700 IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 7, NO. 4, …users.ics.forth.gr/tsakalid/PAPERS/2005-Aug-MM-IEEETran.pdf · 700 IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 7, NO. 4, AUGUST 2005

700 IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 7, NO. 4, AUGUST 2005

Hidden Messages in Heavy-Tails: DCT-DomainWatermark Detection Using Alpha-Stable Models

Alexia Briassouli, Panagiotis Tsakalides, Member, IEEE, and Athanasios Stouraitis, Senior Member, IEEE

Abstract—This paper addresses issues that arise in copyrightprotection systems of digital images, which employ blind water-mark verification structures in the discrete cosine transform (DCT)domain. First, we observe that statistical distributions with heavyalgebraic tails, such as the alpha-stable family, are in many casesmore accurate modeling tools for the DCT coefficients of JPEG-analyzed images than families with exponential tails such as thegeneralized Gaussian. Motivated by our modeling results, we thendesign a new processor for blind watermark detection using theCauchy member of the alpha-stable family. The Cauchy distribu-tion is chosen because it is the only non-Gaussian symmetric alpha-stable distribution that exists in closed form and also because itleads to the design of a nearly optimum detector with robust detec-tion performance. We analyze the performance of the new detectorin terms of the associated probabilities of detection and false alarmand we compare it to the performance of the generalized Gaussiandetector by performing experiments with various test images.

Index Terms—Alpha-stable distributions, discrete cosine trans-form, image watermarking, Neyman–Pearson detector, statisticalmodeling.

I. INTRODUCTION

DURING the last decades, imaging technology and applica-tions have advanced greatly as multimedia information is

acquired, represented, stored, and distributed in digital format.Digital processing and transmission has major advantages, butproblems can arise due to the ease with which digital informa-tion is reproduced and distributed. Original multimedia infor-mation, such as digital images, music or video, is easily copiedwithout any loss of fidelity and it can be readily distributed tounauthorized users. Traditional methods for the prevention ofillegal use of digital multimedia data include the addition of ac-cess control headers, the encryption of the original data, andother copy protection techniques. These methods do not pro-vide an effective way of securing intellectual property rights, asthe original data remains unprotected once the copyright con-trol mechanisms are surpassed. An access control header, for

Manuscript received March 21, 2001; revised January 12, 2004. The work ofP. Tsakalides was supported by the Greek General Secretariat for Research andTechnology under Programs E�ET II, Code 97E� – 152 and E�AN, CodeH�A-011. The associate editor coordinating the review of this manuscript andapproving it for publication was Prof. Tsuhan Chen.

A. Briassouli is with the Beckman Institute Department of Electrical andComputer Engineering, University of Illinois at Urbana-Champaign, Urbana,IL 61801 USA (e-mail: [email protected]).

P. Tsakalides is with the Department of Computer Science, University ofCrete and Institute of Computer Science, FORTH ICS-FORTH, 711 10 Her-aklion, Crete, Greece (e-mail: [email protected]).

A. Stouraitis is with the Department of Electrical and Computer Engineering,University of Patras 261 10 Rio, Greece (e-mail: [email protected]).

Digital Object Identifier 10.1109/TMM.2005.850970

example, may easily be stripped from the data, leaving it com-pletely unprotected. Cryptography, on the other hand, does notallow the use of digital data in its original form and once thedata is decrypted, it is no longer protected. Other security ap-plications, like steganography, aim to effectively hide a mes-sage in the cover data without revealing the methods used forits embedding.

A method for the protection of intellectual property rights thathas been gaining popularity lately is the embedding of a dig-ital watermark in multimedia data [1]. A signal is embedded di-rectly in the data to be marked without altering it significantly,so, in contrast to encrypted data, watermarked signals can stillbe used while remaining protected. Watermarking also differsfrom steganography, as the entire embedding algorithm maybe known, its security depending only on a secret key. It doesnot necessarily prevent the copying of digital data, but shouldrather be seen as a “last line of defense” in a copyright protec-tion scheme, which identifies the original data source and its in-tended legitimate destination, so that copyright violations can atleast be detected. Thus, watermarking should be considered aspart of a general system and not a complete solution by itself. Itsbasic principle is to embed a signal directly into the data, whichserves as a host or a cover for that information. The hiddensignal travels with the data, which thus remains “marked” andprotected, until the intended receiver removes the watermark. Insome applications, such as the digital video disc (DVD), the wa-termark can also prevent the duplication of its contents, as theDVD player will not play an unwatermarked, illegally copiedvideo [2].

Watermarking is very similar to the communicationsproblem of embedding a weak and usually imperceptiblesignal in “host” data, i.e., the problem of information hiding.Security applications often use the hiding of information indigital data in order to protect it or to monitor its usage. Thewatermark is a digital signal, usually containing informationabout the data origin, destination and owner, as well as addi-tional information concerning transaction dates, serial numbersetc., which can be useful when tracking illegal use of the data[1], [3]. In practical applications, there exist both perceptibleand imperceptible watermarks. The former are usually used inpreview images in the World Wide Web or in image databasesto prevent their commercial use. Digital copyright systemsemploy imperceptible, i.e., invisible or inaudible watermarks,since their aim is to protect the copyright of the cover datawithout altering it [3].

Two main types of watermarking schemes exist. A water-marking system may embed a specific piece of information in

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BRIASSOULI et al.: HIDDEN MESSAGES IN HEAVY-TAILS: DCT-DOMAIN WATERMARK DETECTION 701

the data, such as identification numbers used for image trackingor classification and for video distribution. In this case, the em-bedded watermark communicates a message which must be ex-tracted with accuracy. The other type of system, often used incopyright-protection applications, embeds a watermark whichmodifies the original data, but does not necessarily commu-nicate a message. Hence, only the presence of the watermarkneeds to be verified in order to determine whether or not thedata is protected. These two schemes are inherently equivalent,as the detection of a watermark’s presence is equivalent to theextraction of a known message, while the extraction of a water-mark can be seen as an extension of the verification of many1-bit watermarks.

