Abstract

A new method is proposed in this paper for the synthesis and encryption of information with digital holography technique and virtual optics. By using a three-step phase-shifting interferometry, the fused or subtracted digital hologram can be calculated from different interference patterns. To protect the digital data that can be transmitted through communication channel, an encryption approach based on virtual optics is also proposed. The encryption method proposed is based on extended fractional Fourier transforms. Both the encryption and decryption processes are performed in all-digital manner. The encrypted data and the synthesized data reconstructed numerically also can be stored and transmitted in the conventional communication channel. Numerical simulation results are given to verify the proposed idea.

© 2006 Optical Society of America

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References

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  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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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  7. 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]
  8. 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]
  9. E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, "Optoelectronic information encryption with phase-shifting interferometry," Appl. Opt. 39, 2313-2320 (2000).
    [CrossRef]
  10. B. Javidi and T. Nomural, "Optical encryption by double-random phase encoding in the fractional Fourier domain," Opt. Lett. 25, 887-889 (2000).
    [CrossRef]
  11. N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, "Optical encryption using a localized fractional Fourier transform," Opt. Eng. 42, 3566-3571 (2003).
    [CrossRef]
  12. L. Yu and L. Cai, "Multidimensional data encryption with digital holography," Opt. Commun. 215, 271-284 (2003).
    [CrossRef]
  13. X. Peng, L. Yu, and L. Cai, "Digital watermarking in three-dimensional space with a virtual-optics imaging modality," Opt. Commun. 226, 155-165 (2003).
    [CrossRef]
  14. N. K. Nishchal, J. Joseph, and K. Singh, "Fully phase-encrypted memory using cascaded extended fractional Fourier transform," Opt. Lasers Eng. 42, 141-151 (2004).
    [CrossRef]
  15. N. K. Nishchal, J. Joseph, and K. Singh, "Securing information using fractional Fourier transform in digital holography," Opt. Commun. 235, 253-259 (2004).
    [CrossRef]
  16. H. Kim, D. H. Kim, and Y. H. Lee, "Encryption of digital hologram of 3-D object by virtual optics," Opt. Express 12, 4912-4921 (2004).
    [CrossRef]
  17. L. Chen and D. Zhao, "Optical image encryption based on fractional wavelet transform," Opt. Commun. 254, 361-367 (2005).
    [CrossRef]
  18. X. Wang, D. Zhao, and L. Chen, "Image encryption based on extended fractional Fourier transform and digital holography technique," Opt. Commun. (in press).
    [PubMed]
  19. L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps," Opt. Lett. 28, 1808-1810 (2003).
    [CrossRef] [PubMed]
  20. J. Hua, L. Liu, and G. Li, "Extended fractional Fourier transforms," J. Opt. Soc. Am. A 14, 3316-3322 (1997).
    [CrossRef]

2005 (1)

L. Chen and D. Zhao, "Optical image encryption based on fractional wavelet transform," Opt. Commun. 254, 361-367 (2005).
[CrossRef]

2004 (3)

N. K. Nishchal, J. Joseph, and K. Singh, "Fully phase-encrypted memory using cascaded extended fractional Fourier transform," Opt. Lasers Eng. 42, 141-151 (2004).
[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]

H. Kim, D. H. Kim, and Y. H. Lee, "Encryption of digital hologram of 3-D object by virtual optics," Opt. Express 12, 4912-4921 (2004).
[CrossRef]

2003 (4)

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, "Optical encryption using a localized fractional Fourier transform," Opt. Eng. 42, 3566-3571 (2003).
[CrossRef]

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

X. Peng, L. Yu, and L. Cai, "Digital watermarking in three-dimensional space with a virtual-optics imaging modality," Opt. Commun. 226, 155-165 (2003).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps," Opt. Lett. 28, 1808-1810 (2003).
[CrossRef] [PubMed]

2001 (1)

2000 (3)

1997 (1)

1995 (1)

1991 (1)

1986 (1)

1981 (1)

1975 (1)

J. F. Ebersole, "Optical image subtraction," Opt. Eng. 14, 436-447 (1975).

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

X. Peng, L. Yu, and L. Cai, "Digital watermarking in three-dimensional space with a virtual-optics imaging modality," Opt. Commun. 226, 155-165 (2003).
[CrossRef]

Cai, L. Z.

Chen, L.

L. Chen and D. Zhao, "Optical image encryption based on fractional wavelet transform," Opt. Commun. 254, 361-367 (2005).
[CrossRef]

X. Wang, D. Zhao, and L. Chen, "Image encryption based on extended fractional Fourier transform and digital holography technique," Opt. Commun. (in press).
[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]

Hua, J.

Javidi, B.

Joseph, J.

N. K. Nishchal, J. Joseph, and K. Singh, "Fully phase-encrypted memory using cascaded extended fractional Fourier transform," Opt. Lasers Eng. 42, 141-151 (2004).
[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]

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, "Optical encryption using a localized fractional Fourier transform," Opt. Eng. 42, 3566-3571 (2003).
[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]

Khoo, I. C.

Kim, D. H.

Kim, H.

Lee, Y. H.

