1308 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 14, NO. 12, DECEMBER 2004

Optimal Watermark Detection Under Quantizationin the Transform Domain

Alexia Briassouli and Michael G. Strintzis, Fellow, IEEE

Abstract—The widespread use of digital multimedia data hasincreased the need for effective means of copyright protection.Watermarking has attracted much attention, as it allows the em-bedding of a signature in a digital document in an imperceptiblemanner. In practice, watermarking is subject to various attacks,intentional or unintentional, which degrade the embedded infor-mation, rendering it more difficult to detect. A very common, butnot malicious, attack is quantization, which is unavoidable forthe compression and transmission of digital data. The effect ofquantization attacks on the nearly optimal Cauchy watermarkdetector is examined in this paper. The quantization effects onthis detection scheme are examined theoretically, by treating thewatermark as a dither signal. The theory of dithered quantizers isused in order to correctly analyze the effect of quantization attackson the Cauchy watermark detector. The theoretical results areverified by experiments that demonstrate the quantization effectson the detection and error probabilities of the Cauchy detectionscheme.

Index Terms—Cauchy detector, dither theory, quantization at-tacks, watermarking.

I. INTRODUCTION

THE advances in multimedia processing allow text, audio,image, and video data to be represented, stored and dis-

tributed in digital format. However, the facility with which dig-ital information can be reproduced at a high quality has led tonumerous violations of copyright laws and intellectual propertyrights. This issue has attracted great attention, leading to the de-velopment of various copyright protection systems. A methodthat has gained considerable popularity is the watermarking ofdata. Watermarking is essentially the embedding of informationdirectly into the data to be protected, which in turn serves as ahost or a cover for that information. There exists both visibleand invisible watermarks, depending on the intended use of thedigital data. The former are usually used in preview images inthe World Wide Web or in image data bases to prevent theircommercial use. On the other hand, digital copyright systems

Manuscript received March 10, 2001; revised March 2, 2004. This work wassupported in part by the European Program ASPIS: An Authentication and Pro-tection Innovative Software System for DVDROM and Internet, InformationSociety Technologies (IST). This paper was recommended by Associate EditorR. L. Stevenson.

A. Briassouli was with the Informatics and Telematics Institute (ITI/CERTH),Thessaloniki, Greece. She is now with the Beckman Institute, Department ofElectrical and Computer Engineering, University of Illinois at Urbana-Cham-paign, Urbana, IL 61801 USA (e-mail: briassou@vision.ai.uiuc.edu).

M. G. Strintzis is with the Informatics and Telematics Institute (ITI/CERTH),Thessaloniki, Greece, and also with the Department of Electrical and Com-puter Engineering, University of Thessaloniki, Thessaloniki, Greece (e-mail:strintzi@eng.auth.gr).

Digital Object Identifier 10.1109/TCSVT.2004.836753

employ invisible watermarks [1], which are the concern of thepresent paper.

The watermark is a digital signal, usually containing in-formation about the data origin, destination, and owner, oradditional information concerning transaction dates, serialnumbers, etc., which can be useful when tracking illegal use ofdigital data [2], [3]. This information is contained in a secretkey, which is known by the legal owner of the watermarkeddata. Watermarking is different from other copyright protectionmethods, like steganography, as it needs to be robust againstmalicious attacks by unauthorized users, who try to remove orrender the digital signature undetectable. To ensure maximumrobustness, the watermark should be resilient against attackseven if the attacker knows the exact embedding process, butnot the cryptographically secure key. In cryptography, this isknown as Kerkhoff’s Law [3].

Digital data is subject to nonmalicious attacks, such as geo-metric transformations, analog-to-digital conversion, compres-sion, and quantization [4]. In addition to these benign attacks,which are simply caused by the processing and transmission ofthe data, many attacks are intentional, and aim at the removalof the watermark without destroying the host data, to achieveits illegal use. In this paper, we focus on quantization attacks,which are unavoidable, since quantization is essential for thecompression and transmission of digital data [4], [5]. Quantiza-tion partially destroys the watermark, making it more difficultto detect. Thus, it is of particular interest to study the effects ofquantization on watermarked data and examine the limits of per-formance of watermark detection schemes under quantization.

The effects of quantization attacks on watermarking schemeshave also been examined in [6] and [7], where the analysis ofthe detection robustness was based on the traditionally used cor-relator. This detector is widely used, although its performanceis optimal only in the case of Gaussian data [8], [9]. In [6], thedata used are discrete cosine transform (DCT) coefficients ofstill images, which do not follow a Gaussian but a much moreheavy tailed distribution [10]. In our analysis, we will also con-sider still image or video DCT coefficients, since watermarksare often embedded in this domain. These coefficients havebeen modeled more accurately with the generalized Gaussiandistribution by Hernandez [9], who also designed an improveddetector based on this model. However, the most importantimage (or video) DCT coefficients can be even more accuratelymodeled by the symmetric alpha-stable ( ) distribution[11], [12]. Thus, we have preferred to focus our analysis on adetector that is based on this statistical model [13].

It must be noted that the Cauchy detector is used and not ageneral detector, as the Cauchy and Gaussian distributions

1051-8215/04$20.00 © 2004 IEEE

BRIASSOULI AND STRINTZIS: OPTIMAL WATERMARK DETECTION UNDER QUANTIZATION IN THE TRANSFORM DOMAIN 1309

Fig. 1. Additive noise model of a quantizer.

are the only members of the family with a closed from ex-pression. Although the Cauchy detector is not fully optimal forgeneral data, it has proven to be particularly robust, evenwhen the given data deviates from the strictly Cauchy model.In particular, in [14] and [15], it is shown that this detector con-tinues to perform well even with non-Cauchy data, whereas theperformance of the Gaussian detector deteriorates visibly in anon-Gaussian environment. As a consequence, the Cauchy wa-termark detector is expected to remain nearly optimal even forgeneral data. The robustness of this detector and the ac-curacy of the data modeling motivated us to examine theperformance of Cauchy watermark detection under quantizationattacks.

