Abstract

In this paper, we propose a new method for color image coding and synthesis based on fractional Fourier transforms and wavelength multiplexing with digital holography. A color image is divided into three channels and each channel, in which the information is encrypted with different wavelength, fractional orders and random phase masks, is independently encrypted or synthesized. The system parameters are additional keys and this method would improve the security of information encryption. The images are fused or subtracted by phase shifting technique. The possible optical implementations for color image encryption and synthesis are also proposed with some simulation results that show the possibility of the proposed idea.

© 2007 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] [PubMed]
  2. B. Javidi, “Security information with optical technology,” Phys. Today  50, 27–32 (1997).
  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. G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
    [Crossref] [PubMed]
  5. G. Unnikrishnan and K. Singh, “Double random fractional Fourier-domain encoding for optical security,” Opt. Eng. 39, 2853–2859 (2000).
    [Crossref]
  6. B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28, 269–271 (2003).
    [Crossref] [PubMed]
  7. E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, “Optoeletronic information encryption with phase-shifting interferometry,” Appl. Opt. 39, 2313–2320 (2000).
    [Crossref]
  8. S. T. Liu, Q. L. Mi, and B. H. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001).
    [Crossref]
  9. B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25, 28–30 (2000).
    [Crossref]
  10. X. Peng, L. F. Yu, and L. L. Cai, “Double-lock for image encryption with virtual optical wavelength,” Opt. Exp. 10, 41–45(2002).
  11. N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
    [Crossref]
  12. H. Kim, D. H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Exp. 12, 4912–4921 (2004).
    [Crossref]
  13. L. F. Chen and D. M. Zhao, “Optical image encryption with Hartley transforms,” Opt. Lett. 31, 3438–3440 (2006).
    [Crossref] [PubMed]
  14. Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202, 277–285 (2002).
    [Crossref]
  15. G. H. Situ and J. J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1308 (2005).
    [Crossref] [PubMed]
  16. D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, “Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation,” Phys. Lett. 18, 116–118 (1965).
    [Crossref]
  17. J. F. Ebersole, “Optical image subtraction,” Opt. Eng. 14, 436–447 (1975).
  18. G. S. Pati, G. Unnikrishnan, and K. Singh, “Multichannel image addition and subtraction using joint-transform correlator architecture,” Opt. Commun. 150, 33–37 (1998).
    [Crossref]
  19. A. E. Chiou and P. Yeh, “Parallel image subtraction using a phase-conjugate Michelson interferometer,” Opt. Lett. 11, 306–308 (1986).
    [Crossref] [PubMed]
  20. S. H. Lee., S. K. Yao, and A. G. Milines, “Optical image synthesis (complex amplitude addition and subtraction) in real time by a diffraction-grating interferometric method,” J. Opt. Soc. Am. A 60, 1037–1041 (1970).
    [Crossref]
  21. S. T. Wu and F. T. S. Yu, “Image subtraction with an encoded extended incoherent source,” Appl. Opt. 20, 4082–4088 (1981).
    [Crossref] [PubMed]
  22. M. Y. Shih, A. Shishido, and I. C. Khoo, “All-optical image processing by means of a photosensitive nonlinear liquid-crystal film: edge enhancement and image addition-subtraction,” Opt. Lett. 26, 1140–1142 (2001).
    [Crossref]
  23. L. F. Chen and D. M. Zhao, “Optical image addition and encryption by multi-exposure based on fractional Fourier transform hologram,” Chin. Phys. Lett. 23, 603–606 (2006).
    [Crossref]
  24. X. G. Wang, D. M. Zhao, F. Jing, and X. F. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics,” Opt. Exp. 14, 1476–1486 (2006).
    [Crossref]
  25. S. Q. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” John Wiley & Sons Inc. Microwave Opt. Technol. Lett. 21, 318–323 (1999).
    [Crossref]
  26. L. F. Chen and D. M. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Exp. 14, 8552–8560 (2006).
    [Crossref]
  27. W. M. Jin, L. H. Ma, and C. J. Yan, “Real color fractional Fourier transform holograms,” Opt. Commun. 259, 513–516 (2006).
    [Crossref]
  28. I. Yamaguchi, T. Matsumura, and J. Kato, “Phase-shifting color digital holography,” Opt. Lett. 27, 1108–1110 (2002).
    [Crossref]
  29. A.W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [Crossref]
  30. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668–1670 (2004).
    [Crossref] [PubMed]

