Performance Analysis of a 2-D-Multipulse Amplitude Modulation ...

510 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 4, MAY 1998

Performance Analysis of a 2-D-MultipulseAmplitude Modulation Scheme for DataHiding and Watermarking of Still Images

Juan R. Hernandez,Student Member, IEEE, Fernando Perez-Gonzalez, Member, IEEE,Jose Manuel Rodrıguez, and Gustavo Nieto

Abstract—In this paper a watermarking scheme for copyrightprotection of still images is modeled and analyzed. In this schemea signal following a key-dependent two-dimensional multipulsemodulation is added to the image for ownership enforcementpurposes. The main contribution of this paper is the introductionof an analytical point of view to the estimation of performancemeasurements. Two topics are covered in the analysis: the own-ership verification process, also called watermark detection test,and the data-hiding process. In the first case, bounds and ap-proximations to the receiver operating characteristic are derived.These results can be used to determine the threshold associatedto a required probability of false alarm and the correspondingprobability of detection. The data-hiding process is modeled asa communications system and approximations for the bit errorrate are derived. Finally, analytical expressions are contrastedwith experimental results.

Index Terms—Copyright protection, codes, cryptography, de-cision-making, image processing, image communication, informa-tion theory.

I. INTRODUCTION

T HE enormous progress that digital technologies haveexperienced during the last decades has contributed to the

popularization of the use of electronic media for transmissionand storage of documents, images, audio, video, and othertypes of information. Information stored in digital format canbe copied without quality loss and distributed efficiently atfairly low costs. These developments have also increased thepotential for interception, manipulation, misuse, and unau-thorized distribution of information. This is, in fact, one ofthe main impediments to commercial use of communicationnetworks and electronic storage media for distribution ofdigital information. For this reason, the design of techniquesfor preserving the ownership of digital information is thecornerstone of the development of future multimedia services.

Previous research on copyright protection of still images hasresulted in the appearance of several watermarking methods.In all these techniques, the contents of the original image isaltered in a fashion determined by a secret key and, optionally,by a certain amount of information to be hidden into the image.

Manuscript received March 10, 1997; revised July 15, 1997.The authors are with the Departmento de Tecnologıas de las Comu-

nicaciones, E.T.S.I. Telecomunicacion, University of Vigo, Vigo 36200,Pontevedra, Spain (e-mail: [email protected], [email protected]).

Publisher Item Identifier S 0733-8716(98)01107-X.

In [1], a watermarking procedure based on spread spectrum(SS) techniques is proposed for application to multimediadata. The watermark consists of a sequence of independentand identically distributed (i.i.d.) Gaussian random variablesthat are added to the perceptually most significantDCTcoefficients. Placing the watermark in the perceptually relevantcomponents of the original image provides a high level ofrobustness against many signal processing techniques aimedat eliminating noise from the image. However, the mainlimitation of this technique is the need for the original imagein the ownership verification process.

A JPEG-based method for embedding labels into imagesis described in [2]–[4]. In this method, the original image isdivided into 8 8 blocks. A triple is chosen among the DCTcoefficients at the middle frequencies in each block, and itscomponents are modified to encode one bit. This techniqueresembles a frequency-hopping SS scheme, but no perceptualconstraint is imposed to the modifications introduced in theimage. It is also sensitive to attacks such as cropping andaffine transforms, that alter the spatial location of the 88blocks with respect to the borders of the image, as well asadditive noise concentrated in the middle frequencies.

In [5], a watermarking method is presented. It is based onthe addition in the frequency (DCT) domain of an SS signalshaped by a perceptual mask that guarantees that the hiddensignal is invisible. The watermarking process is performedblockwise, and the original image is required in the verificationtest. A data-hiding scheme based on a similar approach isdescribed in [6]. In this case, an alternative spatial-domainwatermarking technique is also proposed. The original imageis segmented into blocks that are modified by single bits ofthe hidden message. For this reason, this data-hiding scheme isnot robust against cropping. The original image is not requiredfor information decoding. Similar techniques are applied forauthentication and distortion measurement of images [7] andfor watermarking of audio signals [8]. The scheme we analyzein this article is, in fact, similar to the spatial-domain data-hiding method described in [6].

In [9], a data-hiding scheme, called Patchwork, is proposed.In this method, one bit is encoded by randomly choosing acertain number of pairs of pixels and modifying the differencein luminance level of each pair. This method is sensitiveto image cropping and affine transforms, because the spatialreference is fundamental for the correct operation of the

0733–8716/98$10.00 1998 IEEE

HERNANDEZ et al.: PERFORMANCE ANALYSIS OF A 2-D-MULTIPLULSE AMPLITUDE MODULATION SCHEME 511

Fig. 1. General model of a data-hiding system.

decoding algorithm. It is also weak against random additivenoise attacks.

Another method for data hiding is proposed in [10]. In thiscase, the image is divided into blocks that are transformedto a different domain (DCT, FFT, Daubechies wavelets). Thecoefficients in this new domain with the highest energy arealtered to encode several bits. Each bit in fact modulates asingle coefficient. This method is not resilient to cropping andaffine transforms, because the performance of the detectionalgorithm relies on the correct segmentation of the image intoblocks.

