Abstract

An optical correlator, believed to be novel, adds an x–y mirror image to the input and filter images to perform convolution involving amplitude and phase. The resulting real Fourier transform filters can be loaded into a liquid-crystal device (LCD). In contrast, a complex filter would require high-resolution film. A Hilbert transform and a point source are applied at the filter plane to reduce filter storage and LCD loading time by a factor of 2. An optional spatial filter removes an offset intensity and squares the result. Filters have only twice the number of pixels of the images. Analysis is verified by computer simulation, and performance is discussed.

© 2000 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  2. B. Javidi, E. Abouzi, “Optical security system with Fourier transform plane encoding,” Appl. Opt. 37, 6247–6255 (1998).
    [CrossRef]
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  5. F. T. S. Yu, D. A. Gregory, “Optical pattern recognition architectures and techniques,” Proc. IEEE 84, 733–752 (1996).
    [CrossRef]
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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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    [CrossRef]
  12. L. Muller, J. Marquard, “The Hilbert transform and its generalization in optics and image processing,” Optik 110, 99–109 (1999).
  13. A. D. McAulay, Optical Computer Architectures (Wiley, New York, 1991).
  14. A. V. Oppenheim, F. W. Schafer, Digital Signal Processing (Prentice-Hall, Engelwood Cliffs, N.J., 1975).
  15. A. Papoulis, Systems and Transforms with Applications in Optics (Krieger, Malabar, Fla., 1981).

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 110, 99–109 (1999).

1998 (1)

1997 (1)

J. P. Karlas, S. A. Mills, J. R. Ryan, R. B. Dydyk, J. Lucas, “Performance of a second-generation miniature ruggedized optical correlator module,” Opt. Eng. 36, 2747–2753 (1997).
[CrossRef]

1996 (1)

F. T. S. Yu, D. A. Gregory, “Optical pattern recognition architectures and techniques,” Proc. IEEE 84, 733–752 (1996).
[CrossRef]

1994 (1)

1993 (1)

A. D. McAulay, J. Wang, “Optical wavelet transform classifier with positive real Fourier transform wavelets,” Opt. Eng. 32, 1333–1339 (1993).
[CrossRef]

1984 (1)

1964 (1)

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

Abouzi, E.

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]

Cohn, R. W.

Dydyk, R. B.

J. P. Karlas, S. A. Mills, J. R. Ryan, R. B. Dydyk, J. Lucas, “Performance of a second-generation miniature ruggedized optical correlator module,” Opt. Eng. 36, 2747–2753 (1997).
[CrossRef]

Gianino, P. D.

Goodman, J. W.

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

Gregory, D. A.

F. T. S. Yu, D. A. Gregory, “Optical pattern recognition architectures and techniques,” Proc. IEEE 84, 733–752 (1996).
[CrossRef]

Horner, J. L.

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]

Karlas, J. P.

J. P. Karlas, S. A. Mills, J. R. Ryan, R. B. Dydyk, J. Lucas, “Performance of a second-generation miniature ruggedized optical correlator module,” Opt. Eng. 36, 2747–2753 (1997).
[CrossRef]

Lucas, J.

J. P. Karlas, S. A. Mills, J. R. Ryan, R. B. Dydyk, J. Lucas, “Performance of a second-generation miniature ruggedized optical correlator module,” Opt. Eng. 36, 2747–2753 (1997).
[CrossRef]

Marquard, J.

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

McAulay, A. D.

A. D. McAulay, J. Wang, “Optical wavelet transform classifier with positive real Fourier transform wavelets,” Opt. Eng. 32, 1333–1339 (1993).
[CrossRef]

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

A. D. McAulay, J. Wang, “Experimental results for matched filter construction and correlation with a single curved element,” in Advances in Optical Information Processing VI, D. R. Pape, ed., Proc. SPIE2240, 265–269 (1994).
[CrossRef]

Mills, S. A.

J. P. Karlas, S. A. Mills, J. R. Ryan, R. B. Dydyk, J. Lucas, “Performance of a second-generation miniature ruggedized optical correlator module,” Opt. Eng. 36, 2747–2753 (1997).
[CrossRef]

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 110, 99–109 (1999).

Oppenheim, A. V.

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

Papoulis, A.

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

Ryan, J. R.

J. P. Karlas, S. A. Mills, J. R. Ryan, R. B. Dydyk, J. Lucas, “Performance of a second-generation miniature ruggedized optical correlator module,” Opt. Eng. 36, 2747–2753 (1997).
[CrossRef]

Schafer, F. W.

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

VanderLugt, A.

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).

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, “Optical wavelet transform classifier with positive real Fourier transform wavelets,” Opt. Eng. 32, 1333–1339 (1993).
[CrossRef]

A. D. McAulay, J. Wang, “Experimental results for matched filter construction and correlation with a single curved element,” in Advances in Optical Information Processing VI, D. R. Pape, ed., Proc. SPIE2240, 265–269 (1994).
[CrossRef]

Yu, F. T. S.

