Abstract

In a feature-based geometrically robust watermarking system, it is a challenging task to detect geometric-invariant regions (GIRs) which can survive a broad range of image processing operations. Instead of commonly used Harris detector or Mexican hat wavelet method, a more robust corner detector named multi-scale curvature product (MSCP) is adopted to extract salient features in this paper. Based on such features, disk-like GIRs are found, which consists of three steps. First, robust edge contours are extracted. Then, MSCP is utilized to detect the centers for GIRs. Third, the characteristic scale selection is performed to calculate the radius of each GIR. A novel sector-shaped partitioning method for the GIRs is designed, which can divide a GIR into several sector discs with the help of the most important corner (MIC). The watermark message is then embedded bit by bit in each sector by using Quantization Index Modulation (QIM). The GIRs and the divided sector discs are invariant to geometric transforms, so the watermarking method inherently has high robustness against geometric attacks. Experimental results show that the scheme has a better robustness against various image processing operations including common processing attacks, affine transforms, cropping, and random bending attack (RBA) than the previous approaches.

© 2009 OSA

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Y. Xin, S. Liao, and M. Pawlak, “Circularly orthogonal moments for geometrically robust image watermarking,” Pattern Recognit. 40(12), 3740–3752 (2007).
[CrossRef]

X. Qi and J. Qi, “A robust content-based digital image watermarking scheme,” Signal Processing 87(6), 1264–1280 (2007).
[CrossRef]

X. Wang, J. Wu, and P. Niu, “A new digital image watermarking algorithm resilient to desynchronization attacks,” IEEE Trans. Info. Forens. Sec. 4,655–663 (2007).
[CrossRef]

X. Zhang, M. Lei, D. Yang, Y. Wang, and L. Ma, “Multi-scale curvature product for robust image corner detection in curvature scale space,” Pattern Recognit. Lett. 28(5), 545–554 (2007).
[CrossRef]

L. Zhang, G. Qian, W. Xiao, and Z. Ji, “Geometric invariant blind image watermarking by invariant Tchebichef moments,” Opt. Express 15(5), 2251–2261 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2251 .
[CrossRef] [PubMed]

2006 (1)

J. Dugelay, S. Roche, C. Rey, and G. Doerr, “Still-image watermarking robust to local geometric distortions,” IEEE Trans. Image Process. 15(9), 2831–2842 (2006).
[CrossRef] [PubMed]

2005 (1)

M. Barni, “Effectiveness of exhaustive search and template matching against watermark desynchronization,” IEEE Signal Process. Lett. 12(2), 158–161 (2005).
[CrossRef]

2004 (4)

K. Mikolajczyk and C. Schmid, “Scale & affine invariant interest point detectors,” Int. J. Comput. Vis. 60(1), 63–86 (2004).
[CrossRef]

J. S. Seo, C. D. Chang, and D. Yoo, “Localized image watermarking based on feature points of scale-space representation,” Pattern Recognit. 37(7), 1365–1375 (2004).
[CrossRef]

M. Hsieh and D. Tseng, “Perceptual digital watermarking for image authentication in electronic commerce,” Electron. Commerce Res. 4(1/2), 157–170 (2004).
[CrossRef]

M. Alghoniemy and A. H. Tewfik, “Geometric invariance in image watermarking,” IEEE Trans. Image Process. 13(2), 145–153 (2004).
[CrossRef] [PubMed]

2003 (3)

C. Tang and H. Hang, “A feature-based robust digital image watermarking scheme,” IEEE Trans. Signal Process. 51(4), 950–959 (2003).
[CrossRef]

H. Kim and H. Lee, “Invariant image watermark using zernike moments,” IEEE Trans. Circuits Syst. Video Technol. 8, 766–775 (2003).

