Optimal Spread Spectrum Watermark Embedding via a Multistep Feasibility Formulation

17 Pages·2012·12.28 MB·English

IEEETRANSACTIONSONIMAGEPROCESSING,VOL.18,NO.2,FEBRUARY2009 371
Optimal Spread Spectrum Watermark Embedding via
a Multistep Feasibility Formulation
H. OktayAltun, Student Member, IEEE, AdemOrsdemir, Student Member, IEEE,
Gaurav Sharma,Senior Member,IEEE,and MarkF. Bocko,Member,IEEE
Abstract—We consider optimal formulations of spread spec- theembeddedwatermark,iscommunicatedtothereceiverover
trumwatermarkembeddingwherethecommonrequirementsof a channel that may include distortion and noise. The receiver
watermarking, such as perceptual closeness of the watermarked
extracts the watermark information for its use, possibly using
imagetothecoveranddetectabilityofthewatermarkinthepres-
acryptographickeysharedwiththeembedderintheextraction
enceofnoiseandcompression,areposedasconstraintswhileone
metricpertainingtotheserequirementsisoptimized.Wepropose process.Watermarkingisoftentheonlysolutionavailablewhen
an algorithmic framework for solving these optimal embedding secureout-bandchannelsareunavailableforcarryingtheaux-
problemsviaamultistepfeasibilityapproachthatcombinespro- iliaryinformationorwhentheinformationissusceptibletore-
jections ontoconvexsets(POCS)basedfeasibility watermarking
moval.Today,watermarkingisemployedinmanyareas[2]–[4],
with a bisection parameter search for determining the optimum
including,butnotlimitedto,copyrightenforcement[5],authen-
value of the objective function and the optimum watermarked
image. The framework is general and can handle optimal wa- tication[6],broadcastmonitoring[7],andfingerprinting[8].
termark embedding problems with convex and quasi-convex The watermark design process is composed of three major
formulations of watermark requirements with assured conver- steps: coding the watermark data, modulating the coded data
gence to the global optimum. The proposed scheme is a natural
with a suitable technique, and embedding the modulated data
extension of set-theoretic watermark design and provides a link
intothecoverfilewhiletakingintoaccounttheperceptualim-
between convex feasibility and optimization formulations for
watermarkembedding.Wedemonstrateanumberofoptimalwa- pactoftheembeddingandpotentialinterferencefromthecover
termark embeddings in the proposed framework corresponding and the channel. For instance, conventionally in spread spec-
to maximal robustness to additive noise, maximal robustness to
trum(SS) embedding [9],ifthe datato beembedded contains
compression, minimal frequency weighted perceptual distortion,
redundancyitmaybecompressedtoreducethedataload,then
and minimal watermark texture visibility. Experimental results
demonstrate that the framework is effective in optimizing the utilizing a key, a pseudo-noise (pn)-sequence is generated for
desired characteristic while meeting the constraints. The results each bit/symbol to be embedded, and these pn-sequences are
also highlight both anticipated and unanticipated competition then embedded into the cover after perceptual shaping. These
betweenthecommonrequirementsforwatermarkembedding.
stepscanbeperformedseparatelyorjointly,andsomeofthem
Index Terms—Optimum watermark design, projections onto maybeentirelyskippeddependingontheneedsoftheapplica-
convexsets(POCS),settheoreticwatermarking,spreadspectrum
tion.
watermarking.
The coding step is used to make the watermark data re-
I. INTRODUCTION silient to perturbations introduced during transmission of the
D IGITAL watermarking is a special form of digital com- watermarkedimagetotheintendedusers(i.e.,errorcorrecting
coding), to reduce the size of the watermark data when there
municationwhereamultimediacoversignalisutilizedas
is redundancy (i.e., compression), or to provide the necessary
thecarrierofauxiliaryinformation.Thiscarrierdata,alongwith
adaptation of the information to the cover (i.e., informed
coding). Error-coding adds carefully designed redundancy to
ManuscriptreceivedJanuary18,2008;revisedSeptember28,2008.Current
versionpublishedJanuary09,2009.PartofthisworkwaspresentedatICIP the watermark data [10]. Compression, on the other hand,
2006.ThisworkwassupportedbytheAirForceOfficeofScientificResearch works in the opposite direction removing the redundancy of
(AFOSR)undergrantnumberFA9550-07-1-0017.Theassociateeditorcoordi-
the watermark data thereby making it more fragile. Coding
natingthereviewofthismanuscriptandapprovingitforpublicationwasProf.
