Abstract

We describe an incoherent correlator, based on the shadow-casting principle, that is able to implement any real-valued linear correlation filter. The correlation filter and the input image are displayed on commercial liquid-crystal television (LCTV) panels. Although it cannot handle high-resolution images, the incoherent correlator is lensless, compact, low cost, and uses a white-light source. A bipolar technique is devised to represent any linear filter, computed from a single reference image or composite, in the correlator. We demonstrate experimentally the efficiency of the design in the case of optimal trade-off (OT) filters and optimal trade-off synthetic discriminant function (OT–SDF) filters.

© 1996 Optical Society of America

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References

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    [CrossRef]
  2. L. Bragg, “Lightning calculations with light,” Nature 154, 69–72 (1944).
    [CrossRef]
  3. E. L. Green, “Diffraction in lensless correlation,” Appl. Opt. 7, 1237–1239 (1968).
    [CrossRef] [PubMed]
  4. G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).
  5. B. V. K. Vijaya Kumar, C. Hendrix, D. W. Carlson, “Tradeoffs in the design of correlation filters,” in Optical Pattern Recognition, J. L. Horner, B. Javidi, eds. (Society of Photo-Optical Instrument Engineers Optical Engineering, Bellingham, Wash., 1992), pp. 191–215.
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    [CrossRef]
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    [CrossRef]
  8. Y. Li, A. Kostrzewski, D. H. Kim, G. Eichmann, “Compact parallel real-time programmable optical morphological image processor,” Opt. Lett. 14, 981–983 (1989).
    [CrossRef] [PubMed]
  9. A. Louri, “Efficient optical implementation method for symbolic substitution logic based on shadow casting,” Appl. Opt. 14, 3264–3267 (1989).
  10. P. L. Jackson, “Correlation function spatial filtering with incoherent light,” Appl. Opt. 6, 1272–1973 (1967).
    [CrossRef] [PubMed]
  11. D. Raj, D. W. Prather, R. A. Athale, J. N. Mait, “Performance analysis of optical shadow-casting correlators,” Appl. Opt. 32, 3108–3112 (1993).
    [CrossRef] [PubMed]
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  13. M. Gedziorowski, T. Szoplik, “Resolution of a lensless shadow casting correlator with partially coherent illumination,” Opt. Commun. 106, 167–172 (1994).
    [CrossRef]
  14. B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [CrossRef]
  15. J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
    [CrossRef] [PubMed]
  16. Ph. Réfrégier, “Application of the stabilizing functional approach to pattern recognition filters,” J. Opt. Soc. Am. A 11, 1243–1251 (1994).
    [CrossRef]
  17. Ph. Réfrégier, V. Laude, B. Javidi, “Basic properties of nonlinear global filtering techniques and optimal discriminant solutions,” Appl. Opt. 34, 3915–3923 (1995).
    [CrossRef] [PubMed]
  18. L. P. Yaroslavsky, “Optical correlators with (−k)th-law nonlinearity: optimal and suboptimal solutions,” Appl. Opt. 34, 3924–3932 (1995).
    [CrossRef] [PubMed]
  19. Ph. Réfrégier, “Filter design for optical pattern recognition: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
    [CrossRef] [PubMed]
  20. J. Figue, Ph. Réfrégier, “On the optimality of trade-off filters,” Appl. Opt. 32, 1933–1935 (1993).
    [CrossRef] [PubMed]
  21. V. Laude, Ph. Réfrégier, “Multicriteria characterization of coding domains with optimal Fourier spatial light modulator filters,” Appl. Opt. 33, 4465–4471 (1994).
    [CrossRef] [PubMed]
  22. V. Laude, S. Maze, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
    [CrossRef]
  23. L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in images,” in Progress in Optics XXXII, (North Holland, Amsterdam, 1993), pp. 145–201.
    [CrossRef]
  24. D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef] [PubMed]
  25. J. Figue, Ph. Réfrégier, “Angle determination of airplanes by multicorrelation techniques with optimal trade-off synthetic discriminant filters,” Opt. Eng. 33, 1821–1828 (1994).
    [CrossRef]

1995 (2)

1994 (4)

V. Laude, Ph. Réfrégier, “Multicriteria characterization of coding domains with optimal Fourier spatial light modulator filters,” Appl. Opt. 33, 4465–4471 (1994).
[CrossRef] [PubMed]