In [11], two data-hiding techniques are described. Thefirst one is used with uncompressed video, and is based ondirect-sequence (DS) SS techniques. The main weakness ofthis method, as it happens with all the techniques based onthe use of spatial pseudorandom sequences, is the spatialsynchronization. Attacks such as cropping, line removal, andaffine transforms may shift and rotate the pseudorandomsequence; as a result, resynchronization is necessary beforecorrelating.

In [12], the authors offer an overview of two data-hidingschemes based on classical SS techniques. One of them usesDS spectrum spreading and the other uses frequency hopping.An attempt is made to apply the concept of channel capacityto data hiding. These methods have the same weaknesses asother SS related schemes.

Even though different proposals for solving the copyrightenforcement problem have been described and tested withdiverse results, previous research in watermarking techniqueshas suffered from the absence of a theoretical approach to thestudy of limits in performance. In this paper a watermarkingscheme is modeled and analyzed. Both watermarking and datahiding are unified into a single model and special emphasis isplaced on the application of concepts from digital transmissiontheory and information theory. The technique we analyze issimilar to schemes described by other authors [6], [11], [12].

Section II offers an overview of the main issues that appearin a watermarking system. In Section III-A, we describe thetwo-dimensional modulation scheme used to generate thewatermark. Equivalent vector channels under different as-sumptions are derived in Sections III-B–III-D. In Section IV,a coding scheme and a detector structure are studied andanalytical approximations to the bit error rate are obtained.In Section V, a watermark detection test structure is pro-posed and analytical approximations to the probability of false

alarm and the probability of detection are derived. Finally, inSection VI, results from simulations are compared to analyticalexpressions.

II. GENERAL MODEL

In a watermarking scheme, an invisible signal carryingcopyright information is added to the image to be protected.In this context, the watermarking approach may be consideredas a steganographic technique [13]. In Fig. 1 a general modelof a watermarking system is represented.

The signal at the output of the host source encoder corre-sponds to the image that must be watermarked. The hiddeninformation source generates a message that identifies boththe issuer and the recipient of the host data, and optionally,additional information. This message is then mapped onto amodulated waveform that is added to the image. One of thegoals of the watermarking scheme is to make it difficult toguess the exact mapping between information and modulatedwaveforms. For this purpose the modulation process will havea secret key as one of its parameters.

The channel models the transformations suffered by theimage during distribution and authorized usage by the intendedrecipient. The delivered image may also be intercepted andmanipulated by an unauthorized agent (or even by the intendedrecipient) to delete or corrupt the watermark and illegallyredistribute the image. The attacks that the watermarked imagemay suffer can be categorized as follows.

1) Attacks aimed at deleting the watermark by extractingan estimate of the hidden signal from the watermarkedimage.

2) Attacks with the purpose of altering the extra informa-tion encoded in the hidden waveform. An example ofthis kind of attack is the use of additive random noise inorder to increase the probability of error in the hiddendata decoding test.

Two tests are available for ownership-verification purposes.The first one, which we will call the watermark detection test,is used to decide whether an image contains a watermarkgenerated with a certain key. The second one, applied onlyif the watermark detection test has been passed, decodes themessage carried by the hidden waveform.

For a given original image, the watermarked images ob-tained for different keys and messages can be considered pointsin a multidimensional space. This set of points must satisfy


certain conditions to be useful. First of all, the watermarkmust be imperceptible. This means that the set of possiblewatermarked images must lie inside a hypersphere defined bycertain perceptually significant distortion metric and whosecenter is the original image.

In order to guarantee that the watermarking process issecure, the hidden signal must be inseparable from the originalimage. In other words, it must be difficult to estimate the orig-inal image from the watermarked image when the secret keyis not known. Otherwise, an attacker could obtain a good es-timate of the watermark and subtract it from the watermarkedimage. In addition, the set of signal points corresponding tovalid watermarked images for a given original image mustbe sparse to achieve a low probability of generating a validwatermark when the secret key is not known. The watermarkmust also be robust against manipulations aimed at forcinga change in the result of the watermark detection test orthe watermark decoding process. These are perhaps the mostchallenging requirements, since an unauthorized person whointercepts a watermarked image may store it and then applyany kind of processing technique in order to delete or corruptthe watermark.

SS techniques have been proposed to achieve securityand robustness against manipulations. However, the maindifference with respect to an SS communication system isthat in a watermarking scheme, the jammer is not limitedto additive noise attacks because he is the channel himself.Nevertheless, the hacker is limited to those manipulations thatdo not severely distort the image contents. The modulationscheme, therefore, must exploit this fact. Returning to thegeometrical approach, the set of hidden message points inthe signal space must lie in the subspace where the originalimage is defined. Forcing the hidden signal to be coupled tothe original image makes it difficult to eliminate the watermarkwithout altering the image. In addition, it guarantees that thewatermark is resilient to compression.

