Practical Capacity of DigitalWatermarks

Practical Capacity of Digital Watermarks Ryo Sugihara Tokyo Research Laboratory,IBM Japan, Ltd. 1623-14, Shimotsuruma, Yamato-shi, Kanagawa-ken 242-8502, Japan [email protected]

Abstract. A practical approach for evaluating the capacity of watermarks is presented. In real applications of watermarks, reliability is one of the most important metrics. The problem focused on in this paper is maximizing the number of embedded bits when there are some constraints on detection reliability. Error rates are formulated under some assumptions about the watermarking scheme, and the capacity can be determined by setting the bounds on each error rate. Experiments are performed to verify the theoretical predictions using a prototype watermarking system which conforms to the assumptions, and the resulting capacity agrees with the theory. Further, the theoretical effects of employing errorcorrecting codes are considered. It is shown that this approach yields the practical capacity of watermarks, as compared with channel capacity in communication theory.

1 Introduction Most of the previous works on watermark capacity [2,8,10,11,12] are based on communication theoretic considerations. They regard watermarking procedures as communication over a noisy channel, and the capacity of the watermark corresponds to the capacity of the “watermark channel”. Based on Shannon’s channel capacity, these studies discuss the amount of information which can be transmitted on a certain communication channel. However, Shannon’s theory does not provide any realistic methods to achieve the capacity. The capacity is satisfied in the ideal case when assuming random coding, which is usually too complex for practical use. Moreover, when regarding watermarking as communications, it is very difficult to handle the case of no watermark embedded in the cover-media. Communication channel has its basis on the premise that there is some input, and it does not treat the case of “no input”. This makes the consideration on false alarms much more difficult. When someone tries to implement a watermarking scheme, the amount of information included in the embedded message usually has to be determined in advance, but it usually cannot be figured out from the channel capacity. For that reason, an estimation procedure for the “practical” capacity is necessary. As mentioned above, the difficulty in channel coding makes the capacity issue less practical. So at first, we do not use any coding method. This makes the watermarking system less efficient, but the implementation and reliability evaluation become much easier to handle. The capacity of a watermark is usually constrained by fidelity, robustness, and reliability [5,6]. Of these constraints, fidelity will not be discussed here, as it is another large I. S. Moskowitz (Ed.): IH 2001, LNCS 2137, pp. 316–330, 2001. c Springer-Verlag Berlin Heidelberg 2001


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topic which requires discussions based on physiology and psychology. Robustness can be considered as equivalent to reliability, as it is a measure related to the probability of correct extraction of an embedded message after the watermarked image has suffered some degradation. In this paper, we discuss the capacity of watermarks where there are some constraints on reliability. In Section 2, we define the measurements of reliability and the constraints on those measurements. In that section, we also consider a simple watermarking scheme and make some assumptions about its attributes. Based on these assumptions, we derive the theoretical probability of detection errors in Section 3. In Section 4, we describe how we built a prototype watermarking system with certain attributes, and we compare the theoretical and experimental results. In Sections 5 and 6, we discuss the results and conclude that this reliability-driven approach is useful for real applications.

2 Problem Statement In short, the problem handled in this paper is to maximize the number of bits embedded in the image, when there are some constraints on the reliability of detection. Reliability can be measured by three metrics: the probability of a false alarm (Pf ), of erroneous extraction (Pe ), and of correct extraction (Pc ). Figure 1 shows the definitions of each of these. The constraints are assumed to be written as follows: Pf ≤ Pf max

(1)

Pe ≤ Pemax

(2)

Pc ≥ Pcmin

(3)

2.1 Assumptions for a Watermarking Scheme In order to solve the problem, we model a simple watermarking scheme. We postulate that the scheme has the following attributes:

probability

Watermark embedded correct extraction

Pc

erroneous extraction

Pe

detect not detect

1 - (Pc+Pe)

correct error

No watermark embedded detect (false alarm)

Pf

error

not detect

1 - Pf

correct

Fig. 1. Definition of errors on watermarking

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extract “0”

not detected

extract “1”

not embedded (cover-image) xi

0

embedded (stego-image) 0

(n) µ max

xi

embedded & degraded

µ (n)

0

xi

at worst acceptable quality 0 −T

(n)

T

(n)

(n) µ min

xi

Fig. 2. A schematic figure for the distribution of the detection statistic xi . It is always assumed to follow a Gaussian distribution with a unit variance. µ(n) (the average of xi ) gets closer to 0 as the embedded image degrades

