Abstract

We propose a multiple-image encryption scheme, based on polarized light encoding and the interference principle of phase-only masks (POMs), in the Fresnel-transform (FrT) domain. In this scheme, each secret image is converted into an intensity image by polarized light encoding, where a random key image and a pixilated polarizer with random angles are employed as keys. The intensity encrypted images produced by different secret images are convolved together and then inverse Fresnel-transformed. Phase and amplitude truncations are used to generate the asymmetric decryption keys. The phase-truncated inverse FrT spectrum is sent into an interference-based encryption (IBE) system to analytically obtain two POMs. To reduce the transmission and storage load on the keys, the chaotic mapping method is employed to generate random distributions of keys for encryption and decryption. One can recover all secret images successfully only if the corresponding decryption keys, the mechanism of FrTs, and correct chaotic conditions are known. The inherent silhouette problem can be thoroughly resolved by polarized light encoding in this proposal, without using any time-consuming iterative methods. The entire encryption and decryption process can be realized digitally, or in combination with optical means. Numerical simulation results are presented to verify the effectiveness and performance of the proposed scheme.

© 2013 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 and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
    [CrossRef]
  2. A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photon. 1, 589–636 (2009).
    [CrossRef]
  3. 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]
  4. S. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001).
    [CrossRef]
  5. Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
    [CrossRef]
  6. G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
    [CrossRef]
  7. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
    [CrossRef]
  8. 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]
  9. A. Alfalou and A. Mansour, “Double random phase encryption scheme to multiplex and simultaneous encode multiple images,” Appl. Opt. 48, 5933–5947 (2009).
    [CrossRef]
  10. 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]
  11. 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]
  12. 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]
  13. W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35, 118–120 (2010).
    [CrossRef]
  14. 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]
  15. 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]
  16. S. K. Rajput and N. K. Nishchal, “Image encryption based on interference that uses fractional Fourier domain asymmetric keys,” Appl. Opt. 51, 1446–1452 (2012).
    [CrossRef]
  17. X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated fractional Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
    [CrossRef]
  18. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33, 2443–2445 (2008).
    [CrossRef]
  19. Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A 11, 125406 (2009).
    [CrossRef]
  20. P. Kumar, J. Joseph, and K. Singh, “Optical image encryption based on interference under convergent random illumination,” J. Opt. 12, 095402 (2010).
    [CrossRef]
  21. 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]
  22. X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51, 686–691 (2012).
    [CrossRef]
  23. Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285, 4294–4301 (2012).
    [CrossRef]
  24. Q. Wang, Q. Guo, L. Lei, and J. Zhou, “Multiple-image encryption based on interference principle and phase-only mask multiplexing in Fresnel transform domain,” Appl. Opt. 52, 6849–6857 (2013).
    [CrossRef]
  25. P. C. Mogensen and J. Gluckstad, “A phase-based optical encryption system with polarization encoding,” Opt. Commun. 173, 177–183 (2000).
    [CrossRef]
  26. J. A. Davis, D. E. McNamara, D. M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. 39, 1549–1554 (2000).
    [CrossRef]
  27. B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39, 2439–2443 (2000).
    [CrossRef]
  28. U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Polarization encoding and multiplexing of two-dimensional signals: application to image encryption,” Appl. Opt. 45, 5693–5700 (2006).
    [CrossRef]
  29. A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35, 2185–2187 (2010).
    [CrossRef]
  30. M. Dubreuil, A. Aflalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using Stokes–Mueller formalism,” J. Opt. 14, 094004 (2012).
    [CrossRef]
  31. S. K. Rajput and N. K. Nishchal, “Image encryption using polarized light encoding and amplitude and phase truncation in the Fresnel domain,” Appl. Opt. 52, 4343–4352 (2013).
    [CrossRef]
  32. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
  33. D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Signal Process. Lett. 16, 853–856 (2009).
    [CrossRef]

2013 (2)

2012 (5)

M. Dubreuil, A. Aflalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using Stokes–Mueller formalism,” J. Opt. 14, 094004 (2012).
[CrossRef]

X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51, 686–691 (2012).
[CrossRef]

Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285, 4294–4301 (2012).
[CrossRef]

