Abstract

We propose a new method using coherent diffractive imaging for optical color-image encryption and synthesis in the Fresnel domain. An optical multiple-random-phase-mask encryption system is applied, and a strategy based on lateral translations of a phase-only mask is employed during image encryption. For the decryption, an iterative phase retrieval algorithm is applied to extract high-quality decrypted color images from diffraction intensity maps (i.e., ciphertexts). In addition, optical color-image synthesis is also investigated based on coherent diffractive imaging. Numerical results are presented to demonstrate feasibility and effectiveness of the proposed method. Compared with conventional interference methods, coherent diffractive imaging approach may open up a new research perspective or can provide an effective alternative for optical color-image encryption and synthesis.

© 2012 OSA

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2011 (6)

2010 (4)

2009 (9)

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]

H. Li, “Image encryption based on gyrator transform and two-step phase-shifting interferometry,” Opt. Lasers Eng. 47(1), 45–50 (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] [PubMed]

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. A, Pure Appl. Opt. 11(7), 075402 (2009).
[CrossRef]

K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17(22), 19662–19673 (2009).
[CrossRef] [PubMed]

X. Yong-Liang, Z. Xin, W. Qiong-Hua, Y. Sheng, and C. Yao-Yao, “Optical image encryption topology,” Opt. Lett. 34(20), 3223–3225 (2009).
[CrossRef] [PubMed]

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(24), 3917–3919 (2009).
[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. 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]

2008 (4)

2007 (4)

L. Chen and D. Zhao, “Color information processing (coding and synthesis) with fractional Fourier transforms and digital holography,” Opt. Express 15(24), 16080–16089 (2007).
[CrossRef] [PubMed]

L. De Caro, C. Giannini, A. Cedola, D. Pelliccia, S. Lagomarsino, and W. Jark, “Phase retrieval in X-ray coherent Fresnel projection-geometry diffraction,” Appl. Phys. Lett. 90(4), 041105 (2007).
[CrossRef]

F. C. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75(4), 043805 (2007).
[CrossRef]

G. Situ, U. Gopinathan, D. S. Monaghan, and J. T. Sheridan, “Cryptanalysis of optical security systems with significant output images,” Appl. Opt. 46(22), 5257–5262 (2007).
[CrossRef] [PubMed]

2006 (7)

2005 (3)

2004 (2)

2003 (3)

L. Yu and L. Cai, “Multidimensional data encryption with digital holography,” Opt. Commun. 215(4-6), 271–284 (2003).
[CrossRef]

B. Zhu, H. Zhao, and S. Liu, “Image encryption based on pure intensity random coding and digital holography technique,” Optik (Stuttg.) 114(2), 95–99 (2003).
[CrossRef]

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
[CrossRef] [PubMed]

2002 (2)

2000 (3)

1999 (3)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[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] [PubMed]

S. Q. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” Microw. Opt. Technol. Lett. 21(5), 318–323 (1999).
[CrossRef]

1998 (1)

1997 (1)

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

1995 (1)

1982 (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 (Stuttg.) 35, 237–246 (1972).

Allen, L. J.

W. McBride, N. L. O’Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. A 61(3), 321–324 (2005).
[CrossRef] [PubMed]

Arcos, S.

Barrera, J. F.

Bolognini, N.

Bunk, O.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100(15), 155503 (2008).
[CrossRef] [PubMed]

Cai, L.

L. Yu and L. Cai, “Multidimensional data encryption with digital holography,” Opt. Commun. 215(4-6), 271–284 (2003).
[CrossRef]

X. Peng, L. Yu, and L. Cai, “Double-lock for image encryption with virtual optical wavelength,” Opt. Express 10(1), 41–45 (2002).
[PubMed]

Cai, L. Z.

Cao, L.

Carnicer, A.

Cedola, A.

L. De Caro, C. Giannini, A. Cedola, D. Pelliccia, S. Lagomarsino, and W. Jark, “Phase retrieval in X-ray coherent Fresnel projection-geometry diffraction,” Appl. Phys. Lett. 90(4), 041105 (2007).
[CrossRef]

Chang, H. T.

Chapman, H. N.

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[CrossRef]

Chen, L.

