Generalized Embedding of Multiplicative Watermarks - IEEE Xplore

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 19, NO. 7, JULY 2009

Generalized Embedding of Multiplicative Watermarks Qiang Cheng

Abstract— This paper constructs a class of generalized embeddings of multiplicative watermarks. Ordinary multiplicative and additive methods are included as special cases. The new watermarks automatically adapt to the local contents of host signals, benefiting the perceptual quality. The decoding makes use of the optimal generalized correlation detector. The host interference is precanceled at the embedder side and very high gains are obtained in terms of decoding capability. We develop performance analysis for this new class of embeddings. It turns out that the plain multiplicative watermark is far outperformed by the new embedding. Further, the multiplicative watermark with host interference rejection is still suboptimal. The best embeddings and configurations are specified for typical scenarios. Our construction and performance analyses of the generalized embedding offer a class of new methods. The construction and analyses are confirmed by empirical experiments. Index Terms— Data hiding and watermarking, generalized correlator, generalized embedding, image modeling, performance analysis.

I. I NTRODUCTION

C

ONSIDERABLE concerns exist over the infringement of intellectual property rights and the security of digital contents, due to the ease of manipulating digital data. Digital watermarking technology has been developed to protect the digital content as well as deliver secondary data through multimedia in an innocuous way. Many novel functionalities can be enabled, e.g., quality assessment for multimedia mobile communications, image indexing, and content-based retrieval in multimedia databases. There has been a thrust of research activities recently. A watermark needs to be inserted or embedded into host multimedia contents by an embedder. For watermark embedding, the Costa scheme [1] has been applied in watermarking research [2]–[4], including several practical realizations of Costa’s scheme, e.g., quantization index modulation [2], scalar Costa scheme [3], and (quasi) orthogonal dirty-paper coding [4]. These schemes are host interference rejective: that is, host signals have no adverse effect on watermark detection or decoding accuracies. Scaling or other attacks may be used by the attacker to limit their performance [5]. Besides Costa-type methods, spread spectrum (SS) techniques have also been widely applied since [7]. SS

Manuscript received January 30, 2008; revised June 20, 2008. First version published April 7, 2009; current version published July 22, 2009. This paper was recommended by Associate Editor Q. Sun. The author is with the Computer Science Department, Southern Illinois University, Carbondale, IL 62901, USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCSVT.2009.2020255

embeds a watermark into many transform-domain coefficients of a host signal. Each component of the watermark signal has a weak embedding strength, while collectively the watermark can have a strong strength with a known secret key for the embedding sequence. Statistical modeling of host data has been performed using generalized Gaussian, Weibull, etc. The statistical property of the host is relatively invariant to various attacks due to the power constraint of the watermarking channel. This may be useful to obtain desirable robustness to attacks. To exploit the host statistics, optimum detection at the receiver end has been considered. The generalized Gaussian model has been applied to image and video watermarking for subband transforms such as discrete cosine transform (DCT), pyramid, and discrete wavelet transform (DWT) [8]. And the Weibull model for DFT magnitudes [9], [10] has also been considered. The resultant detectors may significantly outperform other suboptimal detectors. Additive and multiplicative embedding methods are commonly used at the SS embedder side. Improved transmitter design for SS watermarking has been considered in [5], [11]. Inspired by the Costa scheme, host interference precoding has been studied in communication areas, e.g., in the multiple antenna downlink channel by Caire and Shamai [12]. In watermarking research, informed embedding has been proposed, with the correlation coefficients maximized [11] or the host interference precanceled [5], [6]. The schemes in [5], [11] consider additive SS watermarking and the linear correlator that is optimal only for Gaussian host data. Multiplicative embedding adapts the watermark to a multimedia signal’s local contents. More watermark power may be allocated to busy host regions, which turns out to be beneficial to the perceptual quality or robustness. To further exploit the property, we construct a class of generalized embedding of multiplicative (GEM) watermarks. Ordinarily used multiplicative and additive watermarking methods are special cases of this class. The new GEM watermarks automatically fit themselves to the host content to achieve desirable perceptual quality. For ordinary multiplicative embedding, a class of generalized correlators has been obtained from the locally optimal test [13]. The SS embedding strengths may vary according to human perceptual models. The generalized correlators are robust to the variations of the embedding strengths, and optimal in the sense of the composite hypothesis that tests the positiveness of the embedding depths instead of any individual value. This makes the generalized correlators precisely suitable for the GEM. We construct the GEM in a way that the host interference is precanceled at the embedder side, and the watermarks

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CHENG: GENERALIZED EMBEDDING OF MULTIPLICATIVE WATERMARKS

have different degrees of adaptivity to the host signal. The performance analyses of the class of GEM watermarks are developed. The ordinary multiplicative watermark, even with host interference rejection, is usually suboptimal. For a given host multimedia signal, the best choice of the GEM watermark is specified. The construction of the GEM offers a class of new watermark embedding systems, with the limitations of many existing methods revealed and overcome. Existing SS watermarking systems as well as their resistance to geometrical distortions can be inherited by or extended to the GEM straightforwardly. This makes GEM backward compatible to the deployed SS watermarking systems. The new methods and the performance are confirmed by the theoretical analyses and experiments. The structure of the paper is organized as follows. Section II presents the problem formulation and the generalized correlator. The construction of the generalized embedding of multiplicative SS watermarks is shown in Section III. Performance analyses for GEM watermarks in general are developed in Section IV, while Section V-A analyzes the performance for a special case of GEM. Section V-B considers the performance of general degrees including some special cases of GEM. Several issues in practical applications are discussed in Section VI. Experimental results are provided in Section VII to confirm the effectiveness of theoretical analyses. Section VIII concludes the paper and points out future research lines. The following notations are used. Boldfaced letters denote vectors. Upper case letters denote random variables. R N denotes the set of N -dimensional vectors with real-valued . entries. We use = to denote the first order Taylor expansion. E[·] and V ar (·) denote, respectively, the expectation and variance of random variables. II. P ROBLEM F ORMULATION AND G ENERALIZED C ORRELATORS A. Problem Formulation An N -dimensional vector x from a host multimedia signal is used as a carrier for a message symbol m ∈ M, where M is an alphabet. An embedding function f (·) encodes m into a physical watermark w = f (x, m, κ), where κ is a cryptographic key for high security of the message. The watermarked signal y = x + w is stored or distributed by the digital content creator or distributor, and may be subject to attacks from an attacker who attempts to disrupt the watermark by introducing noise n. It is the resultant signal z = y + n that is available to the receiver. To recover the message from the received signal, a detector g(z, κ) is used for decoding or detection. The commonly used multiplicative SS embedding is [9], [10] (1) yk = xk + ak xk sk , k = 1, . . . , N . That is, the physical watermark is wk = ak xk sk . In the above, N is ak > 0 are gain factors or embedding strengths, and {sk }k=1 a pseudo-random noise sequence (PRNS) seeded by the key κ. B. Statistical Model for the Multimedia Data In this paper, we mainly use the transform-domain coefficients as our host signal. The transform includes DCT,

