Abstract

A three-dimensional to two-dimensional mapping is proposed that permits the reduction of three-dimensional convolutions–correlations to two-dimensional ones and thereby lays a theoretical foundation for their optical implementation.

© 1997 Optical Society of America

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References

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  1. F. K. Knight, D. I. Klick, D. P. Ryan-Howard, B. K. Theriault, J. R. Tussey, A. M. Beckman, “Three-dimensional imaging using a single laser pulse,” in Laser Radar IV, R. J. Becherer, ed., Proc. SPIE1103, 174–189 (1989).
  2. C. C. Aleksoff, “Interferometric two-dimensional imaging of rotating objects,” Opt. Lett. 1, 54–55 (1977).
    [CrossRef] [PubMed]
  3. N. H. Farhat, “Holography, wavelength diversity and inverse scattering,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981), pp. 627–642.
  4. J. C. Marron, K. S. Schroeder, “Three-dimensional lensless imaging using laser frequency diversity,” Appl. Opt. 31, 255–262 (1992).
    [CrossRef] [PubMed]
  5. J. Rose, A. Yariv, “Three-dimensional imaging of random radiation sources,” Opt. Lett. 21, 1011–1013 (1996).
    [CrossRef]
  6. W. W. Stoner, W. J. Miceli, F. A. Horrigan, “One-dimensional to two-dimensional transformations in signal correlation,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. SPIE373, 21–30 (1981).
  7. W. T. Rhodes, “The falling raster in optical signal processing,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. SPIE373, 11–20 (1981).
  8. A. E. Siegman, “Two-dimensional calculations using one-dimensional arrays, or ‘Life on the Skew,’” Comput. Phys. Nov./Dec.74–75 (1988).
  9. J. Hofer-Alfeis, R. Bamler, “Three-dimensional and four-dimensional convolutions by coherent optical filtering,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. SPIE373, 77–87 (1981).
  10. R. Bamler, J. Hofer-Alfeis, “Three-and four-dimensional filter operations by coherent optics,” Opt. Acta 29, 747–757 (1982).
    [CrossRef]
  11. A. B. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  12. D. Casasent, D. Psaltis, “Position-,rotation-, and scale-invariant optical correlator,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  13. D. Asselin, H.-H. Arsenault, “Rotation and scale invariance with polar and log-polar coordinate transformations,” Opt. Commun. 104, 391–404 (1994).
    [CrossRef]
  14. D. Mendelovic, E. Marom, N. Konforti, “Scale-invariant pattern recognition,” in Optical Computing ’88 (Sept. 1988, Toulon, France), P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. SPIE963, 304–310 (1988).
    [CrossRef]

1996 (1)

1994 (1)

D. Asselin, H.-H. Arsenault, “Rotation and scale invariance with polar and log-polar coordinate transformations,” Opt. Commun. 104, 391–404 (1994).
[CrossRef]

1992 (1)

1988 (1)

A. E. Siegman, “Two-dimensional calculations using one-dimensional arrays, or ‘Life on the Skew,’” Comput. Phys. Nov./Dec.74–75 (1988).

1982 (1)

R. Bamler, J. Hofer-Alfeis, “Three-and four-dimensional filter operations by coherent optics,” Opt. Acta 29, 747–757 (1982).
[CrossRef]

1977 (1)

1976 (1)

1964 (1)

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

Aleksoff, C. C.

Arsenault, H.-H.

D. Asselin, H.-H. Arsenault, “Rotation and scale invariance with polar and log-polar coordinate transformations,” Opt. Commun. 104, 391–404 (1994).
[CrossRef]

Asselin, D.

D. Asselin, H.-H. Arsenault, “Rotation and scale invariance with polar and log-polar coordinate transformations,” Opt. Commun. 104, 391–404 (1994).
[CrossRef]

Bamler, R.