Chen, W.

Chen, X.

Cheng, X. C.

David, C.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100(15), 155503 (2008).
[CrossRef] [PubMed]

De Caro, L.

L. De Caro, C. Giannini, A. Cedola, D. Pelliccia, S. Lagomarsino, and W. Jark, “Phase retrieval in X-ray coherent Fresnel projection-geometry diffraction,” Appl. Phys. Lett. 90(4), 041105 (2007).
[CrossRef]

Dierolf, M.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100(15), 155503 (2008).
[CrossRef] [PubMed]

Dong, G. Y.

Dowling, T.

Fienup, J. R.

Gao, M.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
[CrossRef] [PubMed]

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 (Stuttg.) 35, 237–246 (1972).

Giannini, C.

L. De Caro, C. Giannini, A. Cedola, D. Pelliccia, S. Lagomarsino, and W. Jark, “Phase retrieval in X-ray coherent Fresnel projection-geometry diffraction,” Appl. Phys. Lett. 90(4), 041105 (2007).
[CrossRef]

Gopinathan, U.

Gray, J.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100(15), 155503 (2008).
[CrossRef] [PubMed]

Harder, R.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

He, M.

He, Q.

Hennelly, B. M.

Hwang, H. E.

Jark, W.

L. De Caro, C. Giannini, A. Cedola, D. Pelliccia, S. Lagomarsino, and W. Jark, “Phase retrieval in X-ray coherent Fresnel projection-geometry diffraction,” Appl. Phys. Lett. 90(4), 041105 (2007).
[CrossRef]

Javidi, B.

Jefimovs, K.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100(15), 155503 (2008).
[CrossRef] [PubMed]

Jin, G.

Jing, F.

Johnson, I.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100(15), 155503 (2008).
[CrossRef] [PubMed]

Joseph, J.

Juvells, I.

Karim, M. A.

S. Q. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” Microw. Opt. Technol. Lett. 21(5), 318–323 (1999).
[CrossRef]

Kim, D. H.

Kim, H.

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[CrossRef]

Kumar, A.

Kumar, P.

Kuo, C. J.

Lagomarsino, S.

L. De Caro, C. Giannini, A. Cedola, D. Pelliccia, S. Lagomarsino, and W. Jark, “Phase retrieval in X-ray coherent Fresnel projection-geometry diffraction,” Appl. Phys. Lett. 90(4), 041105 (2007).
[CrossRef]

Lai, S.

S. Lai and M. A. Neifeld, “Digital wavefront reconstruction and its application to image encryption,” Opt. Commun. 178(4-6), 283–289 (2000).
[CrossRef]

Lee, Y. H.

Li, H.

H. Li, “Image encryption based on gyrator transform and two-step phase-shifting interferometry,” Opt. Lasers Eng. 47(1), 45–50 (2009).
[CrossRef]

Lie, W. N.

Lin, C.

Liu, S.

Z. Liu, L. Xu, C. Lin, and S. Liu, “Image encryption by encoding with a nonuniform optical beam in gyrator transform domains,” Appl. Opt. 49(29), 5632–5637 (2010).
[CrossRef] [PubMed]

B. Zhu, H. Zhao, and S. Liu, “Image encryption based on pure intensity random coding and digital holography technique,” Optik (Stuttg.) 114(2), 95–99 (2003).
[CrossRef]

Liu, Z.

Lu, W. C.

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] [PubMed]

Matsushima, K.

McBride, W.

W. McBride, N. L. O’Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. A 61(3), 321–324 (2005).
[CrossRef] [PubMed]

Meng, X. F.

Miao, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[CrossRef]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15(6), 1662–1669 (1998).
[CrossRef]

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]

Monaghan, D. S.

Montes-Usategui, M.

Mosso, F.

Nagahara, L. A.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
[CrossRef] [PubMed]

Naughton, T. J.

Neifeld, M. A.

S. Lai and M. A. Neifeld, “Digital wavefront reconstruction and its application to image encryption,” Opt. Commun. 178(4-6), 283–289 (2000).
[CrossRef]

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]

Nugent, K. A.

