Abstract

The earlier proposed interference-based encryption method with two phase-only masks (POMs), which actually is a special case of our method, is quite simple and does not need iterative encoding. However, it has been found recently that the encryption method has security problems and cannot be directly applied to image encryption due to the inherent silhouette problem. Several methods based on chaotic encryption algorithms have been proposed to remove the problem by postprocessing of the POMs, which increased the computation time or led to digital inverse computation in decryption. Here we propose a new method for image encryption based on optical interference and analytical algorithm that can be directly used for image encryption. The information of the target image is hidden into three POMs, and the silhouette problem that exists in the method with two POMs can be resolved during the generation procedure of POMs based on the interference principle. Simulation results are presented to verify the validity of the proposed approach.

© 2012 Optical Society of America

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References

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    [CrossRef]
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  14. W. Chen, X. Chen, and C. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35, 3817–3819 (2010).
    [CrossRef]
  15. P. Clemente, V. Durán, V. Torres-Company, E. Tajahuerce, and J. Lancis, “Optical encryption based on computational ghost imaging,” Opt. Lett. 35, 2391–2393 (2010).
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    [CrossRef]
  23. B. Wang and Y. Zhang, “Double images hiding based on optical interference,” Opt. Commun. 282, 3439–3443 (2009).
    [CrossRef]
  24. N. Zhu, Y. Wang, J. Liu, J. Xie, and H. Zhang, “Optical image encryption based on interference of polarized light,” Opt. Express 17, 13418–13428 (2009).
    [CrossRef]
  25. 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]
  26. C. J. Tay, C. Quan, W. Chen, and Y. Fu, “Color image encryption based on interference and virtual optics,” Opt. Laser Technol. 42, 409–415 (2010).
    [CrossRef]
  27. C. Niu, X. Wang, N. Lv, Z. Zhou, and X. Li, “An encryption method with multiple encrypted keys based on interference principle,” Opt. Express 18, 7827–7834 (2010).
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  28. N. Zhu, Y. Wang, J. Liu, and J. Xie, “Holographic projection based on interference and analytical algorithm,” Opt. Commun. 283, 4969–4971 (2010).
    [CrossRef]
  29. 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]
  30. Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A 11, 125406(2009).
    [CrossRef]
  31. P. Kumar, J. Joseph, and K. Singh, “Optical image encryption based on interference under convergent random illumination,” J. Opt. 12, 095402 (2010).
    [CrossRef]
  32. P. Kumar, J. Joseph, and K. Singh, “Optical image encryption using a jigsaw transform for silhouette removal in interference-based methods and decryption with a single spatial light modulator,” Appl. Opt. 50, 1805–1811 (2011).
    [CrossRef]

2011

2010

J. A. Rodrigo, H. Duadi, T. Alieva, and Z. Zalevsky, “Multi-stage phase retrieval algorithm based upon the gyrator transform,” Opt. Express 18, 1510–1520 (2010).
[CrossRef]

C. Niu, X. Wang, N. Lv, Z. Zhou, and X. Li, “An encryption method with multiple encrypted keys based on interference principle,” Opt. Express 18, 7827–7834 (2010).
[CrossRef]

Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18, 12033–12043 (2010).
[CrossRef]

A. Alfalou and C. Brosseau, “Exploiting root-mean-square time-frequency structure for multiple-image optical compression and encryption,” Opt. Lett. 35, 1914–1916 (2010).
[CrossRef]

P. Clemente, V. Durán, V. Torres-Company, E. Tajahuerce, and J. Lancis, “Optical encryption based on computational ghost imaging,” Opt. Lett. 35, 2391–2393 (2010).
[CrossRef]

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

P. Kumar, J. Joseph, and K. Singh, “Optical image encryption based on interference under convergent random illumination,” J. Opt. 12, 095402 (2010).
[CrossRef]

C. J. Tay, C. Quan, W. Chen, and Y. Fu, “Color image encryption based on interference and virtual optics,” Opt. Laser Technol. 42, 409–415 (2010).
[CrossRef]

N. Zhu, Y. Wang, J. Liu, and J. Xie, “Holographic projection based on interference and analytical algorithm,” Opt. Commun. 283, 4969–4971 (2010).
[CrossRef]