An important issue concerning watermark recovery—detection or extraction—techniques is the availability of theoriginal data. In general, watermark extraction is facilitatedwhen the host data is available, since one can then detect anydistortions in the data and invert them to accurately recover thehidden signal. However, in many practical applications suchas data monitoring or tracking, the unwatermarked data is notalways available. Moreover, in many multimedia settings, e.g.,in video watermarking, the use of the host data is impracticalbecause of its large volume. As a result, it is necessary todesign so-called blind watermarking techniques that do not usethe original data in the extraction process and treat it as noisesuperimposed on the hidden signal [1]. Recent advances in theinformation and game theoretic approach to the watermarkingproblem suggest that more information can be hidden in thehost signal if the latter is not treated as additive noise, but asside information at the encoder, i.e., the embedder [4], [5].Nevertheless, we consider the popular spread spectrum water-marking system, where the original cover signal is still treatedas noise, since these schemes are very often encountered in theliterature [6], [7]. This allows us to implement a system whoseperformance can be compared to that of other systems found inthe literature [8].

In this paper, we address the problem of watermark detec-tion in DCT transformed images, i.e., the problem of verifyingwhether a DCT-domain image has been watermarked. Variousmethods have been proposed for the detection of a watermarkin this domain, but most are based on correlator structures anddo not take into account the actual statistical properties of thedata [4], [9]. A more systematic approach is to treat water-mark verification as a statistical detection problem and to for-mulate a binary hypothesis test, concerning whether an imagewas watermarked or not. This approach leads to a detectionscheme which is more appropriate for DCT-domain images thana simple Gaussian correlator, since it is based on the data non-Gaussian statistical properties. The detector performance can beanalyzed theoretically [10]–[12], by measuring and relating theprobability of detection to the probability of false alarm. It is es-sential to predetermine the false alarm probability and to keepit at low levels, especially in commercial systems, as it directlyaffects their credibility.

The commonly used correlation detectors are optimal onlyin the case of Gaussian data [11]. However, it is well knownthat important DCT coefficients, which correspond to the lowand mid frequencies, do not follow a Gaussian distribution [13].

Hernandez et al. proposed the design of a statistical watermarkdetector that models the DCT coefficients by the commonlyused generalized Gaussian distribution [14]. The resulting de-tector structure based on this model led to considerable im-provements when compared to the correlation receiver [8]. Inthis paper, we claim that the alpha-stable family of distributionsis sufficiently flexible and rich to characterize the DCT coeffi-cients at least as accurately as the generalized Gaussian distri-bution. Motivated by our modeling results, we design a novelstatistical watermark detector based on the Cauchy member ofthe alpha-stable family and we analyze its theoretical perfor-mance. We use the Cauchy model as it is the only non-Gaussianalpha-stable distribution with a closed form probability densityfunction (pdf), but also because it leads to improved, robust de-tection schemes [15], [16]. Finally, we compare the performanceof the Cauchy processor with that of the generalized Gaussiandetector and we show that it achieves better detection than thegeneralized Gaussian scheme in a wide range of watermark-to-document power levels for real images.

The paper is organized as follows. In Section II, we providesome necessary preliminaries on watermark generation, embed-ding, and detection structures. In Section III, we present ourresults on the modeling of DCT coefficients of test images bymeans of alpha-stable distributions. In Section IV, we design ablind watermark detector based on the Cauchy distribution andwe analyze its performance in terms of the associated proba-bilities of detection and false alarm. In Section V, we comparethe performance of our proposed scheme with the performanceof the generalized Gaussian detector in real images. Finally, inSection VI, we discuss some of the issues that need to be ad-dressed in the future.

II. DCT-DOMAIN WATERMARKING FOR STILL IMAGES

In this section, we briefly describe a watermark generationand embedding procedure for still images in the DCT do-main, similar to the one presented in [8]. Many watermarkingsystems are based on additive spread spectrum ideas [6], [7],which are inspired by the spread spectrum modulation schemesused in digital communications in jamming environments [4],[17]. The role of the jammer in the watermarking problemis assumed by the host image (additive noise), as well as amalicious attacker who intentionally tries to destroy or ex-tract the embedded signal [18]. Spread-spectrum techniqueshave gained popularity for watermarking because they exhibitlow probability of intercept (LPI), which in the case of water-marking translates to high imperceptibility of the hidden signal[19]. The anti-jamming properties of these systems increase therobustness of the watermark, while the pseudorandom modu-lation of the hidden information signal increases the achievedsecurity, since knowledge of this sequence is required for suc-cessful watermark retrieval [18].

A. Watermark Generation and Embedding

In the sections that follow, we shall represent an image inthe spatial domain as a discrete 2-D sequence of its lumi-nance component with pixels and its DCT transformas . The watermark can be considered as a 2-D DCT signal


Fig. 1. Watermark embedding block diagram.

, which is added to the DCT coefficients giving the water-marked image (see Fig. 1). Boldface typesetting is usedto represent 2-D indexes like and ,and vector quantities like the vector of message bits, . Thediscrete cosine transform can be applied either to the entireimage or in blocks as in the JPEG standard [20], [21]. We usedthe block-wise DCT not only because of the JPEG standard,but also because its blockwise application is faster than thefull-image DCT and, most importantly in our case, because itallows the statistical characterization of the DCT coefficients.Specifically, the application of the DCT in 8 8 blocks ofthe image leads to 64 DCT coefficients which can be zig zagscanned and thus arranged in order of decreasing importance.Using this ordering, the first coefficients are of the main in-terest as they correspond to the low and mid frequencies thatcarry the most information about the image. By applying theblockwise DCT, we can extract these coefficients from eachblock and estimate their statistical distribution, which is essen-tial in the design of our detection system.

The watermark may or may not contain a message , whichis mapped by an encoder to a -dimensional codeword vector(Fig. 1). In the frequently considered case where the watermarkdoes not contain a message, the codeword vector reduces to

[8]. An expansion process follows, during which every el-ement of the codeword vector is repeated over a set of theimage pixels so as to cover the entire image. This introduces re-dundancy in the watermark, since it is repeated over more pixelsthan necessary for the coding of the message. In this manner itcovers the entire image, so it is present in every coefficient and,at the same time, it achieves a certain degree of inherent robust-ness because of this redundancy.

In order to generate a direct spread spectrum modulatedwatermark, the resulting signal, consisting of the “spread” mes-sage bits, is multiplied by a two-dimensional pseudorandomsequence (PRS) . The sequence is generated by a pseudo-random noise generator, which is initialized with a seed thatdepends on the cryptographic key . This pseudorandom se-quence is the spreading sequence of the system, which spreadsthe spectrum of the original signal over many frequencies,making it difficult to detect [19]. To achieve this, it needs tohave white noise-like properties, so it takes values , ,whose mean is zero and their autocorrelation approximatelyequal to the Kronecker delta function.

