Abstract

A nonlinear color and grayscale images cryptosystem based on phase-truncated fractional Fourier transform and optical superposition principle is proposed. In order to realize simultaneous encryption of color and grayscale images, each grayscale image is first converted into two phase masks by using an optical coherent superposition, one of which is treated as a part of input information that will be fractional Fourier transformed while the other in the form of a chaotic random phase mask (CRPM) is used as a decryption key. For the purpose of optical performance, all the processes are performed through three channels, i.e., red, green, and blue. Different from most asymmetric encryption methods, the decryption process is designed to be linear for the sake of effective decryption. The encryption level of a double random phase encryption based on phase-truncated Fourier transform is enhanced by extending it into fractional Fourier domain and the load of the keys management and transmission is lightened by using CRPMs. The security of the proposed cryptosystem is discussed and computer simulation results are presented to verify the validity of the proposed method.

© 2013 Optical Society of America

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2013 (1)

M. R. Abuturab, “Color information security system using Arnold transform and double structured phase encoding in gyrator transform domain,” Opt. Laser Technol. 45, 525–532 (2013).
[CrossRef]

2012 (5)

X. Wang and D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285, 4280–4284 (2012).
[CrossRef]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

M. R. Abuturab, “Color information security system using discrete cosine transform in gyrator transform domain radial-Hilbert phase encoding,” Opt. Lasers Eng. 50, 1209–1216 (2012).
[CrossRef]

S. K. Rajput and N. K. Nishchal, “Image encryption based on interference that uses fractional Fourier domains asymmetric keys,” Appl. Opt. 51, 1446–1452 (2012).
[CrossRef]

M. R. Abuturab, “Color image security system using double random-structured phase encoding in gyrator transform domain,” Appl. Opt. 51, 3006–3016 (2012).
[CrossRef]

2011 (6)

X. Wang and D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284, 4441–4445 (2011).
[CrossRef]

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
[CrossRef]

X. Wang and D. Zhao, “Image encoding based on coherent superposition and basic vector operations,” Opt. Commun. 284, 945–951 (2011).
[CrossRef]

M. Joshi and K. Singh, “Simultaneous encryption of a color and a gray-scale image using byte-level encoding based on single-channel double random-phase encoding architecture in fractional Fourier domain,” Opt. Eng. 50, 047007 (2011).
[CrossRef]

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

W. Chen and X. Chen, “Optical color image encryption based on an asymmetric cryptosystem in the Fresnel domain,” Opt. Commun. 284, 3913–3917 (2011).
[CrossRef]

2010 (1)

2009 (4)

2008 (4)

X. Cheng, L. Cai, Y. Wang, X. Meng, H. Zhang, X. Xu, X. Shen, and G. Dong, “Security enhancement of double-random phase encryption by amplitude modulation,” Opt. Lett. 33, 1575–1577 (2008).
[CrossRef]

T. J. Naughton, B. M. Hennelly, and T. Dowling, “Introducing secure modes of operation for optical encryption,” J. Opt. Soc. Am. A 25, 2608–2617 (2008).
[CrossRef]

N. Singh and A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46, 117–123 (2008).
[CrossRef]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption for twin images in fractional Fourier domain,” Opt. Commun. 281, 5713–5720 (2008).
[CrossRef]

2007 (4)

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
[CrossRef]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
[CrossRef]

R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15, 16067–16079 (2007).
[CrossRef]

2006 (4)

2005 (1)

2004 (1)

2003 (2)

B. M. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28, 269–271 (2003).
[CrossRef]

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571 (2003).
[CrossRef]

2002 (1)

2001 (1)

2000 (1)

1999 (1)

S. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” Microw. Opt. Technol. Lett. 21, 318–323 (1999).
[CrossRef]

1995 (1)

Abuturab, M. R.

