimate the embedded watermarkfrom the watermarked signal while reducing the host-signal interference at the watermark detector based onunderdetermined ICA framework.

4. PROPOSED ICA BASED WATERMARK DETECTOR

The proposed ICA-based watermark detector consists of two stages: 1) watermark estimation stage, and, 2)watermark decoding and/or detection stage. The watermark estimation stage estimates watermark W from thereceived watermarked image x using underdetermined ICA framework, whereas, the watermark decoding and/ordetection stage decodes and/or detects the embedded watermark using the ML approach. The block diagramof the proposed watermark detector is given in the Figure 2. In general BSS using ICA estimates demixingmatrix B from the observed data x, which is then used to estimate the underlying independent componentss(j), j = 1, 2, · · · , n1. This model is extendable to the watermark estimation problem from the watermarkedsignal, assuming identifiability conditions of the ICA model are satisfied [9]. In order to examine whether theSS-based watermarking satisfies the identifiability constraints of an ICA model, we considered additive SS-basedwatermark embedding scenario in this paper. Let us assume that the watermark is embedded into the host signal

4

W ATERMARK

E STIMATION

USING ICA W ATERMARK

D ECODER

S ECRET K EY

R ECEIVED

W ATERMARKED

S IGNAL

ˆ w

ˆ b

W ATERMARK

D ETECTOR

1 H

K

x

0 H

Figure 2. Block Diagram of the Proposed ICA-based Watermark Detector

according to Eq. (1), as the watermark, W, and the host signal, s, are assumed to be mutually independent for allSS-based watermarking schemes. Therefore, we only need to show non-Gaussian distribution of the underlyingcomponents e.g. W and s. As DCT coefficients of real-world images can be approximated by the Laplaciandistribution [3], therefore, if the watermark, W, is generated based on some non-Gaussian distribution then non-Gaussianity requirement of the ICA model is satisfied. Once the identifiability conditions of the underdeterminedICA model are satisfied, the noisy ICA model can be extended to solve the watermark estimation problem forSS-based watermarking. For blind watermark estimation existing BSS schemes based on degenerate ICA can beused [9–13].

4.1. Watermark EstimationFor watermark estimation, the proposed watermark detector first estimates the watermark-mixing matrix Awhich is then used to estimate the underlying independent components. An estimate of the watermark-mixingmatrix, A, is obtained by optimizing some highly nonlinear function also known as contrast function. Thepseudo-inverse of the estimated watermark-mixing matrix A† is applied to the observed mixture to estimate thehost signal s and the watermarkhatW . As blind watermark estimation, for SS-based watermarking, using ICA framework requires degenerateICA model. Therefore, just estimation of watermark-mixing matrix is insufficient to separate the underlyingindependent components perfectly. The ICA generative model, e.g. x = As, has an affine set of solutions incase of degenerate observation [14]. A preferred solution in this affine set is generally selected using probabilisticprior model of the independent components. The performance of the proposed ICA-based watermark estimatordepends on the separation quality of the separated (estimated) watermark . The separation quality of theseparated source is generally measured in terms of source-to-interference ratio (watermark-to-interference ratio(WIR), in case of watermark estimation), source-to-noise ratio and source-to-artifact ratio (for further details onthese separation quality measures please see [15] and references therein). For performance analysis of the proposedICA detector, only WIR distortion measure is considered here, the estimated watermark can be expressed as

wi = η1iαiwib + sinterfi (12)

where η1i ∈ R : 0 ≤ η1i ≤ 1 and sinterfi is interference due to the host signal. Let sinterf

i = η2isi : η2i ∈ R : 0 ≤η2i ≤ 1 then Eq. (12) can be rewritten as,

wi = η1iαiwib + η2isi (13)

The relative distortion due to interference in the estimated watermark is defined as,

Dinterf = (η1/η2)2 (14)

where WIR = 10 log10

(Dinterf

)

In general, Dinterf > 1 for most of the existing BSS schemes based on ICA framework. Several researchershave proposed elegant BSS algorithms based on underdetermined ICA model [10, 12], these algorithms can beused for watermark estimation. However, simulation results presented in this paper, we used underdeterminedICA based on mean-field theory [13] for blind watermark estimation from the watermarked image. The mean-field theory based ICA scheme first estimates mixing matrix A using maximum a posteriori (MAP) and theunderlying sources are estimated using mean of their posterior distribution. In the following subsections weanalyze the performance of the proposed ICA-based detector in terms of three parameters: (a) detection rate interms of false negatives and true positives, (b) decoding error probability, and (c) maximum watermarking rate.

