IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 3, NO. 1, MARCH 2008 29
New Additive Watermark Detectors Based On AHierarchical Spatially Adaptive Image Model
Antonis K. Mairgiotis, Nikolaos P. Galatsanos, Senior Member, IEEE, and Yongyi Yang, Senior Member, IEEE
Abstract—In this paper, we propose a new family of watermarkdetectors for additive watermarks in digital images. These detec-tors are based on a recently proposed hierarchical, two-level imagemodel, which was found to be beneficial for image recovery prob-lems. The top level of this model is defined to exploit the spatiallyvarying local statistics of the image, while the bottom level is usedto characterize the image variations along two principal directions.Based on this model, we derive a class of detectors for the additivewatermark detection problem, which include a generalized like-lihood ratio, Bayesian, and Rao test detectors. We also proposemethods to estimate the necessary parameters for these detectors.Our numerical experiments demonstrate that these new detectorscan lead to superior performance to several state-of-the-art detec-tors.
ADDITIVE watermark detection can be formulated as ahypothesis testing problem, where one needs to determine
the presence or absence of a known watermark in an image.Within such a formulation, the watermark is treated as theknown signal and the image is treated as the corrupting noise[1] and [2]. To derive a test statistic for this problem, such as thelikelihood ratio test detector, a statistical model for the imagehas to be defined.
A well-known and widely used detector for watermarkingis the correlation detector. It is straightforward to show thatthis detector can be derived using the likelihood ratio test cri-terion under the assumption that the pixels in an image are in-dependent, identically distributed (IID) Gaussian random vari-ables [1]. While such an image model greatly simplifies theproblem, it is often not accurate in characterizing the imageproperties. Alternatively, detectors are proposed based on theuse of a high-pass filter to “pre-whiten” the image before thecorrelation detector is applied [1]. In such detectors, the outputof the high-pass filter rather than the image is modeled by IIDGaussian random variables. While it is an improvement overits predecessor, this model is found to be sometimes inadequate
Manuscript received December 22, 2006; revised August 27, 2007. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Dr. Ingemar Cox.
A. K. Mairgiotis and N. P. Galatsanos are with the Department of ComputerScience, University of Ioannina, Ioannina 45110, Greece (e-mail: [email protected]; [email protected]).
Y. Yang is with the Department of Electrical and Computer Engineering, Illi-nois Institute of Technology, Chicago, IL 60616 USA (e-mail: [email protected]).
Digital Object Identifier 10.1109/TIFS.2007.916290
for characterizing the residuals of the image (i.e., the output ofthe highpass filter). For example, in the vicinity of edges in theimage, large residual values will be produced by the highpassfilter, which will lead to heavy tails in the observed residual sta-tistics [9].
In recent years, there have been considerable efforts in theresearch community on development of image models for im-proved watermark detection. In [3], Hernandez et al. proposedan optimal detector for watermarking based on the assumptionof a generalized Gaussian density (GGD) function for the dis-crete cosine transform (DCT) coefficients of the image. In [4],the Weibull distribution was used for the discrete Fourier trans-form (DFT) coefficients of the image. In [5] and [6], optimal de-tectors were derived for both additive and multiplicative water-marking by assuming GGD distributions for the DCT, DFT, ordiscrete wavelet transform (DWT) coefficients of the image. In[7], Nikolaidis and Pitas proposed asymptotically optimal detec-tors by using GGD models for the DCT and DWT coefficients.In [8], Briassouli et al. used -stable distributions for the DCTcoefficients of the image.
In this paper, we propose the use of a hierarchical, locallyadaptive image model for watermark detection. The top levelof this model is defined to exploit the spatially varying localstatistics of the image. This model can be viewed as a gener-alization of the concept of line process used in the context ofcompound Markov random fields [23] and [24]. The differenceis that a continuous model, rather than binary edges, is used forcharacterizing the local discontinuities in the image. Using thisimage model, we will derive several detectors for additive wa-termarking, including a pseudogeneralized likelihood ratio test(PGLRT), Bayesian, and Rao test detectors. The term “pseudo”is used since we use maximum a posteriori instead of max-imum-likelihood (ML) estimates for the unknown parametersas in the generalized likelihood ratio test (GLRT) [28].
