PERFORMANCE ANALYSIS OF WATERMARKING SCHEMES BASED ON SKEW TENT CHAOTIC SEQUENCES
PERFORMANCE ANALYSIS OF WATERMARKING SCHEMES BASED ON SKEW TENT
CHAOTIC SEQUENCES
A.Tefas,A.Nikolaidis,N.Nikolaidis,V.Solachidis,S.Tsekeridou and I.Pitas
Department of Informatics,Aristotle University of Thessaloniki
Box451,Thessaloniki54006,GREECE,pitas@zeus.csd.auth.gr
ABSTRACT
In this paper,statistical analysis of watermarking schemes based
on correlation detection is presented.Statistical properties of wa-
termark sequences generated by piecewise-linear Markov maps
are exploited,resulting in superior watermark detection reliabil-
ity.Correlation/spectral properties of such sequences are easily
controllable,a fact that re?ects on the watermarking system per-
formance.A family of chaotic maps,namely the skew tent map
family,is proposed for use in watermarking schemes.The pro-
posed chaotic sequences are compared against the widely used
pseudorandom sequences,indicating the superiority of the former
in watermarking applications.
1.INTRODUCTION
The design of robust techniques for copyright protection and con-
tent veri?cation of multimedia data became an urgent necessity
in the last years.This demand has been lately addressed by the
emergence of a variety of watermarking methods.Such methods
target towards hiding an imperceptible and undetectable signal in
the original data,which conveys copyright information about the
owner or authorized user.For a review of existing schemes and
a detailed discussion on the main requirements of a watermarking
scheme,the interested reader may consult[1].
So far,performance evaluation of the existing watermarking
methods has been mostly experimental without any theoretical jus-
ti?cation of their ef?ciency.Only few approaches have attempted
to statistically analyze the performance of image watermarking
schemes in terms of detection reliability by addressing the prob-
lem in a communication framework[2,3,4].In these papers,the
statistical properties of watermarking schemes based on pseudo-
random watermark signals and correlation detectors,among oth-
ers,are derived.In[3],the authors investigate the performance
of white and lowpass-?ltered pseudorandom watermarks conclud-
ing that the former are ideal when no distortions are in?icted on
the image,whereas the latter provide additional robustness against
lowpass distortions.
This paper deals with the statistical analysis of the behavior of
a blind copyright protection watermarking system utilizing chaotic
watermark signals generated by piecewise-linear Markov maps.
An overview of chaotic watermarking techniques can be found in
[5].However,up to now,their performance has been evaluated
solely within an experimental framework.The system is modeled
in a communication framework considering the host signal as in-
In order to decide on the valid hypothesis,is compared against
a suitably selected threshold.For a given threshold the sys-
tem performance can be measured in terms of the probability of false alarm,(i.e.,the probability to detect a watermark in a signal that is not watermarked or is watermarked with a different
watermark)and the probability of false rejection(i.e.,the probability to erroneously neglect the watermark existence in the signal).The plot of versus is called the receiver operating characteristics(ROC)curve of the corresponding watermarking system.This curve conveys all the necessary system performance information.
For the watermark sequences that will be studied in this paper,
i.e.,the sequences generated by piecewise linear Markov maps,the
correlation output is normally distributed(see Section3).Thus,
it can be fully determined in terms of its mean,,and variance,,which can be derived in a straightforward manner:
(3)
where is a constant that controls the range of the watermark se-
quence,and is the unit vector.By substituting(12)in(3)and
(4)and considering that,according to(11), it is straightforward to derive the mean value and the variance of
the correlation.The constant value is usually chosen to be the
mean value of the chaotic sequence x in order to have a DC free
watermark which,according to[4],results in better system perfor-mance.Moreover,by subtracting the test signal mean value prior
to detection,we can decrease the variance of the correlation,thus
obtaining better system performance.By using a DC free water-
mark and subtracting the test signal mean value prior to detection the mean value and the variance of the correlation are given by:
(13)
(14) where is the Dirac delta function,is the mean value of the chaotic sequence and k is given by(5).
Expressions(13)and(14)are suf?ciently broad to include all
events that occur in the watermarking model described in Sec-
tion2,provided that piecewise linear Markov maps are used to generate the watermark sequence.That is,the case of watermark absence(event)is represented by setting the watermark em-bedding factor equal to zero.The case of watermark presence is represented by positive watermark embedding factor and in the case of right watermark presence(event)or in the case of wrong watermark presence(event).The correlation statistics needed for evaluating expressions(13)and(14)can be derived in closed form or evaluated numerically[7].
Although samples of Markov chaotic watermarks are corre-
lated for small,since they posses exponential autocorrela-
tion function and w is a shifted version of w,the Central Limit
Theorem for random variables with small dependency[8]may be used in order to establish that the correlation in eq.(2)attains a Gaussian distribution,even in the case of wrong watermark pres-ence(assuming that is suf?ciently large).Furthermore,under the worst case assumption(event),both and,given by eqs.(13)and(14)respectively,converge to constant values for large.In such a case,substitutes since it is the worst case and it can be estimated using the limit values()
of and.values are estimated using the values of and
for(event).
Moreover,if we examine in detail the mean value of the cor-
relation given by(13)we can notice that the mean value converges
to zero for event.Additionally,for event the mean value
of the detector is equal to the variance of the watermark multiplied by the embedding power.This addresses the fact that the mean value of the correlation depends only on the power and the variance
of the watermark and not on the watermark generator(chaotic or
pseudorandom),or the spectral properties of the watermark signal.
