IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 4, NO. 1, MARCH 2009 25

Hyperbolic RDM for NonlinearValumetric Distortions

Pietro Guccione, Member, IEEE, and Michele Scagliola, Student Member, IEEE

Abstract—Quantization-based watermarking techniques aresensitive to valumetric scaling, a wide class of distortions applied toimages and videos, such as contrast change or gamma correction.Several methods have been proposed to counteract valumetricattacks, but the common approach to the problem is to take onlythe linear ones into account. This paper presents an extension tothe rational dither modulation (RDM) data-hiding scheme whichprovides robustness against nonlinear distortions modelled by apower-law attack. The algorithm makes use of proper mapping ofthe pixel values from the Cartesian to hyperbolic coordinates. Thismapping is able to render the problem similar to the classical RDMscheme, since fixed multiplicative scaling is cancelled out whilethe exponentiation of a nonlinear distortion is transformed intoa gain scaling. The validity of the approach has been confirmedby applying the watermarking scheme to Gaussian host andreal images; experimental results confirm its intrinsic invarianceagainst the power-law attack. Finally, it will be shown that underthe white Gaussian noise addition, the proposed scheme achievesa good bit-error rate (BER). The measured BER is affected bythe properties of the embedding domain, as supported by thetheoretical analysis given in this paper.

Index Terms—Nonadditive watermarking channel, rationaldither modulation (RDM), watermarking.

I. INTRODUCTION

T HE purpose of digital watermarking techniques is to hideinformation into a host signal which can be recovered

also after processing and transformation of that host. The modelof the watermarking problem as a communication channel withside information at the encoder was first settled in [1]. Costashowed that the capacity of this communication channel isequivalent to a channel where the known noise source at theencoder is absent [2]. Since in digital watermarking the sideinformation available at the encoder corresponds to the hostsignal, Costa’s result involves the theoretical possibility ofdeveloping a watermarking system that is able to reject theinterference between the host signal and the hidden message.

A practical implementation of this framework is the classof quantization index modulation (QIM) algorithms, first in-troduced by Chen and Wornell [3]. The main advantages ofthis approach are the robustness to the additive white Gaussiannoise (AWGN) channel and the high capacity capability that canbe reached preserving the host signal fidelity. QIM algorithms

Manuscript received February 21, 2008; revised October 17, 2008. First pub-lished January 23, 2009; current version published February 11, 2009. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Dr. Fernando Perez-Gonzalez.

The authors are with the Dipartimento di Elettrotecnica ed Elettronica—Po-litecnico di Bari, Bari 4-70125, Italy (e-mail: p.guccione@poliba.it;m.scagliola@poliba.it).

Digital Object Identifier 10.1109/TIFS.2008.2011080

embed the hidden message by quantizing the host signal sam-ples with a quantizer chosen among a finite set according to aproper relation between the message and a set of quantizers. Dif-ferent codebooks of quantizers can be used but the simplest andwidely used solution is the adoption of the lattice-based quan-tizers. If a set of scalar and uniform quantizers is used, the QIMscheme resembles the dither modulation (DM) [4], which allowsfor further fidelity improvement.

The main weakness of lattice-based watermarking is its ex-treme sensitivity to valumetric distortions (i.e., any kind of am-plitude scaling or gamma compensation). These kinds of dis-tortions usually do not introduce significant fidelity degradationand are quite common in many applications for audio and video.It has been found [5] that even small changes in the brightnessor contrast (and so an amplitude scaling) of a lattice-based wa-termarked image can severely increase the bit-error rate (BER)because of the mismatch between the encoder and decoder lat-tice volumes.

Several techniques are known in the literature that cope withamplitude scale attacks in the framework of QIM watermarking.A rough classification groups them into three categories. A classof watermarking systems tries to estimate the amplitude scalingfactor; here, the embedding of an auxiliary pilot signal [6] orblind estimation methods [7], [8] can be used. Another approachis the adoption of spherical codewords, together with a correla-tion decoder [9]. The watermarking systems belonging to thethird class define an embedding domain invariant to valumetricscaling. Several methods that reference this class have been pro-posed. Both Oostveen et al. [10] and Li and Cox [11] havepresented some methods in which the quantization step size isscaled by a factor computed according to a perceptual model.Hence, if the image is scaled by a factor , then the perceptualmodel scales similarly and so does the resulting quantizationstep size. Another approach, called AQIM [12], works by quan-tizing the angle formed by the host signal in a hypersphericalcoordinates system, constituting a gain-invariant embedding do-main. Pérez-González et al. have proposed the rational dithermodulation (RDM) [13], where the embedding method can beseen as a quantization operation with a variable step quantizer,whose size is a function of several past watermarked samples.The resulting quantization step size is gain invariant adaptive atthe encoder and decoder.