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pyramid transform, DWT, lapped orthogonal transform (LOT), etc. For natural images and videos, their transform-domain coefficients usually manifest strong non-Gaussianity and have heavy tails. The statistical characteristic of these coefficients can be well modeled by generalized Gaussian distribution (GGD), with the following zero-mean probability density function (p.d.f.) p X (x) for a random variable X ∼ GG D(0, σx ; c): p X (x) = Ae−|βx|

c

(2)

where c is the shape parameter, β = (1/σ X )[(3/c)/ (1/c)]1/2 , σ X is the standard deviation of X , and A = βc/2(1/c). The Gamma function (·) in the ∞ above equations is (x) = 0 u x−1 e−u du. The GGD has been applied in watermarking for additive [8] or multiplicative embedding [13]. The shape parameter can be well estimated from the multimedia data using, for example, the method of minimizing relative entropy [8]. The smaller the c value, the heavier the tails and the more peaky the shape. The shape of the subband coefficients are relatively stable even after a variety of attacks. Due to its accurate modeling capability, we shall employ this model to describe the host statistics. The sequels make use of the following equation: Fact 1: If X ∼ GG D(0, σx ; c), then 2A ν+1 ν E|X | = ν+1 cβ c ν/2 ν+1 ν (1/c) −1 1 . (3) = σX (3/c) c c In particular, if ν = c, then

σxc (1/c) c/2 E|X | = . (4) c (3/c) The above fact is obtained from the definition of expectation by taking integration straightforwardly. c

C. Generalized Correlators for Multiplicative Watermarks For plain multiplicative watermarks in (1) with variable embedding depths ak , a class of generalized correlators has been constructed [13] g(z, κ) =

N 1 |z k |c sk N

(5)

k=1

N where s = {sk }k=1 is a random zero-mean sequence generated by the key κ, and c is the shape parameter of the subband transformed coefficients. The generalized correlators [13] are constructed from the locally optimal test, taking into account the effect of attacks on the embedding strengths. They are designed to detect the watermark optimally for a range of embedding depths instead of specific depths, and thus are robust to the variations of the embedding depths. Therefore, at the receiver side they need only the information of the cryptographic key, and not that of the embedding depths. At the embedder side, the embedding depths can be rather general and may be further changed by the attacker. As long as the embedding strengths are not completely erased by the attacker, the generalized correlators are applicable and optimal. These properties of the

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generalized correlators are useful for the detection of the GEM watermarks. The GGD contains several special cases: Case 1 c = 1: The GGD reduces to Laplacian distribution, and the generalized correlation to correlation (but the first component needs to take absolute value). Case 2 c = 2: The GGD reduces to plain Gaussian distribution, and the generalized correlation to a quadratic detector. Case 3 c = (1/2): The GGD has heavy tails and a high peak. The corresponding generalized correlation is called a square-root detector and identified as a universally optimal detector, in the sense that it is near optimum for most of natural images and the statistical model of the host data need not be estimated explicitly [13]. This is beneficial to image database and video watermarking in particular. III. G ENERALIZED E MBEDDING OF M ULTIPLICATIVE WATERMARKS A. Construction of the GEM As explained previously, it is desirable to have the nice properties of multiplicative watermarks such as the adaptivity to the host contents. At the same time, the embedder needs to take into account the available information of host states so that we may have the host interference rejection capability at the receiver. Assume the host data are independently and identically distributed (i.i.d.) GG D(0, σx ; c). Now we construct a class of GEM watermarks as follows: yk = f (x, b, κ) = xk + ak b|xk |ν sk sgn(xk ) − λξ xk sk , k = 1, . . . , N

(6)

where ν ≥ 0 is the degree of the GEM, ak > 0 is the embedding depths, which satisfy that the magnitude of ak b|xk |ν sk sgn(xk ) are much smaller than |xk | to ensure good perceptual quality of the host signal, b ∈ M is a symbol in a covert message to be transmitted, and for binary message, b N is pseudorandom noise are information bits 1 or −1, {sk }k=1 sequence (PRNS) consisting of ±1 with equal probability and generated from the key κ, and it is statistically independent N , λ ≥ 0 is a precoding gain, sgn(x ) is the from {xk }k=1 k signum function representing the sign of xk , and ξ is the precoding term, which is defined as N |xk |c sk . (7) ξ = k=1 N c k=1 |x k | In the above embedding of the GEM, the first term is the host signal. The second term represents the main part of the self-adaptive watermark to be embedded, where the adaptivity is mainly controlled by a parameter ν. And the third term is designed to cancel the interference from the host signals. Some specific remarks regarding the design of the GEM is as follows. Remark 1: The GEM in (6) is easily seen to have an extended form of the plain multiplicative embedding, which corresponds to setting ν = 1 and λ = 0. And the GEM watermark is wk = sgn(xk )sk (ak b|xk |ν − λξ |xk |), k = 1, . . . , N . (8)

Remark 2: The precoding term ξ is designed to cancel the host interference at the receiver side, with the known information of the host states and statistics, as well as the generalized correlation detectors. Even for moderate N , it is easily seen from Proposition 1 that ξ is close to zero with small variations. Henceforth, the ξ term has little effect on the perceptual quality while significantly increasing the performance of the GEM watermarking systems. Literally, the performance analyses in the following sections manifest that the performance gains can be infinitely large (up to the secondorder approximation residual errors). However, for small N ’s such as those below 10, the precoding term may possibly perturb the perceptual quality. We employ the precoding gain factor λ to control the perceptual effects. This precoding gain factor may as well control the embedding power for each user, when code division scheme is employed to embed watermarks for multiple users in a similar way to multiuser code division multiple access (CDMA) coding [14] in wireless communications. The combined term λξ is usually small. Remark 3: For any ν and xk (we only consider nonzero xk ), constrained by imperceptibility requirement, the embedding depths ak are usually small such that | ak b|xk |ν sk sgn(xk ) | |xk |, or equivalently, ak |b||xk |ν−1 1.

