Performance analysis of sign quantized projection

Available online at www.sciencedirect.comScienceDirectThe Journal of ChinaUniversities of Posts andTelecommunicationsEL SEVIERFebruary 2010, 17(1): 62- 66www.sciencedirect.com/science/joumal10058885www.buptjournal.cn/xbenPerformance analysis of sign quantized projectionYE Tian-yu (2), NIU Xin-xin, MA Zhao-feng, YANG Yi-xianI. Information Security Center, State Key Laboratory of Networking and Switching Technology,Bejjig University of Posts and Telecommuicains, Bejing 100876, China2. Key Laboratory ofNetwork and lfomamo Atack and Defnse Tchnologoy ofMOE,Beijing Univesity of Posts and Tlecommunicatins, Bejjig 100876, China3. National Engineering Laboratory for Disaster Backup and Recovery, Beijing 100876, ChinaAbstractQuantized projection technology (QP) is the version of quantization index modulation (QIM) working in the projection domain.It essentially belongs to a double-sided additive embedder. In this article, the concept of sign quantized projection (SQP) isproposed, which differs from the conventional QP when working as a single- sided embedder. Theoretical analysis reveals thatSQP has the same probability of miss and probability of false alarm as QP. Also, the document-to-watermark ratio (DWR) of SQPis less than its counterpart ofQPby 10lg(1+) dB.Keywords DWR, QR, SQP1 Introductionlength L with zero mean and |s| = L , and keep independentfrom the original host signal x . In QP, the watermarked dataSpread-spectrum (SS) is one of the pioneering water-y[k] is generated from the original host data x[k] throughmarking embedding schemes. It is always classified into[4(,)-r]s[k]additive spread-spectrum (ASS) and multiplicative spread-y[k]=x[]+(1)Lspectrum (MSS). Both ASS and MSS belong to a single-sidedwhere r is the projected original host signal defined asembedder. The host signal always plays the role 0interference with regard to the performances of ASS and MSS.r=Zx{k}p[k]. Moreover, 4() is an Euclidean scalarTo improve their performances, double sided embedder [1]was suggested, which contrarily utilized the interference fomquantizer with step size d, and its centroids are located ati0+0/2, i∈Z. It can be concluded from Eq. (1) that QPhost cover. Afterwards, sign embedder and is improvedversion [2] were proposed. Their advantage over theessentially belongs to a double-sided additive embedder, dueto its quantizer. Can QP works as a single-sided embedder?customary linear, additive embedder in the detecting sense hasbeen validated. It is easy to find out that the sign embedderHow will it perform? This article is devoted to invetigatingsuggested in Ref. [2] is essentially equivalent to a double-these two problems.sided embedder proposed in Ref. [1], although they were put2 SQP and its detection performanceforward from different backgrounds.When QIM [3] was introduced into the projection domain,2.1 Embedding ruleQP was derived [4 5], which took great performanceadvantage over QIM. Let s be a pseudorandom sequence ofIn this section, the authors propose the concept of SQP. The中国煤化工Recceived date: 11-03-20090Coresponding author: YE Tan-yu, E-mail: flystu008@yaboo.com.cnTYHCNMHGDOL: 10.1016/S1005- 88(0960425-8Issue 1YE Tian-yu, et al. / Performance analysis of sigp quantized projection63|x[k]+-([M(,)-rxbtk}; r;≥0and probability of false alarm [5] are measured byPr = Pr{(A(r)-r|≤T|H,}(10)y[k]=<(2)|l观]-()+r]s[k]; r,<0Pm =Pr{4(r)-~r|\>T|H}(11)It turns out that the pdf of R under the hypothesis H。Introducing Eq. (2) into the projection domain leads tois fnp,(r,|Ho)=N(0,o嗜 +σk). Therefore, according to[r+M(r)-r;=A(r); r;≥0 .