Because a data-hiding system can be analyzed as a com-munication system, the concept of channel capacity can beapplied. However, the computation of the channel capacityshould take into account first the input constraints derived fromthe requirements the watermark must satisfy, then a statisticalcharacterization of the host image, which acts as noise since itis assumed to be unknown. It should also consider the worstof the attacks that do not severely distort the image contents.

III. 2-D M ULTIPULSE AMPLITUDE MODULATION

A. Definitions

In the two-dimensional (2-D) multipulse amplitude mod-ulation scheme, the signal carrying information is expressedas a linear combination of a set of orthogonal functions

(1)

where satisfy The signalis added to the original image to obtain

the watermarked version Thecoefficients are used to encode a hidden message.Let be the collection of sets of pointswhere the pulses take nonzero values

(2)

These sets define the spatial shape of the modulation pulses.The 8 8 blocks used in some watermarking methods [6], [7]can be considered as special cases of this model. For the sakeof simplicity, we will assume in the sequel that pulses do notoverlap, i.e. This assumption guaranteesthat pulses are orthogonal.

To meet the security requirements, a different set of modu-lation pulses is generated for each value of the secret keyThe modulation pulses are defined as follows:

ifotherwise

(3)

where is a key-dependent pseudorandom sequencethat can be modeled as a zero-mean i.i.d. random sequencewith marginal pdf whose variance is constrained to beone. The sequence indicates the maximum allowablestandard deviation at each pixel for the pulses to be invisible.We can infer that the coefficients must satisfyfor the watermark to be invisible. The sequence isi.i.d. to provide maximum uncertainty (entropy) for a givenmarginal distribution when the secret key is unknown.This modulation technique is similar to a DS SS scheme.However, as we noted in the introduction, the main differencewith respect to classical SS systems used in communicationsis that in our context, the jammer is not limited to additivenoise attacks. He can in fact play the role of a worst-casechannel especially designed to attack the hidden signal withoutperceptually degrading the image. In Sections III-B–III-D wediscuss the impact of the choice of a marginal distribution

on the watermarking scheme.

B. Equivalent Vector Channel for Nonaltered Images

As we stated in the introduction, we assume that the originalimage is not available in the watermark detectionand decoding processes. Therefore, it must be taken as noisethat introduces uncertainty in the detection problem. When

can be modeled as Gaussian noise, the correlation

coefficients are sufficient statistics for signaldetection. However, the Gaussian model is not suitable forreal world images. Because there is in fact a lack of goodstatistical models for common images [14], we will reducethe observation space to the projection onto the pulsesand assume that the information in the subspace orthogonal tothese pulses can be ignored.

We will now obtain a statistical characterization of thecoefficients for a given image Our approach is toobtain detector structures based on these coefficients and aimedat optimizing the probability of error averaged over the set ofkeys for a given image (see Sections IV and V). Westart by considering the case in which the watermarked imagedoes not suffer any alteration, and fixed sets


are used for all the keys. We assume that if a perceptualanalysis of the watermarked image is performed, agood estimate of the perceptual mask used to generatethe watermark is obtained, because the watermarked imageand the original are perceptually equivalent. This assumptionis in fact supported by simulation results. We will also assumethat the spatial location of the pixels that compose the pulses

is precisely known when the secret key is available.This is not the case when the watermarked image has beencropped or has suffered a geometrical transformation. Theissue of spatial synchronization is addressed in Section V. Ifthe watermarked image has not been manipulated, then

(4)

If we define and (4) can be writtenas

(5)

Both and are random variables that reflect the randomnature of the modulation pulses. Since the key is theonly random variable in the model and is i.i.d., eachcoefficient is the sum of independent random variables.Therefore, if the pulse size is large enough, by the centrallimit theorem, is approximately Gaussian. Let us define

and Now

(6)

where The second-order moments of thezero-mean random variables and are

(7)

(8)

(9)

(10)

where is the Kronecker delta function. We will assumethat is chosen to be symmetrical about the origin. Underthis assumption, and, as a result, and areuncorrelated for all and Therefore, the variance of theaggregate noise is Let us

define the vectors

and the matrices and

Then

(11)

(12)

(13)

These equations describe the equivalent Gaussian vectorchannel represented graphically in Fig. 2. Note that

Fig. 2. Equivalent channel for a 2-D multipulse modulation scheme.

for any pdf since is fixed to one, and thatequality holds only for the two-level discrete distributionin which takes values. Therefore, when thisdistribution is used, and the aggregate noisepower is minimum for a given fixed image and a givenset However, this two-level discrete distribution has lessuncertainty than other discrete and continuous pdf’s. There isa tradeoff between noise variance (and hence, probability oferror) in the equivalent vector channel and uncertainty aboutthe pulse amplitude when the secret key is not known. Thismeans that if we want the hidden signal to be more difficult tointercept, a penalty in performance must be paid. For a givenvalue the maximum entropy distribution hasthe form [15], [16]

(14)

The distribution which maximizes the entropy subject onlyto the second order moment constraint is theGaussian distribution for which

Therefore, this distribution achieves the maximumpossible entropy over all the values of For other values,the maximum entropy distribution is not Gaussian but followsexpression (14).