(9)

From the performance analysis, it can be seen the bigger ak the higher the performance that is represented by the signal to noise ratio of the GEMs. The maximum allowable depths must be determined empirically from multimedia signals using psycho-perceptual experiments. Our experiments show that good visual quality and high performance can be obtained simultaneously with the GEMs by choosing the most suitable self-adaptivity and precoding terms. Remark 4: The degree of self-adaptivity depends on the GEM degree. For example, if ν = (1/2), some (albeit small) amount of watermark energy may be allocated to plain regions; but if ν = 2, watermark energy is mostly allocated to those significant coefficients, while those insignificant almost spared. Proposition 1: The precoding term in (7) has an approximate Gaussian distribution N [0, (1 + c)/(N + c)] and thus approaches zero uniformly w.r.t. host signals, when N becomes large. Proof : The mean is obvious by the statistical independence N N . For the variance, we calculate directly and {sk }k=1 of {xk }k=1 using the distribution of |x|c . Making a change of variable u = |x|c and considering positive x, we have pU (u) = c p X (x)d x/du = (A/c)u 1/c−1 e−β u . By symmetry of x about c a the origin, pu (u) = (2 A/c)u 1/c−1 e−β u , which Nis simply |X k |c = Gamma distribution Gamma(1/c, β c ). Thus k=1 N Uk ∼ Gamma(N (1/c), β c ) [20]. Also, by defining k=1 N c ¯ ¯ Uk = l=1,l =k Ul , we know Uk ∼ Gamma((N − 1)1/c, β ) and is independent of Uk . Defining T = Uk + U¯ k and Rk = (Uk /T ), k = 1, . . . , N , the Jacobian is J [(Uk , U¯ k )/(Rk , T )] = −T , so after simple calculation we obtain the distribution of Rk 1/c−1 r (1 − rk )(N −1)/c−1 (10) p Rk (rk ) = k B(1/c, (N − 1)/c)


where B(·) is the Beta function. Hence, Rk has a Beta distribution Beta(1/c, N − 1/c) [20]. It is easily seen that Rk are mutually independent, and the variance of ξ is N N

Uk sk = V ar E Rk2 V ar (ξ ) = T k=1 k=1 (N − 1)c 1+c 1 =N = + . (11) N 2 (N + c) N2 N +c The third equation uses E[Rk2 ] = V ar (Rk )+ E 2 [Rk ]. Invoking the central limit theorem (CLT) concludes the proposition. QED. B. Special Cases of the GEM The GEM includes several embedding methods as special cases. We list some of them in the following. Special Case 1—Ordinary or Plain Multiplicative Embedding: as pointed out in Remark 1. Special Case 2—Ordinary Additive Embedding: In the case of ν = 0 and λ = 0, we essentially get the additive embedding, with the only difference from the plain additive embedding [7] being the additional term sgn(xk ) multiplied to the watermark. This additional sign term is needed here because of the absolute operations in the generalized correlator. Special Case 3—Automatically Host-Content Adaptive Embedding: In the case of λ = 0, we get the automatically host-content adaptive watermarks [15]. At busy parts such as edges and texture, these watermarks have big embedding strengths, while in plain featureless regions, the embedding strengths are weak. This property benefits the perceptual quality and the robustness because watermarks are essentially integrated into perceptually important parts of the multimedia content. Other Special Cases: ν = 1 − c and ν = 1/2. We analyze these special cases in Corollary 2 in Section VI, and it turns out that both cases have important special properties. When ν = 1 − c and there is no attack, the embedding has optimal performance that is completely interference-free from the host signal. When ν = 1/2, good approximation to 1 − c may be obtained. IV. P ERFORMANCE A NALYSIS IN G ENERAL We analyze the performance of the GEM SS watermarks with the generalized correlators [13]. The employment is based on their optimality property and their robustness to variations of embedding depths. As pointed out after (5), the generalized correlators are designed to detect the watermark optimally for a range of embedding depths, and thus are robust to variations of the embedding depths. At the embedding side, the embedding depths can be rather general and further may be changed by the attacker, while the generalized correlators are still optimal. After attacks, we may apply re-synchronization techniques to restore the synchronizations [7], and we model the attacks using noise model z k = yk + n¯ k , where n¯ k are some attacking noise. We decompose the noise multiplicatively N is assumed to be an i.i.d. as n¯ k = xk n k , where {n k }k=1 sequence with zero mean and variance σn2 . Similar to (9) in

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the embedding process, the noise n¯ k usually is much smaller than xk due to the imperceptibility constraints. That is |n¯ k | |xk |, or equivalently, |n k | 1.

(12)

For the GEM watermark using (6), the decision statistic given by the generalized correlator is g(z, κ) =

N c 1 |xk |c 1 + ak b|xk |ν−1 sk − λξ sk + n k sk N

. 1 = N =

1 N

k=1 N k=1 N k=1

|xk |c 1 + cak b|xk |ν−1 sk − cλξ sk + cn k sk |xk |c sk (1 − cλ + cn k ) + cb

N 1 ak |xk |c+ν−1 . N k=1 (13)

In the above, the first equation uses the generalized correlator (5). The second employs the first-order Taylor series expansion by noting the embedding and attacking constraints (9) and (12). And the third one uses the definition of the precoding term ξ in (7). To make the statistical analysis tractable, we assume that the embedding depths ak , k = 1, . . . , N are drawn from a nonnegative stochastic process N is statistically i.i.d. with mean a and variance σa2 , and {ak }k=1 N N N independent from {xk }k=1 , {sk }k=1 , and {n k }k=1 . The i.i.d. assumption might not be precisely true; however, the effect of any possible mismatch in our analysis is minimized thanks to the employment of the PRNS {sk }1N . The PRNS spreads the signal into different parts of the multimedia host, where the assumption is reasonably valid [16]. Considering the form of the decision statistic in (13), we shall invoke the CLT to examine its performance. For even moderately large N , an SS scheme can usually be well modeled by using the CLT, and a reasonably good approximation can be obtained. From (13), to the first-order approximation, the mean of g(z, κ) is E[g(z, κ)] = cba E|X |c+ν−1

(14)

and the variance can be obtained after some algebras as follows: V ar (g(z, κ)) (1 − cλ)2 + c2 σn2 c2 b2 (σa2 + a 2 ) E|X |2c + E|X |2(c+ν−1) N N 2 c2 b2 a 2 E|X |c+ν−1 − N 2c + 1 (1 − cλ)2 + c2 σn2 2c (1/c) c −1 1 = σx N (3/c) c c c2 b2 2(c+ν−1) (1/c) c+ν−1 −1 1 σ + N x (3/c) c