The invisibility of the alterations introduced to the image bythe watermark is guaranteed by multiplying the pseudorandomsequence by a perceptual mask . This mask is obtained by

using a psychovisual model in the DCT domain which was pro-posed by Ahumada et al. in [22], [23] and takes into accountthe properties of the human visual system (HVS). As in [8],one must estimate the visibility threshold for everyDCT coefficient of each 8 8 block. It is approximated in log-arithmic units by the following function:

(1)

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 modelis only valid for the AC frequencies and not for the DC coef-ficient. Since the value of every AC DCT coefficient changesfrom block to block, the threshold is corrected for everyblock by the formula

(2)

where is the DC coefficient for each block and isthe average screen luminance. These two functions contain pa-rameters which are set to , ,

, and , following[23]. The perceptual mask is then obtained from the visibilitythreshold according to the following equation:

(3)

where , , is the Kroneckerfunction, and is a scaling factor. Equation (3) givesthe final expression for the values of the perceptual maskthat corresponds to every DCT coefficient of each block. Thismask ensures that the watermark will remain imperceptible tothe human eye, but at the same time it allows the maximum pos-sible alteration of the pixel values to achieve increased robust-ness and detectability.

Finally, the resulting watermark is addedto the original image DCT coefficients , giving the water-marked image (Fig. 1). The hidden signalcan be retrieved if one uses a copy of the pseudorandom se-quence that can be reproduced only if one knows the entireprocedure through which it is generated and the cryptographickey that was used as its seed. Consequently, an attacker will notbe able to extract the watermark without knowledge of the se-cret key, even if the entire watermark generation and embeddingprocess is known. Thus, the dependence of the generated water-mark on the cryptographic key and the high degree of redun-dancy achieved through the expansion process result in a partic-ularly robust but invisible watermark.


Fig. 2. Watermark detection block diagram.

B. Watermark Detection as a Binary Hypothesis TestingProblem

As already mentioned in Section I, although the watermark de-tection and extraction problems are often treated separately inthe literature, they are inherently equivalent [1], [12]. Most copy-right-protection applications contain a known watermark, so theverificationof its existence is sufficient.Sincewearedealingwiththe problem of watermark verification, in the following we as-sume that the image can contain only one possible watermark.

In order to verify the existence of a watermark based on thestatistical properties of the given data, one must design a binaryhypothesis test where the two hypotheses concern the existenceor not of a known signal in a given image. The given data forthis test is a DCT transformed image which possibly contains awatermark, but the test is performed without knowledge of theoriginal, unwatermarked coefficients , so this is actuallya blind watermark detection problem (cf. Fig. 2). The two hy-potheses can be formulated as follows:

(4)

In this detection problem, the watermark is the desiredsignal or information, while the DCT domain image playsthe role of unknown additive noise. The goal of the optimal wa-termark detector is to find whether or not there is a watermarkin the received image , based on the statistical properties ofthe image and its watermark . A basic assumption isthat the PDF of the original coefficients does not change withthe embedding of the watermark [8]. This assumption is rea-sonable, since the watermarked image coefficients should havea distribution similar to that of the host in order to remain (per-ceptually) similar to it.

We assume that the data has been modeled by an appropriatestatistical distribution, so the PDFs of the image and its water-mark are known or can be accurately approximated from thegiven data. In that case, the decision rule is formulated as follows:

(5)

with the likelihood ratio , which is compared to thethreshold is defined as

(6)

and the PDFs under each hypothesis are known. In practice, thelog-likelihood ratio is usually preferred to perform hypothesistesting [11]. The log-likelihood ratio is simply defined as the

natural logarithm of the likelihood ratio, , sothe decision rule becomes

(7)

The detection is performed by comparing the quantity to anappropriate threshold . For Neyman–Pearson testing, which isexamined here, the threshold is determined by the probabilityof false alarm , so a reliable detector is designed for prede-fined false alarm rates [11].

III. DATA MODELING

In the previous section, we presented a setup for treating water-mark detection as a binary hypothesis test, based on the statisticalproperties of the data. Obviously, such a structure will lead to re-liable detection results if the data, i.e., the low and mid frequencyDCTcoefficients,aremodeledasaccuratelyaspossible. It isoftenmentioned in the literature that the Gaussian model is not appro-priate for these DCT coefficients [8], [13], as they exhibit heaviertails than the normal distribution. The generalized Gaussian den-sity (GGD) is frequently used to describe these coefficients, asits tails decay at a slower rate than the Gaussian distribution.

However, even the GGD does not always sufficiently charac-terize the low- and mid-frequency DCT coefficients. In [8], it isobserved that some samples in the tails of the empirical distribu-tions have relatively high amplitudes and consequently cannotbe adequately modeled by the exponentially decaying tails ofthe GGD. This fact leads to a loss in detection performance, sothe authors in [8] propose to use what they call “point elimina-tion techniques,” which essentially discard the samples whosemagnitude is above a certain threshold. This threshold cannot bedetermined analytically and it varies depending on the image,so point elimination approaches are heuristic and they cannotbe reliably applied to a wide range of image data. Also, theirperformance cannot be measured objectively since it heavily re-lies on the arbitrary deletion of high-magnitude data samples. Amore systematic and efficient approach to this problem is to con-sider a better model for the DCT coefficients, which can capturetheir heavy tailed nature, i.e., the high-magnitude samples in thetails. The symmetric alpha-stable ( ) distribution provides arich and flexible modeling tool for heavy tailed non-Gaussiandata [24], so it is expected to adequately describe the low- andmid-frequency DCT coefficients.

A. Laplacian and Generalized Gaussian Modeling of DCTCoefficients

The Laplacian and generalized Gaussian distributions haveoften been used in the literature to characterize transform do-main (DCT or wavelet) image coefficients. The Laplacian PDFis given by

(8)

where , . As shown in [25], theLaplacian distribution is inadequate for the heavier tailed sam-ples, because its tails decay at a fast exponential rate. The gen-eralized Gaussian distribution provides a better model for the


low- and mid-frequency DCT coefficients [8], [13], [14], as itstails decay at a slower rate. This distribution, characterized bythe parameter and the standard deviation of the data , has thefollowing PDF:

(9)

whose parameters and can be expressed in terms of andas follows:

(10)

For , the generalized Gaussian distribution reduces tothe Laplacian, while the Gaussian density can be obtained for

. The optimal value of may be different for each DCTcoefficient, but for practical purposes a constant value over allfrequencies suffices. In [8], the value is used in theproposed generalized Gaussian detector, as it has been found toprovide a reasonable model for DCT coefficients at low and midfrequencies [25].