2006 (5)

L. F. Chen and D. M. Zhao, “Optical image encryption with Hartley transforms,” Opt. Lett. 31, 3438–3440 (2006).
[Crossref] [PubMed]

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

W. M. Jin, L. H. Ma, and C. J. Yan, “Real color fractional Fourier transform holograms,” Opt. Commun. 259, 513–516 (2006).
[Crossref]

L. F. Chen and D. M. Zhao, “Optical image addition and encryption by multi-exposure based on fractional Fourier transform hologram,” Chin. Phys. Lett. 23, 603–606 (2006).
[Crossref]

X. G. Wang, D. M. Zhao, F. Jing, and X. F. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics,” Opt. Exp. 14, 1476–1486 (2006).
[Crossref]

2005 (1)

2004 (4)

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
[Crossref]

H. Kim, D. H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Exp. 12, 4912–4921 (2004).
[Crossref]

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

F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668–1670 (2004).
[Crossref] [PubMed]

2003 (1)

2002 (3)

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

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

I. Yamaguchi, T. Matsumura, and J. Kato, “Phase-shifting color digital holography,” Opt. Lett. 27, 1108–1110 (2002).
[Crossref]

2001 (2)

2000 (4)

1999 (1)

S. Q. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” John Wiley & Sons Inc. Microwave Opt. Technol. Lett. 21, 318–323 (1999).
[Crossref]

1998 (1)

G. S. Pati, G. Unnikrishnan, and K. Singh, “Multichannel image addition and subtraction using joint-transform correlator architecture,” Opt. Commun. 150, 33–37 (1998).
[Crossref]

1997 (1)

B. Javidi, “Security information with optical technology,” Phys. Today  50, 27–32 (1997).

1995 (1)

1993 (1)

1986 (1)

1981 (1)

1975 (1)

J. F. Ebersole, “Optical image subtraction,” Opt. Eng. 14, 436–447 (1975).

1970 (1)

S. H. Lee., S. K. Yao, and A. G. Milines, “Optical image synthesis (complex amplitude addition and subtraction) in real time by a diffraction-grating interferometric method,” J. Opt. Soc. Am. A 60, 1037–1041 (1970).
[Crossref]

1965 (1)

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, “Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation,” Phys. Lett. 18, 116–118 (1965).
[Crossref]

Brumm, D.

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, “Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation,” Phys. Lett. 18, 116–118 (1965).
[Crossref]

Cai, L. L.

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

Chen, L. F.

L. F. Chen and D. M. Zhao, “Optical image addition and encryption by multi-exposure based on fractional Fourier transform hologram,” Chin. Phys. Lett. 23, 603–606 (2006).
[Crossref]

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

L. F. Chen and D. M. Zhao, “Optical image encryption with Hartley transforms,” Opt. Lett. 31, 3438–3440 (2006).
[Crossref] [PubMed]

Chiou, A. E.

Ebersole, J. F.

J. F. Ebersole, “Optical image subtraction,” Opt. Eng. 14, 436–447 (1975).

Funkhouser, A.

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, “Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation,” Phys. Lett. 18, 116–118 (1965).
[Crossref]

Gabor, D.

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, “Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation,” Phys. Lett. 18, 116–118 (1965).
[Crossref]

Hennelly, B.

Javidi, B.

Jin, W. M.

W. M. Jin, L. H. Ma, and C. J. Yan, “Real color fractional Fourier transform holograms,” Opt. Commun. 259, 513–516 (2006).
[Crossref]

Jing, F.

X. G. Wang, D. M. Zhao, F. Jing, and X. F. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics,” Opt. Exp. 14, 1476–1486 (2006).
[Crossref]

Joseph, J.

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (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]

Karim, M. A.

S. Q. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” John Wiley & Sons Inc. Microwave Opt. Technol. Lett. 21, 318–323 (1999).
[Crossref]

Kato, J.

Khoo, I. C.

Kim, D. H.

H. Kim, D. H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Exp. 12, 4912–4921 (2004).
[Crossref]

Kim, H.

H. Kim, D. H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Exp. 12, 4912–4921 (2004).
[Crossref]

Lee, Y. H.

H. Kim, D. H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Exp. 12, 4912–4921 (2004).
[Crossref]

Lee., S. H.

S. H. Lee., S. K. Yao, and A. G. Milines, “Optical image synthesis (complex amplitude addition and subtraction) in real time by a diffraction-grating interferometric method,” J. Opt. Soc. Am. A 60, 1037–1041 (1970).
[Crossref]

Liu, S. T.

