Abstract

In this paper, we propose a new method based on a three-dimensional (3D) space-based strategy for the optical image encryption. The two-dimensional (2D) processing of a plaintext in the conventional optical encryption methods is extended to a 3D space-based processing. Each pixel of the plaintext is considered as one particle in the proposed space-based optical image encryption, and the diffraction of all particles forms an object wave in the phase-shifting digital holography. The effectiveness and advantages of the proposed method are demonstrated by numerical results. The proposed method can provide a new optical encryption strategy instead of the conventional 2D processing, and may open up a new research perspective for the optical image encryption.

© 2010 OSA

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. O. Matoba, T. Nomura, E. Perez-Cabre, M. Í. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  7. Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  16. X. Peng, H. Wei, and P. Zhang, “Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain,” Opt. Lett. 31(22), 3261–3263 (2006).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  23. W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
    [CrossRef]
  24. W. Chen and X. Chen, “Quantitative phase retrieval of a complex-valued object using variable function orders in the fractional Fourier domain,” Opt. Express 18(13), 13536–13541 (2010).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  26. I. Yamaguchi, K. Yamamoto, G. A. Mills, and M. Yokota, “Image reconstruction only by phase data in phase-shifting digital holography,” Appl. Opt. 45(5), 975–983 (2006).
    [CrossRef] [PubMed]
  27. B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003).
    [CrossRef] [PubMed]
  28. F. J. Dyson and H. Falk, “Period of a discrete cat mapping,” Am. Math. Mon. 99(7), 603–614 (1992).
    [CrossRef]
  29. W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282(18), 3680–3685 (2009).
    [CrossRef]
  30. Y. Sheng, Z. Xin, M. S. Alam, L. Xi, and L. Xiao-Feng, “Information hiding based on double random-phase encoding and public-key cryptography,” Opt. Express 17(5), 3270–3284 (2009).
    [CrossRef] [PubMed]
  31. X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
    [CrossRef]
  32. C. P. McElhinney, B. M. Hennelly, and T. J. Naughton, “Extended focused imaging for digital holograms of macroscopic three-dimensional objects,” Appl. Opt. 47(19), D71–D79 (2008).
    [CrossRef] [PubMed]
  33. M. Antkowiak, N. Callens, C. Yourassowsky, and F. Dubois, “Extended focused imaging of a microparticle field with digital holographic microscopy,” Opt. Lett. 33(14), 1626–1628 (2008).
    [CrossRef] [PubMed]

2010 (4)

2009 (8)

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
[CrossRef]

M. He, Q. Tan, L. Cao, Q. He, and G. Jin, “Security enhanced optical encryption system by random phase key and permutation key,” Opt. Express 17(25), 22462–22473 (2009).
[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(3), 331–333 (2009).
[CrossRef] [PubMed]

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282(18), 3680–3685 (2009).
[CrossRef]

Y. Sheng, Z. Xin, M. S. Alam, L. Xi, and L. Xiao-Feng, “Information hiding based on double random-phase encoding and public-key cryptography,” Opt. Express 17(5), 3270–3284 (2009).
[CrossRef] [PubMed]

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[CrossRef]

N. Singh and A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009).
[CrossRef]

O. Matoba, T. Nomura, E. Perez-Cabre, M. Í. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
[CrossRef]

2008 (4)

2006 (3)

2005 (1)

2004 (2)

2003 (1)

2002 (1)

2000 (1)

1999 (1)

1997 (2)

B. Javidi, “Securing information with optical technologies,” Phys. Today 50(3), 27–32 (1997).
[CrossRef]

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997).
[CrossRef] [PubMed]

1995 (1)

1992 (1)

F. J. Dyson and H. Falk, “Period of a discrete cat mapping,” Am. Math. Mon. 99(7), 603–614 (1992).
[CrossRef]

Ahmad, M. A.

Alam, M. S.

Antkowiak, M.

Arcos, S.

Cai, L.

Cai, L. Z.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[CrossRef]

X. C. Cheng, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, X. F. Xu, X. X. Shen, and G. Y. Dong, “Security enhancement of double-random phase encryption by amplitude modulation,” Opt. Lett. 33(14), 1575–1577 (2008).
[CrossRef] [PubMed]

Callens, N.

Cao, L.

Carnicer, A.

Chen, H.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Chen, W.

W. Chen and X. Chen, “Quantitative phase retrieval of a complex-valued object using variable function orders in the fractional Fourier domain,” Opt. Express 18(13), 13536–13541 (2010).
[CrossRef] [PubMed]

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
[CrossRef]

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282(18), 3680–3685 (2009).
[CrossRef]

Chen, X.