DanSchonfeld. can also be utilized to provide adaptation to the cover, Trellis
H. O. Altun, A. Orsdemir, and M. F. Bocko are with the Elec- codes [11] and wet paper codes [12] fall in this class. The
trical and Computer Engineering Department, University of Rochester,
modulation step consists of selecting a carrier signal format
Rochester, NY 14627-0126 USA (e-mail: [email protected]; ors-
[email protected];[email protected]). suitable for carrying the watermark information within the
G.SharmaiswiththeElectricalandComputerEngineeringDepartment,Uni- cover.Thischoiceisusuallymadeaccordingtothedistortion,
versityofRochester,Rochester,NY14627-0126USA,andalsowiththeDe-
capacity, and robustness properties desired of the watermark.
partmentofBiostatisticsandComputationalBiology,UniversityofRochester
MedicalCenter,Rochester,NY14642USA(e-mail:gaurav.sharma@rochester. Themostcommonformsofmodulationsarequantizationindex
edu). modulation (QIM) [13] or SS [9] modulation, either of which
This paper has supplementary PDF material available at http://ieeexplore.
maybeutilizedindifferentdomains(e.g.,spatial/frequencyor
ieee.org, provided by the authors. This material is 2.7 MB in file size. The
supplementarymaterialcontainsadditionalimagesthatdemonstratetheperfor- wavelet). The watermark embedding step is the final step to
manceoftheoptimumwatermarkingtechniquesproposedinthispaper. adaptively incorporate the modulated watermark signal within
Colorversionsofoneormoreofthefiguresinthispaperareavailableonline
the cover signal. This step involves the choice of additive or
athttp://ieeexplore.ieee.org.
DigitalObjectIdentifier10.1109/TIP.2008.2008222 multiplicative insertion of the data, perceptual shaping of the
1057-7149/$25.00©2009IEEE
372 IEEETRANSACTIONSONIMAGEPROCESSING,VOL.18,NO.2,FEBRUARY2009
carrier signal, and the interference management between the compression is more important in network broadcast applica-
coverandthewatermarksignals. tions or in other applications when the image is likely to be
In this paper we focus on the watermark embedding step, compressed during transmission. In other applications, water-
specifically for zero-bit spread spectrum watermarking. Em- markdetectabilitywithhighprobabilitymaybedesireddespite
bedding techniques can be classified as optimization formu- being subject to noise in transmission. In order to emphasize
lations, feasibility formulations, and heuristic methodologies. theapplicationspecificcriticalrequirement,watermarkembed-
Heuristic methods are usually practical, computationally effi- dingcanbeformulatedasaconstrainedoptimizationformula-
cient, and intuitive ways of embedding watermarks into cover tionwhereanobjectivefunctioncorrespondingtothecriticalre-
files.Thesemethodsdominatetheliteratureandearlyversions quirementisoptimizedsubjecttootherconstraints.Inthispaper
oftheconventionalmodulationmethodsfallintothiscategory wewillconsidervariousoptimalwatermarkembeddingformu-
[9], [13]. The methods usually do not offer systematic guar- lationsforSSmodulation.
antees for satisfying all the criteria, however, their simplicity SS modulation is a common watermark modulation tech-
makesthemratheruseful.Inoptimizationformulationsofwa- nique. This type of modulation is widely used in communi-
termarkembedding,thewatermarkinsertionisformulatedasa cations due to its wide-band nature and its immunity against
constrainedoptimizationproblem[14]–[16].Therequirements narrow-band interference signals and has been successfully
forthewatermarkedsignalareexpressedinanalyticalformasan adapted to the watermarking field [9]. A key dependent
optimizationofoneperformancemetricwhilemeetingrequire- pseudo-noise (pn) sequence is inserted1 onto the cover
mentsforotherperformancemetrics.Variousscenariosmaybe image and the detection is performed by correlating the
formulated in this framework. For instance, the designer may test image against the same pn-sequence. Among the
wishtominimizethevisualdistortionduetoembeddingwhile many available correlation metrics, we use a mean corrected
maintaining a desired level of watermark embedding strength linear correlation metric [26, p. 127], in our description. Let
androbustnesstocompression.Inotherapplications,themax- denote the image with dimensions
imization of watermark embedding strength or robustness to and denote the pn-sequence. We stack
compressioncanbethekeyobjective.Feasibilityformulations the columns of the matrices together to generate vectors
of watermark embedding [17], [18] provides a middle ground , , and
betweenoptimizationmethodsandheuristicmethods,wherea . Throughout the paper we will use
watermarksignalmeetingdesiredcriteriaisdeterminedwithout this 1-D vector notation and assume that any image operators
explicitly optimizing a performance criterion. Feasibility for- are also represented as matrices conforming to the vector
mulations are typically computationally less demanding com- representation. The mean corrected correlation is represented
paredwiththeoptimizationformulations.Specifically,setthe- as where , and denotes
oreticwatermarking[17]providesaflexibleandreadilyadapt- the sample mean of the entries in the vector . The pres-
able framework in which a wide variety of constraints can be ence/absence of the watermark is determined by comparing
incorporated. the estimated correlation value against a threshold , i.e., if
Inthispaper,weconsiderseveraloptimalwatermarkembed- the watermark is deemed present and if
dingformulationsforazero-bitspreadspectrumwatermarking thewatermarkisdeemedtobeabsentfrom
problem,wherethedetectorisonlyconcernedwithestablishing theimage.2Weconsidernextaspecificoptimalwatermarkem-
the presence/absence of the watermark. We demonstrate that bedding formulation for SS watermarking. Using the notation
these different formulations can be addressed in a common and terminology established in that context we subsequently
frameworkthatcombinesmultistepfeasibilitybasedoptimiza- define three other optimal watermark embedding formulations
tion [19] with projections onto convex sets (POCS) based set fordifferentcriteria.
