Abstract

Electro-optical implementation of a three-dimensional (3-D) spatial correlation is proposed. A 3-D scene of objects, seen from the paraxial zone, is correlated with a reference object. As an example of application, we describe a 3-D joint transform correlator that is capable of recognizing targets in the 3-D space.

© 1998 Optical Society of America

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References

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  1. B. R. Brown, A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. 5, 967 (1966).
    [CrossRef] [PubMed]
  2. J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 7, p. 171.
  3. E. C. Tam, F. T. S. Yu, D. A. Gregory, R. D. Juday, “Au-tonomous real-time object tracking with an adaptive joint transform correlator,” Opt. Eng. 29, 314–320 (1990).
    [CrossRef]
  4. J. Rosen, A. Yariv, “Three-dimensional imaging of random radiation sources,” Opt. Lett. 21, 1011–1013 (1996); J. Rosen, A. Yariv, “Reconstruction of longitudinal distributed incoherent sources,” Opt. Lett. 21, 1803–1805 (1996).
    [CrossRef] [PubMed]
  5. A. Papoulis, System and Transforms with Applications in Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 3, p. 61.
  6. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 4, p. 133.
  7. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  8. J. Rosen, “Three-dimensional optical Fourier transform and correlation,” Opt. Lett. 22, 964–966 (1997).
    [CrossRef] [PubMed]

1997 (1)

1996 (1)

1990 (1)

E. C. Tam, F. T. S. Yu, D. A. Gregory, R. D. Juday, “Au-tonomous real-time object tracking with an adaptive joint transform correlator,” Opt. Eng. 29, 314–320 (1990).
[CrossRef]

1966 (2)

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 4, p. 133.

Brown, B. R.

Goodman, J. W.

C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
[CrossRef] [PubMed]

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 7, p. 171.

Gregory, D. A.

E. C. Tam, F. T. S. Yu, D. A. Gregory, R. D. Juday, “Au-tonomous real-time object tracking with an adaptive joint transform correlator,” Opt. Eng. 29, 314–320 (1990).
[CrossRef]

Juday, R. D.

E. C. Tam, F. T. S. Yu, D. A. Gregory, R. D. Juday, “Au-tonomous real-time object tracking with an adaptive joint transform correlator,” Opt. Eng. 29, 314–320 (1990).
[CrossRef]

Lohmann, A. W.

Papoulis, A.

A. Papoulis, System and Transforms with Applications in Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 3, p. 61.

Rosen, J.

Tam, E. C.

E. C. Tam, F. T. S. Yu, D. A. Gregory, R. D. Juday, “Au-tonomous real-time object tracking with an adaptive joint transform correlator,” Opt. Eng. 29, 314–320 (1990).
[CrossRef]

Weaver, C. S.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 4, p. 133.

Yariv, A.

Yu, F. T. S.

E. C. Tam, F. T. S. Yu, D. A. Gregory, R. D. Juday, “Au-tonomous real-time object tracking with an adaptive joint transform correlator,” Opt. Eng. 29, 314–320 (1990).
[CrossRef]

Appl. Opt. (2)

Opt. Eng. (1)

E. C. Tam, F. T. S. Yu, D. A. Gregory, R. D. Juday, “Au-tonomous real-time object tracking with an adaptive joint transform correlator,” Opt. Eng. 29, 314–320 (1990).
[CrossRef]

Opt. Lett. (2)

Other (3)

A. Papoulis, System and Transforms with Applications in Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 3, p. 61.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 4, p. 133.

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 7, p. 171.

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Figures (8)

Fig. 1
Fig. 1

Illustration of the imaging system for calculating Eq. (1).

Fig. 2
Fig. 2

Schematic of the 3-D spatial correlator. The 3-D scene is observed by an array of cameras distributed on a transverse plane located a distance L from the scene.

Fig. 3
Fig. 3

Schematic of the 3-D JTC. The reference is located on plane P1 at the origin of the axes. The tested objects are distributed along planes I–V, around the point (a, b, c).

Fig. 4
Fig. 4

Five projections, out of 49, of the input space as seen from five different points of view along the camera’s baseline.

Fig. 5
Fig. 5

Intensity distribution on plane P4 obtained by 2-D FT of each projection of Fig. 4.

Fig. 6
Fig. 6

3-D spectral intensity distribution after the coordinate transformation from (u, v, Dn) to (u, v, uDn).

Fig. 7
Fig. 7

Intensity of the cross correlation between the reference with the shape of the cross and the test objects obtained by a 3-D inverse FT of I˜4(u, v, uDn) that is partly shown in Fig. 6. The three high peaks indicate the existence of the three crosses among the group of the tested objects.

Fig. 8
Fig. 8

Same as Fig. 7, except that this time the reference is in the shape of an X.

Equations (11)

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xi=M(Dx+xs)1-zs/L
yi=M(Dy+ys)1-zs/L.
xiM(Dx+xs+zsDx/L+zsxs/L),
yiM(Dy+ys+zsDy/L+zsys/L).
O(u, v, Dx, Dy)=o(xs, ys, zs)×exp[i2πκ(xiu+yiv)]dxsdysdzs.
xiM[Dx+xs+zsDx/L],
yiM[Dy+ys+zsDy/L].
O(u, v, Dx, Dy)=exp[i2πMκ(Dxu+Dyv)]×o(xs, ys, zs)×exp{i2πMκ[xsu+ysv+(zs/L)(Dxu+Dyv)]}dxsdysdzs.
o(xs, ys, zs)=r(xs, ys, zs)+g(xs+a, ys+b, zs+c).
I4(u, v, Dx, Dy)=R(u, v, Dx, Dy)+G(u, v, Dx, Dy)×exp(i2πM/λf)au+bv+cL(Dxu+Dyv)2=|R(u, v, Dx, Dy)|2+|G(u, v, Dx, Dy)|2+G(u, v, Dx, Dy)R*(u, v, Dx, Dy)×exp(i2πM/λf)au+bv+cL(Dxu+Dyv)+G*(u, v, Dx, Dy)R(u, v, Dx, Dy)×exp-(i2πM/λf)au+bv+cL(Dxu+Dyv),
c(xo, yo, zo)=I˜4(ωx, ωy, ωz)exp[-i2π(xoωx+yoωy+zoωz)]dωxdωydωz=rr+gg+(rg) * δ(xo-a, yo-b, zo-c)+(gr) * δ(xo+a, yo+b, zo+c),

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