Cheng, X. C.

Dai, J.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Dong, G. Y.

Dowling, T.

Dubois, F.

Dyson, F. J.

F. J. Dyson and H. Falk, “Period of a discrete cat mapping,” Am. Math. Mon. 99(7), 603–614 (1992).
[CrossRef]

Falk, H.

F. J. Dyson and H. Falk, “Period of a discrete cat mapping,” Am. Math. Mon. 99(7), 603–614 (1992).
[CrossRef]

Gao, Z.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[CrossRef]

Guo, Q.

He, M.

He, Q.

Hennelly, B.

Hennelly, B. M.

Javidi, B.

O. Matoba, T. Nomura, E. Perez-Cabre, M. Í. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
[CrossRef]

O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999).
[CrossRef]

B. Javidi, “Securing information with optical technologies,” Phys. Today 50(3), 27–32 (1997).
[CrossRef]

P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
[CrossRef] [PubMed]

Jin, G.

Joseph, J.

Juvells, I.

Kim, D. H.

Kim, H.

Kumar, A.

Kumar, P.

Lee, Y. H.

Li, A. M.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[CrossRef]

Li, P.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Liu, S.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (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(11), 12033–12043 (2010).
[CrossRef] [PubMed]

Liu, T.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Liu, Z.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (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(11), 12033–12043 (2010).
[CrossRef] [PubMed]

Matoba, O.

O. Matoba, T. Nomura, E. Perez-Cabre, M. Í. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
[CrossRef]

O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999).
[CrossRef]

McElhinney, C. P.

Meng, X. F.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[CrossRef]

X. C. Cheng, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, X. F. Xu, X. X. Shen, and G. Y. Dong, “Security enhancement of double-random phase encryption by amplitude modulation,” Opt. Lett. 33(14), 1575–1577 (2008).
[CrossRef] [PubMed]

Millan, M. Í. S.

O. Matoba, T. Nomura, E. Perez-Cabre, M. Í. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
[CrossRef]

Mills, G. A.

Montes-Usategui, M.

Naughton, T. J.

Nomura, T.

O. Matoba, T. Nomura, E. Perez-Cabre, M. Í. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
[CrossRef]

Peng, X.

Perez-Cabre, E.

O. Matoba, T. Nomura, E. Perez-Cabre, M. Í. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
[CrossRef]

Qin, W.

Quan, C.

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
[CrossRef]

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282(18), 3680–3685 (2009).
[CrossRef]

Refregier, P.

Shen, X. X.

Sheng, Y.

Sheridan, J. T.

Singh, K.

Singh, N.

N. Singh and A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009).
[CrossRef]

Sinha, A.

N. Singh and A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009).
[CrossRef]

Situ, G.

Sun, X.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Tan, Q.

Tay, C. J.

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
[CrossRef]

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282(18), 3680–3685 (2009).
[CrossRef]

Unnikrishnan, G.

Wang, Y. R.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[CrossRef]

X. C. Cheng, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, X. F. Xu, X. X. Shen, and G. Y. Dong, “Security enhancement of double-random phase encryption by amplitude modulation,” Opt. Lett. 33(14), 1575–1577 (2008).
[CrossRef] [PubMed]

Wei, H.

Xi, L.

Xiao-Feng, L.

Xin, Z.

Xu, L.

Xu, X. F.

Yamaguchi, I.

Yamamoto, K.

Yokota, M.

Yourassowsky, C.

Yu, B.

Yu, L.

Zhang, H.

Zhang, J.

Zhang, P.

Zhang, T.

Am. Math. Mon. (1)

F. J. Dyson and H. Falk, “Period of a discrete cat mapping,” Am. Math. Mon. 99(7), 603–614 (1992).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
[CrossRef]

J. Opt. (1)

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[CrossRef]

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

Opt. Commun. (1)

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282(18), 3680–3685 (2009).
[CrossRef]

Opt. Express (6)

Opt. Lasers Eng. (1)

N. Singh and A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009).
[CrossRef]

Opt. Lett. (13)

M. Antkowiak, N. Callens, C. Yourassowsky, and F. Dubois, “Extended focused imaging of a microparticle field with digital holographic microscopy,” Opt. Lett. 33(14), 1626–1628 (2008).
[CrossRef] [PubMed]

B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003).
[CrossRef] [PubMed]

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(13), 1644–1646 (2005).
[CrossRef] [PubMed]

X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31(8), 1044–1046 (2006).
[CrossRef] [PubMed]

