Abstract

We investigated the correlation performance of a joint fractional Fourier-transform correlator (JFRTC) using computer simulation results. We present a mathematical analysis suggesting use of processing techniques based on a nonlinear transformation and fractional-order fractional-power fringe-adjusted filter to attain improved performance in terms of discrimination sensitivity and input space–bandwidth utilization. Optimal noise performance for the JFRTC is predicted in the presence of additive white Gaussian noise. An all-optical implementation scheme based on incoherent erasure in a photorefractive crystal is proposed.

© 2001 Optical Society of America

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  1. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  2. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  3. L. M. Bernardo, O. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
    [CrossRef]
  4. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
    [CrossRef] [PubMed]
  5. 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]
  6. S. H. Song, J.-S. Jeong, S. Park, E. H. Lee, “Planar optical implementation of fractional correlation,” Opt. Commun. 143, 287–293 (1997).
    [CrossRef]
  7. A. M. Almansareh, M. A. Abushagur, “Fractional correlations based on the modified fractional order Fourier transform,” Opt. Eng. 37, 175–184 (1998).
    [CrossRef]
  8. S. Garnieri, M. del Carmen Laspiralla, N. Bolognini, E. E. Sicre, “Space-variant optical correlator based on the fractional Fourier transform: implementation by the use of a photorefractive Bi12GeO20 (BGO) holographic filter,” Appl. Opt. 35, 6951–6954 (1996).
    [CrossRef]
  9. A. W. Lohmann, D. Mendlovic, “Fractional joint transform correlator,” Appl. Opt. 36, 7402–7407 (1997).
    [CrossRef]
  10. C. J. Kuo, Y. Luo, “Generalized joint fractional Fourier transform correlators: a compact approach,” Appl. Opt. 37, 8270–8276 (1998).
    [CrossRef]
  11. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  12. M. S. Alam, M. A. Karim, “Fringe-adjusted joint transform correlation,” Appl. Opt. 32, 4344–4350 (1993).
    [CrossRef] [PubMed]
  13. L. B. Almeida, “Product and convolution theorems for the fractional Fourier transform,” IEEE Signal Process. Lett. 4, 15–17 (1997).
    [CrossRef]
  14. M. S. Alam, O. Perez, M. A. Karim, “Preprocessed multiobject joint transform correlator,” Appl. Opt. 32, 3102–3107 (1993).
    [CrossRef] [PubMed]
  15. S. Zhong, J. Jiang, S. Liu, C. Li, “Binary joint transform correlator based on differential processing of the joint power spectrum,” Appl. Opt. 36, 1776–1780 (1997).
    [CrossRef] [PubMed]
  16. H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
    [CrossRef]
  17. P. Pellat-Finet, “Fresnel diffraction and fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  18. A. R. Weeks, H. R. Myler, H. G. Wenaas, “Computer-generated noise images for the evaluation of image processing algorithms,” Opt. Eng. 32, 982–992 (1993).
    [CrossRef]
  19. Y. Bitran, Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Fractional correlation operation: performance analysis,” Appl. Opt. 35, 297–303 (1996).
    [CrossRef] [PubMed]
  20. R. Tripathi, G. S. Pati, K. Singh, “Photorefractive joint transform correlator using incoherent-erasure in two- and four-wave mixing geometries in a BaTiO3 crystal,” Opt. Eng. 37, 2148–2155 (1998).
    [CrossRef]

1998 (3)

A. M. Almansareh, M. A. Abushagur, “Fractional correlations based on the modified fractional order Fourier transform,” Opt. Eng. 37, 175–184 (1998).
[CrossRef]

C. J. Kuo, Y. Luo, “Generalized joint fractional Fourier transform correlators: a compact approach,” Appl. Opt. 37, 8270–8276 (1998).
[CrossRef]

R. Tripathi, G. S. Pati, K. Singh, “Photorefractive joint transform correlator using incoherent-erasure in two- and four-wave mixing geometries in a BaTiO3 crystal,” Opt. Eng. 37, 2148–2155 (1998).
[CrossRef]

1997 (4)

L. B. Almeida, “Product and convolution theorems for the fractional Fourier transform,” IEEE Signal Process. Lett. 4, 15–17 (1997).
[CrossRef]

A. W. Lohmann, D. Mendlovic, “Fractional joint transform correlator,” Appl. Opt. 36, 7402–7407 (1997).
[CrossRef]

S. Zhong, J. Jiang, S. Liu, C. Li, “Binary joint transform correlator based on differential processing of the joint power spectrum,” Appl. Opt. 36, 1776–1780 (1997).
[CrossRef] [PubMed]

S. H. Song, J.-S. Jeong, S. Park, E. H. Lee, “Planar optical implementation of fractional correlation,” Opt. Commun. 143, 287–293 (1997).
[CrossRef]

