Statistical Analysis of Digital Image FingerprintingBased on Random Projections

Farzad Farhadzadeh, Sviatoslav Voloshynovskiy, Oleksiy Koval, Taras Holotyak and Fokko Beekhof

Computer Science DepartmentUniversity of Geneva

7 route de Drize, CH 1227, Geneva, SwitzerlandEmail: {Farzad.Farhadzadeh, svolos, Oleksiy.Koval, Taras.Holotyak, Fokko.Beekhof}@unige.ch

Abstract—Digital fingerprints have recently received a lot of in-terest in multimedia applications due to their low dimensionalityand security/privacy preserving capabilities. In digital fingerprintextraction, random projections play an important role due totheir approximate distance preserving property. This paper isdedicated to the analysis of the statistical properties of digitalfingerprints obtained based on random projections. In particular,we were able to demonstrate that this sort of mapping guaranteesapproximate decorrelation of its output with high probability.

I. INTRODUCTION

In recent years, many multimedia applications such ascontent based retrieval, content filtering and automatic tagging,content based identification and authentication, and biometricsare using high dimensional multimedia data, that are fre-quently privacy-sensitive. There exist several approaches todeal with these problems, such as robust hashing or digitalfingerprinting. A digital fingerprint represents a short, robustand distinctive content description, allowing fast operations.

Multimedia management and security applications basedon digital fingerprinting have received a lot of interest inthe research community and several different approaches fordigital fingerprinting have been proposed. In most cases,digital fingerprinting consists of a dimensionality reduction byapplying some content independent transform and binarization[1], [2], [3]. For security/privacy reasons, this transform canalso be key-dependent. The main idea behind digital fin-gerprinting approaches is to extract digital fingerprints of alower dimensionality with a maximum possible entropy. Forinstance, in the binary case one expects the bits of digitalfingerprints to be independently and equally likely 0s and 1s.Since multimedia data are correlated, one of the principle tasksof the dimensionality reduction transform is to eliminate thecorrelation between their samples.

The optimal mapper that possesses such properties is theKarhunen-Loeve transform (KLT) [4]. This transform perfectlydecorrelates data as well as optimally compacts their energyinto fewer amount of elements, making dimensionality re-duction straightforward. However, the price to pay for thisoptimality is its data dependence and high computationalcomplexity. The latter issue gains importance due to thefact that computational complexity of this transform can beevaluated as O(N3), where N is the dimensionality of its

input [5]. Besides the above drawbacks, the necessity to sharethe basis vectors for the decoding stage makes this transformunpractical in the privacy-sensitive applications.

In order to relax this dependence, several approximationsof the KLT were proposed. These include, for example,the Discrete Cosine Transform (DCT) and Discrete WaveletTransform (DWT) [4], which demonstrate a nearly optimaldecorrelation of locally correlated data. The basis vectors ofthese transforms are fixed and independent of the statistics oftheir inputs. Due to their decorrelation and energy compactioncapabilities as well as the existence of fast implementationalgorithms, they are a common tool in various signal andimage processing applications. However, similarly to the KLTthe main drawback of such fixed basis transforms consistsin the public disclosure of the basis vectors, which could beunacceptable for multimedia security applications [6].

One solution to overcome this privacy/security shortcomingis a randomized mapper that can be designed based on randomprojections (RP) [2]. The RP have been the object of muchinterest due to their ability to preserve distance betweenvectors after embedding into a lower dimensional space thathas also recently been recognized in the Compressed Sensingcommunity for sparse data [7], [8]. Moreover, by applyingthe RP one can convert an unknown distribution of originaldata to Gaussian [9]. However, in multimedia applications,data are correlated. Although the decorrelation property oforthogonal transforms is well-known [4], the RP are basedon approximately orthogonal bases. The statistics of projecteddata, e.g., the covariance matrix, are not well justified. Onthe other hand, prior knowledge of the statistics of extracteddigital fingerprints to evaluate the performance of contentbased identification and retrieval systems is mandatory.

Therefore, the main goal of this paper is to investigatethe statistical properties of digital fingerprints based on theRP for different classes of image models that include eitherindependent and identically distributed (i.i.d.) models with asymmetric Probability Density Function (PDF), which modelsthe output coefficients of the DCT or DWT, or correlation-basedmodels like the Gauss-Markov processes, which capture imagepixel dependencies directly in the coordinate domain [4]. Atour best knowledge, this problem was not addressed yet by theresearch community in the current formulation. The analysis in

w sign(){0,1}LxBNX LX

K

Fig. 1. Digital fingerprint extraction.