The embedding of an additive watermark in the data fol-lowed by quantization can be considered as a case of ditheredquantization [16]. Dithered quantization is a technique in whicha low-power random or pseudorandom signal is purposefullyadded to a host signal prior to quantization. This intentional dis-tortion leads to a subjectively more pleasing reproduction of thehost signal after quantization and, in some cases, can actuallycause the quantization error to behave in a statistically desirableway. A watermark and a dither signal do not serve the same pur-pose, but their effect on the host data is the same. Consequently,known results for dithered quantization can be used to study wa-termarking, and especially the robustness of watermark detec-tion, under quantization.

This paper is organized as follows. Section II presents thetheory of dithered quantizers, which is used in the analysis ofthe quantization effects on watermarking. In Section III, spreadspectrum watermarking in the DCT-domain is described. Sec-tion IV presents the method of statistical watermark detection inthe transform domain. In particular, the nearly optimal Cauchydetector is presented and its performance under quantization isanalyzed. Section V contains experimental results and, finally,conclusions are drawn in Section VI.

II. DITHERED QUANTIZATION

A very basic step in lossy digital data compression andanalog-to-digital conversion of data is quantization. Thesimplest type of quantizer (Fig. 1) is the uniform quantizer,which maps the input to a collection of equally spacedoutput levels, giving the quantized output . Despitethe simplicity of the uniform quantizer, it has proven difficultto analyze theoretically, because of its inherent nonlinearity[17]. A common misconception is that the quantization error

consists of a sequence of independentidentically distributed (i.i.d.) random variables, uncorrelatedwith each other and with the input signal. Actually, this approx-imation, known as the Bennett approximation [18], approaches

Fig. 2. NSD quantizer.

Fig. 3. SD quantizer.

accuracy only when the the number of quantizer levels is verylarge, or the quantization step size is very small. In manymodern applications, this condition is definitely not met. A wayto force the Bennett conditions and, thus, temper the statisticalproperties of the error signal, is to use a dither signal, which isadded to the host signal before quantization to “break up” thepattern of the quantization noise and render the reconstructedsignal perceptually more pleasing. In this way, the output signalbecomes more smooth, leading to a smoother quantizationerror. More importantly, the resulting quantization error isindependent of the input signal and consequently becomesless audible or visible [16]. The addition of a watermark todigital multimedia data before quantization is very similar tothis procedure [7], so the quantization of watermarked data canindeed be considered as a case of dithered quantization.

Dithered quantizing systems are shown in Figs. 2 and 3. Inboth schemes, the system input is the original signal whilethe quantizer input is . The dither signal is con-sidered to be a strict-sense stationary random process [17], in-dependent of . Fig. 2 represents a nonsubtractively dithered(NSD) system, while a subtractively dithered (SD) system ispresented in Fig. 3, in which the dither signal is subtracted fromthe output of the dithered quantizer. Naturally, in the second caseit is assumed that the dither signal must be available at both endsof the channel (Fig. 3). The total error of these schemes isdefined as the difference between the original system input andthe final output, while the quantization error is the differencebetween the quantizer input and output. These errors are not thesame for subtractively and NSDs. In particular, the total errorof a SD, also referred to as quantization noise in [16], is the dif-ference of the original input and the final output

(1)

In this case, the total error of the SD system happens to be equalto the quantization error of a NSD, a very interesting and usefulresult, as we will show later on. The total error for a NSD is

(2)

which is obviously different from the total error of an SDsystem.

The purpose of the dither signal is to render the quantizationerror uniform and independent of the input signal, which can beachieved if certain conditions, developed by Schuchman [19],

1310 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 14, NO. 12, DECEMBER 2004

are met. Specifically, Schuchman’s conditions require that thequantizer does not overload, meaning that the quantization errornever exceeds , and the characteristic function of the dithersignal, defined as

(3)

has the following property:

(4)

If these conditions are met, Schuchman proved that the totalerror of a SD quantizing system is uniformly distributedon , and statistically independent of the originalinput signal .

Gray and Stockham [16] derived the following necessary andsufficient condition for the th moment of the nonsubtractivequantization error to be independent of the input signal :

(5)

where is a random variable uniformly distributed on, and independent of . This random vari-

able has no special meaning, but it is necessary to formulatecondition (5). As mentioned in [6], these properties of thequantization and the total errors are valid for uniform scalarquantizers where overload does not occur, which is the caseexamined here.

These results are particularly important for the analysis inSection IV, where the effect of the quantization errors on the wa-termark detector are examined. The quantization error of a wa-termarking system under quantization is identical to the quanti-zation error of a NSD quantizer, since the dither signal, which isthe watermark, is not subtracted from the system output. Thus,the quantization error of the watermarked signal is

(6)

However, as (6) shows, the quantization error of an NSD systemis identical to the total error of an SD quantizer given in (1).Consequently, Schuchman’s Conditions hold for the watermarkquantization error that interests us (the quantization error of aNSD quantizer), since it is equal to the total error of an SDsystem. Therefore, the watermarking quantization error will bevalidly considered in the sequel as a sequence of i.i.d. uniformlydistributed random variables in , independent ofthe input signal and white.