We note that the hierarchical image model used in this studywas recently developed for image restoration in [22]. It isinteresting to note that the development of image models hasalso been very important for the classical image denoising andrestoration problems, in which a statistical image model isessential [9] for various estimation methodologies (e.g., max-imum a posteriori estimation. For example, the simultaneousautoregressive (SAR) image prior has been used extensivelyin image restoration (e.g., [10]–[14]); similarly for “edge pre-serving” image priors which are based either on modeling theresiduals of the image or on a decorrelating transform (e.g.,wavelet), for example, [15]–[20].
The rest of this paper is organized as follows. In Section II,we introduce the hierarchical image model and formulate its use
30 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 3, NO. 1, MARCH 2008
for additive watermark detection. In Section III, the PGLRT de-tector is derived and methods to estimate the necessary parame-ters of the model are given. Bayesian and Rao test-based detec-tors are derived in Section IV. In Section V, numerical resultsare given to demonstrate the proposed detectors. Conclusionsare drawn in Section VI. For completeness, in the Appendix,we provide the details on the detectors used for comparison inour experiments, which were based on wavelet transform andGGD modeling.
II. IMAGE MODEL AND ADDITIVE WATERMARKING
The image model we propose to use in this paper is based onthe first-order differences of the image along the two principaldirections. Specifically, consider an image , whose pixels aredenoted by . At pixel location , we define the imagedirectional differences (IDD) along the horizontal and verticaldirections, respectively, as follows:
(1)We assume that these IDDs obey a Gaussian probability den-
sity function (pdf), given by
(2)
where is the variance parameter.For notational convenience, in the rest of this paper,
we will denote the IDDs using a single index as, , 2, where N is the total
number of image pixels. In addition, let , a vectorconsisting of IDDs in both directions.
Assuming independence of the IDDs, we can write the jointpdf
(3)
where , ,, 2, which denotes the corresponding variance parameters.The pdf in (3) allows the flexibility that the local variance
can vary from pixel to pixel. This is desirable for modeling thespatially nonstationary properties of the image (e.g., edges). Un-fortunately, it includes as many variance parameters as thenumber of image pixels. To avoid the problem of overfitting, itis necessary to impose additional constraints on the model tolimit its degrees of freedom. For this purpose, we modelas random variables, and define a hyper-prior on them.
In this paper, we use a Gamma pdf for the hyper-prior [27],which is of the form
(4)where m and l are the parameters of the Gamma distribution.Such a choice is motivated by the fact that the Gaussian and theGamma families are conjugate [26] with respect to the inverse
of the variance of the Gaussian, of which the benefit will be-come clearer later in the estimate of the model parameters. Thiscombination has also been used successfully in sparse Bayesianmodels for machine learning tasks [25].
For the Gamma pdf in (4), we have
Assuming that are independent and identically dis-tributed, then we have
(5)where is a normalization constant.
To illustrate the properties of the proposed image model, weshow an example in Fig. 2 of the estimated values of ,
, 2, from the known image “Lena” (of which the detailof the estimate will be provided in Section V). It can be seenthat the parameters can effectively capture the spatiallyvarying local statistics of the image. Notably, the edge struc-tures along respective directions in the image have been high-lighted by the large values of . In this regard, the parame-ters can be viewed as a generalization of the line processused in the context of compound Markov random fields (CMRF)[23], [24]. More specifically, in the case of CMRFs, the squareof the difference between two adjacent pixels is either used oromitted from the prior depending on the value of the binary lineprocess between these pixels. When the line process has value 1,they are omitted; when the line process has value 0, they are in-cluded. For our model, all differences are included in the prior.However, they are weighted according to the strength of the edgethat lies between them. In the vicinity of edges, this weight issmall while in smooth areas, it is large. Thus, our model doesnot “quantize” image discontinuities.
This two-level model, in essence, characterizes the pdf of theIDDs as an infinite mixture of Gaussians with zero mean anddifferent variances, which is very flexible and captures the IDDsstatistics very accurately. Looking at this model from anotherpoint of view, one can observe that if we marginalize the in-verse variances of this two-level model, the resulting pdf of theIDDs becomes Student-t [25]. This pdf is very flexible and canhave heavy tails, which is very useful for robust modeling. Fur-thermore, depending on the values of its parameters, it can be-have from a Gaussian to a uniform [31]. Such models providea very elegant mechanism that allows us to describe the localimage structure. This is very useful in many low-level imageprocessing applications, such as image recovery [22], [32]–[34]and image watermarking as we demonstrate herein.
In the additive watermark detection problem, one has to de-cide between the following two hypotheses:
(6)
where and are the observed and the original images, respec-tively, and is the watermark signal and is its strength.