M. Gedziorowski, T. Szoplik, “Resolution of a lensless shadow casting correlator with partially coherent illumination,” Opt. Commun. 106, 167–172 (1994).
[CrossRef]

Ph. Réfrégier, “Application of the stabilizing functional approach to pattern recognition filters,” J. Opt. Soc. Am. A 11, 1243–1251 (1994).
[CrossRef]

J. Figue, Ph. Réfrégier, “Angle determination of airplanes by multicorrelation techniques with optimal trade-off synthetic discriminant filters,” Opt. Eng. 33, 1821–1828 (1994).
[CrossRef]

1993 (3)

1992 (1)

1990 (2)

1989 (2)

Y. Li, A. Kostrzewski, D. H. Kim, G. Eichmann, “Compact parallel real-time programmable optical morphological image processor,” Opt. Lett. 14, 981–983 (1989).
[CrossRef] [PubMed]

A. Louri, “Efficient optical implementation method for symbolic substitution logic based on shadow casting,” Appl. Opt. 14, 3264–3267 (1989).

1986 (1)

1984 (2)

Y. Ichioka, J. Tanida, “Optical parallel logic gates using a shadow-casting system for optical digital computing,” Proc. IEEE 72, 787–801 (1984).
[CrossRef]

D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
[CrossRef] [PubMed]

1968 (1)

1967 (1)

1944 (1)

L. Bragg, “Lightning calculations with light,” Nature 154, 69–72 (1944).
[CrossRef]

1943 (1)

M. M. Robertson, “Interpretation of Patterson diagrams,” Nature 152, 411–412 (1943).
[CrossRef]

Athale, R. A.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Bragg, L.

L. Bragg, “Lightning calculations with light,” Nature 154, 69–72 (1944).
[CrossRef]

Carlson, D. W.

B. V. K. Vijaya Kumar, C. Hendrix, D. W. Carlson, “Tradeoffs in the design of correlation filters,” in Optical Pattern Recognition, J. L. Horner, B. Javidi, eds. (Society of Photo-Optical Instrument Engineers Optical Engineering, Bellingham, Wash., 1992), pp. 191–215.

Casasent, D.

Chavel, P.

V. Laude, S. Maze, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

Eichmann, G.

Figue, J.

J. Figue, Ph. Réfrégier, “Angle determination of airplanes by multicorrelation techniques with optimal trade-off synthetic discriminant filters,” Opt. Eng. 33, 1821–1828 (1994).
[CrossRef]

J. Figue, Ph. Réfrégier, “On the optimality of trade-off filters,” Appl. Opt. 32, 1933–1935 (1993).
[CrossRef] [PubMed]

Gedziorowski, M.

M. Gedziorowski, T. Szoplik, “Resolution of a lensless shadow casting correlator with partially coherent illumination,” Opt. Commun. 106, 167–172 (1994).
[CrossRef]

Green, E. L.

Hassebrook, L.

Hendrix, C.

B. V. K. Vijaya Kumar, C. Hendrix, D. W. Carlson, “Tradeoffs in the design of correlation filters,” in Optical Pattern Recognition, J. L. Horner, B. Javidi, eds. (Society of Photo-Optical Instrument Engineers Optical Engineering, Bellingham, Wash., 1992), pp. 191–215.

Horner, J. L.

Ichioka, Y.

Y. Ichioka, J. Tanida, “Optical parallel logic gates using a shadow-casting system for optical digital computing,” Proc. IEEE 72, 787–801 (1984).
[CrossRef]

Jackson, P. L.

Javidi, B.

Kim, D. H.

Kostrzewski, A.

Laude, V.

Li, Y.

Louri, A.

A. Louri, “Efficient optical implementation method for symbolic substitution logic based on shadow casting,” Appl. Opt. 14, 3264–3267 (1989).

Mait, J. N.

Maze, S.

V. Laude, S. Maze, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

Prather, D. W.

Raj, D.

Réfrégier, Ph.

Robertson, M. M.

M. M. Robertson, “Interpretation of Patterson diagrams,” Nature 152, 411–412 (1943).
[CrossRef]

Rogers, G. L.

G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).

Szoplik, T.

