W AmplitudeScaleEstimationforQuantization-BasedWatermarking

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based watermarking context. We concentrate on operations that are common in many applications and at the same time devastating to this class of watermarking schemes, namely, amplitude scaling in combination with additive noise. First we derive the probability density function of the watermarked and attacked data in the ab- sence of subtractive dither. Next we extend these models to incor- porate subtractive dither in the encoder. The dither sequence is primarily used for security purposes, and the dither is assumed to be known also to the decoder. We design the dither signal statistics such that an attacker having no knowledge of the dither cannot decode the watermark. Using an approximation of the probability density function in the presence of subtractive dither, we derive a maximum likelihood procedure for estimating amplitude scaling factors. Experiments are performed with synthetic and real audio signals, showing the feasibility of the proposed approach under re- alistic conditions.

Index Terms—Maximum likelihood estimation, probability of error, quantization, statistics, subtractive dither, watermarking.

I. INTRODUCTION

W

ATERMARKING is the process of imperceptibly embedding a message (watermark) into a host signal (audio, video). The resulting signal is called a watermarked signal. The message should introduce only tolerable distortion to the host signal and it should be recoverable by the intended receiver after signal processing operations on the watermarked data.

Watermarking schemes based on quantization theory have recently emerged as a result of information theoretic analysis [1]–[4]. In terms of additive noise attacks, these schemes have proven to perform better than traditional spread-spectrum watermarking because they can completely cancel the host signal interference, which makes them invariant to the host signal. The existence of good lattices in high dimensions [5] that can be di- rectly and efficiently implemented has made quantization-based schemes of practical interest.

Lattice-based schemes are vulnerable to amplitude scale attacks because these attacks introduce mismatch between the encoder and the decoder lattice volumes. Furthermore, amplitude scaling induces a large amount of distortion with respect to the mean squared error but does not cause significant perceptual

Manuscript received May 30, 2005; accepted January 10, 2006. This work was supported by the Technology Foundation STW, Applied Science Division of NWO, and Technology Program of the Ministry of Economic Affairs. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Anuj Srivastava.

The authors are with the Information and Communication Theory Group, Faculty of Electrical Engineering, Mathematics, and Computer Science, Delft University of Technology, 2628 CD Delft, The Netherlands (e-mail:

i.shterev@gmail.com; r.l.lagendijk@ewi.tudelft.nl).

Digital Object Identifier 10.1109/TSP.2006.881216

and capturing, where the watermarked signal is passed through a digital-to-analog converter, transmitted through an analog noisy channel, captured by a microphone, and converted back to a digital representation. Clearly the microphone will capture a less powerful and degraded watermarked signal, which has led us to model the noisy channel as an amplitude scaling operation followed by additive noise. In this paper, we concentrate on operations consisting of amplitude scaling followed by additive white Gaussian noise (AWGN), often called scale AWGN channel.

Several techniques are known in the literature for combating amplitude scale attacks. One of the approaches is based on de- signing watermarking codes that are invariant to amplitude scale operations, such as modified trellis codes [6], order-preserving lattice codes [7], and rational dither modulation [8]. Another approach is based on estimating the nonadditive operations and in- verting them prior to watermark decoding, using pilot signals [9]

or blind estimation [10]–[12]. More recently, an iterative estimation procedure in combination with error-correcting codes was proposed [13], which proved to perform well even for low watermark-to-noise ratios (WNRs). The advantage of the approach in [9] is the ability to estimate the scaling factor from a small number of signal samples, which makes the estimation procedure applicable in situations where the scaling factor slowly varies. The disadvantage of the method is that the pilot signals consume part of the capacity of the watermarking system. The method proposed in [11] performs well for low WNR but lacks security, in the sense that an attacker knowing the distortion of the embedder is able to estimate the scaling factors and decode the watermark. The methods based on invariant codes give small probability of error with respect to amplitude scale attacks at the expense of increased probability of error [8], [7] with respect to additive noise attacks and reduced payload [6].

In this paper, we propose a maximum likelihood (ML) approach for estimating amplitude scaling factors. Our estimation technique is blind and only assumes knowledge of the watermark message priors. No knowledge of the position of the message bits in the watermark bitstream is required. We also introduce subtractive dither [14] in the encoder. The realization of the dither is assumed to be known to the decoder. An ap- plication of subtractive dither to watermarking appeared first in [15], but with no theoretical analysis of the system security. In this paper, we design the dither statistics such that an attacker without knowing the dither realization is not able to decode the watermark. Thus the dither serves as the key ensuring security of the system.