[10] was focus rather on the performance analysis of forensicfingerprinting based on i.i.d./ correlated fingerprints than onthe statistical analysis of fingerprint generating model.

The outline of the paper is as follows. Section II containsa statistical analysis of data correlation in the RP domain.Experimental validation of our theoretical results is presentedin Section III. Finally, Section IV concludes the paper.

Notations: We use capital letters X to denote scalar randomvariables and X to denote vector random variables. Corre-sponding small letters x and x denote the realizations ofscalar and vector random variables, respectively. All vectorswithout sign tilde are assumed to be of length N and withthe sign tilde of length L, i.e., x = {x1, x2, . . . , xN} andx = {x1, x2, . . . , xL}, where xi, xj are elements of x and x,respectively (1 ≤ i ≤ N, 1 ≤ j ≤ L). Bernoulli(p) indicatesthe Bernoulli distribution with the probability of success p.B(N, p) denotes the Binomial distribution with N trails andprobability of success p. E[·] designates the expectation.

II. STATISTICAL ANALYSIS

This Section is dedicated to the statistical analysis ofextracted digital fingerprints using the RP and binarization.Digital fingerprint extraction, which is illustrated in Fig. 1,consists of two following steps: Dimensionality reductionand Binarization. At the dimensionality reduction stage, thedimensionality of data X is reduced from N to L, L ≤ N ,by applying the RP, which are approximately orthoprojectors,i.e., wwT ≈ IL , where w ∈ 1√

N{±1}L×N with elements

Wij ∼ Bernoulli( 12 ), 1 ≤ i ≤ L and 1 ≤ j ≤ N . Thetransform W can also be generated using some secret keyK. At the binarization stage, L-length binary data are derivedfrom the projected data elements by taking the sign of the pro-jected data, i.e., Bx = {sign(X1), . . . , sign(XL)}, sign(x) ={1,x > 00,x ≤ 0

, where X = wX for a given w.

A. Correlated Data Analysis

In this Section, we investigate the statistics of digital finger-prints obtained by applying the RP in the coordinate domain.We assume that the data X are real zero-mean random vectorsmodeled as the Gauss-Markov process. This is a simple butoften-used model in image processing [4]. For a given w, onehas:

Kxx = E[wXXTwT ] = wKxxwT , (1)

where Kxx is defined by [4]:

Kxx = σ2X

1 ρ . . .ρN−1

ρ 1 . . .ρN−2

......

. . ....

ρN−1ρN−2. . . 1

, (2)

where ρ is the normalized correlation coefficient. We provethe following proposition for statistical modeling of projecteddata.

Proposition 1 (Decorrelation property of RP). Let the ele-ments of the RP matrix, w of size L×N and 1 < L ≤ N , bedrawn from the probability mass function (PMF) Pr{Wij =+ 1√

N} = Pr{Wij = − 1√

N} = 1

2 , and X be a real zero-meanrandom vector modeled as the Gauss-Markov process withvariance σ2

X and normalized correlation coefficient ρ. Then,we have:

Pr

{maxi 6=j|Kij

xx| > βσ2X

}<

1

L, (3a)

Pr{maxi|Kii

xx − σ2X | > ασ2

X

}<

2

L(1ρ ), (3b)

where β =

√12N

(1−ρN1−ρ

)2lnL, α =

√8N ρ(

1−ρN−1

1−ρ

)2lnL.

Proof: At first, we consider off-diagonal elements of Kxx

that can be expanded as follows:

Kijxx=

N∑r=1

N∑c=1

wirKrcxxwjc

=σ2X (wi1wj1 + · · ·+ wiNwjN )︸ ︷︷ ︸

T ij0

+σ2Xρ (wi1wj2 + wi2wj1 + · · ·+ wiN−1wjN + wiNwjN−1)︸ ︷︷ ︸

T ij1

+. . .+ σ2Xρ

N−1 (wi1wjN + wiNwj1)︸ ︷︷ ︸T ijN−1

= σ2X

N−1∑k=0

ρkT ijk . (4)

Due to symmetry of the covariance matrix, we investigateupper off-diagonal elements only, i.e., 1 ≤ i < j ≤ L. Inorder to bound these elements, we evaluate an upper boundfor the probability that the largest upper off-diagonal elementsof Kxx is greater than σ2