D. Zheng, J. Zhao, and A. Saddik, “Rst-invariant digital image watermarking based on log-polar mapping and phase correlation,” IEEE Trans. Circuits Syst. Video Technol. 13(8), 753–765 (2003).
[CrossRef]

2002 (1)

P. Bas, J. Chassery, and B. Macq, “Geometrically invariant watermarking using feature points,” IEEE Trans. Image Process. 11(9), 1014–1028 (2002).
[CrossRef]

2001 (1)

C. Y. Lin, M. Wu, J. Bloom, I. Cox, M. Miller, and Y. Lui, “Rotation, scale, and translation resilient watermarking for images,” IEEE Trans. Image Process. 10(5), 767–782 (2001).
[CrossRef]

2000 (3)

B. Chen and G. W. Wornell, “Preprocessed and postprocessed quantization index modulation methods for digital watermarking,” SPIE 3971, 48–59 (2000).
[CrossRef]

S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Trans. Image Process. 9(6), 1123–1129 (2000).
[CrossRef]

C. Schmid, R. Mohr, and C. Bauckhage, “Evaluation of interest point detectors,” Int. J. Comput. Vis. 37(2), 151–172 (2000).
[CrossRef]

1998 (1)

J. Ruanaidh and T. Pun, “Rotation, scale and translation invariant spread spectrum digital image watermarking,” Signal Processing 66(3), 303–317 (1998).
[CrossRef]

1992 (1)

F. Mokhtarian and A. Mackworth, “A theory of multiscale, curvature-based shape representation for planar curves,” IEEE Trans. Pattern Anal. Mach. Intell. 14(8), 789–805 (1992).
[CrossRef]

1973 (1)

R. M. Haralick, K. Shanmugam, and I. Dinstein, “Textural features for image classification,” IEEE Trans. Syst. Man Cybern. 3(6), 610–621 (1973).
[CrossRef]

Alghoniemy, M.

M. Alghoniemy and A. H. Tewfik, “Geometric invariance in image watermarking,” IEEE Trans. Image Process. 13(2), 145–153 (2004).
[CrossRef] [PubMed]

Barni, M.

M. Barni, “Effectiveness of exhaustive search and template matching against watermark desynchronization,” IEEE Signal Process. Lett. 12(2), 158–161 (2005).
[CrossRef]

Bas, P.

P. Bas, J. Chassery, and B. Macq, “Geometrically invariant watermarking using feature points,” IEEE Trans. Image Process. 11(9), 1014–1028 (2002).
[CrossRef]

Bauckhage, C.

C. Schmid, R. Mohr, and C. Bauckhage, “Evaluation of interest point detectors,” Int. J. Comput. Vis. 37(2), 151–172 (2000).
[CrossRef]

Bloom, J.

C. Y. Lin, M. Wu, J. Bloom, I. Cox, M. Miller, and Y. Lui, “Rotation, scale, and translation resilient watermarking for images,” IEEE Trans. Image Process. 10(5), 767–782 (2001).
[CrossRef]

Chang, C. D.

J. S. Seo, C. D. Chang, and D. Yoo, “Localized image watermarking based on feature points of scale-space representation,” Pattern Recognit. 37(7), 1365–1375 (2004).
[CrossRef]

Chassery, J.

P. Bas, J. Chassery, and B. Macq, “Geometrically invariant watermarking using feature points,” IEEE Trans. Image Process. 11(9), 1014–1028 (2002).
[CrossRef]

Chen, B.

B. Chen and G. W. Wornell, “Preprocessed and postprocessed quantization index modulation methods for digital watermarking,” SPIE 3971, 48–59 (2000).
[CrossRef]

Cox, I.

C. Y. Lin, M. Wu, J. Bloom, I. Cox, M. Miller, and Y. Lui, “Rotation, scale, and translation resilient watermarking for images,” IEEE Trans. Image Process. 10(5), 767–782 (2001).
[CrossRef]

Dinstein, I.

R. M. Haralick, K. Shanmugam, and I. Dinstein, “Textural features for image classification,” IEEE Trans. Syst. Man Cybern. 3(6), 610–621 (1973).
[CrossRef]

Doerr, G.