S. K. Rajput and N. K. Nishchal, “Image encryption based on interference that uses fractional Fourier domain asymmetric keys,” Appl. Opt. 51, 1446–1452 (2012).
[CrossRef]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated fractional Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

2011 (3)

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]

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]

2010 (3)

2009 (4)

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Signal Process. Lett. 16, 853–856 (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]

A. Alfalou and A. Mansour, “Double random phase encryption scheme to multiplex and simultaneous encode multiple images,” Appl. Opt. 48, 5933–5947 (2009).
[CrossRef]

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

2008 (1)

2007 (2)

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
[CrossRef]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

2006 (3)

2005 (1)

2004 (1)

2002 (1)

2001 (1)

2000 (4)

1995 (1)

Aflalou, A.

M. Dubreuil, A. Aflalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using Stokes–Mueller formalism,” J. Opt. 14, 094004 (2012).
[CrossRef]

Alfalou, A.

Alieva, T.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

Arcos, S.

Brosseau, C.

M. Dubreuil, A. Aflalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using Stokes–Mueller formalism,” J. Opt. 14, 094004 (2012).
[CrossRef]

A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35, 2185–2187 (2010).
[CrossRef]

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

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Calvo, M. L.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

Carnicer, A.

Chang, H.

Cottrell, D. M.

Davis, J. A.

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]

Dubreuil, M.

M. Dubreuil, A. Aflalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using Stokes–Mueller formalism,” J. Opt. 14, 094004 (2012).
[CrossRef]

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]

Gluckstad, J.

P. C. Mogensen and J. Gluckstad, “A phase-based optical encryption system with polarization encoding,” Opt. Commun. 173, 177–183 (2000).
[CrossRef]

Gopinathan, U.

Guo, Q.

Javidi, B.

B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39, 2439–2443 (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]

Joseph, J.

Juvells, I.

Kumar, P.

Kuo, C.

Lei, L.

Li, Y.

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

Liu, S.

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
[CrossRef]

S. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001).
[CrossRef]

Liu, Z.

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
[CrossRef]

Lu, W.

Ma, J.

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

Mansour, A.

McNamara, D. E.

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]

Mi, Q.

Mogensen, P. C.

P. C. Mogensen and J. Gluckstad, “A phase-based optical encryption system with polarization encoding,” Opt. Commun. 173, 177–183 (2000).
[CrossRef]

Montes-Usategui, M.

Naughton, T. J.

Nishchal, N. K.

Nomura, T.

B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39, 2439–2443 (2000).
[CrossRef]

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]

Rajput, S. K.

Ran, Q.

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

Refregier, P.

Rodrigo, J. A.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

Sheridan, J. T.

Singh, K.

Situ, G.

Sonehara, T.

Tan, L.

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

Unnikrishnan, G.

Wang, B.

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, Q.

Q. Wang, Q. Guo, L. Lei, and J. Zhou, “Multiple-image encryption based on interference principle and phase-only mask multiplexing in Fresnel transform domain,” Appl. Opt. 52, 6849–6857 (2013).
[CrossRef]

Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285, 4294–4301 (2012).
[CrossRef]

Wang, X.

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated fractional Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51, 686–691 (2012).
[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]

Wei, D.

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

Wei, H.

Yu, B.

Zhang, J.

Zhang, P.

Zhang, Y.

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, D.

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated fractional Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51, 686–691 (2012).
[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, J.

Zhu, B.

Adv. Opt. Photon. (1)

Appl. Opt. (9)

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]

A. Alfalou and A. Mansour, “Double random phase encryption scheme to multiplex and simultaneous encode multiple images,” Appl. Opt. 48, 5933–5947 (2009).
[CrossRef]

S. K. Rajput and N. K. Nishchal, “Image encryption based on interference that uses fractional Fourier domain asymmetric keys,” Appl. Opt. 51, 1446–1452 (2012).
[CrossRef]

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]

X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51, 686–691 (2012).
[CrossRef]

Q. Wang, Q. Guo, L. Lei, and J. Zhou, “Multiple-image encryption based on interference principle and phase-only mask multiplexing in Fresnel transform domain,” Appl. Opt. 52, 6849–6857 (2013).
[CrossRef]