B. Alpha-Stable Modeling of DCT Coefficients

In this paper, we propose the use of the symmetric alpha-stable ( ) family of distributions for the modeling of theheavy-tailed DCT coefficients. The symmetric alpha-stablefamily has recently attracted considerable interest due bothto empirical and theoretical reasons. It has been successful inmodeling heavy-tailed data in various applications [24], suchas underwater acoustic signals, clutter returns in radar [26],financial data, internet traffic [27], as well as transform domainimage or audio signals [28], [29]. The family derives itsname from its stability property, according to which a linearcombination of jointly stable random variables remains stable,i.e., its shape is unchanged (stable) under this transformation.Also, according to the generalized Central Limit Theorem, theonly possible nontrivial limit of a normalized sum of independentidentically distributed (i.i.d.) random variables is stable [30].

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

(11)

which is parameterized by the location parameter (), the scale parameter ( ), known as the dispersion,

and the characteristic exponent ( ). The locationparameter gives the distribution mean for , whilefor it gives the median. The dispersion can beany positive number and corresponds to the spread around thelocation parameter , in other words it behaves like the varianceof the Gaussian distribution.

The characteristic exponent or stability index, , is the mostimportant parameter, as it determines the shape of the distribu-tion. For smaller values of , the tails of the distribution areheavier and the corresponding random process displays high im-pulsiveness, while larger values of correspond to distributionswith more rapidly decaying tails that approach the Gaussian dis-tribution. Closed-form expressions exist for the PDF ofrandom variables only for and , which corre-spond to the Gaussian and Cauchy distributions, respectively.The shape of the PDF resembles that of the Gaussian den-sity as it is bell shaped, however it has much heavier tails, which

Fig. 3. Three standard test images with various degrees of detail, used formodeling: (a)“Moon,” (b)“Lenna,” and (c)“Boat.”.

asymptotically follow an algebraic rate of decay. For arandom variable , as , the tails satisfy

(12)

where , is the Gamma func-tion and we use as to indicate

. Hence, the tail probabilities ofthe random variables follow inverse power laws, whichcoincide with the tail probabilities of the Pareto distribution.The slowly decaying tail behavior for is then said tofollow “stable Paretian laws,” whereas exponentially decreasingtails follow a Gaussian law.

C. Experimental Modeling of DCT Coefficients

In our experiments, we consider each image in the block DCTdomain as it was described in Section II, i.e., as a two-dimen-sional discrete sequence of the DCT coefficients.The low and mid-frequency DCT coefficients are the most im-portant ones [21], so the rest can be discarded without significantimage distortion. Effective lossy image compression is achievedin this manner, which is one of the main reasons why the DCTtransform has been adopted by the JPEG standard [20]. In ourmodeling, we consider the low and mid frequency DCT coeffi-cients that are watermarked in each 8 8 block as in [8], i.e.,the zig zag scanned coefficients 7 to 28 (we use the same groupof coefficients as in [8] for reasons of comparison) and examinetheir distribution over all blocks in each image. We examine awide range of natural standard test images that are frequentlyencountered in the literature (cf. Fig. 3), ranging from simpleimages without many details (e.g., “Moon”) to more detailedones (e.g., “Lenna” and “Boat”), so our modeling results can beconsidered quite general and reliable.

We proceed in two steps. First, we verify whether the DCT co-efficients deviate from the Gaussian distribution. For this pur-pose, we examine the normal probability (p-p) plots and com-pare them with the probability plots for the best-fitting , theCauchy, and the generalized Gaussian distributions. P-P plots be-long to a family of basic diagnostic tools in statistics, especiallyselected to be of high-relevance to the study of extreme valuesand heavy-tailed distributions [31]. They plot probability versusdata values. A strong deviation of the p-p plot from the straightdiagonal line indicates that the given model is incorrect. Hence,an accurate statistical modeling results in p-p plots where thedata follow closely the straight line assumed by the model.

The normal p-p plot for “Lenna,” displayed in Fig. 4(a), clearlyshows that the data deviates from the normal distribution and it is


Fig. 4. Probability plots of the “Lenna” image low and mid frequency DCT coefficients for the Gaussian, optimal S�S, Cauchy, and generalized Gaussian (GGwith c = 0:5) distributions. (a) Normal p-p plot. The “+” marks do not follow a straight line so the data is not Gaussian. (b) Stabilized p-p plot for optimal S�S(� = 1:228) distribution. The empirical probability density (“+” marks) closely follows the straight S�S line so this model gives a very good fit. (c) Cauchy p-pplot; (d) GG (c = 0:5) p-p plot. Note that the “+” marks in (c) and (d) follow a straight line indicating that both the Cauchy and GG distributions are accuratemodels for the “Lenna” DCT coefficients.

heavy-tailed. The “+” marks in the plot show the empirical prob-ability versus the data value for each point in the sample. Themarks are in a curve that does not follow the straight Gaussianline and thus, the normality assumption is violated for this data.The non-Gaussianity of the data does not guarantee, however,that it is stable. Hence, in Fig. 4(b)–(c), we assess the stabilityof the data. First, the characteristic exponent is estimated andthe data sample is fitted with the corresponding distribu-tion. The p-p plots for the optimal and the Cauchy distri-butions, displayed in Fig. 4(b)–(c), show that the general alpha-stable and even the suboptimal Cauchy models indeed give a veryaccurate fit. Fig. 4(d) displays the p-p plot for the generalizedGaussian distribution. It is evident that the data can be accuratelydescribed by this distribution as well. Experiments performedwith the other test images lead to very similar plots, which arenot displayed due to space limitations. The p-p plots thus indi-cate that the alpha-stable family of distributions can describe thelow- and mid-frequency DCT coefficients very precisely.

