Abstract

This paper presents a novel multipurpose scheme for content-based image authentication and copyright protection using a perceptual image hashing and watermarking strategy based on a wave atom transform. The wave atom transform is expected to outperform other transforms because it gains sparser expansion and better representation for texture than other traditional transforms, such as wavelet and curvelet transforms. Images are decomposed into multiscale bands with a number of tilings using the wave atom transform. Perceptual hashes are then extracted from the features of tiling in the third scale band for the purpose of content-based authentication; simultaneously, part of the selected hashes are designed as watermarks, which are embedded into the original images for the purpose of copyright protection. The experimental results demonstrate that the proposed scheme shows great performance in content-based authentication by distinguishing the maliciously attacked images from the nonmaliciously attacked images. Moreover, watermarks extracted from the proposed scheme also achieve high robustness against common malicious and nonmalicious image-processing attacks, which provides excellent copyright protection for images.

© 2012 Optical Society of America

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References

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2012 (1)

F. Liu, H. Y. Leung, L. M. Cheng, and X. Y. Ji, “Content-based image hashing using wave atoms,” Chin. Phys. B 21, 040204 (2012).
[CrossRef]

2011 (2)

L. H. Tian, N. N. Zheng, J. R. Xue, C. Li, and X. F. Wang, “An integrated visual saliency-based watermarking approach for synchronous image authentication and copyright protection,” Signal Process., Image Commun. 26, 427–437 (2011).
[CrossRef]

C. H. Lin and C. Y. Yang, “Multipurpose watermarking based on blind vector quantization (BVQ),” J. Inform. Hiding Multimed. Signal Process. 2, 236–246 (2011).

2010 (2)

F. Khelifi and J. M. Jiang, “Perceptual image hashing based on virtual watermark detection,” IEEE Trans. Image Process. 19, 981–994 (2010).
[CrossRef]

J. Z. Li, “Robust image watermarking scheme against geometric attacks using a computer-generated hologram,” Appl. Opt. 49, 6302–6312 (2010).
[CrossRef]

2008 (1)

C. E. Zhang, L. L. Cheng, Z. D. Qiu, and L. M. Cheng, “Multipurpose watermarking based on multiscale curvelet transform,” IEEE Trans. Inf. Forensics Security 3, 611–619 (2008).
[CrossRef]

2007 (3)

L. Demanet and L. Ying, “Wave atoms and sparsity of oscillatory patterns,” Appl. Comput. Harmon. Anal. 23, 368–387 (2007).
[CrossRef]

T. Addabbo, M. Alioto, A. Fort, A. Pasini, S. Rocchi, and V. Vignoli, “A class of maximum-period nonlinear congruential generators derived from the Rényi chaotic map,” IEEE Trans. Circuits Syst. I, Reg. Papers 54, 816–828 (2007).
[CrossRef]

V. Monga and M. K. Mihcak, “Robust and secure image hashing via non-negative matrix factorizations,” IEEE Trans. Inf. Forensics Security 2, 376–390 (2007).
[CrossRef]

2006 (3)

A. Swaminathan, Y. Mao, and M. Wu, “Robust and secure image hashing,” IEEE Trans. Inf. Forensics Security 1, 215–230 (2006).
[CrossRef]

V. Monga and B. L. Evans, “Perceptual image hashing via feature points: performance evaluation and tradeoffs,” IEEE Trans. Image Process. 15, 3452–3465 (2006).
[CrossRef]

K. Maeno, Q. Sun, S. F. Chang, and M. Suto, “New semi-fragile image authentication watermarking techniques using random bias and nonuniform quantization,” IEEE Trans. Multimedia 8, 32–45 (2006).
[CrossRef]

2005 (3)

Z. M. Lu, D. G. Xu, and S. H. Sun, “Multipurpose image watermarking algorithm based on multistage vector quantization,” IEEE Trans. Image Process. 14, 822–831 (2005).
[CrossRef]

G. S. Spagnolo, C. Simonetti, and L. Cozzella, “Content fragile watermarking based on a computer generated hologram coding technique,” J. Opt. A 7, 333–342 (2005).
[CrossRef]

H. T. Chang and C. L. Tsan, “Image watermarking by use of digital holography embedded in the discrete-cosine-transform domain,” Appl. Opt. 44, 6211–6219 (2005).
[CrossRef]

2003 (1)

C. S. Lu and H. Y. M. Liao, “Structural digital signature for image authentication,” IEEE Trans. Multimedia 5, 161–173 (2003).
[CrossRef]

2002 (1)

2001 (2)

C. S. Lu and H. Y. M. Liao, “Multipurpose watermarking for image authentication and protection,” IEEE Trans. Image Process. 10, 1579–1592 (2001).