The aforementioned remark leads us to the conclusion that,
for watermark signals of the same power and the same variance,
the watermarking system performance is affected only by the vari-ance of the correlation detector.That is,the lower the variance of
the correlation for events and,the better the watermark-
ing system performance.Therefore,the objective is to construct
watermarks that result in small correlation variance.According to(14),this can be achieved by utilizing watermark signals with
suitable?rst,second and third order correlation statistics.
When event holds it can be easily observed that the cor-
relation variance depends only on the watermark autocorrelation function.The autocorrelation function of a signal is di-rectly associated with its power spectral density(psd)which is
given by:
(15) Therefore,the spectral properties of the watermark signal deter-mine the variance of the correlation for the event.Moreover, if we consider the exponential autocorrelation function of Markov chaotic sequences given by(6),it can be easily derived that the correlation variance given by(14)depends on the sum over the samples of the autocorrelation function in the interval, which is minimized for,and maximized https://www.wendangku.net/doc/7511396587.html,-ing(15)one can observe that the two cases correspond to the most highpass and most lowpass signals that can be generated having exponential autocorrelation function.Considering the above dis-cussion,one can conclude that highpass watermarks perform better than lowpass ones,when no attacks on the watermarked signal are considered,since the correlation variance is reduced.
4.THE SKEW TENT MAP
In this section,analysis techniques presented so far are being ex-empli?ed using the skew tent map which is a piecewise linear Markov map.The skew tent map can be expressed as:
(16)
and the?rst order correlation statistic(autocorrelation function)
is given by:
(19)
It can be observed that the autocorrelation function depends only
on the parameter of the skew tent map.Thus,by controlling
the parameter we can generate sequences having any desirable
exponential autocorrelation https://www.wendangku.net/doc/7511396587.html,ing(15),(19),the power
spectral density of the skew tent map sequences can be shown to
be:
(21)
(22)
where is the parameter of the host signal autocorrelation function
given by(6).
5.EXPERIMENTAL RESULTS AND DISCUSSION
In order to experimentally verify the theoretical performance anal-
ysis of a watermarking system based on correlation detection,the
system is fed with a host signal that is compliant with the au-
tocorrelation model in(6).More speci?cally,the system is fed
with a uniformly-distributed zero mean random white signal of
45000samples that has been pre-?ltered with an IIR?lter hav-
ing system function
which is equivalent to the model in(6)for
and?ltered signal variance
the
tent map generates sequences that have the same spectral prop-
erties with the corresponding Bernoulli sequences.It is worth
Fig.2.(a)Comparison between sequences with the same spectral characteristics for use watermarking schemes.(b)Equal error rate of watermarking schemes based on skew tent maps versus the number of watermarked data samples.(c)ROC curves of watermark sequences, having the same spectral properties,after mean?ltering of the watermarked signal using window of size3.
noting here that Bernoulli chaotic maps can generate only low-pass sequences.Lowpass characteristics of Bernoulli sequences weaken as,where the sequences obtain a white spectrum. The most lowpass sequence that can be generated,using Bernoulli maps,is the one obtained for.This is a very crucial lim-itation in the?exibility of Bernoulli maps.On the contrary,tent chaotic watermarks can generate any sequence with exponential autocorrelation function and thus any desirable spectral character-istics.
Experimental results prove that sequences produced by tent maps have superior performance,in terms of the ROC curve,com-pared to white pseudorandom and Bernoulli sequences attaining the same spectral properties.The theoretical ROC curves for white tent(),pseudorandom white,lowpass tent()and lowpass Bernoulli()are plotted in Figure2a.The experi-mental ROCs are also plotted in the same Figure and illustrate the accordance between theoretical and experimental results.
Another important aspect that can be treated by exploiting the theoretical analysis presented in the previous Sections,is the min-imum number of watermarked data samples required for a water-marking scheme based on correlation detection in order to achieve a certain prespeci?ed performance.This number can be estimated by setting the desired and values and using(13),(14).The EER(operating state where)versus the number of wa-termarked data samples is plotted in Figure2b for a system based on tent chaotic watermarks.It can be observed that the number of samples required,for a reliable watermarking scheme(e.g.EER ),is80000for a highpass spectrum watermark and this number increases to190000samples for a white watermark.For a lowpass tent watermark the minimum number is much larger.The issue of the minimum watermark sequence length that achieves a certain performance is very critical in multi-bit watermarking, where this number corresponds to the minimum number of sam-ples needed for encrypting just one bit.Thus,for an-bit message the lower bounds are times the ones derived above.
Experiments were conducted to investigate robustness of the pseudorandom and the chaotic sequences against mean?ltering of window size3.The ROC curves after mean?ltering are plotted in Figure2c.It can be observed that watermarks having lowpass characteristics(Bernoulli with,tent with)attain better performance than white or highpass ones.The best perfor-mance is achieved using lowpass tent watermarks having.
6.CONCLUSIONS
In this paper,chaotic watermarks generated by Markov maps are introduced and their watermarking related statistical properties are investigated.Highpass chaotic watermarks prove to perform bet-ter than white ones whereas lowpass watermarks have the worst performance when no distortion is in?icted on the watermarked signal.Moreover,Markov maps that have appropriate second and third order correlation statistics,like the skew tent map,perform better than sequences with the same spectral properties generated by either Bernoulli or pseudorandom number generators.
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