We then estimate the characteristic exponent to verify thatthe data is in the stable domain of attraction. For this purpose,

TABLE ICHARACTERIZATION OF ACTUAL DCT COEFFICIENT DATA USING

ALPHA-STABLE DISTRIBUTIONS. MAXIMUM LIKELIHOOD PARAMETER

ESTIMATES AND 95% CONFIDENCE INTERVALS FOR THE S�S

CHARACTERISTIC EXPONENT, �. THE DEVIATIONS OF � FROM THE

VALUE � = 2 INDICATE THAT THE DATA IS NON-GAUSSIAN

we perform maximum likelihood (ML) fitting of the DCT co-efficients with distributions and we estimate the corre-sponding values of and the associated 95% confidence inter-vals using the method described by Nolan in [32]. The resultsare displayed in Table I, where it can be seen that the DCT coef-


ficients of the examined images exhibit distinctly non-Gaussiancharacteristics, with values of varying between 0.8 and 1.6,away from the Gaussian point of . It is safe to assume thatthe heavy-tailed nature of the data must definitely be taken intoaccount by the watermark detector in order to ensure reliabledetection performance. We should note here that the associa-tion of the (or any other) model parameters, namely and

, to the underlying physical process that generates the image isa fundamental issue. Such a consideration is beyond the scopeof this manuscript as it requires an in-depth study of the mech-anism that generates a particular image type, e.g., ultrasound,radar, optical, hyperspectral, etc., followed by the relation of thestatistical model parameters to the so-called ground truth of theimage.

Naturally, a question of interest is whether the stable fit de-scribes the data more accurately than other density functionsproposed in the literature. Here, we compare the fits withthose provided by the generalized Gaussian density function.The modeling of the DCT coefficients will be used as a basisfor the development of an improved watermark detector, so weconsider distributions that will actually be employed for the de-tector’s design. Specifically, we model the samples with the gen-eralized Gaussian distribution having parameter thatwas picked in [8] to be a good choice for a fixed, image-inde-pendent parameter. On the other hand, since the Cauchy ( )is the only member of the family (apart from the Gaussian)with a closed-form expression for its PDF, we perform the stablemodeling using the Cauchy, as a robust detector can be designedin a straightforward manner for this distribution.

To model the DCT coefficients, we use the amplitude proba-bility density (APD) function, . The empirical APDcan be plotted directly from the data by simply counting theDCT coefficients for which . The APD can also beevaluated analytically for each density (e.g., the Cauchy or thegeneralized Gaussian) after estimating the corresponding dis-tribution parameters from the data. Due to space limitations,in Fig. 5 we present the APD modeling results for six repre-sentative test images. It should be noted, however, that similarresults were obtained for the other images of Table I as well.It is clear that the Laplacian fit is poor, since the Laplaciantails decay at a fast rate. Comparing the Cauchy and general-ized Gaussian curves, we observe that they both provide fairlysatisfying goodness of fit for most images. In some images, theCauchy APD is the better model, in other images the generalizedGaussian seems to prevail. For example, note that the Cauchydensity provides a better fit to the mode of the “Cameraman” and“Moon” images (cf. Fig. 5(b) and (e), respectively), the general-ized Gaussian PDF is a more accurate fit to the mode of “Bird”[cf. Fig. 5(a)], and the two distributions are equally accurate nearthe mode, in the other cases.

More interestingly from a detection perspective, Fig. 5 re-veals that the actual data tails are almost always heavier thanwhat the generalized Gaussian model assumes, although not asheavy as the tails of the Cauchy APD. In practice, this fact in-dicates that the alpha-stable family will account for high-mag-nitude values in the data tails, which the generalized Gaussiandistribution misses, as reported in [8]. Thus, one can designan improved detector based on the Cauchy assumption, which

will account for high-amplitude data values without resorting toad-hoc techniques such as eliminating high-magnitude samples,as proposed in [8]. Of course, dealing with real images, we seein Fig. 5(f) that for “Boat” the tails of the empirical distributionare closer to the tails of the generalized Gaussian APD than tothose of the Cauchy APD. Not surprisingly, this is the only casewhere the generalized Gaussian processor gives better detectionresults than the Cauchy, as we will see in Section V.

IV. STATISTICAL WATERMARK DETECTION

Blind watermark detectors can be designed based on thebinary hypothesis test presented in Section II-B, as long as thedata probability distribution is known or can be estimated fromthe data. Naturally, these detectors are optimal when the data isaccurately modeled. In [8], a detector based on the generalizedGaussian model was proposed, as this distribution fairly accu-rately characterizes DCT or wavelet image coefficients [14].Motivated by our modeling results in the previous section, wedesign a watermark detector based on the Cauchy distribution.The Cauchy detector better accounts for the high-amplitudeDCT coefficient values in which the watermark is hidden andit is shown to have improved performance for a wide range ofwatermark-to-document power levels and for a large numberof test images.

A. The Cauchy Watermark Detector

A blind watermark detector that considers a model forthe data can be easily designed only for the Cauchy distribution(i.e., for ), the only member of the family with aclosed-form expression for the PDF. Although the Cauchy de-tector is theoretically suboptimal for non-Cauchy data, we willdemonstrate that it performs extremely well in real life appli-cations. In fact, Cauchy-based detectors and estimators wereshown to be particularly robust in heavy-tailed environmentseven under signal mismatch, contrary to traditional detectors[15], [33].

The density function of the Cauchy distribution is given inclosed form by the expression

(13)

where is the data dispersion and is the location parameter. Inthis case, the likelihood ratio of (7) uses the probability densityfunctions

(14)

For simplicity of notation, in the sequel we will omit the 2-DDCT domain index from and . If the set of DCTcoefficients that contain a watermark is denoted as , the likeli-hood ratio becomes


Fig. 5. APD curves P (jXj > n) for the low and mid frequency DCT coefficients of the standard test images: (a) “Bird,” (b) “Cameraman,” (c) “Goldhill,”(d) “Lenna,” (e) “Moon,” and (f) “Boat.” The distributions of interest are the Laplacian, Cauchy, and generalized Gaussian with parameter c = 0:5. In most cases,the modes are accurately captured by the Cauchy and the generalized Gaussian model and the empirical tails lie between the tails of the two distributions.