theoretic watermarking [17] for solving the feasible water-
marking problemat eachstep.ThemethodofPOCS hasbeen A. MaximizationofEmbeddingStrength
successfullyappliedtoanumberofdifferentimageprocessing
In applications where maximal robustness against additive
problems[20]–[25].Ourworkcanbeviewedinthiscontextas
noise is required, it is desirable that the watermarked image
anextensionofPOCSthataddressesthemultiplerequirements
at the embedder maximize the mean corrected correlation
inherent in watermarking applications while simultaneously
, which we refer to as watermark embedding
allowingforoptimizationofonechosenattribute.Theresulting
strength. The maximization of the embedding strength must
methodologyforoptimalwatermarkembeddingisgeneraland
clearly be subject to other constraints. The distortion in the
allows a number of convex optimization criteria under convex
watermarked image with respect to (w.r.t.) the original image
constraints.
should be perceptually tolerable. In addition, in some
applications, it may be desirable that the watermark survive
II. OPTIMALWATERMARKEMBEDDING compression up to a given quality level. In order to account
for these constraints, we formulate the optimal watermark
Dependingontheneedsofthespecificapplication,someper-
formancerequirementsofwatermarkingcanberelativelymore 1Ourformulationrequiresonlyadefinitionofthedetector.Theembeddingis
animplicitoutcomeoftheoptimization.
importantcomparedtoothers.Inmedicalandmilitaryapplica-
2Notethatweassumeobliviousdetection,wherethecoverisunavailableat
tions,thevisualqualityiscriticalsincesensitivevisualinforma-
thedetector.Thenonobliviouscasemaybesimilarlyformulatedandactually
tioniscommunicated.Ontheotherhand,theresilienceagainst representsasimplification.
ALTUNetal.:OPTIMALSPREADSPECTRUMWATERMARKEMBEDDINGVIAAMULTISTEPFEASIBILITYFORMULATION 373
embedding for maximization of embedding strength as the the quantizer in fact consist of distinct scalar quan-
constrainedoptimization tizers foreachofthe transformdomainco-
efficients.Thus,forJPEGcompressionthe term indexesthe
64 different quantizers used for 8 8 block DCT coefficients,
repeatedly tiled so as to cover the entire image. The formula-
tion addresses any transform domain coding scenarios where
subject to (1) thetransformdomaincoefficientsarequantizedviascalarquan-
(2) tizers.
B. MinimizationofFrequencyWeightedPerceptualDistortion
(3)
For applications, where a minimization of the frequency
weighted perceptual distortion metric is desirable, the optimal
where the first two constraints (1) and (2) capture the require-
watermarkembeddingproblemcanbeformulatedas
ments of perceptually tolerable distortion and the third repre-
sentsthecompressionresiliencerequirement.Theseconstraints
andtheassociatedsymbolsintheirdefinitionsaredefinednext.
Constraint (1) is based on the contrast sensitivity of the subject to
human visual system (HVS), where is a linear spatial filter
that represents the characteristics of the HVS. Specifically,
the higher sensitivity of the HVS to lower frequencies com-
pared with higher frequencies is encapsulated in . The term (5)
thus represents a frequency weighted mea-
sure representing the perceptual distortion in the candidate where denotes the desired embedding strength at the em-
watermark image . In the present work, we use the para- bedder,andalltheothertermsareasdefinedearlier.Normally
metric contrast sensitivity model proposed by Mannos and ischosentobelargerthanthedetectionthreshold atthe
Sakrison [27]. For the model, the spatial filter is repre- watermarkdetector.Thegap characterizestherobust-
sentedinthefrequencydomainbyaradiallyisotropicfunction ness of the watermark and can be related to the probability of
, where de- errorforwatermarkdetectionfollowingpostwatermarkingper-
notestheradialfrequencyincyclesperdegree[27]. turbations[2].