J. A. Davis, D. E. McNamara, D. M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. 39, 1549–1554 (2000).
[CrossRef]

U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Polarization encoding and multiplexing of two-dimensional signals: application to image encryption,” Appl. Opt. 45, 5693–5700 (2006).
[CrossRef]

S. K. Rajput and N. K. Nishchal, “Image encryption using polarized light encoding and amplitude and phase truncation in the Fresnel domain,” Appl. Opt. 52, 4343–4352 (2013).
[CrossRef]

IEEE Signal Process. Lett. (1)

D. Wei, Q. Ran, Y. Li, J. Ma, and L. Tan, “A convolution and product theorem for the linear canonical transform,” IEEE Signal Process. Lett. 16, 853–856 (2009).
[CrossRef]

J. Opt. (2)

M. Dubreuil, A. Aflalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using Stokes–Mueller formalism,” J. Opt. 14, 094004 (2012).
[CrossRef]

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 (1)

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. (6)

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated fractional Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[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]

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
[CrossRef]

P. C. Mogensen and J. Gluckstad, “A phase-based optical encryption system with polarization encoding,” Opt. Commun. 173, 177–183 (2000).
[CrossRef]

Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285, 4294–4301 (2012).
[CrossRef]

Opt. Eng. (2)

B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39, 2439–2443 (2000).
[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]

Opt. Lett. (10)

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

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

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

S. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001).
[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]

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

A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35, 2185–2187 (2010).
[CrossRef]

Other (1)

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

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Figures (9)

Fig. 1.
Fig. 1.

Optical implementation setup for polarized light encoding.

Fig. 2.
Fig. 2.

Optoelectronic setup for decryption.

Fig. 3.
Fig. 3.

Secret images: (a) Lena, (b) Baboon, (c) Cameraman, and (d) Livingroom.

Fig. 4.
Fig. 4.

(a) Initial RDF of PWLCM, (b) the random key image, and (c) the angle distribution of pixilated polarizer. Encrypted image of polarized light encoding for (d) Lena, (e) Baboon, (f) Cameraman, and (g) Livingroom. (h) Synthetic intensity image G(u,v).

Fig. 5.
Fig. 5.

Phase distributions of (a) R1 and (b) R2. (c) Amplitude and (d) phase distributions of decryptions key for Lena.

Fig. 6.
Fig. 6.

(a)–(d) Decrypted images obtained with all decryption keys and Fresnel propagation parameters correctly applied. (e)–(h) Decrypted results obtained using only the first POM R1. (i) Recovered image obtained using the encryption key directly for decryption. The data shown within the figures are the corresponding RE values.

Fig. 7.
Fig. 7.

(a)–(d) Decrypted images obtained using wrong wavelengths. (e)–(h) Decrypted results obtained with wrong propagation distance z1. (i)–(l) Decrypted results obtained with wrong propagation distance z2. Data shown within the figures are the corresponding RE values.

Fig. 8.
Fig. 8.

RE curves for Lena image on the variations of (a) wavelength and propagation distances, (b) z1, and (c) z2.

Fig. 9.
Fig. 9.

(a)–(d) Decrypted images obtained with wrong initial RDF used to generate keys for decryption. (e)–(h) Decrypted results obtained using incorrect control parameters to generate keys for decryption. (i)–(l) Decrypted results obtained by choosing the RDF with wrong serial number to form the random key image. (m)–(p) Decrypted results obtained by choosing the RDF with wrong serial number to form the encryption random angles ψrand(u,v). Data shown within the figures are the corresponding RE values.