2009

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]

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

Z. Liu, Q. Li, J. Dai, X. Zhao, Z. Sun, S. Liu, and M. A. Ahmad, “Image encryption based on random scrambling of the amplitude and phase in the frequency domain,” Opt. Eng. 48, 087005 (2009).
[CrossRef]

B. Wang and Y. Zhang, “Double images hiding based on optical interference,” Opt. Commun. 282, 3439–3443 (2009).
[CrossRef]

N. Zhu, Y. Wang, J. Liu, J. Xie, and H. Zhang, “Optical image encryption based on interference of polarized light,” Opt. Express 17, 13418–13428 (2009).
[CrossRef]

H. Hwang, H. Chang, and W. Lie, “Fast double-phase retrieval in Fresnel domain using modified Gerchberg-Saxton algorithm for lensless optical security systems,” Opt. Express 17, 13700–13710 (2009).
[CrossRef]

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

H. Hwang, H. Chang, and W. 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

2006

2005

2004

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
[CrossRef]

Y. Chang, H. Chang, and C. Kuo, “Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain,” Opt. Commun. 236, 245–257 (2004).
[CrossRef]

G. Situ and J. Zhang, “A lenseless optical security system based on computer-generated phase only masks,” Opt. Commun. 232, 115–122 (2004).
[CrossRef]

2003

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]

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

2002

2000

1995

Ahmad, M. A.

Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18, 12033–12043 (2010).
[CrossRef]

Z. Liu, Q. Li, J. Dai, X. Zhao, Z. Sun, S. Liu, and M. A. Ahmad, “Image encryption based on random scrambling of the amplitude and phase in the frequency domain,” Opt. Eng. 48, 087005 (2009).
[CrossRef]

Alfalou, A.

Alieva, T.

Arcos, S.

Brosseau, C.

Carnicer, A.

Chang, H.

Chang, Y.

Y. Chang, H. Chang, and C. Kuo, “Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain,” Opt. Commun. 236, 245–257 (2004).
[CrossRef]

Chen, W.

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

C. J. Tay, C. Quan, W. Chen, and Y. Fu, “Color image encryption based on interference and virtual optics,” Opt. Laser Technol. 42, 409–415 (2010).
[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.

Clemente, P.

Dai, J.

Z. Liu, Q. Li, J. Dai, X. Zhao, Z. Sun, S. Liu, and M. A. Ahmad, “Image encryption based on random scrambling of the amplitude and phase in the frequency domain,” Opt. Eng. 48, 087005 (2009).
[CrossRef]

Dong, Z.

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

Duadi, H.

Durán, V.

Fu, Y.

C. J. Tay, C. Quan, W. Chen, and Y. Fu, “Color image encryption based on interference and virtual optics,” Opt. Laser Technol. 42, 409–415 (2010).
[CrossRef]

Guo, Q.

Hennelly, B.

Hwang, H.

Javidi, B.

Joseph, J.

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

P. Kumar, J. Joseph, and K. Singh, “Optical image encryption based on interference under convergent random illumination,” J. Opt. 12, 095402 (2010).
[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]

Juvells, I.

Kumar, P.

Kuo, C.

Y. Chang, H. Chang, and C. Kuo, “Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain,” Opt. Commun. 236, 245–257 (2004).
[CrossRef]

H. Chang, W. Lu, and C. Kuo, “Multiple-phase retrieval for optical security systems by use of random-phase encoding,” Appl. Opt. 41, 4825–4834 (2002).
[CrossRef]

Lancis, J.

Li, Q.

Z. Liu, Q. Li, J. Dai, X. Zhao, Z. Sun, S. Liu, and M. A. Ahmad, “Image encryption based on random scrambling of the amplitude and phase in the frequency domain,” Opt. Eng. 48, 087005 (2009).
[CrossRef]

Li, X.

Lie, W.

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]

N. Zhu, Y. Wang, J. Liu, and J. Xie, “Holographic projection based on interference and analytical algorithm,” Opt. Commun. 283, 4969–4971 (2010).
[CrossRef]

N. Zhu, Y. Wang, J. Liu, J. Xie, and H. Zhang, “Optical image encryption based on interference of polarized light,” Opt. Express 17, 13418–13428 (2009).
[CrossRef]

Liu, S.