Comesaña and Pérez-González have lately proposed in [14]to embed the watermark in the logarithmic domain by using aquantization-based system. Here, the DM embedding is appliedto the logarithm of the current host sample; then the markedsample is mapped back to the original domain through the in-verse of logarithmic transformation. In this framework, a scalingresistant scheme can be obtained by embedding the watermark

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26 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 4, NO. 1, MARCH 2009

into the difference between two successive samples in the loga-rithmic domain. Moreover, the embedded watermark is intrinsi-cally perceptually shaped since a larger watermark is introducedwhen the host signal takes larger values due to the properties ofthe logarithm.

The aforementioned techniques have been developed to allcope with a gain attack consisting of the multiplication of thehost feature sequence by a gain factor, unknown to the decoder.They lack then any property to counteract nonlinear distortions,which can impair the watermark retrieving. Nonlinear distor-tions are as common as the linear ones in the context of gammacompensation or changes of contrast and/or brightness. Somethods to preserve the watermark information when imagesundergo these kind of distortions are obviously welcome as notmuch literature exists on this problem. Among these, Guerriniet al. [15] present a method which embeds the watermarksamples into the kurtosis of selected image blocks since thisparameter remains approximately constant if the image contentis not modified significantly. However, the proposed decodingmethod is able only to verify the presence of the watermark(1-b watermarking). In [16], Bas proposes a method which usesa proper set of floating quantizers; here, the quantization stepsize is a function of the minimum and the maximum of a tripletof pixels. In this way, the quantizers use a content-dependentquantization grid, whose variations in case of nonlinear distor-tions do not impair the hidden message detection.

In this paper, we propose a new watermarking method thatembeds the hidden message in a domain invariant to gain andpower-law distortions, such as gamma compensation. The pro-posed algorithm retains a high-rate capability. The main ideais based on the properties of mapping Cartesian coordinatesinto hyperbolic ones; this transformation allows changing theexponentiation to a gain scaling with which several methodscan cope. Data transformed in the hyperbolic domain are com-bined with the watermark using RDM (see Section III). The hy-perbolic transformation is performed by using the current hostsample and a proper function of some previously watermarkedsamples. In this way, the system has additional memory require-ments while preserving robustness against AWGN.

This paper is organized as follows. In Section II, after the in-troduction of some notations, the formulation of the power-lawattack problem and some discussion on the host statistics aregiven. Then, the RDM method is revised in Section III. In Sec-tion IV, the full algorithm and its properties are explained. Nu-merical simulations that validate the approach are given in Sec-tion V and, finally, in Section VI, some conclusions are drawn.

II. PROBLEM FORMULATION

Notation

We use uppercase letters to denote random variables, low-ercase letters for their individual values, calligraphic fontsfor sets, and boldface fonts for sequences or vectors (e.g.,

), where refers to the th element.The length of the vector will be clear from the context. More-over, the random vectors modelling the signals are denoted as

.

Fig. 1. Power-law attack.

We assume that the host coefficients are arranged in a 1-Dvector . The watermarked signal is denoted by , so that thewatermark signal is by definition. The average powerof the watermark is defined as the embedding distortion

(1)

where is the expectation operator.

