Abstract

We propose a novel random fractional Fourier transform by randomizing the transform kernel function of the conventional fractional Fourier transform. The random fractional Fourier transform inherits the excellent mathematical properties from the fractional Fourier transform and can be easily implemented in optics. As a primary application the random fractional Fourier transform can be directly used in optical image encryption and decryption. The double phase encoding image encryption schemes can thus be modeled with cascaded random fractional Fourier transformers.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, Fractional Fourier Transform with Applications in Optics and Signal Processing, (Wiley, 2000).
  2. D. Mendlovic, Z. Zalevsky, D. Mas, J. Garcia, and C. Ferreira, Appl. Opt. 36, 4801 (1997).
    [Crossref] [PubMed]
  3. T. H. MacGregor, J. Comput. Appl. Math. 105, 93 (1999).
    [Crossref]
  4. S. C. Pei, C. C. Tseng, M. H. Yeh, and J. J. Shyu, IEEE Trans. Circuits Syst., II: Analog Digital Signal Process. 45, 665 (1998).
    [Crossref]
  5. S. C. Pei and M. H. Yeh, IEEE Trans. Signal Process. 49, 1198 (2001).
    [Crossref]
  6. Y. Zhang, B. Gu, B. Dong, G. Yang, H. Ren, X. Zhang, and S. Liu, Opt. Lett. 22, 1583 (1997).
    [Crossref]
  7. Z. Liu, H. Zhao, and S. Liu, Opt. Commun. 255, 357 (2005).
    [Crossref]
  8. Z. Liu and S. Liu, Opt. Lett. 32, 478 (2007).
    [Crossref] [PubMed]
  9. D. Mendlovic and H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
    [Crossref]
  10. A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
    [Crossref]
  11. P. Pellat-Finet, Opt. Lett. 19, 1388 (1994).
    [Crossref] [PubMed]
  12. S. Liu, J. Wu, and C. Li, Opt. Lett. 20, 1415 (1995).
    [Crossref] [PubMed]
  13. S. C. Pei and M. H. Yeh, Opt. Lett. 22, 1047 (1997).
    [Crossref] [PubMed]
  14. P. Refregier and B. Javidi, Opt. Lett. 20, 767 (1995).
    [Crossref] [PubMed]
  15. G. Unnikrishnan, J. Joseph, and K. Singh, Opt. Lett. 25, 887 (2000).
    [Crossref]

2007 (1)

2005 (1)

Z. Liu, H. Zhao, and S. Liu, Opt. Commun. 255, 357 (2005).
[Crossref]

2001 (1)

S. C. Pei and M. H. Yeh, IEEE Trans. Signal Process. 49, 1198 (2001).
[Crossref]

2000 (2)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, Fractional Fourier Transform with Applications in Optics and Signal Processing, (Wiley, 2000).

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

1999 (1)

T. H. MacGregor, J. Comput. Appl. Math. 105, 93 (1999).
[Crossref]

1998 (1)

S. C. Pei, C. C. Tseng, M. H. Yeh, and J. J. Shyu, IEEE Trans. Circuits Syst., II: Analog Digital Signal Process. 45, 665 (1998).
[Crossref]

1997 (3)

1995 (2)

1994 (1)

1993 (2)

Dong, B.

Ferreira, C.

Garcia, J.

Gu, B.

Javidi, B.

Joseph, J.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, Fractional Fourier Transform with Applications in Optics and Signal Processing, (Wiley, 2000).

Li, C.

Liu, S.

Liu, Z.

Z. Liu and S. Liu, Opt. Lett. 32, 478 (2007).
[Crossref] [PubMed]

Z. Liu, H. Zhao, and S. Liu, Opt. Commun. 255, 357 (2005).
[Crossref]

Lohmann, A. W.

MacGregor, T. H.

T. H. MacGregor, J. Comput. Appl. Math. 105, 93 (1999).
[Crossref]

Mas, D.

Mendlovic, D.

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, Fractional Fourier Transform with Applications in Optics and Signal Processing, (Wiley, 2000).

D. Mendlovic and H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
[Crossref]

Pei, S. C.

S. C. Pei and M. H. Yeh, IEEE Trans. Signal Process. 49, 1198 (2001).
[Crossref]

S. C. Pei, C. C. Tseng, M. H. Yeh, and J. J. Shyu, IEEE Trans. Circuits Syst., II: Analog Digital Signal Process. 45, 665 (1998).
[Crossref]

S. C. Pei and M. H. Yeh, Opt. Lett. 22, 1047 (1997).
[Crossref] [PubMed]

Pellat-Finet, P.