W. McBride, N. L. O’Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. A 61(3), 321–324 (2005).
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D. Wang, J. Zhao, F. Zhang, G. Pedrini, and W. Osten, “High-fidelity numerical realization of multiple-step Fresnel propagation for the reconstruction of digital holograms,” Appl. Opt. 47(19), D12–D20 (2008).
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F. C. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75(4), 043805 (2007).
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L. De Caro, C. Giannini, A. Cedola, D. Pelliccia, S. Lagomarsino, and W. Jark, “Phase retrieval in X-ray coherent Fresnel projection-geometry diffraction,” Appl. Phys. Lett. 90(4), 041105 (2007).
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Peng, X.

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. A, Pure Appl. Opt. 11(7), 075402 (2009).
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X. Peng, L. Yu, and L. Cai, “Double-lock for image encryption with virtual optical wavelength,” Opt. Express 10(1), 41–45 (2002).
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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).
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Renker, D.

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Situ, G.

Tajahuerce, E.

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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).
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Torroba, R.

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M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
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Wang, X.

Wang, Y. R.

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Xu, L.

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L. Yu and L. Cai, “Multidimensional data encryption with digital holography,” Opt. Commun. 215(4-6), 271–284 (2003).
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Zhang, H.

Zhang, J.

Zhang, P.

Zhang, R.

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
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Zhao, H.

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Zhu, B.

B. Zhu, H. Zhao, and S. Liu, “Image encryption based on pure intensity random coding and digital holography technique,” Optik (Stuttg.) 114(2), 95–99 (2003).
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J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
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Acta Crystallogr. A (1)

W. McBride, N. L. O’Leary, K. A. Nugent, and L. J. Allen, “Astigmatic electron diffraction imaging: a novel mode for structure determination,” Acta Crystallogr. A 61(3), 321–324 (2005).
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Appl. Opt. (10)

W. Chen, X. Chen, and C. J. R. Sheppard, “Optical double-image cryptography based on diffractive imaging with a laterally-translated phase grating,” Appl. Opt. 50(29), 5750–5757 (2011).
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Appl. Phys. Lett. (1)

L. De Caro, C. Giannini, A. Cedola, D. Pelliccia, S. Lagomarsino, and W. Jark, “Phase retrieval in X-ray coherent Fresnel projection-geometry diffraction,” Appl. Phys. Lett. 90(4), 041105 (2007).
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W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13(7), 075404 (2011).
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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. A, Pure Appl. Opt. 11(7), 075402 (2009).
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J. Opt. Soc. Am. A (2)

Microw. Opt. Technol. Lett. (1)

S. Q. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” Microw. Opt. Technol. Lett. 21(5), 318–323 (1999).
[CrossRef]

Nature (2)

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
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J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
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S. Lai and M. A. Neifeld, “Digital wavefront reconstruction and its application to image encryption,” Opt. Commun. 178(4-6), 283–289 (2000).
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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).
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H. Li, “Image encryption based on gyrator transform and two-step phase-shifting interferometry,” Opt. Lasers Eng. 47(1), 45–50 (2009).
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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).
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Optik (Stuttg.) (2)

B. Zhu, H. Zhao, and S. Liu, “Image encryption based on pure intensity random coding and digital holography technique,” Optik (Stuttg.) 114(2), 95–99 (2003).
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Phys. Rev. A (1)

F. C. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75(4), 043805 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100(15), 155503 (2008).
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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).
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Science (1)

J. M. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300(5624), 1419–1421 (2003).
[CrossRef] [PubMed]

Other (4)

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

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

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Original color image or plaintext (Peppers): http://sipi.usc.edu/database .

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Figures (9)

Fig. 1
Fig. 1

Schematic optical setup for multiple-random-phase-mask encryption based on lateral translations of a phase-only mask [39]: M, the phase-only mask; CCD, charge-coupled device. Three phase-only masks are used to illustrate the proposed method, and the application with fewer or more random phase-only masks is straightforward.

Fig. 2
Fig. 2

Schematic coordinate system for angular spectrum algorithm.

Fig. 3
Fig. 3

Optical setup for multiple-random-phase-mask encryptions using three channels.

Fig. 4
Fig. 4

Flow chart for retrieval algorithm at each channel during image decryption.