Li, G.

Liu, L.

Liu, Q.

Matoba, O.

Miteva, M.

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]

N. K. Nishchal, J. Joseph, and K. Singh, "Fully phase-encrypted memory using cascaded extended fractional Fourier transform," Opt. Lasers Eng. 42, 141-151 (2004).
[CrossRef]

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, "Optical encryption using a localized fractional Fourier transform," Opt. Eng. 42, 3566-3571 (2003).
[CrossRef]

Nomural, T.

Peng, X.

X. Peng, L. Yu, and L. Cai, "Digital watermarking in three-dimensional space with a virtual-optics imaging modality," Opt. Commun. 226, 155-165 (2003).
[CrossRef]

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]

Shih, M. Y.

Shishido, A.

Singh, K.

N. K. Nishchal, J. Joseph, and K. Singh, "Fully phase-encrypted memory using cascaded extended fractional Fourier transform," Opt. Lasers Eng. 42, 141-151 (2004).
[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]

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, "Optical encryption using a localized fractional Fourier transform," Opt. Eng. 42, 3566-3571 (2003).
[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]

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]

Tai, A.

Tajahuerce, E.

Unnikrishnan, G.

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, "Optical encryption using a localized fractional Fourier transform," Opt. Eng. 42, 3566-3571 (2003).
[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]

Verrall, S. C.

Wang, X.

X. Wang, D. Zhao, and L. Chen, "Image encryption based on extended fractional Fourier transform and digital holography technique," Opt. Commun. (in press).
[PubMed]

Yang, X. L.

Yeh, P.

Yu, F. T. S.

Yu, L.

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

X. Peng, L. Yu, and L. Cai, "Digital watermarking in three-dimensional space with a virtual-optics imaging modality," Opt. Commun. 226, 155-165 (2003).
[CrossRef]

Zhao, D.

L. Chen and D. Zhao, "Optical image encryption based on fractional wavelet transform," Opt. Commun. 254, 361-367 (2005).
[CrossRef]

X. Wang, D. Zhao, and L. Chen, "Image encryption based on extended fractional Fourier transform and digital holography technique," Opt. Commun. (in press).
[PubMed]

Zhivkova, S.

Appl. Opt. (2)

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

Opt. Commun. (5)

L. Chen and D. Zhao, "Optical image encryption based on fractional wavelet transform," Opt. Commun. 254, 361-367 (2005).
[CrossRef]

X. Wang, D. Zhao, and L. Chen, "Image encryption based on extended fractional Fourier transform and digital holography technique," Opt. Commun. (in press).
[PubMed]

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

X. Peng, L. Yu, and L. Cai, "Digital watermarking in three-dimensional space with a virtual-optics imaging modality," Opt. Commun. 226, 155-165 (2003).
[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]

Opt. Eng. (2)

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, "Optical encryption using a localized fractional Fourier transform," Opt. Eng. 42, 3566-3571 (2003).
[CrossRef]

J. F. Ebersole, "Optical image subtraction," Opt. Eng. 14, 436-447 (1975).

Opt. Express (1)

Opt. Lasers Eng. (1)

N. K. Nishchal, J. Joseph, and K. Singh, "Fully phase-encrypted memory using cascaded extended fractional Fourier transform," Opt. Lasers Eng. 42, 141-151 (2004).
[CrossRef]

Opt. Lett. (7)

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.

Phase-shifting digital holography with three-step PSI.

Fig. 2.
Fig. 2.

(a) The schematic of the digital hologram encryption; (b) the virtual optical setup for encryption of the kth segment based on extended FRT.

Fig. 3.
Fig. 3.

(a) The schematic of the digital hologram decryption; (b) the virtual optical setup for decryption of the kth segment.

Fig. 4.
Fig. 4.

(a) and (b) The images to be synthesized and encrypted.

Fig. 5.
Fig. 5.

(a) Real part, (b) imaginary part of the recorded fused digital hologram; (c) real part, (d) imaginary part of one of digital hologram segments; (e) real part, (f) imaginary part of one of the encrypted digital hologram segments; decrypted results of one of the encrypted digital hologram segments with all correct keys (g) real part, (h) imaginary part; (i) real part, (j) imaginary part of decrypted results of the digital hologram segment with incorrect parameters but correct random phase codes; (k) real part, (l) imaginary part of decrypted results of the digital hologram segment with correct parameters but incorrect random phase codes.

Fig. 6.
Fig. 6.

(a) Real part, (b) imaginary part of the decrypted digital hologram with correct codes; reconstructed results from (c) correct decrypted digital hologram, (d) incorrect decrypted digital hologram.

Fig. 7.
Fig. 7.

(a) Real part, (b) imaginary part of the encrypted results of the subtracted digital hologram; (c) reconstructed results from correct decrypted digital hologram.

Fig. 8.
Fig. 8.

(a), (b) and (c) The images to be synthesized and encrypted; (d) real part, (e) imaginary part of encrypted fused digital hologram; (f) real part, (g) imaginary part of encrypted subtracted digital hologram; synthesized results reconstructed from (h) right decrypted fused digital hologram, (i) right decrypted subtracted digital hologram.