This paper is organized as follows. In Section II, we for- mulate the attack channel, watermark encoder, and decoder. In Section III, we derive the probability density function (pdf) of

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Fig. 1. Watermark encoder.

the received data in the absence of dither [11], as a preliminary step. Then, we extend these pdf models to incorporate subtractive dither. In Section IV, we give approximations to the pdf models for the case when the dither variance is much smaller than the host signal variance. In Section V, we give conditions for the dither sequence statistics such that a given level of security is achieved and at the same time the dither variance is as small as possible, using the probability of error of the watermarking system as an objective function. A description of the ML estimation procedure is given in Section VI. Section VII contains experimental results with synthetic and real audio host signals, and Section VIII concludes this paper.

II. MATHEMATICALFORMULATION

In this paper, we focus on the most popular quantization-based watermarking scheme: scalar quantization index modulation (QIM). Throughout this paper, random variables are denoted by capital letters and their realizations by the respective small letters. The notation indicates that the random variable has a pdf .

Fig. 1 shows the watermark encoder, where denotes the message bits that are embedded in the host data, is the host signal itself with a variance , is the watermarked signal, and is the dither sequence with a variance . The statistics of the dither sequence will be derived in Section V. The variable is the output of the quantizer.

denotes uniform quantization with step size . The quantization noise, which is the difference between the quantizer input and output, is defined as

(1) where is a coefficient to be defined later.

From (1), we see that the watermark and the quantization noise are equal. The quantizer input–output characteristic is shown in Fig. 2 for the watermark message . The output of the quantizer can be written as

if

if (2)

where is an integer.

Fig. 2. Quantizer input–output characteristics.

Fig. 3. Attack channel.

The attack channel is shown in Fig. 3. It consists of the con- stant amplitude scale factor and the noise . The noise is independent of and . We choose the co-

efficient as in [4], where is the

variance of . Other choices for are also possible [16].

The attacked (received) signal can be written in the following way:

(3) Using the relation , we obtain the received data in terms of , , and the watermark-bearing signal

(4) The watermark decoder is shown in Fig. 4. From the received signal , the decoder first performs maximum a posteriori probability estimation of the signal , which under mild as-

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Fig. 5. The effect of on probability of error P for D = 0. Experiments are performed forX N (0; 1), N N (0; ), and DWR= 20 dB.

sumptions [1] is equivalent to multiplication by .¹Then the decoder adds the dither , obtaining

(5) The decoder then computes the absolute value of the quantization noise and makes an estimate of the embedded watermark in the following way:

if

if (6)

Throughout this paper, we denote

, and the document-to-watermark

ratio DWR .

Experimental results of the effect of unknown on probability of error are shown in Fig. 5. We can see that the amplitude scale attack is more devastating at high WNR. At low WNR, the effect of the attack is less pronounced because the probability of error is already quite large for .

III. PDF MODELS

In this section, we first derive exact pdf models of the watermarked and attacked signals in the absence of dither ( ).

1Here we assume that we are able to perfectly estimate

occurrence of bit 0 and 1, respectively, and and are the conditional pdfs of the watermarked data corresponding to and , respectively.

Taking and into account and using the fact that for any

, we have , we obtain the

pdf of the received data as

(8) where the convolution follows from the independence between

and .

We derive the expression for . The derivation of follows using similar reasoning.

Let us consider the case where the input to the quantizer is in the th quantization cell, i.e., the output of the quantizer is

. We have

(9) Multiplying all sides by the positive term , we get

(10) Adding to all sides and reorganizing, we obtain

(11) We define the indicator function

if

if (12)

where

(13) Therefore, the pdf of over the support set

is .

Recognizing that is the watermarked data for a particular , we can find the pdf of by summing over

. Thus we have

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Fig. 6. Graph off (x). Chosen settings are X N (0; 1), N N (0; 0:01), = 0:01, and 1 p

0:12.

In the same fashion we can express the pdf of the watermarked

data for as

(15) where

(16) An illustration of (14) is shown in Fig. 6.