5

4.2. Watermark DetectionA watermark detector is generally characterized by two performance measures: the probability of false alarmPF and the probability of detection PD. The probability of detection indicates the probability of detecting awatermark when the received image indeed contains a watermark. The probability of false alarm representschances of detecting a watermark when in fact the received image does not contain a watermark. The watermarkdetection process can be treated as a binary decision problem in the presence or absence of attack-channeldistortions. Consider second embedding case, e.g., one bit is embedded in |ζ| coefficients and watermarked imageexperienced zero attack-channel distortion. The estimated watermark is given by

Wi = η1iαiwib + η2isi : i ∈ ζ, and |ζ| > 1 (15)

In this scenario, the watermark detection can be formulated as hypotheses test:

H1 : Wi = η1iαiwib + η2isi

H0 : Wi = η2isi : i ∈ ζ, and |ζ| > 1 (16)

If the probability distribution function, fs(τ) , of si is known, then the optimum decision tests are given by theNeyman-Pearson rule. In the watermark detection, an optimum detector that maximizes the PD for any valueof PF is determined by

L(W ) = lnfw(W |H1)fw(W |H0)

= ln∑

l

(p(bl)fs(W − η1αwbl)

fs(W )

)H1

≷H0

ξ (17)

where, l ∈ {±1}, ξ is a threshold and defined as si = η2isi. Assuming equally probable messages and Laplaciandistribution for random variable si, the log-likelihood function can be written as:

L(W ) =∑

i∈ζ

βi

(|Wi| − |Wi − η1iαiwi|

)(18)

where βi =√

2/η2iσsi In order to determine the statistical characterization of L(W ) under hypothesis H0 andzero attack-channel distortion, we have:

L(W ) =∑

i∈ζ

βi

(|Wi| − |Wi − η1iαiwi|

)(19)

L(W ) def= βi (qi) (20)

qidef= βi

(|Wi| − |Wi − η1iαiwi|

)(21)

Here L(W ) is a sum of |ζ| statistically independent random variables that can be approximated by the Gaussianrandom variable based on the CLT mean and variance of L(W ) is calculated as follows, assuming wi ∈ {±1}with probability 1

2 :

E{L(W |H0)} =∑

i∈ζ

βiE{qi} (22)

m0def= E{L(W |H0)} (23)

Var{L(W |H0)} =∑

i∈ζ

β2i Var{qi} (24)

σ20

def= Var{L(W |H0)} (25)

E{qi|si,H0} = |si| − 12

(|si|+ η1iαi + ||si| − η1iαi|) (26)

Var{qi|si,H0} =14

(|si|+ η1iαi + ||si| − η1iαi|)2 (27)

6

these equation can be rewritten as

E{qi|si,H0} =

{|si| − η1iαi |si| ≤ η1iαi

0 |si| > η1iαi

(28)

Var{qi|si,H0} =

{|s2

i | |si| ≤ η1iαi

η21iα

2i |si| > η1iαi

(29)

Averaging over si and substituting E{qi|H0}, and Var{L(qi|H0)} in Eq. (27) and Eq. (27)we have

m0 =∑

i∈ζ

(1− e

− η1i√

2η2iλi − η1i

√2

η2iλi

)(30)

σ20 =

∑

i∈ζ

(3− e

− η1i2√

2η2iλi − 2e

− η1i√

2η2iλi

(1 +

2η1i

√2

η2iλi

))(31)

When hypothesis H1 is true, i.e.,Wi = η1iαiwib + si, the log-likelihood function can be written as:

L(Wi) = βi (|wi + η1iαiwi| − |wi|) (32)

Here L(W ) can be approximated by a Gaussian random variable with same set of assumptions as under hypothesisH0. In addition, the distribution of L(W ) under hypothesis H1 is symmetrical to the distribution of under H0

with respect to the origin, i.e., m1 = −E{L(W )|H0} and σ21 = Var{L(W )|H0}. Now false alarm PF and detection

PD probabilities are given as:

PF = Q

(ξ + m1

σ1

)(33)

PD = Q

(ξ −m1

σ1

)(34)

If we define the watermark-to-noise ratio (WNR1 )

WNR1 def=m2

1

σ21

(35)

If we denote Q−1(PF ) the value x ∈ R such that Q(x) = PF then receiver operating characteristics (ROC) ofthe proposed detector is given as:

PD = Q(Q−1(PF )− 2

√WNR1

)(36)

Eq. (36) shows that the detection performance of the proposed detector exclusively depends on the WNR1. Asthe proposed ICA-based detector cancels the host-signal interference before detection, the ICA-based detector isexpected to perform better than the traditional blind detectors [14,15] operating without reducing the host-signalinterference for watermark detection process. Figure 3 gives the theoretical ROC performance of the proposedICA-based detector for different values of WIR and along with ROC performance of an optimal detector proposedin [6, 7]. We can see from Figure 3 that the ROC performance of the proposed ICA-based detector is betterthan the ROC performance of the detector proposed in [6,7], and this improvement is a function of WIR in theestimated watermark.

4.3. Watermark DecodingIn order to evaluate the performance of the proposed detector in terms of decoding error probability, let usconsider SS-based watermark embedding and decoding framework discussed in Section 2. Consider one-bit percoefficient embedding case first, the decoding error probability for the estimated watermark Wi (given by Eq.( 13)) can be expressed as:

Pe ICA =12e−{ η1i

η2i}(√

2/λi) (37)

7

10−15

10−10

10−5

100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PFPD

P. Gonzalex et al’s Detector [17]WIR = 6.0 dBWIR = 14.0 dBWIR = 20.0 dB

Figure 3. ROC Performance of the Proposed Detector and the Detector proposed in [6, 7], for different values of WIR,WSR = 13 dB, and one-bit per five Image Coefficients Embedding, e.g. |ζ| = 5 (theoretical values)

Eq. (37) shows that ICA-based detector improves decoding error performance compared to the optimal decoderthat does not reduce the host-signal interference at the detector (discussed in Section 2). The decoding errorperformance gain for the proposed ICA-based watermark detector over the decoder proposed in [6] is given as:

G =Pe

Pe ICA= e

−√

2λi{1− η1i

η2i} (38)

In general, BSS schemes using ICA have relatively small interference distortion or large WIR value, i.e., η1i/η2i

[14], therefore, G ≥ 0 for WIR > 0 dB. The improvement in the decoding error probability due to ML decoderused in the proposed ICA-based detector over that of the optimal decoder given by Eq. (8) is plotted in Figure4, for different values of WIR and SWR. It shows that for a fixed value of SWR, decoding error probability of theproposed detector improves with the increase in the separation quality of the ICA scheme used for watermarkestimation. Let us now consider one-bit per |ζ| coefficients embedding case i.e., watermarked image is given as:

0 2 4 6 8 10 12 14 16 18 20

100

101

102

103

104

105

106

107

108

WIR (dB)

Gain:

G

SWR = 0.0 dBSWR = 20.0 dBSWR = 13.0 dBSWR = 6.0 dB

Figure 4. Decoding Bit Error Probability Performance Improvement over Correlation-based Detector due to Host-Signal-Interference Cancelation at the Detector (theoretical values)

xi = si + αiwib : i ∈ ζand|ζ| > 1 (39)

In this case, the estimated watermark W using proposed ICA-based watermark detector, under zero attack-channel distortion, can be expressed as,

Wi = η2isi + η1iαiwib : i ∈ ζand|ζ| > 1 (40)