Constraint (2) exploits texture masking by virtue of which
watermarkdistortionintroducedintexturedregionsislessvis- C. MaximizationofRobustnesstoCompression
ible than in smooth regions. The and in (2)
Forscenarioswheremaximalcompressionresilienceisdesir-
represent pixel-wise upper and lower bounds on tolerable dis-
able,analternateoptimalwatermarkembeddingproblemcanbe
tortion that are determined from the cover image . For de-
formulatedas
termining and ,weusethemodelofPereira
et al. [14], which is briefly summarized in Appendix I. Based
onthedevelopmentinAppendixI,wewillfinditnotationally
convenient,intheensuingpresentation,torepresentconstraint subject to
(2)inanalternateformas
(6)
(4)
where is a diagonal matrix determined from the orig- D. MinimizingWatermarkTextureVisibility
inal image using the texture masking model and
In a manner analogous to the preceding formulations, one
representsthe -normofthevector .
mayalsoformulatetheoptimalembeddingproblemtominimize
Theterm representsameasureofthe(worst
ametricrepresentingtheperceptibilityofthetexturenoisein-
case)visibilityofthetexturenoiseintroducedintheembedding
troducedbywatermarkembedding.Theproblemmaybemath-
andtheparameter representsaboundonthismeasure.
ematicallyformulatedas
Constraint(3)representsthecompressionresiliencerequire-
mentfortransform domaincompression,where represents
the forward transform, represents the quantization opera-
tion in the transform domain. represents the inverse trans- subject to
form. The constraint thus enforces a mean corrected correla-
tion greater than in the compressed image. Specifically, in
this paper we consider JPEG compression, where and (7)
represents the forward and inverse discrete cosine transforms
(DCTs),respectively,and istheJPEGquantizationfunction Asindicatedearlier,thedevelopmentofthismetricispresented
at a predetermined quality level.In the transform coding[28], inAppendix I.
374 IEEETRANSACTIONSONIMAGEPROCESSING,VOL.18,NO.2,FEBRUARY2009
III. CONVEXOPTIMIZATIONFRAMEWORKFOROPTIMAL
WATERMARKEMBEDDING
For the optimal watermark embedding formulations in the
precedingsectiontobeuseful,efficientmethodsfordetermining
the solution for the corresponding problems are required. We
observethatwiththeexceptionofcompressionresilience,other
constraints and objective functions in our formulations are in-
herentlyconvex.Notethatthewatermarktextureperceptibility
constraint depends on the cover image in a complex fashion.
However, given the cover image it is manifested simply as an
upperandlowerboundconstraintonpixelvaluesthatisclearly
convex [17], [29]. Since convex optimization problems have
uniqueglobaloptimaandthereareavarietyofwellbehavedop-
timizationalgorithmsfortheproblems,weconsidertheoptimal
watermarkembeddingproblemwhentheobjectivefunctionand
theconstraintsareconvex.
Toaddresscompressionresilience,weintroducetheconvex(in
factlinear)approximation
Fig.1. Bisectionsearchinthemultistepfeasibilityformulationofoptimum
where watermarking.
represents the approximation of obtained by
approximatingthescalarquantizer forthe thtransform
Let denote a real-valued number, and consider the feasi-
domaincoefficientby whichisdefinedas
bilityproblem:
Find
otherwise.
(8) subjectto
Thisapproximatesthecompressionoperationasaprojection (10)
onto the subspace of transform bases that the quantization
operation in compression does not obliterate. The efficacy This feasibility problem is closely related to our optimiza-
of this approximation for JPEG compression has been ex- tionproblemin(9).Inparticularif istheoptimumvaluefor
plored in [17]. Given the cover image, the concatenation the objective function in our constrained optimization (9), the
is also a linear operator, thus feasibility problem(10) has a solution if and only if .
Supposenextthatanalgorithmexistsforsolvingthefeasibility
represents a
problemin(10)thatprovidesafeasible whenoneexistsand
linearand,therefore,aconvexconstraint.
indicatesthatnosolutionexistsotherwise.Thenbycombining
Wenowconsidertheoptimalwatermarkingproblemformu-
thealgorithmwithabisectionsearchon wecanobtainaso-
latedinSectionII-Aanddescribeitinanalternategeneralform
lutiontotheoptimizationproblemasindicatedintheaforemen-
tionedmultistepfeasabilityalgorithm.Theoperationoftheal-
gorithm is illustrated pictorially in Fig. 1 for the case of two
subjectto (9) constraints.Atthestartofiteration , andtheproblem
(10) has a feasible solution for the upper limit and
where , no feasible solution exists for the lower limit . There-
, , and fore, . A solution of the feasibility problem is
,where , , . attemptednextfor themidpoint ofthelowerandupperlimits,
Thisoptimizationproblemnowrepresentsaconvexoptimiza- i.e., .Ifasolutionexists oth-
tionproblem.Following[19,p.159],wenotethatasolutionto erwise .Accordingly,intheformercase is
theproblemcanbeobtainedviaamultistepfeasibilitytechnique setto andinthelattercase issetto .Atthestartofthe
thatwebrieflyoutline. iteration onceagain andtheprocess
isrepeated.Ineachiteration,theinterval isbisectedso
1) Initialize ,andset , suchthattheproblem(9) thatafterexactly iterations,thealgorithm
hasnosolutionfor andhasafeasiblesolutionfor willterminate,where isthetoleranceparameter.
A. MultistepConvexFeasibilityFormulationofWatermarking
2) , , and .