Referring to the above equations, we can now take the scaling factor into account

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(18) where the indicator sets are given as

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(20) An illustration of (17) is shown in Fig. 7. The regular pattern that carries information about the quantity in the pdf of the watermarked data can clearly be seen. Reference [9] exploits similar modeling.

Finally, an illustration of (8) with and is given in Fig. 8.

B. Case

In this section, we assume that the dither is present in the watermarking system as shown in Figs. 1 and 4. Since in the presence of subtractive dither will be perturbed by , it

Fig. 7. Graph off (x). Chosen settings are X N (0; 1), N N (0; 0:01), = 0:01, 1 p

0:12, and = 0:8.

Fig. 8. Graph off (x) with Pr(W = 0) = 1. Chosen settings are X N (0; 1), N N (0; 0:01), = 0:01, and = 1.

is difficult to derive a useful exact mathematical expression for it. That is why we choose to manipulate in a convenient way, having knowledge of , so that we are able to mathematically describe the structure of the pdf of the resulting random variable.

For simplicity, we will assume that only message

is embedded, therefore working only with the first part of (8).

Extension to the more general case of embedding zeros and ones is straightforward: use the whole expression (8). Using the same reasoning and notation as in the previous subsection, we derive the pdf models in the presence of subtractive dither.

Referring to Fig. 1, let us assume that and belongs to the th quantization cell, i.e.,

(21) Multiplying by and adding , we obtain

(22) Recognizing that the leftmost and rightmost parts of (22) are the indicator set as given by (13) and taking into account the fact that

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Equation (24) is the key expression for the estimation procedure in the presence of subtractive dither. We can see that although is perturbed by the dither, if we add the term to the watermarked signal, we are able to obtain a signal that has a pdf with an indicator function equal to that when no dither is used. In other words, we are able to recover the structure of the watermarked signal pdf by the use of the dither.

Taking into account and the additive noise , we now have

(25) where the convolution follows from the independence between

and .

IV. APPROXIMATION TO THEPDF MODELS IN THEPRESENCE OFSUBTRACTIVEDITHER

Since (24) is very complex to implement, we make approximations to it. We can see that there are only two random variables involved in (24): and . Assuming that , we can approximate in the following way:

(26) Note that the output of the quantizer depends both on and , but since the variance of the first is assumed to be much larger, the term is present in the approximation together with . An illustration of , its approximation as given by (26), and is shown in Fig. 9. The difference between and its approximation can hardly be recognized.

We can also see the huge difference between and .

V. DESIGN OF THEDITHERSEQUENCE

In the previous section, we saw that in order for (26) to be an accurate approximation. Approximation is perfect if , but this is unacceptable from security point of view.

In this section, we find sufficient conditions for the dither sequence statistics such that, for as small as possible, an attacker is not able to decode the watermark with an error probability²different than 0.5.

2Since we have one-dimensional one-bit watermarking, the error probability and bit error probability are equal.

Fig. 9. An illustration off (x) (dashed line), f (x) (dotted line),

and its approximation f (x)I (x) (solid line).

Chosen settings areX N (0; 1), = = = 0:01, and = 1.

To derive the conditions, we first need to derive the error probability, which is given by the following theorem.

Theorem 1: When the dither sequence is not known to the decoder, the error probability is given by the expression

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where is an integer.

Proof: The error probability can be expressed as

(28) where the last line follows from the fact that the encoder is a symmetric scheme of two quantizers, that the channel strategy is independent of the embedded message, i.e.,

, and that . Therefore, we

can model the whole watermarking system, together with the attack channel, as a binary symmetric channel with crossover probability .

From (28) and (6), it is straightforward to show that the probability of error when is not added at the decoder can be written as

(29) Observe that for any and scalar quantizer with step size

, we can write the relation

(30) Using (30) in (29), we have

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(32) where denotes the union of two events.

Using (5) and taking into account that , the quantizer lattice, we can write the equation shown at the bottom of the page. Using number theory [17], we can write that for any

and any such that , and any , where

is an integer, the solution to the inequalities (33) (34) is

(35) and

(36) respectively. Therefore, after simple arithmetic, we arrive at (27).

We would like to choose the dither sequence statistics such that the error probability for all choices of the attacker noise . We state the following theorem.

Theorem 2: For the probability of error , it is sufficient to choose the dither uniformly distributed over the base quantization cell,³i.e., .