J. Dugelay, S. Roche, C. Rey, and G. Doerr, “Still-image watermarking robust to local geometric distortions,” IEEE Trans. Image Process. 15(9), 2831–2842 (2006).
[CrossRef] [PubMed]

Dugelay, J.

J. Dugelay, S. Roche, C. Rey, and G. Doerr, “Still-image watermarking robust to local geometric distortions,” IEEE Trans. Image Process. 15(9), 2831–2842 (2006).
[CrossRef] [PubMed]

Hang, H.

C. Tang and H. Hang, “A feature-based robust digital image watermarking scheme,” IEEE Trans. Signal Process. 51(4), 950–959 (2003).
[CrossRef]

Haralick, R. M.

R. M. Haralick, K. Shanmugam, and I. Dinstein, “Textural features for image classification,” IEEE Trans. Syst. Man Cybern. 3(6), 610–621 (1973).
[CrossRef]

Hsieh, M.

M. Hsieh and D. Tseng, “Perceptual digital watermarking for image authentication in electronic commerce,” Electron. Commerce Res. 4(1/2), 157–170 (2004).
[CrossRef]

Ji, Z.

Kim, H.

H. Kim and H. Lee, “Invariant image watermark using zernike moments,” IEEE Trans. Circuits Syst. Video Technol. 8, 766–775 (2003).

Lee, H.

H. Kim and H. Lee, “Invariant image watermark using zernike moments,” IEEE Trans. Circuits Syst. Video Technol. 8, 766–775 (2003).

Lei, M.

X. Zhang, M. Lei, D. Yang, Y. Wang, and L. Ma, “Multi-scale curvature product for robust image corner detection in curvature scale space,” Pattern Recognit. Lett. 28(5), 545–554 (2007).
[CrossRef]

Liao, S.

Y. Xin, S. Liao, and M. Pawlak, “Circularly orthogonal moments for geometrically robust image watermarking,” Pattern Recognit. 40(12), 3740–3752 (2007).
[CrossRef]

Lin, C. Y.

C. Y. Lin, M. Wu, J. Bloom, I. Cox, M. Miller, and Y. Lui, “Rotation, scale, and translation resilient watermarking for images,” IEEE Trans. Image Process. 10(5), 767–782 (2001).
[CrossRef]

Lui, Y.

C. Y. Lin, M. Wu, J. Bloom, I. Cox, M. Miller, and Y. Lui, “Rotation, scale, and translation resilient watermarking for images,” IEEE Trans. Image Process. 10(5), 767–782 (2001).
[CrossRef]

Ma, L.

X. Zhang, M. Lei, D. Yang, Y. Wang, and L. Ma, “Multi-scale curvature product for robust image corner detection in curvature scale space,” Pattern Recognit. Lett. 28(5), 545–554 (2007).
[CrossRef]

Mackworth, A.

F. Mokhtarian and A. Mackworth, “A theory of multiscale, curvature-based shape representation for planar curves,” IEEE Trans. Pattern Anal. Mach. Intell. 14(8), 789–805 (1992).
[CrossRef]

Macq, B.

P. Bas, J. Chassery, and B. Macq, “Geometrically invariant watermarking using feature points,” IEEE Trans. Image Process. 11(9), 1014–1028 (2002).
[CrossRef]

Mikolajczyk, K.

K. Mikolajczyk and C. Schmid, “Scale & affine invariant interest point detectors,” Int. J. Comput. Vis. 60(1), 63–86 (2004).
[CrossRef]

Miller, M.

C. Y. Lin, M. Wu, J. Bloom, I. Cox, M. Miller, and Y. Lui, “Rotation, scale, and translation resilient watermarking for images,” IEEE Trans. Image Process. 10(5), 767–782 (2001).
[CrossRef]

Mohr, R.