Matrices are strongly dependent on the spatial pulseshape. Block pulses, for example, induce highly variablevalues of the matrix coefficients because the statistics ofthe image in different blocks can significantly differ. If wespread the modulation pulses over the whole image, the matrixcoefficients associated with each vector elementwill gathercontributions from all the regions of the image, and, therefore,more homogeneous matrices will result.

C. Equivalent Vector Channel Under Linear Filtering

A similar analysis can be made to include in the vectorchannel model the effects of possible transformations sufferedby the watermarked image. One such transformation, interest-ing to study because of its power as a signal processing tool, isthe finite-impulse response (FIR) space-variant linear filtering.Let be the coefficients of a space-variant linear filterthat is applied to the watermarked image. Let be theresulting filtered image

(15)


Let us define and let us definein the same way and The correlationcoefficient can now be written as

(16)

Now we can see that for a given key, i.e., a given outcomeof intersymbol interference (ISI) is present because

in general when TheGaussian approximation is still applicable if the pulse size islarge enough since can be seen as the sum of independentrandom variables. The equivalent ISI channel in matrix form is

(17)

(18)

(19)

Let be the image filtered by The co-variance matrix of the noise vector has the followingentries

(20)

We can decompose the random matrix as we did inSection III-B

(21)

where

(22)

(23)

(24)

Therefore, is deterministic and is a zero-mean randommatrix. The second-order moments of the elements ofare

(25)

(26)

(27)

where The product(that we will denote by is a zero-mean random vectorwhose components are uncorrelated since the entries of thenoise matrix are zero-mean, uncorrelated and independentof Furthermore, this random vector is uncorrelated with the

noise vector Let be the covariance matrixof Its components are

(28)

Now we can define the aggregate noise whosecovariance matrix is the sum of two diagonal matrices

(29)

Again we can observe that for a given image and fixedsets the elements of the covariance matrix canbe minimized by choosing to be a two-level discretedistribution defined at since in this case

If we use a different distribution in order to increaseuncertainty when the secret key is not known, then a penaltyin noise variance will result. The equivalent channel, aftergrouping the noise contributions together in a single vector is

(30)

It is interesting to note that if the sets are fixed, bothmatrices and are diagonal even when the watermarkedimage has been linearly filtered.

D. Key-Dependent Pulse Spatial Location

The formulas we have obtained so far are conditioned tofixed sets As we stated in previous paragraphs,we propose the use of key-dependent pulses spread overthe image in order to provide high spatial uncertainty whenthe secret key is not known, as well as to achieve higherresilience to cropping. Spread pulses can be generated byrandomly assigning each pixel of the image to one of thesets Maximum uncertainty is achieved if theprobability of assigning the pixel to each set is

and this probability is independent of the assignmentsperformed on the rest of the pixels. In Appendix A, we provethat the matrices and corresponding to the resultingequivalent vector channel are

(31)

(32)

(33)


Note that even though the covariance matrix of the noiseconditioned to a fixed set is diagonal, the covariance matrixconsidering all the possible sets is no longer diagonal.However, simulations show that the cross-covariance terms aresmall compared to the terms in the diagonal, and therefore,they can be neglected.

E. Application to Wiener Filtering

Linear filtering can be used as an attack but it can alsobe useful for improving the performance of the watermarkdetection and decoding processes. In fact, if a minimum meansquare error (MMSE) linear estimate of the original image issubtracted from the watermarked image, the signal-to-noiseratio (SNR) in the coefficients is substantially improved.Assume that the adaptive one-tap Wiener filter presented in[17], normally used for image restoration, is applied. Then,the preprocessed signal is

(34)

The expectation is actually estimated by meansof a moving average filter. Hence, it can be included as partof the resulting filter. The variance is also estimatedfrom the watermarked image as

Therefore, the resulting filter kernel is

(35)

where the summation corresponds to the moving average andis defined on a -pixels region around the origin. Note thatthis is not the optimal MMSE linear estimate ofbecause it implicitly assumes that is white, and thisis not true. However, simulations show that this simple filterimproves the performance of the detection process. The sta-tistics of the resulting equivalent channel can be computed bysubstituting the filter kernel (35) into (31)–(33)

(36)

(37)

(38)

IV. CHANNEL CODING

A. Detector Structure

Once we have obtained a statistical characterization of theobservation vector we can study detector structures todecode hidden information. Assume that each messageinan alphabet of size is mapped onto a vector

whereTherefore, -bit messages can be hidden. The

ML detector for the vector channel analyzed in Sections III-B–III-D chooses the codeword which maximizes the Gauss-ian pdf

(39)

where is the covariance matrix of andThis is equivalent to maximize

(40)

The resulting decision regions are

(41)

As we have seen in previous sections, when the pulselocations defined by the sets are fixed and independent ofthe key the matrices and are diagonal, and the decisionregions of the ML detector for the simple code above proposedare defined by the coordinate hyperplanes. The detector is thusquite simple, since it is equivalent to a bit-by-bit hard decoder.