Refregier, P.

Ren, H.

Shyu, J. J.

S. C. Pei, C. C. Tseng, M. H. Yeh, and J. J. Shyu, IEEE Trans. Circuits Syst., II: Analog Digital Signal Process. 45, 665 (1998).
[Crossref]

Singh, K.

Tseng, C. C.

S. C. Pei, C. C. Tseng, M. H. Yeh, and J. J. Shyu, IEEE Trans. Circuits Syst., II: Analog Digital Signal Process. 45, 665 (1998).
[Crossref]

Unnikrishnan, G.

Wu, J.

Yang, G.

Yeh, M. H.

S. C. Pei and M. H. Yeh, IEEE Trans. Signal Process. 49, 1198 (2001).
[Crossref]

S. C. Pei, C. C. Tseng, M. H. Yeh, and J. J. Shyu, IEEE Trans. Circuits Syst., II: Analog Digital Signal Process. 45, 665 (1998).
[Crossref]

S. C. Pei and M. H. Yeh, Opt. Lett. 22, 1047 (1997).
[Crossref] [PubMed]

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, Fractional Fourier Transform with Applications in Optics and Signal Processing, (Wiley, 2000).

D. Mendlovic, Z. Zalevsky, D. Mas, J. Garcia, and C. Ferreira, Appl. Opt. 36, 4801 (1997).
[Crossref] [PubMed]

Zhang, X.

Zhang, Y.

Zhao, H.

Z. Liu, H. Zhao, and S. Liu, Opt. Commun. 255, 357 (2005).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Circuits Syst., II: Analog Digital Signal Process. (1)

S. C. Pei, C. C. Tseng, M. H. Yeh, and J. J. Shyu, IEEE Trans. Circuits Syst., II: Analog Digital Signal Process. 45, 665 (1998).
[Crossref]

IEEE Trans. Signal Process. (1)

S. C. Pei and M. H. Yeh, IEEE Trans. Signal Process. 49, 1198 (2001).
[Crossref]

J. Comput. Appl. Math. (1)

T. H. MacGregor, J. Comput. Appl. Math. 105, 93 (1999).
[Crossref]

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

Opt. Commun. (1)

Z. Liu, H. Zhao, and S. Liu, Opt. Commun. 255, 357 (2005).
[Crossref]

Opt. Lett. (7)

Other (1)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, Fractional Fourier Transform with Applications in Optics and Signal Processing, (Wiley, 2000).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Optical implementation of the RFrFT based on the Lohmann’s type I FrFT configuration.

Fig. 2
Fig. 2

Optical implementation of the RFrFT based on the Lohmann’s type II FrFT configuration.

Fig. 3
Fig. 3

RFrFT of a one-dimensional rectangular function with fractional orders α = 0.3 .

Fig. 4
Fig. 4

RFrFT for a two-dimensional image with fractional order α = 0.8 . (a) Original input image, (b) output image with the phase functions P = exp ( i r 4 ) , (c) output image with a random phase.

Fig. 5
Fig. 5

MSE curves between the decrypted images and the original image with different fractional order α.

Fig. 6
Fig. 6

Double random phase encoding image encryption scheme can be represented with two cascaded RFrFTs.

Equations (11)

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

F α { f ( x ) } ( u ) = f ( x ) K α ( x ; u ) d x ,
K α ( x ; u ) = A α exp [ i π ( u 2 + x 2 tan θ α 2 x u sin θ α ) ] ,
f o u t ( u ) = P * ( u ) F α [ f i n ( x ) P ( x ) ] .
R α { f ( x ) } ( u ) = f ( x ) P ( x ) K α ( x ; u ) P * ( u ) d x ,
R α { c 1 f ( x ) + c 2 f ( y ) } = c 1 R α { f ( x ) } + c 2 R α { f ( y ) } ,
R α { R β { f ( x ) } } = R α + β { f ( x ) } .
f ( x ) 2 d x = R α { f ( x ) } 2 d u .
R 4 + α { f ( x ) } = R α { f ( x ) } .
MSE = m = 1 M n = 1 N D ( m , n ) O ( m , n ) 2 M × N ,
I o u t = F α 2 { P 2 F α 1 { P 1 I i n } } ,
I o u t = R 2 α 2 { R 1 α 1 { I ( x , y ) } } P 1 P 2 ,

Metrics