The corresponding log-likelihood ratio is

(15)

This analytical expression of the likelihood ratio permits thetheoretical measurement and experimental verification of theCauchy detector performance, as the receiver operating charac-teristic (ROC) curves can be derived from it. The ratio in(15) is a sum of statistically independent random variables, so


according to the CLT, it should follow a Gaussian distributionfor an appropriately large number of samples [15]. Therefore,we consider that it follows an approximately Gaussian distribu-tion with mean and variance under hypotheses and thatcan be estimated from the given data, as shall be done in thesequel.

In order to simplify the theoretical analysis, some realistic as-sumptions are made concerning the watermark, as in [8]. In thefrequently considered case that the watermark does not containhidden information, the codeword is . In this case, thewatermark will be equal to , where is theperceptual mask and the output of the PRS generator. In ourexperiments, the sequence is assumed to be an i.i.d. two-di-mensional random sequence with two equiprobable values ,

, so the watermark will be equal to or at every-pixel. Thus, the mean value of the log-likelihood function (15)

can be computed assuming that under there is no watermark,while under the watermark at each pixel can be either or

, each with probability 1/2. Then, the mean of the log-like-lihood ratio under , having omitted the index , becomes asin (16), shown at the bottom of the page. The variance of thelog-likelihood function under is

(17)

where under we have (18), shown at the bottom of the page.Using the above result, the variance becomes as shown in (19),

at the bottom of the page. In order to simplify the notation, wetemporarily set

(20)

Assuming that the watermark is an i.i.d. random sequence,the variance is

(21)

Substituting and , we get the final expression for the vari-ance of the likelihood ratio

(22)

(16)

(18)

(19)


The likelihood ratio mean under is similarly computed bysubstituting with in (15)

(23)

Finally, following the same procedure as before, the variance ofthe log-likelihood ratio under is found to be . Thus,the log-likelihood ratio under and can be consid-ered as a Gaussian random variable with mean and variance

(24)

These quantities can be computed from the given data, i.e., theDCT coefficients of the image, since they only depend on thedata values and , and the distribution parameters and, which are estimated from the data. Thus, with the mean ( ,

) and variance ( , ) of the log-likelihood ratio knownunder both hypotheses, the probabilities of false alarm and de-tection can be estimated from

(25)

(26)

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

(27)

We consider Neyman–Pearson testing [11], where the thresholdand consequently the detection probability is found for a pre-determined . Then, the threshold is given by the expression[11]

(28)

By replacing this threshold in (26) and by taking into accountthat , we get

(29)

Finally, defining as in [8], [11] and substitutingit in (29), we obtain the following relation between and :

(30)

From the above equation, we conclude that the ROC curve de-pends only on the value of the SNR for a fixed value of .Obviously, larger values of SNR improve the detector perfor-mance, as they will lead to a higher probability of detection fora given .

The proposed Cauchy detector will be compared with thegeneralized Gaussian detector with whose likelihoodratio was derived in [8]

(31)

and its mean and variance under are given by

(32)

(33)

while under , we have and . Using theabove expressions, one can form the ROCs for the generalizedGaussian detector and compare them to the ones of the Cauchyprocessor.

V. EXPERIMENTAL RESULTS

In order to reliably measure the actual performance of the de-tector proposed in Section IV, we conduct experiments with var-ious test images that were also used for modeling. Due to limita-tions of space, we present results for the “Bird,” “Cameraman,”“Goldhill,” “Lenna,” “Moon,” and “Boat” images, which exhibitvarious degrees of detail.

We initially verify the analysis of Section IV for the statis-tics of the two tests by calculating the mean and variance ofthe generalized Gaussian and the Cauchy log-likelihood ratiosboth theoretically and empirically. Then, the actual watermarkdetectors are implemented to measure the performance of thetwo schemes in terms of detection and false alarm probabilitiesnot only theoretically, but also experimentally. Using the expres-sions of the previous section for the theoretical detection andfalse alarm probabilities, as well as their experimental values,we plot and compare the theoretical and empirical ROC curvesfor the two systems. Finally, we examine the performance ofeach processor for invisible watermarks of varying strength.This comparison provides significant insight into the robustnessof the two detectors as it shows how their performance is af-fected by the embedding of weaker or stronger watermarks.

A. Experimental Values of the Likelihood Ratio Mean andVariance

In our experiments, the watermarks used are generated andembedded following the procedure described in Section II. Aperceptual mask is obtained for each image according to(3) to ensure that the DCT coefficients suffer the maximum ad-missible alteration, while the watermark remains invisible to theHuman Visual System. This mask is multiplied by indepen-dently generated pseudorandom sequences to generate therandom watermarks that are necessary for the Monte Carlo ex-periments. At every Monte Carlo iteration, we estimate the like-lihood ratio for the watermarked ( ) and unwatermarked( ) DCT coefficients of each image for both detectors usingexpressions (15) and (31). The experimental mean and varianceof is then estimated from the Monte Carlo simulations foreach detector and for both hypotheses. The theoretical mean andvariance of the likelihood ratio for the Cauchy distribution can


TABLE IIEMPIRICAL AND THEORETICAL VALUES FOR THE MEAN AND VARIANCE OF THE

LOG-LIKELIHOOD RATIO UNDER HYPOTHESES H AND H FOR THE CAUCHY

AND GENERALIZED GAUSSIAN (c = 0:5) DISTRIBUTIONS. NOTE THAT THESE

QUANTITIES ARE VERY ACCURATELY COMPUTED ANALYTICALLY

be computed directly from the data via (16) and (22), and forthe generalized Gaussian detector via (32) and (33).

The results from these experiments, presented in Table II, jus-tify the assumption that follows a Gaussian distribution andshow that its mean and variance can be very accurately com-puted analytically for both detectors. As a consequence, the ex-perimental estimates of the SNR are also exact, since they de-pend only on and . Since the ROC curves depend only onthe SNR ((30)), the experimental ROC curves are also expectedto be very close to the theoretical ones. The performance of thegeneralized Gaussian and the Cauchy detectors can thus be pre-dicted before actual experiments are conducted, by estimatingthe SNR experimentally. Table III shows a few characteristicvalues of the SNR that were derived for the examined test im-ages. In most cases the Cauchy detector leads to a higher SNR,so it is expected to exhibit better performance.

