Abstract

In this paper, we propose a scheme for information security under the basic double random phase encoding framework but with enhanced complexity and immunity against the known-plaintext attack. Modified Gerchberg–Saxton algorithm is used to convert a primary image into a phase-only mask (POM). The POM is used as a Fresnel domain key for encrypting an arbitrary data, called random intensity mask (RIM) bonded with a random phase mask. With phase- and amplitude-truncation, asymmetric keys are generated from the RIM. For decryption, the main target is to get the POM, for which the concept of known-plaintext attack has been used. The conventional schemes for attack use known-plaintext for key generation, but in this study it refers to the asymmetric keys. Obtaining Fresnel transform with the same parameters of the POM gives the primary image. We present the computer simulation results of multiple gray-scale images without any cross talk and also for a color image. The decryption is simple and straightforward and can be done digitally or optically. An optical setup for decryption has been suggested.

© 2013 Optical Society of America

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  1. P. Refregier and B. Javidi, “Optical image encryption based on input plane encoding and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
    [CrossRef]
  2. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33, 2443–2445 (2008).
    [CrossRef]
  3. P. Clemente, V. Durán, V. T. Company, E. Tajahuerce, and J. Lancis, “Optical encryption based on computational ghost imaging,” Opt. Lett. 35, 2391–2393 (2010).
    [CrossRef]
  4. W. Chen and X. Chen, “Space-based optical image encryption,” Opt. Express 18, 27095–27104 (2010).
    [CrossRef]
  5. W. Chen, X. Chen, and J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35, 3817–3819 (2010).
    [CrossRef]
  6. A. Alfalou, and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35, 2185–2187 (2010).
    [CrossRef]
  7. A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Opt. Photonics News 1, 589–636 (2009).
    [CrossRef]
  8. N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
    [CrossRef]
  9. B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28, 269–271 (2003).
    [CrossRef]
  10. G. Situ and J. Zhang, “Double random phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
    [CrossRef]
  11. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
    [CrossRef]
  12. A. Carnicer, M. M. Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of the optical encryption schemes based on double random phase keys,” Opt. Lett. 30, 1644–1646 (2005).
    [CrossRef]
  13. 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]
  14. X. Peng, H. Wei, and P. Zhang, “Chosen-plaintext attack on lensless double random phase encoding in Fresnel domain,” Opt. Lett. 31, 3261–3263 (2006).
    [CrossRef]
  15. 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. 11, 075402 (2009).
    [CrossRef]
  16. Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
    [CrossRef]
  17. P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double random-phase encryption scheme with randomized lens-phase function,” Opt. Lett. 34, 331–333 (2009).
    [CrossRef]
  18. 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, 1575–1577 (2008).
    [CrossRef]
  19. P. Kumar, J. Joseph, and K. Singh, “Vulnerability of the security enhanced double random phase-amplitude encryption scheme to point spread function attack,” Opt. Lasers Eng. 50, 1196–1201 (2012).
    [CrossRef]
  20. W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35, 118–120(2010).
    [CrossRef]
  21. S. K. Rajput and N. K. Nishchal, “Image encryption based on interference that uses fractional Fourier domains asymmetric keys,” Appl. Opt. 51, 1446–1452 (2012).
    [CrossRef]
  22. 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]
  23. X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
    [CrossRef]
  24. G. Situ and J. Zhang, “Multiple image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1308 (2005).
    [CrossRef]
  25. J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption–decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
    [CrossRef]
  26. 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, 3917–3919 (2009).
    [CrossRef]
  27. H. T. Chang, H.-E. Wang, and C.-L. Lee, “Position multiplexing multiple image encryption using cascaded phase only masks in Fresnel transform domain,” Opt. Commun. 284, 4146–4151 (2011).
    [CrossRef]
  28. X. Deng and D. Zhao, “Multiple-image encryption using phase retrieval algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
    [CrossRef]
  29. A. Alfalou and A. Mansour, “Double random phase encryption scheme to multiplex and simultaneous encode multiple image,” Appl. Opt. 48, 5933–5947 (2009).
    [CrossRef]
  30. 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]
  31. A. E. Shortt, T. J. Naughton, and B. Javidi, “Compression of optically encrypted digital holograms using artificial neural networks,” J. Disp. Technol. 2, 401–410 (2006).
    [CrossRef]
  32. L. Chen and D. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14, 8552–8560 (2006).
    [CrossRef]
  33. M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
    [CrossRef]
  34. 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]
  35. H.-E. Hwang, “Optical color image encryption based on the wavelength multiplexing using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 285, 567–573 (2012).
    [CrossRef]
  36. S. K. Rajput and N. K. Nishchal, “Asymmetric color cryptosystem using polarization selective diffractive optical element and structured phase mask,” Appl. Opt. 51, 5377–5386(2012).
    [CrossRef]
  37. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg–Saxton algorithm applied in fractional Fourier or Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [CrossRef]