M. Gedziorowski, T. Szoplik, “Resolution of a lensless shadow casting correlator with partially coherent illumination,” Opt. Commun. 106, 167–172 (1994).
[CrossRef]

Tanida, J.

Y. Ichioka, J. Tanida, “Optical parallel logic gates using a shadow-casting system for optical digital computing,” Proc. IEEE 72, 787–801 (1984).
[CrossRef]

Vijaya Kumar, B. V. K.

B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[CrossRef]

B. V. K. Vijaya Kumar, C. Hendrix, D. W. Carlson, “Tradeoffs in the design of correlation filters,” in Optical Pattern Recognition, J. L. Horner, B. Javidi, eds. (Society of Photo-Optical Instrument Engineers Optical Engineering, Bellingham, Wash., 1992), pp. 191–215.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Yaroslavsky, L. P.

L. P. Yaroslavsky, “Optical correlators with (−k)th-law nonlinearity: optimal and suboptimal solutions,” Appl. Opt. 34, 3924–3932 (1995).
[CrossRef] [PubMed]

L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in images,” in Progress in Optics XXXII, (North Holland, Amsterdam, 1993), pp. 145–201.
[CrossRef]

Appl. Opt. (11)

E. L. Green, “Diffraction in lensless correlation,” Appl. Opt. 7, 1237–1239 (1968).
[CrossRef] [PubMed]

A. Louri, “Efficient optical implementation method for symbolic substitution logic based on shadow casting,” Appl. Opt. 14, 3264–3267 (1989).

P. L. Jackson, “Correlation function spatial filtering with incoherent light,” Appl. Opt. 6, 1272–1973 (1967).
[CrossRef] [PubMed]

D. Raj, D. W. Prather, R. A. Athale, J. N. Mait, “Performance analysis of optical shadow-casting correlators,” Appl. Opt. 32, 3108–3112 (1993).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[CrossRef]

J. L. Horner, “Metrics for assessing pattern-recognition performance,” Appl. Opt. 31, 165–166 (1992).
[CrossRef] [PubMed]

Ph. Réfrégier, V. Laude, B. Javidi, “Basic properties of nonlinear global filtering techniques and optimal discriminant solutions,” Appl. Opt. 34, 3915–3923 (1995).
[CrossRef] [PubMed]

L. P. Yaroslavsky, “Optical correlators with (−k)th-law nonlinearity: optimal and suboptimal solutions,” Appl. Opt. 34, 3924–3932 (1995).
[CrossRef] [PubMed]

J. Figue, Ph. Réfrégier, “On the optimality of trade-off filters,” Appl. Opt. 32, 1933–1935 (1993).
[CrossRef] [PubMed]

V. Laude, Ph. Réfrégier, “Multicriteria characterization of coding domains with optimal Fourier spatial light modulator filters,” Appl. Opt. 33, 4465–4471 (1994).
[CrossRef] [PubMed]

D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (2)

Nature (2)

M. M. Robertson, “Interpretation of Patterson diagrams,” Nature 152, 411–412 (1943).
[CrossRef]

L. Bragg, “Lightning calculations with light,” Nature 154, 69–72 (1944).
[CrossRef]

Opt. Commun. (2)

M. Gedziorowski, T. Szoplik, “Resolution of a lensless shadow casting correlator with partially coherent illumination,” Opt. Commun. 106, 167–172 (1994).
[CrossRef]

V. Laude, S. Maze, P. Chavel, Ph. Réfrégier, “Amplitude and phase coding measurements of a liquid crystal television,” Opt. Commun. 103, 33–38 (1993).
[CrossRef]

Opt. Eng. (1)

J. Figue, Ph. Réfrégier, “Angle determination of airplanes by multicorrelation techniques with optimal trade-off synthetic discriminant filters,” Opt. Eng. 33, 1821–1828 (1994).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

Y. Ichioka, J. Tanida, “Optical parallel logic gates using a shadow-casting system for optical digital computing,” Proc. IEEE 72, 787–801 (1984).
[CrossRef]

Other (4)

G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).