– statistical watermarking, where detection yields n-dimensional values xi (1 ≤ i ≤ n ; n is the number of bits embedded in an image) and they are statistically tested as follows:
watermark detected if |xi | ≥ T (n) (∀i) else no watermark detected where T (n) (> 0) is a predetermined threshold value for n bits embedding. If a watermark is detected, the message is extracted as follows:
xi ≥ T (n) “1” for the i-th bit “0” for the i-th bit xi ≤ −T (n) – xi is i.i.d.(independent and identically distributed) and follows a Gaussian distribution – for the cover-image, xi is distributed according to N(0, 12 ), where N(µ, σ2 ) stands for the Gaussian distribution with mean µ and standard deviation σ (n) – for a stego-image, xi is distributed according to N(µmax , 12 ). Without losing gen(n) erality, we can assume µmax ≥ 0, which means the embedded message is always assumed to be “11 · · · 1”. – When the stego-image is degraded, xi is distributed according to N(µ(n) , 12 ) (0 ≤ (n) µ(n) ≤ µmax ). Note that the standard deviation of xi is assumed to be held constant at 1.0 by an ideal normalization.

Practical Capacity of Digital Watermarks

– µ(n) is proportional to

 1/n, or equivalently  µ(n) = µ(1)

1 n

319

(4)

Whether a watermark is embedded in the spatial domain or frequency domain, there are a finite number of modifiable pieces in an image. If we try to embed more bits, the number of pieces for each bit decreases in proportion to the number of bits. The summation of m i.i.d. Gaussian variables √ yields N(mµ, m), as each variable 2 ). By dividing each variable by m, N(mµ, m) is normalized to be follows N(µ, 1 √ N( mµ, 12 ) – As the lower bound of degradation, there is “worst acceptable quality”. At that (n) (n) point, µ(n) equals µmin . When images have poorer quality (i.e. µ(n) < µmin ), they are assumed to have no commercial value, and are not necessary to be protected. So the constraint on probability of correct extraction (inequality (3)) is not applied in that case. However, that of erroneous extraction (inequality (2)) is valid, as such error should not occur even for the images with poor quality. – Watermark degrades as well as the cover-image does. In other words, any selective attacks on watermark, such as desynchronization by non-linear distortion, are not assumed in this analysis1 . Fig.2 shows the assumptions for the watermarking scheme. It is quite important whether these assumptions are reasonable or not. Later in the experiments, we will use a modified version of the “patchwork algorithm” [3], which is one of the most popular algorithms for watermarking. It can be considered as a watermarking scheme that closely follows the assumptions above. The details of the algorithm and the legitimacy of the assumptions will be discussed in Appendix A.

3 Theoretical Analysis First, we explain how we treat detection error rates theoretically. 3.1 Formulation of Error Rates Basic Formulae Let fµ(n) ,σ(n) denote the probability density function of the detection statistic for each bit. It is assumed to follow a Gaussian distribution written as   1 (x − µ(n))2 fµ(n) ,σ(n) (x) =  exp − (5) 2 2 2σ(n) 2πσ(n) 1

As a positive reason for this, it is important for watermarks not only to survive against various attacks, but also to be reliable on the normal procedures performed by legitimate users. Especially on the industrial use, the latter will be much more important on the viewpoint of product liability. As a negative reason, many selective attacks on watermarks are specialized to each watermarking algorithms, and so it is very difficult to analyze them generally.

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(n)

The probability of correct extraction (pc ) and erroneous extraction (pw ) for each bit are given by

Z

(n)

pc =

∞

T (n)

Z

(n)

pw =

fµ(n) ,1 (x)dx

−T (n)

(6)

fµ(n) ,1 (x)dx

−∞

(7)

where T (n) is the threshold when the number of bits is n, and the standard deviation is always assumed to be 1. The probability of erroneous extraction including an i-bit error (out of n bits) is as follows:    n (n) (n) i (n) n−i pw pc pe (i) = (8) i   n where is the number of combinations for choosing i items out of n. i For Non-Watermarked Images When the image is not watermarked, each detection statistic follows the standard Gaussian distribution N(0, 12 ). The probability of false alarm for one bit is (n)

p0 =

Z

Z

∞

f0,1 (x)dx + T (n)

 T (n) = erfc √ 2

−T (n)

−∞

f0,1 (x)dx (9)

where erfc(x) is the complementary error function defined as follows: Z∞ 2 exp (−t 2 )dt erfc(x)
√ π x (n)