2012

P. Kumar, J. Joseph, and K. Singh, “Vulnerability of the security enhanced double random phase-amplitude encryption scheme to point spread function attack,” Opt. Lasers Eng. 50, 1196–1201 (2012).
[CrossRef]

S. K. Rajput and N. K. Nishchal, “Image encryption based on interference that uses fractional Fourier domains 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 Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieval algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
[CrossRef]

H.-E. Hwang, “Optical color image encryption based on the wavelength multiplexing using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 285, 567–573 (2012).
[CrossRef]

S. K. Rajput and N. K. Nishchal, “Asymmetric color cryptosystem using polarization selective diffractive optical element and structured phase mask,” Appl. Opt. 51, 5377–5386(2012).
[CrossRef]

2011

H. T. Chang, H.-E. Wang, and C.-L. Lee, “Position multiplexing multiple image encryption using cascaded phase only masks in Fresnel transform domain,” Opt. Commun. 284, 4146–4151 (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]

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
[CrossRef]

2010

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

W. Chen and X. Chen, “Space-based optical image encryption,” Opt. Express 18, 27095–27104 (2010).
[CrossRef]

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

A. Alfalou, and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35, 2185–2187 (2010).
[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, “Exploiting root-mean-square time-frequency structure for multiple-image optical compression and encryption,” Opt. Lett. 35, 1914–1916 (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]

2009

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, 3917–3919 (2009).
[CrossRef]

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

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

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. 11, 075402 (2009).
[CrossRef]

P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double random-phase encryption scheme with randomized lens-phase function,” Opt. Lett. 34, 331–333 (2009).
[CrossRef]

2008

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, 1575–1577 (2008).
[CrossRef]

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

2007

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (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]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

2006

A. E. Shortt, T. J. Naughton, and B. Javidi, “Compression of optically encrypted digital holograms using artificial neural networks,” J. Disp. Technol. 2, 401–410 (2006).
[CrossRef]

L. Chen and D. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14, 8552–8560 (2006).
[CrossRef]

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption–decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[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 Fresnel domain,” Opt. Lett. 31, 3261–3263 (2006).
[CrossRef]

2005

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

G. Situ and J. Zhang, “Multiple image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1308 (2005).
[CrossRef]

2004

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

2003

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

1996

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg–Saxton algorithm applied in fractional Fourier or Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef]

1995

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

Alfalou, A.

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

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

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

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.

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

Barrera, J. F.

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption–decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Bolognini, N.

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption–decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Brosseau, C.

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]

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,” Opt. Photonics News 1, 589–636 (2009).
[CrossRef]

Cai, L. Z.

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, 1575–1577 (2008).
[CrossRef]

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.

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

Castro, A.

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
[CrossRef]

Chang, H. T.

H. T. Chang, H.-E. Wang, and C.-L. Lee, “Position multiplexing multiple image encryption using cascaded phase only masks in Fresnel transform domain,” Opt. Commun. 284, 4146–4151 (2011).
[CrossRef]

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, 3917–3919 (2009).
[CrossRef]

Chen, L.

L. Chen and D. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14, 8552–8560 (2006).
[CrossRef]

Chen, W.

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 and X. Chen, “Space-based optical image encryption,” Opt. Express 18, 27095–27104 (2010).
[CrossRef]

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

Chen, X.

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

W. Chen and X. Chen, “Space-based optical image encryption,” Opt. Express 18, 27095–27104 (2010).
[CrossRef]

Cheng, X. C.

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, 1575–1577 (2008).
[CrossRef]

Clemente, P.

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

Company, V. T.