A. Channel

A valumetric distortion can be defined as a generic functionapplied pointwise to all of the image pixels that modi-

fies their original values. The distortion function can be eitherlinear or nonlinear. The typical linear valumetric distortion is again scaling of the pixel amplitude , which yieldsbrightness and contrast change for images and video. If con-stant scaling of amplitude is applied to the whole watermarkedsignal, the so-called fixed-gain attack is performed. Nonlinearfunctions are, however, a more widely used distortion class; forexample, the gamma correction function is especially rel-evant to video watermarking applications in case the signal haspassed through digital-to-analog/analog-to-digital conversions.The Gamma correction function is given by

(2)

where is the maximum value of a pixel.The power-law attack consists of a constant exponentiation

and constant gain scaling of the amplitudes of the watermarkedsignal, as illustrated in Fig. 1; gamma correction is then a par-ticular case of the power-law attack. We assume that on thechannel, a zero-mean additive white noise with varianceand independent of the watermarked signal can be added. In thisway, the context of a watermarked signal affected by a nonlineardistortion can be compared with other methods, even those non-robust against valumetric distortions. Let denote the gainparameter and denote the exponent. The attacked signalis written as

(3)

Nonadditive attacks are not efficiently taken into account bymean square error (MSE)-based models, which are used to eval-uate the introduced distortion between the marked and the at-tacked signal. For the power-law attack, the scale factor and theexponentiation may produce very large MSE values, but the re-sulting perceptual modification on the marked signal cannot besignificant. In this framework, the valumetric distortion is often

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GUCCIONE AND SCAGLIOLA: HYPERBOLIC RDM FOR NONLINEAR VALUMETRIC DISTORTIONS 27

considered imperceptible, then it is reasonable to disregard itwhen the effects of noise addition are evaluated. Hence, we de-fine the attacking noise distortion so that it is independentof and

(4)

B. Host Statistics

Valumetric distortions are typically performed in the spacedomain, where the host samples correspond to the valuesof image pixels. For analytical purposes, the samples canbe considered as generated according to a random variablewhich resembles the pixel image distribution. are assumedindependent and identically distributed (i.i.d.) Gaussian randomvariables with mean and variance , constrained to havevalues within the range .

Embedding in a transformed domain is not considered here.In fact, since nonlinear distortions applied to the signal in thespace domain do not have a straightforward effect in a trans-formed domain, in case of a power-law attack, the modificationson the transformed samples do not have reference to the sameattack and are not trivial to take into account.

III. RATIONAL DITHER MODULATION

RDM is the proposed scheme to achieve invariance againstthe fixed-gain attack [13]. This property is obtained by quan-tizing a rational function of consecutive samples instead of thesamples themselves. Equivalently, the embedding method canbe seen as a quantization operation with a variable step quan-tizer, whose step size is a function of several past watermarkedsamples.

For the typical RDM, a function , isneeded, belonging to the set , with the following property:

(5)

In standard DM watermarking, the th bit to be hiddenindexes the quantizer to be used, so that the

quantized samples belong to the shifted lattices

(6)

where is the fixed step size.In the th-order RDM, the th information bit is embedded

into the th host sample by using the previously watermarkedsamples; the step size of the DM quantizer is scaled by ,with the vector , so that the thwatermarked sample results

(7)

The system needs to be initialized to a state shared by the em-bedder and the decoder.

At the decoder side, the hidden bit is retrieved from byapplying the standard DM decoding procedure to the rational

Fig. 2. Block diagram of �th-order RDM.

function , where .Ideally, we should divide the received sample by torecover the same quantized quantity at the encoder, but, due tothe unavailability of , we use as its estimate. Hence,the hidden information bit is estimated according to

(8)

The embedding/decoding scheme is fully depicted in Fig. 2.The decoder output is intrinsically invariant to scaling applied

to the watermarked signal since the gain is cancelled out bythe ratio; in fact, if , recalling the properties ofthe function , we have

(9)

It is worth noting that the main element in RDM is the func-tion and a significant parameter is the system memory .In [13], the vector norms is adopted as function , given by

(10)

By increasing , the influence of the attacking noise on thedecoding quantization step size decreases [13]; in fact, for large

, and, consequently, the embedding anddecoding step size become very close despite the additive noisesource in the channel.

In [13], the selection of the parameter that determines thevector norm is related with the distribution of . When fol-lows a generalized Gaussian distribution with shape parameter

and variance , there is the suggestion to choosein order to minimize the discrepancy between

and .A weakness of the basic RDM algorithm is that the functions

belonging to generally lack in properties to cope with nonlin-earity, which makes RDM performance poor against these kindsof distortions.

IV. HYPERBOLIC RDM

Starting from the RDM scheme and the hyperbolic mappingof the host samples, we propose an extension of the basic RDMscheme to make the decoder output intrinsically invariant topower-law distortions.