R. Bamler, J. Hofer-Alfeis, “Three-and four-dimensional filter operations by coherent optics,” Opt. Acta 29, 747–757 (1982).
[CrossRef]

J. Hofer-Alfeis, R. Bamler, “Three-dimensional and four-dimensional convolutions by coherent optical filtering,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. SPIE373, 77–87 (1981).

Beckman, A. M.

F. K. Knight, D. I. Klick, D. P. Ryan-Howard, B. K. Theriault, J. R. Tussey, A. M. Beckman, “Three-dimensional imaging using a single laser pulse,” in Laser Radar IV, R. J. Becherer, ed., Proc. SPIE1103, 174–189 (1989).

Casasent, D.

Farhat, N. H.

N. H. Farhat, “Holography, wavelength diversity and inverse scattering,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981), pp. 627–642.

Hofer-Alfeis, J.

R. Bamler, J. Hofer-Alfeis, “Three-and four-dimensional filter operations by coherent optics,” Opt. Acta 29, 747–757 (1982).
[CrossRef]

J. Hofer-Alfeis, R. Bamler, “Three-dimensional and four-dimensional convolutions by coherent optical filtering,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. SPIE373, 77–87 (1981).

Horrigan, F. A.

W. W. Stoner, W. J. Miceli, F. A. Horrigan, “One-dimensional to two-dimensional transformations in signal correlation,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. SPIE373, 21–30 (1981).

Klick, D. I.

F. K. Knight, D. I. Klick, D. P. Ryan-Howard, B. K. Theriault, J. R. Tussey, A. M. Beckman, “Three-dimensional imaging using a single laser pulse,” in Laser Radar IV, R. J. Becherer, ed., Proc. SPIE1103, 174–189 (1989).

Knight, F. K.

F. K. Knight, D. I. Klick, D. P. Ryan-Howard, B. K. Theriault, J. R. Tussey, A. M. Beckman, “Three-dimensional imaging using a single laser pulse,” in Laser Radar IV, R. J. Becherer, ed., Proc. SPIE1103, 174–189 (1989).

Konforti, N.

D. Mendelovic, E. Marom, N. Konforti, “Scale-invariant pattern recognition,” in Optical Computing ’88 (Sept. 1988, Toulon, France), P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. SPIE963, 304–310 (1988).
[CrossRef]

Marom, E.

D. Mendelovic, E. Marom, N. Konforti, “Scale-invariant pattern recognition,” in Optical Computing ’88 (Sept. 1988, Toulon, France), P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. SPIE963, 304–310 (1988).
[CrossRef]

Marron, J. C.

Mendelovic, D.

D. Mendelovic, E. Marom, N. Konforti, “Scale-invariant pattern recognition,” in Optical Computing ’88 (Sept. 1988, Toulon, France), P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. SPIE963, 304–310 (1988).
[CrossRef]

Miceli, W. J.

W. W. Stoner, W. J. Miceli, F. A. Horrigan, “One-dimensional to two-dimensional transformations in signal correlation,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. SPIE373, 21–30 (1981).

Psaltis, D.

Rhodes, W. T.

W. T. Rhodes, “The falling raster in optical signal processing,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. SPIE373, 11–20 (1981).

Rose, J.

Ryan-Howard, D. P.

F. K. Knight, D. I. Klick, D. P. Ryan-Howard, B. K. Theriault, J. R. Tussey, A. M. Beckman, “Three-dimensional imaging using a single laser pulse,” in Laser Radar IV, R. J. Becherer, ed., Proc. SPIE1103, 174–189 (1989).

Schroeder, K. S.

Siegman, A. E.

A. E. Siegman, “Two-dimensional calculations using one-dimensional arrays, or ‘Life on the Skew,’” Comput. Phys. Nov./Dec.74–75 (1988).

Stoner, W. W.

W. W. Stoner, W. J. Miceli, F. A. Horrigan, “One-dimensional to two-dimensional transformations in signal correlation,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. SPIE373, 21–30 (1981).

Theriault, B. K.