MAIRGIOTIS et al.: NEW ADDITIVE WATERMARK DETECTORS 31
Define image directional difference operators , , 2such that with in (1). Applying these operators
, , 2 to the observed image in (6), we obtain
(7)
where , , , , 2.Based on (3) and (5), the pdfs of the observations under the
two hypotheses in (7) can be written as
(8)
where . In what follows, these pdfs will beused to derive the PGLRT, Bayesian, and Rao test detectors.
III. PSEUDOGENERALIZED LIKELIHOOD RATIO DETECTOR
The likelihood ratio test for the hypothesis testing problem in(7) is given by
Unfortunately, the parameters are not known, and the test in(9) cannot be used directly. In such a case, the GLRT is usuallyemployed, where one uses estimates of the unknown parameters[28]. The GLRT is given by
where and are the ML estimates of in (9) underthe two hypotheses. In what follows, since we do not use theML estimates in (10), we call our detectors “pseudo GLRT”(PGLRT).
With the pdfs in (8), the test statistic for the detector in (10)can be written as
(11)
For weak watermarks, it is reasonable to expect that the esti-mates , are approximately equal. Thus, thetest statistic in (11) can be simplified as (upon ignoring themiddle term as it does not depend directly on the data)
(12)
where is a threshold that determines the tradeoff false alarmversus probability of detection of the detector [28].
The simplified test statistic in (12) offers a rather informativeinsight on the PGLRT detector. It assumes essentially the formof a matched filter, where the observation at each pixel is nor-malized by its local variance.
The test statistic in (11) and (12) requires the estimates ofthe parameters . Obviously, the ML estimate here will beproblematic because only one data point is available. Instead,we use a maximum a posteriori estimate, where the hyper-prior
is used to ameliorate this difficulty. By invoking theBayes’ law, this estimate is obtained as
and similarly for . After some algebra, it can be shownthat
and
(13)
It is interesting to examine the effect of the parameter in thisestimate. As , the estimate becomesfor and . That is, the prior dominates the estimate. Onthe other hand, as , the prior parameters disappear in (13)and the estimate simply degenerates to the ML estimate. For
, the prior “regularizes” the estimate , wherethe ML estimate is unstable because of a lack of data.
In our experiments, we tested both the PGLRT detector in(11) and its approximate in (12), and found that their perfor-mance was almost identical. For this purpose in the rest of thispaper, we will report results with the simplified one in (12).
At this point, it is worth pointing out that the teststatistic of the proposed PGLRT detector in (12) fol-lows a Gaussian pdf under both hypotheses. The teststatistic under hypothesis has mean and vari-ance ,
, re-spectively. Similarly, the test statistic distribution underhypothesis has mean and variance
. Thus, thederivation of probability of false alarm and probability ofdetection is then straightforward [28].
32 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 3, NO. 1, MARCH 2008
IV. BAYESIAN AND RAO DETECTORS
An alternative strategy to deal with the unknown parametersin the LRT detector in (9) is to “marginalize” them using the
hyper-prior [28]. This leads to the Bayesian detector given by
(14)
It is important to note that the integration terms in (14) can becomputed in a closed form. This is made possible by the choiceof the Gamma pdf as hyper-prior which is conjugate with theGaussian [26], because the integrals in (14) are Gamma integralsof the form . Indeed, one can show thatthe test statistic for the Bayesian detector is given by
(15)
By comparing the detector above with the PGLRT in (12), wecan observe that in the Bayesian detector, the influence of thevariance parameters of the image model is now exhibitedin the form of the parameters of the hyper-prior.
Interestingly, recall the Bayesian estimate for the variance pa-rameters earlier in (13). One can rewrite the Bayesian detectorin (15) as
(16)
The form above provides an intuitive insight on the Bayesiandetector. The test statistic is formed by comparing the varianceof the IDDs at individual pixels. The presence of a watermarksignal will increase the value of the variance estimate under ,thereby leading to a smaller value of the test statistic.
In deriving the watermark detectors so far, we have consid-ered the situations that the watermark strength is exactly known[i.e., parameter in (6)]. There are also situations where it mightnot be known (e.g., in public watermarking). In such a case,one could treat the same way as other model parameters anduse its estimate in the PGLRT detector. However, this becomesproblematic in applications where the watermark signal is muchweaker than the cover image. Our experiments indicate that thiscan greatly compromise the accuracy of the ML estimate of .In order to address this difficulty, we use the Rao test, which isa locally optimal detector (LOD) with performance close to thatof a clairvoyant GLRT (when is small) [29]. This detector wasfirst introduced to the image watermarking problem in [7].