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28 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 4, NO. 1, MARCH 2009

A. Hyperbolic Coordinates

A point in the first quadrant of a 2-D Cartesian planecan be described by its hyperbolic coordinates representation

. The mapping from the Cartesian to the hyperbolic planeis a function whose coordinates aregiven by

(11)

(12)

where is the hyperbolic angle and is the geometric mean.The inverse mapping to the Cartesian co-

ordinates representation gives

(13)

(14)

For the forthcoming discussion, it can be shown that the hy-perbolic angle, computed from according to (12), exhibitsinteresting behavior against power-law scaling of both Cartesiansamples. In fact, given , the hyperbolic anglecorresponding to results

(15)

B. Encoder

Equation (15) shows that by mapping two samplesin a hyperbolic angle, the gain is cancelled out by the ratioand the exponent becomes a scaling gain; hence, the basicidea is to apply the RDM scheme to a hyperbolic angle repre-sentation of host samples. The key idea of the algorithm is thenenclosed in this two-step rational logarithmic dither modulation,where the first rational step is carried out by the mapping andthe second rational step by RDM. According to this reasoning,a formal representation of the mapping as a transformation in ahyperbolic domain seems to be not essential; however, the au-thors will continue to adopt the proposed name and notation toclearly distinguish this method from the others in the literature.

Similar to RDM, a set of functions ,is needed, which have the property

(16)

The function has to be properly chosen since the wholeprocessing depends on its behavior. An example of function be-longing to the set is the geometric mean

(17)

which will be used in the forthcoming discussion. The authors,however, do not exclude that other, more or less complicatedfunctions could exist. The main weakness of the geometric mean

is its proneness to drift away from the desired valueunder noise addition. Choosing , would result

in , so that the hyperbolic representation of

two successive sample is computed, without the need for thefunction . But as will be shown in Section V, for smallvalues of , the drift caused by noise noticeably reduces thesystem performance.

The proposed embedding method is split in three stages; thefirst one is the hyperbolic mapping, the second one is the actualembedding stage, and the third one is the inverse mapping.

The mapping is performed from Cartesian coordinatesto the hyperbolic angle, according to

(12). The hyperbolic representation of the th sample and of thefunction computed on the previous watermarked samples

results

(18)

Thus, the th information bit is embedded into the hyperbolicangle sample by the RDM encoding rule (7), obtaining therelative quantized sample . Here, the quantization step sizeis scaled by .

Then, the marked sample has to be converted back toCartesian coordinates. If the inverse mapping were performedby using (13) and (14), then the embedded watermark sample

would spread over both Cartesian coordinates. How-ever, the second coordinate cannot vary since it de-pends uniquely on the previously watermarked samples. Sincewe only need to map back the hyperbolic angle coordinate,where the th bit was embedded, the inverse mapping equationthat is actually used is obtained by inserting (14) in (13)

(19)

where is the resulting watermarked sample.Moreover, since embedding is performed only in the hyper-

bolic angle coordinate and since the inverse mapping is per-formed according to (19), it is worth noting that computing thehyperbolic coordinate is not necessary either at the encoderor at the decoder side.

C. Decoder

At the decoder side, the information bits are retrievedby applying the basic RDM decoder to the estimatedmarked samples in hyperbolic space. Thus, with and

, the hyperbolic angle mapping is first computed; then, the RDM decoder

is applied to the hyperbolic angle sample sequence.The whole embedding/decoding scheme is sketched in Fig. 3.Similar to RDM, as explained in Section III, the decoder

should ideally map the Cartesian coordinates intothe hyperbolic space to perform an RDM recovering the markedsample in the hyperbolic domain; however, the unavailability of

forces the use of as an estimate of it.The decoder output is intrinsically invariant to a power-law

attack applied to the watermarked signal. In fact, the gainis cancelled out by the division between the current receivedsample and performed in hyperbolic mapping; thepower exponent becomes gain scaling in the hyperbolic anglerepresentation, which RDM can cope with. Thus, recalling the

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GUCCIONE AND SCAGLIOLA: HYPERBOLIC RDM FOR NONLINEAR VALUMETRIC DISTORTIONS 29

Fig. 3. Block diagram of hyperbolic RDM.

property of and (12), in case of a power-law attack, we have

(20)