Liu, Z.

Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18, 12033–12043 (2010).
[CrossRef]

Z. Liu, Q. Li, J. Dai, X. Zhao, Z. Sun, S. Liu, and M. A. Ahmad, “Image encryption based on random scrambling of the amplitude and phase in the frequency domain,” Opt. Eng. 48, 087005 (2009).
[CrossRef]

Lu, W.

Lv, N.

Montes-Usategui, M.

Nishchal, N. K.

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]

Niu, C.

Peng, X.

Quan, C.

C. J. Tay, C. Quan, W. Chen, and Y. Fu, “Color image encryption based on interference and virtual optics,” Opt. Laser Technol. 42, 409–415 (2010).
[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]

Ran, Q.

Refregier, P.

Rodrigo, J. A.

Sheppard, C.

Sheridan, J. T.

Singh, K.

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

P. Kumar, J. Joseph, and K. Singh, “Optical image encryption based on interference under convergent random illumination,” J. Opt. 12, 095402 (2010).
[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]

Situ, G.

G. Situ and J. Zhang, “A lenseless optical security system based on computer-generated phase only masks,” Opt. Commun. 232, 115–122 (2004).
[CrossRef]

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
[CrossRef]

Sun, Z.

Z. Liu, Q. Li, J. Dai, X. Zhao, Z. Sun, S. Liu, and M. A. Ahmad, “Image encryption based on random scrambling of the amplitude and phase in the frequency domain,” Opt. Eng. 48, 087005 (2009).
[CrossRef]

Tajahuerce, E.

Tay, C. J.

C. J. Tay, C. Quan, W. Chen, and Y. Fu, “Color image encryption based on interference and virtual optics,” Opt. Laser Technol. 42, 409–415 (2010).
[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]

Torres-Company, V.

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, B.

B. Wang and Y. Zhang, “Double images hiding based on optical interference,” Opt. Commun. 282, 3439–3443 (2009).
[CrossRef]

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

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

Wang, X.

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]

N. Zhu, Y. Wang, J. Liu, and J. Xie, “Holographic projection based on interference and analytical algorithm,” Opt. Commun. 283, 4969–4971 (2010).
[CrossRef]

N. Zhu, Y. Wang, J. Liu, J. Xie, and H. Zhang, “Optical image encryption based on interference of polarized light,” Opt. Express 17, 13418–13428 (2009).
[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]

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]

N. Zhu, Y. Wang, J. Liu, and J. Xie, “Holographic projection based on interference and analytical algorithm,” Opt. Commun. 283, 4969–4971 (2010).
[CrossRef]

N. Zhu, Y. Wang, J. Liu, J. Xie, and H. Zhang, “Optical image encryption based on interference of polarized light,” Opt. Express 17, 13418–13428 (2009).
[CrossRef]

Xu, L.

Yu, B.

Zalevsky, Z.

Zhang, H.

Zhang, J.

G. Situ and J. Zhang, “A lenseless optical security system based on computer-generated phase only masks,” Opt. Commun. 232, 115–122 (2004).
[CrossRef]

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
[CrossRef]

Zhang, P.

Zhang, Y.

B. Wang and Y. Zhang, “Double images hiding based on optical interference,” Opt. Commun. 282, 3439–3443 (2009).
[CrossRef]

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

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

Zhao, X.

Z. Liu, Q. Li, J. Dai, X. Zhao, Z. Sun, S. Liu, and M. A. Ahmad, “Image encryption based on random scrambling of the amplitude and phase in the frequency domain,” Opt. Eng. 48, 087005 (2009).
[CrossRef]

Zhou, Z.

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]

N. Zhu, Y. Wang, J. Liu, and J. Xie, “Holographic projection based on interference and analytical algorithm,” Opt. Commun. 283, 4969–4971 (2010).
[CrossRef]

N. Zhu, Y. Wang, J. Liu, J. Xie, and H. Zhang, “Optical image encryption based on interference of polarized light,” Opt. Express 17, 13418–13428 (2009).
[CrossRef]

Adv. Opt. Photon.