3) Solvefeasibilityproblem(10)for . Since the level set of the convex ob-
4) Ifthereexistafeasiblesolution,set ,else . jective function is also a convex set, (10) represents a
5) If goto(2),elsedeclarethelastfeasible convex feasibility problem. The solution to this problem can,
solutionastheoptimumandstop. therefore, be obtained by utilizing the method of POCS [30],
ALTUNetal.:OPTIMALSPREADSPECTRUMWATERMARKEMBEDDINGVIAAMULTISTEPFEASIBILITYFORMULATION 375
[31].Specifically,definethesets , fourdifferentoptimizationformulations.Forourdescriptionwe
and , . Then if assumethat denotesthewatermarkedimageobtainedfrom
, starting from any arbitrary ini- theoptimizationformulationunderdiscussion.
Observing that represents the correlation
tialimage ,thesequenceofimages convergestoa
value of the watermark sequence with itself, we normalize
feasiblesolutionof(10),where
the embedding strength and the upper bound for the embed-
ding strength by this value. Thus, for the optimization of
Section II-A, we report the optimum values of the embedding
and denotestheprojectionof strength by recording , and for
ontotheconvexset . the other formulations we establish the embedding strength
In fact this feasibility problem represents a particular case constraintintermsofthecorrespondingnormalizedparameter
of the set theoretic watermarking formulation [17]. Projection value .Specifically,weutilizethevalues
operators for thesesets were enumerated in [17] and are sum- 1,3,and5inourexperiments.
marized in Appendix II for completeness. Using these projec- Inananalogousfashion,fortheoptimizationformulationof
tionoperators,thefeasibilityproblem(10)canreadilybesolved Section II-C, we report the optimal value in the normalized
usingPOCS.Byemployingthis“feasibilitysolver”inthemul- form
tistepfeasibilityoptimizationofFig.1,weobtainasolutionfor
and for the other formulations the upper-bound parameter for
the optimal watermark embedding formulation of (10) (which
the robustness to compression constraints is given in terms of
isidenticaltoSectionII-A).
.Weusethevalues 1,2,and3inourex-
For the purpose of concrete illustration, we utilized the op-
periments.
timal watermark embedding problem of Section II-A in our
Iftheimagerepresentationhasamaximalvalueof (e.g.,
description of the multistep feasibility formulation. One can
for 8 bit images), the peak signal energy at the
immediately see that the framework presented here is general
outputoftheHVSfilter foranyinputoriginalimagecanbe
andalsoaddressestheoptimalembeddingformulationsofSec-
upper-boundedas
tions II-B–D (with the convex approximation of compression
resilience). Furthermore, additional convex constraints can be
readily incorporated in the watermarking process through the
additionofsuitableconstraintsets. where denotestheFouriertransformoftheHVScontrast
sensitivityfilter and denotesthefrequencyatwhichthe
IV. EXPERIMENTALRESULTS magnitudeofthisFouriertransformattainsamaximum.Using
Wereportexperimentalresultsforeachofthefourdifferent thisupperboundfornormalization,fortheoptimizationformu-
optimal watermark embedding scenarios described in Sec- lation of Section II-B, we report the frequency weighted per-
tionIII.Wefirstoutlineourexperimentalsetupthenwepresent ceptualdistortionmetric intermsofthepeak
results for a sample image that highlight the impact of the visualsignaltowatermarkpowerratio
differentformulations.Finally,wepresentnumericalresultsfor
theoptimizationsintabularformat.Wenotethatthewatermark
detectionperformancecanbeinferredinlargemeasurefromthe
constraintparametersandtheoptimalvalues.Forthefeasibility Correspondingly, for the other optimization frameworks, the
formulation,thishasalsobeenextensivelyinvestigatedin[17]. upper bound for the frequency weighted perceptual distortion
In this paper, we, therefore, focus our attention on the impact isstatedintermsofacorrespondingupperboundforthepeak
ofthe differentoptimizationsand theinteractions betweenthe visualsignaltowatermarkpowerratio
optimalvaluesandtheconstraintparameters.
A. ExperimentalSetup
Werecallthatourcoverimagehas pixelssothat We utilize the values 49.68, 51.44, and 54.45
and are vectors and is a matrix dB in our experiments corresponding to values of 120000,
representing the HVS contrast sensitivity function as a linear 80000, and 40000, respectively (with ).
spatially invariant filter. For our SS watermark , we utilize For the watermarked images, we also report the peak
a randomly generated iid sequence taking values and , signal to watermark power ratio defined as
equiprobably.Fortherobustnesstocompressionconstraints,the .