Proof: For notational simplicity, we make the following substitution:

(37) We can express (27) in the following way:

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By definition, we can write

(39)

3In [14], it was shown that this is sufficient for making the quantization noise independent of the input signal.

Substituting with (39) in (38), we get

(40) where in the second equality we interchanged the order of inte- gration and summation.

From (37), we can write

(41) Therefore, can be written as

(42) From (41), we see that the term affects only the mean of . If we choose the dither to be uniform over the base quantization cell, i.e., , then we can show that (see Fig. 10)

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Therefore

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Note that in the proof of the theorems, we do not need the assumption , and therefore the high resolution quantization assumption [10] is not necessary for the system security.

However, we will assume the low distortion case

because of the approximation assumptions in the previous section.

Using Fig. 10, it can also be shown that , with

equality when and .

Experimental curves for the probability of error as a function of for different values of are shown in Fig. 11. It can be seen that independently of , as long as

.

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Fig. 10. An illustration of f (zjn ; n )dz for D U(0; ).

Fig. 11. Experimental curves forP as a function of for different values of . The solid curve is for = 0, the dashed curve is for = 0:01, and the dotted curve is for = 0:02. Chosen settings are X N (0; 1), D U(0; ), N N (0; ), = 0:01, and = 1.

VI. ML ESTIMATION OF

The pdf models of the watermarked and attacked data have been derived as a function of in the previous sections. We are now able to use these models to estimate from the observed signal .

We assume that the host signal and attack channel noise are independent identically distributed (i.i.d.) vector sources, i.e., we consider all signals to be -dimensional vectors with i.i.d.

components. The ML estimation of is done based on the following relation:

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Representing as a joint distribution,

the ML estimation of the parameter [18] is given as

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Fig. 12. Graph of MLF for different values of. Chosen settings are X N (0; 1), D U(0; 0:01), N N (0; 0:01), and = 0:01.

Fig. 13. Graph of MLF for different values of WNR. Chosen settings areX N (0; 1), D U(0; 0:01), = 0:01, and = 1.

Here the second line follows from the assumption that the received data consist of i.i.d. samples, and therefore the joint pdf can be written as a product of the “ ” marginal pdfs. The last line follows from the monotonicity of the logarithm.

Experimental curves for the maximum likelihood functional (MLF), which is the expression

, are shown in Figs. 12 and 13.

VII. EXPERIMENTS

In this section, we describe experiments carried out to test the estimation accuracy of the proposed technique in terms of WNR and the number of available signal samples . In principle, one aims at developing estimation techniques that require a small amount of data, so that they can be applied in situations where the estimating parameter slowly varies. Since it is difficult to further manipulate (25) (even for Gaussian sources) because of

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Fig. 14. Graphs of ^ as a function of WNR. The crosses represent the mean and the lines the standard deviation in both directions. The chosen settings are X N (0; 1), DWR = 20 dB, = 1, and n = 350 000.

Fig. 15. Graphs of ^ for synthetic host signals as a function of the number of signal samplesn. The crosses represent the mean and the lines the standard deviation in both directions. Chosen settings areX N (0; 1), DWR = 20 dB, WNR= 0 dB, and = 1.

the indicator function in , we do

brute force search for the optimal . A. Synthetic Host Signals

Here we perform experiments with synthetic host signals. We assume that the estimator has perfect knowledge of the host signal variance. In Fig. 14, we present results for as a function of WNR. It can be seen that for WNR dB, the mean of is very close to the true value of , and the standard deviation of

is always smaller than 1%. In Fig. 15, we present results for as a function of number of signal samples. It can be seen that around 100 signal samples are needed for reliable estimation of . Results of as a function of are presented in Fig. 16.

We can see that the standard deviation of is smaller than

1% for .

Fig. 16. Graphs of 0 ^ as a function of . The crosses represent the mean and the lines the standard deviation in both directions. Chosen settings areX N (0; 1), DWR = 20 dB, WNR = 0 dB, n = 350 000.

Fig. 17. Graphs of ^ for real audio signals as a function of WNR. The crosses represent the estimation mean and the lines the standard deviation in both directions. Chosen settings are DWR= 20 dB, = 1, and n = 350 000 . . . 500 000.

B. Real Host Signals

In this subsection, we describe experiments with real audio signals (audio and speech with sampling frequency 48 kHz).