Equations (27)

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

Sfi(u,v)=[Sfi0(u,v)000]T,
Sk(u,v)=[Sk0(u,v)000]T,
MRet(θ,ϕ)=(10000cos2(2θ)+cos(ϕ)sin2(2θ)12sin(4θ)12sin(4θ)cos(ϕ)sin(2θ)sin(ϕ)012sin(4θ)12sin(4θ)cos(ϕ)sin2(2θ)+cos(ϕ)cos2(2θ)cos(2θ)sin(ϕ)0sin(2θ)sin(ϕ)cos(2θ)sin(ϕ)cos(ϕ)),
MPol(ψ)=12(1cos(2ψ)sin(2ψ)0cos(2ψ)cos2(2ψ)12sin(4ψ)0sin(2ψ)12sin(4ψ)sin(2ψ)00000).
Sfi(u,v)=MRet1(θ,ϕ)MPol1(ψ)Sfi(u,v)=(α1Sfi0(u,v)β1Sfi0(u,v)γ1Sfi0(u,v)δ1Sfi0(u,v))T,
Sk(u,v)=MRet2(θ,ϕ)MPol2(ψ)Sk(u,v)=(α2Sk0(u,v)β2Sk0(u,v)γ2Sk0(u,v)δ2Sk0(u,v))T,
α=1/2,
β=1/2{cos(2θ)cos[2(θψ)]+cosϕsin(2θ)sin[2(θψ)]},
γ=1/2{sin(2θ)cos[2(θψ)]cosϕcos(2θ)sin[2(θψ)]},
δ=1/2sinϕsin[2(θψ)].
SRi(u,v)=(α1Sfi0(u,v)+α2Sk0(u,v)β1Sfi0(u,v)+β2Sk0(u,v)γ1Sfi0(u,v)+γ2Sk0(u,v)δ1Sfi0(u,v)+δ2Sk0(u,v)).
SCi(u,v)=[SCi0(u,v)SCi1(u,v)SCi2(u,v)0]T,
SCi0(u,v)=12[(α1Sfi0(u,v)+α2Sk0(u,v))+(β1Sfi0(u,v)+β2Sk0(u,v))cos(2ψrand(u,v))+(γ1Sfi0(u,v)+γ2Sk0(u,v))sin(2ψrand(u,v))],
G(u,v)=E1(u,v)E2(u,v)Ei(u,v),
g(ξ,η)=Fλ,z11{G(u,v)}=i=1Nei(ξ,η),
ec(ξ,η)=PT{g(ξ,η)}=i=1N|ei(ξ,η)|,
ec(ξ,η)=ec(ξ,η)exp[jφ(ξ,η)],
di(ξ,η)=AT{ei(ξ,η)}n=1,niN|en(ξ,η)|exp[jφ(ξ,η)],
M(x,y)=Fλ,z21{ec(ξ,η)}=R1+R2,
r1=arg{M(x,y)}arcos[|M(x,y)|/2],
r2=arg{M(x,y)R1(x,y)},
ζt+1(m,n)=Fp{ζt(m,n)}={ζt(m,n)/q0ζt(m,n)<q[ζt(m,n)q]/(0.5q)qζt(m,n)<0.5Fp{1ζt(m,n)}0.5ζt(m,n)<1,
Ei(u,v)=|Fλ,z1{Fλ,z2{R1+R2}di(ξ,η)}|2,
Sdfi(u,v)=12(1cos(2ψdec(u,v))sin(2ψdec(u,v))0cos(2ψdec(u,v))cos2(2ψdec(u,v))12sin(4ψdec(u,v))0sin(2ψdec(u,v))12sin(4ψdec(u,v))sin(2ψdec(u,v))00000)×(SCi0(u,v)000)=14[(α1Sfi0(u,v)+α2Sk0(u,v))+(β1Sfi0(u,v)+β2Sk0(u,v))cos(2ψrand(u,v))+(γ1Sfi0(u,v)+γ2Sk0(u,v))sin(2ψrand(u,v))]×(1cos(2ψdec(u,v))sin(2ψdec(u,v))0).
Sdfi(u,v)=18(Sfi0(u,v)+Sk0(u,v))[1+cos(2ψrand(u,v))]×(1cos(2ψdec(u,v))sin(2ψdec(u,v))0).
Sfi0(u,v)=2SCi0(u,v)Sk0(u,v)[α2+β2cos(2ψrand(u,v))+γ2sin(2ψrand(u,v))]α1+β1cos(2ψrand(u,v))+γ1sin(2ψrand(u,v)).
RE=m=1Mn=1N||I(m,n)||I(m,n)||2m=1Mn=1N|I(m,n)|2,

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