quantizerapproximation wasdeterminedcorrespondingto The parameterforthetexturemaskingmodeldenotesthe
aJPEGqualityfactorof70. maximum allowable deviation at any pixel and does not re-
1) Normalized Constraint Parameter Values and Objective quire normalization. We use the values 10, 20, and 30
Functions: Inordertoallowmoreintuitiveunderstandingofthe for the constraints. Note that this deviation is infact allowable
numericalvaluesusedinthesimulationsandthenumericalre- onlyinthe limitinextremelybusyregionsofthecoverimage
sultsoftheoptimizations,weintroducenormalizedversionsof (wherethenoisevisibilityfunctiontendsto1).Intheoptimiza-
oursimulationparametersthatdefinetheconstraintsandcorre- tion formulation of Section II-D the optimum value of
spondingnormalizedversionsoftheobjectivefunctionsforour isreporteddirectlyandthisrepresentsthe
376 IEEETRANSACTIONSONIMAGEPROCESSING,VOL.18,NO.2,FEBRUARY2009
Fig.2. Originalimages.(a)Boatimage.(b)Goldhillimage.(c)Lenaimage.(d)Tankimage.
maximumnoisevisibilityweightedabsolutedeviationfromthe resultsfortheBoatimageforeachofthefouroptimalembed-
coverimageatanypixellocation. dingformulations.3Sincethegoalistohighlightdifferencesin-
2) Test Images: We test our algorithms using four images ducedbydifferentobjectivecriteria,constraintparametervalues
from USC image database [32], specifically: Lena, Barbara, arechosencorrespondingtothemostrelaxedconstraintsfrom
Tank, and Goldhill. These are 8-bit gray-scale images with a among the values indicated earlier. This allows the optimiza-
sizeof512 512pixels.Theoriginalversionsoftheseimages tion choice over a larger set of possibilities making the im-
are shown in Fig. 2. We performed the optimal embedding pactoftheobjectivefunctionmoresalient.Specifically,weuse
for different parameter selections for each of the four images , , , and for
for each of the four optimality formulations. In each case, definingtherequiredconstraints.4Optimalwatermarkedimages
all the different values of the constraint parameters listed in correspondingtothefourdifferentembeddingformulationsare
the preceding section were utilized, resulting in 27 different showninFig.3.Thespatialdomainwatermarksignal
combinations of constraint parameters for each optimization foreachoftheseoptimalembeddingsisshowninFig.4,where
formulation.Duetothecomputationallyintensivenatureofthe forthepurposeofclearervisualpresentation,thesignalhasbeen
algorithm, our evaluation is limited to this relatively small set scaled by a factor of 4 and translated about a mid gray value
ofimages.Ourresults,however,servethedesiredobjectiveof of 128 to allow both positive and negative values to be repre-
highlighting the effect of the different optimality formulations sented. In order to show the distribution of watermark energy
and the interaction between the constraints and the optimal inthefrequencydomain,themagnitudeofthediscreteFourier
valuesoftheobjectivefunctions. transform (DFT) of the watermark signal for each
B. WatermarkedImageforDifferentOptimalEmbeddings 3Correspondingresultsfortheotherimagesareincludedinsupplementary
materialspostedonline.
Inordertodemonstratehowthechoiceoftheobjectivefunc-
4Noteeachoptimizationformulationonlyrequiresasubsetofthreeofthese
tion impacts the watermarked image, we begin by presenting parametervalues.
ALTUNetal.:OPTIMALSPREADSPECTRUMWATERMARKEMBEDDINGVIAAMULTISTEPFEASIBILITYFORMULATION 377
Fig.3. WatermarkedversionsoftheBoatimageobtainedforthefourdifferentoptimizationformulations.(a)Maximizingembeddingstrength.PSWRofthe
imageis24.60dB.(b)Minimizingfrequencyweightedperceptualdistortion.PSWRoftheimageis37.71dB.(c)Maximizingrobustnesstocompression.PSWR
oftheimageis24.79dB.(d)Minimizingthewatermarktexturevisibility.PSWRoftheimageis37.28dB.
of these optimal embeddings is shown in Fig. 5. For each of watermark largely resembles white noise with the power dis-
thesefigures,subfigures(a)–(d)correspond,respectively,tothe tributedoverallthefrequencybands.Whenmaximizingtheem-
watermarked images with maximal embedding strength (Sec- beddingstrengththewhitenoisenatureofthespreadspectrum
tionII-A),withminimalfrequencyweightedperceptualdistor- watermark dominates over the other constraints for this cover
tion (Section II-B), maximal robustness to compression (Sec- image.
tionII-C),andminimalwatermarktexturenoisevisibility(Sec- The watermarked Boat image generated by minimizing fre-
tion II-D). quencyweightedperceptualdistortionisshowninFig.3(b).The
ThewatermarkedBoatimagegeneratedbymaximizingem- PSWR of the image is 37.71 dB. The watermarked image is
bedding strength is shown in Fig. 3(a). The PSWR of the wa- visually pleasing. The difference image between watermarked
termarkimageis24.60dB.Thedifferenceimagebetweenwa- image and the original image in spatial domain is shown in
termarked image and the original image, representing the em- Fig.4(b).Theadaptationoftheembeddedwatermarksignalto
beddedwatermarksignalisshowninFig.4(a)inthespatialdo- thecoverimageisstillvisible;however,thecontrastofthedif-
main.Thisfigureclearlyshowstheadaptationoftheembedded ferenceimageinspatialdomainislowercomparedtotheother
watermarksignaltothecoverimage.Thetexturedregionsofthe algorithms.Theeffectofthealgorithmcanbebetterobservedby
image, such as the sandy shore displayed at the bottom of the examiningthewatermarkinfrequencydomaininFig.5(b).The
image,hostmorewatermarkpowercomparedtothe smoother watermarkpowerispushedoutofthevisiblefrequencyregion.
