Abstract

Many proposed image watermarking techniques are sensitive to geometric distortions such as rotation, scaling, and translation. Geometric distortion, even by a slight amount, can disable a watermark decoder. In this study, a geometric distortion-invariant watermarking technique is designed by utilizing Tschebycheff moments of the original image to estimate the geometric distortion parameters of corrupted watermarked images. The Tschebycheff moments of an original image can be used as a private key for watermark extraction. The embedding process is a closed-loop system that modifies the embedding intensity according to the results of the performance analysis. The convergence of the closed-loop system is proved. Different from early heuristic methods, the optimal blind watermark detector is designed with the introduction of dual-channel detection utilizing high-order spectra detection and likelihood detection. Even with a small signal-to-noise ratio (SNR), the detector can still get a satisfying detection probability if there is enough high-order spectra information. When the high-order spectra are small, this dual-channel detection system will become a likelihood detection system. The watermark decoder extracts a watermark by blindly utilizing independent component analysis (ICA). The computational aspects of the proposed watermarking technique are also discussed in detail. Experimental results demonstrate that the proposed watermarking technique is robust with respect to attacks performed by the popular watermark benchmark, StirMark.

©2007 Optical Society of America

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  1. S. Voloshynovskiy, S. Pereira, T. Pun, J. J. Eggers, and J. K. Su, “Attacks on digital watermarks: classification, estimation-based attacks, and benchmarks,” IEEE Commun. Mag. 8, 2–10 (2001).
  2. L. Zhang, G.-B. Qian, X.W.-W. Xiao, and Z. Ji, “Geometric invariant blind image watermarking by invariant Tschebycheff moments,” Opt. Express 15, 2251–2261 (2007).
  3. p. Dong, J. G. Brankov, N. P. Galatsanos, Y. Yang, and F. Davoine “Digital watermarking robust to geometric distortions,” IEEE Trans. Image Process. 14, 2140–2150 (2005).
  4. M. Alghoniemy and A. H. Tewfik, “Geometric invariants in image watermarking,” IEEE Trans. Image Process. 13, 145–153 (2004).
  5. Y. Xin, S. Liao, and M. Pawlak, “A multibit geometrically robust image watermark based on Zernike moments,” International Conference on Pattern Recognition 4, 861–864 (2004).
  6. S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Trans. Image Process. 9, 1123–1129 (2000).
  7. M. Kutter, “Performance improvement of spread spectrum based image watermarking schemes through M-ary modulation,” Lect. Notes Comput. Sci. 1728, 238–250 (1999).
  8. P. Dong and N. P. Galasanos, “Affine transform resistant watermarking based on image normalization,” in Proceedings of IEEE International Conference on Image Processing, 3, 489–492 (2002).
  9. P. Bas, J.-M. Chassery, and B. Macq, “Geometrically invariant watermarking using feature points,” in Proceedings of IEEE International Conference on Image Processing, 11, 1014–1028 (2002).
  10. J. O’Ruanaidh and T. Pun, “Rotation, scale, and translation invariant spread spectrum digital image watermarking,” Signal Process. 66, 303–317 (1998).
  11. H. S. Kim and H. K. Lee, “Invariant image watermark using Zernike moments,” IEEE Trans. Circuits Syst. Vid. Technol. 13, 766–775 (2003).
  12. R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tschebycheff moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).
  13. R. Mukundan, “Some computational aspects of discrete orthonormal moments,” IEEE Trans. Image Process. 13, 1055–1059 (2004).
  14. P.-T. Yap and R. Paramesran, “Local watermarks based on Krawtchouk moments,” in IEEE Region 10 Conference pp. 73–76 (2002).
  15. M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Information Theory IT-8, 179–187, (1962).
  16. A. Hyvarinen and E. Oja, “Independent component analysis: a tutorial,” in Notes for International Joint Conference on Neural Networks (1999), http://www.cis.hut.fi/projects/ica/.
  17. G. Sundaramorthy, M. R. Raghuveer, and S. A. Diana, “Bispectral reconstruction of signal in noise amplitude reconstruction issues,” IEEE Trans. Acoust. Speech Signal Process. 38, 1297–1300 (1990).

2007 (1)

2005 (1)

p. Dong, J. G. Brankov, N. P. Galatsanos, Y. Yang, and F. Davoine “Digital watermarking robust to geometric distortions,” IEEE Trans. Image Process. 14, 2140–2150 (2005).