C. Y. Lin and S. F. Chang, “A robust image authentication system distinguishing JPEG compression from malicious manipulation,” IEEE Trans. Circuits Syst. Video Technol. 11, 153–168 (2001).
[CrossRef]

2000 (1)

E. Lin, C. Podilchuk, and E. Delp, “Detection of image alterations using semi-fragile watermarks,” Proc. SPIE 3971, 152–163 (2000).
[CrossRef]

1996 (1)

J. P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Process. 52, 259–281 (1996).
[CrossRef]

Addabbo, T.

T. Addabbo, M. Alioto, A. Fort, A. Pasini, S. Rocchi, and V. Vignoli, “A class of maximum-period nonlinear congruential generators derived from the Rényi chaotic map,” IEEE Trans. Circuits Syst. I, Reg. Papers 54, 816–828 (2007).
[CrossRef]

Alioto, M.

T. Addabbo, M. Alioto, A. Fort, A. Pasini, S. Rocchi, and V. Vignoli, “A class of maximum-period nonlinear congruential generators derived from the Rényi chaotic map,” IEEE Trans. Circuits Syst. I, Reg. Papers 54, 816–828 (2007).
[CrossRef]

Antoine, J. P.

J. P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Process. 52, 259–281 (1996).
[CrossRef]

Baldoza, A. C.

J. Fridrich, M. Goljan, and A. C. Baldoza, “New fragile authentication watermark for images,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2000), pp. 446–449.

Bhattacharjee, S.

S. Bhattacharjee and M. Kutter, “Compression tolerant image authentication,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 1998), pp. 435–439.

Chang, H. T.

Chang, S. F.

K. Maeno, Q. Sun, S. F. Chang, and M. Suto, “New semi-fragile image authentication watermarking techniques using random bias and nonuniform quantization,” IEEE Trans. Multimedia 8, 32–45 (2006).
[CrossRef]

C. Y. Lin and S. F. Chang, “A robust image authentication system distinguishing JPEG compression from malicious manipulation,” IEEE Trans. Circuits Syst. Video Technol. 11, 153–168 (2001).
[CrossRef]

M. Schneider and S. F. Chang, “A robust content based digital signature for image authentication,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 1996), pp. 227–230.

Cheng, L. L.

C. E. Zhang, L. L. Cheng, Z. D. Qiu, and L. M. Cheng, “Multipurpose watermarking based on multiscale curvelet transform,” IEEE Trans. Inf. Forensics Security 3, 611–619 (2008).
[CrossRef]

Cheng, L. M.

F. Liu, H. Y. Leung, L. M. Cheng, and X. Y. Ji, “Content-based image hashing using wave atoms,” Chin. Phys. B 21, 040204 (2012).
[CrossRef]

C. E. Zhang, L. L. Cheng, Z. D. Qiu, and L. M. Cheng, “Multipurpose watermarking based on multiscale curvelet transform,” IEEE Trans. Inf. Forensics Security 3, 611–619 (2008).
[CrossRef]

Cozzella, L.

G. S. Spagnolo, C. Simonetti, and L. Cozzella, “Content fragile watermarking based on a computer generated hologram coding technique,” J. Opt. A 7, 333–342 (2005).
[CrossRef]

Delp, E.

E. Lin, C. Podilchuk, and E. Delp, “Detection of image alterations using semi-fragile watermarks,” Proc. SPIE 3971, 152–163 (2000).
[CrossRef]

Demanet, L.

L. Demanet and L. Ying, “Wave atoms and sparsity of oscillatory patterns,” Appl. Comput. Harmon. Anal. 23, 368–387 (2007).
[CrossRef]

Evans, B. L.