When key-dependent pulse locations are assumed,is nolonger diagonal. Therefore, the decision regions are intricateand the computational complexity increases. We can, alterna-tively, use a bit-by-bit hard decoder. The use of this detectorstructure is reasonable because, even though it is suboptimalin this case, it does not considerably degrade the performancesince the cross-covariance terms inare in practice smallcompared to the terms in the diagonal. Furthermore, it issimple to implement and independent ofand

Let be any of the elements in the diagonal ofandrespectively. Considering the structure of the binary antipodalcode proposed above, we can prove that in both the key-dependent and the key-independent pulse location cases theprobability of bit error averaged over all the keys for a givenimage and the bit-by-bit detector is given by

(42)

where

(43)


If we analyze (31) and (32) we can see that for a fixed imagesize, the ratio (which may be regarded as an SNR),decreases with In other words, decreases with the pulsesize. Therefore, in order to achieve a certaina minimumpulse size is required, or equivalently, a maximum number ofbits per pixel is allowed.

B. Attacks

We can use (31)–(33) to analyze the effects of differentkinds of attacks on the watermarking scheme. When thewatermarked image is cropped, for example, some pointsof the modulation pulses will be lost. Hence, the SNR foreach will decrease and the probability of bit error willdegrade. If spread pulses with key-dependent pulse locationsare used, then the probability of bit error will increase in thesame amount for all bits and errors affecting different bits areapproximately independent. These characteristics facilitate thedesign of good binary coding schemes at bit level.

The watermarked image could also be attacked by addingzero-mean white noise. If the noise variance at pixelis and the noise is added before the linear filteringoperation, then we can analyze the effect of this attack just byadding to (32) the term

(44)

A worst-case attack of this kind is the addition of noiseshaped by the perceptual mask used to generate the watermark,i.e. This attack is studied inSection VI.

V. SYNC RECOVERY AND WATERMARK DETECTION

A. Detector Structure

Throughout the analysis in Sections III and IV we haveassumed that the exact location of the pulses was known.However, several kinds of attacks such as cropping and affinetransforms may change the spatial location of the watermark.The synchronization recovery algorithm is in fact intimatelyrelated to the watermark detection test. When it succeeds/failsto acquire synchronization, we can infer that the image iswatermarked/not watermarked with the given key. In thesequel we will consider both tests as equivalent processes.

Suppose that the watermarked image may have suffered ageometric transformation with unknown parametersfor which we do not assume anya priori distribution. Thewatermark detection test can be formulated as the binaryhypothesis test

(45)

The performance of this test is measured by the probabilityof false alarm and the probability of detectionThe former is the probability of an arbitrary nonwatermarkedimage yielding a positive result in the watermark detectiontest and the latter is the probability of getting a positive result

with an image that is in fact watermarked. It is crucial for thecredibility of the watermarking system to fix a very lowIn fact, the goodness of the system can be measured in termsof the guaranteed for a certain

As we have already stated, we will reduce the observationspace to the projection onto the subspace spanned by thepulses Therefore, the watermark detection test will bebased on the coefficients and we will use the modelsdefined in previous sections in our derivations. A uniformlymost powerful test (UMP) [18] does not exist for the binaryhypothesis test in (45). Alternatively, we can perform the testindependently for each value ofand finally decide if thetest yielded a positive result for at least one of those values.This technique can be expressed as the following test:

(46)

where is the pdf of assumingthat the image is watermarked and has suffered the trans-formation When a geometric transformationis applied to the watermarked image, the perceptual masksuffers approximately the same transformation, i.e.,

For this reason, if is obtained fromthe image under test, the components of are actuallycomputed as

As we did in previous sections, we assume that the keyis the only random variable in the watermarking model

and that all the keys are equiprobable. Therefore, we willmeasure as the probability of getting a key that yieldsa positive result in the watermark detection test when theimage has been watermarked with that key, and as theprobability of getting a key that yields a positive result whenthe image has not been watermarked. A detection test will bedesigned specifically for each image, fixing a threshold valuethat guarantees a desired However, the achievable ineach case depends on the characteristics of the image undertest. In fact, indicates the suitability of each image forbeing watermarked. Small images, for instance, will lead topoor values and will be bad candidates for watermarking.For any image under test, the decision should be accompaniedwith the corresponding which should be considered as ameasure of the confidence level for that decision, assumingthat a certain is guaranteed. Our goal in this section isto provide expressions that can be used both to fix thresholdsto achieve a desired for any image and to measure fromeither the original or a watermarked image the that canbe expected in each test.

