Abstract

We have investigated the multiplicity and complexity in eigenvalues of the fractional Fourier transform and found that the ambiguity of the eigenvalues may indicate randomness. We have therefore proposed a method to randomize the Fourier transform. Such a random Fourier transform can be applied in the field of image encryption and decryption.

© 2007 Optical Society of America

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References

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  1. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).
  2. C. C. Shih, Opt. Commun. 118, 495 (1995).
    [CrossRef]
  3. S. Liu, J. Zhang, and Y. Zhang, Opt. Commun. 133, 50 (1997).
    [CrossRef]
  4. B. Zhu, S. Liu, and Q. Ran, Opt. Lett. 25, 1159 (2000).
    [CrossRef]
  5. S. C. Pei and W. L. Hsue, IEEE Signal Process. Lett. 13, 329 (2006).
    [CrossRef]
  6. C. C. Shih, Opt. Lett. 20, 1178 (1995).
    [CrossRef] [PubMed]
  7. G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).
    [CrossRef]
  8. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, and R. G. Dorsch, Opt. Commun. 125, 18 (1996).
    [CrossRef]
  9. S. Li and Y. Wang, Inf. Sci. (N.Y.) 177, 192 (2007).
  10. M. S. Baptista, S. Boccaletti, K. Josic, and I. Leyva, Phys. Rev. E 69, 056228 (2004).
    [CrossRef]
  11. S. C. Pei and M. H. Yeh, Opt. Lett. 22, 1047 (1997).
    [CrossRef] [PubMed]
  12. C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Signal Process. Lett. 48, 1329 (2000).
    [CrossRef]
  13. Z. Liu, H. Zhao, and S. Liu, Opt. Commun. 255, 357 (2005).
    [CrossRef]

2007 (1)

S. Li and Y. Wang, Inf. Sci. (N.Y.) 177, 192 (2007).

2006 (1)

S. C. Pei and W. L. Hsue, IEEE Signal Process. Lett. 13, 329 (2006).
[CrossRef]

2005 (1)

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

2004 (1)

M. S. Baptista, S. Boccaletti, K. Josic, and I. Leyva, Phys. Rev. E 69, 056228 (2004).
[CrossRef]

2000 (3)

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Signal Process. Lett. 48, 1329 (2000).
[CrossRef]

G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).
[CrossRef]

B. Zhu, S. Liu, and Q. Ran, Opt. Lett. 25, 1159 (2000).
[CrossRef]

1997 (2)

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

S. Liu, J. Zhang, and Y. Zhang, Opt. Commun. 133, 50 (1997).
[CrossRef]

1996 (1)

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, and R. G. Dorsch, Opt. Commun. 125, 18 (1996).
[CrossRef]

1995 (2)

Baptista, M. S.

M. S. Baptista, S. Boccaletti, K. Josic, and I. Leyva, Phys. Rev. E 69, 056228 (2004).
[CrossRef]

Boccaletti, S.

M. S. Baptista, S. Boccaletti, K. Josic, and I. Leyva, Phys. Rev. E 69, 056228 (2004).
[CrossRef]

Candan, C.

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Signal Process. Lett. 48, 1329 (2000).
[CrossRef]

Cariolaro, G.

G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).
[CrossRef]

Dorsch, R. G.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, and R. G. Dorsch, Opt. Commun. 125, 18 (1996).
[CrossRef]

Erseghe, T.

G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).
[CrossRef]

Hsue, W. L.

S. C. Pei and W. L. Hsue, IEEE Signal Process. Lett. 13, 329 (2006).
[CrossRef]

Josic, K.

M. S. Baptista, S. Boccaletti, K. Josic, and I. Leyva, Phys. Rev. E 69, 056228 (2004).
[CrossRef]

Kraniauskas, P.

G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).
[CrossRef]

Kutay, M. A.

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Signal Process. Lett. 48, 1329 (2000).
[CrossRef]

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

Laurenti, N.

G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).
[CrossRef]

Leyva, I.

M. S. Baptista, S. Boccaletti, K. Josic, and I. Leyva, Phys. Rev. E 69, 056228 (2004).
[CrossRef]

Li, S.