V. Monga and B. L. Evans, “Perceptual image hashing via feature points: performance evaluation and tradeoffs,” IEEE Trans. Image Process. 15, 3452–3465 (2006).
[CrossRef]

V. Monga, D. Vats, and B. L. Evans, “Image authentication under geometric attacks via structure matching,” in Proceedings of IEEE International Conference on Multimedia and Expo (IEEE, 2005), pp. 229–232.

Fort, A.

T. Addabbo, M. Alioto, A. Fort, A. Pasini, S. Rocchi, and V. Vignoli, “A class of maximum-period nonlinear congruential generators derived from the Rényi chaotic map,” IEEE Trans. Circuits Syst. I, Reg. Papers 54, 816–828 (2007).
[CrossRef]

Fridrich, J.

J. Fridrich and M. Goljan, “Robust hash functions for digital watermarking,” in Proceedings of IEEE International Conference on Information Technology: Coding and Computing (IEEE, 2000), pp. 178–183.

J. Fridrich, M. Goljan, and A. C. Baldoza, “New fragile authentication watermark for images,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2000), pp. 446–449.

Goljan, M.

J. Fridrich, M. Goljan, and A. C. Baldoza, “New fragile authentication watermark for images,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2000), pp. 446–449.

J. Fridrich and M. Goljan, “Robust hash functions for digital watermarking,” in Proceedings of IEEE International Conference on Information Technology: Coding and Computing (IEEE, 2000), pp. 178–183.

Jakubowski, M. H.

R. Venkatesan, S. M. Koon, M. H. Jakubowski, and P. Moulin, “Robust image hashing,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2000), pp. 664–666.

Ji, X. Y.

F. Liu, H. Y. Leung, L. M. Cheng, and X. Y. Ji, “Content-based image hashing using wave atoms,” Chin. Phys. B 21, 040204 (2012).
[CrossRef]

Jiang, J. M.

F. Khelifi and J. M. Jiang, “Perceptual image hashing based on virtual watermark detection,” IEEE Trans. Image Process. 19, 981–994 (2010).
[CrossRef]

Kailasanathan, C.

C. Kailasanathan and R. S. Naini, “Image authentication surviving acceptable modifications using statistical measures and k-mean segmentation,” presented at the IEEE-EURASIP Workshop on Nonlinear Signal Image Processing, Baltimore, Md., USA, 3–6 June 2001.

Khelifi, F.

F. Khelifi and J. M. Jiang, “Perceptual image hashing based on virtual watermark detection,” IEEE Trans. Image Process. 19, 981–994 (2010).
[CrossRef]

Koon, S. M.

R. Venkatesan, S. M. Koon, M. H. Jakubowski, and P. Moulin, “Robust image hashing,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2000), pp. 664–666.

Kutter, M.

S. Bhattacharjee and M. Kutter, “Compression tolerant image authentication,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 1998), pp. 435–439.

Leung, H. Y.

F. Liu, H. Y. Leung, L. M. Cheng, and X. Y. Ji, “Content-based image hashing using wave atoms,” Chin. Phys. B 21, 040204 (2012).
[CrossRef]

Li, C.

L. H. Tian, N. N. Zheng, J. R. Xue, C. Li, and X. F. Wang, “An integrated visual saliency-based watermarking approach for synchronous image authentication and copyright protection,” Signal Process., Image Commun. 26, 427–437 (2011).
[CrossRef]

Li, J. Z.

Liao, H. Y. M.

C. S. Lu and H. Y. M. Liao, “Structural digital signature for image authentication,” IEEE Trans. Multimedia 5, 161–173 (2003).
[CrossRef]

C. S. Lu and H. Y. M. Liao, “Multipurpose watermarking for image authentication and protection,” IEEE Trans. Image Process. 10, 1579–1592 (2001).

Lin, C. H.

C. H. Lin and C. Y. Yang, “Multipurpose watermarking based on blind vector quantization (BVQ),” J. Inform. Hiding Multimed. Signal Process. 2, 236–246 (2011).

Lin, C. Y.

C. Y. Lin and S. F. Chang, “A robust image authentication system distinguishing JPEG compression from malicious manipulation,” IEEE Trans. Circuits Syst. Video Technol. 11, 153–168 (2001).
[CrossRef]

Lin, E.