2004 (3)

M. Alghoniemy and A. H. Tewfik, “Geometric invariants in image watermarking,” IEEE Trans. Image Process. 13, 145–153 (2004).

Y. Xin, S. Liao, and M. Pawlak, “A multibit geometrically robust image watermark based on Zernike moments,” International Conference on Pattern Recognition 4, 861–864 (2004).

R. Mukundan, “Some computational aspects of discrete orthonormal moments,” IEEE Trans. Image Process. 13, 1055–1059 (2004).

2003 (1)

H. S. Kim and H. K. Lee, “Invariant image watermark using Zernike moments,” IEEE Trans. Circuits Syst. Vid. Technol. 13, 766–775 (2003).

2001 (2)

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tschebycheff moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).

S. Voloshynovskiy, S. Pereira, T. Pun, J. J. Eggers, and J. K. Su, “Attacks on digital watermarks: classification, estimation-based attacks, and benchmarks,” IEEE Commun. Mag. 8, 2–10 (2001).

2000 (1)

S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Trans. Image Process. 9, 1123–1129 (2000).

1999 (1)

M. Kutter, “Performance improvement of spread spectrum based image watermarking schemes through M-ary modulation,” Lect. Notes Comput. Sci. 1728, 238–250 (1999).

1998 (1)

J. O’Ruanaidh and T. Pun, “Rotation, scale, and translation invariant spread spectrum digital image watermarking,” Signal Process. 66, 303–317 (1998).

1990 (1)

G. Sundaramorthy, M. R. Raghuveer, and S. A. Diana, “Bispectral reconstruction of signal in noise amplitude reconstruction issues,” IEEE Trans. Acoust. Speech Signal Process. 38, 1297–1300 (1990).

1962 (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Information Theory IT-8, 179–187, (1962).

Alghoniemy, M.

M. Alghoniemy and A. H. Tewfik, “Geometric invariants in image watermarking,” IEEE Trans. Image Process. 13, 145–153 (2004).

Bas, P.

P. Bas, J.-M. Chassery, and B. Macq, “Geometrically invariant watermarking using feature points,” in Proceedings of IEEE International Conference on Image Processing, 11, 1014–1028 (2002).

Brankov, J. G.

p. Dong, J. G. Brankov, N. P. Galatsanos, Y. Yang, and F. Davoine “Digital watermarking robust to geometric distortions,” IEEE Trans. Image Process. 14, 2140–2150 (2005).

Chassery, J.-M.

P. Bas, J.-M. Chassery, and B. Macq, “Geometrically invariant watermarking using feature points,” in Proceedings of IEEE International Conference on Image Processing, 11, 1014–1028 (2002).

Davoine, F.

p. Dong, J. G. Brankov, N. P. Galatsanos, Y. Yang, and F. Davoine “Digital watermarking robust to geometric distortions,” IEEE Trans. Image Process. 14, 2140–2150 (2005).

Diana, S. A.

G. Sundaramorthy, M. R. Raghuveer, and S. A. Diana, “Bispectral reconstruction of signal in noise amplitude reconstruction issues,” IEEE Trans. Acoust. Speech Signal Process. 38, 1297–1300 (1990).

Dong, p.

p. Dong, J. G. Brankov, N. P. Galatsanos, Y. Yang, and F. Davoine “Digital watermarking robust to geometric distortions,” IEEE Trans. Image Process. 14, 2140–2150 (2005).

P. Dong and N. P. Galasanos, “Affine transform resistant watermarking based on image normalization,” in Proceedings of IEEE International Conference on Image Processing, 3, 489–492 (2002).

Eggers, J. J.

S. Voloshynovskiy, S. Pereira, T. Pun, J. J. Eggers, and J. K. Su, “Attacks on digital watermarks: classification, estimation-based attacks, and benchmarks,” IEEE Commun. Mag. 8, 2–10 (2001).

Galasanos, N. P.

P. Dong and N. P. Galasanos, “Affine transform resistant watermarking based on image normalization,” in Proceedings of IEEE International Conference on Image Processing, 3, 489–492 (2002).

Galatsanos, N. P.

p. Dong, J. G. Brankov, N. P. Galatsanos, Y. Yang, and F. Davoine “Digital watermarking robust to geometric distortions,” IEEE Trans. Image Process. 14, 2140–2150 (2005).

