Abstract

Recently, optical interpretations of the fractional-Fourier-transform operator have been introduced. On the basis of this operator the fractional correlation operator is defined in two different ways that are both consistent with the definition of conventional correlation. Fractional correlation is not always a shift-invariant operation. This property leads to some new applications for fractional correlation as shift-variant image detection. A bulk-optics implementation of fractional correlation is suggested and demonstrated with computer simulations.

© 1995 Optical Society of America

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References

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  1. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).
    [CrossRef]
  2. A. W. Lohmann, H. W. Werlich, “Incoherent matched filter with Fourier holograms,” Appl. Opt. 7, 561–563 (1968).
  3. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  4. L. M. Deen, J. F. Walkup, M. O. Hagler, “Representations of space-variant optical systems using volume holograms,” Appl. Opt. 14, 2438–2446 (1975).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. J. A. Davis, D. M. Cottrell, N. Nestorovic, S. M. Highnote, “Space-variant Fresnel transform optical correlator,” Appl. Opt. 31, 6889–6893 (1992).
    [CrossRef] [PubMed]
  7. G. G. Mu, X. M. Wang, Z. Q. Wang, “A new type of holographic encoding filter for correlation: a lensless intensity correlator,” in Holographic Applications, J. Ke, R. J. Pryputniewicz, eds. Proc. Soc. Photo-Opt. Instrum. Eng. 673, 546–549 (1986).
  8. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
    [CrossRef]
  9. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  10. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  11. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  12. H. M. Ozaktas, D. Mendlovic, “Fourier transform of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  13. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  14. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner distribution functions, and fractional Fourier transform,” Appl. Opt. 33, 6182–6187 (1994).
    [CrossRef]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 7.
  16. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  17. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  18. H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  19. T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution: a tool for time-frequency signal analysis: Part 1: Continuous time signals,” Philips J. Res. 35, 217–250 (1980).
  20. T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution: a tool for time-frequency signal anslysis: Part 2: Discrete time signals,” Philips J. Res. 35, 276–300 (1980).
  21. M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).
  22. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  23. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domain and their relation to chirp and wavelet transform, J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]

1994 (2)

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner distribution functions, and fractional Fourier transform,” Appl. Opt. 33, 6182–6187 (1994).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domain and their relation to chirp and wavelet transform, J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

1993 (4)

1992 (1)

1987 (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1980 (4)

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution: a tool for time-frequency signal analysis: Part 1: Continuous time signals,” Philips J. Res. 35, 217–250 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution: a tool for time-frequency signal anslysis: Part 2: Discrete time signals,” Philips J. Res. 35, 276–300 (1980).

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
[CrossRef]

1978 (1)

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

1977 (1)

1975 (1)

1968 (1)

1967 (1)

1966 (1)

1964 (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Abramovitz, M.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

Barshan, B.

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

Brenner, K.-H.

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Claasen, T. A. C. M.

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution: a tool for time-frequency signal anslysis: Part 2: Discrete time signals,” Philips J. Res. 35, 276–300 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution: a tool for time-frequency signal analysis: Part 1: Continuous time signals,” Philips J. Res. 35, 217–250 (1980).

Cottrell, D. M.

Davis, J. A.

Deen, L. M.

Goodman, J. W.

Hagler, M. O.

Highnote, S. M.

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Krile, T. F.

Lohmann, A. W.

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner distribution functions, and fractional Fourier transform,” Appl. Opt. 33, 6182–6187 (1994).
[CrossRef]

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

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

A. W. Lohmann, H. W. Werlich, “Incoherent matched filter with Fourier holograms,” Appl. Opt. 7, 561–563 (1968).

A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
[CrossRef] [PubMed]

Marks, R. J.

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mecklenbraucker, W. F. G.

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution: a tool for time-frequency signal anslysis: Part 2: Discrete time signals,” Philips J. Res. 35, 276–300 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution: a tool for time-frequency signal analysis: Part 1: Continuous time signals,” Philips J. Res. 35, 217–250 (1980).

Mendlovic, D.

Mu, G. G.

G. G. Mu, X. M. Wang, Z. Q. Wang, “A new type of holographic encoding filter for correlation: a lensless intensity correlator,” in Holographic Applications, J. Ke, R. J. Pryputniewicz, eds. Proc. Soc. Photo-Opt. Instrum. Eng. 673, 546–549 (1986).