Inserting (19) in (20) and neglecting the noise contribution,we have

(21)

as was expected.It is worth noting that by extending the RDM scheme, we

need to initialize the inner and the outer memory vectors, whichfeed the functions and , respectively. The wholeinitial state has to be shared by the encoder and the decoder. Ifthe basic RDM decoder needs to receive samples to updateevery element of the memory vector, in the proposed scheme,the latency will be greater. For the decoder memory to be up-dated, not only must the outer memory be filled, but also theinner memory vector as well and updated with the received sam-ples. Consequently, the state of the system uniquely depends onthe received samples after samples have beenprocessed. It will be shown, however, that reasonable values ofthe inner and outer memory lengths do not impair too much onthe system performance, since the length of the feature vector istypically much greater than .

D. Embedding Distortion

According to the analysis carried out in [13], for large andhigh DWRs, RDM behaves similar to a classical DM with stepsize , where ; thereby, under thesehypothesis, given , the watermarking signal in thehyperbolic domain can be assumed independent of the hostsignal and its steady-state variance is .Hence, we can write

(22)

We are interested in establishing the power of the watermarksignal in the host domain which, for highDWRs and then for , can be written as1

(23)

As a consequence, the becomes in-dependent of the power of the original host, for any distributionof the host signal [14].

The value in (23) can be computed from the knowl-edge of the distribution of the hyperbolic angle random variable.Under the hypothesis of embedding with high DWRs, the pdf of

should reasonably resemble that of . The hyperbolic angle,according to (18), can be written as

(24)

Assuming and , the logarithmcan be approximated by using series expansion

(25)

so that truncating the series expansion to the second term andassuming , we have

(26)

1Note that� � � in steady state, according to the fact that the watermarksignal can be assumed stationary.

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30 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 4, NO. 1, MARCH 2009

Fig. 4. Alternative scheme for memory reduction at the encoding side.

where denotes a chi square variate with degrees offreedom. With , it is reasonable to neglect the chisquare term since it is scaled by .

Using the approximation just shown and exploiting the inde-pendence of the original host components, the random variablerepresenting the hyperbolic angle is now given by

(27)

The knowledge of the distribution of the hyperbolic anglerandom variable enables computing , whose value can benumerically evaluated. The results of the analytical approxima-tion derived before are presented in Section V.

E. Alternative Scheme for Memory Reduction

Actually, it is possible to simplify the proposed scheme byremoving the inner memory vector by a modification to thescheme so that the memory need is reduced. In the RDM en-coder and decoder, instead of using the function computed onthe previously marked samples in the hyperbolic angle domain,a function can be used that is fed by the same previouslymarked samples that feed the function .

The function , , must obey the fol-lowing property:

(28)

Thus, in the RDM encoder and decoder, the sample to bemarked is divided by the function , computed onthe previously marked samples collected in the outer memoryvector. The aforementioned proposed encoding scheme issketched in Fig. 4. The advantage of this alternative schemelies in the need of only the outer memory vector. An exampleof function having the previously explained property is

(29)where .

V. EXPERIMENTAL RESULTS

In this section, the effectiveness of the proposed methodagainst some attacks is exposed. In Section V-A, we present the

experimental results for a Gaussian host, while in Section V-B,the host signals are real images.

To evaluate the fidelity of the watermarked signal, thedocument-to-watermark ratio (DWR) is used. In all of theexperiments, the DWR was fixed at 25 dB. The strength of thenoise addition attack is measured by the watermark-to-noiseratio (WNR), which is the ratio between the embedding distor-tion and the attacking noise distortion .

A. Gaussian Host

First, the embedding distortion of the proposed method isdiscussed. Then, we illustrate how the memory sizes and

affect the system performance against noise addition; af-terwards, we outline the robustness against the power-law at-tack and noise addition, comparing the empirical performancewith the classical DM and RDM schemes. Moreover, the empir-ically measured error probabilities of the proposed scheme arecompared with the analytical approximation of the achievableBER for memory sizes going to infinity and high DWRs, whosederivation is described in the Appendix. In order to obtain a faircomparison, the RDM and hyperbolic RDM embedding and de-coding schemes do not use the distortion compensation exten-sion [3].

The measured error probabilities have been computed by av-eraging the results obtained with ten simulations; in each sim-ulation, the host vector is composed of i.i.d.Gaussian random samples, constrained to have a value withinthe range . In all experiments, one bit per sample has beenembedded.

