Abstract

We propose a method for designing a correlator for achieving rotation-invariant and controllable space-variant optical correlation. The design concept is based on a combination of fractional correlation and circular-harmonic decomposition of the reference object. The suggested method is described and analyzed in detail. Numerical simulations show that this new correlator might provide potential applications in practice.

© 1998 Optical Society of America

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References

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1998 (1)

1997 (3)

1996 (3)

Z. Zalevsky, I. Ouzieli, D. Mendlovic, “Wavelet-transform-based composite filters for invariant pattern recognition,” Appl. Opt. 35, 3141–3147 (1996).
[CrossRef] [PubMed]

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

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

1995 (2)

1994 (1)

1993 (4)

1987 (1)

1982 (1)

1980 (1)

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

1966 (1)

1964 (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–140 (1964).

Arikan, O.

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

Arizaga, R.

Arsenault, H. H.

Barshan, B.

Bozdag, G.

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

Chen, L.

Dong, B.-Z.

Garcia, J.

Goodman, J. W.

Granieri, S.

Gu, B.-Y.

Hsu, Y. N.

Kutay, M. A.

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

Li, C.

Liu, S.

Lohmann, A. W.

Mendlovic, D.

Namias, V.

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

Onural, L.

Ouzieli, I.

Ozaktas, H. M.

Sheng, Y.

Sicre, E. E.

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–140 (1964).

Weaver, C. S.

Xu, J.

Yang, G.-Z.

Zalevsky, Z.

Zhang, Y.

Appl. Opt. (6)

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–140 (1964).

IEEE Trans. Signal Process. (1)

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

J. Inst. Math. Appl. (1)

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

J. Opt. Soc. Am. A (7)

Opt. Commun. (2)

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[CrossRef]

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

Opt. Lett. (1)

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Figures (8)

Fig. 1
Fig. 1

Scheme of the optical fractional correlator.

Fig. 2
Fig. 2

Reference object.

Fig. 3
Fig. 3

Intensity distribution on the output plane with the original reference object as the input object.

Fig. 4
Fig. 4

Intensity distribution on the output plane for the letter E rotated by 135° at the center of the input plane.

Fig. 5
Fig. 5

False-alarm object: the letter P.

Fig. 6
Fig. 6

Intensity distribution on the output plane with the false-alarm object as the input object.

Fig. 7
Fig. 7

Mixed-input image for rotation-invariant and controllable space-variant correlation.

Fig. 8
Fig. 8

Intensity distribution on the output plane with the object shown in Fig. 7 as the input object.

Tables (2)

Tables Icon

Table 1 Intensities of FC Peaks for P = 0.8

Tables Icon

Table 2 Intensities of FC Peaks for P = 0.3

Equations (15)

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

U 2 x 2 = F P U 1 x 1 = 1 - i   cot   α 1 / 2     U 1 x 1 exp i π x 1 2 + x 2 2 cot   α - i 2 π x 1 x 2 csc   α d x 1 ,
C 12 P 1 , P 2 , P 3 x = F P 3 { F P 1 t 1 x 1 F P 2 * t 2 x 1 } ,
C 12 P 1 , P 2 , P 3 = exp i π x 2 cot   ϕ 3 ×   t 1 x 1 exp i π x 1 2 cot   ϕ 1 d x 1 ×   t 2 * x 1 exp - i π x 1 2 cot   ϕ 2 d x 1 ×   exp i π x 1 2 cot   ϕ 1 - cot   ϕ 2 + cot   ϕ 3 × exp - 2 π ix 1 x   csc   ϕ 3 + x 1   csc   ϕ 1 - x 1   csc   ϕ 2 d x 1 ,
ϕ 1 = P 1 π / 2 ,     ϕ 2 = P 2 π / 2 ,     ϕ 3 = P 3 π / 2 ,
1 tan   ϕ 1 - 1 tan   ϕ 2 + 1 tan   ϕ 3 = 0 .
F P U 1 x 1 + k = exp - i 2 π k x 2 + k   cos   α / 2 sin   α × U 2 x 2 + k   cos   α ,
f r ,   θ = N = -   f N r exp iN θ ,
f N r = 1 2 π 0 2 π   f r ,   θ exp - iN θ d θ .
F P g r ,   θ = 0 0 2 π   g r ,   θ exp i π ρ 2 + r 2 cot   α - 2 i π ρ r   csc   α   cos θ - ϕ r d r d θ = - i m exp im ϕ H m α g m r ,
H m α g m r = 2 π   0 2 π   g m r J m 2 π r ρ   csc   α r d r ,
exp - iM ϕ H M α * g M r .
F P f r ,   θ = N - i N exp iN ϕ H N α f N r .
D ρ ,   ϕ = N - i N H N α f N r H M α * g M r × exp i N - M ϕ .
E M = 0 R 0 2 π   D ρ ,   ϕ ρ d ρ d ϕ ,
E M = 2 π - i M 0 R   H M α f M r H M α * g N r ρ d ρ .

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