The Rao test for the observations in (7) is given by
(17)
where is the derivative of the pdf with re-spect to the observations [29].
It is noted in (17) that it is only the watermark shape (notthe parameter ) that is necessary for the Rao detector.
Substituting the pdfs into (17), we obtain
(18)
where is the estimated value of under hypothesis .Interestingly, the Rao detector assumes the form of a nor-
malized correlation detector, where the watermark shape is cor-related with the normalized observations. One may recall thatthe earlier PGLRT detector in its simplified form in (12) alsoassumes the form of a correlator. The Rao detector in (18) isinvariant with respect to the strength of the watermark. Theparameter estimates are also obtained by the MAPmethodology as for the PGLRT detector in (13).
V. NUMERICAL EXPERIMENTS
Numerical experiments are used to test the performance of thedetectors based on the proposed image model. At first, four com-monly used test images (“Barbara,” “Boat,” “Bridge,” “Lena”)of size 512 512 in image processing tasks were used in ourexperiments to demonstrate our detectors. Then, in order to es-tablish statistical significance of our results, we used 200 rep-resentative images (ten from each of the 20 categories) of theMicrosoft Image Recognition database [30]. These images wereinterpolated to size 512 512.
To quantify the power of the watermark in our experiments,the so-called watermark-to-document ratio (WDR) is used,which is defined as
(19)
To quantify the detection performance, the receiver operatingcharacteristics (ROC) curves are used. In particular, the areaunder the ROC (AUROC) curve for the false alarm probabilityrange [0–0.1] is used to quantify the performance of the de-tector at low false alarm rates; the total area under the ROCcurve is also computed to quantify the overall performance ofthe detector. These two metrics are referred to as AUROC1 andAUROC2, respectively, in the rest of this paper. ROC curveswere obtained using two approaches.
In the first approach, a set of 100 different randomly gener-ated 1-bit spread-spectrum watermarks [1] were used for eachimage in Fig. 1 at a specified WDR. For each watermark, thetest statistic was evaluated twice, once with the watermarkedand once with the unwatermarked image. The histograms of thetest statistic for the two cases are then computed based on whichthe ROC curve is generated using a moving threshold. For this
MAIRGIOTIS et al.: NEW ADDITIVE WATERMARK DETECTORS 33
Fig. 1. Watermark embedding in the three detail subbands of the second levelof a two-level DWT was used.
Fig. 2. Values of log(a (i)) for the first-order differences along: (a) hori-zontal and (b) vertical directions.
approach, “random watermarks” were used to obtain ROCs forfixed images.
In the second approach, the 200 images were added with thesame watermark. Then, the test statistic was evaluated for the200 images with the watermark and without the watermark. His-tograms of the test statistic for the two cases were created fromwhich ROC curves were generated. For this approach, “randomimages” were used to obtain ROCs for fixed watermarks.
For comparison purposes, we considered detectors that arebased on wavelet transform and GGD modeling. These detec-tors represent the state-of-the-art methods for additive water-mark detection. To the best of our knowledge, a nonadaptivewavelet model is typically used in the existing work in the liter-ature (e.g., [5] and [7]), where a single GGD model is assumedfor all wavelet bands. In our experiments, we considered an“adaptive” wavelet GGD model, in which we used a differentGGD model for each wavelet band. Our experiments demon-strated that an adaptive wavelet GGD model can lead to betterdetection performance than a nonadaptive model. Thus, we re-port our comparison results based on this adaptive wavelet GGDmodel. For completeness, we provide the detail of these waveletGGD-based detectors in the Appendix. Hereafter, these detec-tors are referred to as wavelet GGD detectors.
For fairness to the wavelet GGD detectors, watermarking wasperformed in the wavelet domain of the images. The watermarkwas embedded in the second level of the DWT, as illustrated inFig. 1. The Daubechies-8 2-D separable filters were used [35].
TABLE I(a)–(d)AUROC1 AND AUROC2 FOR THE PGLRT DETECTOR WITH: (a) “Barbara”,
(b) “Boat”, (c) “Bridge”, AND (d) “Lena” IMAGES
In all of the experiments, the watermarked images were firstquantized using 8-bpp accuracy in the spatial domain beforewatermark detection.
For the wavelet GGD detectors, the watermark detectionwas performed directly in the wavelet domain using thewatermarked coefficients. For our proposed detectors, thewavelet-domain watermark was first transformed back to thespatial domain before the detection was performed.
