Abstract

A novel fast double-phase retrieval algorithm for lensless optical security systems based on the Fresnel domain is presented in this paper. Two phase-only masks are efficiently determined by using a modified Gerchberg-Saxton algorithm, in which two cascaded Fresnel transforms are replaced by one Fourier transform with compensations to reduce the consumed computations. Simulation results show that the proposed algorithm substantially speeds up the iterative process, while keeping the reconstructed image highly correlated with the original one.

© 2009 Optical Society of America

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  1. H. T. Chang, "Image encryption using separable amplitude-based virtual image and iteratively retrieved phase information," Opt. Eng. 40, 2165-2171 (2001)
    [CrossRef]
  2. G.-S. Lin, H. T. Chang, W.-N. Lie, and C.-H. Chuang, "A public-key-based optical image cryptosystem based on data embedding techniques," Opt. Eng. 42, 2331-2339 (2003)
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  6. E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, "Optoelectronic information encryption with phase shifting interferometry," Appl. Opt. 39, 2313-2320 (2000)
    [CrossRef]
  7. 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]
  8. B. Zhu, S. Liu, and Q. Ran, "Optical image encryption based on multifractional Fourier transforms," Opt. Lett. 25, 1159-1161 (2000)
    [CrossRef]
  9. Y. Zhang, C.-H. Zheng, and N. Tanno, "Optical encryption based on iterative fractional Fourier transform," Opt. Commun. 202, 277-285 (2002)
    [CrossRef]
  10. B. Hennelly and J. T. Sheridan, "Optical image encryption by random shifting in fractional Fourier domains," Opt. Lett. 28, 269-271 (2003)
    [CrossRef] [PubMed]
  11. P. C. Mogensen and J. Glückstad, "Phase-only optical encryption," Opt. Lett. 25, 566-568 (2000)
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  12. X. Peng, L. Yu, and L. Cai, "Double-lock for image encryption with virtual optical wavelength," Opt. Express 10, 41-45 (2002)
    [PubMed]
  13. S. Sinzinger, "Microoptically integrated correlators for security applications," Opt. Commun. 209, 69-74 (2002)
    [CrossRef]
  14. R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical secutity," Opt. Eng. 35, 2464-2469 (1996)
    [CrossRef]
  15. Y. Li, K. Kreske, and J. Rosen, "Security and encryption optical systems based on a correlator with significant output image," Appl. Opt. 39, 5295-5301 (2000)
    [CrossRef]
  16. H. T. Chang, W. C. Lu, and C. J. Kuo, "Multiple-phase retrieval for optical security systems by use of random-phase encoding," Appl. Opt. 41, 4825-4834 (2002)
    [CrossRef] [PubMed]
  17. Y. C. Chang, HsuanT. Chang, and C. J. Kuo, "Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain," Opt. Commun. 236, 245-257 (2004)
    [CrossRef]
  18. G. Situ, and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik 114, 473-477 (2003)
    [CrossRef]
  19. L. G. Neto and Y. Sheng, "Optical implementation of image encryption using random phase encoding," Opt. Eng. 35, 2459-2463 (1996)
    [CrossRef]
  20. G. Situ and J. Zhang, "A lensless optical security system based on computer-generated phase only masks," Opt. Commun. 232, 115-122 (2004)
    [CrossRef]
  21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968)
  22. H.P. Herzig (Ed.), Micro-Optics (Taylor & Francis, London, UK, 1996)
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    [CrossRef] [PubMed]
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  25. X. Tan, O. Matoba, T. Shimura, and K. Kuroda, "Improvement in holographic storage capacity by use of double-random phase encryption," Appl. Opt. 40, 4721-4727 (2001)
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  27. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972)
  28. S. Deng, L. Liu, H. Lang, W. Pan, and D. Zhao, "Hiding an image in cascaded Fresnel digital holograms," Chin. Opt. Lett. 4, 268-271 (2006)
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2006 (1)

2005 (1)

2004 (2)

Y. C. Chang, HsuanT. Chang, and C. J. Kuo, "Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain," Opt. Commun. 236, 245-257 (2004)
[CrossRef]

Y. C. Chang, HsuanT. Chang, and C. J. 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 lensless optical security system based on computer-generated phase only masks," Opt. Commun. 232, 115-122 (2004)
[CrossRef]

2003 (3)