E. Lin, C. Podilchuk, and E. Delp, “Detection of image alterations using semi-fragile watermarks,” Proc. SPIE 3971, 152–163 (2000).
[CrossRef]

Liu, F.

F. Liu, H. Y. Leung, L. M. Cheng, and X. Y. Ji, “Content-based image hashing using wave atoms,” Chin. Phys. B 21, 040204 (2012).
[CrossRef]

Lu, C. S.

C. S. Lu and H. Y. M. Liao, “Structural digital signature for image authentication,” IEEE Trans. Multimedia 5, 161–173 (2003).
[CrossRef]

C. S. Lu and H. Y. M. Liao, “Multipurpose watermarking for image authentication and protection,” IEEE Trans. Image Process. 10, 1579–1592 (2001).

Lu, Z. M.

Z. M. Lu, D. G. Xu, and S. H. Sun, “Multipurpose image watermarking algorithm based on multistage vector quantization,” IEEE Trans. Image Process. 14, 822–831 (2005).
[CrossRef]

Maeno, K.

K. Maeno, Q. Sun, S. F. Chang, and M. Suto, “New semi-fragile image authentication watermarking techniques using random bias and nonuniform quantization,” IEEE Trans. Multimedia 8, 32–45 (2006).
[CrossRef]

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed.(Academic, 1999).

Mao, Y.

A. Swaminathan, Y. Mao, and M. Wu, “Robust and secure image hashing,” IEEE Trans. Inf. Forensics Security 1, 215–230 (2006).
[CrossRef]

Mifune, Y.

Mihcak, M. K.

V. Monga and M. K. Mihcak, “Robust and secure image hashing via non-negative matrix factorizations,” IEEE Trans. Inf. Forensics Security 2, 376–390 (2007).
[CrossRef]

Monga, V.

V. Monga and M. K. Mihcak, “Robust and secure image hashing via non-negative matrix factorizations,” IEEE Trans. Inf. Forensics Security 2, 376–390 (2007).
[CrossRef]

V. Monga and B. L. Evans, “Perceptual image hashing via feature points: performance evaluation and tradeoffs,” IEEE Trans. Image Process. 15, 3452–3465 (2006).
[CrossRef]

V. Monga, D. Vats, and B. L. Evans, “Image authentication under geometric attacks via structure matching,” in Proceedings of IEEE International Conference on Multimedia and Expo (IEEE, 2005), pp. 229–232.

Moulin, P.

R. Venkatesan, S. M. Koon, M. H. Jakubowski, and P. Moulin, “Robust image hashing,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2000), pp. 664–666.

Murenzi, R.

J. P. Antoine and R. Murenzi, “Two-dimensional directional wavelets and the scale-angle representation,” Signal Process. 52, 259–281 (1996).
[CrossRef]

Naini, R. S.

C. Kailasanathan and R. S. Naini, “Image authentication surviving acceptable modifications using statistical measures and k-mean segmentation,” presented at the IEEE-EURASIP Workshop on Nonlinear Signal Image Processing, Baltimore, Md., USA, 3–6 June 2001.

Pasini, A.

T. Addabbo, M. Alioto, A. Fort, A. Pasini, S. Rocchi, and V. Vignoli, “A class of maximum-period nonlinear congruential generators derived from the Rényi chaotic map,” IEEE Trans. Circuits Syst. I, Reg. Papers 54, 816–828 (2007).
[CrossRef]

Podilchuk, C.

E. Lin, C. Podilchuk, and E. Delp, “Detection of image alterations using semi-fragile watermarks,” Proc. SPIE 3971, 152–163 (2000).
[CrossRef]

Qiu, Z. D.

C. E. Zhang, L. L. Cheng, Z. D. Qiu, and L. M. Cheng, “Multipurpose watermarking based on multiscale curvelet transform,” IEEE Trans. Inf. Forensics Security 3, 611–619 (2008).
[CrossRef]

Rocchi, S.

T. Addabbo, M. Alioto, A. Fort, A. Pasini, S. Rocchi, and V. Vignoli, “A class of maximum-period nonlinear congruential generators derived from the Rényi chaotic map,” IEEE Trans. Circuits Syst. I, Reg. Papers 54, 816–828 (2007).
[CrossRef]

Schneider, M.