B. V. K. Vijaya Kumar, C. Hendrix, D. W. Carlson, “Tradeoffs in the design of correlation filters,” in Optical Pattern Recognition, J. L. Horner, B. Javidi, eds. (Society of Photo-Optical Instrument Engineers Optical Engineering, Bellingham, Wash., 1992), pp. 191–215.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in images,” in Progress in Optics XXXII, (North Holland, Amsterdam, 1993), pp. 145–201.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Shadow-casting principle. In the first plane, P 1 , is a diffuse source. In the second, P 2 , is a transparency. In the third, P 3 , is a screen or a camera.

Fig. 2
Fig. 2

Several possibilities for the diffuse source in plane P 1 . (a) CRT, (b) LED or laser diode array, (c) SLM followed by a diffuser.

Fig. 3
Fig. 3

Principle of OT filters.

Fig. 4
Fig. 4

Example of bipolar filter design. (a) Reference image. (b) OT filter designed from reference image. Since this filter has positive and negative values, the zero value is offset to grey level 127. (c) Positive part of bipolar filter. (d) Negative part of bipolar filter.

Fig. 5
Fig. 5

Shadow-casting correlator. Filter h and input image x are displayed on Epson LCTV SLM’s. A, analyzer; P, polarizer; D, diffuser; RGB, red, green, and blue inputs to the Epson video projector.

Fig. 6
Fig. 6

Example of a bipolar filtering result. (a) Input image, from which the reference image of Fig. 4(a) was extracted; (b) three-dimensional view of the measured correlation of the bipolar filter of Fig. 4 and the input image; (c) three-dimensional view of the simulated correlation of the bipolar filter of Fig. 4 and the input image.

Fig. 7
Fig. 7

Example of a bipolar filtering result. (a) Input image, slightly different from that of Fig. 6(a); (b) three-dimensional view of the measured correlation of the bipolar filter of Fig. 4 and the input image.

Fig. 8
Fig. 8

Example of a bipolar filtering result. (a) Positive part of bipolar filter; (b) negative part of bipolar filter; (c) input image from which the reference image was extracted; (d) three-dimensional view of the measured correlation of the bipolar filter and the input image.

Fig. 9
Fig. 9

Example of a OT–SDF filter designed for attitude recognition (see text) and its bipolar decomposition. (a) A pattern of the learning base; (b) OT–SDF filter designed from learning base; (c) positive part of bipolar OT–SDF filter; (d) negative part of bipolar OT–SDF filter.

Fig. 10
Fig. 10

Experimental response of the bipolar OT–SDF filter of Fig. 9 when the airplane is viewed on a black or a cloudy background. (a) Example of a test image without background and the airplane rotated at 28°; (b) example of a test image with a cloudy background and the airplane rotated at 28°; (c) normalized response of OT–SDF filter.

Equations (22)

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r 2 d = r 1 d + r 3 p ,
1 d = 1 d + 1 p .
E ( r 3 ) = P 1 d r 1 ξ ( r 1 , r 3 ) M 1 ( r 1 ) M 2 [ d ( r 1 d + r 3 p ) ] .
G = d | d | = | d + p p | .
ρ 1 ρ 2 = αλ d ,
S 1 S 2 = 4 α 2 λ 2 d 2 ,
N 1 = A 1 / S 1 , N 2 = A 2 / S 2 ,
N 1 N 2 = A 1 A 2 4 α 2 λ 2 d 2 .
N 1 = N 2 115 × 115 .
[ h x ] k = c k 1 N m = 1 N h m * x m + k .
c ˆ n = h ˆ n * x ˆ n .
MSE ( h ) h ˆ S ˆ h ˆ
SNR ( h ) | h ˆ r ˆ | 2 h ˆ S ˆ h ˆ ,
h ˆ n = r ˆ n S ˆ n n .
PCE ( h ) | h ˆ r ˆ | 2 h ˆ D ˆ h ˆ ,
h ˆ n = r ˆ n | r ˆ n | 2 .
h ˆ n OT = r ˆ n ( 1 μ ) S ˆ n n + μ D ˆ n n .
h = h + h , with h + , h [ 0 , 1 ] N .
| c | 2 = | c + c | 2 = | h + x h x | 2 = | h x | 2 .
h = [ h + + f ] [ h + f ] , with [ h + + f ] , [ h + f ] [ 0 , 1 ] N .
h = DFT 1 ( γ h ˆ OT ) .
h m + = max ( 0 , h m ) , h m = max ( 0 , h m ) .

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