G. Situ, and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik 114, 473-477 (2003)
[CrossRef]

G.-S. Lin, H. T. Chang, W.-N. Lie, and C.-H. Chuang, "A public-key-based optical image cryptosystem based on data embedding techniques," Opt. Eng. 42, 2331-2339 (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] [PubMed]

2002 (4)

X. Peng, L. Yu, and L. Cai, "Double-lock for image encryption with virtual optical wavelength," Opt. Express 10, 41-45 (2002)
[PubMed]

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

Y. Zhang, C.-H. Zheng, and N. Tanno, "Optical encryption based on iterative fractional Fourier transform," Opt. Commun. 202, 277-285 (2002)
[CrossRef]

S. Sinzinger, "Microoptically integrated correlators for security applications," Opt. Commun. 209, 69-74 (2002)
[CrossRef]

2001 (2)

H. T. Chang, "Image encryption using separable amplitude-based virtual image and iteratively retrieved phase information," Opt. Eng. 40, 2165-2171 (2001)
[CrossRef]

X. Tan, O. Matoba, T. Shimura, and K. Kuroda, "Improvement in holographic storage capacity by use of double-random phase encryption," Appl. Opt. 40, 4721-4727 (2001)
[CrossRef]

2000 (6)

1999 (3)

1997 (1)

1996 (2)

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical secutity," Opt. Eng. 35, 2464-2469 (1996)
[CrossRef]

L. G. Neto and Y. Sheng, "Optical implementation of image encryption using random phase encoding," Opt. Eng. 35, 2459-2463 (1996)
[CrossRef]

1995 (1)

1993 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972)

1971 (1)

R. W. Gerchberg and W. O. Saxton, "Phase determination for image and diffraction plane pictures in the electron microscope," Optik 34, 275-284 (1971)

Abookasis, D.

Abramson, O.

Cai, L.

Chang, H. T.

G.-S. Lin, H. T. Chang, W.-N. Lie, and C.-H. Chuang, "A public-key-based optical image cryptosystem based on data embedding techniques," Opt. Eng. 42, 2331-2339 (2003)
[CrossRef]

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

H. T. Chang, "Image encryption using separable amplitude-based virtual image and iteratively retrieved phase information," Opt. Eng. 40, 2165-2171 (2001)
[CrossRef]

Chang, Y. C.

Y. C. Chang, HsuanT. Chang, and C. J. Kuo, "Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain," Opt. Commun. 236, 245-257 (2004)
[CrossRef]

Chatwin, C.

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical secutity," Opt. Eng. 35, 2464-2469 (1996)
[CrossRef]

Chen, N. X.

Chuang, C.-H.

G.-S. Lin, H. T. Chang, W.-N. Lie, and C.-H. Chuang, "A public-key-based optical image cryptosystem based on data embedding techniques," Opt. Eng. 42, 2331-2339 (2003)
[CrossRef]

Cong, W. X.

Deng, S.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972)

R. W. Gerchberg and W. O. Saxton, "Phase determination for image and diffraction plane pictures in the electron microscope," Optik 34, 275-284 (1971)

Glückstad, J.

Gu, B.Y.

Hennelly, B.

Hsuan, Y. C.

Y. C. Chang, HsuanT. Chang, and C. J. Kuo, "Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain," Opt. Commun. 236, 245-257 (2004)
[CrossRef]

Javidi, B.

Joseph, J.

Kreske, K.

Kuo, C. J.

Kuroda, K.

Lang, H.

Li, J.

Li, Y.

Lie, W.-N.

G.-S. Lin, H. T. Chang, W.-N. Lie, and C.-H. Chuang, "A public-key-based optical image cryptosystem based on data embedding techniques," Opt. Eng. 42, 2331-2339 (2003)
[CrossRef]

Lin, G.-S.

G.-S. Lin, H. T. Chang, W.-N. Lie, and C.-H. Chuang, "A public-key-based optical image cryptosystem based on data embedding techniques," Opt. Eng. 42, 2331-2339 (2003)
[CrossRef]

Liu, L.

Liu, S.

Lu, W. C.

Matoba, O.

Mogensen, P. C.

Montal, O.

Neto, L. G.

L. G. Neto and Y. Sheng, "Optical implementation of image encryption using random phase encoding," Opt. Eng. 35, 2459-2463 (1996)
[CrossRef]

Pan, W.