III. DCT-DOMAIN WATERMARKING

In this section, we briefly describe the watermark generationand embedding process used for image or video data in the DCTdomain [20]. Most watermarking systems are inspired by thespread spectrum modulation schemes used in digital commu-nications in jamming environments [21]–[23]. The role of thejammer in the watermarking problem is assumed by the hostsignal, which is equivalent to additive noise, and by the at-tacker who unintentionally or intentionally tries to destroy orextract the embedded watermark [2], while the watermark is thehidden information signal. Spread spectrum techniques are so

Fig. 4. Watermark embedding process.

popular for watermarking because they exhibit low probabilityof intercept (LPI), which in the case of watermarking translatesto high imperceptibility [8]. Increased robustness of the water-marking system is ensured by the anti-jamming properties ofspread spectrum systems. The pseudorandom modulation of thewatermark, described analytically in Section III-A, increases theachieved security, since knowledge of this sequence is requiredfor successful watermark retrieval [2].

A. Watermark Generation and Embedding

If an image is represented as a discrete two-dimensional (2-D)sequence of its luminance component with pixels,the watermark can be considered as a 2-D DCT signal ,which is added to the DCT coefficients. Indexes in boldfacetypesetting represent the corresponding 2-D indexes, so

and . The DCT can either be applied to theentire image or in blocks. We use the blockwise DCT because itis used much more widely in practice and because the blockwiseapplication of the DCT is faster than the full-image DCT. In ourcase, it also happens to allow the statistical characterization ofeach DCT coefficient.

The watermark may or may not contain a message ,which is mapped by an encoder to an -dimensional codewordvector (Fig. 4). In the frequently considered case where thewatermark does not contain a message the codeword vector,reduces to [9]. Every element of the codeword vectoris then repeated in a different set of image or video pixelsin the DCT domain, in an expansion process, which spreads itover all the pixels. It must be noted that this repetition of theinformation bits introduces a certain degree of redundancy inthe watermark, which is desirable, as it increases its robustness.In the case of watermark detection, there is no hidden message,so the element repeated over all the pixels is simply .

A direct spread spectrum modulated watermark is then cre-ated from the signal resulting from the above expansion process,by multiplying it in a pixel-by-pixel manner with an appro-priate 2-D pseudorandom sequence . The security of theentire system is ensured by the initializing seed to the pseudo-random noise generator: this key is cryptographically secure andis known only by the legal owner of the watermarked document.Without knowledge of the secret key, the generation of the wa-termark at the receiver is impossible.

The spreading sequence must have noise-like proper-ties, in order to spread the spectrum of the input signal ,

BRIASSOULI AND STRINTZIS: OPTIMAL WATERMARK DETECTION UNDER QUANTIZATION IN THE TRANSFORM DOMAIN 1311

so its mean should be approximately zero and its auto-correla-tion should approach the delta function. For the system consid-ered here, the pseudorandom sequence is bipolar, i.e. it takes thevalues , with relative frequencies 1/2 each. The multi-plication of the signal with this sequence spreads its power overa wide bandwidth, increasing its invisibility and robustness.

In order to guarantee the invisibility of the alterations intro-duced by the watermark, the spread signal is multiplied by a per-ceptual mask , that takes into account the properties of thehuman visual system [24], [25]. In order to design it, one mustestimate the visibility threshold for every DCT co-efficient of each 8 8 block, which is approximated in loga-rithmic units by the following function

(7)

where and are the vertical and horizontal spatial fre-quencies, respectively, (in cycles/degree) of the DCT basis func-tions and is the minimum value of the quadratic function

associated with . It must be noted that this model isvalid for the ac frequencies only and not for the dc coefficient.The threshold is corrected for every block through thefollowing equation:

(8)

where is the dc coefficient for each block and isthe average luminance of the screen. These two functions con-tain parameters which are set to , ,

, cycles/degrees and ,following [25] and the perceptual mask is obtained from the vis-ibility threshold according to the following equation:

(9)

where mod 8, mod 8, is the Kroneckerfunction and is a scaling factor. This mask ensures thatthe watermark will remain imperceptible to the human eye, butat the same time it will alter the pixel values as much as possiblein order to achieve maximum robustness.

The resulting watermark, for the case where , isand is added to the original DCT coefficients ,

giving the watermarked signal (Fig. 4).The only way for the attacker to retrieve the hidden signal is tohave a copy of the pseudorandom sequence, in other words itsseed and the procedure through which it is generated. This way,the attacker cannot extract the watermark without knowledge ofthe secret key (the pseudorandom sequence seed), even if theentire watermark generation and embedding process is known.Thus, the dependence of the generated watermark on the secretcryptographic key and the spread spectrum embedding proce-dure result in a robust watermark, which is resilient against stan-dard signal processing and manipulation as well as intentionalattacks.

IV. WATERMARK DETECTION AFTER QUANTIZATION

A. Statistical Watermark Detection

Traditionally, detectors correlate the given data with theknown watermark. Alternatively, the detection problem can beformulated as a binary hypothesis test [26]. Although the givendata for this test possibly contains a watermark, the actualtest is performed without knowledge of the original, unwater-marked data. Thus, this technique actually is a blind watermarkdetection method. The analysis that follows can be applied toany kind of data, but we will focus on image and video DCTcoefficients, in which watermarks are often embedded. The twohypotheses for the test are formulated as follows:

(10)

In this detection problem, the watermark is the desiredsignal or information, while the DCT image or video coeffi-cients , i.e. host signal, play the role of unknown additivenoise with known statistical properties. The goal of the water-mark detector is to verify whether or not a watermark exists inthe given data, based on the statistical properties of the (pos-sibly) watermarked signal that is received, and the known wa-termark. A basic assumption is that the statistical distribution ofthe DCT coefficients is not significantly altered after the inser-tion of a watermark. This assumption is intuitively justified bythe fact that the watermarked signal should follow a statisticaldistribution similar to that of the original signal, if the embeddedsignal is imperceptible.

