Abstract

We propose a new method for image encryption based on Hartley transforms that is a real transform and can be realized by spatially incoherent or coherent illumination. The proposed optical implementation is based on a Michelson-type interferometer in which the pure random intensity is distributed at the Hartley plane in encryption. Computer simulations prove it is possible. A Hartley hologram method is also given and described to resolve the sign ambiguity problem that would be encountered in image reconstruction.

© 2006 Optical Society of America

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2006

2005

G. Situ and J. Zhang, Opt. Lett. 30, 1306 (2005).
[CrossRef] [PubMed]

L. F. Chen and D. M. Zhao, Opt. Commun. 254, 361 (2005).
[CrossRef]

2004

2003

B. Hennelly and J. T. Sheridan, Opt. Lett. 28, 269 (2003).
[CrossRef] [PubMed]

B. H. Zhu, H. F. Zhao, and S. T. Liu, Optik (Stuttgart) 114, 95 (2003).
[CrossRef]

2002

X. Peng, L. F. Yu, and L. L. Cai, Opt. Express 10, 41 (2002).
[PubMed]

Y. Zhang, C. H. Zheng, and N. Tanno, Opt. Commun. 202, 277 (2002).
[CrossRef]

2001

2000

1997

B. Javidi, Phys. Today 50(3), 27 (1997).
[CrossRef]

1995

1994

1989

1988

J. D. Villasenor and R. N. Bracewell, Nature 335, 617 (1988).
[CrossRef]

1987

J. D. Villasenor and R. N. Bracewell, Nature 330, 735 (1987).
[CrossRef]

1985

Andrés, P.

Bartelt, H.

Bates, R. H. T.

R. H. T. Bates and M. J. McDonnel, Image Restoration and Reconstruction (Clarendon, 1986).

Bracewell, R. N.

J. D. Villasenor and R. N. Bracewell, Nature 335, 617 (1988).
[CrossRef]

J. D. Villasenor and R. N. Bracewell, Nature 330, 735 (1987).
[CrossRef]

R. N. Bracewell, H. Bartelt, A. W. Lohmann, and N. Streibl, Appl. Opt. 24, 1401 (1985).
[CrossRef] [PubMed]

Cai, L. L.

Chen, L. F.

L. F. Chen and D. M. Zhao, Chin. Phys. Lett. 23, 603 (2006).
[CrossRef]

L. F. Chen and D. M. Zhao, Opt. Commun. 254, 361 (2005).
[CrossRef]

Dorsch, R. G.

George, N.

Hennelly, B.

Ichioka, Y.

Itoh, K.

Javidi, B.

Jing, F.

Joseph, J.

N. K. Nishchal, J. Joseph, and K. Singh, Opt. Commun. 235, 253 (2004).
[CrossRef]

G. Unnikrishnan, J. Joseph, and K. Singh, Opt. Lett. 25, 887 (2000).
[CrossRef]

Kim, D. H.

Kim, H.

Konforti, N.

Lancis, J.

Lee, Y. H.

Liu, S. T.

B. H. Zhu, H. F. Zhao, and S. T. Liu, Optik (Stuttgart) 114, 95 (2003).
[CrossRef]

S. T. Liu, Q. L. Mi, and B. H. Zhu, Opt. Lett. 26, 1242 (2001).
[CrossRef]

Lohmann, A. W.

Matoba, O.

McDonnel, M. J.

R. H. T. Bates and M. J. McDonnel, Image Restoration and Reconstruction (Clarendon, 1986).

Mendlovic, D.

Mi, Q. L.

Nishchal, N. K.

N. K. Nishchal, J. Joseph, and K. Singh, Opt. Commun. 235, 253 (2004).
[CrossRef]

Nomura, T.

Peng, X.

Pohit, M.

G. Unnikrishnan, M. Pohit, and K. Singh, Opt. Commun. 185, 25 (2000).
[CrossRef]

Refregier, P.

Sheridan, J. T.

Singh, K.

N. K. Nishchal, J. Joseph, and K. Singh, Opt. Commun. 235, 253 (2004).
[CrossRef]

G. Unnikrishnan, M. Pohit, and K. Singh, Opt. Commun. 185, 25 (2000).
[CrossRef]

G. Unnikrishnan, J. Joseph, and K. Singh, Opt. Lett. 25, 887 (2000).
[CrossRef]

Situ, G.

Streibl, N.

Tajahuerce, E.

Tanno, N.

Y. Zhang, C. H. Zheng, and N. Tanno, Opt. Commun. 202, 277 (2002).
[CrossRef]

Unnikrishnan, G.

G. Unnikrishnan, M. Pohit, and K. Singh, Opt. Commun. 185, 25 (2000).
[CrossRef]

G. Unnikrishnan, J. Joseph, and K. Singh, Opt. Lett. 25, 887 (2000).
[CrossRef]

Verrall, S. C.

Villasenor, J. D.

