Abstract

A new type of electro-optical three-dimensional (3-D) correlator is proposed and demonstrated. A 3-D object scene, observed by multiple cameras from several points of view, is correlated with a 3-D complex computer-generated function. This correlator is a hybridization of the joint transform and the VanderLugt correlators, and, as such, it allows correlations to be made between 3-D real-world objects and 3-D general complex functions. Experimental results are presented.

© 2000 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2000

1999

1998

1997

1992

1991

J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
[CrossRef]

U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of complex reference functions in a joint-transform correlator,” Opt. Lett. 16, 330–332 (1991).
[CrossRef] [PubMed]

1990

1984

1964

A. B. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Esteve-Taboada, J. J.

Garcia, J.

Gianino, P. D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 8, p. 237.

Horner, J. L.

Kim, T.

Konforti, N.

Kotzer, T.

J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
[CrossRef]

Li, Y.

Mahlab, U.

Marom, E.

Mas, D.

Mendlovic, D.

Poon, T. C.

Rosen, J.

Shamir, J.

J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
[CrossRef]

U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of complex reference functions in a joint-transform correlator,” Opt. Lett. 16, 330–332 (1991).
[CrossRef] [PubMed]

VanderLugt, A. B.

A. B. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Vijaya Kumar, B. V. K.

Appl. Opt.

IEEE Trans. Inf. Theory

A. B. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

J. Opt. Soc. Am. A

Opt. Commun.

J. Rosen, T. Kotzer, J. Shamir, “Optical implementation of phase extraction pattern recognition,” Opt. Commun. 83, 10–14 (1991).
[CrossRef]

Opt. Lett.

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 8, p. 237.

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Figures (11)

Fig. 1
Fig. 1

Schematic of the hybrid 3-D correlator. DFT, discrete Fourier transform.

Fig. 2
Fig. 2

Three images of thirty-one of the scene as observed from different points of view from plane P2 at angles (a) θ = -15°, (b) θ = 0°, and (c) θ = 15°.

Fig. 3
Fig. 3

Three electronic Fourier holograms of the three images in Fig. 2, as recorded by the CCD at plane P4.

Fig. 4
Fig. 4

Three examples of thirty-one of POF’s computed from a single race car observed by the CCD at plane P2 from angles (a) θ = -15°, (b) θ = 0°, and (c) θ = 15°.

Fig. 5
Fig. 5

Nine of sixty-seven holograms displayed on SLM2 for several values of longitudinal axis z 0.

Fig. 6
Fig. 6

Nine of the output correlation planes obtained by optical FT of the nine holograms shown in Fig. 5.

Fig. 7
Fig. 7

(a) Intensity distributions about the first diffraction orders of nine correlation planes for several values of longitudinal axis z 0 as a result of the first experiment with the 3-D correlator. (b) Top view of the observed scene in the first experiment.

Fig. 8
Fig. 8

Three images of twenty-one of the input scene in the second experiment, as observed from different points of view from plane P2 at angles (a) θ = -10°, (b) θ = 0°, and (c) θ = 10°.

Fig. 9
Fig. 9

(a) Intensity distribution of nine correlation planes, for different values of z 0, that resulted from the second experiment with the 3-D correlator. (b) Top view of the observed scene in the second experiment.

Fig. 10
Fig. 10

Three images of twenty-one of the input scene in the third experiment, as observed from different points of view from plane P2 at angles (a) θ = -50°, (b) θ = 0°, and (c) θ = 50°.

Fig. 11
Fig. 11

(a) Intensity distribution of nine correlation planes, for several values of z 0 that resulted from the third experiment with the 3-D correlator. (b) Top view of the observed scene in the third experiment.

Equations (15)

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xi, yi=Mx cos θ+z sin θ, y,
I4u, v, θ=A expi 2πλ v sin ψ+-- g3xi, yi; θ×expi 2πλfuxi+vyidxidyi2,
u4u, v, θ=A expi 2πλ v sin ψ+g1x, y, z×expi 2πλfuxix, z, θ+vyiyΔxΔyΔz.
u4u, v, θ=A expi 2πλ v sin ψ+g1x, y, z×expi 2πMλfux cos θ+vy+uz sin θΔxΔyΔz.
u4u, v, θ=A expi 2πλ v sin ψ+G1u, v; θ.
G1u, v; θ=--- g1x, y, zexpi 2πMλfux cos θ+vy+uz sin θdxdydz.
I4u, v, θ=A expi 2πλ v sin ψ+G1u, v; θ2
Hu, v; θ=F*u, v; θ/|Fu, v; θ|,
Fu, v; θ=--- fx, y, zexpi 2πMλfux cos θ+vy+uz sin θdxdydz.
T4u, v, θ=Hu, v, θI4u, v, θ-|A|2.
T˜4u cos θ, v, u sin θ=|G˜1u cos θ, v, u sin θ|2H˜u cos θ, v, u sin θ+A*G˜1u cos θ, v, u sin θH˜u cos θ, v, u sin θexp-i 2πλ v sin ψ+AG˜1*u cos θ, v, u sin θH˜u cos θ, v, u sin θexpi 2πλ v sin ψ,
T˜4u cos θ, v, u sin θ=2 ReT˜4u cos θ, v, u sin θexpi4πvsin ψ/λ=|G˜1u cos θ, v, u sin θ|2H˜u cos θ, v, u sin θexpi4πvsin ψ/λ+|G˜1u cos θ, v, u sin θ|2H˜*u cos θ, v, u sin θexp-i4πvsin ψ/λ+A*G˜1u cos θ, v, u sin θH˜u cos θ, v, u sin θexpi2πvsin ψ/λ+AG˜1*u cos θ, v, u sin θH˜*u cos θ, v, u sin θexp-i2πvsin ψ/λ+AG˜1*u cos θ, v, u sin θH˜u cos θ, v, u sin θexpi6πvsin ψ/λ+A*G˜1u cos θ, v, u sin θH˜*u cos θ, v, u sin θexp-i6πvsin ψ/λ.
T˜5u cos θ, v, z0=- T˜4u cos θ, v, u sin θ×exp-i 2πMλfz0u sin θ×du sin θ.
cx0, y0, z0  --- G˜1u cos θ, v, u sin θ×H˜u cos θ, v, u sin θexpi2πvsin ψ/λ×exp-i 2πMλfz0u sin θdu sin θ×exp-i 2πMλfx0u cos θ+y0vdu cos θdv=--- g1x, y, zhx0-x, y0-y, z0-zdxdydz * δx0, y0-fsin ψ/M, z0.
x0, y0, z0=0, fsin ψ/M, 0.

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