(10)

(n)

Using p0 , the probability of false alarm (Pf ) is given as follows: (n)

Pf

=



(n) n

p0

(11)

For Watermarked Images When the image is watermarked, the probabilities of cor(n) (n) rect extraction (Pc ) and erroneous extraction (Pe ) can be written: (n)

Pc

(n)

Pe

(n)

= pe (0) =

n

(n)

∑ pe

i=1

(i)

(12) (13)

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321

6 Tf(n)

5

max{Te(n)}

Threshold

4 3 2 1 0 0

10

20

30

40

50

60

n (Number of bits)

 (n) (n) ) when Pf max = 10−6 , Pemax = 10−4, 0 ≤ µ(n) . Fig. 3. Threshold (T f , maxµ(n) Te For each n, T (n) is the larger of these two threshold values

3.2 Theoretical Capacity If the number of bits in the watermark is fixed, the probability of detection error is totally controlled by the threshold value. In other words, we have to figure out the threshold value to satisfy the constraints on error rates. As for the probability of false alarm, the threshold value is derived from Equations (9) and (11) as follows √ (n) (n) T f = 2 erfc−1 (p0 )


1  √ (n) n −1 Pf = 2 erfc (14) (n)

where T f

is the threshold value constrained by the probability of false alarm, and (n)

−1

erfc (x) is the inverse function of the complementary error function. Note that T f only satisfies the constraint on false alarm, and does not necessarily satisfy that on erroneous extraction. For the erroneous extraction, the problem is much more difficult because it calls for solving Equation (13) analytically. However, we can find an approximate threshold (n) value Te for each distribution by using an iterative algorithm such as the Newton (n) method.As the probability of the erroneous extraction must not exceed Pemax in any µ , (n) becomes the threshold value that satisfies the constraint on erroneous maxµ(n) Te extraction. The above two constraints must be satisfied at the same time, and so the threshold T (n) is given as    (n) (n) (n) T = max T f , max Te (15) µ(n)

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0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Pc

(n)

action) (Probability of correct extr

1.0

0.0

n be um (N

20 30

ro

40

fb

50

) its

60 0

5

10

15

20

25

30

35

40

45

50

) of x i when n=1 µ(1) (Mean value

(n)

Fig. 4. Theoretically derived probability of correct extraction (Pc ), when Pf max = 10−6 , Pemax = 10−4. Here µ(n) is represented by µ(1) , since we assume µ(n) = µ(1) 1/n Figure 3 shows the threshold values as a function of n. By using T (n) , the probability of correct extraction can be calculated for certain n and µ(n) as shown in Figure 4. As we assumed there are the lower limits on µ(n) and P(n) (n) (n) (µmin and Pcmin , respectively), the maximum number of bits can be determined, and that is the capacity in this study.

4 Experiments As we discussed in the previous section, we can estimate the capacity of the watermark theoretically, if the requirements on reliability (Pf max , Pemax , Pcmin ) and the limit (n) (n) only for any particuof degradation, µmin , are given. Note that µmin needs to be known  lar n and not necessarily for all n, as we assume µ(n) = µ(1) 1/n. For the verification of the analytical results, we implemented a prototype watermarking system and performed experiments. We used the patchwork algorithm, and modified it to realize multiple bit embedding. 4.1 Conditions The constraints were as follows: – reliability : Pf max = 10−6 , Pemax = 10−4 , Pcmin = 0.5 – worst acceptable quality : JPEG compression (Quality: 80)2 2

The conversions were performed using “Nconvert” (http://perso.wanadoo.fr/pierre.g/indexgb.html). The compression ratio was approximately 1/14(7.026%) on average.

