Abstract

We propose an image encryption technique based on the interference principle and phase-truncation approach in the fractional Fourier domain. The proposed scheme offers multiple levels of security with asymmetric keys and is free from the silhouette problem. Multiple input images bonded with random phase masks are independently fractional Fourier transformed. Amplitude truncation of obtained spectrum helps generate individual and universal keys while phase truncation generates two phase-only masks analytically. For decryption, these two phase-only masks optically interfere, and this results in the phase-truncated function in the output. After using the correct random phase mask, universal key, individual key, and fractional orders, the original image is retrieved successfully. Computer simulation results with four gray-scale images validate the proposed method. To measure the effectiveness of the proposed method, we calculated the mean square error between the original and the decrypted images. In this scheme, the encryption process and decryption keys formation are complicated and should be realized digitally. For decryption, an optoelectronic scheme has been suggested.

© 2012 Optical Society of America

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References

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  1. P. Refregier and B. Javidi, “Optical image encryption based on input plane encoding and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
    [CrossRef]
  2. O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762–764 (1999).
    [CrossRef]
  3. G. Situ and J. Zhang, “Multiple image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1308 (2005).
    [CrossRef]
  4. J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
    [CrossRef]
  5. H.-E. Hwang, H. T. Chang, and W.-N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34, 3917–3919 (2009).
    [CrossRef]
  6. 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]
  7. B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28, 269–271 (2003).
    [CrossRef]
  8. N. Singh and A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46, 117–123 (2008).
    [CrossRef]
  9. A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Opt. Photon. 1, 589–636 (2009).
    [CrossRef]
  10. B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 25, 1159–1161 (2000).
    [CrossRef]
  11. N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
    [CrossRef]
  12. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33, 2443–2445 (2008).
    [CrossRef]
  13. Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A: Pure Appl. Opt. 11, 125406 (2009).
    [CrossRef]
  14. Y. Han and Y. Zhang, “Optical image encryption based on two beams interference,” Opt. Commun. 283, 1690–1692 (2010).
    [CrossRef]
  15. C. H. Niu, X. L. Wang, N. G. Lv, Z. H Zhou, and X. Y. Li, “An encryption method with multiple encrypted keys based on interference principle,” Opt. Express 18, 7827–7834(2010).
    [CrossRef]
  16. D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284, 2485–2487 (2011).
    [CrossRef]
  17. P. Kumar, J. Joseph, and K. Singh, “Optical image encryption using jigsaw transform for silhouette removal in interference-based methods and decryption with single spatial light modulator,” Appl. Opt. 50, 1805–1811 (2011).
    [CrossRef]
  18. B. Yang, Z. Liu, B. Wang, Y. Zhang, and S. Liu, “Optical stream-cipher-like system for image encryption based on Michelson interferometer,” Opt. Express 19, 2634–2642 (2011).
    [CrossRef]
  19. S. Yuan, S.-X. Yao, Y.-H. Xin, and M.-T. Liu, “Information hiding based on the optical interference principle,” Opt. Commun. 284, 5078–5083 (2011).
    [CrossRef]
  20. W. Stallings, Cryptography and Network Security: Principles and Practice, 5th ed. (Prentice-Hall, 2011).
  21. X. Peng, H. Wie, and P. Zhang, “Asymmetric cryptography based on wavefront sensing,” Opt. Lett. 31, 3579–3581 (2006).
    [CrossRef]
  22. W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35, 118–120 (2010).
    [CrossRef]
  23. 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]
  24. 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]
  25. X. Wang and D. Zhao, “Security enhancement of a phase-truncation based image encryption algorithm,” Appl. Opt. 50, 6645–6651 (2011).
    [CrossRef]
  26. A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of the optical encryption schemes based on double random phase keys,” Opt. Lett. 30, 1644–1646 (2005).
    [CrossRef]
  27. X. Peng, P. Chang, H. Wei, and B. Yu, “Known plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31, 1044–1046 (2006).
    [CrossRef]
  28. 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]

2011

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

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284, 2485–2487 (2011).
[CrossRef]

P. Kumar, J. Joseph, and K. Singh, “Optical image encryption using jigsaw transform for silhouette removal in interference-based methods and decryption with single spatial light modulator,” Appl. Opt. 50, 1805–1811 (2011).
[CrossRef]

B. Yang, Z. Liu, B. Wang, Y. Zhang, and S. Liu, “Optical stream-cipher-like system for image encryption based on Michelson interferometer,” Opt. Express 19, 2634–2642 (2011).
[CrossRef]

S. Yuan, S.-X. Yao, Y.-H. Xin, and M.-T. Liu, “Information hiding based on the optical interference principle,” Opt. Commun. 284, 5078–5083 (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]

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]

X. Wang and D. Zhao, “Security enhancement of a phase-truncation based image encryption algorithm,” Appl. Opt. 50, 6645–6651 (2011).
[CrossRef]

2010

2009

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A: Pure Appl. Opt. 11, 125406 (2009).
[CrossRef]

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Opt. Photon. 1, 589–636 (2009).
[CrossRef]

H.-E. Hwang, H. T. Chang, and W.-N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34, 3917–3919 (2009).
[CrossRef]

2008

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

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33, 2443–2445 (2008).
[CrossRef]

2007

2006

2005

2003

2000

1999

1995

Alfalou, A.