2ν − 1 2ν − 1 2ν − 1 2 2 +1 × (σa + a ) c c c ν 1 2 . (15) − a2ν2 c c =

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We define the signal energy to noise variance ratio of the GEM (GSNR) as follows: GSN R =

E 2 (g(z, κ)) . V ar (g(z, κ))

(16)

Invoking the CLT the performance of the GEM is determined by the GSNR, and the performance is given as the probability of errors in decoding the watermark symbol √ Pe = Q(− G S N R) (17) where Q(·) is the cumulative distribution function (c.d.f.) of unit Gaussian distribution. We have exploited the CLT to examine the performance in the above. While the CLT offers good approximation capabilities, more accurate performance analysis and more insights may be obtained by using the large deviation theory [17], which has been applied to additive watermarking in [8]. One future research line is to apply the large deviations techniques to the GEMs. Using (14) and (15), we have the following expression of GSNR: (1 − cλ)2 + c2 σn2 E|X |2c GSN R = N c2 b2 a 2 E 2 |X |c+ν−1 −1 σa2 E|X |2(c+ν−1) + 1+ 2 −1 . (18) a E 2 |X |c+ν−1 Therefore, the performance of the GEM is determined by the discriminant defined as follows: Dc (ν, λ, a, σa , σn ) =

+ c2 σn2 c2 b2 a 2

(1 − cλ)2

E|X |2c

E 2 |X |c+ν−1 σ 2 E|X |2(c+ν−1) + 1 + a2 . (19) a E 2 |X |c+ν−1

That is GSN R =

N . Dc (ν, λ, a, σa , σn ) − 1

(20)

For convenience of discussion, we shall refer to Dc,1 as the first additive term in Dc (ν, λ, a, σa , σn ), and Dc,2 as the second one. The GSNR needs to be maximized to optimize the performance of GEMs, thus, the discriminant Dc (ν, λ, a, σa , σn ) needs to be minimized. Judging from (19) of the discriminant, we can arrive at the following conclusion. Proposition 2: The performance of GEM watermarks monotonically increases as the value of a increases, and monotonically decreases as σa increases. The functional relationship is given by (20). As a consequence, we need to maximize the mean value of embedding depths while keeping their variance small. Actually, we can say more about the effect of a and σa on the performance. Notice the existence and significance of a in both Dc,1 and Dc,2 , and it is clear that a plays a more important role than σa . This becomes even more so in the presence of attacks, which will be discussed in a subsequent paper. For better performance and robustness, we need to maximize each ak under the allowable perceptual quality constraint, even if σa also increases.

We shall determine the effects of λ and ν on the performance of the GEMs in the following section. Due to the limitation of space, we mainly deal with the watermarked signals without attacks in this paper, and discuss how to choose proper GEM parameters and especially how to minimize the effects of attacking noise in the presence of attacks in subsequent work. Without attacks, n k = 0 and the discriminant is simply denoted as Dc (ν, λ, a, σa ). V. P ERFORMANCE OF GEM W ITH D IFFERENT D EGREES Because the GEM with ν = 1 represents a simple and familiar case, we first consider this special case in the next section, and then we move to more general ν and λ to pinpoint their optimal choices. A. GEM With Degree ν = 1 When ν = 1, we have a very familiar case, and especially, if λ = 0 in addition, we have the plain multiplicative embedding in (1). We shall examine the performance for this case in detail in the following. With ν = 1, the discriminant is (1 − cλ)2 σa2 E|X |2c +1+ 2 Dc (ν, λ, a, σa ) = c2 b2 a 2 a E 2 |X |c 2 (1 − cλ) σa2 = 2 2 2 (c + 1) + 1 + 2 (c + 1). c b a a (21) In the above, the second equation uses the fact in (3). A direct conclusion from (21) is Proposition 3: For multiplicative embedding of degree ν = 1, for any nonnegative λ, the variance of the host data has no effect on the performance (up to the first-order), but the shape parameter c does. The corresponding discriminant is given in (21). We proceed by considering several cases under the condition of ν = 1. Case 1: There is no host-interference cancellation, that is, λ = 0. This case corresponds to the plain multiplicative embedding in its simplest form. The discriminant has an expression Dc (ν, λ, a, σa ) = Dc (1, 0, a, σa ) 1 σa2 (c + 1). = + 1 + c2 b2 a 2 a2

(22)

Case 2: The precoding term ξ is small. According to Proposition 1, the precoding term ξ is small with large probability when N is large. By choosing λ = 1/c, the first additive term in (21) vanishes and the interference from the host is rejected. Clearly, the discriminant is dependent on c linearly, and also on σa2 /a 2 linearly, because 1 σ2 Dc (ν, λ, a, σa ) = Dc 1, , a, σa = 1 + a2 (c + 1). c a (23) Proposition 4: For multiplicative embedding of degree ν = 1, when the host interference is rejected with λ = 1/c, the smaller the shape parameter c of the host data, the better the


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10 Minimizer c*

GSNR

12 600 500 400 300 200 100 0 0

8 6 4 2

0.5 1 1.5 σ2 /a2

2 2.5

2

1 1.5 Shape c

0.5

0 0

0

0.5

a

1.5

2

1.5

2

(a)

Fig. 1. GSNR for different c and σa2 /a 2 for plain multiplicative watermarking with host-interference canceled (ν = 1 and λ = 1/c).

GAIN2 . (24) GAIN1 The above analysis leads to the following fact: Fact 2: For plain multiplicative embedding in (1), its discriminant is given in (22), and the GSNR is GAIN = 10 log10

N

. (25) σ2 c + (1 + c) c2 b12 a 2 + aa2 The gain, using the GEM with host interference rejection (with degree ν = 1 and λ = 1/c), is ⎛ ⎞ 1+c ⎠ . (26) GAIN = 10 log10 ⎝1 + σa2 c + a 2 (c + 1) c2 b2 a 2 For natural images, the mid-frequency subbands often show a shape parameter around 0.5. Using this as an example, for c = 0.5, a = 0.1, σa2 = 0, the gain is around 30.80 dB by using the GEM with degree ν = 1 and λ = 1/c. This shows the superiority of the GEM to the plain multiplicative watermarks. We are interested in finding, for a large number of images in an image database or a video, what the relationship of c, b, and a is for achieving the maximal performance. To see how the performance of the plain multiplicative watermarks varies with the images, we minimize the discriminant with respect to (w.r.t.) c. The minimizer satisfies 1 2 σ2 (27) 1 + a2 c3 − 2 2 c − 2 2 = 0. a a b a b The minimizer c is plotted for different a’s in Fig. 2, first with σa2 = 0 and then 0.5. Since usually c ∈ [0.3, 1] for natural images, it is seen that the image with a higher c offers a