1996 (3)

1995 (2)

1994 (3)

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

L. M. Bernardo, O. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
[CrossRef]

P. Pellat-Finet, “Fresnel diffraction and fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[CrossRef] [PubMed]

1993 (4)

A. R. Weeks, H. R. Myler, H. G. Wenaas, “Computer-generated noise images for the evaluation of image processing algorithms,” Opt. Eng. 32, 982–992 (1993).
[CrossRef]

M. S. Alam, M. A. Karim, “Fringe-adjusted joint transform correlation,” Appl. Opt. 32, 4344–4350 (1993).
[CrossRef] [PubMed]

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

M. S. Alam, O. Perez, M. A. Karim, “Preprocessed multiobject joint transform correlator,” Appl. Opt. 32, 3102–3107 (1993).
[CrossRef] [PubMed]

1989 (1)

Abushagur, M. A.

A. M. Almansareh, M. A. Abushagur, “Fractional correlations based on the modified fractional order Fourier transform,” Opt. Eng. 37, 175–184 (1998).
[CrossRef]

Alam, M. S.

Almansareh, A. M.

A. M. Almansareh, M. A. Abushagur, “Fractional correlations based on the modified fractional order Fourier transform,” Opt. Eng. 37, 175–184 (1998).
[CrossRef]

Almeida, L. B.

L. B. Almeida, “Product and convolution theorems for the fractional Fourier transform,” IEEE Signal Process. Lett. 4, 15–17 (1997).
[CrossRef]

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Bernardo, L. M.

L. M. Bernardo, O. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
[CrossRef]

Bitran, Y.

Bolognini, N.

Bozdagi, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

del Carmen Laspiralla, M.

Dorsch, R. G.

Garnieri, S.

Javidi, B.

Jeong, J.-S.

S. H. Song, J.-S. Jeong, S. Park, E. H. Lee, “Planar optical implementation of fractional correlation,” Opt. Commun. 143, 287–293 (1997).
[CrossRef]

Jiang, J.

Karim, M. A.

Kuo, C. J.

C. J. Kuo, Y. Luo, “Generalized joint fractional Fourier transform correlators: a compact approach,” Appl. Opt. 37, 8270–8276 (1998).
[CrossRef]

Kutay, M. A.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Lee, E. H.

S. H. Song, J.-S. Jeong, S. Park, E. H. Lee, “Planar optical implementation of fractional correlation,” Opt. Commun. 143, 287–293 (1997).
[CrossRef]

Li, C.

Liu, S.

Lohmann, A. W.

Luo, Y.

C. J. Kuo, Y. Luo, “Generalized joint fractional Fourier transform correlators: a compact approach,” Appl. Opt. 37, 8270–8276 (1998).
[CrossRef]

Mendlovic, D.

Myler, H. R.

A. R. Weeks, H. R. Myler, H. G. Wenaas, “Computer-generated noise images for the evaluation of image processing algorithms,” Opt. Eng. 32, 982–992 (1993).
[CrossRef]

Ozaktas, H. M.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Fractional correlation,” Appl. Opt. 34, 303–309 (1995).
[CrossRef] [PubMed]

Park, S.

S. H. Song, J.-S. Jeong, S. Park, E. H. Lee, “Planar optical implementation of fractional correlation,” Opt. Commun. 143, 287–293 (1997).
[CrossRef]

Pati, G. S.

R. Tripathi, G. S. Pati, K. Singh, “Photorefractive joint transform correlator using incoherent-erasure in two- and four-wave mixing geometries in a BaTiO3 crystal,” Opt. Eng. 37, 2148–2155 (1998).
[CrossRef]

Pellat-Finet, P.

Perez, O.

Sicre, E. E.

Singh, K.

R. Tripathi, G. S. Pati, K. Singh, “Photorefractive joint transform correlator using incoherent-erasure in two- and four-wave mixing geometries in a BaTiO3 crystal,” Opt. Eng. 37, 2148–2155 (1998).
[CrossRef]

Soares, O. D.

L. M. Bernardo, O. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
[CrossRef]

Song, S. H.

S. H. Song, J.-S. Jeong, S. Park, E. H. Lee, “Planar optical implementation of fractional correlation,” Opt. Commun. 143, 287–293 (1997).
[CrossRef]

Tripathi, R.

R. Tripathi, G. S. Pati, K. Singh, “Photorefractive joint transform correlator using incoherent-erasure in two- and four-wave mixing geometries in a BaTiO3 crystal,” Opt. Eng. 37, 2148–2155 (1998).
[CrossRef]

Weeks, A. R.

A. R. Weeks, H. R. Myler, H. G. Wenaas, “Computer-generated noise images for the evaluation of image processing algorithms,” Opt. Eng. 32, 982–992 (1993).
[CrossRef]

Wenaas, H. G.