For the proposed model, the parameters and l of theGamma hyper-prior were determined as follows in our ex-periments. As mentioned earlier, as we have from(13) , which corresponds to a stationarymodel. In such a model, we can easily find the ML esti-mate of the residual variance aswith , which is the “average”of the two IDDs. Then, the parameter is estimated as
. The parameter was selected empirically insuch a way that the histogram of the resulting normalized IDDs
, , 2, and wouldbest fit a standard Gaussian pdf. The procedure in [20] and [21]
34 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 3, NO. 1, MARCH 2008
TABLE II(a)–(d)AUROC1 AND AUROC2 RESULTS OF BAYESIAN DETECTOR WITH: (a)
“Barbara”, (b) “Boat”, (c) “Bridge”, AND (d) “Lena” IMAGES
was used to fit the histograms. The example mentioned earlierin Fig. 2 was obtained using this procedure, where the estimatedparameters were shown for the “Lena” image.
In what follows, we present four experiments where we testthe proposed detectors. In the first three, averaging over randomwatermarks is used while in the last one, averaging over imagesis used to test different detectors.
Experiment I: The simplified PGLRT detector in (12) wastested. The AUROC1 and AUROC2 metrics for this detector aresummarized in Table I(a)–(d) for various WDR levels for thefour known images, respectively. For comparison, the results ofthe adaptive wavelet domain PGLRT detector [in (A.2)] are alsogiven. From these results, we observe that the proposed PGLRTdetector outperforms the wavelet GGD detector.
Experiment II: The Bayesian detector in (15) was com-pared with the PGLRT detector. The AUROC1 and AUROC2of the Bayesian detector are summarized in Table II(a)–(d)for the same set of test images. Compared with the resultsTable I(a)–(d), it can be seen that the performance of the
Bayesian detector is inferior to that of the PGLRT detector;nevertheless, the overall performance of the Bayesian detectoris close to that of the wavelet GGD detector. Among the fourtest images, the two methods were about the same in two im-ages (“Barbara” and “Lena”), while the Bayesian detector wasbetter in the “Bridge” image and worse in the “Boat” image.
Experiment III: The Rao test detector in (18) was tested.The AUROC1 and AUROC2 results are summarized inTable III(a)–(d) for the four test images. For comparison, re-sults are also furnished for the corresponding Rao test detectorbased on the “adaptive” wavelet GGD model in (A.3). Fromthese results, we observe that the proposed Rao test detectoroutperforms the wavelet GGD Rao test detector. Furthermore,by comparing the earlier results in Table I(a)–(d), we observethat the PGLRT detector outperforms the Rao test detector.This is not unexpected because the Rao test does not assumeknowledge of the watermark power.
Experiment IV: In this experiment, we tested all three pro-posed detectors for a number of WDRs using a set of 200 images
MAIRGIOTIS et al.: NEW ADDITIVE WATERMARK DETECTORS 35
TABLE IV(a)–(c)(a) AUROC RESULTS FOR PGLRT DETECTORS USING 200 IMAGES, (b)
AUROC RESULTS FOR BAYESIAN DETECTOR USING 200 IMAGES, AND (c)AUROC RESULTS FOR RAO TEST DETECTORS USING 200 IMAGES
and the same watermark as explained before. More, specifically,in Table IV(a)–(c) we summarize AUROC1 and AUROC2 re-sults for different WDRs for the proposed detectors and GGDwavelet detectors, the Bayesian detector, and the Rao test usingboth GGD wavelet, and the proposed prior. In Fig. 3(a)–(d),we show the ROCs for the proposed detectors and the corre-sponding wavelet GGD-based detectors. From these results, thesuperiority of the detectors based on the proposed prior for thisexperiment is clear. It is worth noting at this point, for the sakeof simplicity, the same value of the Gamma hyper prior param-eter l was used for all test images. Thus, in this experiment thedetectors based on the proposed prior were not applied to theirfull potential as compared to the previous experiments wherethis parameter was adapted to the statistics of each image. Inspite of this, the proposed detectors worked out well in this ex-periment.
To test the robustness of the proposed PGLRT detector, JPEGcompression attacks were used. Detection performance of ourdetector degraded gracefully as the quality factor decreased. InFig. 4(a) and (b), we show for two different images (“Lena”,“Barbara”) the ROC curves for the PGLRT detector without andwith JPEG compression attacks (quality 95% and 80%).