M. R. Abuturab, “Color information security system using Arnold transform and double structured phase encoding in gyrator transform domain,” Opt. Laser Technol. 45, 525–532 (2013).
[CrossRef]

M. R. Abuturab, “Color image security system using double random-structured phase encoding in gyrator transform domain,” Appl. Opt. 51, 3006–3016 (2012).
[CrossRef]

M. R. Abuturab, “Color information security system using discrete cosine transform in gyrator transform domain radial-Hilbert phase encoding,” Opt. Lasers Eng. 50, 1209–1216 (2012).
[CrossRef]

Alfalou, A.

Alligood, K. T.

K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer, 2001).

Arcos, S.

Brosseau, C.

Cai, L.

Carnicer, A.

Castro, A.

Chang, H.

Chen, H.

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Chen, L.

Chen, W.

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
[CrossRef]

W. Chen and X. Chen, “Optical color image encryption based on an asymmetric cryptosystem in the Fresnel domain,” Opt. Commun. 284, 3913–3917 (2011).
[CrossRef]

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282, 3680–3685 (2009).
[CrossRef]

Chen, X.

W. Chen and X. Chen, “Optical color image encryption based on an asymmetric cryptosystem in the Fresnel domain,” Opt. Commun. 284, 3913–3917 (2011).
[CrossRef]

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
[CrossRef]

Cheng, X.

Dong, G.

Dowling, T.

Frauel, Y.

Gopinathan, U.

Hennelly, B. M.

Javidi, B.

Joseph, J.

Joshi, M.

M. Joshi and K. Singh, “Simultaneous encryption of a color and a gray-scale image using byte-level encoding based on single-channel double random-phase encoding architecture in fractional Fourier domain,” Opt. Eng. 50, 047007 (2011).
[CrossRef]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption for twin images in fractional Fourier domain,” Opt. Commun. 281, 5713–5720 (2008).
[CrossRef]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

Juvells, I.

Karim, M. A.

S. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” Microw. Opt. Technol. Lett. 21, 318–323 (1999).
[CrossRef]

Kumar, A.

Kumar, P.

Kuo, C.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Li, P.

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Lin, C.

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Liu, S.

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
[CrossRef]

S. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001).
[CrossRef]

Liu, T.

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Liu, Z.

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
[CrossRef]

Lu, W.

Mansour, A.

Meng, X.

Mi, Q.

Monaghan, D. S.

Montes-Usategui, M.

Naughton, T. J.

Nishchal, N. K.

S. K. Rajput and N. K. Nishchal, “Image encryption based on interference that uses fractional Fourier domains asymmetric keys,” Appl. Opt. 51, 1446–1452 (2012).
[CrossRef]

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571 (2003).
[CrossRef]

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Peng, X.

Qin, W.

Quan, C.

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282, 3680–3685 (2009).
[CrossRef]

Rajput, S. K.

Refregier, P.

Sauer, T. D.

K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer, 2001).

Shakher, C.

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption for twin images in fractional Fourier domain,” Opt. Commun. 281, 5713–5720 (2008).
[CrossRef]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

Shen, X.

Sheridan, J. T.

Singh, K.

M. Joshi and K. Singh, “Simultaneous encryption of a color and a gray-scale image using byte-level encoding based on single-channel double random-phase encoding architecture in fractional Fourier domain,” Opt. Eng. 50, 047007 (2011).
[CrossRef]

P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double-random-phase encryption scheme with randomized lens-phase functions,” Opt. Lett. 34, 331–333 (2009).
[CrossRef]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption for twin images in fractional Fourier domain,” Opt. Commun. 281, 5713–5720 (2008).
[CrossRef]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571 (2003).
[CrossRef]

G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double random phase encoding in the fractional Fourier domain,” Opt. Lett. 25, 887–889 (2000).
[CrossRef]

Singh, N.

N. Singh and A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46, 117–123 (2008).
[CrossRef]

Sinha, A.

N. Singh and A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46, 117–123 (2008).
[CrossRef]

Situ, G.

Tao, R.

Tay, C. J.

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282, 3680–3685 (2009).
[CrossRef]

Unnikrishnan, G.

