Abstract

In this paper, an image encryption scheme based on polarized light encoding and a phase-truncation approach in the Fresnel transform domain is proposed. The phase-truncated data obtained by an asymmetric cryptosystem is encrypted and decrypted by using the concept of the Stokes–Mueller formalism. Image encryption based on polarization of light using Stokes–Mueller formalism has the main advantage over Jones vector formalism that it manipulates only intensity information, which is measurable. Thus any intensity information can be encrypted and decrypted using this scheme. The proposed method offers several advantages: (1) a lens-free setup, (2) flexibility in the encryption key design, (3) use of asymmetric keys, and (4) immunity against special attack. We present numerical simulation results for gray-scale and color images in support of the proposed security scheme. The performance measurement parameters relative error and correlation coefficient have been calculated to check the effectiveness of the scheme.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  41. N. Zhou, Y. Wang, L. Gong, H. He, and J. Wu, “Novel single channel color image encryption based on chaos and fractional Fourier transform,” Opt. Commun. 284, 2789–2796 (2011).
    [CrossRef]
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    [CrossRef]
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2013 (2)

2012 (6)

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

S. K. Rajput and N. K. Nishchal, “Asymmetric color cryptosystem using polarization selective diffractive optical element and structured phase mask,” Appl. Opt. 51, 5377–5386 (2012).
[CrossRef]

P. Kumar, J. Joseph, and K. Singh, “Vulnerability of the security enhanced double random phase-amplitude encryption scheme to point spread function attack,” Opt. Lasers Eng. 50, 1196–1201 (2012).
[CrossRef]

M. Dubreuil, A. Aflalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using Stokes-Mueller formalism,” J. Opt. 14, 094004 (2012).
[CrossRef]

X. Deng and D. Zhao, “Single channel color image encryption based on asymmetric cryptosystem,” Opt. Laser Technol. 44, 136–140 (2012).
[CrossRef]

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

2011 (6)

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

N. Zhou, Y. Wang, L. Gong, H. He, and J. Wu, “Novel single channel color image encryption based on chaos and fractional Fourier transform,” Opt. Commun. 284, 2789–2796 (2011).
[CrossRef]

X. Wang and D. Zhao, “Multiple image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284, 148–152 (2011).
[CrossRef]

W. Qin, X. Peng, X. Meng, and B. Gao, “Universal and special keys based on phase-truncated Fourier transform,” Opt. Eng. 50, 080501 (2011).
[CrossRef]

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

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
[CrossRef]

2010 (6)

2009 (3)

W. Qin and X. Peng, “Vulnerability to known-plaintext attack of optical encryption schemes based on two fractional Fourier transform order keys and double random phase keys,” J. Opt. 11, 075402 (2009).
[CrossRef]

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Trans. Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

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

2008 (2)

2007 (3)

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]

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

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

2006 (3)

2005 (2)

2004 (3)

2001 (1)

2000 (4)

J. A. Davis, D. E. McNamara, D. M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. 39, 1549–1554 (2000).
[CrossRef]

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25–31 (2000).
[CrossRef]

P. C. Mogensen and J. Gluckstad, “A phase-based optical encryption system with polarization encoding,” Opt. Commun. 173, 177–183 (2000).
[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]

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)

Aflalou, A.

M. Dubreuil, A. Aflalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using Stokes-Mueller formalism,” J. Opt. 14, 094004 (2012).
[CrossRef]

Alfalou, A.

Alieva, T.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

Arcos, S.

Barrera, J. F.

Biener, G.

Bolognini, N.

Brosseau, C.

M. Dubreuil, A. Aflalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using Stokes-Mueller formalism,” J. Opt. 14, 094004 (2012).
[CrossRef]

A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35, 2185–2187 (2010).
[CrossRef]

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Cai, L. Z.

Calvo, M. L.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

Carnicer, A.

Castro, A.

Chen, G.

Chen, M.-L.

H.-Y. Tu, C.-J. Cheng, and M.-L. Chen, “Optical image encryption based on polarization encoding by liquid crystal spatial light modulator,” J. Opt. A 6, 524–528 (2004).
[CrossRef]

Chen, W.