Hu, M. K.

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Information Theory IT-8, 179–187, (1962).

Hyvarinen, A.

A. Hyvarinen and E. Oja, “Independent component analysis: a tutorial,” in Notes for International Joint Conference on Neural Networks (1999), http://www.cis.hut.fi/projects/ica/.

Ji, Z.

Kim, H. S.

H. S. Kim and H. K. Lee, “Invariant image watermark using Zernike moments,” IEEE Trans. Circuits Syst. Vid. Technol. 13, 766–775 (2003).

Kutter, M.

M. Kutter, “Performance improvement of spread spectrum based image watermarking schemes through M-ary modulation,” Lect. Notes Comput. Sci. 1728, 238–250 (1999).

Lee, H. K.

H. S. Kim and H. K. Lee, “Invariant image watermark using Zernike moments,” IEEE Trans. Circuits Syst. Vid. Technol. 13, 766–775 (2003).

Lee, P. A.

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tschebycheff moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).

Liao, S.

Y. Xin, S. Liao, and M. Pawlak, “A multibit geometrically robust image watermark based on Zernike moments,” International Conference on Pattern Recognition 4, 861–864 (2004).

Macq, B.

P. Bas, J.-M. Chassery, and B. Macq, “Geometrically invariant watermarking using feature points,” in Proceedings of IEEE International Conference on Image Processing, 11, 1014–1028 (2002).

Mukundan, R.

R. Mukundan, “Some computational aspects of discrete orthonormal moments,” IEEE Trans. Image Process. 13, 1055–1059 (2004).

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tschebycheff moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).

O’Ruanaidh, J.

J. O’Ruanaidh and T. Pun, “Rotation, scale, and translation invariant spread spectrum digital image watermarking,” Signal Process. 66, 303–317 (1998).

Oja, E.

A. Hyvarinen and E. Oja, “Independent component analysis: a tutorial,” in Notes for International Joint Conference on Neural Networks (1999), http://www.cis.hut.fi/projects/ica/.

Ong, S. H.

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tschebycheff moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).

Paramesran, R.

P.-T. Yap and R. Paramesran, “Local watermarks based on Krawtchouk moments,” in IEEE Region 10 Conference pp. 73–76 (2002).

Pawlak, M.

Y. Xin, S. Liao, and M. Pawlak, “A multibit geometrically robust image watermark based on Zernike moments,” International Conference on Pattern Recognition 4, 861–864 (2004).

Pereira, S.

S. Voloshynovskiy, S. Pereira, T. Pun, J. J. Eggers, and J. K. Su, “Attacks on digital watermarks: classification, estimation-based attacks, and benchmarks,” IEEE Commun. Mag. 8, 2–10 (2001).

S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Trans. Image Process. 9, 1123–1129 (2000).

Pun, T.

S. Voloshynovskiy, S. Pereira, T. Pun, J. J. Eggers, and J. K. Su, “Attacks on digital watermarks: classification, estimation-based attacks, and benchmarks,” IEEE Commun. Mag. 8, 2–10 (2001).

S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Trans. Image Process. 9, 1123–1129 (2000).

J. O’Ruanaidh and T. Pun, “Rotation, scale, and translation invariant spread spectrum digital image watermarking,” Signal Process. 66, 303–317 (1998).

Qian, G.-B.

Raghuveer, M. R.

G. Sundaramorthy, M. R. Raghuveer, and S. A. Diana, “Bispectral reconstruction of signal in noise amplitude reconstruction issues,” IEEE Trans. Acoust. Speech Signal Process. 38, 1297–1300 (1990).

Su, J. K.

S. Voloshynovskiy, S. Pereira, T. Pun, J. J. Eggers, and J. K. Su, “Attacks on digital watermarks: classification, estimation-based attacks, and benchmarks,” IEEE Commun. Mag. 8, 2–10 (2001).

Sundaramorthy, G.

G. Sundaramorthy, M. R. Raghuveer, and S. A. Diana, “Bispectral reconstruction of signal in noise amplitude reconstruction issues,” IEEE Trans. Acoust. Speech Signal Process. 38, 1297–1300 (1990).

Tewfik, A. H.

M. Alghoniemy and A. H. Tewfik, “Geometric invariants in image watermarking,” IEEE Trans. Image Process. 13, 145–153 (2004).