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In the experiment, 1000 images (resolution: 640 × 426) were used. We tested for n between 1 and 64 bits. Embedding was performed on the luminance components, and the change for each pixel was a constant ±5, except for solid regions that were not changed3. See Appendix A for more details on the watermarking algorithm. 4.2 Preliminary Experiment Before the consideration of capacity, we performed a preliminary experiment to measure how much the watermarks were degraded by the JPEG compression. In the preliminary experiment, n was set to 1. The resulting averages for the detection statistics over 1000 images were 36.619 for the uncompressed images, and 18.663 for the JPEG(1) compressed images, respectively. Based on these results, we used µmax = 36.619 for (1) uncompressed images, and µmin = 18.663 for JPEG-compressed images for the theoretical analysis. 4.3 Theoretical Result Figure 5 shows the theoretically derived probability of correct detection. It is an excerpt of Figure 4 for the region where 18.663 ≤ µ(1) ≤ 36.619, which is the observed region of watermark degradation. From this figure, the theoretical capacity is found to be 31 bits for the same conditions as in the experiment. 4.4 Experiment For each n from 1 to 64, we collected data of the average of xi over the 1000 images and the number of images from which the watermark is correctly extracted. The experiments were done both on the watermarked images without degradation, and after JPEG-compression. Figure 6 shows the relationship between the number of bits and the average detection statistic. The experimental data are plotted over the theoretical curves, and this  shows that µ(n) = µ(1) 1/n is a valid assumption. Figure 7 shows the relationship between the number of bits and the probability of correct detection. As seen from this figure, the capacity is approximately 31 bits according to the experiments. This agrees well with the theoretical result of the previous section.

5 Discussion We have shown that the experimental results agree quite well with the theoretical calculations, which shows that our underlying model is sound. In Figure 7, there are dis(n) crepancies between theory and experiment in cases where Pc is very high or very low. They are presumably caused by the variation between each image. The amount of change caused by embedding varies between images, and so the detection statistic xi is 3

The resulting SNR (Signal-to-Noise Ratio) was 28.56 dB on average for the 1000 images.

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1.0

0.8 0.7 0.6 0.5 0.4 0.3

µ(1)max = 36.619

0.2

Pc

(n)

) (Probability of correct extraction

0.9

35 0.1 30 0.0 10

25 20

30 31

n (Num

40

ber of bit

20

50

60

s)

) (1

µ

(M

n ea

of

)) =1 (n

xi

µ(1)min = 18.663

(n)

Fig. 5. Theoretically derived probability of correct extraction (Pc ), when Pf max = 10−6 , Pemax = 10−4 , 18.663 ≤ µ(1) ≤ 36.619. If Pcmin = 0.5, the capacity is approximately 31 bits

µ(n) (Mean value of xi )

40 experiment (uncompressed) experiment (JPEG-compressed) (1) theory (µ = 36.619) theory (µ(1) = 18.663)

30

20

10

0 0

10

20

30

40

50

60

n (Number of bits)

Fig. 6. The number of bits and µ(n) , the average detection statistic. Experiments were performed on 1000 images, and the results for uncompressed images and JPEGcompressed images are shown. The corresponding theoretical curves are also shown

Practical Capacity of Digital Watermarks

325

Pc(n) (Probability of correct extraction)

1.0 0.9 experiment (JPEG-compressed) theory (µ(1) = 18.663)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 31

0.0 0

10

20

30

40

50

60

n (Number of bits) (n)

Fig. 7. Probability of correct detection (Pc ) and the number of bits. Experimental data and theoretical curve are shown together. If Pcmin = 0.5, the capacity from the experiment is 31 bits

not expected to follow exactly the same distribution over all images. Furthermore, there is also a significant variation in how much the image (and the watermark) degrades from the same degree of compression. The capacity value derived from our method might seem to be very small, compared with other studies on channel capacity, or even compared with commercial watermark products. This is mainly because the watermarking system used in the experiment was not specifically designed to be robust against compression. The watermark pattern was concentrated at high spatial frequency, which is susceptible to degradation in most of the image compression algorithms. If it were designed to be robust, for example by using larger “patches” for the patchwork algorithm, the value of µ(n) would not be reduced by so much. The resulting capacity becomes larger when the value of µ(n) is large, as seen from Figure 5. 5.1 Remarks on Error-Correcting Codes (ECC) Until now, the use of ECC has not been discussed in this paper, since we have been focusing on the practical capacity which can be easily obtained. However, as communication theory and some works on watermark [7] mention, the use of ECC is expected to improve the capacity. Here we offer some considerations on applying ECC to our analysis. In our approach, ECC can easily be taken into account in the same way as in the previous section. If an original message is encoded to be a codeword by using the ECC which has the capability of correcting t-bit errors, Equations (12) and (13) are rewritten

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n (Number of bits)

50 Capacity (length of info. symbols) Code length

40 30 20 10 0 0

1

2

3

4

5

6

7

8

9

10

Error-correcting capability (bit)

Fig. 8. Theoretical capacity when ECC is used. Constraints are the same as in Figure 5: Pf max = 10−6, Pemax = 10−4, Pcmin = 0.5, 18.663 ≤ µ(1) ≤ 36.619. The maximum capacity is approximately 35 bits, which is achieved by using (n, k) = (45, 35) code with a 2-bit error-correcting capability

as follows: (n)

