Abstract

A fractional correlator that is based on the anamorphic fractional Fourier transform is defined. This new, to our knowledge, correlator has been extended to work with multiple filters. The novelty introduced by the suggested system is the possibility of the simultaneous detection of several objects in different parts of the input scene (when anamorphic optics are dealt with), thereby permitting an independent degree of space invariance in two perpendicular directions. Computer experiments as well as experimental optical implementation are presented.

© 1996 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  22. D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Localized fractional Fourier transform,” Opt. Commun. (to be published).
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1996

1995

A. Sahin, H. M. Osaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in two dimensions,” J. Opt. Soc. Am. A 120, 134–138 (1995).

1994

1993

1992

1990

1989

1987

1986

1969

1967

1964

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

1937

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Almeida, L. B.

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

L. B. Almeida, An Introduction to the Angular Fourier Transform (IEEE, Minneapolis, 1993).

Barshan, B.

Bernardo, L. M.

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

Bitran, Y.

Y. Bitran, Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Performance analysis of the fractional correlation operation,” Appl. Opt. 35, 297–303 (1996).
[CrossRef] [PubMed]

D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garćia, H. M. Ozaktas, “Anamorphic fractional Fourier transforming—optical implementation and applications,” Appl. Opt. (to be published).

Casasent, D.

Casasent, D. P.

Cauldfield, H. J.

Chang, W. T.

Condon, E. U.

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Cottrell, D. M.

Davis, J. A.

Dorsch, R.

J. Garcia, R. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Adjustable fractional Fourier correlator and fractional spatial filtering,” Opt. Commun. (to be published).

Dorsch, R. G.

Y. Bitran, Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Performance analysis of the fractional correlation operation,” Appl. Opt. 35, 297–303 (1996).
[CrossRef] [PubMed]

D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garćia, H. M. Ozaktas, “Anamorphic fractional Fourier transforming—optical implementation and applications,” Appl. Opt. (to be published).

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Localized fractional Fourier transform,” Opt. Commun. (to be published).

Ferreira, C.

D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garćia, H. M. Ozaktas, “Anamorphic fractional Fourier transforming—optical implementation and applications,” Appl. Opt. (to be published).

J. Garcia, R. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Adjustable fractional Fourier correlator and fractional spatial filtering,” Opt. Commun. (to be published).

Garcia, J.

J. Garcia, R. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Adjustable fractional Fourier correlator and fractional spatial filtering,” Opt. Commun. (to be published).

D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garćia, H. M. Ozaktas, “Anamorphic fractional Fourier transforming—optical implementation and applications,” Appl. Opt. (to be published).

Gregory, D. A.

Highnote, S. M.

Jin, Y.

Konforti, N.

Lohmann, A. W.

Mahalanobis, A.

Maloney, W.

Marom, E.

Mendlovic, D.

Y. Bitran, Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Performance analysis of the fractional correlation operation,” Appl. Opt. 35, 297–303 (1996).
[CrossRef] [PubMed]

A. Sahin, H. M. Osaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in two dimensions,” J. Opt. Soc. Am. A 120, 134–138 (1995).

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

D. Mendlovic, H. Ozaktas, A. W. Lohmann, “Graded index fibers, Wigner distribution functions and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994).
[CrossRef] [PubMed]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: Part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: Part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

D. Mendlovic, N. Konforti, E. Marom, “Shift and projection invariant pattern recognition using logarithmic harmonics,” Appl. Opt. 29, 4784–4789 (1990).
[CrossRef] [PubMed]

D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garćia, H. M. Ozaktas, “Anamorphic fractional Fourier transforming—optical implementation and applications,” Appl. Opt. (to be published).

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Localized fractional Fourier transform,” Opt. Commun. (to be published).

Nestorovic, N.

Onural, L.

Osaktas, H. M.

A. Sahin, H. M. Osaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in two dimensions,” J. Opt. Soc. Am. A 120, 134–138 (1995).

Ozaktas, H.

Ozaktas, H. M.

Paris, D. P.

Pellat-Finet, P.

Sahin, A.

A. Sahin, H. M. Osaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in two dimensions,” J. Opt. Soc. Am. A 120, 134–138 (1995).

Soares, O. D. D.

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

VanderLugt, A.

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

Vijaya Kumar, B. V. K.

Yu, F. T. S.

Zalevsky, Z.

Y. Bitran, Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Performance analysis of the fractional correlation operation,” Appl. Opt. 35, 297–303 (1996).
[CrossRef] [PubMed]

J. Garcia, R. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Adjustable fractional Fourier correlator and fractional spatial filtering,” Opt. Commun. (to be published).

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Localized fractional Fourier transform,” Opt. Commun. (to be published).

Zhang, C.

Appl. Opt.

IEEE Trans. Inf. Theory

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

IEEE Trans. Signal Process.

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

J. Opt. Soc. Am. A

Opt. Commun.

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

Opt. Lett.

Proc. Natl. Acad. Sci. USA

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Other

L. B. Almeida, An Introduction to the Angular Fourier Transform (IEEE, Minneapolis, 1993).

D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garćia, H. M. Ozaktas, “Anamorphic fractional Fourier transforming—optical implementation and applications,” Appl. Opt. (to be published).

J. Garcia, R. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Adjustable fractional Fourier correlator and fractional spatial filtering,” Opt. Commun. (to be published).

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Localized fractional Fourier transform,” Opt. Commun. (to be published).

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Figures (8)

Fig. 1
Fig. 1

Optical setup for performing a FRT operation.

Fig. 2
Fig. 2

Algorithm for obtaining a generalized fractional correlation.

Fig. 3
Fig. 3

Experimental setup for obtaining the anamorphic fractional correlation.

Fig. 4
Fig. 4

Input image used for computer simulations and optical experiments.

Fig. 5
Fig. 5

Numerical calculation of the correlation, showing the detection of an F18 target in the upper part of the image and of a Tornado target in the lower part.

Fig. 6
Fig. 6

Input image used for computer simulations.

Fig. 7
Fig. 7

Numerical calculation of the correlation, showing the detection of an F18 target with invariance to the one-dimensional scale in two axes.

Fig. 8
Fig. 8

Experimental results for multiple anamorphic FRT correlations with the image in Fig. 4.

Equations (8)

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

F P ( x ) = P { f ( x ) } = - + f ( x ) × exp ( i π x 2 + x 2 T ) exp ( - i 2 π x x S ) d x ,
T = λ f 1 tan ϕ , S = λ f 1 sin ϕ , ϕ = P π 2 ,
C P 1 , P 2 , P 3 ( x ) = P 3 [ P 1 { f ( x ) } P 2 { g ( x ) } ] ,
P 1 = P , P 2 = P , P 3 = 1 ,
F P x , P y ( x , y ) = - + f ( x , y ) exp [ i π ( x 2 + x 2 T x + y 2 + y 2 T y ) ] × exp [ - i 2 π ( x x S x + y y S y ) ] d x d y ,
T x = λ f 1 x tan ϕ x , S x = λ f 1 x sin ϕ x , ϕ x = P x π 2 , T y = λ f 1 y tan ϕ y , S y = λ f 1 y sin ϕ y , ϕ y = P y π 2 ,
Z = Z x = f 1 x sin P x π 2 = Z y = f 1 y sin P y π 2 .
AR = f 1 x f 1 y = sin ( P y π / 2 ) sin ( P x π / 2 ) .

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