Voloshynovskiy, S.

S. Voloshynovskiy, S. Pereira, T. Pun, J. J. Eggers, and J. K. Su, “Attacks on digital watermarks: classification, estimation-based attacks, and benchmarks,” IEEE Commun. Mag. 8, 2–10 (2001).

Xiao, X.W.-W.

Xin, Y.

Y. Xin, S. Liao, and M. Pawlak, “A multibit geometrically robust image watermark based on Zernike moments,” International Conference on Pattern Recognition 4, 861–864 (2004).

Yang, Y.

p. Dong, J. G. Brankov, N. P. Galatsanos, Y. Yang, and F. Davoine “Digital watermarking robust to geometric distortions,” IEEE Trans. Image Process. 14, 2140–2150 (2005).

Yap, P.-T.

P.-T. Yap and R. Paramesran, “Local watermarks based on Krawtchouk moments,” in IEEE Region 10 Conference pp. 73–76 (2002).

Zhang, L.

IEEE Commun. Mag. (1)

S. Voloshynovskiy, S. Pereira, T. Pun, J. J. Eggers, and J. K. Su, “Attacks on digital watermarks: classification, estimation-based attacks, and benchmarks,” IEEE Commun. Mag. 8, 2–10 (2001).

IEEE Trans. Acoust. Speech Signal Process. (1)

G. Sundaramorthy, M. R. Raghuveer, and S. A. Diana, “Bispectral reconstruction of signal in noise amplitude reconstruction issues,” IEEE Trans. Acoust. Speech Signal Process. 38, 1297–1300 (1990).

IEEE Trans. Circuits Syst. Vid. Technol. (1)

H. S. Kim and H. K. Lee, “Invariant image watermark using Zernike moments,” IEEE Trans. Circuits Syst. Vid. Technol. 13, 766–775 (2003).

IEEE Trans. Image Process. (5)

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tschebycheff moments,” IEEE Trans. Image Process. 10, 1357–1364 (2001).

R. Mukundan, “Some computational aspects of discrete orthonormal moments,” IEEE Trans. Image Process. 13, 1055–1059 (2004).

S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Trans. Image Process. 9, 1123–1129 (2000).

p. Dong, J. G. Brankov, N. P. Galatsanos, Y. Yang, and F. Davoine “Digital watermarking robust to geometric distortions,” IEEE Trans. Image Process. 14, 2140–2150 (2005).

M. Alghoniemy and A. H. Tewfik, “Geometric invariants in image watermarking,” IEEE Trans. Image Process. 13, 145–153 (2004).

IRE Trans. Information Theory (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Information Theory IT-8, 179–187, (1962).

Lect. Notes Comput. Sci. (1)

M. Kutter, “Performance improvement of spread spectrum based image watermarking schemes through M-ary modulation,” Lect. Notes Comput. Sci. 1728, 238–250 (1999).

Opt. Express (1)

Signal Process. (1)

J. O’Ruanaidh and T. Pun, “Rotation, scale, and translation invariant spread spectrum digital image watermarking,” Signal Process. 66, 303–317 (1998).

Other (5)

A. Hyvarinen and E. Oja, “Independent component analysis: a tutorial,” in Notes for International Joint Conference on Neural Networks (1999), http://www.cis.hut.fi/projects/ica/.

Y. Xin, S. Liao, and M. Pawlak, “A multibit geometrically robust image watermark based on Zernike moments,” International Conference on Pattern Recognition 4, 861–864 (2004).

P. Dong and N. P. Galasanos, “Affine transform resistant watermarking based on image normalization,” in Proceedings of IEEE International Conference on Image Processing, 3, 489–492 (2002).

P. Bas, J.-M. Chassery, and B. Macq, “Geometrically invariant watermarking using feature points,” in Proceedings of IEEE International Conference on Image Processing, 11, 1014–1028 (2002).

P.-T. Yap and R. Paramesran, “Local watermarks based on Krawtchouk moments,” in IEEE Region 10 Conference pp. 73–76 (2002).

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Figures (5)

Fig. 1.
Fig. 1. Traditional watermark embedding system.

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Fig. 2.
Fig. 2. Closed-loop watermark embedding system.

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Fig. 3
Fig. 3 Watermark detection with dual-channel detection.

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Fig. 4.
Fig. 4. Results of estimation rotation angle.

