Abstract

Fractional correlation was introduced recently. We generalize the architecture of a joint (Fourier) transform correlator (JTC) to achieve the joint fractional (Fourier) transform correlator (JFrTC) such that fractional correlation can be obtained. Here the Fourier transform in the JTC is replaced by the fractional Fourier transform, and four different JFrTC architectures can be implemented. The mathematical derivations for these JFrTC architectures are given, together with the simulation verifications. The JFrTC can provide a correlation signal similar to a delta function but with a small discrimination ratio, such that it is insensitive to additive noise. In a conventional JTC the distance between the two desired correlation signals at the output plane is fixed and depends on the distance between the input and the reference signals. However, with a given fractional order and an additional phase mask the separation distance between the two correlation signals at the output plane of a JFrTC can be larger or smaller than that of a JTC. This property is useful for the applications of real-time target tracking. Unlike in a previous approach [Appl. Opt. 36, 7402 (1997)], we need only two fractional Fourier transformations instead of three to achieve fractional correlation.

© 1998 Optical Society of America

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References

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  1. F. T. S. Yu, I. C. Khoo, Principles of Optical Engineering (Wiley, New York, 1990).
  2. A. W. Lohmann, D. Mendlovic, “Fractional joint transform correlator,” Appl. Opt. 36, 7402–7407 (1997).
    [CrossRef]
  3. D. Mendlovic, Y. Bitran, R. G. Dorsch, A. W. Lohmann, “Optical fractional correlation: experimental results,” J. Opt. Soc. Am. A 12, 1665–1670 (1995).
    [CrossRef]
  4. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]
  5. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  6. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
    [CrossRef]
  7. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation. I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  8. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation. II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  9. B. Javidi, C. J. Kuo, “Joint transform image correlation using a binary spatial light modulator at the Fourier plane,” Appl. Opt. 27, 663–665 (1988).
    [CrossRef] [PubMed]
  10. C. J. Kuo, “Theoretical expression for the correlation signal of nonlinear joint-transform correlators,” Appl. Opt. 31, 6264–6271 (1992).
    [CrossRef] [PubMed]
  11. Q. Tang, B. Javidi, “Technique for reducing the redundant and self-correlation terms in joint transform correlators,” Appl. Opt. 32, 1911–1918 (1993).
    [CrossRef] [PubMed]

1997 (1)

1995 (2)

1993 (3)

1992 (1)

1988 (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 (1)

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

Bitran, Y.

Dorsch, R. G.

Javidi, B.

Kerr, F. H.

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

Khoo, I. C.

F. T. S. Yu, I. C. Khoo, Principles of Optical Engineering (Wiley, New York, 1990).

Kuo, C. J.

Lohmann, A. W.

McBride, A. C.

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

Mendlovic, D.

Namias, V.

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

Ozaktas, H. M.

Tang, Q.

Yu, F. T. S.

F. T. S. Yu, I. C. Khoo, Principles of Optical Engineering (Wiley, New York, 1990).

Appl. Opt. (5)

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 order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980).
[CrossRef]

J. Opt. Soc. Am. A (3)

Other (1)

F. T. S. Yu, I. C. Khoo, Principles of Optical Engineering (Wiley, New York, 1990).

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Figures (4)

Fig. 1
Fig. 1

Generalized JFrTC architecture: CCD, charge-coupled device; FTL’s, FT lenses; SLM, spatial light modulator. z = f*[1 -cos(aπ/2)], where f is the focal length and a is the fractional order.

Fig. 2
Fig. 2

Images for the correlation test: (a) building, (b) jet plane.

Fig. 3
Fig. 3

Autocorrelations of the four JFrTC’s: (a) from FT–FT architecture; (b) from FT–FrT architecture with a 2 = 1/3; (c) from FrT–FT architecture with a 1 = (2/π)tan-1(128/100) and p = 100; (d) from FrT–FrT architecture with a 1= (2/π)tan-1(128/100), a 2 = 1/3, and p = 100.

Fig. 4
Fig. 4

Cross correlation of the four JFrTC’s with the building and the jet plane as input and reference signals, respectively: (a) from FT–FT architecture; (b) from FT–FrT architecture with a 2 = 1/3; (c) from FrT–FT architecture with a 1 = (2/π)tan-1(128/100) and p = 100; (d) from FrT–FrT architecture with a 1= (2/π)tan-1(128/100), a 2 = 1/3, and p = 100.