If we assume that all hidden messages have same priori probability, then an ML decoder will minimize thedecoding error probability satisfying the following condition:

lnfw(W |b+)fw(W |b−)

= lnfs(W − η1αw)fs(W + η1αw)

> 0 (41)

8

The ML sufficient statistics assuming s obeys Laplacian distribution and can be written as:

r =∑

i∈ζ

βi

(|Wi + η1iαiwi| − |Wi − η1iαiwi|

)(42)

if b = 1 was embedded, then r can be expressed as

r =∑

i∈ζ

βi (|si + 2η1iαiwi| − |si|) (43)

r def=∑

i∈ζ

βizi (44)

zi = (|si + 2η1iαiwi| − |si|) (45)

The ML decoder is a bit-by-bit hard decoderb = sgn(r) (46)

In order to determine the bit error probability for this ML decoder, a statistical characterization of r is required.Here r is sum of |ζ| i.i.d. random variables. Therefore, by applying CLT, r can be approximated by a Gaussianrandom variable, with mean and variance computed as:

E{r} def=∑

i∈ζ

βiE{zi} (47)

Var{r} def=∑

i∈ζ

β2i Var{zi} (48)

In zi, variables wi and si are the only random variables and if wi ∈ {±1} with probability 12 , then by integrating

zi over zi condition to the selected host indices si : i ∈ ζ we have:

E{zi|si} =12

(|si|+ 2η1iαi + ||si| − 2η1iαi|)− |si| (49)

Var{L(zi|si,H0)} =14

(|si|+ 2η1iαi + ||si| − 2η1iαi|)2 (50)

these equation can be rewritten as

E{zi|si} =

{−|si|+ 2η1iαi |si| ≤ 2η1iαi

0 |si| > 2η1iαi

(51)

Var{zi|si} =

{|s2

i | |si| ≤ 2η1iαi

4η21iα

2i |si| > 2η1iαi

(52)

Averaging over si and then substituting E{zi}, and Var{zi} in Eq. (48) and Eq. (48) we have

E{r} =∑

i∈ζ

(e− η1i2

√2

η2iλi +2η1i

√2

η2iλi− 1

)(53)

Var{r} =∑

i∈ζ

(3− e

− η1i4√

2η2iλi − 2e

− η1i2√

2η2iλi

(1 +

4η1i

√2

η2iλi

))(54)

Thus decoding bit error probability for an ICA-based detector is given as:

Pe ICA = Q

(|E{r}|√Var{r}

)(55)

9

Eq. (53), (54) and (55) show that the decoding error probability of the ML decoder applied to the estimatedwatermark is a function of WIR and SWR. The decoding error probability performance of the ML decoder usedfor the proposed ICA-based detector is compared against decoding error probability performance of the MLdecoder operating without reducing the host-signal interference. The performance of the proposed ICA-baseddetector given by Eq. (55) and the optimal decoder given by [6] for different values of WIR and SWR is plotted inFigure 5. The analytical results shown in Figure 5 depict that the proposed detector exhibits superior decoding

0 2 4 6 8 10 1210

−300

10−250

10−200

10−150

10−100

10−50

100

WSR (dB)

P e

P. Gonzalez et al’s Detector [17]WIR = 6.0 dBWIR = 14.0 dBWIR = 20.0 dB

0 2 4 6 8 10 1210

−300

10−250

10−200

10−150

10−100

10−50

100

WSR (dB)

P e

P. Gonzalez et al’s Detector [17]WIR = 6.0 dBWIR = 14.0 dBWIR = 20.0 dB

Figure 5. Decoding Bit Error Probability Performance of the Proposed ICA-based Detector for different values of WIRand one-bit per |ζ| Image Coefficients Embedding, i.e. |ζ| = 5 (left) , and |ζ| = 10 (right) (theoretical values)

error probability compared to the existing detectors [1,2,4]. This superior performance can be attributed to thehost-signal interference cancelation capability of the proposed ICA-based detector. In addition, the decodingperformance of the proposed ICA-based detector is a function of the WIR performance of the BSS scheme usedfor watermark estimation. When WIR = 0 dB in the estimated watermark, the proposed detector is equivalentto the optimal detector proposed in [6, 7].