X

(1−ρN1−ρ

)ζ, where ζ is a positive

real value. This probability is given by:

Pr

{maxi 6=j|Kij

xx| > σ2X

(1− ρN

1− ρ

)ζ

}(a)

≤ L(L− 1)

2Pr

{|Kij

xx| > σ2X

(1− ρN

1− ρ

)ζ

}=L(L− 1)

2Pr

{1

σ2X

(1− ρ1− ρN

)|Kij

xx| > ζ

}(b)

≤L(L− 1) exp(−sζ)E[exp

(s

1

σ2X

(1− ρ1− ρN

)Kij

xx

)](c)=L(L− 1) exp(−sζ)E

[exp

(s

N−1∑k=0

κkTijk

)](5)

where (a) follows from the fact that there are only L(L−1)2

such i.i.d. random variables [11], (b) follows form the gener-alized Chebyshev inequality [12] for s ≥ 0, (c) holds form(4), and κk = ρk

(1−ρ1−ρN

)and

∑N−1k=0 κk = 1. Then, we have:

Pr

{maxi 6=j|Kij

xx| > σ2X

(1− ρN

1− ρ

)ζ

}(a)

≤L(L− 1) exp(−sζ)

[N−1∑k=0

κkE[exp

(sT ijk

)]](b)=L(L− 1)

N−1∑k=0

κk

[exp(−sζ)

Nk∏l=1

E[exp

(sV ijl

)]](c)

≤L(L− 1)

N−1∑k=0

κk

[exp(−sζ)

Nk∏l=1

exp

(s2(

1N −

−1N

)28

)](d)=L(L− 1)

N−1∑k=0

κk

[exp

(−N

2ζ2

2Nk

)]

=L(L− 1)

[κ0 exp

(−Nζ

2

2

)+

N−1∑k=1

κk exp

(− N2ζ2

(N − k)4

)]

<L(L− 1)

[κ0 exp

(−Nζ

2

2

)+

N−1∑k=1

κk exp

(−Nζ

2

4

)]

≤L(L− 1) exp

(−Nζ

2

4

), (6)

where (a) holds due to the convexity of exp(·), (b) followsfrom the fact that E

[exp(sT ijk )

]is the moment generating

function of T ijk that is the sum of Nk ∈ {N, 2(N −1), . . . , 2}i.i.d. Bernoulli(0.5) random variables V ij = WirWjc ∈{+1N , −1N }, (c) holds due to the fact that V ij is a bounded

random variable [13], and (d) holds by choosing s = N2ζNk

.

By setting β =(

1−ρN1−ρ

)ζ, the probability can be bounded

by:

Pr

{maxi6=j|Kij

xx| > σ2Xβ

}≤ exp

(−N

4

(1− ρ1− ρN

)2

β2

).

(7)

By substituting β =

√12N

(1−ρN1−ρ

)2lnL, (3a) is obtained.

For the diagonal elements of Kxx we have:

Kiixx=σ

2X +

N∑r=1

N∑c=1

r 6=c

wirKrcxxwic

=σ2X + 2σ2

Xρ (wi1wi2 + · · ·+ wiN−1wiN )︸ ︷︷ ︸Dii1

+2σ2Xρ

2 (wi1wi3 + · · ·+ wiN−2wiN )︸ ︷︷ ︸Dii2

+. . .+ 2σ2Xρ

N−1 (wi1wiN )︸ ︷︷ ︸DiiN−1

= σ2X + 2σ2

X

N−1∑k=1

ρkDiik ,

(8)

Similar to (5), we evaluate an upper bound for the probabilitythat the largest deviation of diagonal elements of Kxx fromσ2X is greater than 2σ2

X

(ρ−ρN1−ρ

)ε, where ε is a positive real

value. This probability is given by:

Pr

{maxi|Kii

xx − σ2X | > 2σ2

X

(ρ− ρN

1− ρ

)ε

}(a)

≤LPr{|Kii

xx − σ2X | > 2σ2

X

(ρ− ρN

1− ρ

)ε

}(b)=2L exp(−sε)E

[exp

(s

N−1∑k=1

λkDiik

)](c)

≤2L exp(−sε)

[N−1∑k=1

λkE[exp

(sDii

k

)]](d)

