Abstract

An improved method for implementing correlation filters in the joint transform correlator architecture is proposed. We derived the method from computer-generated holography techniques. It allows us to use any correlation filters, especially ones that provide an optimal trade-off between noise robustness, peak sharpness, and optical efficiency, with any spatial light modulator (SLM). This method also allows for an objective comparison of the performance of the coding domains of various SLM’s.

© 1999 Optical Society of America

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References

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  1. A. Maréchal, P. Croce, “Un filtre de fréquences spatiales pour l’amélioration du contraste des images optiques,” C. R. Acad. Sci. 237, 607–609 (1953).
  2. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
    [CrossRef]
  3. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  4. H. J. Caulfield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
    [CrossRef] [PubMed]
  5. Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  6. Ph. Réfrégier, “Filter design for optical pattern recognition: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
    [CrossRef] [PubMed]
  7. B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  8. B. V. K. Vijaya Kumar, R. D. Juday, K. P. Rajan, “Saturated filters,” J. Opt. Soc. Am. A 9, 405–412 (1992).
    [CrossRef]
  9. R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
    [CrossRef] [PubMed]
  10. V. Laude, Ph. Réfrégier, “Multicriteria characterization of coding domains with optimal Fourier SLM filters,” Appl. Opt. 33, 4465–4471 (1994).
    [CrossRef] [PubMed]
  11. B. V. K. Vijaya Kumar, D. W. Carlson, A. Mahalanobis, “Optimal trade-off synthetic discriminant function filters for arbitrary devices,” Opt. Lett. 19, 1556–1558 (1994).
    [CrossRef]
  12. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  13. L. P. Yaroslavsky, “Is the phase-only filter and its modifications optimal in terms of the discrimination capability in pattern recognition?” Appl. Opt. 31, 1677–1679 (1992).
    [CrossRef] [PubMed]
  14. Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  15. J. L. de Bougrenet de la Tocnaye, L. Dupont, “Complex amplitude modulation by use of liquid-crystal spatial light modulators,” Appl. Opt. 36, 1730–1741 (1997).
    [CrossRef]
  16. U. Mahlab, J. Shamir, “Iterative optimization algorithms for filter generation in optical correlators: a comparison,” Appl. Opt. 31, 1117–1125 (1992).
    [CrossRef] [PubMed]
  17. M. G. Roe, K. L. Schehrer, R. Dobson, L. Schirber, “Distortion-invariant optical pattern recognition using composite binary filters,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE1959, 203–213 (1993).
    [CrossRef]
  18. Y. Pétillot, G. Keryer, J. L. de Bougrenet de la Tocnaye, “Real-time distortion-invariant joint transform correlator using ferroelectric liquid crystal spatial light modulators,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, Ph. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Wash., 1994), pp. 267–274.
  19. B. V. K. Vijaya Kumar, D. W. Carlson, A. Mahalanobis, “Efficient determination of the optimum gain and angle in the design of optical correlation filters,” Opt. Eng. 37, 132–137 (1998).
    [CrossRef]
  20. J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filters,” Proc. IEEE 55, 599–601 (1967).
    [CrossRef]
  21. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3865–3870 (1989).
    [CrossRef]
  22. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  23. L. M. Bregman, “Finding the common point of convex sets by the method of successive projections,” Dokl. Akad. Nauk SSSR 162, 487–490 (1965).
  24. H. Stark, W. C. Catino, J. L. LoCicero, “Design of phase gratings by generalized projections,” J. Opt. Soc. Am. A 8, 566–571 (1991).
    [CrossRef]
  25. H. Stark, M. I. Sezan, “Image processing using projection methods,” in Real-Time Optical Information Processing, B. Javidi, J. L. Horner, eds. (Academic, San Diego, Calif., 1994), pp. 185–232.
  26. E. Zhang, S. Noehte, C. H. Dietrich, R. Männer, “Gradual and random binarization of gray-scale holograms,” Appl. Opt. 34, 5987–5995 (1995).
    [CrossRef] [PubMed]
  27. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]
  28. J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. SPIE884, 2–9 (1988).
    [CrossRef]
  29. B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
    [CrossRef]
  30. L. Legeard, Ph. Réfrégier, P. Ambs, “Multicriteria optimality for iterative encoding of computer generated holograms,” Appl. Opt. 36, 7444–7449 (1997).
    [CrossRef]
  31. I. Juvells, A. Carnicer, S. Vallmitjana, J. Campos, “Implementation of real filters in a joint transform correlator using a positive-only display,” J. Opt. 25, 33–40 (1994).
    [CrossRef]
  32. M. Taniguchi, K. Matsuoka, Y. Ichioka, “Computer-generated multiple-object discriminant correlation filters: design by simulated annealing,” Appl. Opt. 34, 1379–1385 (1995).
    [CrossRef] [PubMed]
  33. W.-H. Lee, “Sampled Fourier transform hologram generated by computer,” Appl. Opt. 9, 639–643 (1970).
    [CrossRef] [PubMed]
  34. S. Mazé, P. Réfrégier, “Optical correlation: influence of the coding of the input image,” Appl. Opt. 33, 6788–6796 (1994).
    [CrossRef] [PubMed]
  35. S.-T. Wu, “Nematic liquid crystal,” in Spatial Light Modulator Technology: Materials, Devices, and Applications, U. Efron, ed. (Marcel Dekker, New York, 1995), pp. 1–31.
  36. S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, “Programmable multiple-level phase modulation using ferroelectric liquid crystal spatial light modulators,” Appl. Opt. 34, 6652–6665 (1995).
    [CrossRef] [PubMed]