4.4. Maximum Watermarking-Rate

The maximum watermarking-rate is another watermarking performance measure which indirectly depends onthe detector structure. In order to evaluate the maximum watermarking-rate performance of the proposed ICA-based watermark detector against informed and blind watermark detectors, we assume one-bit per coefficientembedding scenario and received watermarked coefficient is corrupted by the independent Gaussian noise, withmean zero and variance σ2

ni. In this case, the received watermarked sample xi can be approximated by a

Gaussian random variable based on the CLT. Now based on the independence of s,w, and n, variance of thereceived watermarked signal xi is given as:

σ2xi

= σ2Wi

+ σ2si

+ σ2ni

(56)

In this case, the estimated watermark sample at the detector in the presence of additive noise is given as:

Wi = η1iαiwib + η2isi + η3ini (57)

where η3ini is noise contribution (interference) in the estimated watermark, and η3i ∈ R : 0 ≤ η3i ≤ 1. Byapplying CLT the watermark estimate Wi can be approximated by the Gaussian random variable with meanzero and variance given as:

σ2Wi

= η21iσ

2Wi

+ η22iσ

2si

+ η23iσ

2ni

(58)

The maximum watermarking-rate of SS-based watermarking using blind watermark detector, operating withoutreducing the host-signal interference, can be calculated by the capacity of an additive white Gaussian noisechannel [2], i.e.

RCor =12

log2

(1 +

σ2Wi

σ2si

+ σ2ni

)(59)

10

Similarly for an informed detector can be expressed as:

RInformed =12

log2

(1 +

σ2Wi

σ2ni

)(60)

and for the proposed ICA-based watermark detector is given as:

RICA =12

log2

(1 +

η21iσ

2Wi

η22iσ

2si

+ η23iσ

2ni

)(61)

Since existing BSS using ICA schemes exhibit good source separation performance i.e. low interference and noisedistortions in the estimated sources [14,15], therefore we can claim

σ2Wi

σ2ni

≤ η21iσ

2Wi

η22iσ

2si

+ η23iσ

2ni

≤ σ2Wi

σ2si

+ σ2ni

⇒ RCor ≤ RICA ≤ RCor (62)

The above results show that the proposed ICA-based watermark detector performs better than the conven-tional watermark decoders and detectors [1, 2, 4] operating without reducing the host-signal interference.

5. SIMULATION RESULTS

In order to test performance of the proposed ICA-based detector, a gray scale image of Lenna of 256x256 pixels iswatermarked using Podilchunk et al’s image adaptive watermarking scheme [4] and subjected to several commonsignal manipulations and distortions. For watermark embedding, original image is segmented into four non-overlapping blocks each of size 128x128 pixels. Watermark of size 128x128 samples is generated based on Gaussianpseudo-random sequence generator. Watermark is embedded in in each block in DCT domain using embeddingscheme presented in [4], and watermarked image is obtained by merging watermarked blocks. The original(left) and the watermarked image Lenna (right) is given in Figure 6. In order to detect embedded information,

Figure 6. Original Image Lenna (left) and Watermarked Image (right)

watermarked image, subjected to various degradations like addition of white noise, lossy compression, andresizing, is applied to the proposed ICA-based detector. For the sake of comparison, same watermarked imageis also applied to the correlation-based detector [1]. The maximum of four correlation values obtained fromeach watermarked segment is used to compare it with the correlation value for ICA-based detector. Detectionperformance of both detectors against addition of white Gaussian noise for different noise powers is given inFigure 7 (left). Figure 7 (left) shows that the propose detector outperforms the correlation based detector.Performance against JPEG compression attack for various Q (quality) factors is plotted in Figure 7 (middle),Figure 7 (middle) shows that for the ICA-based detector performs better than correlation based detector for highQ-values and almost similar rigorous compression attack. The detection performance against resizing attack fordifferent resizing ratio is given in Figure 7 (right).