C. Schmid, R. Mohr, and C. Bauckhage, “Evaluation of interest point detectors,” Int. J. Comput. Vis. 37(2), 151–172 (2000).
[CrossRef]

Mokhtarian, F.

F. Mokhtarian and A. Mackworth, “A theory of multiscale, curvature-based shape representation for planar curves,” IEEE Trans. Pattern Anal. Mach. Intell. 14(8), 789–805 (1992).
[CrossRef]

Niu, P.

X. Wang, J. Wu, and P. Niu, “A new digital image watermarking algorithm resilient to desynchronization attacks,” IEEE Trans. Info. Forens. Sec. 4,655–663 (2007).
[CrossRef]

Pawlak, M.

Y. Xin, S. Liao, and M. Pawlak, “Circularly orthogonal moments for geometrically robust image watermarking,” Pattern Recognit. 40(12), 3740–3752 (2007).
[CrossRef]

Pereira, S.

S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Trans. Image Process. 9(6), 1123–1129 (2000).
[CrossRef]

Pun, T.

S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Trans. Image Process. 9(6), 1123–1129 (2000).
[CrossRef]

J. Ruanaidh and T. Pun, “Rotation, scale and translation invariant spread spectrum digital image watermarking,” Signal Processing 66(3), 303–317 (1998).
[CrossRef]

Qi, J.

X. Qi and J. Qi, “A robust content-based digital image watermarking scheme,” Signal Processing 87(6), 1264–1280 (2007).
[CrossRef]

Qi, X.

X. Qi and J. Qi, “A robust content-based digital image watermarking scheme,” Signal Processing 87(6), 1264–1280 (2007).
[CrossRef]

Qian, G.

Rey, C.

J. Dugelay, S. Roche, C. Rey, and G. Doerr, “Still-image watermarking robust to local geometric distortions,” IEEE Trans. Image Process. 15(9), 2831–2842 (2006).
[CrossRef] [PubMed]

Roche, S.

J. Dugelay, S. Roche, C. Rey, and G. Doerr, “Still-image watermarking robust to local geometric distortions,” IEEE Trans. Image Process. 15(9), 2831–2842 (2006).
[CrossRef] [PubMed]

Ruanaidh, J.

J. Ruanaidh and T. Pun, “Rotation, scale and translation invariant spread spectrum digital image watermarking,” Signal Processing 66(3), 303–317 (1998).
[CrossRef]

Saddik, A.

D. Zheng, J. Zhao, and A. Saddik, “Rst-invariant digital image watermarking based on log-polar mapping and phase correlation,” IEEE Trans. Circuits Syst. Video Technol. 13(8), 753–765 (2003).
[CrossRef]

Schmid, C.

K. Mikolajczyk and C. Schmid, “Scale & affine invariant interest point detectors,” Int. J. Comput. Vis. 60(1), 63–86 (2004).
[CrossRef]

C. Schmid, R. Mohr, and C. Bauckhage, “Evaluation of interest point detectors,” Int. J. Comput. Vis. 37(2), 151–172 (2000).
[CrossRef]

Seo, J. S.

J. S. Seo, C. D. Chang, and D. Yoo, “Localized image watermarking based on feature points of scale-space representation,” Pattern Recognit. 37(7), 1365–1375 (2004).
[CrossRef]

Shanmugam, K.

R. M. Haralick, K. Shanmugam, and I. Dinstein, “Textural features for image classification,” IEEE Trans. Syst. Man Cybern. 3(6), 610–621 (1973).
[CrossRef]

Tang, C.

C. Tang and H. Hang, “A feature-based robust digital image watermarking scheme,” IEEE Trans. Signal Process. 51(4), 950–959 (2003).
[CrossRef]

Tewfik, A. H.

M. Alghoniemy and A. H. Tewfik, “Geometric invariance in image watermarking,” IEEE Trans. Image Process. 13(2), 145–153 (2004).
[CrossRef] [PubMed]

Tseng, D.