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Opt. Photon. 1, 589–636 (2009).
[CrossRef]

Arcos, S.

Barrera, J. F.

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Bolognini, N.

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Brosseau, C.

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Opt. Photon. 1, 589–636 (2009).
[CrossRef]

Carnicer, A.

Castro, A.

Chang, H. T.

Chang, P.

Dong, Z.

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A: Pure Appl. Opt. 11, 125406 (2009).
[CrossRef]

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]

Han, Y.

Y. Han and Y. Zhang, “Optical image encryption based on two beams interference,” Opt. Commun. 283, 1690–1692 (2010).
[CrossRef]

Henao, R.

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Hennelly, B.

Hwang, H.-E.

Javidi, B.

Joseph, J.

Juvells, I.

Kumar, P.

Li, X. Y.

Lie, W.-N.

Liu, J.

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284, 2485–2487 (2011).
[CrossRef]

Liu, M.-T.

S. Yuan, S.-X. Yao, Y.-H. Xin, and M.-T. Liu, “Information hiding based on the optical interference principle,” Opt. Commun. 284, 5078–5083 (2011).
[CrossRef]

Liu, S.

Liu, Z.

Lv, N. G.

Matoba, O.

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]

Montes-Usategui, M.

Naughton, T. J.

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

Nishchal, N. K.

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

Niu, C. H.

Peng, X.

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]

Ran, Q.

Refregier, P.

Sheridan, J. T.

Singh, K.

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.

Stallings, W.

W. Stallings, Cryptography and Network Security: Principles and Practice, 5th ed. (Prentice-Hall, 2011).

Tebaldi, M.

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Torroba, R.

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Unnikrishnan, G.

Wang, B.

Wang, X.

X. Wang and D. Zhao, “Security enhancement of a phase-truncation based image encryption algorithm,” Appl. Opt. 50, 6645–6651 (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]

Wang, X. L.

Wang, Y.

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284, 2485–2487 (2011).
[CrossRef]

Wei, H.

Weng, D.

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284, 2485–2487 (2011).
[CrossRef]

Wie, H.

Xie, J.

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284, 2485–2487 (2011).
[CrossRef]

Xin, Y.-H.

S. Yuan, S.-X. Yao, Y.-H. Xin, and M.-T. Liu, “Information hiding based on the optical interference principle,” Opt. Commun. 284, 5078–5083 (2011).
[CrossRef]

Yang, B.

Yao, S.-X.

S. Yuan, S.-X. Yao, Y.-H. Xin, and M.-T. Liu, “Information hiding based on the optical interference principle,” Opt. Commun. 284, 5078–5083 (2011).
[CrossRef]

Yu, B.

Yuan, S.

S. Yuan, S.-X. Yao, Y.-H. Xin, and M.-T. Liu, “Information hiding based on the optical interference principle,” Opt. Commun. 284, 5078–5083 (2011).
[CrossRef]

Zhang, J.

Zhang, P.

Zhang, Y.

B. Yang, Z. Liu, B. Wang, Y. Zhang, and S. Liu, “Optical stream-cipher-like system for image encryption based on Michelson interferometer,” Opt. Express 19, 2634–2642 (2011).
[CrossRef]

Y. Han and Y. Zhang, “Optical image encryption based on two beams interference,” Opt. Commun. 283, 1690–1692 (2010).
[CrossRef]

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A: Pure Appl. Opt. 11, 125406 (2009).
[CrossRef]

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33, 2443–2445 (2008).
[CrossRef]

Zhao, D.

X. Wang and D. Zhao, “Security enhancement of a phase-truncation based image encryption algorithm,” Appl. Opt. 50, 6645–6651 (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]

Zhou, Z. H

Zhu, B.

Zhu, N.

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284, 2485–2487 (2011).
[CrossRef]

Appl. Opt.

J. Opt. A: Pure Appl. Opt.

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A: Pure Appl. Opt. 11, 125406 (2009).
[CrossRef]

Opt. Commun.

Y. Han and Y. Zhang, “Optical image encryption based on two beams interference,” Opt. Commun. 283, 1690–1692 (2010).
[CrossRef]

S. Yuan, S.-X. Yao, Y.-H. Xin, and M.-T. Liu, “Information hiding based on the optical interference principle,” Opt. Commun. 284, 5078–5083 (2011).
[CrossRef]

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284, 2485–2487 (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]

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

Opt. Eng.

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]

Opt. Express

Opt. Lasers Eng.