regions of the image such as the sky region. The effect of the Thecircularregionseeninthemiddleofthespectrummatches
maximizationofthedetectabilityalgorithmcanbeobservedby thefrequencyregionoverwhichthefilterrepresentingtheHVS
examining the frequency domain representation of the water- contrastsensitivitytakesonsignificantvalues.
mark in Fig. 5(a). In the frequency domain, despite the water- The watermarked Boat image generated by maximizing ro-
mark power shaping seen in the spatial domain, the designed bustness to compression is shown in Fig. 3(c). The PSWR of
378 IEEETRANSACTIONSONIMAGEPROCESSING,VOL.18,NO.2,FEBRUARY2009
Fig.4. Embeddedwatermarksignalscorrespondingtothedifference(cid:0) (cid:0)(cid:0) betweenthewatermarkedBoatimageandtheoriginalBoatimageforthefour
differentoptimizationformulations.(a)Maximizingembeddingstrength.(b)Minimizingfrequencyweightedperceptualdistortion.(c)Maximizingrobustnessto
compression.(d)Minimizingthewatermarktexturevisibility.Valueshavebeenscaledbyafactorof4andtranslatedtoamidgrayvalueof128tomakedifferences
clearerandtoallowrepresentationofbothpositiveandnegativevalues.
this image is 24.79 dB. This optimization scenario introduced this metric, therefore, causes the watermark power to be dis-
visible artifacts in the watermark image that are similar toag- tributedwithinthesebases.Thefrequency domainrepresenta-
gressiveJPEGcompressionartifacts.Thisismainlyduetothe tionofthewatermarkpowerinFig.5(c)illustratesthatthemax-
natureoftheobjectivefunction.Inordertoprovideresilienceto imizationofrobustnesstocompressiontendstopushthewater-
JPEG compression the maximization of robustness causes the markpowertowardthelowfrequencyregion.
watermarksignalpowertobedistributedinlowerfrequencyre- The watermarked Boat image generated by minimizing the
gionsoftheimage,whichusuallyfallsintothevisibleregions. watermark texture visibility metric is shown in Fig. 3(d). The
Itisimportanttonotethattheseartifactscanbeminimizedby PSWR of the image is 37.28 dB. The watermarked image is
tightening the visual quality constraints. However, we utilize visually pleasing. The difference image between watermarked
ratherrelaxedvisualqualityconstraintsinordertodemonstrate imageandtheoriginalimageisshowninFig.4(d).Thewater-
the dominant distortion encountered in this optimization. The markpowerinthespatialdomainisconcentratedinregionsof
differenceimagebetweenwatermarkedimageandtheoriginal high spatial activity where the embedded watermarked signal
image(i.e.,embeddedwatermarksignal)isshowninFig.4(c). isbettermaskedbythecontentinthecoverimage.Thespatial
This figure illustrates the adaptation of the watermark to the distributionofenergyinthewatermarksignalresemblesthedis-
coverimage.Inaddition,someblockingisseenintheembedded tributionobtainedbymaximizingrobustnesstocompressionin
watermarksignalinlesstexturedregionsoftheimage.Thero- Fig.5(c),however,thescalesarequitedifferent,theoveralldis-
bustnesstocompressionmetricmeasuresthewatermarkpower tortionbeingmuchlowerinthepresentcase.Thesimilarityin
distributed in the DCT bases that are preserved in the original thespatialdistributionofwatermarkenergyexistsbecausemore
image in the compressed representation. The maximization of ofthehighfrequencyDCTbasesarepreservedinregionswith
ALTUNetal.:OPTIMALSPREADSPECTRUMWATERMARKEMBEDDINGVIAAMULTISTEPFEASIBILITYFORMULATION 379
Fig.5. FrequencydomaindistributionofwatermarkpowerfortheoptimallywatermarkedBoatimages.Thefiguresrepresentthemagnitudesoftheembedded
watermarksignalsforthefourdifferentoptimizationformulations.(a)Maximizingembeddingstrength.(b)Minimizingfrequencyweightedperceptualdistortion.