≤N−1∑k=1

λk exp

(− N2ε2

2(N − k)

)≤ exp

(−Nε

2

2

), (9)

where λk = ρk(

1−ρρ−ρN

)and

∑N−1k=1 λk = 1, (a) follows from

the fact that there are only L such random variables which areidentically distributed [11], (b) follows form the generalizedChebyshev inequality [12] for s ≥ 0, (d) results from theconvexity of exp(·), and (e) holds following the same resultsof parts (c) and (d) in (6) and by choosing s = N2ζ

2(N−k) . By

setting α = 2(ρ−ρN1−ρ

)ε, we have:

Pr{maxi|Kij

xx − σ2X | > σ2

Xα}≤ exp

(−N

8

(1− ρρ− ρN

)2

α2

).

(10)

By substituting α =

√8N ρ(

1−ρN−1

1−ρ

)2lnL, (3b) is obtained.

Remark 1. For a sufficiently large N and L, L ≤ N , α→ 0and β → 0, Kxx asymptotically converges to σ2

XIL withhigh probability. Moreover, from the fact that the contentsource is the Gauss-Markov process, which implies that thecontent vector x is jointly Gaussian, and RP is a lineartransform, the projected data x follows the jointly Gaussiandistribution, i.e., X ∼ N (0,Kxx). Therefore, since elementsof x are asymptotically uncorrelated, Kxx ≈ σ2

XIL, one canconclude that x are asymptotically i.i.d. In addition, the digitalfingerprint extracted from x consists of L bits that are i.i.d.Bernoulli( 12 ) due to symmetry of the Gaussian distribution.

B. Independent Identically Distributed Data

As mentioned in Section I, multimedia data can be modeledas i.i.d., i.e., X ∼ p(x) =

∏Ni=1 p(xi), where p(xi) is a

symmetric distribution with zero mean and the variance σ2X

in certain domains. For example, p(xi) can be approximatedby a Generalized Gaussian distribution, due to the propertyof DCT or DWT coefficients of multimedia data [14], [15].Therefore in this section we investigate the covariance matrixof projected data in order to justify the correlation induced bythe RP to the i.i.d. data.

Collary 1 (uncorrelatedness preservation property of RP).Let the elements of the RP matrix, W, be generated as inProposition 1, and X is drawn i.i.d. from a common stationarydistribution with variance σ2

X . Then, the diagonal elements ofcovariance matrix of the projected noise X = WX are equalto σ2

X , i.e., ∀i,Kiixx = σ2

X , and all off-diagonal elements ofKxx satisfies:

Pr

{maxi 6=j|Kij

xx| > δσ2X

}<

1

L, (11)

where δ =√

12N lnL.

Proof: This is a corollary of Proposition 1, where ρ→ 0.For the off-diagonal elements of Kxx, we can easily derive(11) by substituting ρ = 0. For the diagonal elements, α|ρ=0 =

0. Thus, Pr{maxi|Kiixx − σ2

X | > 0} < limρ→01

L(1ρ )

= 0 for

all L > 1, which implies that ∀i, 1 ≤ i ≤ L,Kiixx = σ2

X .

Remark 2. For a sufficiently large N and L, L ≤ N ,δ → 0 and Kxx converges to σ2

XIL with high probability.Then, one can conclude that the samples of projected data areasymptotically uncorrelated. Moreover, in the case that thedata are generated i.i.d. from the Gaussian distribution, onecan conclude that the binary digital fingerprint extracted from aprojected content consists of L bits that are i.i.d. Bernoulli( 12 ),due to the fact that the Gaussian distribution is symmetric,and uncorrelated and identically distributed Bernoulli randomvariables are i.i.d.

III. SIMULATION RESULTS

In this Section we present experimental validation of ourtheoretical findings for synthetic data and real images.

First we apply the RP generated according to the setupdiscussed in Section II to the i.i.d. zero mean unit varianceGaussian data of length N = 210. Fig. 2 illustrates the impactof the RP on the the projected data with the same dimension-ality (Fig. 2(a)) and the reduced dimensionality (Fig. 2(b)). Itis possible to observe that such a transform nearly preservesthe uncorrelatedness of its i.i.d. input according to the upperbound (11) defined in Proposition 1.