11

100

101

102

103

10

20

30

40

50

60

70

Noise Variance

Cor

rela

tion

ICA_DetectorCorr_Detector

01020304050607080901001100

10

20

30

40

50

60

70

Q (Qualiry) Factor

Cor

rela

tion

ICA_DetectorCorr_Detector

505560657075808590951000

10

20

30

40

50

60

70

Resize Ratio

Cor

rela

tion

ICA_DetectorCOrr_Detector

Figure 7. Detection Performance Against Addition of White Noise (left), JPEG Compression (middle), and Resize Attack(right)

6. CONCLUSION

A new watermark detector for SS-based watermarking is presented in this paper. The proposed detector iscapable of canceling the host-signal interference at the detector using ICA theory. Bind watermark detection,lower host-signal interference at the detector, improved decoding, detections and watermarking-rate performancesare the salient features of the proposed detector. Bind watermark detection, lower host-signal interference atthe detector, improved decoding, detections and watermarking-rate performances are the salient features of theproposed detector.

REFERENCES1. I. J. Cox, M. L. Miller, and J. A. Bloom, Digital Watermarking, Morgan Kaufmann, 2001.

2. J. Eggers, and B. Girod, Informed Watermarking, Kluwer Academic Publisher, 2002.

3. R. J. Clarke, Transform Coding of Images, New York: Academic, 1985.

4. R. B. Wolfgang, C. I. Podilchuk, and E. J. Delp, Perceptual Watermarks for Digital Images and Video, Proc. of IEEE,vol. 87(7), pp. 1108–1126, July, 1999.

5. B. Chen, and G. W. Wornell, Quantization Index Modulation: A Class of Provably Good Methods for DigitalWatermarking and Information Embedding, IEEE Trans. Information Theory, vol. 47(4), pp. 1423–1443, May, 2001.

6. F. Prez-Gonzlez, F. Balado, and J. R. Hernndez, Performance Analysis of Existing and New Methods for Data Hidingwith Known-Host Information in Additive Channels, IEEE Trans. Signal Processing, vol. 51(4), pp. 960–980, April,2003.

7. J. R. Hernndez, M. Amado, and F. Prez-Gonzlez, DCT-domain watermarking techniques for still images: Detectorperformance analysis and a new structure, IEEE Trans. on Image Processing, vol. 9(1), pp. 55–68, January, 2000.

8. H. Malik, A. Khokhar, and R. Ansari, An Improved Detector/Decoder for Spread Spectrum based Watermarkingusing Independent Component Analysis, ACM Fifth Workshop on Digital Rights Management (DRM’05), November7, 2005.

9. A. Hyvrinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley & Sons, 2001.

10. A. Cichocki, S. Douglas, and S. Amari, Robust Techniques for Independent Component Analysis (ICA) with NoisyData, Neurocomputing, Elsevier Science, vol. 22, pp. 113–129, 1998.

11. P. Pajunen, Blind Separation of Binary Sources With Less Sensors Than Sources, Proc. Int. Conf. on Neural Networks(ICNN’97), pp. 1994–1997, 1997.

12. P. Bofill, and M. Zibulevsky, Underdetermined Blind Source Separation using Sparse Representations, Signal Process-ing, Elsevier Science, vol.81(11), pp.2353–2362, 2001.

13. P. Hojen-Sorensen, O. Winther, and L. K. Hansen, Mean-Field Approaches to Independent Component AnalysisNeural Computation, MIT Press Journals, 14, pp. 889–918, 2002.

14. R. Gribonval, L. Benaroya, E. Vincent, C. Fvotte, Proposals for Performance Measurement in Source Separation, 4th

Int. Sym. Independent Component Analysis and Blind Source Separation (ICA’03), April, 2003.

15. Y. Li, D. Powers, and J. Peach, Comparison of Blind Source Separation Algorithms, Advances in Neural Networks

and Applications N. Mastorakis (Ed.), WSES, pp. 18–21, 2000.

12