We assume that the data has been modeled by an appropriatestatistical distribution, so the probability density functions(pdfs) of the host signal and its watermark are known or can bevery accurately approximated from the given data. In that case,the detection rule is

(11)

where the likelihood ratio is defined as

(12)

In practice, the log-likelihood ratio is usually preferred to per-form hypothesis testing. The log-likelihood ratio is simply de-fined as the natural logarithm of the likelihood ratio,

, given by

(13)

Neyman–Pearson testing is performed, where is comparedto a threshold , which is determined by the probability of false-alarm , i.e. the probability of detecting a watermark in unwa-termarked data (Fig.5). The exact relation of this threshold to theoriginal data can be formulated analytically and is presented inthe following section.

B. Cauchy Detector

The hypothesis test described above depends on the statisticalmodel used for the data, so it is necessary to model the data

1312 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 14, NO. 12, DECEMBER 2004

Fig. 5. Watermark detection process.

by a suitable pdf to be used in the design of the detector. Theparameters of this model can be estimated with accuracy fromthe given data [27], as it will become clear later on.

It is well known that DCT image or video coefficients, are notaccurately modeled by the Gaussian distribution, since they dis-play heavy tails [28], [29]. The use of more appropriate models,such as the generalized Gaussian distribution [30] has been sug-gested in the literature. However, other heavy tailed distribu-tions, like the members of the alpha-stable family, have beenfound to be more suitable. The family is often used todescribe non-Gaussian signals, which are characterized by animpulsive nature [11]. Because of their impulsiveness, thesesignals exhibit distributions with heavy tails, like the low-fre-quency DCT coefficients.

The stable distribution can be best described by its character-istic function

(14)

where

forfor (15)

The stable distribution is completely described by the followingfour parameters [31]: the location parameter

, the scale parameter , which is known as thedispersion, the index of skewness , and thecharacteristic exponent . We will examine the

distribution, which has . The location parametergives the distribution mean for , while forit gives the median. The dispersion can be any positive numberand corresponds to the spread around the location parameter, in other words it behaves like the variance of the Gaussian

distribution [32].The characteristic exponent determines the shape of the dis-

tribution as it measures the “thickness” of the tails of the pdf.For smaller values of , the tails of the distribution are heavierand the corresponding random process displays high impulsive-ness, while increasing values of correspond to distributionswith rapidly decaying tails, that approach the Gaussian case.There are closed-form expressions for the pdf of randomvariables only for and , which correspond to theGaussian and Cauchy distributions, respectively [11]. A Cauchydetector has been designed for the case that in [13].Naturally, this detector is not optimal for all cases, since thedata is generally and not specifically Cauchy. However,the Cauchy detector has proven to be particularly robust even

if the input data has a shape factor that deviates from[14], [15]. It is certainly better than the traditional correlator, asit correctly assumes a heavy tailed model for the data.

The Cauchy model has the following pdf:

(16)

where is the data dispersion and is the location parameter.The corresponding Cauchy detector can be derived by replacing(16) in (13). In this case, the hypothesis test presented in (10)uses the pdfs

(17)

In the sequel, the 2-D index will be omitted for notational sim-plicity. If the set of DCT coefficients that contain a watermarkis denoted by , the likelihood ratio becomes

and the corresponding log-likelihood ratio is

(18)

This ratio will be compared against a suitably chosen thresholdin order to determine the existence of a watermark. The low- andmid-frequency DCT coefficients are very accurately modeled by

distributions which may deviate slightly from the Cauchymodel. Despite this, it has been shown [14] that the performanceof this detector is not significantly affected by such deviations,so its results remain very accurate.

C. Effect of Quantization on the Cauchy Detector

Quantized data is corrupted by quantization errors and, corresponding to watermarked and unwatermarked data,

respectively. These errors are defined as the difference betweenthe quantizer input and output

(19)

which satisfy Schuchman’s conditions, as seen in Section II.Consequently, the watermarking quantization error of (19)will be statistically independent of the input signal, i.i.d., anduniformly distributed on . These properties of the

BRIASSOULI AND STRINTZIS: OPTIMAL WATERMARK DETECTION UNDER QUANTIZATION IN THE TRANSFORM DOMAIN 1313

quantization error are of essential importance for the analysisthat follows.

Taking the quantization errors into account, the binary hy-pothesis test of (10) becomes

(20)

Thus, the corresponding log-likelihood ratio under is

(21)

and under

(22)

The likelihood functions (21) and (22) are sums of many i.i.d.random variables, so according to the central limit theorem, theycan be approximated by a Gaussian variable of mean and vari-ance that can be estimated from the data. This approximationis accurate in practice, since a large number of coefficients aresummed in the log-likelihood ratio test.

As already detailed in Section III-A, the watermark isequal to the value of the perceptual mask multiplied bythe pseudorandom sequence . Thepseudorandom sequence used is modeled as an i.i.d. two—di-mensional random process with a two-level equiprobablediscrete marginal distribution, . The perceptualmask, which defines the watermark amplitude , is designedaccording to the properties of the human visual system asdescribed in Section III-A. Thus, the expected value of thewatermark at every pixel is or , with probability1/2 each. It must also be noted that the watermark is supposedto be embedded in a set S of the DCT coefficients, so only thesecoefficients are taken into account.