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571 (2003).
[CrossRef]

G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double random phase encoding in the fractional Fourier domain,” Opt. Lett. 25, 887–889 (2000).
[CrossRef]

Wang, S. W.

Wang, X.

X. Wang and D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285, 4280–4284 (2012).
[CrossRef]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

X. Wang and D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284, 4441–4445 (2011).
[CrossRef]

X. Wang and D. Zhao, “Image encoding based on coherent superposition and basic vector operations,” Opt. Commun. 284, 945–951 (2011).
[CrossRef]

Wang, Y.

Wei, H.

Xie, J.

Xin, Y.

Xu, L.

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Xu, X.

Yorke, J. A.

K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer, 2001).

Yuan, S.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Zhang, H.

Zhang, J.

Zhang, P.

Zhang, S.

S. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” Microw. Opt. Technol. Lett. 21, 318–323 (1999).
[CrossRef]

Zhao, D.

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

X. Wang and D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285, 4280–4284 (2012).
[CrossRef]

X. Wang and D. Zhao, “Image encoding based on coherent superposition and basic vector operations,” Opt. Commun. 284, 945–951 (2011).
[CrossRef]

X. Wang and D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284, 4441–4445 (2011).
[CrossRef]

L. Chen and D. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14, 8552–8560 (2006).
[CrossRef]

Zhou, X.

Zhu, B.

Adv. Opt. Photon. (1)

Appl. Opt. (5)

J. Opt. (1)

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
[CrossRef]

J. Opt. Soc. Am. A (1)

Microw. Opt. Technol. Lett. (1)

S. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” Microw. Opt. Technol. Lett. 21, 318–323 (1999).
[CrossRef]

Opt. Commun. (10)

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

W. Chen and X. Chen, “Optical color image encryption based on an asymmetric cryptosystem in the Fresnel domain,” Opt. Commun. 284, 3913–3917 (2011).
[CrossRef]

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282, 3680–3685 (2009).
[CrossRef]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption for twin images in fractional Fourier domain,” Opt. Commun. 281, 5713–5720 (2008).
[CrossRef]

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
[CrossRef]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

X. Wang and D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284, 4441–4445 (2011).
[CrossRef]

X. Wang and D. Zhao, “Image encoding based on coherent superposition and basic vector operations,” Opt. Commun. 284, 945–951 (2011).
[CrossRef]

X. Wang and D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285, 4280–4284 (2012).
[CrossRef]

Opt. Eng. (2)

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571 (2003).
[CrossRef]

M. Joshi and K. Singh, “Simultaneous encryption of a color and a gray-scale image using byte-level encoding based on single-channel double random-phase encoding architecture in fractional Fourier domain,” Opt. Eng. 50, 047007 (2011).
[CrossRef]

Opt. Express (4)

Opt. Laser Technol. (1)

M. R. Abuturab, “Color information security system using Arnold transform and double structured phase encoding in gyrator transform domain,” Opt. Laser Technol. 45, 525–532 (2013).
[CrossRef]

Opt. Lasers Eng. (2)

M. R. Abuturab, “Color information security system using discrete cosine transform in gyrator transform domain radial-Hilbert phase encoding,” Opt. Lasers Eng. 50, 1209–1216 (2012).
[CrossRef]

N. Singh and A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46, 117–123 (2008).
[CrossRef]

Opt. Lett. (10)

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Other (2)

K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer, 2001).

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Figures (12)

Fig. 1.
Fig. 1.

Flowchart of the nonlinear encryption procedure of the proposed cryptosystem in the green channel.

Fig. 2.
Fig. 2.

Flowchart of the linear decryption procedure of the proposed cryptosystem in the green channel.

Fig. 3.
Fig. 3.

Images to be encoded: (a) Cameraman, (b) Lena, (c) Rice, and (d) Football.

Fig. 4.
Fig. 4.

CRPMs and their corresponding phase-encoded images. (a) Cr (μ=3.94, x0=0.12, m=20), (b) P1r, (c) Cg (μ=3.9, x0=0.24, m=30), (d) P1g, (e) Cb (μ=3.86, x0=0.36, m=40), and (f) P1b.

