Abstract

Novel real-time joint transform, Hilbert transform optical correlators are proposed in which only a half or a quarter of the Fourier plane is written onto an optically addressable spatial light modulator. A point source is used to recover the result for the whole plane. As a result, images with a two- or four-times larger space–bandwidth product can be matched in amplitude and phase. The effect of truncating the transform plane is explained with two- and one-dimensional Hilbert transform analysis. Results of computer simulation are shown.

© 2001 Optical Society of America

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References

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  1. A. L. VanderLugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT10, 139–145 (1964).
  2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  3. A. D. McAulay, J. Wang, “Joint transform correlator for optical outer product associative memory with electron trapping materials,” in Optical Pattern Recognition VI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 360–364 (1995).
    [CrossRef]
  4. A. D. McAulay, “Hilbert transform and mirror-image optical correlators,” Appl. Opt. 39, 2300–2309 (2000).
    [CrossRef]
  5. Q. Tang, B. Javidi, “Multiple-object detection with a chirp-encoded joint transform correlator,” Appl. Opt. 32, 5079–5088 (1993).
    [CrossRef] [PubMed]
  6. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  7. R. W. Cohn, J. L. Horner, “Effects of systematic phase errors in phase-only correlators,” Appl. Opt. 33, 5432–5439 (1994).
    [CrossRef] [PubMed]
  8. R. D. Juday, R. S. Barton, S. E. Monroe, “Experimental optical results with minimum Euclidean distance optimal filters, coupled modulators and quadratic metrics,” Opt. Eng. 38, 302–312 (1999).
    [CrossRef]
  9. A. Papoulis, Systems and Transforms with Applications in Optics (Krieger, Malabar, Fla., 1981).
  10. A. V. Oppenheim, F. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  11. S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, Boston, Mass., 1996).
  12. L. Muller, J. Marquard, “The Hilbert transform and its generalization in optics and image processing,” Optik (Stuttgart) 110, 99–109 (1999).
  13. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).
  14. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, U.K., 1999).
  15. M. N. Nabighian, “Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transforms: fundamental relations,” Geophysics 49, 780–786 (1984).
    [CrossRef]
  16. A. D. McAulay, J. Wang, “Logic and arithmetic with luminescent rebroadcasting devices,” in Advances in Optical Information Processing III, R. D. Pape, ed., Proc. SPIE936, 321–326 (1988).
    [CrossRef]
  17. A. D. McAulay, Optical Computer Architectures (Wiley, New York, 1991).

2000 (1)

1999 (2)

R. D. Juday, R. S. Barton, S. E. Monroe, “Experimental optical results with minimum Euclidean distance optimal filters, coupled modulators and quadratic metrics,” Opt. Eng. 38, 302–312 (1999).
[CrossRef]

L. Muller, J. Marquard, “The Hilbert transform and its generalization in optics and image processing,” Optik (Stuttgart) 110, 99–109 (1999).

1994 (1)

1993 (1)

1984 (2)

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

M. N. Nabighian, “Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transforms: fundamental relations,” Geophysics 49, 780–786 (1984).
[CrossRef]

1964 (1)

A. L. VanderLugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT10, 139–145 (1964).

Barton, R. S.

R. D. Juday, R. S. Barton, S. E. Monroe, “Experimental optical results with minimum Euclidean distance optimal filters, coupled modulators and quadratic metrics,” Opt. Eng. 38, 302–312 (1999).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, U.K., 1999).

Cohn, R. W.

Gianino, P. D.

Goodman, J. W.

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

Hahn, S. L.

S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, Boston, Mass., 1996).

Horner, J. L.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

Javidi, B.

Juday, R. D.

R. D. Juday, R. S. Barton, S. E. Monroe, “Experimental optical results with minimum Euclidean distance optimal filters, coupled modulators and quadratic metrics,” Opt. Eng. 38, 302–312 (1999).
[CrossRef]

Marquard, J.