Peng, X.

Ran, Q.

Réfrégier, P.

Rosen, J.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972)

R. W. Gerchberg and W. O. Saxton, "Phase determination for image and diffraction plane pictures in the electron microscope," Optik 34, 275-284 (1971)

Sheng, Y.

L. G. Neto and Y. Sheng, "Optical implementation of image encryption using random phase encoding," Opt. Eng. 35, 2459-2463 (1996)
[CrossRef]

Sheridan, J. T.

Shimura, T.

Singh, K.

Sinzinger, S.

S. Sinzinger, "Microoptically integrated correlators for security applications," Opt. Commun. 209, 69-74 (2002)
[CrossRef]

Situ, G.

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

G. Situ, and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik 114, 473-477 (2003)
[CrossRef]

Tajahuerce, E.

Tan, X.

Tanno, N.

Y. Zhang, C.-H. Zheng, and N. Tanno, "Optical encryption based on iterative fractional Fourier transform," Opt. Commun. 202, 277-285 (2002)
[CrossRef]

Unnikrishnan, G.

Verrall, S. C.

Wang, R. K.

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical secutity," Opt. Eng. 35, 2464-2469 (1996)
[CrossRef]

Watson, I. A.

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical secutity," Opt. Eng. 35, 2464-2469 (1996)
[CrossRef]

Yu, L.

Zhang, G.

Zhang, J.

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

G. Situ, and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik 114, 473-477 (2003)
[CrossRef]

Zhang, Y.

Y. Zhang, C.-H. Zheng, and N. Tanno, "Optical encryption based on iterative fractional Fourier transform," Opt. Commun. 202, 277-285 (2002)
[CrossRef]

Zhao, D.

Zheng, C.-H.

Y. Zhang, C.-H. Zheng, and N. Tanno, "Optical encryption based on iterative fractional Fourier transform," Opt. Commun. 202, 277-285 (2002)
[CrossRef]

Zhu, B.

Appl. Opt. (8)

Chin. Opt. Lett. (1)

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

Opt. Commun. (4)

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

Y. Zhang, C.-H. Zheng, and N. Tanno, "Optical encryption based on iterative fractional Fourier transform," Opt. Commun. 202, 277-285 (2002)
[CrossRef]

Y. C. Chang, HsuanT. Chang, and C. J. Kuo, "Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain," Opt. Commun. 236, 245-257 (2004)
[CrossRef]

S. Sinzinger, "Microoptically integrated correlators for security applications," Opt. Commun. 209, 69-74 (2002)
[CrossRef]

Opt. Eng. (4)

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical secutity," Opt. Eng. 35, 2464-2469 (1996)
[CrossRef]

H. T. Chang, "Image encryption using separable amplitude-based virtual image and iteratively retrieved phase information," Opt. Eng. 40, 2165-2171 (2001)
[CrossRef]

G.-S. Lin, H. T. Chang, W.-N. Lie, and C.-H. Chuang, "A public-key-based optical image cryptosystem based on data embedding techniques," Opt. Eng. 42, 2331-2339 (2003)
[CrossRef]

L. G. Neto and Y. Sheng, "Optical implementation of image encryption using random phase encoding," Opt. Eng. 35, 2459-2463 (1996)
[CrossRef]

Opt. Express (1)

Opt. Lett. (7)

Optik (3)

G. Situ, and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik 114, 473-477 (2003)
[CrossRef]

R. W. Gerchberg and W. O. Saxton, "Phase determination for image and diffraction plane pictures in the electron microscope," Optik 34, 275-284 (1971)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972)

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968)

H.P. Herzig (Ed.), Micro-Optics (Taylor & Francis, London, UK, 1996)

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Figures (5)

Fig. 1.
Fig. 1.

The optical setup of an optical security system with dual POMs.

Fig. 2.
Fig. 2.

The block diagram of the MGSA.

Fig. 3.
Fig. 3.

(a) The Baboon image and (b) the Lena image used as the target image for computer simulation.

Fig. 4.
Fig. 4.

(a) and (d): the phase distribution ψ 2(x 2, y 2) ; (b) and (e): the phase distribution ψ 1(x 1, y 1) ; (c) and (f): the decrypted images with correct keys, whose correlation coefficients ρ are at least 0.999.

Fig. 5.
Fig. 5.