A. R. Weeks, H. R. Myler, H. G. Wenaas, “Computer-generated noise images for the evaluation of image processing algorithms,” Opt. Eng. 32, 982–992 (1993).
[CrossRef]

Zalevsky, Z.

Zhong, S.

Appl. Opt. (1)

C. J. Kuo, Y. Luo, “Generalized joint fractional Fourier transform correlators: a compact approach,” Appl. Opt. 37, 8270–8276 (1998).
[CrossRef]

Appl. Opt. (8)

IEEE Signal Process. Lett. (1)

L. B. Almeida, “Product and convolution theorems for the fractional Fourier transform,” IEEE Signal Process. Lett. 4, 15–17 (1997).
[CrossRef]

IEEE Trans. Signal Process. (1)

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

IEEE Trans. Signal Process. (1)

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

J. Opt. Soc. Am. A (1)

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

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

A. M. Almansareh, M. A. Abushagur, “Fractional correlations based on the modified fractional order Fourier transform,” Opt. Eng. 37, 175–184 (1998).
[CrossRef]

A. R. Weeks, H. R. Myler, H. G. Wenaas, “Computer-generated noise images for the evaluation of image processing algorithms,” Opt. Eng. 32, 982–992 (1993).
[CrossRef]

Opt. Commun. (2)

L. M. Bernardo, O. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
[CrossRef]

S. H. Song, J.-S. Jeong, S. Park, E. H. Lee, “Planar optical implementation of fractional correlation,” Opt. Commun. 143, 287–293 (1997).
[CrossRef]

Opt. Eng. (1)

R. Tripathi, G. S. Pati, K. Singh, “Photorefractive joint transform correlator using incoherent-erasure in two- and four-wave mixing geometries in a BaTiO3 crystal,” Opt. Eng. 37, 2148–2155 (1998).
[CrossRef]

Opt. Lett. (1)

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Figures (15)

Fig. 1
Fig. 1

Block diagram for a JFRTC.

Fig. 2
Fig. 2

Surface plots of the autocorrelation signals that we obtained using a circular disk object with a diameter of 50 pixels in the JFRTCs corresponding to (a) p 1 = 1, p 2 = 1, (b) p 1 = 0.6, p 2 = 1.

Fig. 3
Fig. 3

Variation of the FWHM of the correlation signal with fractional order p 1 in the JFRTC corresponding to (a) p 2 = 1, (b) p 2 = 0.8.

Fig. 4
Fig. 4

(a) Binary test objects: (i) reference and (ii) nontarget of size 50 × 28 pixels. (b) Gray-level test objects: (i) reference of size 80 × 42 pixels and (ii) nontarget of size 94 × 49 pixels.

Fig. 5
Fig. 5

Correlation results obtained from (a) JTC and (b) and (c) JFRTC corresponding to the FRT orders indicated in the graphs by use of an input image containing the binary test objects shown in Fig. 4(a).

Fig. 6
Fig. 6

Intensity profiles of the correlation output shown in Figs. 5(b) and 5(c). (a) and (c) were obtained corresponding to y = 115. (b) and (d) are profiles showing C rt 1 and C rt 2 signals.

Fig. 7
Fig. 7

Correlation results obtained for binary test objects shown in Fig. 4(a) for the JFRTCs corresponding to p 1 = 0.55 and p 2 = 1 by (a) and (b) nonlinear transformation and (c) and (d) FOFPFAF.

Fig. 8
Fig. 8

Correlation results obtained by gradient preprocessing in the JFRTC corresponding to the FRT orders indicated. (a) p 1 = 0.75, (b) p 1 = 0.55.

Fig. 9
Fig. 9

Correlation results obtained by an input image containing the gray-scale test objects shown in Fig. 4(b): (a) conventional JTC and (b) JFRTC.

Fig. 10
Fig. 10

Correlation results obtained for the gray-scale test objects shown in Fig. 4(b) for JFRTCs corresponding to p 1 = 0.5 and p 2 = 1 by (a) and (b) nonlinear transformation and (c) and (d) FOFPFAF.

Fig. 11
Fig. 11

(a) and (b) FC signals obtained from the JFRTC corresponding to p 1 = 0.5 and p 2 = 1 for gray-scale objects cluttered with additive white Gaussian noise.

Fig. 12
Fig. 12

Correlation results showing the performance of the JFRTC for gray-scale objects cluttered with additive white Gaussian noise corresponding to p 1 = 0.5 and p 2 = 1 with (a) nonlinear transformation and (b) FOFPFAP.

Fig. 13
Fig. 13

Variation of SNRout with SNRinp in the presence of additive white Gaussian noise.