2 1.8 Minimizer c*

performance of the corresponding GEM. The corresponding discriminant is given in (23). To visualize the performance, we plot the GSNR and Pe using (18) for different c and σa2 /a 2 in Fig. 1, using N = 60. Compared to Case 1 above, the additional term [(1 + c)/ c2 b2 a 2 ] in (22) comes from the host interference, which makes the discriminant larger, thus degrades the performance by (20). Let us define the performance gain (in dB) when comparing the GSNR of the second system against the first as

GSNR =

1 Value a

1.6 1.4 1.2 1 0.8

0

0.5

1 Value a (b)

Fig. 2. Minimizer c for plain multiplicative watermarking (ν = 1 and λ = 0) with different a’s. (a) The variance σa2 = 0 and (b) σa2 = 0.5.

better performance. Also, when the depths ak are nonconstant but small, the Gaussian host data may provide a favorable performance, and standard Gaussian analysis and modeling techniques can be straightforwardly applied [18]. In summary, we have Corollary 1: When the plain multiplicative embedding in (1) is used, the best shape parameter that attains the highest performance satisfies (27). For multimedia host data with c ∈ [0.3, 1], the image with a higher c offers a better performance. Case 3: The precoding gain is bounded by 0 < λ ≤ λmax ≤ 1/c. When the precoding term is not small, or when multiple users embed their individual watermarks with a code division scheme, to guarantee the visual quality of multimedia host signal, the precoding gain λ cannot be as large as 1/c. Then λ is bounded by λ ≤ λmax . This upper limit can be determined by the mean squared error (MSE) of the GEM watermarks in (8), or better yet, by the empirical psychovisual experiments for good perceptual quality. The minimum discriminant is achieved when λ = λmax (1 − cλmax )2 (c + 1) 2 2 2 c b a 2 σ + 1 + a2 (c + 1). a

Dc (1, λmax , a, σa ) =

(28)

The performance is much improved in contrast to Case 1; at the same time, since the host interference cannot be deeply canceled, the GSNR of GEM is not as high as Case 2. In a way similar to Case 2 above, we use the minimum discriminant to

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Minimizer c*

1.2 1 0.8 0.6 0.4 2

1 1.5

1

0.5 Value a

1.5 2

λ max

Fig. 3. Minimizer c for different λmax and a for multiplicative watermarking with host-interference rejection (GEM with ν = 1 and λ = λmax ).

find what c is more favorable. By doing so, for different images in an image database or a video, we can get the relationship of c, a, and λmax for achieving the maximal performance. For example, we can see what ranges of a and λmax work the best for a fixed image. Minimizing Dc (1, λmax , a, σa ) w.r.t. c, we see the minimizer c satisfies the following equation: a 2 b2 + σa2 b2 + λ2max c3 + (2λmax − 1)c − 2 = 0. (29) To visualize the performance, the minimizer c is plotted in Fig. 3 for different λmax and a constant depth a. It is seen that c matches the shape parameters of the natural images. B. GEMs With Optimal Degrees In this section we consider more general GEMs with degrees that are often different from ν = 1. For these GEMs, we decide on the optimal choices of ν and λ. Because we have to satisfy the hard constraints of perceptual quality, the values of λ are used to adjust the precoding term for this purpose. For different ranges of λ we consider the optimal values of ν. For the first term Dc,1 of Dc (ν, λ, a, σa ), after some algebra, we have an expression as 1c 2 + 1c (1 − λc)2 Dc,1 = (30) 2(ν−1) (1/c) ν−1 . c2 b2 a 2 σx 2 c+ν (3/c) c If ν = 1, the above equation reduces to the situations in the previous Section V-A. The second term Dc,2 of Dc (ν, λ, a, σa ) is (2ν − 1 + c)(2ν − 1) σa2 1c 2ν−1 c Dc,2 = 1 + 2 , a ν 2 2 νc 1 if ν = (31) 2 and 1c 1 σa2 (32) Dc,2 = 1 + 2 , if ν = . 1 2 a 2 1 + 2c Since the precoding gain λ exists only in Dc,1 but not in Dc,2 , and it is critically important for perceptual quality, we proceed by discussing the precoding effects for different ranges of λ (or ξ ). 1) Small Precoding Term ξ , or λ = 1/c: If the precoding term ξ is small, the allowable range of λ is large, without

the host-signal precoding λξ degrading the perceptual quality. For example, ξ is close to zero with high probability when N becomes large according to Proposition 1. We may choose λ = 1/c so that Dc,1 vanishes, and the performance of GEM depends only on Dc,2 . We need to minimize, in this case, Dc (ν, λ, a, σa ) = Dc (ν, 1/c, a, σa ) = Dc,2 w.r.t. ν. When c ∈ [0, 1], invoking the Holder ¨ inequality [19], we can easily see that the minimum of Dc,2 is achieved by ν = 1 − c, and the minimal discriminant is σ2 (33) minν {Dc (ν, λ, a, σa )} = 1 + a2 . a It is clear that the performance of GEM with properly configured parameters, including the degree ν and the precoding factor λ, is independent of the host signal. In contrast, the discriminant in (23) for special case of ν = 1 is still dependent on c. Due to the following inequality: σ2 minν {Dc (ν, λ, a, σa )} = 1 + a2 < Dc (1, λ, a, σa ) a 2 σ (34) = 1 + a2 (1 + c) a the performance of GEM, with properly configured parameters as specified in the above, outperforms the best achievable performance of multiplicative embedding (with or without hostinterference rejection). Therefore, we arrive at the following proposition: Proposition 5: The multiplicative embedding, corresponding to GEM with ν = 1, with or without host-interference rejection, is suboptimal. We summarize the previous discussions and results in the following theorem for better reference. Theorem 1: Assume the host data are i.i.d. GG D(0, σx ; c) with c ∈ [0, 1]. For generalized embedding of multiplicative watermarks (GEM) defined in (6), by using the degree ν = 1 − c and the precoding gain λ = 1/c, the performance of GEM is host-interference free, and depends only on the watermark embedding strengths through σa2 /a 2 . To the firstorder approximation, the achievable minimum discriminant defined in (19) is (1 + σa2 /a 2 ), and the achievable maximum GSNR of GEM is N a 2 /σa2 . Remark: In the above theorem, the achievable maximum GSNR of GEM watermarks is N a 2 /σa2 , which can be theoretically infinitely large when σa2 = 0, (thus error-free in decoding). This result is up to the first-order approximation. That is, by using first-order Taylor series expansion for the highly nonlinear embedding and decoding, the higher order terms w.r.t. embedding strengths and host-signal precoding λξ are omitted. These omitted higher order residuals may perturb the performance if the host-signal precoding is not small, for example, with small N ’s. Proposition 1 and our empirical experiments show that even with moderate N , the higher order residuals are indeed negligible: and the above results are confirmed. Thus, we have: Corollary 2: With degree ν = 1 − c, precoding gain λ = 1/c, and σa2 = 0, for even moderate N , the GEM in (6) is error-free in decoding by using the optimal generalized correlation detector.

