Capacity of Text Marking Channel - Semantic Scholar

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CS and EE Departments, Caltech, Pasadena, CA 91125 slow@caltech.edu. Nicholas F. Maxemchuk. AT&T Laboratories - Research, Florham Park, NJ 07932.
Capacity of Text Marking Channel

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Steven H. Low CS and EE Departments, Caltech, Pasadena, CA 91125 [email protected] Nicholas F. Maxemchuk AT&T Laboratories - Research, Florham Park, NJ 07932 [email protected]

Abstract We have proposed earlier watermarking text documents by slightly shifting certain text lines. Such a text line represents a noisy channel and marking represents the transmission of a signal through this channel. The power of the signal represents the size of the shift and must be small for the marks to be imperceptible. In this paper we formulate the channel capacity under a constraint on individual signal power. We shows that to achieve the capacity the shifts should be normally distributed, have maximum power, and adjacent shifts should be negatively correlated. I. Introduction

Many image watermarking methods have been proposed to protect motion and still pictures (see e.g. [5], [6] and references therein). Most of these works, however, are suitable for images with rich greyscale. For text documents, whose image is often binary, marks placed by some of these methods can be easily removed by binarization. We have proposed in [1], [7], [2] a di erent approach for marking text that is immune to the binarization attack. In our approach a text line may be shifted slightly up or down, e.g., by 1 or 2 pixels, to encode a `1' or a `0'. This method turns out to be remarkably robust against severe distortions introduced by processes such as printing, photocopying, scanning, and lossy compression as in facsimile transmission [7]. Text watermarking, using any technique including ours, can always be defeated by retyping the document, possibly with the help of character recognition devices. For possible applications, attacks and countermeasures, see [2]. Each line that is marked represents a noisy channel and marking represents the transmission of a signal through this channel. In this paper we consider the channel capacity when detection is made using text line centroids [7]. In xII we present a simple model of document pro le, marking and noise process. In xIII we describe To appear IEEE Signal Processing Letters, December 2000

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the centroid measurements that will be available for detection. In xIV we formulate our problem as determining the capacity of K parallel Gaussian channels with correlated noise under individual signal power constraints. The signal represents line shifts and the constraint represents the requirement that the shifts be imperceptible. We show that to achieve the capacity, the shifts should be Gaussian distributed with adjacent shifts negatively correlated. II. Profile and Marking

Upon digitization the image of a page is represented by a function f (x; y) 2 [0; 1], x = 0; 1; : : : ; W , y = 0; 1; : : : ; L, where W and L, whose values depend on the scanning resolution, are the width and length of the page, respectively. The value of f (x; y) represents the greyscale at position (x; y). The P horizontal pro le h(y) = W x=0 f (x; y) is the projection of f (x; y) along the horizontal direction. It measures the total intensity along the horizontal scan-lines y. Lines that are marked (shifted) are always surrounded by two control lines that are not shifted. Marking shifts the middle pro le block slightly to the left (up) or right (down), while leaves the neighboring control blocks unchanged. We assume that, after compensating for the structural distortions estimated from the control blocks, the remaining distortion can be modeled by additive white Gaussian noise (see [7] for an empirical justi cation). Hence if the pro le of an uncorrupted document is h(y), then that of the corrupted document is h(y) + G(y), where G(y) are zero-mean independent Gaussian random variables with variance 2 . If the image is binary, i.e., f (x; y) 2 f0; 1g, then the Gaussian noise assumption amounts to that the pro le h(y) and standard deviation of the noise is large compared with 1 pixel, which seems reasonable. III. Centroid detection

For the ith block de ned in the region [bi ; ei ] with pro le h(y), y = bi ; bi + 1; : : : ; ei , its centroid is Pei yh(y) ci = Pye=i bi h(y) : y=bi When the pro le is corrupted by additive white Gaussian noise G(y) to become h(y) + G(y), the centroid becomes Pei y(h(y) + G(y)) 0 ci = Pye=i bi (h(y) + G(y)) : y=bi We de ne the di erence Ni := c0i ? ci as the centroid noise. Ni are independent since G(y) is white. In Figure 1 the even blocks, c2i , carry information and are shifted by Xi and the odd blocks are control blocks that are not shifted. Xi is negative if block 2i is shifted to the left, and positive

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c 1+ N 1

c 3+ N 3 c 2+ X 1+ N 2

c 5+ N 5

c2k+1 + N 2k+1

c 4+ X 2+ N 4

Fig. 1. K parallel marking channels: pro le of a document page with K marked blocks (shaded). Here c is centroid of block i on original unmarked and uncorrupted copy, X is the shift of block i on a marked copy, and N is centroid noise on a corrupted copy. i

i

i

otherwise. We perform di erential detection by measuring the distances between the marked block and the neighboring control blocks:

Li := (c02i ? c02i?1 ) ? (c2i ? c2i?1 ) = Xi + (N2i ? N2i?1) Ri := (c02i+1 ? c02i ) ? (c2i+1 ? c2i ) = ?Xi + (N2i+1 ? N2i ):

(1) (2)

and using the di erence Y = (Yi = Li ? Ri ; i = 1; : : : ; K ) to make a decision. Di erential decoding eliminates the e ects of translation, a common distortion in photocopying and facsimile transmission. IV. Marking Channel Capacity