The second set of tests was dedicated to the experimentaljustification of the decorrelation capabilities of the RP. Forthis purpose, we carried out a number of experiments, wherethe input data are generated from a Gauss-Markov processwith the normalized correlation coefficient 0.5 ≤ ρ ≤ 0.95.The results are demonstrated only for ρ = 0.95 and 0.75(Fig. 3). These results confirm the theoretical finding ofProposition 1 stating that the residual correlation is accuratelyupper bounded by (3a).

The third set of tests is assigned to the RP uncorrelatednesspreservation and decorrelation capabilities for real images inDCT and spatial domains. The tests have been performed for aset of 1180 images of African animals with size 256×384 fromCorel image collection provided by Duygulu et al. [16]. Fig. 4illustrates the RP application to the real images in differentdomains. The results presented in Fig. 5(a) and Fig. 6(a)

0 200 400 600 800 100010

−4

10−3

10−2

10−1

100

j

Mag

nit

ude(

dB

)

K1jxx

K1jxx

δ

(a) N = 210, L = 210

0 20 40 60 80 100 12010

−4

10−3

10−2

10−1

100

jM

agnit

ude(

dB

)

K1jxx

K1jxx

δ

(b) N = 210, L = 27

Fig. 2. I.i.d. preservation property of RP: (a) the first row of Kxx andKxx, and (b) the first 128 elements of the first row of Kxx and Kxx. xis generated from the i.i.d. Gaussian process with σ2

X = 1. δ represents theupper bound on the maximum value of non-diagonal elements of Kxx.

demonstrate the average two dimensional autocorrelation for1180 real images in DCT and spatial domains, the last 512elements of the average autocorrelation of projected DCTcoefficients (Fig. 5(b)) and real images (Fig. 6(b)) using RP,and finally the last 512 elements of the average autocorrelationof the extracted digital fingerprints (Fig. 5(c) and Fig. 6(c)).The obtained results demonstrate that elements of extractedbinary fingerprints are almost uncorrelated.

The obtained results justify the accuracy of Proposition1 applied to real data in the coordinate and DCT transformdomains. Furthermore, the results from Fig. 5 and Fig. 6indicate that the correlation between the samples of the digitalfingerprints extracted from images in DCT and spatial domainare approximately the same under the explained RP. Conse-quently, one can conclude that by directly using RP in spatialdomain instead of joint usage of the DCT and RP, the digitalfingerprints with the same properties can be extracted with alower complexity.

IV. CONCLUSIONS

In this paper, we presented a statistical analysis of dig-ital fingerprint extraction using the RP. We succeeded todemonstrate that the output of the RP for this type of inputhas an asymptotically diagonal covariance matrix with highprobability.

0 50 100 150 200 25010

−4

10−3

10−2

10−1

100

j

Mag

nit

ude(

dB

)

K1jxx

K1jxx

β

(a) ρ = 0.95

0 50 100 150 200 25010

−6

10−5

10−4

10−3

10−2

10−1

100

j

Mag

nit

ude(

dB

)

K1jxx

K1jxx

β

(b) ρ = 0.75

Fig. 3. The first 256 elements of the first row of Kxx and Kxx, where xis generated from the Gauss-Markov process with σ2

X = 1. x and x have thelength of N = 215 and L = 28, respectively. δ represents the upper boundon the maximum value of non-diagonal elements of Kxx.

Since the RP are only approximate orthoprojectors, weanalyzed the correlation that such a transform might induceinto i.i.d. models of images in some domains like DCT andDWT. Using a similar statistical arguments we showed that thedata in the RP domain will converge to a diagonal covariancematrix with high probability. Our theoretical findings weresuccessfully confirmed by a set of experimental results forsynthetic data and real images.

ACKNOWLEDGMENT

The authors would like to thank M. Meckes and V. Bal-akirsky for the stimulating discussions. This paper was par-tially supported by SNF projects 200020-134595.

REFERENCES

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[2] J. Fridrich, “Robust bit extraction from images,” in Proc. of MCS, vol. 2,july 1999, pp. 536 –540.

[3] J. Haitsma and T. Kalker, “A highly robust audio fingerprinting system.”in International Conference Music Information Retrieval, 2002.

[4] A. K. Jain, Fundamentals of digital image processing. Upper SaddleRiver, NJ, USA: Prentice-Hall, Inc., 1989.

[5] G. Golub and C. van Loan, Matrix Computations. Oxford, UK: NorthOxford Academi, 1983.