Appl. Opt.

J. Opt.

P. Kumar, J. Joseph, and K. Singh, “Optical image encryption based on interference under convergent random illumination,” J. Opt. 12, 095402 (2010).
[CrossRef]

J. Opt. A

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

Opt. Commun.

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]

Y. Chang, H. Chang, and C. Kuo, “Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain,” Opt. Commun. 236, 245–257 (2004).
[CrossRef]

G. Situ and J. Zhang, “A lenseless optical security system based on computer-generated phase only masks,” Opt. Commun. 232, 115–122 (2004).
[CrossRef]

B. Wang and Y. Zhang, “Double images hiding based on optical interference,” Opt. Commun. 282, 3439–3443 (2009).
[CrossRef]

N. Zhu, Y. Wang, J. Liu, and J. Xie, “Holographic projection based on interference and analytical algorithm,” Opt. Commun. 283, 4969–4971 (2010).
[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]

Opt. Eng.

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]

Z. Liu, Q. Li, J. Dai, X. Zhao, Z. Sun, S. Liu, and M. A. Ahmad, “Image encryption based on random scrambling of the amplitude and phase in the frequency domain,” Opt. Eng. 48, 087005 (2009).
[CrossRef]

Opt. Express

Opt. Laser Technol.

C. J. Tay, C. Quan, W. Chen, and Y. Fu, “Color image encryption based on interference and virtual optics,” Opt. Laser Technol. 42, 409–415 (2010).
[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 and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
[CrossRef]

B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 251159–1161 (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, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
[CrossRef]

A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30, 1644–1646 (2005).
[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 the Fresnel domain,” Opt. Lett. 31, 3261–3263 (2006).
[CrossRef]

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

A. Alfalou and C. Brosseau, “Exploiting root-mean-square time-frequency structure for multiple-image optical compression and encryption,” Opt. Lett. 35, 1914–1916 (2010).
[CrossRef]

P. Clemente, V. Durán, V. Torres-Company, E. Tajahuerce, and J. Lancis, “Optical encryption based on computational ghost imaging,” Opt. Lett. 35, 2391–2393 (2010).
[CrossRef]

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

H. Hwang, H. Chang, and W. 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]

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of the optical decryption system. BE, beam expander.

Fig. 2.
Fig. 2.

(a) Target image. The phase distributions of (b) P1 and (c) P2. (d) Reconstructed image.

Fig. 3.
Fig. 3.

Images reconstructed from (a) P1, (b) P2, (c) exp(iθ), and (d) exp[iarccos(M(x,y)/2)].

Fig. 4.
Fig. 4.

Phase distributions of the three POMs, (a) P1, (b) P2, and (c) P3, obtained by using our method. (d) Decrypted image.

Fig. 5.
Fig. 5.

Images reconstructed from (a) P1, (b) P2, and (c) P3.

Fig. 6.
Fig. 6.

Decryption results with the wrong POM: (a) P1, (b) P2, and (c) P3.

Equations (13)

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O(x,y)=FT1[O(u,υ)exp[i2πrand(u,υ)]],
O(u,υ)=|FT{[P1(x,y)+P2(x,y)]P3(x,y)}|2,
[P1(x,y)+P2(x,y)]P3(x,y)=M(x,y)exp[iθ(x,y)],
[P1(x,y)]*[P1(x,y)]=[P2(x,y)]*[P2(x,y)]=1,
p1(x,y)=φ(x,y)arccos(M(x,y)/2),
p2(x,y)=arg{M(x,y)exp[iφ(x,y)]P1(x,y)},
p3(x,y)=θ(x,y)φ(x,y),
p1(x,y)=θ(x,y)arccos(M(x,y)/2),
p2(x,y)=θ(x,y)+arccos(M(x,y)/2).
RE=Σn=1NΣm=1MR(m,n)O(m,n)2Σn=1NΣm=1NO(m,n)2,
p1(x,y)=2πrand(x,y)arccos(M(x,y)/2),
p2(x,y)=2πrand(x,y)+arccos(M(x,y)/2),
p3(x,y)=θ(x,y)2πrand(x,y).

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