(3)[r -[A(r)+r,]=-Mr,); r<0Eq. (10), the probability of false alarm is given by(i+1/2)+TAccording to Eq. (3), r, is always greater than 0 regardless2 J112 fqm,(t!Ho)dr,=of r. Therefore, it works as a single-sided embedder inessence. For convenience of ilustration, an indicator I isdefined aslg-s(12)√元+m2 Jσ+σ1=I; R.≥0(4)[0; Rx<0Furthermore, the pdf of Rz under the hypothesis H，The probability density function (pd) of Ry, denoted asdenoted as fe,H, (r:|H), can be obtained from Lemma 2 (infe,(,), can be obtained fom Lemma 1 (in AppendixA).Appendix A). Therefore, according to Eq (11), the probabilityof miss is given by2.2 Probability of false alarm and probability of missp=1-. z J4144The goal of watermark detection is to judge whether or nota watrmark exists, therefore it can be formulated as a binary( kd+T(13)hypothesis test. Given that the input signal of the detector,σ )」denoted as z, may have been corrupted by zero meanThus, SQP has the same probability of miss and probabilityadditive white Gaussian noise (AWGN) n , the two bypothesesof false alarm as QP.for the test are formulated as follows:Ho:z[k]=x[k]+n[k](5)2.3 Document-to watermark ratio|(K]+(k}=xk}+ [M()-rbs[k1+m[k]; r,≥0IFor further discussion, the embedding distortion is definedH;:z[k]=|x)+Hx)=x] [41+,1.+n[k]; r,<0asL(6)D,一之E(w[]} .(14)Introducing Eqs. (5) and (6) into the projection domainFurthermore, for convenience of performance comparison,yieldsthe DWR is defined asHi:r= Zklk}$[k]=之x{k}s[k]+ 2n[k}[k]=r,+r, (7)R= 101g(15)Pw[r+M(,)-r, +r,=4(4)+r; r,≥0Thus, the embedding distortion ofQP [5] isH:r=l -[4(r)+r,]+r,=-A(r)+r; r.<0(8)D, o=(16)where r, is the projected noise definedas r= 2 n[}][k].whereNote that the variance of r, is =Lσ: . The detector吸= ["[4(r)-rJhn(,)r=adopted here is the same as that suggested in Ref. [5], whichis based on the quantization error of r. The detector [5] is(17)formulated asloputting Ea, 16) into Ea. (15) vields the DWR of QP, i.e,d()s{t M2)-IST(9)R中国煤化工s embedding distorion0; |4(r)-r\>TandMHCN MH Gma 3 (in Apendix A).where T is the threshold. Therefore, the probability of missTherefore, the difference between the embedding distortion of64The Jourmal of China Universities of Posts and Telecommunications2010SQP and its counterpart of QP denoted as T, is given byFig. 3 plots the relationship between h and L when 4 isequal to 5. It is shown in Fig. 3 that h monotonouslyt=Dw _sop-Dr_o=- f" 4A(r)rJk, (r)dr=decreases along with L. Fig. 4 describes the relationship(18)between the ratio of N's absolute value occupying Ropand L when J is equal to 5. It shows that the ratio of A'sSince A(r,)r,>0 is always reasonable regardless of r,absolute value occupying Rop monotonously decreasesit follows that T> 0. Therefore, the embedding distortion ofalong with L. For example, when L is equal to 80, the ratio ofSQP is larger than its counterpart of QP under the same initialA's absolute value occupying Rp is 0.569 3.conditions. The reason lies in that the embedding distortion0.80rtriggered by the r,<0 part of Eq. (2) is greater than itscounterpart of Eq, (1), although the embedding distortions.75-triggered by their r,≥0 parts are identical. Likewise, thedifference between DWR of SQP and its counterpart of QP,引点0.70-denoted as h , is given byA=Roe -Rp =-101g1+_'- =-101g1+)(19)0.65Dw_.c)where0.60之4Flg. 2 Relationship between the ratio of h 's absolute valueis+ζ-r, |re(r)d,occupying Rp and 0(L= 20)is always greater than 0. Thus, the DWR of SQP is less thanits counterpart of QP by 10lg(1+y). Fig. 1 ilustrates the16Frelationship between n and 0 when L is equal to 20. It isshown in Fig. 1 that h monotonously increases along with4. The minus sign represents that the DWR of SQP is less思-18than its counterpart of QP. Fig. 2 demonstrates the