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

Deng, X.

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieval algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
[CrossRef]

Dong, G. Y.

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, 1575–1577 (2008).
[CrossRef]

Dorsch, R. G.

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg–Saxton algorithm applied in fractional Fourier or Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef]

Durán, V.

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

Frauel, Y.

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
[CrossRef]

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]

Henao, R.

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption–decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Hennelly, B.

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

Hwang, H.-E.

H.-E. Hwang, “Optical color image encryption based on the wavelength multiplexing using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 285, 567–573 (2012).
[CrossRef]

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, 3917–3919 (2009).
[CrossRef]

Javidi, B.

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
[CrossRef]

A. E. Shortt, T. J. Naughton, and B. Javidi, “Compression of optically encrypted digital holograms using artificial neural networks,” J. Disp. Technol. 2, 401–410 (2006).
[CrossRef]

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

Joseph, J.

P. Kumar, J. Joseph, and K. Singh, “Vulnerability of the security enhanced double random phase-amplitude encryption scheme to point spread function attack,” Opt. Lasers Eng. 50, 1196–1201 (2012).
[CrossRef]

P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double random-phase encryption scheme with randomized lens-phase function,” Opt. Lett. 34, 331–333 (2009).
[CrossRef]

Joshi, M.

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

Juvells, I.

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

Kumar, A.

P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double random-phase encryption scheme with randomized lens-phase function,” Opt. Lett. 34, 331–333 (2009).
[CrossRef]

Kumar, P.

P. Kumar, J. Joseph, and K. Singh, “Vulnerability of the security enhanced double random phase-amplitude encryption scheme to point spread function attack,” Opt. Lasers Eng. 50, 1196–1201 (2012).
[CrossRef]

P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double random-phase encryption scheme with randomized lens-phase function,” Opt. Lett. 34, 331–333 (2009).
[CrossRef]

Lancis, J.

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

Lee, C.-L.

H. T. Chang, H.-E. Wang, and C.-L. Lee, “Position multiplexing multiple image encryption using cascaded phase only masks in Fresnel transform domain,” Opt. Commun. 284, 4146–4151 (2011).
[CrossRef]

Lie, W. N.

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, 3917–3919 (2009).
[CrossRef]

Mansour, A.

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

Mendlovic, D.

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg–Saxton algorithm applied in fractional Fourier or Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef]

Meng, X. F.

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, 1575–1577 (2008).
[CrossRef]

Naughton, T. J.

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
[CrossRef]

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
[CrossRef]

A. E. Shortt, T. J. Naughton, and B. Javidi, “Compression of optically encrypted digital holograms using artificial neural networks,” J. Disp. Technol. 2, 401–410 (2006).
[CrossRef]

Nishchal, N. K.

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

S. K. Rajput and N. K. Nishchal, “Asymmetric color cryptosystem using polarization selective diffractive optical element and structured phase mask,” Appl. Opt. 51, 5377–5386(2012).
[CrossRef]

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
[CrossRef]

Peng, X.

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

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. 11, 075402 (2009).
[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 Fresnel domain,” Opt. Lett. 31, 3261–3263 (2006).
[CrossRef]

Qin, W.

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

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. 11, 075402 (2009).
[CrossRef]

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]

Rajput, S. K.

S. K. Rajput and N. K. Nishchal, “Asymmetric color cryptosystem using polarization selective diffractive optical element and structured phase mask,” Appl. Opt. 51, 5377–5386(2012).
[CrossRef]

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

Refregier, P.

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

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]

Shakher, C.

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

Shen, X. X.

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, 1575–1577 (2008).
[CrossRef]

Sheppard, J. R.

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

Sheridan, J. T.

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

Shortt, A. E.

A. E. Shortt, T. J. Naughton, and B. Javidi, “Compression of optically encrypted digital holograms using artificial neural networks,” J. Disp. Technol. 2, 401–410 (2006).
[CrossRef]

Singh, K.

P. Kumar, J. Joseph, and K. Singh, “Vulnerability of the security enhanced double random phase-amplitude encryption scheme to point spread function attack,” Opt. Lasers Eng. 50, 1196–1201 (2012).
[CrossRef]

P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double random-phase encryption scheme with randomized lens-phase function,” Opt. Lett. 34, 331–333 (2009).
[CrossRef]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

Situ, G.