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

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

W. Chen, X. Chen, and J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35, 3817–3819 (2010).
[CrossRef]

W. Chen and X. Chen, “Space-based optical image encryption,” Opt. Express 18, 27095–27104 (2010).
[CrossRef]

Chen, X.

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

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

W. Chen, X. Chen, and J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35, 3817–3819 (2010).
[CrossRef]

W. Chen and X. Chen, “Space-based optical image encryption,” Opt. Express 18, 27095–27104 (2010).
[CrossRef]

Cheng, C.-J.

H.-Y. Tu, C.-J. Cheng, and M.-L. Chen, “Optical image encryption based on polarization encoding by liquid crystal spatial light modulator,” J. Opt. A 6, 524–528 (2004).
[CrossRef]

Cheng, X. C.

Clemente, P.

Company, V. T.

Cottrell, D. M.

Davis, J. A.

Deng, X.

X. Ding, X. Deng, K. Song, and G. Chen, “Security improvement for asymmetric cryptosystem based on spherical wave illumination,” Appl. Opt. 52, 467–473 (2013).
[CrossRef]

X. Deng and D. Zhao, “Single channel color image encryption based on asymmetric cryptosystem,” Opt. Laser Technol. 44, 136–140 (2012).
[CrossRef]

Ding, X.

Dong, G. Y.

Dubreuil, M.

M. Dubreuil, A. Aflalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using Stokes-Mueller formalism,” J. Opt. 14, 094004 (2012).
[CrossRef]

Durán, V.

Frauel, Y.

Gao, B.

W. Qin, X. Peng, X. Meng, and B. Gao, “Universal and special keys based on phase-truncated Fourier transform,” Opt. Eng. 50, 080501 (2011).
[CrossRef]

Gluckstad, J.

P. C. Mogensen and J. Gluckstad, “A phase-based optical encryption system with polarization encoding,” Opt. Commun. 173, 177–183 (2000).
[CrossRef]

Gong, L.

N. Zhou, Y. Wang, L. Gong, H. He, and J. Wu, “Novel single channel color image encryption based on chaos and fractional Fourier transform,” Opt. Commun. 284, 2789–2796 (2011).
[CrossRef]

Gopinathan, U.

Hasman, E.

He, H.

N. Zhou, Y. Wang, L. Gong, H. He, and J. Wu, “Novel single channel color image encryption based on chaos and fractional Fourier transform,” Opt. Commun. 284, 2789–2796 (2011).
[CrossRef]

Ide, M.

Javidi, B.

Joseph, J.

Joshi, M.

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]

Kleiner, V.

Kumar, A.

Kumar, P.

P. Kumar, J. Joseph, and K. Singh, “Vulnerability of the security enhanced double random phase-amplitude encryption scheme to point spread function attack,” Opt. Lasers Eng. 50, 1196–1201 (2012).
[CrossRef]

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

Kuroda, K.

Lancis, J.

Li, Y.

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Trans. Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

Ma, J.

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Trans. Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

Matoba, O.

McNamara, D. E.

Meng, X.

W. Qin, X. Peng, X. Meng, and B. Gao, “Universal and special keys based on phase-truncated Fourier transform,” Opt. Eng. 50, 080501 (2011).
[CrossRef]

Meng, X. F.

Mogensen, P. C.

P. C. Mogensen and J. Gluckstad, “A phase-based optical encryption system with polarization encoding,” Opt. Commun. 173, 177–183 (2000).
[CrossRef]

Naughton, T. J.

Nishchal, N. K.

Niv, A.

Okada-Shudo, Y.

Peng, X.

W. Qin, X. Peng, X. Meng, and B. Gao, “Universal and special keys based on phase-truncated Fourier transform,” Opt. Eng. 50, 080501 (2011).
[CrossRef]

W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35, 118–120 (2010).
[CrossRef]

W. Qin and X. Peng, “Vulnerability to known-plaintext attack of optical encryption schemes based on two fractional Fourier transform order keys and double random phase keys,” J. Opt. 11, 075402 (2009).
[CrossRef]

X. Peng, P. Zhang, H. Wei, and B. Yu, “Known plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31, 1044–1046 (2006).
[CrossRef]

X. Peng, H. Wei, and P. Zhang, “Chosen-plaintext attack on lensless double random phase encoding in Fresnel domain,” Opt. Lett. 31, 3261–3263 (2006).
[CrossRef]

Pohit, M.