L. Muller, J. Marquard, “The Hilbert transform and its generalization in optics and image processing,” Optik (Stuttgart) 110, 99–109 (1999).

McAulay, A. D.

A. D. McAulay, “Hilbert transform and mirror-image optical correlators,” Appl. Opt. 39, 2300–2309 (2000).
[CrossRef]

A. D. McAulay, J. Wang, “Joint transform correlator for optical outer product associative memory with electron trapping materials,” in Optical Pattern Recognition VI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 360–364 (1995).
[CrossRef]

A. D. McAulay, J. Wang, “Logic and arithmetic with luminescent rebroadcasting devices,” in Advances in Optical Information Processing III, R. D. Pape, ed., Proc. SPIE936, 321–326 (1988).
[CrossRef]

A. D. McAulay, Optical Computer Architectures (Wiley, New York, 1991).

Monroe, S. E.

R. D. Juday, R. S. Barton, S. E. Monroe, “Experimental optical results with minimum Euclidean distance optimal filters, coupled modulators and quadratic metrics,” Opt. Eng. 38, 302–312 (1999).
[CrossRef]

Muller, L.

L. Muller, J. Marquard, “The Hilbert transform and its generalization in optics and image processing,” Optik (Stuttgart) 110, 99–109 (1999).

Nabighian, M. N.

M. N. Nabighian, “Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transforms: fundamental relations,” Geophysics 49, 780–786 (1984).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, F. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (Krieger, Malabar, Fla., 1981).

Schafer, F. W.

A. V. Oppenheim, F. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Tang, Q.

VanderLugt, A. L.

A. L. VanderLugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT10, 139–145 (1964).

Wang, J.

A. D. McAulay, J. Wang, “Joint transform correlator for optical outer product associative memory with electron trapping materials,” in Optical Pattern Recognition VI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 360–364 (1995).
[CrossRef]

A. D. McAulay, J. Wang, “Logic and arithmetic with luminescent rebroadcasting devices,” in Advances in Optical Information Processing III, R. D. Pape, ed., Proc. SPIE936, 321–326 (1988).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, U.K., 1999).

Appl. Opt. (4)

Geophysics (1)

M. N. Nabighian, “Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transforms: fundamental relations,” Geophysics 49, 780–786 (1984).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. L. VanderLugt, “Signal detection by complex filtering,” IEEE Trans. Inf. Theory IT10, 139–145 (1964).

Opt. Eng. (1)

R. D. Juday, R. S. Barton, S. E. Monroe, “Experimental optical results with minimum Euclidean distance optimal filters, coupled modulators and quadratic metrics,” Opt. Eng. 38, 302–312 (1999).
[CrossRef]

Optik (Stuttgart) (1)

L. Muller, J. Marquard, “The Hilbert transform and its generalization in optics and image processing,” Optik (Stuttgart) 110, 99–109 (1999).

Other (9)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, U.K., 1999).

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

A. D. McAulay, J. Wang, “Joint transform correlator for optical outer product associative memory with electron trapping materials,” in Optical Pattern Recognition VI, D. P. Casasent, T.-H. Chao, eds., Proc. SPIE2490, 360–364 (1995).
[CrossRef]

A. D. McAulay, J. Wang, “Logic and arithmetic with luminescent rebroadcasting devices,” in Advances in Optical Information Processing III, R. D. Pape, ed., Proc. SPIE936, 321–326 (1988).
[CrossRef]

A. D. McAulay, Optical Computer Architectures (Wiley, New York, 1991).

A. Papoulis, Systems and Transforms with Applications in Optics (Krieger, Malabar, Fla., 1981).

A. V. Oppenheim, F. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, Boston, Mass., 1996).

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Figures (17)

Fig. 1
Fig. 1

Proposed joint transform, Hilbert transform optical correlator that uses a half of the Fourier plane. LD, laser diode; LCD, liquid-crystal device; LCLV, liquid-crystal light valve; BS, beam splitter; CCD, charge-coupled device.