The pdf under hypothesis can be decomposed as

(47)

We will limit our analysis to transformations consisting ininteger shifts (e.g., cropping). Then, for everyeach condi-tional pdf can be approximated by a Gaussian pdf with mean

and covariance matrix The pdf under hypothesisis approximated by a zero-mean Gaussian vector pdf with


covariance matrix where

(48)

We will assume that pulses are reserved for synchroniza-tion purposes and are thus modulated by known coefficients(assume

(49)

If we neglect the cross-covariance terms inwe get, aftersome algebra, the following expression for the log maximumlikelihood function

(50)

where and are the same as in (42). We will approximateby the probability of exceeding the threshold assuming

is true and the estimate ofis correct. When is very low,the probability of exceeding the threshold for other values of

when is true is negligible. Therefore, the approximationis reasonable. We will define as the probability of anonwatermarked image exceeding the threshold fori.e., when no transformation is assumed. When the image isnot watermarked, any of the points examined during themaximization in (46) can lead to a false alarm event. Hence,the actual can be approximated by further multiplying the

obtained in the derivations by the size of the search space.Let Then

(51)

where It is assumed that forThe Chernoff bound for probabilities and

is [18]

(52)

(53)

and where is the first derivative of withrespect to Using central limit theorem arguments a tighterapproximation is [18]

(54)

(55)

where is the second derivative of with respect toWhen only affine transforms consisting of integer shifts

are considered, the maximization in (46) can be implementedby using a brute force search algorithm. This is actuallya computationally complex technique, considering that thepulses are spread over the whole image. The use of sequentialdetection algorithms in the synchronization recovery processis left as an open research line.

When affine transforms that include scaling and rotationsare considered, other issues appear. If is i.i.d., itsautocorrelation function is a delta function. This means thatthe peak in the function (50) is very narrow and may bevery difficult to find by a brute force searching algorithm.The peak can be smoothed if is nonwhite or, in otherwords, it has some redundancy. A smoother function reducesthe uncertainty when the key is not known. However, it allowsthe use of a synchronization recovery algorithm in two steps:first, during acquisition, a sequential search is performed overa grid defined in the space of unknown parameters; then a fineadjustment of these parameters is performed by means of aniterative algorithm (e.g., a gradient algorithm). The design ofsequences beneficial to the synchronization algorithmis an open research line.

B. Attacks

When an image is cropped, the ratios anddecrease. Therefore, decreases for a fixed For amaximum cropping factor, it is possible to obtain a minimumpulse size (maximum number of pulses) in order to guaranteeresilience to cropping.

Line (or column) removal can considerably reduce theperformance of the watermark detection test with little effort.If one line is removed, for instance, two peaks will appear inthe function as a function of (see Section V-A). In fact,the amplitude of the original peak will be distributed betweenthose two peaks. One solution is doubling the pulse size inorder to guarantee the required in watermark detectionand BER in data decoding. Work is in progress to developsynchronization schemes and pulse shapes robust against thiskind of attacks.

Additive noise will clearly decrease for a given thresh-old. The noise power that a hacker can add to a watermarkedimage is limited. Hence a minimum pulse size can be fixed inorder to guarantee a minimum for a required

VI. EXPERIMENTAL RESULTS AND COMPARISONS

In this section we compare results from simulations to theanalytical expressions derived in previous sections. We haveperformed all the experiments using the gray-level images ofdifferent sizes shown in Fig. 3. In Fig. 4, we can see examplesof watermarked images. The perceptual mask isbased on a visibility function defined in [17]. In Fig. 5,we show the perceptual mask corresponding to the imagesunder study. The marginal distribution of used inthe experiments is a symmetrical discrete distribution takingvalues in


(a) (b)

Fig. 3. Original images used in the experiments.

(b) (a)

Fig. 4. Watermarked images.

(a) (b)

Fig. 5. Perceptual masks.


Fig. 6. BER versus modulation pulse size when estimation of the original image using an adaptive Wiener filter is performed prior to detection.

A. Data Decoding

The first group of figures evaluate the performance of thedata-hiding scheme using the detector discussed in Section IV.In all the cases studied we have obtained the empiricalcurves taking 100 keys at random. In Fig. 6, we can seeplots of the BER obtained both empirical and analyticallyfor different pulse sizes when the watermarked image isunaltered and Wiener filtering is applied for noise reductionas a preprocessing step (Section III-E). The empirical BERis slightly above the theoretical BER due to the additionalvariance term that in practice appears because the coefficientsof the Wiener filter are computed from the watermarked image,which is key-dependent. Work is in progress for contemplatingthis effect and obtaining better estimates of the statistics ofthe equivalent channel.

In Fig. 7, we show similar plots when low pass filteringrather than Wiener filtering is performed before detectionfor estimation of the original image. We can see that theperformance has substantially degraded with respect to theprevious plots. In this case, the filter used in demodulation isindependent of the key, and therefore the additional variancediscussed in the previous paragraph does not appear.

In Fig. 8, we can observe the effect of an attack based onworst case additive Gaussian noise shaped by the perceptualmask. Wiener filtering is performed prior to detection for noisereduction. The BER has increased slightly with respect to thefirst plots. These curves can be used to choose a conservativepulse size that provides robustness against additive noise. Theincrease in variance due to the key-dependent Wiener filteralso appears in these plots.

In Fig. 9, plots of the BER are shown for a Wiener filterattack that can be considered as a worst case linear filtering

attack aimed at deleting the watermark. As we can see, theBER has not substantially degraded with respect to the nonat-tacked case (see Fig. 6). This effect is due to the fact that inideal conditions, linear MMSE estimation of the hidden signalleads to the same results when it is done after performinglinear MMSE estimation of the original image (which isactually the attack). We can see that the difference between thetheoretical BER and the empirical BER has increased becausethere is a greater additional variance term not consideredin the theoretical derivations, resulting from the use of twoconsecutive key-dependent filtering operations.