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25–31 (2000).
[CrossRef]

Qin, W.

W. Qin, X. Peng, X. Meng, and B. Gao, “Universal and special keys based on phase-truncated Fourier transform,” Opt. Eng. 50, 080501 (2011).
[CrossRef]

W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35, 118–120 (2010).
[CrossRef]

W. Qin and X. Peng, “Vulnerability to known-plaintext attack of optical encryption schemes based on two fractional Fourier transform order keys and double random phase keys,” J. Opt. 11, 075402 (2009).
[CrossRef]

Rajput, S. K.

Ran, Q.

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Trans. Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

Refregier, P.

Rodrigo, J. A.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

Shakher, C.

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. X.

Sheppard, J. R.

Sheridan, J. T.

Shimura, T.

Singh, K.

P. Kumar, J. Joseph, and K. Singh, “Vulnerability of the security enhanced double random phase-amplitude encryption scheme to point spread function attack,” Opt. Lasers Eng. 50, 1196–1201 (2012).
[CrossRef]

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

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]

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25–31 (2000).
[CrossRef]

Situ, G.

Sonehara, T.

Song, K.

Tajahuerce, E.

Tan, L.

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Trans. Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

Tan, X.

Tebaldi, M.

Torroba, R.

Tu, H.-Y.

H.-Y. Tu, C.-J. Cheng, and M.-L. Chen, “Optical image encryption based on polarization encoding by liquid crystal spatial light modulator,” J. Opt. A 6, 524–528 (2004).
[CrossRef]

Unnikrishnan, G.

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25–31 (2000).
[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]

Usategui, M. M.

Vargas, C.

Wang, B.

Wang, X.

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

X. Wang and D. Zhao, “Multiple image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284, 148–152 (2011).
[CrossRef]

Wang, Y.

N. Zhou, Y. Wang, L. Gong, H. He, and J. Wu, “Novel single channel color image encryption based on chaos and fractional Fourier transform,” Opt. Commun. 284, 2789–2796 (2011).
[CrossRef]

Wang, Y. R.

Wei, D.

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Trans. Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

Wei, H.

Wu, J.

N. Zhou, Y. Wang, L. Gong, H. He, and J. Wu, “Novel single channel color image encryption based on chaos and fractional Fourier transform,” Opt. Commun. 284, 2789–2796 (2011).
[CrossRef]

Xu, X. F.

Yu, B.

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]

Zhang, Y.

Zhao, D.

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

X. Deng and D. Zhao, “Single channel color image encryption based on asymmetric cryptosystem,” Opt. Laser Technol. 44, 136–140 (2012).
[CrossRef]

X. Wang and D. Zhao, “Multiple image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284, 148–152 (2011).
[CrossRef]

Zhou, N.

N. Zhou, Y. Wang, L. Gong, H. He, and J. Wu, “Novel single channel color image encryption based on chaos and fractional Fourier transform,” Opt. Commun. 284, 2789–2796 (2011).
[CrossRef]

Appl. Opt. (8)

IEEE Trans. Signal Process. Lett. (1)

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Trans. Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

J. Opt. (3)

M. Dubreuil, A. Aflalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using Stokes-Mueller formalism,” J. Opt. 14, 094004 (2012).
[CrossRef]

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Other (1)

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

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Figures (9)

Fig. 1.
Fig. 1.

Schematic diagram of the polarized-light-based image encryption scheme. R, retarder; P, polarizer; BS, beam splitter; SLM, spatial light modulator; M, mirror; CCD, charge-coupled device camera.

Fig. 2.
Fig. 2.

Schematic diagram for the phase-truncated FrT-based decryption scheme.

Fig. 3.
Fig. 3.

Block diagram of proposed single-channel color image for (a) encryption and (b) decryption.

Fig. 4.
Fig. 4.

Simulation results. (a) Lena image, (b) phase-truncated image, (c) random intensity image, and (d) encrypted image.