The mean of (21) can then be found from

(23)

where is the joint probability density function of a multi-dimensional random variable . The above should be under-stood as a short notation for a multivariable integral andas a short notation for . In our case, there are tworandom variables, the watermark and the quantization error. Thepdf of the latter is since the quantization error hasbeen considered to be uniformly distributed in .The watermark is or for every pixel , with proba-bility 1/2 each. Incorporating this in (21) and omitting the index

gives

(24)

It can be found that

(25)

In the interval , the above integral becomes

(26)

In view of (26), with replaced by , ,or as appropriate, (24) becomes

(27)From this equation, the mean of the log-likelihood ratiocan be computed under hypothesis . The variance

can be similarly computed from

(28)

where

(29)Considering, as before, that the watermark is a sequence of i.i.d.random variables with the equiprobable valuesfor every pixel , and also that the quantization error is a uni-formly distributed random variable on , we obtain

(30)

Since the quantization error is uniformly distributed on, the following integral will need to be computed:

(31)

This integral is computed numerically in the experiments and isleft as is in the equations that follow. Under the same assump-tions as before for the watermark, the variance is finally givenby

(32)

1314 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 14, NO. 12, DECEMBER 2004

With these quantities known and since follows a Gaussiandistribution, the probabilities of detection and false alarm for theratio are given by

(33)

(34)

where is the threshold against which the data are comparedand is defined as

(35)

For a given probability of false-alarm , we can compute therequired threshold for a watermark to be detected

(36)

By using this known threshold, we can find the relation betweenand , which gives the receiver operating characteristic

(ROC) curves

(37)

These curves give a measure of the performance of the proposeddetector, both in the absence and in the presence of quantiza-tion. The probability of error for the detector can also be definedas the probability of false alarm and the probability of not de-tecting an existing watermark [7]. Using this error metric, onecan measure the performance of the detector under quantiza-tion attacks as the probability of error versus the embedded wa-termark strength or the quantizer step size. Thus, one can ex-amine the effect of quantization of varying strength on the wa-termarked data for a given probability of false alarm in terms ofdetection or error probability.

V. EXPERIMENTAL RESULTS

The theoretical analysis of the previous section can be usedto examine the influence of quantization on watermarked imageor video data. It is possible to measure the performance of thedetector in the presence of quantization for watermarks of in-creasing strength. The robustness of the Cauchy detector in thepresence of quantization attacks can be predicted and the min-imum watermark strength needed to overcome the quantiza-tion error can be estimated. Additionally, in the experimentsthat will be described, the theoretical results of Section IV-Cwill be verified. This is very important, as these theoretical re-sults allow the accurate prediction of the performance of water-marking schemes under quantization attacks.

The analysis of the previous sections can be applied toany kind of host and watermark signals, characterized by the

heavy-tailed Cauchy or distribution described in Sec-tion IV-B. The blockwise DCT transform is applied to the hostdata in 8 8 blocks. The coefficients corresponding to everyfrequency are arranged in order of importance using the zig-zagscan and are examined separately. In particular, the DCTcoefficients for the same frequency over all the image blocksare grouped in a sample sequence, which can be considereda subsignal. This allows the focus of the experimental anal-ysis on the most important coefficients, which are the low- andmid-frequency ones. For our experiments we use low-frequencyDCT coefficients of an MPEG-4 video object, and in particularthe fifth DCT coefficients of the foreground object from anI-frame of the Akiyo video sequence. These coefficients, orsubchannels, follow a heavy-tailed distribution for which thedetector presented in Section IV-B is nearly optimal.

The analysis of the quantization attacks on the host data is car-ried out assuming uniform scalar quantization of each data set.Such quantization is common in practice and is in fact used forthe JPEG standard for still images, as well as the MPEG stan-dards for video sequences, which assign different but equal stepsizes to every DCT coefficient. The performance of an actualdetector for all the DCT coefficients is expected to be very sim-ilar to that of the detector for each subchannel, and especiallythe low and mid frequency ones examined here, since they arethe most important ones.

A. Detector Robustness for Varying Quantization Attacks

Initially, the expected mean and variance of the log-likelihoodratio (18) under quantization are computed from (27) and (32).The probability of false alarm is set to to reduce thenumber of free parameters. Using the estimated values of ,

, and the , the detection threshold is computed via (36). Fi-nally, the probability of detection is computed from (34). Thisis repeated for increasing quantization step sizes and for a par-ticular watermark. The watermark-to-document ratio (WDR) isdefined by

(38)

where and represent the variance or energy of the wa-termark and the host document respectively. This ratio is set to

23.6163 dB, so that the watermark power is low rel-ative to the document energy [6].

The calculation of the detection probability for decreasingquantization step sizes leads to the curves depicted in Fig. 6(a)and (b) on a linear and logarithmic scale respectively. As thesefigures show, the detection probability indeed decreases whenthe data is very coarsely quantized, but increases as the quanti-zation step becomes smaller. Thus the detection probability in-creases for higher values of , which correspond to smallerquantization step sizes, and decreases in the opposite case. Asthese figures show, the probability of detection is above 0.90 forvalues of above 0.75. The detector can consequently beconsidered very reliable for these quantization step sizes, sincethe detection probability is very high, although the probabilityof false-alarm 10 is very low.

BRIASSOULI AND STRINTZIS: OPTIMAL WATERMARK DETECTION UNDER QUANTIZATION IN THE TRANSFORM DOMAIN 1315

Fig. 6. Probability of detection for varying quantization step sizes, fixed P ,and WDR for the fifth DCT coefficient of Akiyo.