B. Sync Recovery and Watermark Detection

In the last group of figures, we show bounds and approxima-tions to the receiver operating characteristic (ROC) when thewatermarked images do not suffer any attack. Curves obtainedfrom simulation results are also shown. The is so smallthat it cannot be estimated through experimentation. Hencethe empirical curves actually represent the empirical andthe analytical approximation to evaluated over a range ofthreshold values. In all the cases, the experiments have beenperformed taking 400 keys at random. The parameterscomputed and used in the tests correspond to pessimisticSNR values. Hence, the theoretical obtained from theseparameters is an upper bound with respect to the actual

In Fig. 10, we plot the Chernoff bound and an approxi-mation to the theoretical ROC for the Tiger image for 20pulses, none of them reserved for synchronization purposes,when low-pass filtering is used to estimate the original imagebefore detection. Note that the approximation for is veryclose to the empirical


Fig. 7. Bit error rate versus modulation pulse size when estimation of the original image using a low pass filter is performed prior to detection.

Fig. 8. Bit error rate versus modulation pulse size when the watermarked image is attacked with worst case additive noise and Wiener filtering isperformed prior to detection.

In Fig. 11, we show an approximation to the theoreticalROC for the Tiger image when only one pulse coveringthe whole image is used as a watermark, and assuming thatWiener filtering is used to estimate the original image beforedetection. In this simple case the theoretical and can

be computed exactly for the parameters estimatedusing the analytical expressions, becauseis one dimensional.Even though the distance between the theoretical and empiricalcurves has increased due to the additional variance termintroduced by the Wiener filter (Section VI-A), the theoretical


Fig. 9. BER versus modulation pulse size when the watermarked image is attacked with Wiener filtering and Wiener filtering is performed prior to detection.

Fig. 10. Bound and approximation to the ROC of the watermark detection test with low-pass filtering before detection.

is still an upper bound, since the parametersare computed pessimistically. We can see that the hassubstantially improved because a better estimation of theoriginal image is performed and only one pulse with noadditional information is used.

Using the approximations to the ROC, it is possible to fixadequate values of the threshold for a desired Similarapproximations can be used to obtain values of andachievable under different kinds of attacks. In fact, since theattacker should not considerably degrade the image, a worst-


Fig. 11. Approximation to the ROC of the watermark detection test with pure watermarking and Wiener filtering before detection.

case ROC exists that can be considered as the achievableperformance for any attack.

VII. CONCLUSIONS AND FURTHER WORK

In this paper we have introduced the theoretical analysis ofa data hiding and watermarking system. As a result of thisanalysis, we have derived detector structures and expressionsfor performance measures such as the BER and the ROCassociated with a given original image when alterations suchas additive noise, cropping, and linear filtering are possible.These expressions can be used to fix parameters such asthe number of pulses and the watermark detection thresholdnecessary to achieve a desired level of performance. Moreover,performance measures can be obtained prior to watermarkingand can thus be used to estimate the capacity of an image forinformation hiding purposes.

Promising lines of research are the design of channelcodes for small pulse sizes in order to approach the overallinformation capacity of the image, the use of sequentialdetection techniques [19] and new pulse shapes to improvethe watermark detection algorithm in terms of computationalcomplexity and robustness against scaling and rotations, andthe application of the analytical approach discussed in thispaper to transformed domains (DCT, DFT, wavelets, etc.) andcolor images.

The most challenging research line is the definition of atheoretical framework in terms of Shannon information theory.In a watermarking system, problems related to source coding,channel coding and cryptographic security appear. Hence, acareful analysis should combine these three fields.

APPENDIX

A. Equivalent Channel for Key-Dependent Pulse Locations

In this appendix we derive the first- and second-ordermoments of the equivalent vector channel when spread pulseswith key-dependent pulse locations are used and the water-marked image is linearly filtered. Assume that the codeword

is hidden. Then

(56)

(57)

(58)

where and correspond to the formulasfor and shown in Section III-C. The expected value ofthe th component of the received vectoris

(59)

Hence

(60)


The variance of is

(61)

hence

(62)

Following similar arguments, the cross-covariance terms are

(63)

therefore

(64)

REFERENCES

[1] I. J. Cox, J. Kilian, T. Leighton, and T. Shamoon, “Secure spreadspectrum watermarking for multimedia,” NEC Res. Inst., Princeton, NJ,Tech. Rep. 95-10, 1995.

[2] E. Koch, J. Rindfrey, and J. Zhao, “Copyright protection for multi-media data,” inDigital Media and Electronic Publishing. New York:Academic, 1996, pp. 203–213.

[3] E. Koch and J. Zhao, “Toward robust hidden image copyright label-ing,” in Proc. 1995 IEEE Workshop on Nonlinear Signal and ImageProcessing, Neos Marmaras, Greece, June 1995, pp. 452–455.