M. Hsieh and D. Tseng, “Perceptual digital watermarking for image authentication in electronic commerce,” Electron. Commerce Res. 4(1/2), 157–170 (2004).
[CrossRef]

Wang, X.

X. Wang, J. Wu, and P. Niu, “A new digital image watermarking algorithm resilient to desynchronization attacks,” IEEE Trans. Info. Forens. Sec. 4,655–663 (2007).
[CrossRef]

Wang, Y.

X. Zhang, M. Lei, D. Yang, Y. Wang, and L. Ma, “Multi-scale curvature product for robust image corner detection in curvature scale space,” Pattern Recognit. Lett. 28(5), 545–554 (2007).
[CrossRef]

Wornell, G. W.

B. Chen and G. W. Wornell, “Preprocessed and postprocessed quantization index modulation methods for digital watermarking,” SPIE 3971, 48–59 (2000).
[CrossRef]

Wu, J.

X. Wang, J. Wu, and P. Niu, “A new digital image watermarking algorithm resilient to desynchronization attacks,” IEEE Trans. Info. Forens. Sec. 4,655–663 (2007).
[CrossRef]

Wu, M.

C. Y. Lin, M. Wu, J. Bloom, I. Cox, M. Miller, and Y. Lui, “Rotation, scale, and translation resilient watermarking for images,” IEEE Trans. Image Process. 10(5), 767–782 (2001).
[CrossRef]

Xiao, W.

Xin, Y.

Y. Xin, S. Liao, and M. Pawlak, “Circularly orthogonal moments for geometrically robust image watermarking,” Pattern Recognit. 40(12), 3740–3752 (2007).
[CrossRef]

Yang, D.

X. Zhang, M. Lei, D. Yang, Y. Wang, and L. Ma, “Multi-scale curvature product for robust image corner detection in curvature scale space,” Pattern Recognit. Lett. 28(5), 545–554 (2007).
[CrossRef]

Yoo, D.

J. S. Seo, C. D. Chang, and D. Yoo, “Localized image watermarking based on feature points of scale-space representation,” Pattern Recognit. 37(7), 1365–1375 (2004).
[CrossRef]

Zhang, L.

Zhang, X.

X. Zhang, M. Lei, D. Yang, Y. Wang, and L. Ma, “Multi-scale curvature product for robust image corner detection in curvature scale space,” Pattern Recognit. Lett. 28(5), 545–554 (2007).
[CrossRef]

Zhao, J.

D. Zheng, J. Zhao, and A. Saddik, “Rst-invariant digital image watermarking based on log-polar mapping and phase correlation,” IEEE Trans. Circuits Syst. Video Technol. 13(8), 753–765 (2003).
[CrossRef]

Zheng, D.

D. Zheng, J. Zhao, and A. Saddik, “Rst-invariant digital image watermarking based on log-polar mapping and phase correlation,” IEEE Trans. Circuits Syst. Video Technol. 13(8), 753–765 (2003).
[CrossRef]

Electron. Commerce Res. (1)

M. Hsieh and D. Tseng, “Perceptual digital watermarking for image authentication in electronic commerce,” Electron. Commerce Res. 4(1/2), 157–170 (2004).
[CrossRef]

IEEE Signal Process. Lett. (1)

M. Barni, “Effectiveness of exhaustive search and template matching against watermark desynchronization,” IEEE Signal Process. Lett. 12(2), 158–161 (2005).
[CrossRef]

IEEE Trans. Circuits Syst. Video Technol. (2)

D. Zheng, J. Zhao, and A. Saddik, “Rst-invariant digital image watermarking based on log-polar mapping and phase correlation,” IEEE Trans. Circuits Syst. Video Technol. 13(8), 753–765 (2003).
[CrossRef]

H. Kim and H. Lee, “Invariant image watermark using zernike moments,” IEEE Trans. Circuits Syst. Video Technol. 8, 766–775 (2003).