G. Situ and J. Zhang, “Multiple image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1308 (2005).
[CrossRef]

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

Tajahuerce, E.

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

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]

Tebaldi, M.

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption–decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Torroba, R.

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption–decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

Usategui, M. M.

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

Wang, B.

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

Wang, H.-E.

H. T. Chang, H.-E. Wang, and C.-L. Lee, “Position multiplexing multiple image encryption using cascaded phase only masks in Fresnel transform domain,” Opt. Commun. 284, 4146–4151 (2011).
[CrossRef]

Wang, X.

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

Wang, Y. R.

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, 1575–1577 (2008).
[CrossRef]

Wei, H.

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

Xu, X. F.

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, 1575–1577 (2008).
[CrossRef]

Yu, B.

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]

Zalevsky, Z.

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg–Saxton algorithm applied in fractional Fourier or Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef]

Zhang, H.

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, 1575–1577 (2008).
[CrossRef]

Zhang, J.

G. Situ and J. Zhang, “Multiple image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1308 (2005).
[CrossRef]

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

Zhang, P.

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 Fresnel domain,” Opt. Lett. 31, 3261–3263 (2006).
[CrossRef]

Zhang, Y.

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

Zhao, D.

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieval algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
[CrossRef]

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

L. Chen and D. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14, 8552–8560 (2006).
[CrossRef]

Appl. Opt.

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

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

S. K. Rajput and N. K. Nishchal, “Asymmetric color cryptosystem using polarization selective diffractive optical element and structured phase mask,” Appl. Opt. 51, 5377–5386(2012).
[CrossRef]

J. Disp. Technol.

A. E. Shortt, T. J. Naughton, and B. Javidi, “Compression of optically encrypted digital holograms using artificial neural networks,” J. Disp. Technol. 2, 401–410 (2006).
[CrossRef]

J. Opt.

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. 11, 075402 (2009).
[CrossRef]

Opt. Commun.

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

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
[CrossRef]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

H. T. Chang, H.-E. Wang, and C.-L. Lee, “Position multiplexing multiple image encryption using cascaded phase only masks in Fresnel transform domain,” Opt. Commun. 284, 4146–4151 (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]

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

J. F. Barrera, R. Henao, R. Torroba, M. Tebaldi, and N. Bolognini, “Multiplexing encryption–decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006).
[CrossRef]

H.-E. Hwang, “Optical color image encryption based on the wavelength multiplexing using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 285, 567–573 (2012).
[CrossRef]

Opt. Express

L. Chen and D. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14, 8552–8560 (2006).
[CrossRef]

W. Chen and X. Chen, “Space-based optical image encryption,” Opt. Express 18, 27095–27104 (2010).
[CrossRef]

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
[CrossRef]

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]

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieval algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
[CrossRef]

Opt. Lasers Eng.

P. Kumar, J. Joseph, and K. Singh, “Vulnerability of the security enhanced double random phase-amplitude encryption scheme to point spread function attack,” Opt. Lasers Eng. 50, 1196–1201 (2012).
[CrossRef]

Opt. Lett.

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

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, 3917–3919 (2009).
[CrossRef]

G. Situ and J. Zhang, “Multiple image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1308 (2005).
[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. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double random-phase encryption scheme with randomized lens-phase function,” Opt. Lett. 34, 331–333 (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, 1575–1577 (2008).
[CrossRef]

A. Carnicer, M. M. Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of the 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 Fresnel domain,” Opt. Lett. 31, 3261–3263 (2006).
[CrossRef]

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

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

P. Refregier and B. Javidi, “Optical image encryption based on input plane encoding 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]

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

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg–Saxton algorithm applied in fractional Fourier or Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef]

Opt. Photonics News

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

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Figures (12)

Fig. 1.
Fig. 1.

Block diagram of modified G-S phase retrieval algorithm in Fresnel transform domain.

Fig. 2.
Fig. 2.

(a) Schematic diagram of encryption, (b) block diagram representing asymmetric cryptosystem, and (c) schematic diagram of proposed optical setup for decryption. AT, amplitude truncation; PT, phase truncation, SLM, spatial light modulator; CCD, charge-coupled device camera; PC, personal computer; z=free space propagation distance.