Lohmann, A.W.

Ma, L. H.

W. M. Jin, L. H. Ma, and C. J. Yan, “Real color fractional Fourier transform holograms,” Opt. Commun. 259, 513–516 (2006).
[Crossref]

Matoba, O.

Matsumura, T.

Mi, Q. L.

Milines, A. G.

S. H. Lee., S. K. Yao, and A. G. Milines, “Optical image synthesis (complex amplitude addition and subtraction) in real time by a diffraction-grating interferometric method,” J. Opt. Soc. Am. A 60, 1037–1041 (1970).
[Crossref]

Nishchal, N. K.

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
[Crossref]

Nomura, T.

Pati, G. S.

G. S. Pati, G. Unnikrishnan, and K. Singh, “Multichannel image addition and subtraction using joint-transform correlator architecture,” Opt. Commun. 150, 33–37 (1998).
[Crossref]

Peng, X.

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

Refregier, P.

Restrick, R.

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, “Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation,” Phys. Lett. 18, 116–118 (1965).
[Crossref]

Sheridan, J. T.

Shih, M. Y.

Shishido, A.

Singh, K.

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
[Crossref]

G. Unnikrishnan and K. Singh, “Double random fractional Fourier-domain encoding for optical security,” Opt. Eng. 39, 2853–2859 (2000).
[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]

G. S. Pati, G. Unnikrishnan, and K. Singh, “Multichannel image addition and subtraction using joint-transform correlator architecture,” Opt. Commun. 150, 33–37 (1998).
[Crossref]

Situ, G.

Situ, G. H.

Stroke, G. W.

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, “Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation,” Phys. Lett. 18, 116–118 (1965).
[Crossref]

Tajahuerce, E.

Tanno, N.

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

Unnikrishnan, G.

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]

G. Unnikrishnan and K. Singh, “Double random fractional Fourier-domain encoding for optical security,” Opt. Eng. 39, 2853–2859 (2000).
[Crossref]

G. S. Pati, G. Unnikrishnan, and K. Singh, “Multichannel image addition and subtraction using joint-transform correlator architecture,” Opt. Commun. 150, 33–37 (1998).
[Crossref]

Verrall, S. C.

Wang, X. G.

X. G. Wang, D. M. Zhao, F. Jing, and X. F. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics,” Opt. Exp. 14, 1476–1486 (2006).
[Crossref]

Wei, X. F.

X. G. Wang, D. M. Zhao, F. Jing, and X. F. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics,” Opt. Exp. 14, 1476–1486 (2006).
[Crossref]

Wu, S. T.

Yamaguchi, I.

Yan, C. J.

W. M. Jin, L. H. Ma, and C. J. Yan, “Real color fractional Fourier transform holograms,” Opt. Commun. 259, 513–516 (2006).
[Crossref]

Yao, S. K.

S. H. Lee., S. K. Yao, and A. G. Milines, “Optical image synthesis (complex amplitude addition and subtraction) in real time by a diffraction-grating interferometric method,” J. Opt. Soc. Am. A 60, 1037–1041 (1970).
[Crossref]

Yaroslavsky, L. P.

Yeh, P.

Yu, F. T. S.

Yu, L. F.

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

Zhang, F.

Zhang, J.

Zhang, J. J.

Zhang, S. Q.

S. Q. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” John Wiley & Sons Inc. Microwave Opt. Technol. Lett. 21, 318–323 (1999).
[Crossref]

Zhang, Y.

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

Zhao, D. M.

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

X. G. Wang, D. M. Zhao, F. Jing, and X. F. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics,” Opt. Exp. 14, 1476–1486 (2006).
[Crossref]

L. F. Chen and D. M. Zhao, “Optical image addition and encryption by multi-exposure based on fractional Fourier transform hologram,” Chin. Phys. Lett. 23, 603–606 (2006).
[Crossref]

L. F. Chen and D. M. Zhao, “Optical image encryption with Hartley transforms,” Opt. Lett. 31, 3438–3440 (2006).
[Crossref] [PubMed]

Zheng, C. H.

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

Zhu, B. H.