IEEE Trans. Image Process. (5)

S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Trans. Image Process. 9(6), 1123–1129 (2000).
[CrossRef]

P. Bas, J. Chassery, and B. Macq, “Geometrically invariant watermarking using feature points,” IEEE Trans. Image Process. 11(9), 1014–1028 (2002).
[CrossRef]

C. Y. Lin, M. Wu, J. Bloom, I. Cox, M. Miller, and Y. Lui, “Rotation, scale, and translation resilient watermarking for images,” IEEE Trans. Image Process. 10(5), 767–782 (2001).
[CrossRef]

J. Dugelay, S. Roche, C. Rey, and G. Doerr, “Still-image watermarking robust to local geometric distortions,” IEEE Trans. Image Process. 15(9), 2831–2842 (2006).
[CrossRef] [PubMed]

M. Alghoniemy and A. H. Tewfik, “Geometric invariance in image watermarking,” IEEE Trans. Image Process. 13(2), 145–153 (2004).
[CrossRef] [PubMed]

IEEE Trans. Info. Forens. Sec. (1)

X. Wang, J. Wu, and P. Niu, “A new digital image watermarking algorithm resilient to desynchronization attacks,” IEEE Trans. Info. Forens. Sec. 4,655–663 (2007).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

F. Mokhtarian and A. Mackworth, “A theory of multiscale, curvature-based shape representation for planar curves,” IEEE Trans. Pattern Anal. Mach. Intell. 14(8), 789–805 (1992).
[CrossRef]

IEEE Trans. Signal Process. (1)

C. Tang and H. Hang, “A feature-based robust digital image watermarking scheme,” IEEE Trans. Signal Process. 51(4), 950–959 (2003).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

R. M. Haralick, K. Shanmugam, and I. Dinstein, “Textural features for image classification,” IEEE Trans. Syst. Man Cybern. 3(6), 610–621 (1973).
[CrossRef]

Int. J. Comput. Vis. (2)

C. Schmid, R. Mohr, and C. Bauckhage, “Evaluation of interest point detectors,” Int. J. Comput. Vis. 37(2), 151–172 (2000).
[CrossRef]

K. Mikolajczyk and C. Schmid, “Scale & affine invariant interest point detectors,” Int. J. Comput. Vis. 60(1), 63–86 (2004).
[CrossRef]

Opt. Express (1)

Pattern Recognit. (2)

J. S. Seo, C. D. Chang, and D. Yoo, “Localized image watermarking based on feature points of scale-space representation,” Pattern Recognit. 37(7), 1365–1375 (2004).
[CrossRef]

Y. Xin, S. Liao, and M. Pawlak, “Circularly orthogonal moments for geometrically robust image watermarking,” Pattern Recognit. 40(12), 3740–3752 (2007).
[CrossRef]

Pattern Recognit. Lett. (1)

X. Zhang, M. Lei, D. Yang, Y. Wang, and L. Ma, “Multi-scale curvature product for robust image corner detection in curvature scale space,” Pattern Recognit. Lett. 28(5), 545–554 (2007).
[CrossRef]

Signal Processing (2)

X. Qi and J. Qi, “A robust content-based digital image watermarking scheme,” Signal Processing 87(6), 1264–1280 (2007).
[CrossRef]

J. Ruanaidh and T. Pun, “Rotation, scale and translation invariant spread spectrum digital image watermarking,” Signal Processing 66(3), 303–317 (1998).
[CrossRef]

SPIE (1)

B. Chen and G. W. Wornell, “Preprocessed and postprocessed quantization index modulation methods for digital watermarking,” SPIE 3971, 48–59 (2000).
[CrossRef]

Other (7)

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, “Digital Image Processing Using MATLAB”, in Prentice Hall, (New Jersey, 2003).

M. Alghoniemy, and A. Tewfik, “Image watermarking by moment invariants”, in Proceedings of IEEE International Conference on Image Processing (Vancouver, BC, Canada,2000), pp.73–76.