Finally, as far as computational complexity concerned, forcomputing the test statistic for 200 images (100 with valid wa-termarks and 100 with invalid watermarks), it took approxi-mately 20, 6, and 35 min, respectively, for the proposed PGLRT,Bayesian, and Rao test detectors. For the wavelet GGD detec-tors, it took approximately 6.5 and 6.2 min, respectively, forthe GLRT and Rao test. These detectors were implemented in
Fig. 3. (a) and (b) ROC curves for PGLRT and Bayesian detectors and (c) and(d) ROC curves for Rao test-based detectors, using 200 images.
Matlab on a Pentium-4 3.2-GHz PC. The aformentioned re-ported run time also includes the time needed for estimating themodel parameters in each case.
36 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 3, NO. 1, MARCH 2008
Fig. 4. ROC curves for the PGLRT detector before and after JPEG compression(quality factor 95% and 80%) for (a) “Lena” and (b) “Barbara” images.
VI. CONCLUSION AND FUTURE WORK
In this paper, we presented a new class of detectors based ona spatially adaptive image prior. This prior has the ability to ex-plicitly model the local image discontinuities (edges) based onthe variance parameters of the local IDDs of the image.The derived detectors were demonstrated to show notable im-provement over their counterparts derived from the state-of-the-art wavelet-based image models. Also, the PGLRT and Raotest detectors were demonstrated to achieve better performancethan the Bayesian detector.
We note that in addition to the results presented in this paper,we also tested these detectors with several other images, in-cluding JPEG-compressed watermark images. In all of thesetests, similar results were obtained with those presented in thispaper. These results were not included here in favor of space.The proposed detectors are somewhat more computationally ex-pensive than the wavelet GGD detectors, a tradeoff with betterdetection performance.
The performance of the proposed detectors depends on the es-timates of the parameters and the hyper-prior parameters
and . In this paper, we used a maximum a posteriori (MAP)approach to estimate and an empirical method to esti-mate and . We expect that additional gains can be achievedif a Bayesian methodology is used to estimate these parame-ters. One possible direction might be to explore a simultaneousimage segmentation and estimation of these parameters. Also,another important issue that was not addressed in this paper is
to determine the detection performance limits of the proposedscheme.
APPENDIX
WAVELET GGD-BASED DETECTORS
In wavelet-based GGD models, the wavelet coefficientsof the image, where denotes the spatial index of the waveletcoefficients, are modeled as an IID GGD random variable witha pdf given by
(A.1)
where is known as the shape parameter of the distri-bution, and , with
and .Here, the unknown parameters are . When the shape pa-rameter , the GGD becomes the well-known Gaussian.
In our comparison, we considered an “adaptive” GGDwavelet model, in which we used a different GGD model foreach wavelet band. The test statistics for the GLRT and the Raotest detectors are given, respectively, by
(A.2)
(A.3)
where denotes the th coefficient in the thband, is the total number of coefficients in thband, and is the number of bands. The vector
in-cludes all of the wavelet coefficients of the image;
is thewatermark when the watermark power is known; and
is the watermarkshape. The vectors anddenote the GGD model parameters in the bands. TheML estimates of were used for the GLRT and Raodetectors, which were determined for each wavelet bandseparately. In our experiments, and for
, 2, 3 were used for images of size .
REFERENCES
[1] I. Cox, M. Miller, and J. Bloom, Digital Watermarking. San Mateo,CA: Morgan Kaufman, 2002.
[2] G. C. Langelaar, I. Setyawan, and R. L. Lagendijk, “Watermarking dig-ital image and video data: A state-of-the-art overview,” IEEE SignalProcess. Mag., vol. 17, no. 5, pp. 20–46, Sep. 2000.
[3] J. R. Hernandez, M. Amado, and F. Perez-Gonzalez, “DCT-domainwatermarking techniques for still images: Detector performance anal-ysis and a new structure,” IEEE Trans. Image Process., vol. 9, no. 1,pp. 55–68, Jan. 2000.
MAIRGIOTIS et al.: NEW ADDITIVE WATERMARK DETECTORS 37
[4] M. Barni, F. Bartolini, A. D. Rosa, and A. Piva, “Optimum decodingand detection of multiplicative watermarks,” IEEE Trans. SignalProcess., vol. 51, no. 4, pp. 1118–1123, Apr. 2003.
[5] Q. Cheng and T. S. Huang, “An additive approach to transform-do-main information hiding and optimum detection structure,” IEEETrans. Multimedia, vol. 3, no. 3, pp. 273–284, Sep. 2001.
