Threshold Selection for Correlation-Based

Threshold Selection for Correlation-Based Watermark Detection A. Piva, M. Barni, F. Bartolini, V. Cappellini Dipartimento di Ingegneria Elettronica, Universita di Firenze, via S. Marta 3, 50139, Firenze, Italy | e-mail [email protected] .it Phone +39 55 4796380 { Fax: +39 55 494569

Abstract In this paper, the problem of how a decision threshold for a correlation-based watermark detection algorithm can be properly chosen is addressed; it is demonstrated that, if a watermarked image is attacked through signal processing techniques, the statistic of the correlation values changes, resulting in an higher false negative error probability than was expected. To cope with this eect, a new threshold based on a xed constraint on the maximum probability of false positive errors is proposed. Results con rm the robustness of the decoder against various kinds of attacks.

1 Introduction

Here, the watermark consists of a set of n normally distributed samples x1 ; x2 ; : : : ; xn which are used to modify a selected set V of full-frame DCT coecients. In particular, the coecients from the (k + 1)th to the (k + n)th in the zig-zag ordering of the DCT spectrum are selected and modi ed, according to the following rule: f

vX;i = vi +  vi xi; j

j

g

(1)

where vi is the original DCT coecient, xi the watermark sample, vX;i the modi ed coecients, and  is a properly chosen parameter tuning the watermark energy: the higher the , the more robust and the more visible the watermark. By denoting the component by component multiplication of vectors X and V by XV , equation 1 can be put in the more compact form VX = V + X V . During detection, the correlation between the marked and possibly corrupted coecients V 0 and the watermark itself is computed and used as a measure of the presence of a given mark X . More precisely, given a mark X and a set of possibly corrupted and watermarked DCT coecients V 0 , the correlation (X; V 0 ) between X and V 0 , de ned as

Digital watermarking represents an eective solution to the increasing demand for copyright protection mechanisms. A codemark is indissolubly embedded into the image to be protected, to allow the identi cation of its creator, owner, authorized author, and so on. Several systems have been proposed which hide some information in the spatial domain [5, 7] or in a transformed domain of the image [1, 3, 4, 6, 8], trying to ful ll the contrasting requirements of robustness and unobtrusiveness. In this paper, n 0 1 X V the problem of how a decision threshold for a 0 (X; V ) = n = n xivi0 ; (2) correlation-based watermark detection system i=1 [1] can be selected, is discussed; the threshold is chosen according to a xed constraint on the can be used to determine whether a given mark maximum probability of false positive errors in is present or not, by simply comparing it to a prede ned threshold. watermark detection. j




j

X

2 Statistical analysis threshold selection

for

It is important for a correct behaviour of the watermark detection system, to properly choose the decision threshold. Let us consider a scenario in which the decoder is presented a grey level image and it is asked to decide whether a given mark X is embedded in the image. Only one of the following situations is possible: Hp.A: V 0 = V i.e. the image is not marked; Hp.B: V 0 = V + Y V i.e. a mark Y = X is present; Hp.C: V 0 = V + X V i.e. the mark X is present. j

of Hp:0 and Hp:1. By assuming that Hp:0 and Hp:1 are equiprobable, and by taking into account the particular decoding strategy, eq.(3) can be put in the form: Pe = 21 P ( < T 1) + P ( > T 0) : (4) where  = (X; V 0 ). To go on with the computation of Pe , the statistic of (X; V 0 ) must be considered. Let us note rst that by invoking the central limit theorem,  can be assumed to be normally distributed. It is possible to demonstrate [1] that, when 2 1, if hypothesis Hp:0 holds

h

j

i



j

jHp: = 0

(5)

0

6

j

j

and

j

jHp: = (1 +  ) nv 2

2

2

0

(6)

For the present application, i. e. to decide whether the mark X has been embedded in the and if hypothesis Hp:1 holds, we set image or not, hypothesis A and B are equiva(7) jHp:1 =  jvj lent, and can be grouped together: and Hp.0 = Hp.A or Hp.B: the image is not marked with X ; 2 2 2 2 v 2 jv j jHp:1 = (1 + 2 ) n +  n (8) Hp.1 = Hp.C: the image is marked with X . where n 1 To discriminate between Hp:0 and Hp:1, the jvj = n E [ vi ] (9) decoder computes (X; V 0 ) and compares it i=1 with a threshold T , if (X; V 0 ) is lower than is the average value of  over the set of T then the decoder decides the image is not marked coecients, and jvij marked with X , whereas if (X; V 0 ) is above the threshold, the decoder assumes the image 1 n E [v2 ] 2 = (10)  is marked with X . To determine the value of i v n i =1 T, the decoder error probability can be taken into account. The error probability Pe , i.e. the is the average value of 2 over the set of probability of deciding for the wrong hypothe- marked coecients. If 2 vi 1, we can write: sis, can be written as:

X

j

j

X



Pe = P (0 1)P (1) + P (1 0)P (0); j

j

(3)

where P (0 1) is the probability of missing the presence of the mark (false negative), and P (1 0) the probability of revealing the presence of X when X is not actually present (false positive), P (0) and P (1) the a priori probability j

j

jHp: 2

0

'

2

v ,  :  n 2

jHp:

'

1

2

(11)

In g.1 the pdf's of  under hypotheses 0 and 1 are shown. In order to minimize the error probability, a threshold T has to be chosen such that P (0 1) + P (1 0) is minimum. Under the condition  1, P (0 1) = P (1 0), so that j

j



j

j

stated that in presence of attacks the (X;V ) and (X;VY ) should remain approximately the same, whereas (X;VX ) is likely to increase signi cantly. Therefore, we can say that because of attacks, two gaussians are still present, but the one centered in jvj has now a signi cantly larger variance. This suggested that T should be set closer to zero, instead of midway between zero and jvj (see g.2), so that T has Figure 1: The pdf's of  under hypotheses 0 been xed [1] to and 1. Attacks are not considered. (14) T = 3 jvj

3 A new threshold selection

Figure 2: The pdf's of  under hypotheses 0 and 1. Attacks are now considered. the optimum threshold is midway between zero and jHp:1, that is

T = 2 jvj

(12)

For the above analysis to be successfully applied to practical situations, two considerations are in order. On the basis of statistical analysis [2], we have assumed jvj = 0:7; however, if an image is attacked this value can be considerably dierent so that an error is likely to occur when comparing (X; V 0 ) with T . In practical applications, then, it is better for the decoder to use a threshold T which is estimated on the marked image, i.e. we assume

jvj

'

1

X v0 n

ni

=1

j

ij

(13)

However, experimental results have shown that when the watermarked image is attacked the proposed threshold leads to a higher watermark missing rate than was expected. In particular, the probability of missing an embedded watermark results to be considerably higher than the probability of false positive detection, i.e. the probability to detect a watermark which is not really present in the image. This can be explained by the fact that under attacks it usually happens that jHp:1 <  jvj (see g.3). To solve this problem, a dierent approach for threshold selection has been found. In this case, instead of trying to minimize the error probability Pe , it is chosen to x a constrain on the maximum false positive probability (e.g. 10?6 ), so that the threshold is moved leftmost (see again g.3). In particular, given Pf = P ( > T 0) = 12 erfc( T22 ) = 10?6 , the following relation holds: p

j

qT2 



2



3:3

(15)

in such a way that a new threshold is obtained:

q

s

The second consideration concerns the choice of T when the image has been corrupted 2(1 + 2 )v 2 (16) 2 = 3:3 2  T = 3 : 3   by intentional or unintentional attacks. In n such a case, the analysis carried out so far is no longer valid, since both the mean value Once again, the threshold can be evaluated diand the variance of (X; V 0 ) may be altered rectly on the watermarked and possibly corbecause of attacks. In general, it can be rupted image: the value (1 + 2 )v 2 corre-

sponds, in fact, to v 2 so that we have: 0

T = 3:3

s 2

v

n

0

2

(17)

Figure 3: The new choice for the threshold, Figure 5: Watermarked image `Lenna' with pabased on a constrain on the maximum false pos- rameter =0.2, k=16000 and n=16000. itive probability.

4 Experimental results In the following, experimental results will be shown proving that the new threshold choice improves the characteristics of robustness against attacks of the proposed watermarking system, and con rming the aforementioned analysis. In particular, the thresholds T = 3 jv j and T0 = 3:3 2nv 2 will be compared, highlighting the improvement of performance of the system with respect to the correct detection of the watermark. The standard 0

q

0

Figure 4: Original image `Lenna'.