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

Opt. Lett.

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]

P. Refregier and B. Javidi, “Optical image encryption based on input plane encoding and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
[CrossRef]

O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762–764 (1999).
[CrossRef]

B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 25, 1159–1161 (2000).
[CrossRef]

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

G. Situ and J. Zhang, “Multiple image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1308 (2005).
[CrossRef]

A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of the optical encryption schemes based on double random phase keys,” Opt. Lett. 30, 1644–1646 (2005).
[CrossRef]

X. Peng, P. Chang, 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. Wie, and P. Zhang, “Asymmetric cryptography based on wavefront sensing,” Opt. Lett. 31, 3579–3581 (2006).
[CrossRef]

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33, 2443–2445 (2008).
[CrossRef]

H.-E. Hwang, H. T. Chang, and W.-N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34, 3917–3919 (2009).
[CrossRef]

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

Opt. Photon.

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Opt. Photon. 1, 589–636 (2009).
[CrossRef]

Other

W. Stallings, Cryptography and Network Security: Principles and Practice, 5th ed. (Prentice-Hall, 2011).

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Figures (9)

Fig. 1.
Fig. 1.

Block diagram for optical image encryption based on amplitude truncation and phase truncation in FRT domain. AT: amplitude truncation, PT: phase truncation.

Fig. 2.
Fig. 2.

Optical interference-based encryption-decryption scheme. M1 and M2 are analytically generated phase-only masks; L: Lens, HM: half mirror, l: distance between the phase masks and the output plane, S: screen, SLM: spatial light modulator, CCD: charge-coupled camera device, d: distance parameter to be adjusted according to desired fractional order, and PC: personal computer.

Fig. 3.
Fig. 3.

Block diagram for decryption based on amplitude truncation and phase truncation in FRT domain.

Fig. 4.
Fig. 4.

Gray-scale input images to be used for encryption. Images of (a) Lena, (b) Barbara, (c) Cameraman and (d) Lady.

Fig. 5.
Fig. 5.

Simulation results: (a) common decryption key, (b) encrypted image, (c) analytically generated first phase-only mask, and (d) analytically generated second phase-only mask.

Fig. 6.
Fig. 6.

Decrypted images obtained after the use of all correct keys; (a) Lena, (b) Barbara, (c) Cameraman and (d) Lady.

Fig. 7.
Fig. 7.

Simulation results to check the silhouette problem. Decrypted image obtained after using only the first phase-only mask; (a) Lena, (b) Barbara, (c) Cameraman, and (d) Lady.

Fig. 8.
Fig. 8.

Simulation results to check the silhouette problem. Decrypted image obtained after using only the second phase-only mask; (a) Lena, (b) Barbara, (c) Cameraman and (d) Lady.

Fig. 9.
Fig. 9.

Decrypted images obtained after using encryption keys; (a) Lena, (b) Barbara, (c) Cameraman and (d) Lady.

Equations (23)

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

F1(u,v)=K{f1(x,y)×exp[i2πr1(x,y)]}×exp[jπx2+y2+u2+v2tanα12jπxyuvsinα1]dxdy,
e1(u,v)=PT[F1(u,v)].
e2(u,v)=PT[F2(u,v)],
en(u,v)=PT[Fn(u,v)].
E(u,v)=i=1n{ei(u,v)}.
G(ξ,η)=K{E(u,v)×exp[i2πR(u,v)]}×exp[jπu2+v2+ξ2+η2tanβ2jπuvξηsinβ]dudv.
H(ξ,η)=PT[G(ξ,η)],
A(ξ,η)=AT[G(ξ,η)].
a1(u,v)=AT[F1(u,v)],
an(u,v)=AT[Fn(u,v)].
km=am(u,v)Πi=1imn[ei(u,v)].
H(ξ,η)=H(ξ,η)exp[iπφ(ξ,η)],
H(ξ,η)=exp[iM1(x,y)]*h(x,y,l)+exp[iM2(x,y)]*h(x,y,l),
h(x,y,l)=exp[i2πl/λ)ilλexp[iπλl(x2+y2)]
exp[iM1(x,y)]+exp[iM2(x,y)]=Iγ{Iγ[H(ξ,η)]Iγ[h(x,y,l)]},
D=Iλ{Iγ[H(ξ,η)]Iγ[h(x,y,l)]}.
exp[iM2(x,y)]=Dexp[iM1(x,y)].
|Dexp[iM1(x,y)]|2={Dexp[iM1(x,y)]}{Dexp[iM1(x,y)]}*=1.
M1(x,y)=arg(D)+cos1{|D|2},
M2(x,y)=arg{Dexp(iM1)},
E(u,v)=PT[Iβ{H(ξ,η)×A(ξ,η)}],
fi(x,y)=Iαi{E(u,v)×km}.
MSE=xy[fm(x,y)dm(x,y)]2M×N,

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