We choose more realistic settings than in the case of synthetic hosts, in which the estimator does not have a perfect knowledge of the host signal variance. The assumed pdf model of the host signal at the detection side is a zero-mean Laplacian pdf with variance equal to the variance of the received signal, i.e., . This is a realistic assumption because the decoder has access to the received data and can estimate its variance. Furthermore, in practice, most audio signals have a marginal pdf that resembles the Laplacian pdf [19]. Ex- perimental results in terms of WNR are shown in Fig. 17. It can be seen that the standard deviation of is smaller than 1% for . Experimental results of as a function of the number of signal samples are shown in Fig. 18. It can be seen

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Fig. 18. Graphs of ^ for real audio signals as a function of the number of signal samplesn. The crosses represent the mean and the lines the standard deviation in both directions. Chosen settings are DWR= 20 dB, WNR = 0 dB, and = 1.

Fig. 19. Graphs of 0 ^ for real audio signals as a function of . The crosses represent the mean and the lines the standard deviation in both directions. Chosen settings are DWR = 20 dB, WNR = 0 dB, and n = 350 000 . . . 500 000.

that reliable estimation of is possible for samples.

In Fig. 19, we plot experimental results of as a function of for different audio signals. It can be seen that the standard deviation of is smaller than 1% for .

The experimental results with real signals are generally worse than in the case of synthetic signals. There are several reasons for that. First, the experimental settings are different. For real signals the estimator has access only to the received signal. The variance of the received signal differs from the variance of the host signal, and the difference is especially pronounced when deviates from one. This causes a difference between the pdf of the host signal and the pdf assumed by the estimator. Second, real signals are nonstationary and exhibit correlation between the samples, which is not captured by our pdf models.

subtractive dither into the watermarking system and gave sufficient conditions for the dither sequence to achieve a given level of security. The estimation approach needs a small amount of signal samples for estimating reliably in the case of synthetic host signals but a relatively large amount of signal samples in the case of real audio host signals. Experiments showed that the proposed approach performs well under realistic conditions.

ACKNOWLEDGMENT

The authors wish to acknowledge the helpful suggestions of the reviewers.

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Ivo D. Shterev received the M.Sc. degree in elec- tronics from the Technical University of Sofia, Plovdiv, Bulgaria, in 1999. He is currently pursuing the Ph.D. degree in the Information and Communica- tion Theory Group, Delft University of Technology, Delft, The Netherlands.

He did a summer 2004 internship at the Univer- sity of Illinois at Urbana-Champaign. His research interests include probability theory, information theory, and statistics.

Reginald L. Lagendijk (S’87–M’90–SM’97) received the M.Sc. and Ph.D. de- grees in electrical engineering from Delft University of Technology, Delft, The Netherlands, in 1985 and 1990, respectively.

He became an Assistant Professor at Delft University of Technology in 1987.

He was a Visiting Scientist in the Electronic Image Processing Laboratories, Eastman Kodak Research, Rochester, NY, in 1991 and Visiting Professor at Mi- crosoft Research and Tsinghua University, Beijing, China, in 2000 and 2003, respectively. Since 1999, he has been full Professor in the Information and Communication Theory Group, Delft University of Technology. He is author of Iterative Identification and Restoration of Images (Norwell, MA: Kluwer, 1991) and a coauthor of Motion Analysis and Image Sequence Processing (Nor- well, MA: Kluwer, 1993) and Image and Video Databases: Restoration, Water- marking, and Retrieval (Amsterdam, The Netherlands: Elsevier, 2000). He was Associate Editor of Signal Processing: Image Communication. He has been in- volved in the conference organizing committees of ICIP2001, 2003, 2006, and 2011. At present his research interests include signal processing and communication theory, with emphasis on visual communications, compression, analysis, searching, and security. He is currently leading and actively involved in a number of projects in the field of intelligent information processing and data hiding for ad hoc and peer-to-peer multimedia communications.

Prof. Lagendijk was Associate Editor of the IEEE TRANSACTIONS ON IMAGE PROCESSING and IEEE TRANSACTIONS ON SIGNAL PROCESSING’s Supplement on Secure Digital Media. He is currently Associate Editor of the IEEE TRANSACTIONS ONINFORMATIONFORENSICS ANDSECURITY. He was a member of the IEEE Signal Processing Society’s Technical Committee on Image and Multidimensional Signal Processing.