In [7] we have derived the distribution of the centroid noise Ni and have shown that, for a typical pro le, it is accurately approximated by a Gaussian distribution with zero mean and with variance

2 (3) i2 = Hw2 i (i2 + (wi2 ? 1)=12) i P where Hi = ebii h(y) is the `weight' of block i, wi = ei ? bi + 1 is the width of block i, and i = ci ? (ei + bi)=2 the deviation of the centroid from the middle of block i. Then from (1{2) the decision variable Y is given by Yi = Si + Zi ; i = 1; : : : ; K , where the signal Si = 2Xi . Here, the noise Zi = 2N2i ? N2i?1 ? N2i+1 are zero-mean Gaussian. Hence we have K parallel Gaussian channels whose inputs are vector S = (S1 ; : : : ; SK ) and whose outputs are vector Y = (Y1 ; : : : ; YK ). The noise vector Z = (Zi ; i = 1; : : : ; K ) has a subdiagonal covariance matrix CZ : 8 > > > > > > > < (CZ )ij = > > > > > > > :

where i2 is given by (3).

422i + 22i?1 + 22i+1 if i = j 22i+1 if j = i + 1 if i = j + 1 22j+1 0 else

(4)

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Our goal is to choose the signal S so as to maximize the mutual information I (S ; Y ) of the K parallel channels subject to individual power constraint: maxFS I (S ; Y ) subject to ESi2  pi , i = 1; : : : ; K . Here the maximization is over the joint distribution FS of signal vector S . The maximum maxFS I (S ; Y ) is called the channel capacity. This is similar to a classical problem where the aggregate signal power over all K channels is constrained (see, e.g., in [4], [3]). From [3, Chapter 10.5] the capacity can be achieved by choosing the inputs Si to be joint zero-mean Gaussian random variables with covariance matrix CS that maximizes the determinant of the matrix CS + CZ , subject to the power constraint. Hence the problem is equivalent to nding a CS that max C S

subject to

det(CS + CZ )

(5)

(CS )ii  pi ; i = 1; : : : ; K

(6)

where detM denotes the determinant of a matrix M . Let the maximum value be A. Then the channel capacity is 21 log2 detACZ bits. Q

By Hadamard's inequality, det(CS + CZ )  i ((CS )ii +(CZ )ii ), with equality if and only if CS + CZ is diagonal. Hence the unique maximizer CS of (5{6) has diagonal terms maximized and o -diagonal terms being the negative of those of CZ . From (6) and (4) we have 8 > > pi > > > > > < ? 2  (CS )ij = > 22i+1 > ?2j+1 > > > > > :

0

if i = j if j = i + 1 if i = j + 1 else

Hence to achieve the channel capacity, the line shifts should be Gaussian distributed with maximum power and adjacent shifts should be negatively correlated. Moreover the correlation should be of the same magnitude as the noise variance of the centroid that separates the two shifts. Q

The maximum value of (5{6) is then A = i (pi + 422i + 22i?1 + 22i+1 ), where i2 given by (3) depend on the pro le parameters. Hence the channel capacity is Q 1 log i (pi + 422i + 22i?1 + 22i+1 ) bits (7) 2 2 detCZ where CZ is the subdiagonal covariance matrix given by (4). We close by presenting an example from [7] where two pages of document were watermarked. The rst page contained 8 marked lines and the second page contained 11, for a total of K = 19 marked lines. The size of the shift used in the experiment was 2 pixels, so we take the power constraint to be p = 4 pixels2 for all lines i = 1; : : : ; 19. The centroid noice variance was empirically measured to be

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 2 = 0:0781 pixels2 . Assume that all lines have the same centroid noice variance. Then the 19  19 noise covariance matrix is:

2 6 2  2 6 6 6  2 6 2 CZ = 666 6 4

0

0 

0 0

2     0 0    6 2

3 7 7 7 7 7 7 7 5

Using (7) the channel capacity is 31 bits. References

[1] J. Brassil, S. Low, N. Maxemchuk, and L. O'Gorman. Electronic marking and identi cation techniques to discourage document copying. IEEE Journal on Selected Areas in Communications, 13(8), October 1995. [2] J. T. Brassil, S. H. Low, and N. F. Maxemchuk. Copyright protection for the electronic distribution of text documents. Proceedings of the IEEE, 87(7):1181{1196, July 1999. [3] Thomas M. Cover and Joy A. Thomas. Elements of Information Theory. John Wiley & Sons, 1991. [4] R. G. Gallager. Information Theory and Reliable Communications. John Wiley & Sons, 1968. [5] IEEE. Journal on Selected Areas in Communications, volume 16(4). IEEE Communications Society, May 1998. Special issue on Copyright and Privacy Protection. [6] IEEE. Proceedings of the IEEE, volume 87(7). IEEE Communications Society, July 1999. Special issue on Identi cation and Protection of Multimedia Information. [7] S. H. Low, N. F. Maxemchuk, and A. M. Lapone. Document Identi cation for Copyright Protection using Centroid Detection. IEEE Transactions on Communications, 46(3):372{383, March 1998.