F. K. Knight, D. I. Klick, D. P. Ryan-Howard, B. K. Theriault, J. R. Tussey, A. M. Beckman, “Three-dimensional imaging using a single laser pulse,” in Laser Radar IV, R. J. Becherer, ed., Proc. SPIE1103, 174–189 (1989).

Tussey, J. R.

F. K. Knight, D. I. Klick, D. P. Ryan-Howard, B. K. Theriault, J. R. Tussey, A. M. Beckman, “Three-dimensional imaging using a single laser pulse,” in Laser Radar IV, R. J. Becherer, ed., Proc. SPIE1103, 174–189 (1989).

VanderLugt, A. B.

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

Yariv, A.

Appl. Opt. (2)

Comput. Phys. Nov./Dec. (1)

A. E. Siegman, “Two-dimensional calculations using one-dimensional arrays, or ‘Life on the Skew,’” Comput. Phys. Nov./Dec.74–75 (1988).

IEEE Trans. Inf. Theory (1)

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

Opt. Acta (1)

R. Bamler, J. Hofer-Alfeis, “Three-and four-dimensional filter operations by coherent optics,” Opt. Acta 29, 747–757 (1982).
[CrossRef]

Opt. Commun. (1)

D. Asselin, H.-H. Arsenault, “Rotation and scale invariance with polar and log-polar coordinate transformations,” Opt. Commun. 104, 391–404 (1994).
[CrossRef]

Opt. Lett. (2)

Other (6)

D. Mendelovic, E. Marom, N. Konforti, “Scale-invariant pattern recognition,” in Optical Computing ’88 (Sept. 1988, Toulon, France), P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. SPIE963, 304–310 (1988).
[CrossRef]

J. Hofer-Alfeis, R. Bamler, “Three-dimensional and four-dimensional convolutions by coherent optical filtering,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. SPIE373, 77–87 (1981).

F. K. Knight, D. I. Klick, D. P. Ryan-Howard, B. K. Theriault, J. R. Tussey, A. M. Beckman, “Three-dimensional imaging using a single laser pulse,” in Laser Radar IV, R. J. Becherer, ed., Proc. SPIE1103, 174–189 (1989).

N. H. Farhat, “Holography, wavelength diversity and inverse scattering,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1981), pp. 627–642.

W. W. Stoner, W. J. Miceli, F. A. Horrigan, “One-dimensional to two-dimensional transformations in signal correlation,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. SPIE373, 21–30 (1981).

W. T. Rhodes, “The falling raster in optical signal processing,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. SPIE373, 11–20 (1981).

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Equations (54)