Namias, V.

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
[CrossRef]

Nestorovic, N.

Onural, L.

Ozaktas, H. M.

Paris, D. P.

Stegun, I. A.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).
[CrossRef]

Walkup, J. F.

Wang, X. M.

G. G. Mu, X. M. Wang, Z. Q. Wang, “A new type of holographic encoding filter for correlation: a lensless intensity correlator,” in Holographic Applications, J. Ke, R. J. Pryputniewicz, eds. Proc. Soc. Photo-Opt. Instrum. Eng. 673, 546–549 (1986).

Wang, Z. Q.

G. G. Mu, X. M. Wang, Z. Q. Wang, “A new type of holographic encoding filter for correlation: a lensless intensity correlator,” in Holographic Applications, J. Ke, R. J. Pryputniewicz, eds. Proc. Soc. Photo-Opt. Instrum. Eng. 673, 546–549 (1986).

Weaver, C. S.

Werlich, H. W.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Appl. Opt. (7)

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).
[CrossRef]

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Inst. Math. Its Appl. (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (3)

H. M. Ozaktas, D. Mendlovic, “Fourier transform of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Philips J. Res. (2)

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution: a tool for time-frequency signal analysis: Part 1: Continuous time signals,” Philips J. Res. 35, 217–250 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbraucker, “The Wigner distribution: a tool for time-frequency signal anslysis: Part 2: Discrete time signals,” Philips J. Res. 35, 276–300 (1980).

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 7.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970).

G. G. Mu, X. M. Wang, Z. Q. Wang, “A new type of holographic encoding filter for correlation: a lensless intensity correlator,” in Holographic Applications, J. Ke, R. J. Pryputniewicz, eds. Proc. Soc. Photo-Opt. Instrum. Eng. 673, 546–549 (1986).

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Figures (13)

Fig. 1
Fig. 1

Bulk-optics setup for performing a fractional Fourier transform of order P. f and z depend on P.

Fig. 2
Fig. 2

Two optical fractional Fourier transformers in cascade for performing the fractional correlation. The first fractional Fourier transformer is for order P 1, and the second is for P 2. The output is C P 1,P 2 (x).

Fig. 3
Fig. 3

Input signal (dashed curve) and its conventional autocorrelation signal (solid curve) C 1.

Fig. 4
Fig. 4

Fractional autocorrelation of order P 1 = 0.9 according to the first definition special case.

Fig. 5
Fig. 5

Same as Fig. 4 but for P 1 = 0.5.

Fig. 6
Fig. 6

Same as Fig. 4 but for P 1 = 0.

Fig. 7
Fig. 7

Fractional correlation of order P 1 = 0.9 according to the first definition special case with a shift of 150 pixels of the input signal.

Fig. 8
Fig. 8

Same as Fig. 7 but with P 1 = 0.5.

Fig. 9
Fig. 9

Fractional autocorrelation of order P 1 = 0.9 according to the second definition special case.

Fig. 10
Fig. 10

Same as Fig. 9 but with P 1 = 0.5.

Fig. 11
Fig. 11

Same as Fig. 9 but with P 1 = 0.2.

Fig. 12
Fig. 12

Fractional correlation of order P 1 = 0.5 according to the second definition special case with a shift of 150 pixels of the input signal.

Fig. 13
Fig. 13

Same as Fig. 12 but with P 1 = 0.2.

Equations (33)