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Fig. 5.
Fig. 5. Results of estimating of scaling factor and comparison.

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Tables (10)

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Table 1. The relation between the probabilities and the embedded watermark

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Table 2. Translation parameters estimation translated by StirMark

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Table 3. Rotation angle and scaling factor estimation in different domain

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Table 4. Translation parameter and scaling factor estimated by StirMark

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Table 5. Robust to additive noise (where μ and σ 2 are the mean and variance, respectively)

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Table 6. Robust to JPEG compression produced by StirMark

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Table 7. Robustness against attacks produced by StirMark

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Table 8. The probabilities of the detector

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Table 9. Experimental results compared to Ref. [6] and two commercial watermarking techniques

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Table 10. Experimental data

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Equations (72)

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T pq = 1 ρ ( p , N ) ρ ( q , N ) ∑ x = 0 N − 1 ∑ y = 0 N − 1 t p ( x ) t p ( y ) f ( x , y ) , p , q = 0,1,2 , ⋯ , N − 1 ,
t n ( x ) = n ! ∑ k = 0 n ( − 1 ) n − k N − 1 − k n − k n + k n x k .
f ( x , y ) = ∑ p = 0 N − 1 ∑ q = 0 N − 1 T pq t p ( x ) t q ( y ) .
m pq = ∑ x = 0 N − 1 ∑ y = 0 N − 1 x p y q f ( x , y ) ,
T pq = 1 ρ ˜ ( p , N ) ρ ˜ ( q , N ) ∑ i = 0 p ∑ j = 0 q c j , q , N c i , p , N m i , j .
T 00 = m 00 N 2 , T 10 = 6 m 10 + 3 ( 1 − N ) m 00 N ( N 2 − 1 ) , T 01 = 6 m 01 + 3 ( 1 − N ) m 00 N ( N 2 − 1 ) ,
T 20 = 30 m 20 + 30 ( 1 − N ) m 10 + 5 ( 1 − N ) ( 2 − N ) m 00 ( N 2 − 1 ) ( N 2 − 2 ) , T 02 = 30 m 02 + 30 ( 1 − N ) m 01 + 5 ( 1 − N ) ( 2 − N ) m 00 ( N 2 − 1 ) ( N 2 − 2 ) ,
T 11 = 36 m 11 + 18 ( 1 − N ) ( m 10 + m 01 ) + 9 ( 1 − N 2 ) m 00 ( N 2 − 1 ) 2 .