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i1j1+λz1 cos α=i2j2+λz2 cos α,
m1n1+λz1 sin α=m2n2+λz2 sin α.
tan α=m1n1-m2/n2i1/j1-i2/j2,
i1j1=i2j2.
z1=z2.
m1n1=m2n2.
fx, y, z*gx, y, za, b, c=fx, y, zg×a-x, b-y, c-zdxdydz
fx, y, z=Fx+λz cos α, y+λz sin α
gx, y, z=Gx+λz cos α, y+λz sin α,
fx, y, zga-x, b-y, c-zdxdydz=Fx+λz cos α, y+λz sin α×Ga-x+λc-zcos α, b-y+λc-zsin αdxdydz.
u=×+λz cos α, v=y+λz sin α, w=z.
Fx+λz cos α, y+λz sin αGa-x+λc-zcos α, b-y+λc-zsin αdxdydz=Fu, vGa+λc cos α-u, b+λc sin α-v×Dx, y, zDu, v, wdudvdw,
10-λ cos α01-λ sin α001,
fx, y, zga-x, b-y, c-zdxdydz= Fu, vGa+λc cos α-u, b+λ sin α-vdudvdw.
Fu, vGa+λc cos α-u, b+λc sin α-vdudvdw=zmax-zminFu, vGa+λc cos α-u,×b+λc sin α-vdudv,
fx, y, zga-x, b-y, c-z0.
fx, y, z=Fx+λz cos α, y+λz sin α,gx, y, z=Gx+λz cos α, y+λz sin α
fx, y, z*gx, y, za, b, c=Kf, g×F x, y*Gx, ya+λc cos α, b+λc sin α,
fx, y, z=Fx+λz cos α, y+λz sin α,
gx, y, z=Gx+λz cos α, y+λz sin α
f˜x, y, z=i, jfiH,jH, z+Δix)×rectx-iH2hrecty-jH2h,
g˜x, y, z=i, jgiH,jH, z+Δix×rectx-iH2hrecty-jH2h,
Δix=x-iHλ cos α,
λ=H1-maxzmax, zmin,
f˜x, y, z=f˜x+λHz cos α, y+λHz sin α,
g˜x, y, z=G˜x+λHz cos α, y+λHz sin α,
F˜x1+λHz1 cos α, y1+λHz1 sin αF˜x2+λHz2 cos α, y2+λHz2 sin α,
x1+λHz1 cos α=x2+λHz2 cos α,
y1+λHz1 sin α=y2+λHz2 sin α.
F˜x1+λHz1 cos α, y1+λHz1 sin α=f˜x1, y1, z1=fiH, jH, z1+Δix1,
F˜x2+λHz2 cos α, y2+λHz2 sin α=f˜x2, y2, z2=fiH, jH, z2+Δix2,
z1+Δix1z2+Δix2.
x1+λHz1 cos αx2+λHz2 cos α,
x1+λHz1 cos α=x2+λHz2 cos α.
f˜x, y, z*g˜x, y, za, b, c=K1f˜x, y*G˜x, y×a+λHc cos α, b+λHc sin α.
f˜x, y, z*g˜x, y, za, b, cK2fx, y, z*gx, y, za, b, c,
fx, y, z*gx, y, za, b, c=limH0K1K2f˜x, y*G˜x, y×a+λHc cos α, b+λHc sin α.
f˜x, y, z*g˜x, y, za, b, cK2fx, y, z*gx, y, za, b, c.
f˜x, y, z=i,jfiH, jH, z+Δix×rectx-iH2hrecty-jH2h=i,jiH, jH, zrectx-iH2h×recty-jH2h+i,jfziH,jH,z˜Δix×rectx-iH2hrecty-jH2h=Sfx, y, z+ofx, y, z,
Sfx, y, z=i,jfiH, jH, zrectx-iH2h×recty-jH2h,
ofx, y, z=i,jfziH,jH,z˜Δixrectx-iH2h×recty-jH2h.
f˜*g˜=Sf*Sg+Sf*og+Sg*of+og*of.
Sf*ogmax Sf max og4h2DxDyH20,
hH=k,
Sg*of0,
og*of0,
f˜*g˜Sf*Sg.
Sf*Sga, b, c=rectx-iH2h*rectx-mH2ha×recty-jH2h*recty-nH2hb×fiH, jH, zgmH, nH, c-zdz=i,j,m,nmax0, 2h-a-iH-mH×max0, 2h-b-jH-nH×fiH, jH, zgmH ,nH, c-zdz.
Sf*Sga, b, c=4h2i+m=p;j+n=qfiH, jH, z×gmH, nH, c-zdz=4h2i,jfiH, jH, z×ga-iH, b-jH, c-zdz.
f*ga, b, c=limH0ijH2fiH, jH, z×ga-iH, b-jH, c-zdz.
f*ga, b, c=limH0H24h2Sf*Sga, b, c=limH0H24h2f˜*g˜a, b, c.
f*ga, b, c=limH0f*gaHH,bHH, c,
f*ga, b, c=limH0H24h2f˜*g˜a, b, c.
fx, y, z*gx, y, za, b, c=limH0K1H24h2F˜x, y*G˜x, y×a+λHc cos α, b+λHc sin α.

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