[6] Q. Cheng and T. S. Huang, “Robust optimum detection of transformdomain multiplicative watermarks,” IEEE Trans. Signal Process., vol.51, no. 4, pp. 906–924, Apr. 2003.
[7] A. Nikolaidis and I. Pitas, “Asymptotically optimal detection for addi-tive watermarking in the DCT and DWT domains,” IEEE Trans. ImageProcess., vol. 12, no. 5, pp. 563–571, May 2003.
[8] A. Briassouli, P. Tsakalides, and A. Stouraitis, “Hidden messagesin heavy-tails: DCT-domain watermark detection using alpha-stablemodels,” IEEE Trans. Multimedia, vol. 7, no. 4, pp. 700–715, Aug.2005.
[9] P. Maragos, R. Schafer, and R. Mersereau, “Two-dimensional linearprediction and its application to adaptive predictive coding of images,”IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-32, no. 6, pp.1213–1229, Dec. 1984.
[10] G. Demoment, “Image restoration and reconstruction: Overview ofcommon estimation structures and problems,” IEEE Trans. Acoust.Speech Signal Process., vol. 37, no. 12, pp. 2024–2036, Dec. 1989.
[11] N. P. Galatsanos and A. K. Katsaggelos, “Methods for choosing theregularization parameter and estimating the noise variance in imagerestoration and their relation,” IEEE Trans. Image Process., vol. 1, no.7, pp. 322–336, Jul. 1992.
[12] R. Molina, “On the hierarchical Bayesian approach to image restora-tion: Applications to astronomical images,” IEEE Trans. Pattern Anal.Mach. Intell., vol. 16, no. 11, pp. 1122–1128, Nov. 1994.
[13] R. Molina, A. K. Katsaggelos, and J. Mateos, “Bayesian and regular-ization methods for hyper-parameter estimation in image restoration,”IEEE Trans. Image Process., vol. 8, no. 2, pp. 231–246, Feb. 1999.
[14] R. Molina and B. D. Ripley, “Using spatial models as priors in astro-nomical images analysis,” J. Appl. Stat., vol. 16, pp. 193–206, 1989.
[15] C. Bouman and K. Sauer, “A generalized Gaussian image model foredge-preserving MAP estimation,” IEEE Trans. Image Process., vol.2, no. 3, pp. 296–310, Jul. 1993.
[16] M. A. T. Figueiredo and R. D. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process., vol. 12, no. 8,pp. 866–881, Aug. 2003.
[17] M. Belge, M. Kilmer, and E. Miller, “Wavelet domain image restora-tion with adaptive edge preserving regularization,” IEEE Trans. ImageProcess., vol. 9, no. 4, pp. 597–608, Apr. 2000.
[18] S. Chang, B. Yu, and M. Vetterli, “Spatially adaptive wavelet thresh-olding with context modeling for image denoising,” IEEE Trans. ImageProcess., vol. 9, no. 9, pp. 1522–1531, Sep. 2000.
[19] P. Moulin and J. Liu, “Analysis of multiresolution image denoisingschemes using generalized Gaussian and complexity priors,” IEEETrans. Inf. Theory, vol. 45, no. 3, pp. 909–919, Apr. 1999.
[20] M. N. Do and M. Vetterli, “Wavelet-based texture retrieval using gen-eralized Gaussian density and Kullback-Leibler distance,” IEEE Trans.Image Process., vol. 11, no. 2, pp. 146–158, Feb. 2002.
[21] M. Pi, “Improve maximum likelihood estimation for subband GGDparameters,” Pattern Recognit. Lett., vol. 27, no. 14, pp. 1710–1713,Oct. 2006.
[22] G. Chantas, N. P. Galatsanos, and A. Likas, “Bayesian restoration usinga new nonstationary edge preserving image prior,” IEEE Trans. ImageProcess., vol. 15, no. 10, pp. 2987–2997, Oct. 2006.
[23] F.-C. Jeng and J. W. Woods, “Compound Gauss-Markov fields forimage estimation,” IEEE Trans. Signal Process., vol. 39, no. 3, pp.683–697, Mar. 1991.
[24] R. Molina, A. K. Katsaggelos, J. Mateos, A. Hermoso, and C. A. Segall,“Restoration of severely blurred high range images using stochasticand deterministic relaxation algorithms in compound Gauss-Markovrandom fields,” Pattern Recognit., vol. 33, pp. 555–571, 2000.