X. Peng, H. Wei, and P. Zhang, “Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain,” Opt. Lett. 31(22), 3261–3263 (2006).
[CrossRef] [PubMed]

P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
[CrossRef] [PubMed]

O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999).
[CrossRef]

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29(14), 1584–1586 (2004).
[CrossRef] [PubMed]

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997).
[CrossRef] [PubMed]

G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000).
[CrossRef]

X. C. Cheng, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, X. F. Xu, X. X. Shen, and G. Y. Dong, “Security enhancement of double-random phase encryption by amplitude modulation,” Opt. Lett. 33(14), 1575–1577 (2008).
[CrossRef] [PubMed]

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(3), 331–333 (2009).
[CrossRef] [PubMed]

W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35(2), 118–120 (2010).
[CrossRef] [PubMed]

Phys. Today (1)

B. Javidi, “Securing information with optical technologies,” Phys. Today 50(3), 27–32 (1997).
[CrossRef]

Proc. IEEE (1)

O. Matoba, T. Nomura, E. Perez-Cabre, M. Í. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

U. Schnars, and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer, New York, 2005).

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

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Figures (5)

Fig. 1
Fig. 1

(a) The conventional schematic experimental setup for optical image encryption in the Fresnel domain: BSC, Beam splitter cube; (b) a schematic experimental setup for the proposed space-based optical image encryption; and (c) a schematic process for the axial translation of a particle: d '       and         d ' ' , new particle locations after the axial translation. More phase-only masks can be placed in the object wave path.

Fig. 2
Fig. 2

Flow charts of the proposed space-based optical (a) encryption and (b) decryption processes in the Fresnel domain: WP, wave propagation.

Fig. 3
Fig. 3

(a) An original input image (Lena); (b) a typical one of the recorded digital holograms; (c) a distance map d ( x , y ) for the plaintext; (d) a scrambled distance map after the ART; and (e) a decrypted image using correct security keys.

Fig. 4
Fig. 4

The decrypted images at one section when the distance d of (a) 10 mm, (b) 50 mm and (c) 100 mm is directly applied; (d) a decrypted image without an inverse ART of the scrambled distance map; decrypted images using (e) an error of 5 mm in the distance z, (f) an error of 5 nm in the wavelength and (g) a totally-wrong phase-only mask M2; decrypted images when the ciphertexts are contaminated by (h) random noise and (i) 25% occlusions.

Fig. 5
Fig. 5

Decrypted images at one section when a distance d of (a) 10 mm, (b) 50 mm and (c) 100 mm is directly applied; decrypted images using (d) a wrong distance z with an error of 5 mm, (e) a wrong wavelength with an error of 5 nm, and (f) a totally-wrong phase-only mask M2; decrypted images when the ciphertexts are contaminated by (g) random noise and (h) 25% occlusions; (i) a decrypted image using correct security keys. In this case, a binary image (“NUS OPTICS”) is studied.

Equations (7)

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O       ( μ , ν ) = WP z { [ ξ , η ( WP d ( x , y ) { O       ( x , y )         exp [ j     P ( x , y ) ] } ) ]       exp [ j     Q ( ξ ,     η ) ] }     ,
WP d ( x , y ) { O       ( x , y )         exp [ j     P ( x , y ) ] } = j λ + + O       ( x , y )         exp [ j     P ( x , y ) ]     exp ( j     k ρ ) ρ         d x     d y ,
I n ( μ , ν ) = O ( μ , ν )     O * ( μ , ν ) + R n ( μ , ν ) R n * ( μ , ν ) + R n * ( μ , ν ) O ( μ , ν ) + R n ( μ , ν )     O * ( μ , ν ) ,
O ( μ , ν ) = 1 j 4     R     {     I 1 ( μ , ν ) I 2 ( μ , ν ) + j       [ I 2 ( μ , ν ) I 3 ( μ , ν ) ]     } ,
ART     [ d ( x , y ) , N ] = { [ v , ( x ' , y ' ) ]           |           ( x ' , y ' )     T = W         ( x , y )     T     ( mod       N ) ,         [ v , ( x , y ) ]           d ( x , y ) }     ,
O '     ( x , y ) = | ( x , y ) [ WP IART [ d ( x ' , y ' ) ] ( { WP z [ O       ( μ , ν ) ] } { exp [ j     Q ( ξ ,     η ) ] } * ) ] |       ,
CC = x y ( O x , y O ¯ )     ( O x , y ' O ' ¯ )         [ x y ( O x , y O ¯ ) 2 ]       [ x y ( O x , y ' O ' ¯ ) 2 ]     ,

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