B. Detector Performance in Terms of Receiver OperatingCharacteristic Curves

The theoretical ROC curves are derived for the generalizedGaussian and the proposed Cauchy detector using (32), (33) and(16), (22), respectively, where and . Theprobability of false alarm is set to the range to andthe ROCs are derived from (30). The experimental ROC curvesare found by measuring the performance of the actual water-mark detection systems over all Monte Carlo runs. For eachDCT-transformed image, we design a generalized Gaussian anda Cauchy detector as detailed in Section IV. Experiments arethen conducted by comparing the likelihood ratio of each pro-cessor with the corresponding threshold for each value of thefalse alarm probability and for randomly generated water-marks. If the likelihood ratio is above the threshold under ,

TABLE IIISNR = 10 log(m =� )(IN DB) FOR THE CAUCHY AND GENERALIZED

GAUSSIAN (c = 0:5) DISTRIBUTIONS. HIGHER VALUES OF THE SNRCORRESPOND TO IMPROVED DETECTION PERFORMANCE

the watermark is detected, while if it is above the thresholdunder , a false alarm occurs. In this manner, the experimentalfalse alarm and detection probabilities and the correspondingempirical ROC curves are derived. Note that the SNRs for whichwe plot the ROC curves are in the range 6 to 9 dB (as shown inTable III). It is possible to embed watermarks that lead to highervalues of the SNR, such as 16 dB (for “Bird”) or 29 dB (for“Lenna” in [8]), which are still invisible, but for these values thedetection probability is equal to 1 for both detection schemes forthe whole range of , so it is meaningless to plot the ROCs.

Fig. 6(a)–(f) clearly show that the empirically obtained ROCcurves verify the theoretical ones for both detectors. The ROCsin Fig. 6(a)–(e) indicate that the Cauchy detector outperformsthe generalized Gaussian detector. This is in agreement with themodeling results of Section III-C, where it was shown that forthese images the model accounts better for the heavy datatails. Fig. 6(f) shows a case, the “Boat” image, where the gen-eralized Gaussian detector performs better than the Cauchy pro-cessor. For this image, the generalized Gaussian distribution pro-vided a closer fit to the DCT coefficients than the model, asshown in Fig. 5(f). This kind of behavior is to be expected, sinceeach image has different characteristics, as it was also noted in[8], where the Laplacian detector led to better detection resultsthan the generalized Gaussian scheme for the “Tiger” image.

C. Detector Performance for Watermarks of Varying Strength

The performance of the two detection schemes is also com-pared when watermarks of varying strength are embedded in agiven set of DCT coefficients. For this purpose, we define the“Watermark to Document Ratio” (WDR) as in [34]

(34)

with

(35)

where is the number of data samples and “document” refersto the original host data, i.e., the unwatermarked DCT imagecoefficients. Watermarks of varying strength are em-bedded in the DCT coefficients, leading to different values of


Fig. 6. Empirical and theoretical ROC curves for low- and mid-frequency DCT coefficients of the test images for the Cauchy (solid line) and generalized Gaussian(c = 0:5) (dashed line) detectors. The Cauchy detector has better performance in all cases except the “Boat” image (f), as it accounts for the high magnitude sampleshidden in the tails, which the generalized Gaussian detector misses.

and WDR. Naturally, the watermark strength is increasedwhile taking into account the properties of the HVS, in order to

remain invisible. Obviously, its power can be decreased to thepoint where it is still detectable.


Fig. 7. Probability of detection for watermarks of varying strength parameterized by the Watermark to Document Ratio WDR = 10 log(� =� ) (in decibels)for the Cauchy (solid line) and generalized Gaussian (c = 0:5) (dashed line) detectors. The false alarm probability is set toP = 10 . (a) “Lenna.” (b) “Goldhill.”(c) “Moon.”

For three of our test images, namely “Goldhill,” “Lenna,” and“Moon”, we embed watermarks for WDR values ranging from

to . For these watermarks, we determine the sta-tistics of the two detectors, as in Section IV. We then considertwo cases: first, we fix the probability of false alarm and we es-timate the corresponding probabilities of detection; then, we fixthe probability of detection and we find the corresponding prob-abilities of false alarm. In the former case, we set the false alarmprobability to the reasonable value and in the lattercase we consider . From the statistics of the likeli-hood ratios for the two detectors and the predetermined and

, we then estimate the corresponding thresholds for the like-lihood ratio tests. For fixed we use (28) and for a fixed thethreshold for each detector can be immediately derived from (26)

(36)

Finally, from (26) and (25), we find the probabilities of detectionand false alarm for all the WDR values and we plot them inFigs. 7 and 8, respectively.

Note in Fig. 7 that, depending on the particular image, aWDR value greater than (for “Lenna”) or (for“Moon”) corresponds to a relatively “strong” watermark thatcan be detected with probability equal to 1 by both detectors.Also, note that the performance of each detector depends notonly on the WDR, but on the specific characteristics of the imageas well. For example, comparing Fig. 7(a) and (c), we observethat a stronger watermark must be embedded in a detailed imagesuch as “Lenna” in order to be reliably detected, whereas in asimple image such as “Moon” a weaker (by about 4 dB) wa-termark is needed to obtain the same probability of detection.This result makes intuitive sense, since it is easier to detect asignal hidden in a smooth image than in an image with manydetails. Similarly, in Fig. 8, it can be seen that more false alarmstend to occur in the more detailed images for the same andWDR. This can also be explained intuitively, since in a smoothand simple image there are not as many areas with a lot of in-formation that may be mistaken for a watermark. Consequently,the false alarm probability in simple images is expected to belower than that in more complicated ones.


Fig. 8. Probability of false alarm for watermarks of varying strength parameterized by the Watermark to Document Ratio WDR = 10 log(� =� ) (in decibels)for the Cauchy (solid line) and generalized Gaussian (c = 0:5) (dashed line) detectors. The detection probability is set to P = 0:95. (a) “Lenna.” (b) “Goldhill.”(c) “Moon.”

Figs. 7 and 8 show that for reasonable values of andand for (invisible) watermarks of varying strength, the perfor-mance of the Cauchy detector is better than that of the gener-alized Gaussian. Indeed, as the power of the hidden signal isdecreased (WDR decreases), the performance of both schemesdeteriorates, but the Cauchy processor outperforms the gener-alized Gaussian. Thus, the Cauchy detector can reliably detecta watermark in low WDR margins, i.e., even if the watermarkhas very low strength or it is subject to an attack that partiallyeliminates it, thus decreasing its power .