[25] M. E. Tipping, “Sparse Bayesian learning and the relevance vector ma-chine,” J. Mach. Learning Res., vol. 1, pp. 211–244, 2001.
[26] J. Berger, Statistical Decision Theory and Bayesian Analysis. Berlin,Germany: Springer-Verlag, 1985.
[27] N. Galatsanos, V. N. Mesarovic, R. M. Molina, J. Mateos, and A. K.Katsaggelos, “Hyper-parameter estimation using gamma hyper-priorsin image restoration from partially-known blurs,” Optical Eng., vol. 41,no. 8, pp. 1845–1854, August 2002.
[28] S. M. Kay, Fundamental of Statistical Signal Processing: DetectionTheory. Upper Saddle River, NJ: Prentice-Hall, 1998, vol. 2.
[29] S. M. Kay, “Asymptotically optimal detection in incompletely char-acterized non-Gaussian noise,” IEEE Trans. Acoust., Speech, SignalProcess., vol. 37, no. 5, pp. 627–633, May 1989.
[30] Microsoft Research Cambridge object recognition image database.[Online]. Available: http://www.research.microsoft.com/downloads.
[31] C. M. Bishop, Pattern Recognition and Machine Learning. NewYork: Springer, 2006.
[32] S. Roth and M. J. Black, “Fields of experts: A framework for learningimage priors,” in Proc. IEEE Conf. Computer Vision Pattern Recogni-tion, Jun. 2005, vol. II, pp. 860–867.
[33] D. Tzikas, A. Likas, and N. Galatsanos, “Variational Bayesian blindimage deconvolution with student-T priors,” presented at the IEEE Int.Conf. Image Processing, San Antonio, TX, 2007.
[34] J. Chantas, N. P. Galatsanos, and N. Woods, “Super resolution basedon fast registration and maximum a posteriori reconstruction,” IEEETrans. Image Process., vol. 16, no. 7, pp. 1821–1830, Jul. 2007.
[35] I. Daubechies, “Orthonormal bases of compactly supported wavelets,”Commun. Pure Appl. Math., vol. 41, pp. 909–996, Nov. 1988.
Antonis K. Mairgiotis received the Diploma degreein computer science and the M.S. degree from theUniversity of Ioannina, Ioannina, Greece, in 1999 and2002, respectively, where he is currently pursuing thePh.D. degree in computer science.
His research interests are in the area of statisticalimage processing, including image watermarking,multimedia security, and Bayesian methods.
Nikolaos P. Galatsanos (SM’95) received theDiploma of Electrical Engineering from the NationalTechnical University of Athens, Athens, Greece,in 1982 and the M.S.E.E. and Ph.D. degrees inelectrical and computer engineering from the Uni-versity of Wisconsin, Madison, in 1984 and 1989,respectively.
From 1989 to 2002, he was on the faculty of theElectrical and Computer Engineering Department,Illinois Institute of Technology, Chicago. Currently,he is with the Department of Computer Science,
University of Ioannina, Ioannina, Greece. His research interests center aroundBayesian methods for image processing, medical imaging, bioinformatics, andvisual communications applications. He was an Associate Editor for the IEEETRANSACTIONS ON IMAGE PROCESSING, the IEEE Signal Processing Magazine,the Journal of Electronic Imaging, and is an Associate Editor for IEEE SIGNAL
PROCESSING LETTERS. He coedited a book Image Recovery Techniques forImage and Video Compression and Transmission (Kluwer, 1998).
Yongyi Yang (M’97–SM’03) received the B.S.E.E.and M.S.E.E. degrees from Northern JiaotongUniversity, Beijing, China, in 1985 and 1988, re-spectively. He received the M.S. degree in appliedmathematics and the Ph.D. degree in electricalengineering from the Illinois Institute of Technology(IIT), Chicago, in 1992 and 1994, respectively.
He is currently on the faculty of the Departmentof Electrical and Computer Engineering at the Illi-nois Institute of Technology, where he is an Asso-ciate Professor. Previously, he was a faculty member
with the Institute of Information Science, Northern Jiaotong University. His re-search interests are in signal and image processing, medical imaging, machinelearning, pattern recognition, and biomedical applications. He is a co-authorof Vector Space Projections: A Numerical Approach to Signal and Image Pro-cessing, Neural Nets, and Optics (Wiley, 1998).
Dr. Yang is an Associate Editor for the IEEE TRANSACTIONS ON IMAGE PRO-