Fig. 3.
Fig. 3.

Simulation results. (a) Primary image to be encrypted, (b) generated POM, and (c) plot between number of iterations and MSE values for generating POM.

Fig. 4.
Fig. 4.

Simulation results. (a) Image of RIM, (b) phase-truncated value of RIM, and (c) intensity of encrypted image.

Fig. 5.
Fig. 5.

Simulation results. (a) Decrypted image obtained after using all correct keys, (b) decrypted image obtained after using wrong asymmetric key, (c) decrypted image obtained after using wrong RIM, (d) decrypted image obtained after using different Fresnel propagation distance, and (e) decrypted image obtained after using wrong wavelength.

Fig. 6.
Fig. 6.

Simulation results for known plaintext attack. (a) Decrypted image obtained when RIM is known and RPM and POM are retrieved using modified G-S algorithm and (b) decrypted image obtained when POM is known and RIM and RPM are generated.

Fig. 7.
Fig. 7.

Multiple gray-scale images used for encryption. (a) Lena, (b) Barbara, (c) Cameraman, and (d) Baboon.

Fig. 8.
Fig. 8.

Simulation results. (a) Multiplexed POM and (b) encrypted image.

Fig. 9.
Fig. 9.

Decrypted images obtained after using all correct keys. (a) Lena, (b) Barbara, (c) Cameraman, and (d) Baboon.

Fig. 10.
Fig. 10.

Decrypted images obtained after using wrong asymmetric keys. (a) Lena, (b) Barbara, (c) Cameraman, and (d) Baboon.

Fig. 11.
Fig. 11.

Simulation results for color image. (a) Color image, (b) red component, (c) green component, (d) blue component, (e) phase-truncated image, and (f) encrypted image.

Fig. 12.
Fig. 12.

(a) Decrypted color image obtained after using all correct keys and (b) decrypted color image obtained after using wrong asymmetric keys.

Equations (22)

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

fn(x,y)=|f(x,y)|×exp{i2πrn(x,y)}.
Fn+1(u,v)=FrTλz[fn(x,y)]=exp{i2πzλ}iλzfn(x,y)×exp[iπλz((xu)2+(yv)2)]dxdy=|Fn+1(u,v)|×exp{iφn(u,v)},
Fn+1(u,v)=1×exp{iφn(u,v)}.
Fn+1(x,y)=FrTλz[Fn+1(u,v)]=|Fn+1(x,y)|×exp{iφn(x,y)}.
fn+1(x,y)=|f(x,y)|×exp(iφn(u,v)=|f(x,y)|×exp{irn+1(u,v)}.
MSE=x=0N1y=0N1{|Fn+1(x,y)||f(x,y)|}2N×N.
E1(u,v)=FrTλz1[e(x,y)×exp{i2πR(x,y)}].
E(x,y)=FrTλz2[E1(u,v)×exp{iφn(u,v)}].
e1(u,v)=PT{FrTλz3[e(x,y)×exp{i2πR1(x,y)}]},
p1(u,v)=AT{FrTλz3[e(x,y)×exp{i2πR1(x,y)}]},
e(x,y)=PT{FrTλz3[e1(u,v)×p1(u,v)]}.
exp{iφn(u,v)}=FrTλz2[E(x,y)]FrTλz1[|e(x,y)|×exp{i2πR(x,y)}].
f(x,y)=FrTλz[exp{iφn(u,v)}].
exp{iψ(u,v)}=Πi=1kexp{iφnk(u,v)}.
q1(u,v)=exp{iψ(u,v)}Πi=2i1kexp{iφnk(u,v)},
q2(u,v)=exp{iψ(u,v)}Πi=1i2kexp{iφnk(u,v)}.
qm(u,v)=exp{iψ(u,v)}Πi=1imkexp{iφnk(u,v)}.
fm(x,y)=FrTλz[exp{iψ(u,v)}×qm(u,v)].
E1r(u,v)=FrTλz1[er(x,y)×exp{i2πRr(x,y)}],
Er(x,y)=FrTλz2[E1r(u,v)×exp{iφnr(u,v)}].
fr(x,y)=FrTλz[exp{iφnr(u,v)}].
MSErgb=MSEr+MSEg+MSEb3.

Metrics