F. T. S. Yu, D. A. Gregory, “Optical pattern recognition architectures and techniques,” Proc. IEEE 84, 733–752 (1996).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Inf. Theory (1)

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

Opt. Eng. (3)

J. P. Karlas, S. A. Mills, J. R. Ryan, R. B. Dydyk, J. Lucas, “Performance of a second-generation miniature ruggedized optical correlator module,” Opt. Eng. 36, 2747–2753 (1997).
[CrossRef]

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]

A. D. McAulay, J. Wang, “Optical wavelet transform classifier with positive real Fourier transform wavelets,” Opt. Eng. 32, 1333–1339 (1993).
[CrossRef]

Optik (1)

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

Proc. IEEE (1)

F. T. S. Yu, D. A. Gregory, “Optical pattern recognition architectures and techniques,” Proc. IEEE 84, 733–752 (1996).
[CrossRef]

Other (6)

A. VanderLugt, Optical Signal Processing (Wiley, New York, 1992).

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

A. D. McAulay, J. Wang, “Experimental results for matched filter construction and correlation with a single curved element,” in Advances in Optical Information Processing VI, D. R. Pape, ed., Proc. SPIE2240, 265–269 (1994).
[CrossRef]

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

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

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

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Figures (15)

Fig. 1
Fig. 1

Current optical correlators: (a) complex filter, (b) output when an offset reference filter is used. FT, Fourier transform.

Fig. 2
Fig. 2

Proposed mirror-imaging optical correlator. FT, Fourier transform.

Fig. 3
Fig. 3

Proposed Hilbert transform optical correlator.

Fig. 4
Fig. 4

Shows that an image plus its xy mirror image has a real Fourier transform: (a) image plus xy mirror image, (b) cos(k x x + k y y), (c) sin(k x x + k y y).

Fig. 5
Fig. 5

Shows that the Fourier transform for an image plus its xy mirror image is real for a discretized case: (a) image plus xy mirror image, (b) even case, (c) odd case.

Fig. 6
Fig. 6

Mirror system to generate and add an xy mirror image. BS, beam splitter.

Fig. 7
Fig. 7

Partial optical correlator showing the effect of Hilbert transform masking on output. FT, Fourier transform.

Fig. 8
Fig. 8

Optical spatial filter to remove intensity offset. FT, Fourier transform.

Fig. 9
Fig. 9

Correlator inputs: (a) input image, (b) modified image by addition of xy mirror image.

Fig. 10
Fig. 10

Images at filter plane for mirror image correlator, shown in Fig. 2: (a) Real part of Fourier transform before passing through filter (imaginary part is zero), (b) image after passing through filter for a matched filter.

Fig. 11
Fig. 11

Output intensity for mirror-image optical correlator, shown in Fig. 2, after taking absolute values squared of Fourier transform of Fig. 10(b).

Fig. 12
Fig. 12

Half-plane mask of Fig. 10(b) to represent a 1D Hilbert transform.

Fig. 13
Fig. 13

Fourier transform of Fig. 12 for partial Hilbert transform correlator, shown in Fig. 7: (a) real part, (b) imaginary part.

Fig. 14
Fig. 14

Output intensity for Hilbert transform correlator, shown in Figure 3.

Fig. 15
Fig. 15

Output of spatial filter shown in Fig. 8.

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

gx, y=fx, y+f-x, -y.
Gkx, ky=-- gx, yexp-jkxx+kyydxdy=-- gx, ycoskxx+kyydxdy-j-- gx, ysinkxx+kyydxdy.
ux, y=g1x, y * g2x, y,
ux, y=f1x, y * f2x, y+f1-x, -y * f2-x, -y+f1x, y * f2-x, -y+f1-x, -y * f2x, y=2f1x, y * f2x, y+f1x, y  f2x, y+f2x, y  f1x, y,
Hkx=-jfor kx>0jfor kx<0 or Hkx=-j sgnkx,
Hkx=exp-jπ/2for kx>0expjπ/2for kx<0.
hx=1πx,
1πx=-1πxexp-jkxxdx=-1πxcoskxxdx-j -1πxsinkxxdx=-j sgnkx,
zx, y=ux, y+juˆx, y,
Zkx, ky=Ukx, ky+jHkxUkx, ky=Ukx, ky+j-j sgnkxUkx, ky=2Ukx, kyfor kx>00for kx<0.
Zkx, ky  zx, y.
0.5zx, y=0.5ux, y+j0.5uˆx, y.
0.5ux, y=0.5g1x, y * g2x, y.
Qkx, ky=0.5Zkx, ky+Aδ0, 0=Ukx, kyfor kx>0,kyeverywhereAδ0, 0for kx=0,ky=00for kx<0,kyeverywhere,
qx, y=0.5ux, y+j0.5uˆx, y+A.
|qx, y|2=0.5ux, y+A+0.5juˆx, y0.5ux, y+A-0.5juˆx, y=0.25u2x, y+A2+Aux, y+0.25uˆ2x, y.
Aux, y  u2x, yfor all x and y,Aux, y  uˆ2x, yfor all x and y.
|qx, y|2=A2+Aux, y.
|qx, y|2=A2+Ag1x, y * g2x, y.
E=0.5up2x, yA2+Aup2x, y+0.5up2x, y,
E12A2.
C=max-minmax+min=Aup2A2+Aup=upup+2A.
C=12A+1.
qx, y=|Aux, y+A2|2=A2u2x, y+A4+2A3ux, y=A4+A22Aux, y+u2x, y.
qx, y=A4+2A3ux, y.
Qfx, fy=A4δ0, 0+2A3Ufx, fy.
Qffx, fy=2A3Ufx, fy.
qfx, y=2A3ux, y.
|qfx, y|2=4A6|ux, y|2.
|qfx, y|2=4A6g1x, y * g2x, y2,
f1(x, y)=γ1 exp-2(x-μx,1)αx,12+(y-μy,1)αy,12+(x-μx,1)αx,1(y-μy,1)αy,1σ12+γ2 exp-2(x-μx,2)αx,22+(y-μy,2)αy,22+(x-μx,2)αx,2(y-μy,2)αy,2σ22.

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