Another way to examine the detector performance is throughthe evaluation of the error probability. The error probability isdefined as the probability of detecting a nonexistent watermark

, known as the probability of false alarm plus the probabilityof “missing” a watermark, multiplied by the respective apriori probabilities for and . In the case examined here,both hypotheses have equal a priori probabilities 1/2, so the errorprobability is

(39)

This quantity is computed using (33) and (34) and plottedagainst decreasing quantization step sizes, or increasing

, in Fig. 7(a) and (b). The false-alarm probabilityis set to 10 and the same watermark, with

23.6163 dB, is embedded in the host signal.In this case, the error probability is very high for high quantiza-tion step sizes or low normalized watermark standard deviation.As before, the error probability becomes very small when

is above 0.75. Fig. 7(a) and (b) shows that the proba-bility of error becomes close to 0 when the quantization stepsize decreases and the corresponding normalized watermarkstandard deviation increases.

Fig. 7. Probability of error for varying quantization step sizes, fixed P , andWDR for the fifth DCT coefficient of Akiyo.

B. Detector Robustness for Watermarks of Varying Strength

The detector performance is also examined for a spe-cific quantization attack and watermarks of varying strength.Fig. 8(a) and (b) depicts the detection probabilities for valuesof ranging from about 35 to 10 dB and for fixedto 10 . The probability of detection is quite low for smallvalues of WDR, which is logical since the embedded water-mark is very weak in these cases. However, for WDR aboveabout 18 dB the detection probability is very high, above0.90. These values of WDR are quite usual, as watermarks withsimilar values of WDR have been used in [6]. Thus, the detectorcan be considered reliable in the presence of quantization sinceits performance is very good for watermarks often used in prac-tice. This is also evident in Fig. 9(a) and (b), where the errorprobability of (39) is plotted for the same simulation settings.The error probability is very high, close to 0.50, for very weakwatermarks, but it becomes very low, below 0.10, when theWDR is above 18 dB, as expected from the previous results.

C. Experimental Values of and

The results of (27) and (32) can be verified experimentally bycalculating the mean and variance of the log-likelihood ratio ex-

1316 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 14, NO. 12, DECEMBER 2004

Fig. 8. Probability of detection for watermarks of varying strength and fixedP for the fifth DCT coefficient of Akiyo.

perimentally as well as theoretically. In particular, Monte Carloexperiments are carried out by embedding a large number ofpseudorandomly generated watermarks in the host data with theprocedure described in Section III. For the Monte Carlo tests,200 watermarks are generated from the corresponding pseudo-random sequences. The log-likelihood ratio is then calculatedfor every one of these experiments and its mean and varianceare finally computed and compared against the theoretically es-timated values. To verify the robustness of the results and toform a better picture of the accuracy of the proposed estima-tors, we conduct such Monte Carlo experiments for watermarksof varying strength. In particular, different watermarks are em-bedded in the host data prior to quantization and the log-likeli-hood ratio is computed with (18). The mean and variance of thelog-likelihood ratios of every watermark are found experimen-tally from the Monte Carlo runs, as well as theoretically from(27) and (32). In Fig. 10(a) and (b), it can be seen that the the-oretical estimates of the log-likelihood ratio mean and varianceare accurate for watermarks of varying strength. This can alsobe seen in Table I, which shows the values represented graphi-cally in Fig. 10. These results show that the estimator is almostalways very accurate, since very small differences between thetheoretical and experimental estimates are observed.

Fig. 9. Probability of error for watermarks of varying strength and fixed Pfor the fifth DCT coefficient of Akiyo.

The accuracy of the estimates of and is especially im-portant for the verification of the curves of Figs. 6–9 and 11.The probabilities of false-alarm, detection and consequently theerror probability too, all depend on the values of these parame-ters, as can be seen from (33), (34), and (37). Thus, if the numberof free parameters is reduced, by predefining the probability offalse alarm for example, these probabilities depend solely onthe ratio mean and variance. Since the experimentally computedmeans and variances practically coincide with their theoreti-cally computed values, it is obvious that the experimental errorand detection probabilities will be very close to their theoreticvalues. Consequently, the accuracy of (27) and (32) permits thetheoretic investigation of the detector performance under quan-tization and the comparison of various watermarking schemes.

D. ROC Curves Under Quantization

The relation of the detection probability with the probabilityof false alarm is depicted in the ROC curves of Fig. 11(a) and(b). The WDR is set to 28.689 dB in this case, whilethe takes values from 10 to 10 . For the very smallvalues of , close to 10 , the detection probability for un-quantized data is relatively high, remaining above 0.75–0.80,but the effects of quantization are quite serious, as the values of

BRIASSOULI AND STRINTZIS: OPTIMAL WATERMARK DETECTION UNDER QUANTIZATION IN THE TRANSFORM DOMAIN 1317

Fig. 10. Theoretical and experimental values of the log-likelihood ratio meanand variance.

TABLE ITHEORETICAL AND EXPERIMENTAL VALUES OF THE LOG-LIKELIHOOD RATIO

MEAN AND VARIANCE

for quantized data are below 0.70. However, this is the caseonly for very small values of . As Fig. 11(a) and (b) shows,

Fig. 11. ROC curves: probability of detection versus probability of false alarmfor quantized and unquantized data from the fifth DCT coefficient of Akiyo.

the detection probability is above 0.80 for above ,where the corresponding detection probability without quan-tization is above 0.90. Very accurate values of , even inthe presence of quantization, are finally obtained for above

, where the probability of false alarm for quantized datais above 0.90.

From these results one can conclude that quantization doesaffect the detector performance, as there is a decrease of the de-tection probability in comparison to the detection probability forunquantized data. However, as it is obvious from the previousdiscussion and Fig. 11( a) and (b), the detector is not rendereduseless in the presence of quantization. On the contrary, veryhigh detection probabilities can still be obtained, as long as theprobability of false alarm is not very low. As seen in the previousexperiments, the performance of the detector is very good evenfor values of as low as 10 . Thus, its results can be con-sidered reliable in the presence of quantization, even for quiterestrictive probabilities of false alarm. Naturally, this increasesthe detector’s usefulness and reliability, as its performance doesnot deteriorate under quantization. This analysis finally permitsus to estimate the detector performance a priori and predict thedetection results under various restrictions.