Pc

(n)

Pe

= =

t

(n)

∑ pe

i=0 n

∑

(i)

(n)

pe (i)

(16) (17)

i=t+1

With ECCs, there are some constraints on the relation between the code length, the number of information symbols, and the minimum distance. As a simple example, here we use Hamming bound [9], which is written as the following inequality for (n, k) block codes with t error-correcting capability:     t n i (18) n − k ≥ logq ∑ (q − 1) i i=0 where q is the number of symbols, which is two in the current discussion. Figure 8 shows the relation between the theoretical capacity and the error-correcting capability under the same constraints as in the case of Figure 5. From this figure, an ECC which corrects 2-bit errors can improve the capacity from 31 bits to 35 bits. In fact, the ideal code that satisfies the equality in (18), which is called a perfect code, does not exist in general [9]. But the conventional codes such as BCH Codes can be evaluated by changing only the inequality of the upper bound. In the conventional codes, it also should be noted that there is a case where a received word, which

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327

is extracted from an image, is not decodable, because of errors exceeding the errorcorrecting capability. Such a case usually results in “error-detection”, and it changes the probability of false alarms and also the threshold value, though it is a very small effect. In order to be precise, such cases should be considered. 5.2 Practicality of the Analysis In order to be more practical, this kind of analysis should be easy to perform. It should not take too much time, computational power, and other kinds of resources. When we calculate the capacity theoretically, our current method requires the average value of the detection statistic from the degraded images. Therefore we had to perform the preliminary experiment for a particular number of bits, as mentioned in the previous section. It is one of the most time-consuming processes and therefore not very practical. However this defect can be removed by estimating the degradation. According to the theory of quantization [13], degradations by quantization-based compressions such as JPEG can be estimated to some content. As another practical consideration, the analysis should be easy to extend. It will not be very practical if the analysis is only applicable for this specific watermarking scheme. We have used a modified version of the patchwork algorithm in the experiment. Though it is a watermark in the spatial domain, it causes no differences in our analysis if the watermark is embedded in the frequency domain. Recently, many kinds of watermarking algorithms have been proposed, and the performance (including reliability) is difficult to analyze in some cases. However, such algorithms are often intended for steganographic use, which is more conscious of the method itself or the size of the message rather than the reliability. In other words, they are not necessarily analyzed precisely for reliability, and so this kind of problem is not important in these cases.

6 Conclusion and Future Works We have described one approach for the theoretical analysis of watermark capacity. We considered a simple statistical watermarking scheme, and formulated the probability of detection errors theoretically based on some simple assumptions. We used the method to calculate the theoretical capacity, and it was successfully verified by experiments. It was also shown theoretically that ECCs can improve the capacity, and was implied that our approach can be extended for other simple watermarking algorithms. There remain several problems and questions. Although we have emphasized the importance of reliability in watermarking, the probabilities of false alarm and erroneous extraction have not been verified by experiments yet. Another major problem is that our assumptions for the detection statistic are too strict and idealistic. An appropriate modelling of the cover-image and detection statistic, such as the generalized Gaussian distribution for DCT coefficients [1], is necessary for accurate evaluation of reliability. Also the experimental watermarking algorithm is too crude for actual use, as it does not protect the fidelity of the image well enough, and it is too susceptible to damage. Also, it is very important to compare our results with those based on channel capacity. Such topics remain to be investigated in our future work.

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Acknowledgements The author thanks Shuichi Shimizu, Ryuki Tachibana, Taiga Nakamura, and other colleagues at IBM Tokyo Research Laboratory for helpful discussions and comments.