Fig. 5
Fig. 5

(a) An input color image (i.e., plaintext); (b) original and (c) the first laterally-translated phase-only masks M3; typical diffraction intensity maps (i.e., ciphertexts) at the (d) red, (e) green and (f) bluechannels.

Fig. 6
Fig. 6

(a) A relationship between the number of iterations and CC values using correct security keys at three channels (red color, red channel; green color, green channel; blue color, blue channel); decrypted images at (b) red, (c) green and (d) blue channels; (e) a decrypted color image after the incorporation.

Fig. 7
Fig. 7

Decrypted images with (a) a distance error of 2 mm for z 3 at red channel; (b) a distance error of 2 mm for z 3 at all three channels; (c) a wavelength error of 10 nm for λ 1 ; (d) a wavelength error of 10 nm at all three channels; (e) a wrong phase-only mask M3 only at the red channel; (f) a wrong phase-only mask M3 at all three channels.

Fig. 8
Fig. 8

Information synthesis with addition operation: (a), (b) two input color images for addition operation; (c), (d) real and imaginary parts of the fused complex amplitude just before the M3 plane at red channel; (e), (f) relationships between iteration number and CC values using correct security keys for two input color images at three channels; (g) a fused color image decrypted at the output plane.

Fig. 9
Fig. 9

Information synthesis with subtraction operation: (a), (b) two input color images for subtraction operation; (c), (d) real and imaginary parts of the subtracted complex amplitude just before the M3 plane at red channel; (e), (f) relationships between the iteration number and CC values using correct security keys for two input color images at three channels; (g) the subtracted color image decrypted at the output plane.

Equations (11)

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O( x 1 , y 1 )= j λ + + O(ξ,η) M 1 (ξ,η) exp( jkρ ) ρ dξdη,
O( x 1 , y 1 )= j λ z 1 exp( j 2π λ z 1 )exp[ j π λ z 1 ( x 1 2 + y 1 2 ) ] × + + O(ξ,η) M 1 (ξ,η)exp[ j π λ z 1 ( ξ 2 + η 2 ) ] ×exp[ j 2π λ z 1 ( x 1 ξ+ y 1 η ) ]dξdη.
O( x 1 , y 1 )=F T 1 ( ( k ξ , k η ; z 1 ){ FT[ O(ξ,η) M 1 (ξ,η) ] } ),
( k ξ , k η ; z 1 )=exp( j z 1 k z )=exp( j z 1 k 2 k ξ 2 k η 2 ),
( f ξ , f η ; z 1 )={ exp{ jk z 1 [ 1 ( λ f ξ ) 2 ( λ f η ) 2 ] 1/2 }if ( f ξ 2 + f η 2 ) 1/2 <( 1/λ ) 0Otherwise ,
(m,n; z 1 )=exp( jk z 1 { 1 [ λ( m M 2 ) MΔξ ] 2 [ λ( n N 2 ) NΔη ] 2 } 1/2 ),
I(μ,ν)= | FW P z 3 { [ FW P z 2 ( { FW P z 1 [ O(ξ,η) M 1 (ξ,η) ] } M 2 ( x 1 , y 1 ) ) ] M 3 ( x 2 , y 2 ) } | 2 ,
O'(ξ,η)=| FW P z 1 ( { FW P z 2 [ O n 4 ( x 2 , y 2 ) ¯ ¯ ] } [ M 2 ( x 1 , y 1 ) ] * ) |.
CC= [ cov( O,O' ) ] / ( σ O × σ O' ) ,
O Add ' (ξ,η)=| FW P z 1 ( { FW P z 2 [ O n 4 ( x 2 , y 2 ) ¯ ¯ (1) + O n 4 ( x 2 , y 2 ) ¯ ¯ (2) ] } [ M 2 ( x 1 , y 1 ) ] * ) |,
O Sub ' (ξ,η)=| FW P z 1 ( { FW P z 2 [ O n 4 ( x 2 , y 2 ) ¯ ¯ (1) O n 4 ( x 2 , y 2 ) ¯ ¯ (2) ] } [ M 2 ( x 1 , y 1 ) ] * ) |,

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