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F ( ν ) = - f ( x ) exp ( - i 2 π ν x ) d x ,
f ( x ) = - F ( ν ) exp ( i 2 π x ν ) d ν .
F ( ν ) = F f ( x ) .
ϕ = P ( π / 2 )
F P [ u 0 ( x ) ] = u P ( x ) = - u 0 ( x 0 ) exp [ i π ( x 2 + x 0 2 T ) ] × exp ( - i 2 π x x 0 S ) d x 0 ,
T = λ f 1 tan ( ϕ ) ,             S = λ f 1 sin ( ϕ ) ,
C 1 ( x ) = - u 0 ( x 0 ) v 0 * ( x - x ) d x 0 = - u 1 ( ν ) v 1 * ( ν ) exp ( i 2 π ν x ) d ν ,
u 1 ( ν ) = F 1 u 0 ( x ) ,             v 1 ( ν ) =             F 1 ν 0 ( x ) .
if P = 1 ,             C P ( x ) C 1 ( x ) .
if v = u ,             C P ( 0 ) = C 1 ( 0 ) = - u 0 ( x 0 ) 2 d x 0 .
if P = 0 ,             C 0 ( x ) = u 0 ( x ) v 0 * ( x ) .
F P v 0 * ( - x ) = v - P * ( - x ) = v 2 - P * ( x ) .
C P 1 , P 2 ( x ) = - - - u 0 ( x 0 ) v 0 * ( ± x ˜ 0 ) × exp [ i π Ψ 1 ( x , x 0 , x ˜ 0 , y ) ] × exp [ - i 2 π Ψ 2 ( x , x 0 , x ˜ 0 , y ) ] d x 0 d x ˜ 0 d y ,
Ψ 1 ( x , x 0 , x ˜ 0 , y ) = x 2 + y 2 T 2 + x 0 2 + y 2 T 1 x ˜ 0 2 + y 2 T 1 ,
Ψ 2 ( x , x 0 , x ˜ 0 , y ) = y ( x 0 x ˜ 0 S 1 + x S 2 ) ,
T 1 = λ f 1 tan ( ϕ 1 ) ,             T 2 = λ f 1 tan ( ϕ 2 ) ,
S 1 = λ f 1 sin ( ϕ 1 ) ,             S 2 = λ f 1 sin ( ϕ 2 ) ,
ϕ 1 = P 1 ( π / 2 ) ,             ϕ 2 = P 2 ( π / 2 ) ,
C P 1 , P 2 ( x ) = - - u 0 ( x 0 ) v 0 * ( x ˜ 0 ) exp ( - i π / 4 ) ( | 1 T 2 + 1 1 T 1 | ) 1 / 2 × exp { i π [ ( x 2 T 2 + x 0 2 x ˜ 0 2 T 1 ) + ( x 0 x ˜ 0 S 1 + x S 2 ) 2 1 T 2 + 1 1 T 1 ] } × d x 0 d x ˜ 0 .
T 2 = - T 1 ,             S 2 = - S 1 ,
C P 1 , P 2 ( x ) = - - u 0 ( x 0 ) v 0 * ( x ˜ 0 ) exp ( - i π / 4 ) ( 1 / T 1 ) 1 / 2 × exp { i π [ ( - x 2 + x 0 2 x ˜ 0 2 T 1 ) T 1 ( x 0 x ˜ 0 - x S 1 ) 2 ] } d x 0 d x ˜ 0 .
1 / T 2 = 0 ,
P 2 = 1 ,             S 2 = λ f .
- exp [ - i 2 π y ( x 0 - x ˜ 0 S 1 + x S 2 ) ] d y = δ ( x 0 - x ˜ 0 + S 1 S 2 x ) ,
C P 1 , 1 ( x ) = exp [ - i π ( S 1 / S 2 ) 2 T 1 x 2 ] - u 0 ( x 0 ) v 0 * × ( x 0 + S 1 S 2 x ) exp ( - i 2 π T 1 S 1 S 2 x x 0 ) d x 0
C P 1 , 1 ( 0 ) = - u 0 ( x 0 ) u 0 * ( x 0 ) d x 0 = - u 0 ( x 0 ) 2 d x 0 .
T 2 = - T 1 / 2 ,
- exp [ - i 2 π y ( x 0 + x ˜ 0 S 1 + x S 2 ) ] d y = δ ( x 0 + x ˜ 0 + S 1 S 2 x ) .
C P 1 , P 2 ( x ) = exp [ - i π ( S 1 / S 2 ) 2 T 1 x 2 ] - u 0 ( x 0 ) v 0 * × ( - x 0 - S 1 S 2 x ) exp ( - i 2 π T 1 S 1 S 2 x x 0 ) d x 0 .
if u = v ,             C P 1 , P 2 ( 0 ) = C 1 ( x ) = - u 0 ( x 0 ) u 0 * ( - x 0 ) d x .
if P 1 = P 2 = 0 ,             C 0 , 0 ( x ) = u 0 ( x ) v 0 * ( - x ) .
H 1 ( x ) = { F P 1 v ( x ) } *
H 2 ( x ) = F P 1 v * ( - x )

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