[4] J. Zhao and E. Koch, “Embedding robust labels into images forcopyright protection,” inProc. Int. Congr. Intellectual Property Rightsfor Specialized Information, Knowledge and New Technologies, R. Old-enbourg, Ed., Vienna, Austria, Aug. 21–25, 1995, pp. 242–251.

[5] M. D. Swanson, B. Zhu, and A. H. Tewfik, “Transparent robust imagewatermarking,” inProc. IEEE Int. Conf. on Image Processing, vol. III,Lausanne, Switzerland, Sept. 1996, pp. 211–214.

[6] M. D. Swanson, B. Zhu, and A. H. Tewfik, “Robust data hiding forimages,” in Proc. IEEE Digital Signal Processing Workshop, Loen,Norway, Sept., 1996, pp. 37–40.

[7] B. Zhu, M. D. Swanson, and A. H. Tewfik, “A transparent robustauthentication and distortion measurement technique for images,” inProc. IEEE Digital Signal Processing Workshop, Loen, Norway, Sept.,1996, pp. 45–48.

[8] L. Boney, A. H. Tewfik, and K. N. Hamdy, “Digital watermarks foraudio signals,” inEUSIPCO-96, VIII European Signal Proc. Conf.,Trieste, Italy, Sept., 1996, pp. 1697–1700.

[9] W. Bender, D. Gruhl, and N. Morimoto, “Techniques for data hiding,”in Proc. SPIE, San Jose, CA, Feb. 1995, pp. 2420–2440.

[10] F. M. Boland, J. J. K. O. Ruanaidh, and C. Dautzenberg, “Water-marking digital images for copyright protection,” inIEE Int. Conf. onImage Processing and its Applications, Edinburgh, Scotland, 1995, pp.326–330.

[11] F. Hartung and B. Girod, “Digital watermarking of raw and compressedvideo,” in Digital Compression Technologies and Systems for VideoCommunications, N. Ohta, Ed., vol. 2952, SPIE Proceedings Series,Oct. 1996, pp. 205–213.

[12] J. R. Smith and B. O. Comiskey, “Modulation and information hidingin images,” inProc. Int. Workshop on Information Hiding, Cambridge,UK, May 1996, pp. 207–226.

[13] R. Anderson, “Stretching the limits of steganography,” inProc. Int.Workshop on Information Hiding, Cambridge, U.K., May 1996, pp.39–48.

[14] A. N. Netravali and B. G. Haskell,Digital Pictures. Representation,Compression and Standards. New York: Plenum, 1995.

[15] T. M. Cover and J. A. Thomas,Elements of Information Theory. NewYork: Wiley, 1991.

[16] A. Papoulis,Probability, Random Variables, and Stochastic Processes.New York: McGraw-Hill, 1984.

[17] J. S. Lim,Two-Dimensional Signal and Image Processing. EnglewoodCliffs, NJ: Prentice-Hall, 1990.

[18] H. L. V. Trees, Detection, Estimation and Modulation Theory, Pt. I.New York: Wiley, 1968.

[19] M. D. Srinath, P. K. Rajasekaran, and R. Viswanathan,Introduction toStatistical Signal Processing with Applications. Englewood Cliffs, NJ:Prentice-Hall, 1996.

Juan R. Hernandez (S’97) was born in Sala-manca, Spain, on February 12, 1970. He receivedthe Ingeniero de Telecomunicacion degree from theUniversity of Vigo, Spain, in 1993 and the M.S.degree in electrical engineering from Stanford Uni-versity, Stanford, CA, in 1996. Since 1996 he hasbeen working toward the Ph.D. degree at StanfordUniversity, Stanford, CA.

From 1993 to 1995, he was a member of the De-partment of Communication Technologies, Univer-sity of Vigo, Spain, where he worked on hardware

for digital signal processing and access control systems for digital television.His research interests include digital communications and copyright protectionin multimedia.


Fernando Perez-Gonzalez (M’93) received the In-geniero de Telecomunicaci´on degree and the Ph.D.degree in telecommunications engineering from theUniversities of Santiago, Spain, in 1990, and Vigo,Spain in 1993.

He joined the Faculty of the School of Telecom-munications Engineering as an Assistant Professorin 1990 and is currently Associate Professor in thesame institution. He has visited the University ofNew Mexico, Albuquerque, for different periodsspanning ten months. His research interests lie in

the areas of digital communications, adaptive algorithms, robust control andcopyright protection. He has been the project manager of different projectsconcerned with digital television, both for satellite and terrestrial broadcasting.He is co-editor of the bookIntelligent Methods in Signal Processing andCommunications(Cambridge, MA: Birkhauser, 1997) and has been guesteditor of a special section ofSignal Processing, devoted to Signal Processingfor Communications.

Jose Manuel Rodrıguez was born on April 30,1973, in Spain. He is currently working towardthe Ingeniero de Telecomunicacion degree at theUniversity of Vigo, Spain.

His current research interest is in image water-marking.

Gustavo Nieto was born on January 9, 1973 inSpain. He is currently working toward the Ingenierode Telecomunicacion degree at the University ofVigo, Spain.

His current research interest is in image water-marking.

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