Equations (19)

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U 1 ( x , y ) = exp ( i 2 π d 0 λ ) d 0 U 0 ( x 0 , y 0 ) exp [ i π λ d 0 [ ( x x 0 ) 2 + ( y y 0 ) 2 ] d x 0 d y 0 ,
R j ( x , y ) = A r exp [ i ( φ r + δ j ) ] .
I 1 ( x , y ) = A 0 2 ( x , y ) + A r 2 + 2 A 0 ( x , y ) A r cos [ φ 0 ( x , y ) φ r δ 1 ] ,
I 2 ( x , y ) = A 0 2 ( x , y ) + A r 2 + 2 A 0 ( x , y ) A r cos [ φ 0 ( x , y ) φ r δ 2 ] ,
I 3 ( x , y ) = A 0 2 ( x , y ) + A r 2 + 2 A 0 ( x , y ) A r cos [ φ 0 ( x , y ) φ r δ 3 ] ,
U 1 ( x , y ) = 1 4 sin [ ( δ 3 δ 2 ) 2 ] × { exp [ i ( δ 1 + δ 2 ) 2 ] sin [ ( δ 3 δ 1 ) 2 ] ( I 1 I 3 ) exp [ i ( δ 1 + δ 3 ) 2 ] sin [ ( δ 2 δ 1 ) 2 ] ( I 1 I 2 ) } ,
U 2 ( x , y ) = 1 4 sin [ ( δ 3 δ 2 ) 2 ] × { exp [ i ( δ 1 + δ 2 ) 2 ] sin [ ( δ 3 δ 1 ) 2 ] ( I 1 I 3 ) exp [ i ( δ 1 + δ 3 ) 2 ] sin [ ( δ 2 δ 1 ) 2 ] ( I 1 I 2 ) } ,
U f ( x , y ) = U 1 ( x , y ) + U 2 ( x , y ) = 1 4 sin [ ( δ 3 δ 2 ) 2 ] × { exp [ i ( δ 1 + δ 2 ) 2 ] sin [ ( δ 3 δ 1 ) 2 ] ( I 1 + I 1 I 3 I 3 ) exp [ i ( δ 1 + δ 3 ) 2 ] sin [ ( δ 2 δ 1 ) 2 ] ( I 1 + I 1 I 2 I 2 ) } ,
U s ( x , y ) = U 1 ( x , y ) U 2 ( x , y ) = 1 4 sin [ ( δ 3 δ 2 ) 2 ] × { exp [ i ( δ 1 + δ 2 ) 2 ] sin [ ( δ 3 δ 1 ) 2 ] ( I 1 I 3 I 1 + I 3 ) exp [ i ( δ 1 + δ 3 ) 2 ] sin [ ( δ 2 δ 1 ) 2 ] ( I 1 I 2 I 1 + I 2 ) } ,
U f ( x , y ) = 1 4 { ( I 1 + I 1 I 3 I 3 ) + i [ ( 2 I 2 I 1 I 3 ) + ( 2 I 2 I 1 I 3 ) ] } .
U s ( x , y ) = 1 4 { ( I 1 I 3 I 1 + I 3 ) + i [ ( 2 I 2 I 1 I 3 ) ( 2 I 2 I 1 I 3 ) ] } .
S k e ( x 2 ) = g k ( x 1 ) exp [ i ϕ k ( x 1 ) ] exp [ i π ( a k 2 x 1 2 + b k 2 x 2 2 ) tan φ k i 2 π a k b k sin φ k x 1 x 2 ] d x 1 ,
g k ( x 1 ) = K S k ( x ) exp [ ( x ) ] exp [ i π ( a k 2 x 2 + b k 2 x 1 2 ) tan φ k i 2 π a k b k sin φ k x x 1 ] d x .
a k 2 = 1 λ k ( f k 1 d k 2 ) 1 2 ( f k 1 d k 1 ) 1 2 1 [ f k 1 2 ( f k 1 d k 1 ) ( f k 1 d k 2 ) ] 1 2 ,
φ k = arccos [ ( f k 1 d k 1 ) 1 2 ( f k 1 d k 2 ) 1 2 f k 1 ] ,
b k 2 = 1 λ k ( f k 1 d k 1 ) 1 2 ( f k 1 d k 2 ) 1 2 1 [ f k 1 2 ( f k 1 d k 1 ) ( f k 1 d k 2 ) ] 1 2 ,
h k d ( x ) = exp [ ( x ) ] × g k d ( x 1 ) exp [ i ϕ k ( x 1 ) ] exp [ i π ( b k 2 x 1 2 + a k 2 x 2 ) tan φ k i 2 π b k a k sin φ k x 1 x ] d x 1 ,
g k d ( x 1 ) = S k ( x 2 ) exp [ i π ( b k 2 x 2 2 + a k 2 x 1 2 ) tan φ k i 2 π b k a k sin φ k x 2 x 1 ] d x 2 .
MSE ( I 1 , I 2 ) = 1 N × N i = 1 N j = 1 N I 2 ( i , j ) I 1 ( i , j ) 2

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