∣ x 2 y 2 ∣ = ∣ a b c d ∣ × ∣ x 1 y 1 ∣ + ∣ e f ∣ ,
[ x ′ y ′ ] = [ cos θ − sin θ sin θ cos θ ] × [ x y ] ,
m ′ 20 = ( 1 + cos θ 2 ) m 20 − ( sin 2 θ ) m 11 + ( 1 − cos θ 2 ) m 02 , m ′ 02 = ( 1 − cos θ 2 ) m 20 + ( sin 2 θ ) m 11 + ( 1 + cos θ 2 ) m 02 ,
m ′ 11 = ( sin 2 θ 2 ) m 20 + ( cos 2 θ ) m 11 − ( sin 2 θ 2 ) m 02 ,
m 20 + m 02 ′ = m 20 + m 02 .
T ′ 10 = T 10 cos θ − m 01 sin θ + 3 N N + 1 ( cos θ − sin θ + 1 ) ,
T ′ 01 = T 10 sin θ − T 01 cos θ + 3 N N + 1 ( cos θ + sin θ − 1 ) .
T 10 ′ cos θ + T 01 ′ sin θ = T 10 + 3 ( 1 − N ) N 2 N ( N 2 − 1 ) ( cos θ + sin θ − 1 ) T 00 ,
m pq ′ = a p + q + 2 m pq .
m 00 ′ = a 2 m 00 .
m ′ pq ( m ′ 00 ) ( p + q + 2 ) 2 = m pq ( m 00 ) ( p + q + 2 ) 2 .
a = N ′ ( N ′ 2 − 1 ) T 10 ′ − 3 ( 1 − N ′ ) N ′ 2 T 00 ′ N ( N 2 − 1 ) T 10 − 3 ( 1 − N ) N 2 T 00 , 3
{ x ′ = x + c y ′ = y + d ,
T 10 = 6 ∑ x = 0 N − 1 ∑ y = 0 N − 1 xf ( x , y ) + 3 ( 1 − N ) ∑ x = 0 N − 1 ∑ y = 0 N − 1 f ( x , y ) N ( N 2 − 1 ) , T 01 = 6 ∑ x = 0 N − 1 ∑ y = 0 N − 1 yf ( x , y ) + 3 ( 1 − N ) ∑ x = 0 N − 1 ∑ y = 0 N − 1 f ( x , y ) N ( N 2 − 1 ) .
T ′ 10 = 6 ∑ x = 0 N − 1 ∑ y = 0 N − 1 ( x + c ) f ( x , y ) + 3 ( 1 − N ) ∑ x = 0 N − 1 ∑ y = 0 N − 1 f ( x , y ) N ( N 2 − 1 ) , T ′ 10 = 6 ∑ x = 0 N − 1 ∑ y = 0 N − 1 ( y + d ) f ( x , y ) + 3 ( 1 − N ) ∑ x = 0 N − 1 ∑ y = 0 N − 1 f ( x , y ) N ( N 2 − 1 )
T 10 ′ = 6 ∑ x = 0 N − 1 ∑ y = 0 N − 1 xf ( x , y ) + 3 ( 1 − N ) ∑ x = 0 N − 1 ∑ y = 0 N − 1 f ( x , y ) N ( N 2 − 1 ) + 6 ∑ x = 0 N − 1 ∑ y = 0 N − 1 c f ( x , y ) N ( N 2 − 1 ) = T 10 + 6 cN T 00 N 2 − 1 ,
T 10 ′ = 6 ∑ x = 0 N − 1 ∑ y = 0 N − 1 yf ( x , y ) + 3 ( 1 − N ) ∑ x = 0 N − 1 ∑ y = 0 N − 1 f ( x , y ) N ( N 2 − 1 ) + 6 ∑ x = 0 N − 1 ∑ y = 0 N − 1 d f ( x , y ) N ( N 2 − 1 ) = T 01 + 6 dN T 00 N 2 − 1 .
c = ( N 2 − 1 ) ( T 10 ′ − T 10 ) ( 6 NT 00 ) , d = ( N 2 − 1 ) ( T 01 ′ − T 01 ) ( 6 NT 00 ) .
{ x 2 = x ′ cos θ − y ′ sin θ = x 1 a cos θ − y 1 a sin θ y 2 = x ′ sin θ + y ′ cos θ = x 1 a sin θ + y 1 a cos θ .
{ N ″ ( N ″ 2 − 1 ) T 10 ″ − 3 ( 1 − N ″ ) N ″ 2 T 00 ″ = a 3 [ N ( N 2 − 1 ) T 10 − 3 ( 1 − N ) N 2 T 00 ] N ″ ( N ″ 2 − 1 ) T 01 ″ − 3 ( 1 − N ″ ) N ″ 2 T 00 ″ = a 3 [ N ( N 2 − 1 ) T 01 − 3 ( 1 − N ) N 2 T 00 ] N ′ ( N ′ 2 − 1 ) T 10 ′ − 3 ( 1 − N ′ ) N ′ 2 T 00 ′ = N ″ ( N ″ 2 − 1 ) ( T 10 ″ cos θ − T 01 ″ sin θ ) − 3 ( 1 − N ″ ) N ″ 2 T 00 ″ ( cos θ − sin θ ) N ′ ( N ′ 2 − 1 ) T 01 ′ − 3 ( 1 − N ′ ) N ′ 2 T 00 ′ = N ″ ( N ″ 2 − 1 ) ( T 10 ″ sin θ + T 01 ″ cos θ ) − 3 ( 1 − N ″ ) N ″ 2 T 00 ″ ( cos θ + sin θ ) .