S. Li and Y. Wang, Inf. Sci. (N.Y.) 177, 192 (2007).

Liu, S.

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

B. Zhu, S. Liu, and Q. Ran, Opt. Lett. 25, 1159 (2000).
[CrossRef]

S. Liu, J. Zhang, and Y. Zhang, Opt. Commun. 133, 50 (1997).
[CrossRef]

Liu, Z.

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

Lohmann, A. W.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, and R. G. Dorsch, Opt. Commun. 125, 18 (1996).
[CrossRef]

Mendlovic, D.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, and R. G. Dorsch, Opt. Commun. 125, 18 (1996).
[CrossRef]

Ozaktas, H. M.

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Signal Process. Lett. 48, 1329 (2000).
[CrossRef]

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

Pei, S. C.

S. C. Pei and W. L. Hsue, IEEE Signal Process. Lett. 13, 329 (2006).
[CrossRef]

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

Ran, Q.

Shih, C. C.

Wang, Y.

S. Li and Y. Wang, Inf. Sci. (N.Y.) 177, 192 (2007).

Yeh, M. H.

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, and R. G. Dorsch, Opt. Commun. 125, 18 (1996).
[CrossRef]

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

Zhang, J.

S. Liu, J. Zhang, and Y. Zhang, Opt. Commun. 133, 50 (1997).
[CrossRef]

Zhang, Y.

S. Liu, J. Zhang, and Y. Zhang, Opt. Commun. 133, 50 (1997).
[CrossRef]

Zhao, H.

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

Zhu, B.

IEEE Signal Process. Lett. (2)

S. C. Pei and W. L. Hsue, IEEE Signal Process. Lett. 13, 329 (2006).
[CrossRef]

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Signal Process. Lett. 48, 1329 (2000).
[CrossRef]

IEEE Trans. Signal Process. (1)

G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).
[CrossRef]

Inf. Sci. (N.Y.) (1)

S. Li and Y. Wang, Inf. Sci. (N.Y.) 177, 192 (2007).

Opt. Commun. (4)

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, and R. G. Dorsch, Opt. Commun. 125, 18 (1996).
[CrossRef]

C. C. Shih, Opt. Commun. 118, 495 (1995).
[CrossRef]

S. Liu, J. Zhang, and Y. Zhang, Opt. Commun. 133, 50 (1997).
[CrossRef]

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

Opt. Lett. (3)

Phys. Rev. E (1)

M. S. Baptista, S. Boccaletti, K. Josic, and I. Leyva, Phys. Rev. E 69, 056228 (2004).
[CrossRef]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Eigenvalue distributions of the Fourier transform, FrFT, and other variations.

Fig. 2
Fig. 2

DRFT of a 1D rectangle function with a set of random eigenvalues.

Fig. 3
Fig. 3

Results of image encryption and decryption with DRFT: (a) original image, (b) encrypted image, (c) decrypted image with incorrect eigenvalues, (d) decrypted image with correct eigenvalues.

Fig. 4
Fig. 4

Optical setup for the implementation of the RFT.

Equations (15)

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F α [ f ( x ) ] = n C n λ F α , n ϕ n ( x ) ,
ϕ n ( x ) = 1 2 n π n ! H n ( x ) exp ( x 2 2 )
C n = + f ( x ) ϕ n ( x ) d x .
λ F , n = exp [ i π 2 mod ( n , 4 ) ] ,
λ F α , n = { [ exp ( i n π 2 ) ] α : n = 0 , 1 , 2 , }
= { exp ( i n π α 2 ) : n = 0 , 1 , 2 , } .
λ F S α , n = exp [ i α π 2 mod ( n , 4 ) ]
λ F L α , n = exp [ i α π 2 mod ( n , 4 k ) ] .
λ F α , n = [ exp ( i π n 2 ) ] α = exp [ i P ( π n 2 + 2 k π ) Q ] ,
α = r r P Q .
R [ f ( x ) ] = n C n λ n R ϕ n ( x ) ,
λ n R = exp [ i π Rand ( n ) ]
X R = R x ,
R = V D λ R V t .
R x I R y t = V D λ , x R V t I V D λ , y R V t .

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