Appl. Opt. (2)

Chin. Phys. Lett. (1)

L. F. Chen and D. M. Zhao, “Optical image addition and encryption by multi-exposure based on fractional Fourier transform hologram,” Chin. Phys. Lett. 23, 603–606 (2006).
[Crossref]

J. Opt. Soc. Am. A (2)

A.W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[Crossref]

S. H. Lee., S. K. Yao, and A. G. Milines, “Optical image synthesis (complex amplitude addition and subtraction) in real time by a diffraction-grating interferometric method,” J. Opt. Soc. Am. A 60, 1037–1041 (1970).
[Crossref]

John Wiley & Sons Inc. Microwave Opt. Technol. Lett. (1)

S. Q. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” John Wiley & Sons Inc. Microwave Opt. Technol. Lett. 21, 318–323 (1999).
[Crossref]

Opt. Commun. (4)

W. M. Jin, L. H. Ma, and C. J. Yan, “Real color fractional Fourier transform holograms,” Opt. Commun. 259, 513–516 (2006).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
[Crossref]

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

G. S. Pati, G. Unnikrishnan, and K. Singh, “Multichannel image addition and subtraction using joint-transform correlator architecture,” Opt. Commun. 150, 33–37 (1998).
[Crossref]

Opt. Eng. (2)

J. F. Ebersole, “Optical image subtraction,” Opt. Eng. 14, 436–447 (1975).

G. Unnikrishnan and K. Singh, “Double random fractional Fourier-domain encoding for optical security,” Opt. Eng. 39, 2853–2859 (2000).
[Crossref]

Opt. Exp. (4)

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

H. Kim, D. H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Exp. 12, 4912–4921 (2004).
[Crossref]

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

X. G. Wang, D. M. Zhao, F. Jing, and X. F. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics,” Opt. Exp. 14, 1476–1486 (2006).
[Crossref]

Opt. Lett. (12)

M. Y. Shih, A. Shishido, and I. C. Khoo, “All-optical image processing by means of a photosensitive nonlinear liquid-crystal film: edge enhancement and image addition-subtraction,” Opt. Lett. 26, 1140–1142 (2001).
[Crossref]

I. Yamaguchi, T. Matsumura, and J. Kato, “Phase-shifting color digital holography,” Opt. Lett. 27, 1108–1110 (2002).
[Crossref]

F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29, 1668–1670 (2004).
[Crossref] [PubMed]

L. F. Chen and D. M. Zhao, “Optical image encryption with Hartley transforms,” Opt. Lett. 31, 3438–3440 (2006).
[Crossref] [PubMed]

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

A. E. Chiou and P. Yeh, “Parallel image subtraction using a phase-conjugate Michelson interferometer,” Opt. Lett. 11, 306–308 (1986).
[Crossref] [PubMed]

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

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

B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25, 28–30 (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] [PubMed]

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]

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

Phys. (1)

B. Javidi, “Security information with optical technology,” Phys. Today  50, 27–32 (1997).

Phys. Lett. (1)

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, “Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation,” Phys. Lett. 18, 116–118 (1965).
[Crossref]

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Figures (8)

Fig. 1.
Fig. 1.

Color image decomposition. R: Red; G: Green; B: Blue.

Fig. 2.
Fig. 2.

Real color image decomposition: (a) an original color image with 800×600 pixels; (b) its red part; (c) its green part; (d) its blue part.

Fig. 3.
Fig. 3.

Encryption implementation with fractional Fourier hologram. BE: beam expander, BSs: beam splitters, Ms: mirrors, RPMs: random phases, I: input plane, O: output plane.

Fig. 4.
Fig. 4.

Color image encryption and decryption with on-axis fractional Fourier hologram. (a) An original color image with 1024×768 pixels; (b) its color encrypted fractional hologram; (c) key hologram; (d) recovered color image.

Fig. 5.
Fig. 5.

Color image encryption and decryption with off-axis fractional Fourier hologram. (a) Retrieved encrypted term; (b) retrieved random key; (c) wrong decryption result with all fractional orders simultaneously incorrect; (d) wrong decryption result with all random phases incorrect; (e) its wrong decryption result with both fractional orders and random phases incorrectly selected; (f)–(h) results with one channel incorrectly decrypted; (i)–(k) results with only one channel correctly decrypted; (l) correct decryption result.

Fig. 6.
Fig. 6.

Color image addition and subtraction. (a), (b) Original images with 1024×768 pixels; (c) their addition image; (d).addition image after post-processing; (e), (f) the other two original images with 257×255 pixels; (g) their subtraction image; (h).subtraction image after post-processing.

Fig. 7.
Fig. 7.