{ a = ( T 00 ′ N ′ 2 ) ( T 00 N 2 ) N ′ ( N ′ 2 − 1 ) ( T 10 ′ cos θ − T 01 ′ sin θ ) − 3 ( 1 − N ′ ) N ′ 2 T 00 ′ ( cos θ + sin θ ) = a 3 [ N ( N 2 − 1 ) T 10 − 3 ( 1 − N ) N 2 T 00 ] .
{ x 2 = x 1 a + m y 2 = y 1 a + n .
a = ( T 00 ′ N ′ 2 ) ( T 00 N 2 )
c = [ N ′ ( N ′ 2 − 1 ) T 10 ′ − a 3 N ( N 2 − 1 ) T 10 − 3 ( 1 − N ′ ) N ′ 2 T 00 ′ + 3 ( 1 − N ) N 2 T 00 ] ( 6 N 2 T 00 )
d = [ N ′ ( N ′ 2 − 1 ) T 01 ′ − a 3 N ( N 2 − 1 ) T 01 − 3 ( 1 − N ′ ) N ′ 2 T 00 ′ + 3 ( 1 − N ) N 2 T 00 ] ( 6 N 2 T 00 ) .
T ̂ m i , n i = T om i , n i + N wk ( m i , n i ) T wm i , n i .
PSNR k = 10 log { x max 2 [ N − 2 ∑ i = 1 N ∑ i = 1 N ( x i , j − y i , j ( k ) ) 2 ] } ,
N wk = N w 0 − ∑ i = 0 k 2 ρ 2 ( i − 1 ) q 0 ( ( n 2 i − 1 + 1 ) − ρ ( m 2 i − 1 + 1 ) ) ,
N wk + 1 − N wk = ρ k q 0 ( m k + 1 ) .
N wk = N w 0 − ∑ i = 0 ( k − 1 ) 2 ρ 2 ( i − 1 ) q 0 ( ( n 2 i − 1 + 1 ) − ρ ( m 2 i − 1 + 1 ) ) − ρ k − 1 q 0 ( n k + 1 ) ,
N wk + 1 − N wk = ρ k q 0 ( m k + 1 + 1 ) .
lim k → ∞ ( N wk + 1 − N wk ) = 0 .
N wk + 1 = N wk + ρ ( n k q k − ρ m k q k ) = N w 0 + ( n 0 − m 0 ρ ) q 0 + ρ ( n 1 − ρ m 1 ) q 1 + ⋯ + ρ ( n k − ρ m k ) q k
= N wo + [ ( n 0 − ρ m 0 ) + ρ 2 ( n 1 − ρ m 1 ) + ⋯ + ρ 2 k ( n k − ρ m k ) ] q 0 .
N wk + 1 − N wk = ρ 2 k ( n k − ρ m k ) q 0 < ρ 2 k u q 0 → 0 .
H 0 : Y ( i , j ) = X ( i , j ) = watermark absent , H 1 : Y ( i , j ) = X ( i , j ) − W ( i , j ) watermark present .
sim ( w , w ′ ) = ∑ w ( n ) w ′ ( n ) w 2 ( n ) w ′ 2 ( n ) .
P dE = Prob { sim ( w , w ′ ) ≥ η H 1 } ,
p fE = Prob { sim ( w , w ′ ) ≥ η H 0 } ,
sim ( w , w ′ ) = ∑ w ( n ) w ′ ( n ) N w = ∑ k ( n ) N w .
p fE = P ( sim ( w , w ′ ) > η H 0 ) = p ( ∑ k ( n ) > N w η H 0 ) .
p fE = P ( ∑ k ( n ) > N w η H 0 ) = ∑ m = ⌈ N w ( η + 1 ) 2 ⌉ N w p ( ∑ k ( n ) = − N w + 2 m H 0 ) ,
P ( ∑ k ( n ) = − N w + 2 m H 0 ) = N w ! m ! ( N w − m ) ! p 0 N w − m ( 1 − p 0 ) m .
p fE = ∑ m = ⌈ N w ( η + 1 ) 2 ⌉ N w N w ! m ! ( N w − m ) ! 0.5 N w .