H. Lee, I. Kang, H. Lee, and Y. Suh, “Evaluation of feature extraction techniques for robust watermarking”, in Proceedings of 4th Int. Workshop on Digital Watermarking(Siena, Italy, 2005), pp. 418–431.

A. Tinku, and K. Ajoy, “Image processing principles and applications”, John Wiley and Sons Inc., (New Jersey, 2005).

F. A. P. Petitcolas, and R. J. Anderson, “Evaluation of copyright marking systems”, in Proceedings of IEEE Multimedia Systems (Florence, Italy, 1999), pp. 574–579.

M. Kutter, S. K. Bhattacharjee, and T. Ebrahimi, “Towards second generation watermarking schemes”, in Proceedings of IEEE International Conference on Image Processing (Kobe, Japan, 1999), pp. 320–323.

J. Weinheimer, X. Qi, and J. Qi, “Towards a robust feature-based watermarking scheme”, in Proceedings of IEEE International Conference on Image Processing (Atlanta, GA, USA, 2006), pp. 1401–1404.

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Figures (5)

Fig. 1
Fig. 1

Watermark embedding framework.

Fig. 2
Fig. 2

Exceptive edge contours.

Fig. 3
Fig. 3

Performance of GIRs detection. (a)Original Lena image. (b)GIRs of (a). (c)Rotated by 10 degree plus cropping and scaling. (d)GIRs of (c). (e)Rotated by 45 degrees plus cropping. (f)GIRs of (e). (g)Rotated by −10 degrees. (h)GIRs of (g). (i)Removed 17 rows and 5 columns. (j)GIRs of (i). (k)StirMark RBA. (l)GIRs of (k). (m)Original Baboon image. (n)GIRs of (m). (o)Original Peppers image. (p)GIRs of (o).

Fig. 4
Fig. 4

The GIR partition.

Fig. 5
Fig. 5

The watermarked images

Tables (5)

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Table 3 The comparisons among the proposed method, Tang’s method and Qi’s method under different geometric distortions.

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Table 1 Several images texture dependent parameters

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Table 2 The comparisons among the proposed method, Tang’s method and Qi’s method under different common processing attacks. a

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Table 4 The comparisons of the proposed method and Tang’s method under different common processing attacks. The length of the watermark sequence is 16 bits

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Table 5 The comparisons of the proposed method and Tang’s method under different geometric distortions. The length of the watermark sequence is 16 bits.

Equations (14)

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k ( u , σ ) = X u ( u , σ ) Y u u ( u , σ ) X u u ( u , σ ) Y u ( u , σ ) ( X u ( u , σ ) 2 + Y u ( u , σ ) 2 ) 1.5
P N ( u ) = j = 1 N k ( u , σ j )
L o G ( x , y , δ i ) = δ i 2 | L x x ( x , y , δ i ) + L y y ( x , y , δ i ) |
R = k δ
k = k 0 W H I L E k δ < R min k = k + 1 E N D W H I L E k δ > R max k = k 1 E N D R = k δ
R min = l o w e r min ( h e i g h t , w i d t h )
R max = u p p e r max ( h e i g h t , w i d t h )
tan α = k l k l 0 1 + k l k l 0
f w ( x , y ) = Q ( f ( x , y ) ; w n )
w ^ n = { 1 , i f N u m n ( 1 ) N u m n ( 0 ) 0 , i f N u m n ( 1 ) < N u m n ( 0 )
P F G I R = r = t s 2 N ( 1 2 ) 2 N ( ( 2 N ) ! r ! ( 2 N r ) ! )
P F i m a g e = i = m N G I R ( P F G I R ) i ( 1 P F G I R ) N G I R i ( N G I R i )
P S G I R = r = t s 2 N P r
P M i m a g e = 1 i = m N G I R ( P S G I R ) i ( 1 P S G I R ) N G I R i ( N G I R i )

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