Abstract

Optical implementation of a three-dimensional (3-D) Fourier transform is proposed and demonstrated. A spatial 3-D object, as seen from the paraxial zone, is transformed to the 3-D spatial frequency space. Based on the new procedure, a 3-D joint transform correlator is described that is capable of recognizing targets in the 3-D space.

© 1997 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 7, p. 141.
  2. F. T. S. Yu and S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks, 1st ed. (Wiley, New York, 1992), Chap. 6, p. 203.
  3. C. S. Weaver and J. W. Goodman, Appl. Opt. 5, 1248 (1966).
    [CrossRef] [PubMed]
  4. J. Rosen and A. Yariv, Opt. Lett. 21, 1011, 1803, (1996).

1996 (1)

1966 (1)

Goodman, J. W.

C. S. Weaver and J. W. Goodman, Appl. Opt. 5, 1248 (1966).
[CrossRef] [PubMed]

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 7, p. 141.

Jutamulia, S.

F. T. S. Yu and S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks, 1st ed. (Wiley, New York, 1992), Chap. 6, p. 203.

Rosen, J.

Weaver, C. S.

Yariv, A.

Yu, F. T. S.

F. T. S. Yu and S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks, 1st ed. (Wiley, New York, 1992), Chap. 6, p. 203.

Appl. Opt. (1)

Opt. Lett. (1)

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, New York, 1968), Chap. 7, p. 141.

F. T. S. Yu and S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks, 1st ed. (Wiley, New York, 1992), Chap. 6, p. 203.

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Figures (2)

Fig. 1
Fig. 1

Schematic of the optical system for the 3-D FT.

Fig. 2
Fig. 2

Computer simulation of the system shown in Fig.  1. (a) Projections of the input image observed from various cameras' translations D. (b) Intensity distribution on plane P3. (c) The same intensity distribution after the coordinates have been transformed. (d) Cross-correlation results between the reference and the tested objects obtained by a 2-D FT of the pattern in (c).

Equations (7)

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xi=MDx+xs/1-zs/L, yi=MDy+ys/1-zs/L.
Ou, v, Dx, Dy=oxs, ys, zsexpi2πxiu+yiv/λfdxsdysdzs.
xiMDx+xs+zsDx/L, yiMDy+ys+zsDy/L.
Ou, v, Dx, Dy=expi2πM/λfDxu+Dyv×oxs, ys, zsexpi2πM/λfxsu+ysv+zs/LDxu+Dyvdxsdysdzs.
oxs, ys, zs=rxs, ys, zs+gxs+a, ys+b, zs+c.
I3u, v, Dx, Dy=Ru, v, Dx, Dy+Gu, v, Dx, Dy×expi2πM/λfau+bv+c/LDxu+Dyv 2=Ru, v, Dx, Dy 2+Gu, v, Dx, Dy2+Gu, v, Dx, DyR*u, v, Dx, Dyexpi2πM/λfau+bv+c/L×Dxu+Dyv+G*u, v, Dx, DyRu, v, Dx, Dy×exp-i2πM/λfau+bv+c/LDxu+Dyv,
cx0, y0, z0=I˜3ωx, ωy, ωzexp-i2πx0ωx+y0ωy+z0ωzdωxdωydωz=rr+gg+rg*δx0-a, y0-b, z0-c+gr*δx0+a, y0+b, z0+c,

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