References 1. M. Barni, F. Bartolini, A. Piva, and F. Rigacci, “Statistical modelling of full frame dct coefficients”, Proceedings of EUSIPCO’98 , pp. 1513-1516, 1998 327 2. M. Barni, F. Bartolini, A. De Rosa, and A. Piva, “Capacity of the Watermark-Channel: How Many Bits Can Be Hidden Within a Digital Image?”, Proceedings of SPIE , Vol. 3657, pp. 437-448, 1999 316 3. W. Bender, D. Gruhl, N. Morimoto, and A. Lu, “Techniques for data hiding”, IBM Systems Journal , Vol. 35, No. 3/4, pp. 313-336, 1996 319 4. I. J. Cox, J. Kilian, F. T. Leighton, and T. Shamoon, “Secure Spread Spectrum Watermarking for Multimedia”, IEEE Transactions on Image Processing , Vol. 6, No. 12, pp. 1673-1687, 1997 5. J. Hernandez and F. Perez-Gonzalez, “Statistical analysis of watermarking schemes for copyright protection on images”, Proceedings of the IEEE, Vol. 87, No. 7, pp. 1142-1166, 1999 316 6. S. Katzenbeisser and F. A. P. Petitcolas, Information Hiding Techniques for Steganography and Digital Watermarking, Artech House, 2000 316 7. G. C. Langelaar, R. L. Lagendijk, and J. Biemond, “Watermarking by DCT coefficient removal: a statistical approach to optimal parameter settings”, Proceedings of SPIE, Vol. 3657, pp. 2-13, 1999 325 8. P. Moulin and J. A. O’Sullivan, “Information-theoretic analysis of watermarking”, Proceedings of ICASSP’2000, pp. 3630-3633, 2000 316 9. W. W. Peterson and E. J. Weldon, Error-Correcting Codes, 2nd ed., MIT Press, 1972 326 10. M. Ramkumar and A. N. Akansu, “Theoretical Capacity Measures for Data Hiding in Compressed Images”, Proceedings of SPIE, Vol. 3528, pp. 482-492, 1998 316 11. S. D. Servetto, C. I. Podilchuk, and K. Ramachandran, “Capacity Issues in Digital Watermarking”, Proceedings of ICIP, Vol. 1, pp. 445-448, 1998 316 12. J. R. Smith and B. O. Comiskey, “Modulation and Information Hiding in Images”, Information Hiding : First International Workshop, Vol. 1174 of Lecture Notes in Computer Science, pp. 207-226, 1996 316 13. B. Widrow, I. Kollar, and M.-C. Liu, “Statistical theory of quantization”, IEEE Trans. on Instrumentation and Measurement, Vol. 45, No. 6, pp. 353-361, 1995 327

A

A Modified Version of the Patchwork Algorithm

Figure 9 shows the schematic view of the modified version of the patchwork algorithm, which is used for the experiment. The largest difference from the original algorithm is to embed multiple bits by dividing the image into multiple regions. The embedding pattern for each bit contains equal numbers of patches4 of “+1” and “-1”, and they are scattered pseudorandomly. The patches in the i-th embedding pattern 4

In the experiment, a patch consisted of one pixel.

Practical Capacity of Digital Watermarks

coverimage

stegoimage

329

image

(b) 1st bit 2nd bit

n-th bit : +1 :-1 (none) : 0

(a) (a)

(a)

1st bit

(c)

x1

2nd bit

(c)

x2

n-th bit

(c)

xn

inner product of two vectors

message

extracting (detection)

embedding (a) (b) (c)

if (message)i = “0”, swap and (no change if (message)i = “1”) transparency control normalization

for i-th bit

Fig. 9. The modified version of the Patchwork algorithm used in the experiment. The entire image is used for embedding and extracting the watermark. In order to realize multiple-bit embedding, the pixels are equally divided to represent each bit, and so the degree of accumulation is in proportion to 1/n, the inverse of the number of embedded bits

do not overlap with those in the j-th ( j = i) embedding pattern. If we accumulate the embedding patterns for all bits, it contains no zeros. Before accumulating, the sign of the embedding pattern is flipped when the message is “0” ((a) in the figure). After making the watermark pattern, it is modulated globally and/or locally in order to preserve fidelity ((b) in the figure). This should be considered in conjunction with the human visual system, but we have not done that in this experiment. The whole watermark pattern is amplified by 5 times, except for the solid regions which are suppressed to zero. When extracting a watermark from an image, the same embedding patterns are used for calculating the inner product with the image. As the pattern includes equal numbers of “+1” and “-1”, the value of the inner product is expected to be the summation of the differences between the two pixel values. If the image is not watermarked, the difference is expected to be distributed around zero, but should have some positive or negative value if it is watermarked. As the patches are pseudorandomly scattered, the difference can be considered to be independent of one another. Therefore the inner product can be seen as the summation of i.i.d. variables. According to the central limit theorem, the resulting value follows a Gaussian distribution when the number of patches for each bit is large enough. For standardizing each value ((c) in the figure), we just divided by a constant. The constant is set so as to make the variance of xi equal to 1 when the image is not water-

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marked. Therefore it is not assured that the variance is expected to be 1 if the image is watermarked.