First, in Figs. 5 and 6, the DWRs evaluated for different valuesof and for different values of and are shown, respec-tively. The experimentally evaluated DWRs are compared withthe analytical approximation derived in Section IV-D. As ex-pected, the DWR decreases linearly with the logarithm of ;a similar linear relation was obtained for the DM in the loga-rithmic domain in [14]. Moreover, the DWR increases if bigger

and are used, even if this behavior saturates very quickly.Embedding techniques that use a varying step size, such as

RDM, may suffer a considerable peak embedding distortion,since if the step size is momentarily large, the watermark samplewill also be large. In [13], the so-called peak-to-average ratio(PAR) is used to evaluate this effect

(30)

where is the magnitude exceeded by 1% of the watermarksamples. For DM, since the watermarks are uniformly dis-tributed in , the PAR is 4.68 dB, while for DM in thelogarithmic domain, we have empirically measured the PARand it is about 6.8 dB. This is motivated by the fact that thestep size reported in the original domain increases with themagnitude of the host sample. On the other hand, this behaviorcauses perceptual shaping of the watermark [14], since a largerwatermark amplitude is introduced when the host sample takesa larger value. Obviously, this effect cannot be taken intoaccount by the PAR index.

Fig. 7 shows the embedding PAR for hyperbolic RDM as afunction of , for Gaussian host, DWR of 25 dB and 100.

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GUCCIONE AND SCAGLIOLA: HYPERBOLIC RDM FOR NONLINEAR VALUMETRIC DISTORTIONS 31

Fig. 5. Comparison of the empirical values of DWR for different values of �with the analytical approximation.

Fig. 6. Comparison of the empirical values of DWR for different values of �with the analytical approximation.

The empirical results in Fig. 7 demonstrate as increasing , thePAR for hyperbolic RDM approaches the PAR for DM in thelogarithmic domain. As expected, for large , the peak embed-ding distortion is due mainly to the hyperbolic transformation;on the other hand, the watermark samples embedded by hyper-bolic RDM are perceptually shaped too.

As a second step, we have evaluated the effects of memoryincreasing. For this purpose, the error probability of the wholeencoding/decoding scheme was measured under the noise at-tack by using a fixed quantization step size. We did not per-form this test under the power-law attack since the proposedscheme is intrinsically invariant to the corresponding modifica-tions. The error probability obtained for different WNRs anddifferent values of and are exhibited in Figs. 8 and 9, re-spectively, and compared with the analytical approximation ofthe BER (see the Appendix) for 25 dB.

Fig. 7. Empirical embedding PAR, as a function of memory size � for aGaussian host and � � 100.

Fig. 8. Analytical approximation and empirical values of the error probabilityfor different values of � (here � � 50).

The increase of the RDM memory size with values largerthan makes the performance increase slightly. In Fig. 9,it is shown that the BER is affected by unimportant changes forvalues larger than 25. For the forthcoming experiments,we have then empirically chosen values 50 and 25as a tradeoff between memory size and robustness to noise.

Afterwards, we have compared the empirical and the analyt-ical BER of hyperbolic RDM w.r.t. DM and RDMunder noise addition. In Fig. 10, the results are depicted. Thebehavior of DM and RDM is similar since RDM approachesDM for memory size going to infinity [13], and outperformsthe hyperbolic RDM. This is motivated by the fact that DM andRDM have been designed to cope with the AWGN channel. Theperformance loss of hyperbolic RDM can be ascribed first tothe hyperbolic transformation itself. The white Gaussian noiseis indeed added to the watermarked signal and then the noisysamples are mapped to the hyperbolic angle prior to RDM de-

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32 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 4, NO. 1, MARCH 2009

Fig. 9. Analytical approximation and empirical values of the error probabilityfor different values of � (here � � 50).

Fig. 10. Analytical approximation and empirical values of the error proba-bility for DM, RDM, and DM in the logarithmic domain and hyperbolic RDM���� � 25 dB).

coding; in this way, the noise at the RDM decoder input is nolonger Gaussian and, above all, no longer additive (see the Ap-pendix). The lower performance can also be ascribed, in minorpart, to the drift of the geometric mean at the receiverfrom at the encoder caused by the noise when memorysizes are not very high.