F ( N w ) = ∑ m = ⌈ ( N w + 1 ) ( η + 1 ) 2 ⌉ N w + 1 ( N w + 1 ) ! m ! ( N w + 1 − m ) ! 0.5 N w + 1 − ∑ m = ⌈ N w ( η + 1 ) 2 ⌉ N w N w ! m ! ( N w − m ) ! 0.5 N w .
F ( k ) = ∑ m = ⌈ ( k + 1 ) ( η + 1 ) 2 ⌉ k + 1 ( k + 1 ) ! m ! ( k + 1 − m ) ! 0.5 k + 1 − ∑ m = ⌈ k ( η + 1 ) 2 ⌉ k k ! m ! ( k − m ) ! 0.5 k < 0 .
F ( k + 1 ) = ∑ m = ⌈ ( k + 2 ) ( η + 1 ) 2 ⌉ k + 2 ( k + 2 ) ! m ! ( k + 2 − m ) ! 0.5 k + 2 − ∑ m = ⌈ k + 1 ( η + 1 ) 2 ⌉ k + 1 ( k + 1 ) ! m ! ( k + 1 − m ) ! 0.5 k + 1
= ∑ m = ⌈ ( k + 2 ) ( η + 1 ) 2 ⌉ k + 1 ( k + 2 ) ! m ! ( k + 2 − m ) ! 0.5 k + 2 − ∑ m = ⌈ k + 1 ( η + 1 ) 2 ⌉ k ( k + 1 ) ! m ! ( k + 1 − m ) ! 0.5 k + 1
< ∑ m = ⌈ ( k + 1 ) ( η + 1 ) 2 ⌉ k + 1 ( k + 1 ) ! m ! ( k + 1 − m ) ! k + 1 k + 1 − m 0.5 k + 2 − ∑ m = ⌈ k ( η + 1 ) 2 ⌉ k k ! m ! ( k − m ) ! k + 1 k + 1 − m 0.5 k + 1
< ∑ m = ⌈ ( k + 1 ) ( η + 1 ) 2 ⌉ k + 1 ( k + 1 ) ! m ! ( k + 1 − m ) ! 0.5 k + 1 − ∑ m = ⌈ k ( η + 1 ) 2 ⌉ k k ! m ! ( k − m ) ! 0.5 k < 0 .
p dE = Prob { sim ( w , w ′ ) ≥ η H 1 } ∫ η ∞ 1 8 π N w 45 e ( t − 1 3 ) 2 4 N w 45 dt .
H 0 : B y ( w 1 , w 2 ) = B x ( w 1 , w 2 ) watermark absent , H 1 : B y ( w 1 , w 2 ) = B w ( w 1 , w 2 ) + B x ( w 1 , w 2 ) present .
T = 2 ∑ P ∣ B ( n ) ( w i , w j ) ∣ 2 [ N KL 2 p y ( w i ) p y ( w j ) p y ( w i + w j ) ] 1 2 .
p ( T H 0 ) = 1 ( 2 πσ 2 ) N 2 exp [ − ∑ i = 1 N ( T ( i ) − μ 0 ) 2 2 σ 2 ] , p ( T H 1 ) = 1 ( 2 πσ 2 ) N 2 exp [ − ∑ i = 1 N ( T ( i ) − μ 1 ) 2 2 σ 2 ] .
λ ( T ) = p ( T H 0 ) p ( T H 1 ) = exp [ μ 1 − μ 0 2 σ 2 ∑ i = 1 N T ( i ) − N ( μ 1 2 − μ 0 2 ) 2 σ 2 ] < H 0 > H 1 η .
∑ i = 1 N T ( i ) < H 0 > H 1 σ 2 μ 1 − μ 0 [ ln η + N ( μ 1 2 − μ 0 2 ) 2 σ 2 ] .
p fB = ∫ η ∞ p ( T ˜ H 0 ) d T ˜ = ∫ η ∞ 1 ( 2 π σ 2 ) N 2 exp [ − ∑ i = 1 N ( T ˜ ( i ) − μ 0 ) 2 2 σ 2 ] d T ˜ .
p dB = ∫ η ∞ p ( T ˜ H 1 ) d T ˜ = ∫ η ∞ 1 ( 2 π σ 2 ) N 2 exp [ − ∑ i = 1 N ( T ˜ ( i ) − μ 1 ) 2 2 σ 2 ] d T ˜ .
p fB = 1 − erf ( 2 η − 4 N − 1 ) = α .
η = 1 2 ( erf − 1 ( 1 − α ) + 4 N − 1 ) 2 .
p dB = 1 − erf ( η − 2 N − B 2 N + B ) .
p D = p dE + p dB − p dE p dB .
T pq = 1 ρ ~ p N ρ ~ q N ∑ x = 0 N 2 − 1 ∑ y = 0 N 2 − 1 t ~ p ( x ) t ~ q ( y ) { f ( x , y ) + ( − 1 ) p f ( N − 1 − x , y ) + ( − 1 ) q f ( x , N − 1 − y ) + ( − 1 ) p + q f ( N − 1 − x , N − 1 − y ) } .
NC = ∑ i , j w i j w ´ i j ∑ i , j ( w i j ) 2 .

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