In order to verify the ROC curves, we conduct experimentswith 200 pseudorandomly generated watermarks and compare

1318 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 14, NO. 12, DECEMBER 2004

Fig. 12. Experimental and theoretical ROC curves for quantized data from thefifth DCT coefficient of Akiyo.

the actual detection probabilities with their theoretical values.This number of Monte Carlo runs actually permits the verifica-tion of probabilities down to , which is why there arefewer experimental results for low values of . Fig. 12(a) and(b) depicts the theoretically estimated ROC curves for quantizeddata as well as the corresponding experimental results. The the-oretical curves are derived from (37) using (27) and (32) forand , respectively. The empirical ROC curves are computedusing the experimental values of the likelihood ratio mean andvariance found from the Monte Carlo runs. As it is obvious fromFig. 12( a) and (b), the experimental results follow the theoret-ical ROC curves quite closely. Thus, the theoretically predictedROC curves and in general the prediction of error probabilitiesin the presence of quantization can be considered reliable, sincethey coincide with the actual empirical values. Finally, these re-sults confirm the theoretical analysis of the effects of quantiza-tion on improved watermark detection of Section IV.

VI. CONCLUSION

In this paper, we examined the effects of the very commonand nonmalicious quantization attack on watermarkingschemes. We focused on the effects of the often used uni-form scalar quantization on the detection of such a watermark.

Specifically, we analyzed the effects of quantization on thenearly optimal Cauchy watermark detector, which is moreaccurate than the traditionally used correlation detector. Forour analysis, we also used certain results from the theory ofdithered quantizers and, thus, derived theoretical expressionsfor the effect of quantization on the improved watermarkdetector. These theoretical expressions of the dependenciesbetween the quantization strength and the error and detectionprobabilities were used to examine the effects of a specificquantization to watermarks of varying embedding strength andthe effects of varying quantization step sizes. It was shown thatin the presence of quantization there is a slight degradation ofthe detector performance, which depends on the watermarkstrength and the quantization step size. Actually, it was foundthat if the watermark is not very weak and the quantizationis not very coarse, the detector remains reliable, giving quitehigh detection probabilities. The theoretical analysis was alsoverified experimentally, providing empirical measures of theCauchy detector performance in the presence of quantization.

Consequently, one can accurately predict the detector perfor-mance in the presence of quantization, even before actual ex-periments are conducted. Since the theoretical results are reli-able, they can be used to predict the expected detection and errorprobabilities for a given watermark. Finally, it is also possible toknow a priori the watermark strength necessary to achieve a cer-tain probability of detection, false alarm and/or error. Such re-liable theoretical results are very important, as the performanceof most watermarking schemes under various attacks, such asthe quantization attack examined here, is usually evaluated onlyempirically.

REFERENCES

[1] M. D. Swanson, B. Zhu, and A. H. Tewfik, “Transparent robust imagewatermarking,” in Proc. IEEE Int. Conf. Image Processing, Lausanne,Switzerland, 1996, pp. 211–214.

[2] F. Hartung and M. Kutter, “Multimedia watermarking techniques,” Proc.IEEE, vol. 87, pp. 1079–1107, July 1999.

[3] R. B. Wolfgang, C. I. Podilchuk, and E. J. Delp, “Perceptual watermarksfor digital images,” Proc. IEEE, vol. 87, pp. 1108–1126, July 1999.

[4] M. G. Strintzis and D. Tzovaras, “Optimal construction of subbandcoders using lloyd-max quantizers,” IEEE Trans. Image Processing,vol. 7, pp. 649–668, May 1998.

[5] , “Optimal pyramidal decomposition for progressive multi-dimen-sional signal coding using optimal quantizers,” IEEE Trans. Signal Pro-cessing, vol. 46, pp. 1054–1069, Apr. 1998.

[6] J. J. Eggers and B. Girod, “Quantization effects on digital watermarks,”Signal Processing, vol. 81, no. 3, 2001.

[7] , “Watermark detection after quantization attacks,” presented at the3rd Workshop on Information Hiding, Dresden, Germany, Sept./Oct.1999.

[8] J. G. Proakis, Digital Communications. New York: McGraw-Hill,1995.

[9] J. R. Hernandez, M. Amado, and F. Perez-Gonzalez, “DCT-domain wa-termarking techniques for still images: detector performance analysisand a new structure,” IEEE Trans. Image Processing, vol. 9, pp. 55–68,Jan. 2000.

[10] R. C. Reininger and J. D. Gibson, “Distributions of the two-dimensionalDCT coefficients for images,” IEEE Trans. Commun., vol. COMM-31,pp. 835–839, 1983.

[11] M. Shao and C. L. Nikias, “On Symmetric Stable Models for ImpulsiveNoise,” Univ. Southern California, Los Angeles, Tech. Rep. USC-SIPI-231, Feb. 1993.

[12] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Pro-cesses: Stochastic Models With Infinite Variance. New York: Chapmanand Hall, 1994.

BRIASSOULI AND STRINTZIS: OPTIMAL WATERMARK DETECTION UNDER QUANTIZATION IN THE TRANSFORM DOMAIN 1319

[13] A. Briassouli, P. Tsakalides, and A. Stouraitis, “Hidden messagesin heavy-tails: DCT-domain watermark detection using alpha-stablemodels,” IEEE Trans. Multimedia, to be published.