Tables (1)

Tables Icon

Table 1 Values of Parameters of JFrTC’s

Equations (17)

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U w x ,   w y = F w x ,   w y + G w x ,   w y exp - jbw x ,
E w x ,   w y | U w x ,   w y | 2 = | F w x ,   w y | 2 + | G w x ,   w y | 2 + F * w x ,   w y G w x ,   w y exp - jw x b + F w x ,   w y G * w x ,   w y exp jw x b ,
C x ,   y = f x ,   y f * x ,   y + g x ,   y g * x ,   y + f x ,   y g * x - b ,   y + f * x ,   y g x + b ,   y ,
a u 0 x 0 = u a x = -   u 0 x 0 exp j π x 2 + x 0 2 T × exp - j 2 π   xx 0 S d x 0 ,
a f x 0 + b = exp - jb w + b 2 cos π a 2 sin π a 2 × F a f x 0 w + b   cos π a 2 .
a exp jbx 0 f x 0 = exp jb w + b 2 sin π a 2 sin π a 2 × F a f x 0 w + b   sin π a 2 .
C a 1 , a 2 x a 2 a 1 u x 0 a 1 * v x 0 .
a 1 g x - b ,   y = exp jb w x - b 2 cos π a 1 2 sin π a 1 2 × F a 1 g x ,   y w x - b   cos π a 1 2 ,   w y .
a 1 exp jpx g x - b ,   y = exp jp w x + p 2 sin π a 1 2 cos π a 1 2 × F a 1 g x - b ,   y w x + p   sin π a 1 2 ,   w y = exp j p w x + p 2 sin π a 1 2 cos π a 1 2 + b w x - b 2 cos π a 1 2 sin π a 1 2 × F a 1 g x ,   y w x - b   cos π a 1 2 + p   sin π a 1 2 ,   w y .
- b   cos π a 1 2 + p   sin π a 1 2 = 0     or     a 1 = 2 π tan - 1 b p ,
a 1 exp jpx g x - b ,   y = exp j p 2 - b 2 4 sin   π a 1 + w x p   cos π a 1 2 + b   sin π a 1 2 × F a 1 g x ,   y w x ,   w y .
| F a 1 | f x ,   y w x ,   w y + exp j p 2 - b 2 4 sin   π a 1 + w x p   cos π a 1 2 + b   sin π a 1 2 F a 1 g x ,   y w x ,   w y | 2 = F a 1 f x ,   y w x ,   w y F a 1 * f x ,   y w x ,   w y + F a 1 g x ,   y w x ,   w y F a 1 * g x ,   y w x ,   w y + F a 1 * f x ,   y w x ,   w y F a 1 g x ,   y w x ,   w y × exp j p 2 - b 2 4 sin   π a 1 + w x p   cos π a 1 2 + b   sin π a 1 2 + F a 1 f x ,   y w x ,   w y F a 1 * g x ,   y w x ,   w y × exp - j p 2 - b 2 4 sin   π a 1 + w x p   cos π a 1 2 + b   sin π a 1 2 .
a 2 F a 1 * f x ,   y w x ,   w y F a 1 g x ,   y w x ,   w y × exp j p 2 - b 2 4 sin   π a 1 + w x p   cos π a 1 2 + b   sin π a 1 2 = exp j   p 2 - b 2 4 sin   π a 1 × a 2 F a 1 * f x ,   y w x ,   w y F a 1 g x ,   y w x ,   w y × exp jw x p   cos π a 1 2 + b   sin π a 1 2 = exp j   p 2 - b 2 4 sin   π a 1 exp j p   cos π a 1 2 + b   sin π a 1 2 cos π a 2 2 x + 1 2 p   cos π a 1 2 + b   sin π a 1 2 sin π a 2 2 C a 1 , a 2 x + p   cos π a 1 2 + b   sin π a 1 2 sin π a 2 2 ,   y .
exp - j p   cos π a 1 2 + b   sin π a 1 2 × cos π a 2 2 x - 1 2 p   cos π a 1 2 + b   sin π a 1 2 sin π a 2 2 × exp - j   p 2 - b 2 4 sin   π a 1 C a 1 , a 2 x - p   cos π a 1 2 + b   sin π a 1 2 sin π a 2 2 ,   y .
exp jb x + b 2 sin π a 2 2 cos π a 2 2 C 1 , a 2 x + b   sin π a 2 2 ,   y
exp j   p 2 - b 2 4 sin   π a 1 × C a 1 , 1 x + p   cos π a 1 2 + b   sin π a 1 2 ,   y .
- 2 p 2   +   b 2 2 p   cos π a 1 2 + b   sin π a 1 2 2 p 2   +   b 2 ,

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