Fig. 5.
Fig. 5.

Simulation results. (a) Decrypted image using all correct keys, (b) decrypted image using a different intensity image, (c) decrypted image using incorrect encryption keys, and (d) decrypted image obtained using an incorrect angle of pixilated polarizer.

Fig. 6.
Fig. 6.

Simulation results for special attack: (a) relation between number of iterations and MSE during key generation, (b) corresponding decrypted image, and (c) decrypted image using incorrect angles of pixilated polarizer during attack.

Fig. 7.
Fig. 7.

Simulation results. (a) Barbara image, (b) red component, (c) green component, and (d) blue component.

Fig. 8.
Fig. 8.

(a) Convolved image in FrT domain, (b) random intensity image, and (c) encrypted image.

Fig. 9.
Fig. 9.

(a) Decrypted color image obtained after using all correct keys, (b) decrypted color image obtained after using an incorrect intensity key image, (c) decrypted color image obtained after using an encryption key, and (d) decrypted color image obtained after using an incorrect angle of pixelated polarizer.

Equations (46)

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F(u,v)=exp{i2πzλ}iλz{f(x,y)×exp[i2πr(x,y)]}×exp[iπλz((xu)2+(yv)2)]dxdy.
E(u,v)=PT{F(u,v)},
p1(u,v)=AT{F(u,v)},
MRet(ϕ,θ)=P(θ)[1000010000cos(ϕ)sin(ϕ)00sin(ϕ)cos(ϕ)]Rot(θ),
MPol(ψ)=12[1cos(2ψ)sin(2ψ)0cos(2ψ)cos2(2ψ)12sin(4ψ)0sin(2ψ)12sin(4ψ)sin2(2ψ)00000],
Rot(θ)=[100001cos(θ)sin(θ)00sin(θ)cos(θ)0000].
SP(i,j)=[SP0(i,j)000]T,
Sk(i,j)=[Sk0(i,j)000]T.
SP(i,j)=MRet(ϕ,θ)MPol(ψ)SP(i,j)=[αPSP0(i,j)βPSP0(i,j)γPSP0(i,j)δPSP0(i,j)]T,
Sk(i,j)=MRet(ϕ,θ)MPol(ψ)Sk(i,j)=[αkSk0(i,j)βkSk0(i,j)γkSk0(i,j)δkSk0(i,j)]T.
α=1/2,
β=1/2{cos(2θ)cos(2(θψ))+cos(ϕ)sin(2(θψ))},
γ=1/2{sin(2θ)cos(2(θψ))cos(ϕ)sin(2(θψ))},
δ=1/2sin(ϕ)sin(2(θψ)).
SR(i,j)=[αPSP0(i,j)+αkSk0(i,j)βPSP0(i,j)+βkSk0(i,j)γPSP0(i,j)+γkSk0(i,j)δPSP0(i,j)+δkSk0(i,j)]T,
Sc(i,j)=[Pc0(i,j)Pc1(i,j)Pc2(i,j)0]T.
Pc0(i,j)=12[(αPSP0(i,j)+αkSk0(i,j))+(βPSP0(i,j)+βkSk0(i,j))cos(2ψrand(i,j))+(γPSP0(i,j)+γkSk0(i,j))sin(2ψrand(i,j))],
Pc1(i,j)=12[(αPSP0(i,j)+αkSk0(i,j))×cos(2ψrand(i,j))+(βPSP0(i,j)+βkSk0(i,j))×cos2(2ψrand(i,j))+12(γPSP0(i,j)+γkSk0(i,j))×sin(4ψrand(i,j))],