A further confirmation of the aforementined explanationsabout the system performance against noise addition can beinferred from the comparison of hyperbolic RDM, its analyticalapproximation, and the DM in the logarithmic domain [14]since both use logarithmic mapping. In Fig. 10, the empiricalerror probability is also evaluated for DM in the logarithmicdomain embedding/decoding scheme under noise addition andcompared with the analytical approximation of BER. As shown,the hyperbolic RDM approaches DM in the logarithmic domainfor memory sizes going to infinity; for both plots, however,

Fig. 11. Empirical values of the error probability for a power-law attack with� � 0.7 and � � ��� ���� � 25 dB).

the achievable BER is lower than that of classical DM. Thissupports the conclusion that noise sensitivity can be ascribedmainly to the logarithmic transformation.

Finally, since the hyperbolic RDM is intrinsically invariant topower-law distortions, we expect superior performances w.r.t.DM, RDM and DM in the logarithmic domain if the water-marked signal undergoes such attacks. Experimental results sus-tain this hypothesis. In Fig. 11, the empirical error probabili-ties for different watermarked signals under a power-law attack

with and are shown. Underthis attack, the other DM-based decoders fail and the relativeerror probabilities are roughly 1/2 because these schemes to-tally lack in properties to cope with nonlinearity.

The alternative scheme for memory reduction, described inSection IV-E, was tested too. The measured error probabilitiesof the whole encoding/decoding scheme for different memorylengths and DWR of 25 dB are depicted in Fig. 12; here, it isshown that the BER suffers minor changes for bigger than25. Moreover, by comparison of the results in Fig. 8 and inFig. 12, it is noticeable that the performance of the alternativescheme for the selected values of are approximately equiva-lent to the ones of the basic scheme with two memory vectors.

B. Real Images

In order to apply the proposed watermarking method to realimages, embedding was performed in the space domain, in-serting one bit per pixel. Prior to the embedding, the 2-D hosthas to be arranged in a 1-D vector . If the arrangement order islexicographical, subsequent are highly correlated and the rel-ative geometric mean and norm will follow the signal trend.The resulting low variability of the hyperbolic angle reduces thesystem performance.

Then, before embedding is performed, the host vector isscrambled, changing the sample order. Now successivewill result in being highly uncorrelated and the relative geo-metric mean and norm will have small variations aroundthe respective global mean values. By embedding the message

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GUCCIONE AND SCAGLIOLA: HYPERBOLIC RDM FOR NONLINEAR VALUMETRIC DISTORTIONS 33

Fig. 12. Empirical values of the error probability for the alternative scheme anddifferent values of � .

Fig. 13. Empirical values of the error probability for different ordering of aLena host vector for hyperbolic RDM and RDM ���� � 25 dB).

into the scrambled image vector, the system performance isapproximately similar to that obtained in the ideal case of aGaussian host vector.

The measured error probabilities have been computed by av-eraging the results obtained by using ten simulations and thestandard images Lena and Baboon as the host signal. In all ex-periments, one bit per sample has been embedded.

Fig. 13 depicts the system performance in case the imagesamples are arranged in lexicographical order and in randomorder. Here, the BER of hyperbolic RDM applied to a Gaussianhost and the BER of RDM applied to the scrambled Lena imageare also depicted by comparison. It is noticeable that by embed-ding the watermark in the scrambled image vector, the measurederror probability under noise addition is considerably lower.

The robustness of the proposed scheme under gamma com-pression, defined in (2), was tested in the case of a real imagehost vector. Since gamma compression resembles a power-law

Fig. 14. Empirical values of the error probability for the scrambled Lena hostvector and scrambled Baboon host vector under gamma compression with � ���� ���� � 25 dB).

attack, the empirical error probability is expected to be notinfluenced by the nonlinear distortion. Into the scrambled Lenavector and into the scrambled Baboon vector, the watermarkhas been embedded by using DM, RDM, and hyperbolic RDMschemes; the error probabilities evaluated from these water-marked signals, after gamma compression with , aredepicted in Fig. 14. Only hyperbolic RDM exhibits robustnessagainst this attack as it was predictable.