(c)Maximizingrobustnesstocompression.(d)Minimizingthewatermarktexturevisibility.
highspatialactivityintheimage.Thefrequencydomainrepre- C. OptimumValuesasaFunctionofConstraintParameters
sentationofthewatermarksignalinFig.5(d)illustratesthatthe
watermarkpowertendstoconcentrateatlowerfrequencies.This The optimum value of the objective function achievable in
is because the spatial domainshaping of the embeddedwater- each of the constrained optimization formulations depends on
marksignalalsocausesthewatermarksignalpowerdistribution the constraint parameters and on the cover image content. In
toapproximatethelowpasscharacteristicofthecoverimage. general,changesinparametervaluesexertavaryinginfluence
Toillustratetheimpactofthecoverimagecontent,wepresent ontheoptimumvalue.Weillustratethisbyconsideringeachof
inFigs.6–8thewatermarkedimages,theembeddedwatermark theoptimalformulationsandtabulatingtheoptimumvaluesob-
signalsinthespatialdomain,andthemagnitudeofthediscrete tained for the objective functions for each of the 4 images for
Fouriertransformsfortheembeddedwatermarksignalsforthe the27possibleparametervaluesfortheconstraintsenumerated
Lena image for the different optimality formulations. The re- earlier.Thevariationintheoptimalvalueswithchangeincon-
sults largely follow the trends observed for the Boat image, straintparametervaluesnotonlyindicatetheinfluenceofeach
though, of course the shaping of the watermark power adapts of the constraint parameters on the optimum but also serve to
to the different content for this image. Specifically comparing establishlimitsforwhatisattainableundertheconstraints.
Fig.8(c)and(d)withFig.5(c)and(d),respectively,weseethat Table I lists the maximum embedding strength obtained
thesincethepowerintheLenaimageconcentratesinlowerfre- fortheformulationofSectionII-Aforvariousparametervalues.
quenciesthanintheBoatimage,sodoesthewatermarksignal Themaximumembeddingstrengthfollowsexpectedtrendswith
power for the optimality formulations for maximizing robust- the change in the constraint parameter values: increasing with
ness to compression and minimizing watermark texture visi- an increase in values of and , and decreasing with an in-
bility.Theresultsforthetwootherimages(includedinthesup- creasein .Forsomecases,however,therelaxationofacon-
plementarymaterials)supportthesedeductions. straintdoesnotchangetheoptimalvaluebecausetheconstraint
380 IEEETRANSACTIONSONIMAGEPROCESSING,VOL.18,NO.2,FEBRUARY2009
Fig.6. WatermarkedversionsoftheLenaimageobtainedforthefourdifferentoptimizationformulations.(a)Maximizingembeddingstrength.PSWRofthe
imageis26.37dB.(b)Minimizingfrequencyweightedperceptualdistortion.PSWRoftheimageis37.08dB.(c)Maximizingrobustnesstocompression.PSWR
oftheimageis26.14dB.(d)Minimizingthewatermarktexturevisibility.PSWRoftheimageis36.71dB.
isinactiveattheoptima.Ingeneral,thewatermarktexturevis- lowedbytherobustnesstocompressionconstraintparameter
ibilityconstraintparameter exertsthemostinfluenceonthe andthenbytheembeddingstrengthparameter ,indecreasing
optimalvalueoftheembeddingstrength,thisisfollowedinde- order of impact. The strong influence of reflects the com-
creasingorderbytherobustnesstocompressionconstraintpa- plementary nature of the two visual constraints. Whereas, the
rameter andbythefrequencyweightedperceptualdistortion maximization of the frequency weighted measure of percep-
bound . The maximal values of also exhibit vari- tual distortion , which is based on the HVS contrast
ation along expected lines based on the content of the cover sensitivity, attempts to move the watermark signal power into
image:themaximalvaluesforthesameparametervaluesforthe highfrequencies,itisconstrainedfrommovingthepowerinto
constraintsincreaseprogressivelyasthecoverimagebecomes smoothregionsoftheimagebythewatermarktexturevisibility
busier(intheorderLena,Tank,Boat,andGoldhill). constraints.
TableIIsummarizestheminimumfrequencyweighteddistor- The maximum values for the robustness to compression
tionobtainedfortheformulationofSectionII-Busingthepro- metric areenumeratedinTableIIIforthe4imagesandeach
posedoptimizationmethodologywheretheoptimalvaluesare of the 27 values of the constraint parameters. The resulting
reportedintermsofthepeakvisualsignaltowatermarkpower optimum values of are higher when the visual constraints
ratio definedearlier.Thepeakvisualsignaltowater- are relaxed, i.e., for higher and higher ; however, they
mark power ratio increases as the constraints are relaxed but diminish with a higher embedding strength constraint, i.e.,
once again demonstrates several scenarios where a relaxation higher . While this initially appears counter-intuitive, it can
of one or two of the constraints does not change the optimal beunderstoodfromthefactthattherobustnesstocompression
value due to the constraints being nonbinding. The numerical and embedding strength compete for the same “perceptual
valuesindicatethatthevisualtexturemodelconstraintparam- bandwidth” controlled by the visual constraint sets. Among
eter exerts the strongest influence on the optimal value fol- the constraint parameters, the watermark texture visibility