Fig. 2
Fig. 2

Proposed joint transform, Hilbert transform optical correlator that uses a quarter of the Fourier plane.

Fig. 3
Fig. 3

Input image for correlation: (a) contour plot and (b) three-dimensional (3-D) view.

Fig. 4
Fig. 4

Input image for convolution: (a) contour plot and (b) 3-D view.

Fig. 5
Fig. 5

Input image for correlation in y and convolution in x: (a) contour plot and (b) 3-D view.

Fig. 6
Fig. 6

Intensity for the Fourier transform of Fig. 3 for correlation with a matched case: (a) contour plot and (b) 3-D view.

Fig. 7
Fig. 7

Result of the masking out of negative k y in Fig. 6 for correlation with a matched case: (a) contour plot and (b) 3-D view.

Fig. 8
Fig. 8

Real part of the inverse Fourier transform of Fig. 7 for correlation with a matched case: (a) contour plot and (b) 3-D view.

Fig. 9
Fig. 9

Absolute value of the inverse Fourier transform of the whole Fourier plane of Fig. 6 for comparison with Fig. 8 for correlation with a matched case: (a) contour plot and (b) 3-D view.

Fig. 10
Fig. 10

Intensity for the Fourier transform of Fig. 4 for convolution with a matched case: (a) contour plot and (b) 3-D view.

Fig. 11
Fig. 11

Result of the masking out of negative k y in Fig. 10 for convolution with a matched case: (a) contour plot and (b) 3-D view.

Fig. 12
Fig. 12

Real part of the inverse Fourier transform of Fig. 11 for convolution with a matched case: (a) contour plot and (b) 3-D view.

Fig. 13
Fig. 13

Absolute value of the inverse Fourier transform of the whole Fourier plane of Fig. 10 for comparison with Fig. 12 for convolution with a matched case: (a) contour plot and (b) 3-D view.

Fig. 14
Fig. 14

Intensity for the Fourier transform of Fig. 5 for convolution and correlation with a matched case: (a) contour plot and (b) 3-D view.

Fig. 15
Fig. 15

Result of the masking out of negative k y in Fig. 14 for convolution and correlation with a matched case: (a) contour plot and (b) 3-D view.

Fig. 16
Fig. 16

Real part of the inverse Fourier transform of Fig. 15 for convolution and correlation with a matched case: (a) contour plot and (b) 3-D view.

Fig. 17
Fig. 17

Absolute value of the inverse Fourier transform of the whole Fourier plane of Fig. 14 for comparison with Fig. 16 for convolution and correlation with a matched case: (a) contour plot and (b) 3-D view.

Equations (30)