VI. CONCLUSION

In this paper, we have presented an extension to the classicalRDM scheme, whose results are intrinsically invariant to non-linear power-law attacks. Mapping the host sample values intothe hyperbolic angle coordinate enables performing the embed-ding and decoding in a gain-invariant domain where the ex-ponent becomes gain scaling, of which several existing water-marking methods can cope with. This behavior is obtained in asimple way by increasing the memory with respect to the RDMsystem.

Experimental results confirmed the expected properties of theproposed scheme against the power-law attack; on the otherside, the analytical approximation of the achievable BER againstAWGN was validated. However, the algorithm has shown per-formance loss against the noise addition with respect to the clas-sical DM-based methods. The design of a function toreject the noise interference could be a solution to this problem.The authors are still working in these directions.

APPENDIX

We want to outline a statistical analysis of the achievable BERfor hyperbolic RDM against an AWGN channel. At first, sincethe proposed scheme is intrinsically invariant to gain and expo-nentiation attacks, we assume and in the BERanalysis, so that . The host signal samples are as-sumed i.i.d. Gaussian random variables with mean and vari-ance .

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34 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 4, NO. 1, MARCH 2009

In [13], the error probability of RDM for large has beenanalytically derived, resulting in

(31)

where denotes the pdf of , withthe random vector mod-

elling the signal which feeds the RDM decoder.In (31), denotes the error probability of the DM

for step-size , which was shown to be a function of the pdfof the additive noise

(32)

which generally cannot be put in closed form.Our purpose is to exploit the expression of error probability

in (31) within the proposed method, properly rearranging the re-ceived hyperbolic angle samples. To this aim, we need to definean equivalent channel where the additive noise source is movedjust ahead of the RDM decoder. Due to the logarithm properties,it is convenient to represent the difference between the encodedand the received samples with a multiplicative factor

(33)

so that results in

(34)

The hyperbolic angle samples at the decoder can be written as

(35)

Equation (35) exhibits a straight relation between the RDMencoded samples and the received ones with two additive noiseterms in the hyperbolic angle space.

The noise terms can be simplified byusing the series expansion of the logarithm function in(25), being reasonably . Hence, we can ap-proximate the logarithm, truncating the series at the firstterm ; the noise term

is the ratio of two independent Gaussianrandom variables, whose pdf results in

(36)

Here, the approximation is admitted under the hypothesis ofhigh DWRs. If the DWR is large and the original host compo-nents are independent, the noise samples are indepen-dent too.

Then, the received sample in the hyperbolic angle domain canbe written as

(37)

where the last approximation is motivated by the fact that forlarge , we can invoke the Central Limit Theorem for thesum of i.i.d. components so that this noise source can be as-sumed approximately Gaussian with zero mean and variance

.By properly rearranging (31), the error probability of the hy-

perbolic RDM can be evaluated by

(38)

where is computed for the equivalent additive noisepdf in the hyperbolic angle domain.

Recalling the considerations in [13] and under the high DWRhypothesis, can be approximated for large as

(39)

with , being the param-eter depending on the chosen vector normand .

Since the pdf of the random variable is known (see Sec-tion IV-D), the analytical approximation for large andof the error probability for the hyperbolic RDM encoding/de-coding scheme can be numerically computed from (38).

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Pietro Guccione (M’07) received the M.S. and Ph.D.degrees in electronic engineering from the Diparti-mento di Elettrotecnica ed Elettronica, Politecnico diBari, Bari, Italy, in 1996 and 1999, respectively.

Currently, he is with the Dipartimento di Elet-trotecnica ed Elettronica, Politecnico di Bari. He hasparticipated in many contracts with the EuropeanSpace Agency for the research and development offocusing processors for the Environmental SatelliteAdvanced Synthetic Aperture Radar sensor andfor the Cryosat delay/Doppler altimeter. His main

interests include SAR processing, interferometry, motion compensation, radaraltimetry, image processing, and digital watermarking.

Michele Scagliola (S’08) received the B.S. and M.S.degrees in telecommunications engineering fromthe Dipartimento di Elettrotecnica ed Elettronica,Politecnico di Bari, Bari, Italy, in 2004 and in 2006,respectively, where he is currently pursuing thePh.D. degree in watermarking and data hiding.

He is involved in the Scuola Interpolitecnica diDottorato, a joint Ph.D. program managed by the Po-litecnico di Torino, the Politecnico di Milano, and thePolitecnico di Bari.

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