Pc2(i,j)=12[(αPSP0(i,j)+αkSk0(i,j))×sin(2ψrand(i,j))+12(βPSP0(i,j)+βkSk0(i,j))×sin(4ψrand(i,j))+(γPSP0(i,j)+γkSk0(i,j))×sin2(2ψrand(i,j))].
S0decry(i,j)=MPol(ψdecry(i,j))×[Pc0(i,j)000].
S0decry(i,j)=12[1cos(2ψdecry(i,j))sin(2ψdecry(i,j))0cos(2ψdecry(i,j))cos2(2ψdecry(i,j))12sin(4ψdecry(i,j))0sin(2ψdecry(i,j))12sin(4ψdecry(i,j))sin2(2ψdecry(i,j))00000]×[12[(αPSP0(i,j)+αkSk0(i,j))+(βPSP0(i,j)+βkSk0(i,j))×cos(2ψrand(i,j))+(γPSP0(i,j)+γkSk0(i,j))×sin(2ψrand(i,j))]000]=14[(αPSP0(i,j)+αkSk0(i,j))+(βPSP0(i,j)+βkSk0(i,j))×cos(2ψrand(i,j))+(γPSP0(i,j)+γkSk0(i,j))×sin(2ψrand(i,j))]×[1cos(2ψdecry(i,j))sin(2ψdecry(i,j))0],
S0decry(i,j)=14[12(SP0(i,j)+Sk0(i,j))+12(SP0(i,j)+Sk0(i,j))×cos(2ψrand(i,j))+(6.123×1017)(SP0(i,j)+Sk0(i,j))×sin(2ψrand(i,j))]×[1cos(2ψdecry(i,j))sin(2ψdecry(i,j))0]=18[(SP0(i,j)+Sk0(i,j)){1+cos(2ψrand(i,j))}]×[1cos(2ψdecry(i,j))sin(2ψdecry(i,j))0],
SP0(i,j)=2Pc0(i,j)Sk0(i,j){αk+βkcos(2ψrand(i,j))+γksin(2ψrand(i,j))}αP+βPcos(2ψrand(i,j))+γPsin(2ψrand(i,j)),
SP0(i,j)=2Pc1(i,j)Sk0(i,j){αkcos(2ψrand(i,j))+βkcos2(2ψrand(i,j))+12γksin(4ψrand(i,j))}αPcos(2ψrand(i,j))+βPcos2(2ψrand(i,j))+12γPsin(4ψrand(i,j)),
SP0(i,j)=2Ic2(i,j)Sk0(i,j){αksin(2ψrand(i,j))+12βksin(4ψrand(i,j))+γksin2(2ψrand(i,j))}αPsin(2ψrand(i,j))+12βPsin(4ψrand(i,j))+γPsin2(2ψrand(i,j)).
f(x,y)=PT{FrTλz[E(u,v)×p1(u,v)]}.
Irm(x,y)=Ir(x,y)Rr(x,y)Igm(x,y)=Ig(x,y)Rg(x,y)Ibm(x,y)=Ib(x,y)Rb(x,y)}.
c(x,y)=Irm(x,y)Igm(x,y)Ibm(x,y),
C(u,v)=FrTλz[c(x,y)].
F(u,v)=Frm(u,v)×Fgm(u,v)×Fbm(u,v),
k(u,v)=AT[F(u,v)],
e(u,v)=PT[F(u,v)].
e(u,v)=|Frm(u,v)|·|Fgm(u,v)|·|Fbm(u,v)|.
pr=AT[Frm(u,v)]|Fgm(u,v)|×|Fbm(u,v)|,
pg=AT[Fgm(u,v)]|Frm(u,v)|×|Fbm(u,v)|,
pb=AT[Fbm(u,v)]|Frm(u,v)|×|Fgm(u,v)|.
Ii(x,y)=FrTλz[e(u,v)×pi(u,v)].
fn(x,y)=PT{FrTλz[E0(u,v)×Rn(u,v)]},
En+1(u,v)=PT{FrTλz[fn(x,y)×R(x,y)]},
kn+1(u,v)=AT{FrTλz[fn(x,y)×R(x,y)]},
MSE=u=0N1v=0N1{|En+1(u,v)||E(u,v)|}2N×N.
RE=x=1Ny=1N{|d(x,y)||f(x,y)|}2x=1Ny=1N{|f(x,y)|}2.
CC=COV(f(x,y),d(x,y))σf(x,y)σd(x,y).
COV(f(x,y),d(x,y))=Ave{[f(x,y)Ave{f(x,y)}]×[d(x,y)Ave{d(x,y)}]},
RErgb=REr+REg+REb3.
CCrgb=CCr+CCg+CCb3.

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