[6] S. Voloshynovskiy, O. Koval, F. Beekhof, F. Farhadzadeh, andT. Holotyak, “Information-theoretical analysis of private content identi-fication,” in Proc. of ITW, Dublin, Ireland, 2010.

[7] M. A. Davenport, P. T. Boufounos, M. B. Wakin, and R. G. Baraniuk,“Signal processing with compressive measurements,” IEEE Journal ofSelected Topics in Signal Processing, vol. 4, no. 2, pp. 445–460, 2010.

[8] W. B. Johnson and J. Lindenstrauss, “Extensions of lipschitz mappinginto hilbert space,” in Conf. in MAP, ser. Contemporary Mathematics,vol. 26. AMS, 1984, pp. 189–206.

[9] O. Koval and S. Voloshynovskiy, “Multimodal object authentication withrandom projections: a worst-case approach,” in Proceedings of SPIE /Media Forensics and Security XII, San Jose, USA, January 21–24 2010.

[10] A. L. Varna, A. Swaminathan, and M. Wu, “A decision theoreticframework for analyzing hash-based content identification systems,” inACM Digital Rights Management Workshop, October 2008, pp. 67–76.

[11] G. Caraux and O. Gascuel, “Bounds on distribution functions of orderstatistics for dependent variates,” Statistics & Probability Letters, vol. 14,no. 2, pp. 103 – 105, 1992.

[12] R. G. Gallager, Information Theory and Reliable Communication. NewYork, NY, USA: John Wiley & sons, 1968.

[13] W. Hoeffding, “Probability inequalities for sums of bounded randomvariables,” Journal of the American Statistical Association, vol. 58, no.301, pp. 13–30, 1963.

[14] S. G. Mallat, “A theory for multiresolution signal decomposition: thewavelet representation,” IEEE Transactions on Pattern Analysis andMachine Intelligence, vol. 11, pp. 674–693, 1989.

[15] S. LoPresto, K. Ramchandran, and M. Orchard, “Image coding basedon mixture modeling of wavelet coefficients and a fast estimation-quantization framework,” in Proc. of Conf. on Data Compression.Washington, DC, USA: IEEE Computer Society, 1997.

[16] P. Duygulu, K. Barnard, J. F. G. d. Freitas, and D. A. Forsyth, “Objectrecognition as machine translation: Learning a lexicon for a fixedimage vocabulary,” in Proceedings of the 7th European Conference onComputer Vision-Part IV, ser. ECCV ’02, 2002, pp. 97–112.

X1W

2W

LW

,

X

X

1X

2X

LX

X,

,

(a)

X1W

2W

LW

,

X

X

1X

2X

LX

X,

,

(b)

Fig. 4. Two dimensional data projection by the RP: (a) DCT domain, and(b) spatial domain.

−2000

200

−200

0

200

0

0.2

0.4

0.6

0.8

1

cr

Mag

nit

ude

(a)

0 100 200 300 400 500 60010

−3

10−2

10−1

100

j

Mag

nit

ude(

dB

)

Rjxx

(b)

0 100 200 300 400 500 60010

−3

10−2

10−1

100

j

Mag

nit

ude(

dB

)

Rjbxbx

(c)

Fig. 5. I.i.d. preservation property of RP for real images: (a) the averagetwo dimensional autocorrelation of the 2D-DCT of real images, (b) the last512 elements of the average autocorrelation of projected DCT coefficientsusing RP, and (c) the last 512 elements of the average autocorrelation ofthe extracted digital fingerprints. Rj

xx and Rjbxbx

denote the jth elementof autocerrelation of a projected data and an extracted digital fingerprint,respectively.

−2000

200

−200

0

200

0

0.2

0.4

0.6

0.8

1

cr

Mag

nit

ude

(a)

0 100 200 300 400 500 60010

−3

10−2

10−1

100

j

Mag

nit

ude(

dB

)

Rjxx

(b)

0 100 200 300 400 500 60010

−3

10−2

10−1

100

j

Mag

nit

ude(

dB

)

Rjbxbx

(c)

Fig. 6. Decorrelation property of RP for real images: (a) the averagetwo dimensional autocorrelation of the set of real images, (b) the last 512elements of the average autocorrelation of projected images using RP, and(c) the last 512 elements of the average autocorrelation of the extracted digitalfingerprints. Rj

xx and Rjbxbx

denote the jth element of autocerrelation of aprojected data and an extracted digital fingerprint, respectively.