image "Lenna" in Figure 4 was signed with parameter  = 0.2, with a watermarking random sequence of length n = 16000, and skipping the rst k = 16000 coecients in the zig-zag scan, to obtain a watermarked copy shown in Figure 5. The detector was applied to this copy: as shown in Figure 6, the new threshold T0 is lower than the old threshold T , but much higher again with respect to the responses to the other watermarks, so that no false positive watermark detection could happen. To verify the eectiveness of the new threshold in the watermark detection of attacked images, some modi cations of "Lenna" were carried out: rst, a JPEG compression with quality factor of 5% was applied; the results of the decoder are shown in Figure 7: the new threshold T0 is lower than the correct response, whereas the old threshold T is higher than the response, so that using T a missed detection would occur. Then, the sequence of JPEG compression with quality factor of 15%, despeckle ltering, and dithering to only 4 grey levels, was applied, obtaining the image in Figure 8. Once again, by using the threshold T0 the embedded watermark is correctly detected, whereas by using T it would be missed (see Figure 9). Similar results were obtained also carrying out other attacks, like median and low pass ltering, sharpening, blurring.

0.05

Detector Response

0.04 0.03

Tρ

0.02

T’ρ

0.01 0 -0.01 0

100

200

300

400 500 600 Watermarks

700

800

900

1000

Figure 6: Detector response of the watermaked image in Figure 5 to 1000 randomly generated watermarks. Only watermark number 100 matches that embedded. As shown, the new threshold T0 is lower than the old threshold T, but still much higher with respect to the responses of the other watermarks. 0.02

Tρ

Detector Response

0.015

T’ρ

0.01

0.005 0

-0.005 -0.01 0

100

200

300

400 500 600 Watermarks

700

800

900

1000

Figure 7: Detector response of the watermaked image in Figure 5 after a JPEG compression with quality factor of 5%.

5 Conclusions

on a xed constraint on the maximum probability of false positive errors is proposed. Results con rm the robustness of the decoder against In this paper, the problem of how a decision the attacks. threshold for a correlation-based watermark detection algorithm can be properly chosen is addressed; to determine whether a given watermark is present or not, the correlation (X; V ) between the watermark X and a set of selected DCT coecients is computed, and compared to Acknowledgments a threshold T . It has been demonstrated that, if a watermarked image is attacked through signal processing techniques, the statistic of the This work was partially supported by the \Procorrelation changes, resulting in an higher false getto Finalizzato Beni Culturali" (Finalized negative error probability than was expected. Project for Cultural Heritage) of the Italian To cope with this eect, a new threshold based National Research Council (CNR).

0.02

Tρ

Detector Response

0.015 0.01

T’ρ

0.005 0

-0.005 -0.01 0

100

200

300

400 500 600 Watermarks

700

800

900

1000

Figure 9: Detector response of the watermaked image in Figure 5 after JPEG compression of 15%, despeckle ltering and dithering with reduction to 4 grey levels. [3] A. Bors and I. Pitas, "Image watermarking using DCT domain constraints," Proc. ICIP'96, Int. Conf. on Image Processing, vol.III, pp.231{234, Lausanne, September 1996. [4] I.J. Cox, J. Kilian, T. Leighton and T. Shamoon, "Secure spread spectrum watermarking for multimedia," IEEE Trans. Image Processing, vol.6, no.12, pp. 1673{ 1687, December 1997. [5] I. Pitas, "A method for signature casting on digital images," Proc. ICIP'96, Int. Conf. on Image Processing, vol.III, pp. 215{218, Lausanne, September 1996. Figure 8: Watermarked image `Lenna' after JPEG compression of 15%, despeckle ltering [6] J.J.K. O Ruanaidh, F.M. Boland and W.J. Dowling, "Phase watermarking of digital and dithering with reduction to 4 grey levels. images," Proc. ICIP'96, Int. Conf. on Image Processing, vol.III, pp. 239{242, LauReferences sanne, September 1996. [1] M. Barni, F. Bartolini, V. Cappellini, A. [7] R. Wolfgang and E.J. Delp "A watermark for digital images," Proc. ICIP'96, Int. Piva, "A DCT-domain system for robust Conf. on Image Processing, vol. III, pp. image watermarking", Signal Processing, 219{222, Lausanne, September 1996. vol.66, no.3, pp. 357{372, May 1998. [8] J. Zhao and E. Koch, "Embedding robust [2] M. Barni, F. Bartolini, A. Piva and F. labels into images for copyright protecRigacci, "Statistical Modelling of Full tion," Proc. of the Int. Congress on IntelFrame DCT Coecients", to appear in lectual Property Rights for Specialized InProc. European Signal Processing Conformation, Knowledge and New Technoloference EUSIPCO 98, Island of Rhodes, gies, pp. 242-251, Vienna, August 1995. Greece, September 8-11, 1998.