M. Schneider and S. F. Chang, “A robust content based digital signature for image authentication,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 1996), pp. 227–230.

Simonetti, C.

G. S. Spagnolo, C. Simonetti, and L. Cozzella, “Content fragile watermarking based on a computer generated hologram coding technique,” J. Opt. A 7, 333–342 (2005).
[CrossRef]

Spagnolo, G. S.

G. S. Spagnolo, C. Simonetti, and L. Cozzella, “Content fragile watermarking based on a computer generated hologram coding technique,” J. Opt. A 7, 333–342 (2005).
[CrossRef]

Sun, Q.

K. Maeno, Q. Sun, S. F. Chang, and M. Suto, “New semi-fragile image authentication watermarking techniques using random bias and nonuniform quantization,” IEEE Trans. Multimedia 8, 32–45 (2006).
[CrossRef]

Sun, S. H.

Z. M. Lu, D. G. Xu, and S. H. Sun, “Multipurpose image watermarking algorithm based on multistage vector quantization,” IEEE Trans. Image Process. 14, 822–831 (2005).
[CrossRef]

Suto, M.

K. Maeno, Q. Sun, S. F. Chang, and M. Suto, “New semi-fragile image authentication watermarking techniques using random bias and nonuniform quantization,” IEEE Trans. Multimedia 8, 32–45 (2006).
[CrossRef]

Swaminathan, A.

A. Swaminathan, Y. Mao, and M. Wu, “Robust and secure image hashing,” IEEE Trans. Inf. Forensics Security 1, 215–230 (2006).
[CrossRef]

Takai, N.

Tian, L. H.

L. H. Tian, N. N. Zheng, J. R. Xue, C. Li, and X. F. Wang, “An integrated visual saliency-based watermarking approach for synchronous image authentication and copyright protection,” Signal Process., Image Commun. 26, 427–437 (2011).
[CrossRef]

Tsan, C. L.

Vats, D.

V. Monga, D. Vats, and B. L. Evans, “Image authentication under geometric attacks via structure matching,” in Proceedings of IEEE International Conference on Multimedia and Expo (IEEE, 2005), pp. 229–232.

Venkatesan, R.

R. Venkatesan, S. M. Koon, M. H. Jakubowski, and P. Moulin, “Robust image hashing,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2000), pp. 664–666.

Vignoli, V.

T. Addabbo, M. Alioto, A. Fort, A. Pasini, S. Rocchi, and V. Vignoli, “A class of maximum-period nonlinear congruential generators derived from the Rényi chaotic map,” IEEE Trans. Circuits Syst. I, Reg. Papers 54, 816–828 (2007).
[CrossRef]

Wang, X. F.

L. H. Tian, N. N. Zheng, J. R. Xue, C. Li, and X. F. Wang, “An integrated visual saliency-based watermarking approach for synchronous image authentication and copyright protection,” Signal Process., Image Commun. 26, 427–437 (2011).
[CrossRef]

Wu, M.

A. Swaminathan, Y. Mao, and M. Wu, “Robust and secure image hashing,” IEEE Trans. Inf. Forensics Security 1, 215–230 (2006).
[CrossRef]

Xu, D. G.

Z. M. Lu, D. G. Xu, and S. H. Sun, “Multipurpose image watermarking algorithm based on multistage vector quantization,” IEEE Trans. Image Process. 14, 822–831 (2005).
[CrossRef]

Xue, J. R.

L. H. Tian, N. N. Zheng, J. R. Xue, C. Li, and X. F. Wang, “An integrated visual saliency-based watermarking approach for synchronous image authentication and copyright protection,” Signal Process., Image Commun. 26, 427–437 (2011).
[CrossRef]

Yang, C. Y.

C. H. Lin and C. Y. Yang, “Multipurpose watermarking based on blind vector quantization (BVQ),” J. Inform. Hiding Multimed. Signal Process. 2, 236–246 (2011).

Ying, L.

L. Demanet and L. Ying, “Wave atoms and sparsity of oscillatory patterns,” Appl. Comput. Harmon. Anal. 23, 368–387 (2007).
[CrossRef]

Zhang, C. E.

C. E. Zhang, L. L. Cheng, Z. D. Qiu, and L. M. Cheng, “Multipurpose watermarking based on multiscale curvelet transform,” IEEE Trans. Inf. Forensics Security 3, 611–619 (2008).
[CrossRef]

Zheng, N. N.