Fig. 14
Fig. 14

(a) Distribution of the object and noise power spectrum in the FRT domain corresponding to p 1 = 0.5 for a target object shown in the inset. (b) and (c) Correlation results obtained from the JTC and the JFRTC, respectively.

Fig. 15
Fig. 15

Proposed architecture for an all-optical implementation of the JFRTC: BE, beam expander; BS, beam splitter; S, shutter; PBS, polarizing beam splitter; L, lens; f, focal length of lens; z, free-space distance associated with the FRT.

Tables (2)

Tables Icon

Table 1 Calculated DR Values and Comparison (in dB) of the DS of JFRTCs with a Conventional JTC for the Binary (Object 1) and Gray-Level (Object 2) Test Objects of Fig. 4

Tables Icon

Table 2 Performance of JFRTCs in the Presence of Additive White Gaussian Noise in the Input Scene of the Target Object in Terms of SNR and dc/I oc , where I oc is the Intensity in the Autocorrelation FC Signal

Equations (18)

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I x ,   y = r x ,   y + n = 1 N   t n x - x n ,   y exp - jm n x .
F p 1 I x ,   y u ,   v = F p 1 r x ,   y u ,   v + n = 1 N exp j x n 2 - m n 2 4 sin π p 1 - u x n   sin π p 1 2 + m n   cos π p 1 2 × F p 1 t n x ,   y u - x n   cos π p 1 2 + m n   sin π p 1 2 ,   v ,
m n = x n   cot π p 1 2 ,     n = 1 ,   2 , N .
E u ,   v = | F p 1 r x ,   y | 2 + n = 1 N | F p 1 t n x ,   y | 2 + F p 1 * r x ,   y F p 1 t n x ,   y × exp j x n 2 - m n 2 4 sin π p 1 - u x n   sin π p 1 2 + m n   cos π p 1 2 + c . c + n , l = 1 n l N F p 1 * t n x ,   y F p 1 t l x ,   y × exp x n - l 2 - m n - l 2 4 sin π p 1 - u x n - l   sin π p 1 2 + m n - l   cos π p 1 2 + c . c . ,
C p 1 , p 2 x ,   y = C r , r p 1 , p 2 + n = 1 N   C t n , t n p 1 , p 2 + n = 1 N   C r , t n p 1 , p 2 × x ± x n   sin π p 1 2 + m n × cos π p 1 2 sin π p 2 2 ,   y × exp ± j ϕ n x ,   y + n , l = 1 n l N   C t n , t l p 1 , p 2 x ± x n - l   sin π p 1 2 + m n - l   cos π p 1 2 sin π p 2 2 ,   y × exp ± j ϕ n - l x ,   y ,
ϕ n x ,   y = x n 2 - m n 2 4 sin π p 1 - x n   sin π p 1 2 + m n   cos π p 1 2 x + 1 2 x n   sin π p 1 2 + m n   cos π p 1 2 sin π p 2 2 × cos π p 2 2 ,
C f , g p 1 , p 2 x ,   y = F p 2 F p 1 * f x ,   y F p 1 g x ,   y ,
± x n   sin π p 1 2 + m n   cos π p 1 2 ,   0
± x n - l   sin π p 1 2 + m n - l   cos π p 1 2 ,   0
t k E = E u ,   v k ,     0 k 1 .
t k E = v = 1 Γ k + 1 | F p 1 r x ,   y |   | F p 1 t 1 x ,   y | k 2 k - 1 Γ 1 - ν - k 2 Γ 1 - ν + k 2 × cos ν x 1   sin π p 1 2 + m 1   cos π p 1 2 + ν x 1 2 - m 1 2 4 sin π p 1 + ν ϕ r p 1 - ν ϕ t 1 p 1 ,
E FOFPFAF u ,   v = B A + | F p 1 r x ,   y u ,   v | 2 p × E u ,   v ;     0 p 2 ,
FOFPFAF 1 - A | F p 1 r x ,   y | p × circ   d u 2 + v 2 1 / 2 ,
C p 1 , p 2 x ,   y = 1 2 π csc 2 π p 2 2 exp j x 2 + y 2 2 × cot π p 2 2 , C p 1 , p 2 x ,   y = exp - j x 2 + y 2 2 cot π p 2 2 FT FOFPFAF x   csc π p 2 2 ,   y   csc π p 2 2 ,
F p 1 I x ,   y u ,   v = u F p 1 I x ,   y u ,   v cos π p 1 2 + ju F p 1 I x ,   y u ,   v × sin π p 1 2 ,
t 1 u ,   v = 1 1 + f m E n u ,   v , m = I e I c , I c = I 1 + I 2 + I 3 ,
t 2 u ,   v = 1 1 + f m 1 E n u ,   v + f m 2 E rn u ,   v ,
m 1 = I e 1 I c ,   m 2 = I e 2 I c ,   I c = I 1 + I 2 + I 3 .

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