J. D. Villasenor, Proc. IEEE 82, 391 (1994).
[CrossRef]

J. D. Villasenor, Appl. Opt. 28, 2671 (1989).
[CrossRef] [PubMed]

J. D. Villasenor and R. N. Bracewell, Nature 335, 617 (1988).
[CrossRef]

J. D. Villasenor and R. N. Bracewell, Nature 330, 735 (1987).
[CrossRef]

Wang, S.

Wang, X. G.

Wei, X. F.

Yoshimura, Y.

Yu, L. F.

Zalevsky, Z.

Zhang, J.

Zhang, Y.

Y. Zhang, C. H. Zheng, and N. Tanno, Opt. Commun. 202, 277 (2002).
[CrossRef]

Zhao, D. M.

X. G. Wang, D. M. Zhao, F. Jing, and X. F. Wei, Opt. Express 14, 1476 (2006).
[CrossRef] [PubMed]

L. F. Chen and D. M. Zhao, Chin. Phys. Lett. 23, 603 (2006).
[CrossRef]

L. F. Chen and D. M. Zhao, Opt. Commun. 254, 361 (2005).
[CrossRef]

Zhao, H. F.

B. H. Zhu, H. F. Zhao, and S. T. Liu, Optik (Stuttgart) 114, 95 (2003).
[CrossRef]

Zheng, C. H.

Y. Zhang, C. H. Zheng, and N. Tanno, Opt. Commun. 202, 277 (2002).
[CrossRef]

Zhu, B. H.

B. H. Zhu, H. F. Zhao, and S. T. Liu, Optik (Stuttgart) 114, 95 (2003).
[CrossRef]

S. T. Liu, Q. L. Mi, and B. H. Zhu, Opt. Lett. 26, 1242 (2001).
[CrossRef]

Appl. Opt.

Chin. Phys. Lett.

L. F. Chen and D. M. Zhao, Chin. Phys. Lett. 23, 603 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Nature

J. D. Villasenor and R. N. Bracewell, Nature 330, 735 (1987).
[CrossRef]

J. D. Villasenor and R. N. Bracewell, Nature 335, 617 (1988).
[CrossRef]

Opt. Commun.

N. K. Nishchal, J. Joseph, and K. Singh, Opt. Commun. 235, 253 (2004).
[CrossRef]

G. Unnikrishnan, M. Pohit, and K. Singh, Opt. Commun. 185, 25 (2000).
[CrossRef]

Y. Zhang, C. H. Zheng, and N. Tanno, Opt. Commun. 202, 277 (2002).
[CrossRef]

L. F. Chen and D. M. Zhao, Opt. Commun. 254, 361 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Optik (Stuttgart)

B. H. Zhu, H. F. Zhao, and S. T. Liu, Optik (Stuttgart) 114, 95 (2003).
[CrossRef]

Phys. Today

B. Javidi, Phys. Today 50(3), 27 (1997).
[CrossRef]

Proc. IEEE

J. D. Villasenor, Proc. IEEE 82, 391 (1994).
[CrossRef]

Other

R. H. T. Bates and M. J. McDonnel, Image Restoration and Reconstruction (Clarendon, 1986).

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Figures (3)

Fig. 1
Fig. 1

Optical implementation of the Hartley transform.

Fig. 2
Fig. 2

Computer simulation results: (a) original image with 256 × 256   pixels , (b) encryption result, (c) incorrect decryption image, (d) correct decryption image, (e) correct random intensity key, (f) incorrect random intensity key used in (c).

Fig. 3
Fig. 3

Optical implementation of the Hartley hologram technique.

Equations (10)

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

H ( u , v ) = f ( x , y ) cas [ 2 π ( u x + v y ) ] d x d y ,
f ( x , y ) = H ( u , v ) cas [ 2 π ( u x + v y ) ] d u d v ,
H ( u , v ) = f ( x , y ) cas [ 2 π ( u x + v y ) ] d x d y = exp ( i π 4 ) 2 [ F ( u , v ) + exp ( i π 2 ) F ( u , v ) ] .
H ( u , v ) = t ( u , v ) f ( x , y ) cas [ 2 π ( u x + v y ) ] d x d y ,
f ( x , y ) = H ( u , v ) 1 t ( u , v ) cas [ 2 π ( u x + v y ) ] d u d v ,
MSE = m = 1 M n = 1 N [ f ( m Δ x , n Δ y ) f 0 ( m Δ x , n Δ y ) 2 ] m = 1 M n = 1 N [ f ( m Δ x , n Δ y ) 2 ] ,
P ( u , v ) 2 = t ( u , v ) [ H ( u , v ) + A ] 2 ,
g ( x , y ) = exp ( i π 4 ) 2 { F [ P ( u , v ) 2 1 t 2 ( u , v ) ] ( x , y ) + exp ( i π 2 ) F 1 [ P ( u , v ) 2 1 t 2 ( u , v ) ] ( x , y ) } .
F [ P ( u , v ) 2 1 t 2 ( u , v ) ] ( x , y ) 2 A F [ H ( u , v ) ] ( x , y ) + F [ A 2 ] ( x , y ) ,
g ( x , y ) = 2 A f ( x , y ) + 2 A 2 δ ( x , y ) .

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