relationshipbetween the ratio of A's absolute value occupying Rop19-and s whenL is equal to 20. It shows that the ratio of A'sabsolute value occupying Rop monotonously decreases alongwith A. For example, when s is equal to 5, the ratio of30405060708090100A's absolute value occupying Rop is 0.680 7.Fig. 3 Relationship between h and L(0= s)-10r0.70p-15-0.66--20三|2 0.62--25-300.58-8 To4中国煤化工708090100Flg. 1 Relationship between h and 0(L = 20).DYHC N M H Gf h's absolute valueoccupying Rop ana L(0= 2)Issue 1YE Tan-y, et al /Perfomance analysis of sign quantized projection653 Conclusions:20-(+小+In this article, the concept of SQP is proposed, whichdistinguishes itself from the conventional QP by working as a(A.3):(12()7.+(+1)4]single-sided embedder. Theoretical analysis reveals that SQPhas the same probability of miss and probability of falsewhere P(,)= fe, (r)dr, u() denotes the identityalarm as QP. Also, the DWR of SQP is less than itsfunction, and 8() denotes the Dyrac's delta function. Bcounterpart ofQPby 101g(1+/) dB.resorting to the central limit theorem (CLT)， r, can beapproximated to follow a Gaussian distribution such thatAcknowledgements()=/(V2TOk, )e-/)20) , where呢=No; . Tus,This work was sppored by the Natioal Basic Research ProgramP(c)=Q(4/O%, )-2[(+1)4/0n] where ()=/√2zr.of China (2007CB311203), the National Natural Science Foundationof China (60821001, U0835001, 60803157), the Specialized，eriI2d . Afer conducting complicated computations forResarch Fund for the Doctoral Program of Higher EducationEq. (A.3), one derives Lemma 1.(20070013007), the National 111 Project (B08004), the NationalLemma 2 The probability density functionof Rz under theStandard Development Planning (20080200-T-339)， and theStandardization Public Industry Special Foundation of Chinabyothesis H is fqm(r[HM)=2Zht[-(+12)4)P().(10-126).ProofAppendix AJuun(r|1=1,H)=_dFeyrun(r]r=1,H)drLemma 1 The probability density function of Ry is .dP(Rr≤r|I =1,H)fn(5)= 2pl(,)8[,(+/2)2].dr.P(I=)'[P(R≤r,I=咽)= PEDd P[(Rr+vSr,Rx=0}Q (.4)frnpu(,|=1)=dEpu(s,|=1)_ dP(R,≤,|l=1_Similarly, the following can be obtained:dr,Jryuo.un(r!=0,H)=d-[P(Rp,≤r,I =1]=(A.5)P(I=1) dr,IC.P(R)Ss,Rx≥0] (A.)Since Jgmn, (.|H.)= fepmeun, (x!r=1H)P(=1)+P(I=1) d,frImo.n, (r|I=0,H)P( =0), we haveSimilarly, one hasJrvw(,JI=0)=P(1=0) dr,F-M(R,)<,R2ol+Since. ()=xwp_(,|r =1)P( =1)+ fnvro(r|I=0)-。P[-A(Rx)+Rv≤r,Rr <0](A.6)P(I=0), we haveAfterwards, the result for the first term of Eq. (A.6) can befn(;)=。P[(R,)>r,Rx≥0]+derived at first. It follows二P(R, +R,≤r,R,≥0)= [[ f(,h)d,dr,=二P-4(Ry))+thg]2007, 2(3): 297-3102. Merhav N, Sabbag E. Optimal watermark embedding and detectionstrategies under limited detection resources. EEE Transactions onE=(4(;)+r)B=0|(A. 10)Information Theory, 2008, 54(1): 255- 274Chen B, Wornell G W. Quantization index modulation: a class ofSince Ew[k}]= E[w[k]I =I]P( =1)+ E[w[l]I=0]provably good methods for digtal watermarking and iinformationembedding, IEE Transactions on Information Theory, 2001, 47(4):P(I =0) , it gencrates from Eq. (14)1423-14434. Perez-Gonzalez F, Balado F, Hermnandez J R. Performance analysis ofexisting and new methods for data hidig with known-host iformatio inD.so=(E()-y吊=1P(=1)+ E(,)+r外aditive channels. IEEE Transactions on Signal Processng. 2003, 51(4):1=0]P(I =0)}(A.11)960- -9805. Perez-Freire L， Comesafa P, Perez-Gonzalez F. Detection iwherequantization-basedwatermarkiperformance and security isesE{[4(x)-rI|r=1}= [[Cr>-yJFJ.vc.(rI[=I)r,=Proceedings of Conference on Security, Steganogmapby, Watermarking ofMultimedia Contents VII, Jan 16- -20, 2005, San Jose, CA, USA. SPIE[M(,)-,P' 40Pu=n. (A.2)5681. Bllingham, WA, USA: SPIE, 200S: 721-733(Editor: ZHANG Ying)中国煤化工MYHCNMHG