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uhxx, y=ux, y+jHxsux, y,
Uhxkx, ky=2Ukx, kyfor kx>00for kx<0,
Hxkkx=-jfor kx>0jfor kx<0,  or  Hxkkx=-jsgnkx,
Hxykkx, ky=HxkkxHykky=-jsgnkx-jsgnky=-sgnkxsgnky.
Uq1kx, ky=4Ukx, kyfor kx>0 and ky>00elsewhere.
uq1x, y=ux, y+jHxysux, y
Uq1kx, ky=1+sgnkxsgnky+j-jsgnkx+-jsgnkyUkx, ky.
40U100=U4U1U3U2+-U4U1U3-U2+U4U1-U3-U2+-U4U1-U3U2.
Uq1kx, ky=Ukx, ky-HxykUkx, ky+jHxkUkx, ky+HykUkx, ky.
uq1x, y=ux, y-Hxysux, y+jHxsux, y+Hysux, y.
uq4x, y=ux, y+Hxysux, y+jHxsux, y-Hys[ux, y.
gx, y=f1x-sx, y+f2x+sx, y.
Gkx, ky=F1kx, kyexp-jkxsx+F2kx, kyexpjkxsx,
Ukx, ky=|Gkx, ky|2=F1kx, kyexp-jkxsx+F2kx, kyexpjkxsx×F1kx, kyexp-jkxsx+F2kx, kyexpjkxsx*=|F1kx, ky|2+|F2kx, ky|2+F1kx, kyF2*kx, kyexp-j2kxsx+F1*kx, kyF2kx, kyexpj2kxsx.
|ux, y|2=A2+A-1Ukx, ky=A2+Af1x, y   f1x, y+f2x, y   f2x, y+f1x, y   f2-x+2sx, -y+f1x, y   f2x+2sx, y,
gcx, y=f1x-sx, y+f2-x-sx, -y.
Uckx, ky=F1kx, kyexp-jkxsx+F2*kx, kyexpjkxsx×F1*kx, kyexpjkxsx+F2kx, kyexp-jkxsx=|F1kx, ky|2+|F2kx, ky|2+F1kx, kyF2kx, kyexp-j2kxsx+F1*kx, kyF2*kx, kyexpj2kxsx.
|ux, y|2=A2+Af1x, y   f1x, y+f2x, y   f2x, y+f1x, y ** f2-x-2sx, -y+f1x, y ** f2x-2sx, y,
gx, y=f1x-sx, y+f2-x-sx, y.
Fxkx, ky=-- f-ξ, ηexp-jkxξ+kyηdξdη=--- fξ, ηexpjkxξ-kyηdξdη=-- fξ, ηexpjkxξ-kyηdξdη.
Fykx, ky=-- f-ξ, ηexp-jkxξ+kyηdξdη=--- fξ, ηexp-jkxξ-kyηdξdη=-- fξ, ηexp-jkxξ-kyηdξdη.
Gkx, ky=F1kx, kyexp-jkxsx+F2xkx, kyexpjkxsx.
Fykx, ky=Fxkx, ky*
Umkx, ky=|Gkx, ky|2=F1kx, kyexp-jkxsx+F2xkx, kyexpjkxsx×F1*kx, kyexpjkxsx+F2ykx, kyexp-jkxsx=|F1kx, ky|2+F2xkx, kyF2ykx, ky+F1kx, kyF2ykx, kyexp-j2kxsx+F1*kx, kyF2xkx, kyexpj2kxsx.
um,3x, y=-1F1kx, kyF2ykx, kyexp-j2kxsx=-1F1kx, ky-- f2ξ, η×exp-jkxξ-kyηexp-j2kxsxdξdη=-- -1F1kx, kyexp-jkxξ-kyη×exp-j2kxsxf2ξ, ηdξdη=-- f1x-2sx-ξ, y+ηf2ξ, ηdξdηf2x, y *  f1x-2sx, y.
um,4x, y=-1F1*kx, kyF2ykx, kyexpj2kxsx=-1F1*kx, ky-- f2ξ, η×expjkxξ-kyη×expj2kxsxdξdη=-- -1F1*kx, kyexpjkxξ-kyη×expj2kxsxf2ξ, ηdξdη=-- f1-x-2sx-ξ, -y+ηf2ξ, ηdξdηf2x, y *  f1-x-2sx, -y.
|umx, y|2=A2+Af1x, y ** f1x, y+f2x, y   f2x, y+f2x, y *  f1x-2sx, y+f2x, y *  f1-x-2sx, -y,
gx,y=fx-sx, y+f-x-sx, y.
Umkx, ky=|Fkx, ky|2+|Fxkx, ky|2+Fkx, kyFxkx, ky*exp-j2kxsx+F*kx, kyFxkx, kyexpj2kxsx.
Um-kx, ky=|Fkx, ky|2+|Fkx, ky|2+F*kx, kyFxkx, kyexpj2kxsx+Fkx, kyFxkx, ky*exp-j2kxsx,

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