5 c = 0.1 c = 0.3 4.5 c = 0.5 4 c = 0.7 3.5 3 2.5 2 1.5 1 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Value ν

985

Value of Dc

Dc/(1+(σa/a)2)


20 18 16 14 12 10 8 6 4 2 0

c = 0.1 c = 0.3 c = 0.5 c = 0.7 c = 0.9

0

0.5

1 1.5 Value of ν

2

Fig. 4. Normalized discriminant values for different ν’s, with c = 0.1, 0.3, 0.5, and 0.7.

Fig. 5. Discriminant for different c’s and ν’s for GEM with λmax = 3, σx = 12, and a = 0.3.

The range of c ∈ [0, 1] is particularly adopted for practical applications, because the shape parameters of most natural images are in this range. In the case of c > 1, because we restrict ν ≥ 0, one choice is ν = 0. For several values of c = 0.1, 0.3, 0.5, and 0.7, we plot the normalized discriminant Dc (by 1 + σa2 /a 2 ) in Fig. 4. It is seen that the minimal values of ν are obtained at 1 − c as stated in the above theorem. Also, it is noted from Fig. 4 that the discriminant is flat for a rather large range of values around the minimizer. This offers us a nice property that near optimum performance can be obtained in the presence of estimation bias of the shape parameter. When we choose λ = 1/c, the decision statistic from (13) is

for the minimizer ν for a given c. For a parameter setting with σx = 12, λmax = 3, a = 0.3, and σa = 0, we plot the discriminant values for different c’s and ν’s in Fig. 5. Because λmax = 3, the discriminant is minimized numerically when the values of c ≤ 1/3; in contrast, the minimizer is always 1 − c when c > 1/3 according to Theorem 1. When c < 0.1, the discriminant increases dramatically.

g(z, κ) = cb

N 1 ak = cba + e. N

(35)

k=1

In the above, the term e represents the error which comes from the mean approximation to the embedding strengths and the higher order terms of Taylor series expansion. Thanks to the law of large numbers and negligible high-order terms, the error term is usually very small without attacks. In the presence of attacks, the attacking noise is also present in this term. It is clear the mean decision statistics are essentially constant cba. 2) Case of 0 < λ ≤ λmax < 1/c: In the case of λ ≤ λmax < 1/c, the Dc,1 term cannot be deeply eliminated. We have (1 − cλ)2 E|X |2c c2 b2 a 2 E 2 |X |c+ν−1 σ 2 E|X |2(c+ν−1) + 1 + a2 a E 2 |X |c+ν−1

Dc (ν, λ, a, σa ) =

=

(1 − cλ)2 2(ν−1)

c2 b2 a 2 σx

1

2c+1

c ((1/c)/ (3/c))ν−1 2 1 + νc σa2 1c 2c+2ν−1 c + 1+ 2 . (36) a 2 1 + νc

×

c

The λ term only exists in the first term Dc,1 . Since λ ≤ λmax < 1/c, the value of λ needs to take λmax to minimize Dc (ν, λ, a, σa ). Because the above equation of the discriminant is highly nonlinear in ν, we resort to numerical methods

VI. D ISCUSSIONS FOR P RACTICAL A PPLICATIONS In our experiments, we embed the watermarks in the DCT and DWT domains. For the domains of pyramid, LOT, or other subband transforms, similar results can be obtained by choosing proper coefficients, typically near mid-frequency ones. The GEM is very applicable to these domains because they have similar statistical characteristics when applied to watermarking and are well modeled by the GGD [8]. In practice, the GEM is naturally suitable when the image and video coding make use of DCT, such as JPEG, MPEG-1, 2, 4, H.261, or H.263. The same can be said for those using DWT such as JPEG 2000 and texture coding in MPEG-4. If the current watermarking systems are spread spectrum-type, the GEM is a backward-compatible extension to them, and may retain their structures and be straightforwardly deployed. When N is small, for example, N = 4, 5, . . . , 10, high payload may be obtained. In this case, the precoding term ξ may be not as small for each chip sequence sk , k = 1, . . . , N . To avoid potential degradation of perceptual quality, we need to use the precoding gain factor λ to control the embedding power. That is, we may use λmax < 1/c and rely upon the analyses given in previous sections. For a given multimedia host, we may employ a single λmax for the whole data. Alternatively, we may control individual ξ which has a value larger than a predefined threshold. When only a fraction of ξ ’s has larger values, the latter control method can be more effective and lead to higher performance. As shown in our analysis above, the higher the average embedding strength a, the better the performance of GEMs. Compared to the variance σa2 of ak , the mean strength a is more important. In practice, the perceptual quality and the MSE may not coincide, and the proper ak have to be determined by using psychovisual experiments. In our experiments in Section VII, for the purpose of comparing the performance, we simply choose a’s for different ν’s so that

986


original

GEM degree = 1

(a)

(b)

GEM degree = 0.5

GEM degree = 1−c

(c)

(d)

(a)