Color image encryption and addition. (a) Encrypted result of three added images; (b), (c) their incorrect decryption images; (d) result with one channel correctly decrypted; (e) result with two channels correctly decrypted; (f) their fused image with correct decryption.

Fig. 8.
Fig. 8.

Color image encryption and subtraction. (a) Encrypted result of two subtracted images; (b), (c) their incorrect decryption images; (d) result with one channel correctly decrypted; (e) result with two channels correctly decrypted; (f) their subtraction image with correct decryption.

Equations (20)

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g ( x 1 , y 1 ) = FrFT [ f ( x , y ) P ( x , y ) ] = f ( x , y ) P ( x , y ) exp ( i π x 2 + x 1 2 λ f s 1 tan ϕ 1 2 π i xx 1 λ f s 1 sin ϕ 1 )
× exp ( i π y 2 + y 1 2 λ f s 2 tan ϕ 2 2 π i yy 1 λ f s 2 sin ϕ 2 ) d x d y ,
I ( u , v ) = 1 + g ( x 1 , y 1 ) 2 + g * ( x 1 , y 1 ) Q ( u , v ) + g ( x 1 , y 1 ) Q * ( u , v ) .
h ( u , v ) = Q ( u , v ) + Q * ( u , v ) .
f ( x , y ) = IFrFT [ g ( x 1 , y 1 ) ] = g ( x 1 , y 1 ) Q * ( u , v ) Q ( u , v ) exp ( i π x 2 + x 1 2 λ f s 1 tan ( ϕ 1 ) 2 π i xx 1 λ f s 1 sin ( ϕ 1 ) )
× exp ( i π y 2 + y 1 2 λ f s 2 tan ( ϕ 2 ) 2 π i yy 1 λ f s 2 sin ( ϕ 2 ) ) d x 1 d y 1
= f ( x , y ) P ( x , y ) .
g ( x 1 , y 1 ) = FrFT [ f ( x , y ) ] = f ( x , y ) exp ( i π x 2 + x 1 2 λ f s 1 tan ϕ 1 2 π i xx 1 λ f s 1 sin ϕ 1 )
× exp ( i π y 2 + y 1 2 λ f s 2 tan ϕ 2 2 π i yy 1 λ f s 2 sin ϕ 2 ) d x d y .
I 1 ( u , v ) = 1 + g 1 ( x 1 , y 1 ) 2 + g 1 * ( x 1 , y 1 ) Q 1 ( u , v ) + g 1 ( x 1 , y 1 ) Q 1 * ( u , v ) ,
I 2 ( u , v ) = 1 + g 2 ( x 1 , y 1 ) 2 + g 2 * ( x 1 , y 1 ) Q 1 ( u , v ) exp ( i π ) + g 2 ( x 1 , y 1 ) Q 1 * ( u , v ) exp ( i π ) .
g 12 ( x 1 , y 1 ) = Q 1 * ( u , v ) Q 1 ( u , v ) [ g 1 ( x 1 , y 1 ) + g 2 ( x 1 , y 1 ) exp ( i π ) ]
= g 1 ( x 1 , y 1 ) + g 2 ( x 1 , y 1 ) ,
f 12 ( x , y ) = IFrFT [ g 12 ( x 1 , y 1 ) ] = g 12 ( x 1 , y 1 ) exp ( i π x 2 + x 1 2 λ f s 1 tan ( ϕ 1 ) 2 π i xx 1 λ f s 1 sin ( ϕ 1 ) )
× exp ( i π y 2 + y 1 2 λ f s 2 tan ( ϕ 2 ) 2 π i yy 1 λ f s 2 sin ( ϕ 2 ) ) d x 1 d y 1
= IFrFT [ g 1 ( x 1 , y 1 ) ] IFrFT [ g 2 ( x 1 , y 1 ) ]
= f 1 ( x , y ) f 2 ( x , y ) .
g 12 ( x 1 , y 1 ) = Q 1 * ( u , v ) Q 1 ( u , v ) [ g 1 ( x 1 , y 1 ) + g 2 ( x 1 , y 1 ) ] = g 1 ( x 1 , y 1 ) + g 2 ( x 1 , y 1 ) ,
f 12 ( x , y ) = IFrFT [ g 12 ( x 1 , y 1 ) ] = IFrFT [ g 1 ( x 1 , y 1 ) ] + IFrFT [ g 2 ( x 1 , y 1 ) ]
= f 1 ( x , y ) + f 2 ( x , y ) .

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