VI. CONCLUSIONS AND FUTURE WORK

In this paper, we proposed and analyzed an improved blindwatermark detector for DCT-transformed images based onthe statistical properties of the DCT coefficients. The new de-tector considers the Cauchy statistical model, a member of theheavy-tailed family, which accurately describes the DCT

coefficients of interest. The accuracy of this model was veri-fied through experimental modeling of the data, which showedthat it can better account for the heavy-tailed DCT coefficientsthan the generalized Gaussian model. Then, the performancesof the Cauchy and the generalized Gaussian detectors werecompared in terms of their corresponding ROC curves. Ex-perimental results with real images verified the theoreticalanalysis and provided insight into how the performance of thetwo detectors is affected by embedding watermarks of varyingstrength.

Our current work involves the design of a maximum likeli-hood watermark decoder based on the modeling of theDCT coefficients. This decoder could actually be the secondpart of a watermark verification and extraction system that alsoincludes a watermark detector. Considering the improved per-formance of the proposed Cauchy detector, such a decoder isalso expected to give good results, thus leading to the design ofan integrated watermark verification and extraction system withbetter performance than the currently existing schemes.


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Alexia Briassouli received the Diploma in electricalengineering from the National Technical Universityof Athens (NTUA), Athens, Greece, in 1999 andthe M.Sc. degree in image and signal processingfrom the University of Patras, Patras, Greece, in2000. She is currently pursuing the Ph.D. degree inelectrical engineering at the University of Illinois atUrbana-Champaign.

From 2000 to 2001, she was a Research Assistantat the Informatics and Telematics Institute, Center ofResearch and Technology Hellas (CERTH), Thessa-

loniki, Greece, participating in a European-funded research project. Her re-search interests lie in the fields of statistical signal and image processing. Shehas worked on the design of optimal watermark embedding and detection sys-tems for images and video that are robust to various attacks. Her current researchinterests lie in the areas of statistical image processing and computer vision, andinclude problems like motion estimation and segmentation for video

Panagiotis Tsakalides (M’95) received the Ph.D.degree in electrical engineering from the Universityof Southern California (USC) in 1995 and theDiploma in Electrical Engineering from the AristotleUniversity of Thessaloniki, Thessaloniki, Greece, in1990.

He is an Associate Professor of Computer Scienceat the University of Crete, Crete, Greece, and aResearcher with the Institute of Computer Science,Foundation for Research and Technology-Hellas(ICS-FORTH), Hellas, Greece. From 1999 to 2002,

he was with the Department of Electrical Engineering, University of Patras,Patras, Greece. From 1996 to 1998, he was a Research Assistant Professor withthe Signal and Image Processing Institute, USC, and he consulted for the U.S.Navy and Air Force. His research interests lie in the field of statistical signalprocessing with emphasis in non-Gaussian estimation and detection theory, andapplications in wireless communications, imaging, and multimedia systems.He has co-authored over 50 technical publications in these areas, including 15journal papers.

Dr. Tsakalides was awarded IEE’s A. H. Reeve Premium in 2002 for the paper(co-authored with P. Reveliotis and C. L. Nikias) “Scalar quantization of heavy-tailed signals,” published in the October 2000 issue of the IEE Proceedings –Vision, Image and Signal Processing.


Thanos Stouraitis (SM’97) received the B.S. degreein physics from the University of Athens, Athens,Greece, in 1979, the M.S. degree in electronicautomation from the University of Athens, Athens,Greece, in 1981, the M.Sc. degree in electrical andcomputer engineering from the University of Cincin-nati, Cincinnati, OH, in 1983, and the Ph.D. degreein electrical engineering from the University ofFlorida, Gainesville, in 1986 (for which he receivedthe Outstanding Ph.D. Dissertation Award).

He is a Professor of Electrical and Computer En-gineering at the University of Patras, Patras, Greece, where he directs the Inter-national Graduate Studies Program on Digital Signal Processing Systems. Heis also a member of the Administrative Committee of the University of StereaHellas. He has served on the faculty of The Ohio State University and has vis-ited the University of Florida and Polytechnic University. His current researchinterests include signal and image processing systems, application-specific pro-cessor technology and design, computer arithmetic, and design and architectureof optimal digital systems. He has led several DSP processor design projectsfunded by the European Union, American organizations, and the Greek gov-ernment and industry and has served as a Consultant for various industries. Hehas authored and co-authored over 140 technical papers. He holds one patenton DSP processor design. He has authored Digital Signal Processing (Patras,Greece: University of Patras Press) and co-authored Digital Filter Design Soft-ware for the IBM PC (New York: Marcel Dekker), in addition to several bookchapters.

Dr. Stouraitis serves as the Regional Editor for Europe for the Journal ofCircuits, Systems, and Computers, has served as Associate Editor for the IEEETRANSACTIONS ON CIRCUITS AND SYSTEMS II, Associate Editor for the IEEETRANSACTIONS ON VERY LARGE SCALE INFORMATION SYSTEMS, Editor forthe IEEE Interactive Magazines, an Editor-at-Large for Marcel Dekker Inc.,and Associate Editor for the Journal of Circuits, Systems and Computers.He regularly reviews for the IEEE TRANSACTIONS ON SIGNAL PROCESSING,IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, IEEE TRANSACTIONS ON

COMPUTERS, and IEEE TRANSACTIONS ON EDUCATION, for IEE Proceedings(Parts E, F, and G), and for conferences like IEEE ISCAS, ICASSP, VLSI,SiPS, Computer Arithmetic, Euro DAC, etc. He also reviews proposals for theNational Science Foundaiton, the European Commission, and other agencies.He has served as the Chair of the VLSI Systems and Applications (VSA) Tech-nical Committee and as a member of the DSP and the Multimedia TechnicalCommittees of the IEEE Circuits and Systems (CAS) Society. He is a founderand the Chair of the IEEE Signal Processing chapter in Greece. He was theGeneral Chair of the 1996 IEEE International Conference on Electronics,Circuits, and Systems (ICECS) and the Technical Program Chair of Eusipco’98 and ICECS ’99. He has served as Chair or as a member of the TechnicalProgram Committees of a multitude of IEEE Conferences, including ISCAS(Program Committee Track Chair). He is the General Chair of ISCAS 2006.He received the 2000 IEEE CAS Society Guillemin-Cauer Award (best paper)for the paper ”Multi-Function Architectures for RNS Processors.”

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