(b)

they have approximately the same MSE, and for each ν, ak are a constant. In practical applications, the highest allowable value of a should be chosen to satisfy the perceptual quality constraint for each ν. With a constant a, the perceptual quality is already appealing due to the self-adaptability to the host content. To make a better balance between the performance and perceptual quality, each ak may be maximized according to more sophisticated human perceptual characteristics. To this end, a research line to develop psycho-perceptual model would turn out beneficial. VII. E XPERIMENTAL R ESULTS We conduct experiments to validate the effectiveness of GEM and the performance analysis. For the Lena image, we embed the watermark in the DWT domain. For DCT domain or other subband transforms, similar results are obtained by choosing proper coefficients. Biorthogonal spline wavelet filters and three-level deposition are used for DWT. The near mid-frequency bands are employed as hosts for the watermarks. Specifically, they include the horizontal and vertical subbands at the first level, and the horizontal, vertical, and diagonal subbands at the second level. The watermark information is a bitstream and the bits are represented with bipolar symbols b = ±1. Each chip sequence sk , k = 1, . . . , N , carries one symbol. The shape parameter is estimated by using the minimal relative entropy method [8], which is around 0.58 for Lena. The GEMs are constructed with different N ’s and ν’s. First, we study the perceptual quality of the GEMs. For N = 8, we embed 22 528 bits in the above-mentioned subbands. We adopt a = 0.03, 0.13, and 0.12, respectively, for ν = 1, 1 − c, and 0.5, such that the resultant MSEs are almost the same for a given N and a given chip sequence. The original and

Fig. 7. Magnitudes of the difference, amplified by a factor of 20 for viewing purpose, between original and watermarked Lena using GEMs with degrees of (a) ν = 1. (b) ν = 0.5. (c) ν = 1 − c.

10

Prob. of Errors

Fig. 6. Original and watermarked Lena using GEMs with different degrees: (a) Original. (b) ν = 1. (c) ν = 0.5. (d) ν = 1 − c.

(c)

0 Plain multi. watermarking GEM degree = 1

10−1 10−2 10−3 10−4 10

15

20 25 30 Value of N

35

40

Fig. 8. Probabilities of decoding errors using plain multiplicative watermark and GEM with degree 1 and host-interference rejection. The GEM is superior to the plain multiplicative watermark.

watermarked Lena using GEMs with degrees 1, 1 − c, and 0.5 are shown in Fig. 6. The magnitudes of the difference images, amplified by a factor of 20, are shown in Fig. 7. The selfadaptability is clear from the difference images. Then we validate the performance analyses using the experiments. To show the performance of GEMs, we compare the decoding capabilities. Each chip sequence carries one information symbol b. The decision on b is made according to sgn(g(z, κ)), where sgn(·) is the sign function, and g(z, κ) employs the optimal generalized correlation detectors. To show the effectiveness of embedder-side precoding, we embed the watermarks using plain multiplicative watermark (ν = 1 and λ = 0), and the corresponding GEM (ν = 1 and λ = 1/c). For comparison purposes, we fix the MSEs of these two embeddings. And the corresponding embedding strengths are 0.18 and 0.03, respectively, for plain multiplicative watermark and the GEM with degree 1 and λ = 1/c. For the Lena image, the probabilities of decoding errors are plotted in


GEM degree = 1 GEM degree = 1−c GEM degree = 0.5

10−1 10−2 10−3 10−4

0

5

10 15 20 25 30 35 40 Value of N

Fig. 9. Probabilities of decoding errors using GEMs with different degrees of 1, 0.5, and 1 − c.

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 90

0.24 0.22 Prob. of Errors

0

Prob. of Errors

Prob. of Errors

10

987


80

70 60 50 40 30 Quality Factor of JPEG

20

0.2 GEM degree = 1 GEM degree = 1−c GEM degree = 0.5

0.18 0.16 0.14 0.12 0.1 3

4 5 6 7 Window Size of Wiener Filer

(a)

8

(b)

Fig. 11. Probabilities of decoding errors using the GEMs with degrees of ν = 1, ν = 0.5, and ν = 1 − c, with N = 400, (a) after JPEG compressions with different quality factors and (b) after Wiener filtering with different window sizes.

x 10−3 2.5


Prob. of Errors

2 1.5 1 0.5 0 40

60

80 100 Value of N

120

140

Fig. 10. Probabilities of decoding errors using GEMs with different degrees of 1, 0.5, and 1 − c for larger N .

Fig. 8. It can be seen that the GEM with degree 1 significantly outperforms the plain multiplicative watermarking. This is consistent with our performance analysis given in Propositions 4 and 5. Furthermore, the GEM with degree 1, even after the hostinterference cancellation, is suboptimal, and the optimal one is the GEM with degree 1 − c. We plot the probabilities of decoding errors in Fig. 9 for the GEMs with degrees of ν = 1, 0.5, and 1 − c. The MSEs are fixed for comparison purposes. We use a range of N ’s from 4 to 40. With N = 4, a total number of 45 056 bits are embedded. The empirical probability of decoding errors is 0.049. With N = 20, a total number of 9011 bits are embedded, and the resultant empirical probability of decoding errors is 0.0054. For larger N , the probabilities of decoding errors are also plotted in Fig. 10. Since these probabilities are small, we plot them separately from those with smaller N in Fig. 9. It is clear for moderate N , the probabilities of decoding errors are vanishingly small or error-free. As stated at the beginning of this section, we fix the MSEs for comparison purposes. The PSNR, defined as 20 log10 (256/M S E), is 41.1 dB. It is observed that the GEMs with degrees of 0.5 and 1 − c outperform that with degree 1. The GEMs with degrees of 0.5 and 1 − c have similar performance because the degrees are so close. From Fig. 4 it is noted that the discriminant is flat for a rather large range of values around the minimizer. This offers us a nice property that near-optimum performance can be obtained in the presence of estimation bias of the shape parameter. Further, when the

shape parameter of multimedia is close to 0.5, or for a large image database, we may simply elect 0.5 in the GEM for near optimal performance. This is in a similar vein to the use of the square-root detector as a near-universally optimal detector in [13]. We develop the GEMs and the analysis without explicitly modeling the attacking in this paper, and leave attack modeling to a subsequent work. To show the GEMs can still work under attacks, we conduct some experiments with certain modifications of watermarked contents. Using the GEMs with ν = 1, 1 − c, and 0.5, we choose N = 400 and otherwise similar settings to the above experiments, so that 450 bits are embedded with PSNR = 41.32 dB. Then we compress the watermarked images with JPEG compression with different quality factors ranging from 90 to 20. The lower the quality factor, the lower the quality of compressed image. We also apply Wiener filtering to the watermarked images with different window sizes ranging from 3 × 3 to 8 × 8. The bigger the window size, the larger the filtering effect. For the modified compressed images, we apply similar detectors to the above experiments. The probabilities of decoding errors are plotted in Fig. 11. It can be seen that, without explicitly modeling the attacks, the GEMs have certain robustness to attacks or noise even though they cannot completely reject the noise. VIII. C ONCLUSION This paper has constructed a class of generalized embedding of multiplicative spread spectrum watermarks, including as special cases the ordinary multiplicative and additive watermarking methods. The class of new embeddings automatically adapts to the local contents of the host signal, benefiting the perceptual quality and decoding or detection capability. The perceptual quality of the watermarked signal can be well controlled, configured, and customized with the new embedding. The decoding of these watermarks exploits the optimal generalized correlation detectors. The host interference is precanceled at the embedder side taking into account the host states, the host statistics, and the detector. The precancellation offers a big gain of the decoding performance compared to the ordinary embedding; at the same time, the imperceptibility is retained. We have developed performance analysis for this new class of embeddings that is host interference rejective. This paper