Fig. 5.
Fig. 5.

Simulation results of the proposed method in the green channel. (a) The encrypted result Eg, (b) phase key Pg, correctly decrypted images of (c) fr, and (d) Lena.

Fig. 6.
Fig. 6.

Simulation results of the proposed method. (a) The final encrypted image E, (b) phase key P, and (c) final correctly decrypted color image. (d)–(f) are the three decrypted grayscale images.

Fig. 7.
Fig. 7.

Simulation results. (a) The wrong phase key used as Pg for decryption, (b) the decrypted green component of color image, and (c) the decrypted grayscale image.

Fig. 8.
Fig. 8.

(a) The CRPM with a wrong parameter μ=3.901 (Δμ=0.001). (b) The decrypted green component of color image and (c) grayscale image by using the wrong CRPM shown in (a). (d) The CRPM with a wrong parameter x0=0.241 (Δx=0.001). (e) The decrypted green component of the color image and (f) the grayscale image with the wrong CRPM shown in (b). (g) The CRPM with a wrong parameter m=31 (Δm=1). (h) The decrypted green component of color image and (i) grayscale image with the wrong CRPM shown in (g).

Fig. 9.
Fig. 9.

MSE (between the decrypted image and its original component of the color image) versus the deviation of chaotic function parameter (a) μ, (b) x0, and (c) m.

Fig. 10.
Fig. 10.

MSE (between the decrypted grayscale image and its original grayscale image) versus the deviation of chaotic function parameter (a) μ, (b) x0, and (c) m.

Fig. 11.
Fig. 11.

Incorrectly decrypted results with a wrong fractional order αg=1.16. (a) Recovered green component of the color image and (b) decrypted grayscale image.

Fig. 12.
Fig. 12.

MSE versus fractional orders of FRT with respect to the decryption for (a) color images and (b) grayscale images.

Equations (17)

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Fα[f(x)](u)=+Kα(x,u)f(x)dx,
Kα(x,u)={Aexp[iπ(x2cotϕ2xucscϕ+u2cotϕ)]ϕnπδ(xu)ϕ=2nπδ(x+u)ϕ=(2n+1)π,
Fα1,α2[f(x,y)](u,υ)=+Kα1,α2(x,y;u,υ)f(x,y)dxdy,
Kα1,α2(x,y;u,υ)=Kα1(x,u)Kα2(y,υ),
xn+1=μ·xn·(1xn),
I2(x,y)=PT{P1g(x,y)+Cg(x,y)},
P1g(u,υ)=exp[iφg(x,y)]
Cg(u,υ)=exp[i2πSg(x,y)],
φg(x,y)=2πSg(x,y)+πarccos(1I22(x,y)2).
fg(u,υ)=PT{Fαg[fg(x,y)·P1g(x,y)]},
P0g(u,υ)=PR{Fαg[fg(x,y)·P1g(x,y)]},
Eg(x,y)=PT{Fαg[fg(u,υ)·P2g(u,υ)]},
Pg(x,y)=PR{Fαg[fg(u,υ)·P2g(u,υ)]},
P2g(u,υ)=P0g(u,υ)·Cg*(u,υ),
fg(x,y)=PT{Fαg[Fαg[Eg(x,y)·Pg(x,y)]·Cg(x,y)]},
I2(x,y)=PT{Cg(x,y)+P1g(x,y)}=PT{Cg(x,y)+PR{Fαg[Fαg[Eg(x,y)·Pg(x,y)]·Cg(x,y)]}}.
Fαg[Fαg[Eg(x,y)·Pg(x,y)]·Cg(x,y)]=Fαg[fg(u,υ)·P2g(u,υ)·Cg(x,y)]=Fαg[fg(u,υ)·P0g(u,υ)·Cg*(u,υ)·Cg(x,y)]=Fαg[fg(u,υ)·P0g(u,υ)]=fg(x,y)·P1g(x,y),

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