(a) and (d): the phase distributions ψ 2(x 2, y 2) ; (b) and (e): the phase distributions ψ 1(x 1, y 1) ; (c) and (f): the decrypted images with correct keys, whose correlation coefficients ρ are at least 0.999, all corresponding to the method of phase pre-scrambling.

Tables (2)

Tables Icon

Table 1. Results of correlation coefficient ρ for MGSA and POCSA with different numbers of FFT and IFFT operations for the Baboon image.

Tables Icon

Table 2. Results of correlation coefficient ρ for MGSA and POCSA with different numbers of FFT and IFFT operations for the Lena image.

Equations (29)

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

F (μ1,v1) =
exp(j2πzλ)jλz f (μ0,v0)exp {jπλz[(μ1μ0)2+(v1v0)2]}dμ0dv0,
IFT[g(x0,y0);λ;z1]=1λz1g(x0,y0)exp{j2π(x1x0+y1y0)λz1}dx0dy0
=G(x1,y1)exp[jψt(x1,y1)],
FT{exp[jψ2(x2,y2)];λ;z2}= exp [jψ2(x2,y2)]·exp{j2π(x2x1+y2y1)λz2}dx2dy2
=Ĝ (x1,y1)exp[jψG(x1,y1)],
ĝ(x0,y0;λ;z1)=FT{Ĝ(x1,y1)·exp[jψt(x1,y1)];λ;z1}.
FrT {exp[jψ2(x2,y2)];λ;z2}
=exp(j2πz2λ)jλz2 exp [j2πz2λ+jψ2(x2,y2)jπ(x22+y22)λz2]
× exp [jπλz2(x2x1)2+jπλz2(y2y1)2] dx2 dy2
=1jλz2exp[jπ(x12+y12)λz2]exp[jψ2(x2,y2)]exp{j2π(x2x1+y2y1)λz2}dx2dy2
=exp [jπ(x12+y12)λz2]FT{exp[jψ2(x2,y2)];λ;z2}
=exp [jπ(x12+y12)λz2]Ĝ(x1,y1)exp[jψG(x1,y1)].
Ĝ(x1,y1)=FrT{exp[jψ2(x2,y2)];λ;z2}exp[jψG(x1,y1)]·exp[jπ(x12+y12)λz2].
ψ1(x1,y1)=2πz1λ+ψt(x1,y1)ψG(x1,y1)π(x12+y12)λ(1z2+1z1).
FrT(FrT{exp[jψ2(x2,y2)];λ;z2}exp[jψ1(x1,y1)];λ;z1)
=FrT{exp[jπ(x12+y12)λz2]Ĝ(x1,y1)exp[jψG(x1,y1)]×exp[jψ1(x1,y1)];λ;z1}
=FrT{exp(j2πz1λ)exp[jπ(x12+y12)λz1]Ĝ(x1,y1)exp[jψt(x1,y1)];λ;z1}
=exp[j2πz1λ]jλz1exp(j2πz1λ)exp[jπ(x12+y12)λz1]Ĝ(x1,y1)exp[jψt(x1,y1)]×exp[jπλz1(x1x0)2+jπλz1(y1y0)2]dx1dy1
=1jλz1exp[jπ(x02+y02)λz1]Ĝ(x1,y1)exp[jψt(x1,y1)]×exp[j2π(x1x0+y1y0)λz1]dx1dy1
=exp[jπ(x02+y02)λz1]FT{Ĝ(x1,y1)exp[jψt(x1,y1)]}
=ĝ(x0,y0).
ρ=E{[gE[g]][ĝE[ĝ]]}{E{[gE[g]]2}E{[ĝE[ĝ]]2}}12,
G(x1,y1)exp[jψt(x1,y1)]=IFT[g(x0,y0);λ;z1],
ψ2x2y2=MGSA{[G(x1,y1);λ;z2]},
ψ2(x2,y2)=2πz2λ+ψ2(x2,y2)π(x22+y22)λz2,
ψG(x1,y1)=arg (FT{exp[jψ2(x2,y2)];λ;z2}),
ψ1(x1,y1)=2πz1λ+ψt(x1,y1)ψG(x1,y1)π(x12+y12)λ(1z2+1z1).
ĝ(x0,y0)=FrT(FrT{exp[jψ2(x2,y2)];λ;z2}·exp[jψ1(x1,y1)];λ;z1).

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