988


has focused on the analysis without attacks, and we have discussed that with attacks in a subsequent paper. It turns out that the new embeddings are far superior to the ordinary multiplicative watermark. Furthermore, even after the hostinterference rejection, the ordinary multiplicative watermark is still suboptimal. For a given host multimedia signal, the best selection of the watermark is specified. Different scenarios for the generalized embedding are studied, and the optimal configuration of the embedding is attained for each scenario. In sharp contrast to the ordinary multiplicative embedding, even with moderate lengths of the chip sequence, the optimal GEM is free of errors in decoding. Our construction offers a class of powerful new methods, and our performance analysis discloses the limitation of the existing methods as well as offers optimal properties of the new construction. The construction and the performance analysis have been confirmed by empirical experiments. Future research lines may include the performance analysis using large deviation theory and the development of dedicated perceptual models.

[11] I. J. Cox and M. L. Miller. “Watermarking as communications with side information,” in Proc. IEEE, vol. 87, no. 7, pp. 1127–1141, Jul. 1999. [12] G. Caire and S. Shamai, “On the achievable throughput in multiple antenna Gaussian broadcast channel,” IEEE Trans. Info. Theory, vol. 49, no. 7, pp. 1691–1706, Jul. 2003. [13] Q. Cheng and T. S. Huang, “Robust optimum detection of transformdomain multiplicative watermarks,” IEEE Trans. Signal Process., vol. 51, no. 4, pp. 906–924, Apr. 2003. [14] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. New York: Cambridge Univ. Press, 2005. [15] Q. Cheng, Y. Wang, and T. S. Huang, “Maximizing efficacy for efficient watermarking systems,” in Proc. Int. Conf. Image Process., Barcelona, Spain, Sep. 2003, pp. II-747–II-750. [16] J. G. Proakis, Digital Communication. 4th ed. New York: McGraw-Hill, 2001. [17] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. 2nd ed. New York: Springer-Verlag, 1998. [18] M. H. Hayes, Statistical Digital Signal Processing and Modeling. Hoboken, NJ: Wiley, 1996. [19] H. L. Royden, Real Analysis. 3rd ed. New York: MacMillan, 1988. [20] G. Casella and R. L. Berger, Statistical Inference. Stamford, CT: Thomson Learning, 2001.

R EFERENCES

Qiang Cheng received the B.S. and M.S. degrees from Peking University, Beijing, China, and the Ph.D. degree from the Department of Electrical and Computer Engineering at the University of Illinois, Urbana-Champaign (UIUC), 2002. During the summer of 2000, he was with the IBM T.J. Watson Research Center, Yorktown Heights, NY. In June 2002, he joined the Electrical and Computer Engineering Department, Wayne State University, Detroit, Michigan, as an Assistant Professor. From May to August 2005, he was an AFOSR Faculty Fellow at the Air Force Research Laboratory, Wright-Patterson, OH. From August 2005 to June 2007, he was a Senior Researcher, Senior Research Scientist with Siemens Medical Solutions, Siemens Corporate Research, Princeton, NJ. Since August 2007, he has been an Assistant Professor with the Computer Science Department, Southern Illinois University, Carbondale. He has more than 40 technical publications and also a number of international patents issued or pending with the IBM T.J. Watson Research Laboratory and Siemens Medical. His research interests include signal and image processing, pattern recognition, learning theory, and applications to medicine and healthcare. Prof. Cheng is the recipient of various awards and privileges, including the Achievement Awards from Siemens Corporation from 2006 to 2007, the University Research/Enhancement Awards from Wayne State University in 2003, 2004, 2005, the Invention Achievement Award from the IBM T.J. Watson Research Center, 2001, a University Fellowship at UIUC from 1997 to 1998, a Guanghua University Fellowship at Peking University, China, from 1995 to 1996, and the Antai Award for Academic Excellence from Peking University, China, 1995.

[1] M. Costa, “Writing on dirty paper,” IEEE Trans. Info. Theory, vol. 29, no. 3, pp. 439–441, May 1983. [2] B. Chen and G. W. Wornell, “Quantization index modulation: A class of provably good methods for digital watermarking and information embedding,” IEEE Trans. Info. Theory, vol. 47, no. 4, pp. 1423–1443, May 2001. [3] J. Eggers, R. Bauml, R. Tzchoppe, and B. Girod, “Scalar costa scheme for information embedding,” IEEE Trans. Signal Process., vol. 51, no. 4, pp. 1003–1019, Apr. 2003. [4] A. Abrardo and M. Barni, “Informed watermarking by means of orthogonal and quasi-orthogonal dirty paper coding,” IEEE Trans. Signal Process., vol. 53, no. 2, pp. 824–833, Feb. 2005. [5] H. S. Malvar and D. A. Florencio, “Improve spread spectrum: a new modulation technique for robust watermarking,” IEEE Trans. Signal Process., vol. 51, no. 4, pp. 895–905, Apr. 2003. [6] J. Zhong and S. Huang, “An enhanced multiplicative spread spectrum watermarking scheme,” IEEE Trans. Circuits Syst. Video Technol., vol. 16, no. 12, pp. 1491–1506, Dec. 2006. [7] I. Cox, J. Kilian, T. Leighton, and T. Shamoon, Digital Watermarking. San Mateo, CA: Morgan Kaufman, 2001. [8] Q. Cheng and T. S. Huang, “An additive approach to transform-domain information hiding and optimum detection structure,” IEEE Trans. Multimedia, vol. 3, no. 3, pp. 273–284, Sep. 2001. [9] M. Barni, F. Bartolini, A. De Rosa, and A. Piva, “A new decoder for the optimum recovery of nonadditive watermarks,” IEEE Trans. Image Process., vol. 10, no. 5, pp. 755–766, May 2001. [10] Q. Cheng and T. S. Huang, “Optimum detection and decoding of DFT domain watermarking,” in Proc. ICASSP ’02, Orlando, FL, pp. IV-3477– IV-3480.