[14] P. Tsakalides, “Array Signal Processing With Alpha-Stable Distribu-tions,” Ph.D. dissertation, Univ. Southern California, Los Angeles, CA,Dec. 1995.

[15] G. A. Tsihrintzis and C. L. Nikias, “Data adaptive algorithms for signaldetection in subgaussian impulsive interference,” IEEE Trans. SignalProcessing, vol. 45, pp. 1873–1878, July 1997.

[16] R. M. Gray, “Dithered quantizers,” IEEE Trans. Inform. Theory, vol. 39,pp. 805–812, Mar. 1993.

[17] R. A. Wannamaker, S. P. Lipshitz, J. Vanderkooy, and J. N. Wright, “Atheory of nonsubtractive dither,” IEEE Trans. Signal Processing, vol. 48,pp. 499–516, Feb. 2000.

[18] W. R. Bennett, “Spectra of quantized signals,” Bell Syst. Tech. J., vol.27, pp. 446–472, July 1948.

[19] L. Schuchman, “Dither signals and their effect on quantization noise,”IEEE Trans. Commun. Technol., vol. COMM-12, pp. 162–165, 1964.

[20] D. Tzovaras, N. Karagiannis, and M. G. Strintzis, “Robust imagewatermarking in the subband or discrete cosine transform domain,” inProc. Eur. Signal Processing Conf. (EUSIPCO), Greece, Sept. 1998,pp. 2285–2288.

[21] I. J. Cox, J. Kilian, F. T. Leighton, and T. Shamoon, “Secure spreadspectrum perceptual watermarking for images, audio and video,” IEEETrans. Image Processing, vol. 6, pp. 1673–1687, Dec. 1997.

[22] I. J. Cox, M. L. Miller, and A. McKellips, “Watermarking as commu-nications with side information,” Proc. IEEE, vol. 87, pp. 1127–1141,July 1999.

[23] J. R. Hernandez, F. Perez-Gonzalez, J. M. Rodriguez, and G. Nieto, “Per-formance analysis of a 2D-multipulse modulation scheme for data hidingand watermarking of still images,” IEEE J. Select. Areas Commun., vol.16, pp. 510–524, May 1998.

[24] J. A. Solomon, A. B. Watson, and A. J. Ahumada, “Visibility of DCTbasis functions: Effects of contrast masking,” in Proc. Data CompressionConf., Snowbird, UT, 1994, pp. 361–370.

[25] A. J. Ahumada and H. A. Peterson, “Luminance-model-based DCTquantization for image compression,” Proc. SPIE, vol. 1666, pp.365–374, 1992.

[26] A. Papoulis, Probability, Random Variables, and Stochastic Processes,2nd ed. New York: McGraw-Hill, 1987.

[27] J. P. Nolan, “Maximum likelihood estimation and diagnostics for stabledistributions,” Dept. of Math. and Stat., American Univ., Washingon,DC, June 1999.

[28] J. R. Hernandez and F. Perez-Gonzalez, “Statistical analysis of water-marking schemes for copyright protection of still images,” Proc. IEEE,vol. 87, pp. 1142–1166, July 1999.

[29] K. A. Birney and T. R. Fischer, “On the modeling of DCT and subbandimage data for compression,” IEEE Trans. Image Processing, vol. 4, pp.186–193, Feb. 1995.

[30] F. Müller, “Distribution shape of two-dimensional DCT coefficients ofnatural images,” Electron. Lett., vol. 29, pp. 1935–1936, Oct. 1993.

[31] E. F. Fama and R. Roll, “Some properties of symmetric stable distribu-tions,” J. Amer. Statist. Assoc., vol. 63, pp. 817–836, 1968.

[32] S. Cambanis, G. Samorodnitsky, and M. S. Taqqu, Eds., Stable Processesand Related Topics. Boston, MA: Birkhauser, 1991.

Alexia Briassouli recevied the B.S. degree in electrical engineering from theNational Technical University of Athens, Athens, Greece, in 1999 and the M.S.degree in image and signal processing from the University of Patras, Patras,Greece, in 2000. She is currently working toward the Ph.D. degree in electricalengineering at the University of Illinois at Urbana-Champaign, Urbana.

From 2000 to 2001, she worked as a Research Assistant at the Informatics andTelematics Institute, Thessaloniki [Center of Research and Technology Hellas(CERTH)], Greece, participating in a European-funded research project. Herresearch interests lie in the fields of statistical signal processing and image pro-cessing. She has worked on the design of optimal watermark embedding anddetection systems for images and video that are robust to various attacks. Hercurrent research interests include statistical image processing and computer vi-sion, and problems such as motion estimation and segmentation for video.

Michael G. Strintzis (M’70–SM’80–F’04) received the Diploma degreein electrical engineering from the National Technical University of Athens,Athens, Greece, in 1967, and the M.A. and Ph.D. degrees in electricalengineering from Princeton University, Princeton, NJ, in 1969 and 1970,respectively.

He then joined the Electrical Engineering Department, University of Pitts-burgh, Pittsburgh, PA, where he served as an Assistant Professor (1970–1976)and an Associate Professor (1976–1980). Since 1980, he has been a Professor ofelectrical and computer engineering with the University of Thessaloniki, Thes-saloniki, Greece, and, since 1999, Director of the Informatics and TelematicsResearch Institute, Thessaloniki. His current research interests include 2-D and3-D image coding, image processing, biomedical signal and image processing,and DVD and Internet data authentication and copy protection.

Dr. Strintzis has been serving as an Associate Editor of the IEEETRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY since1999. In 1984, he was the recipient of a Centennial Medal of the IEEE.