L. H. Tian, N. N. Zheng, J. R. Xue, C. Li, and X. F. Wang, “An integrated visual saliency-based watermarking approach for synchronous image authentication and copyright protection,” Signal Process., Image Commun. 26, 427–437 (2011).
[CrossRef]

Appl. Comput. Harmon. Anal. (1)

L. Demanet and L. Ying, “Wave atoms and sparsity of oscillatory patterns,” Appl. Comput. Harmon. Anal. 23, 368–387 (2007).
[CrossRef]

Appl. Opt. (3)

Chin. Phys. B (1)

F. Liu, H. Y. Leung, L. M. Cheng, and X. Y. Ji, “Content-based image hashing using wave atoms,” Chin. Phys. B 21, 040204 (2012).
[CrossRef]

IEEE Trans. Circuits Syst. I, Reg. Papers (1)

T. Addabbo, M. Alioto, A. Fort, A. Pasini, S. Rocchi, and V. Vignoli, “A class of maximum-period nonlinear congruential generators derived from the Rényi chaotic map,” IEEE Trans. Circuits Syst. I, Reg. Papers 54, 816–828 (2007).
[CrossRef]

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

C. Y. Lin and S. F. Chang, “A robust image authentication system distinguishing JPEG compression from malicious manipulation,” IEEE Trans. Circuits Syst. Video Technol. 11, 153–168 (2001).
[CrossRef]

IEEE Trans. Image Process. (4)

V. Monga and B. L. Evans, “Perceptual image hashing via feature points: performance evaluation and tradeoffs,” IEEE Trans. Image Process. 15, 3452–3465 (2006).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Wave atom tiling of the frequency plane (only the first quadrant is shown).

Fig. 2.
Fig. 2.

Wave atom coefficient distributions in each scale band.

Fig. 3.
Fig. 3.

Processes of proposed multipurpose scheme.

Fig. 4.
Fig. 4.

Distribution of NHDs under different kinds of malicious attacks in the pure image hashing strategy.

Fig. 5.
Fig. 5.

FAR and FRR under different values of threshold θ in the pure image hashing strategy.

Fig. 6.
Fig. 6.

ROC curve under different values of threshold θ in the pure image hashing strategy.

Fig. 7.
Fig. 7.

Relationship between image quality and parameters α and β.

Fig. 8.
Fig. 8.

NC distribution under different malicious attacks of pure image watermarking strategy.

Fig. 9.
Fig. 9.

NHD distribution under different malicious attacks of combined multipurpose scheme.

Fig. 10.
Fig. 10.

NC distribution under different malicious attacks of combined multipurpose scheme.

Tables (2)

Tables Icon

Table 1. Average Values of NHD and NC under Different Nonmalicious Manipulations

Tables Icon

Table 2. Average Values of NHD and NC under Different Percentages of Malicious Replacing Manipulations

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

xμ=2jn,wμ=π2jm,C12j|mi|i=1,2maxC22j,
ψm,nj(x)=ψmj(x2jn)=2j/2ψm0(2jxn),
ψm0(w)=eiw/2[eiαmg(ϵm(wπ(m+1/2)))+eiαmg(ϵm+1(w+π(m+1/2)))],
Cj,m,nD=k=2π(2j/2+112j/2)ei2jnkp2πZψ^mj(k+2jp)¯u^(k+2jp).
φμ+(x1,x2)=ψm1j(x12jn1)ψm2j(x22jn2).
φμ(x1,x2)=Hψm1j(x12jn1)Hψm2j(x22jn2).
φu(1)=φu++φu2,φu(2)=φu+φu2.
Meani=k=(1,1)(l,l)C(3,mi,nk)l×l,
Vari=k=(1,1)(l,l)(C(3,mi,nk)Meani)2l×l1,
R(x)=(λ2pux+2vx)mod2p,
h1(i)={1,ifMeani>Meani+10,otherwise,
h2=G(Pmax_1)G(Pmax_2)G(Pmax_10)G(Pmin_1)G(Pmin_2)G(Pmin_10),
NC=i=1q[W(i)×W(i)]/i=1q[W(i)]2,

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