1998 (1)

B. V. K. Vijaya Kumar, D. W. Carlson, A. Mahalanobis, “Efficient determination of the optimum gain and angle in the design of optical correlation filters,” Opt. Eng. 37, 132–137 (1998).
[CrossRef]

1997 (2)

1995 (3)

1994 (4)

1993 (1)

1992 (4)

1991 (2)

1990 (1)

1989 (2)

F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3865–3870 (1989).
[CrossRef]

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

1987 (1)

1984 (1)

1982 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1970 (1)

1969 (1)

1967 (1)

J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filters,” Proc. IEEE 55, 599–601 (1967).
[CrossRef]

1966 (1)

1965 (1)

L. M. Bregman, “Finding the common point of convex sets by the method of successive projections,” Dokl. Akad. Nauk SSSR 162, 487–490 (1965).

1964 (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

1953 (1)

A. Maréchal, P. Croce, “Un filtre de fréquences spatiales pour l’amélioration du contraste des images optiques,” C. R. Acad. Sci. 237, 607–609 (1953).

Allebach, J. P.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
[CrossRef] [PubMed]

J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. SPIE884, 2–9 (1988).
[CrossRef]

Ambs, P.

Arsenault, H. H.

Bregman, L. M.

L. M. Bregman, “Finding the common point of convex sets by the method of successive projections,” Dokl. Akad. Nauk SSSR 162, 487–490 (1965).

Broomfield, S. E.

Burch, J. J.

J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filters,” Proc. IEEE 55, 599–601 (1967).
[CrossRef]

Campos, J.

I. Juvells, A. Carnicer, S. Vallmitjana, J. Campos, “Implementation of real filters in a joint transform correlator using a positive-only display,” J. Opt. 25, 33–40 (1994).
[CrossRef]

Carlson, D. W.

B. V. K. Vijaya Kumar, D. W. Carlson, A. Mahalanobis, “Efficient determination of the optimum gain and angle in the design of optical correlation filters,” Opt. Eng. 37, 132–137 (1998).
[CrossRef]

B. V. K. Vijaya Kumar, D. W. Carlson, A. Mahalanobis, “Optimal trade-off synthetic discriminant function filters for arbitrary devices,” Opt. Lett. 19, 1556–1558 (1994).
[CrossRef]

Carnicer, A.

I. Juvells, A. Carnicer, S. Vallmitjana, J. Campos, “Implementation of real filters in a joint transform correlator using a positive-only display,” J. Opt. 25, 33–40 (1994).
[CrossRef]

Catino, W. C.

Caulfield, H. J.

Croce, P.

A. Maréchal, P. Croce, “Un filtre de fréquences spatiales pour l’amélioration du contraste des images optiques,” C. R. Acad. Sci. 237, 607–609 (1953).

de Bougrenet de la Tocnaye, J. L.

J. L. de Bougrenet de la Tocnaye, L. Dupont, “Complex amplitude modulation by use of liquid-crystal spatial light modulators,” Appl. Opt. 36, 1730–1741 (1997).
[CrossRef]

Y. Pétillot, G. Keryer, J. L. de Bougrenet de la Tocnaye, “Real-time distortion-invariant joint transform correlator using ferroelectric liquid crystal spatial light modulators,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, Ph. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Wash., 1994), pp. 267–274.

Dietrich, C. H.

Dobson, R.

M. G. Roe, K. L. Schehrer, R. Dobson, L. Schirber, “Distortion-invariant optical pattern recognition using composite binary filters,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE1959, 203–213 (1993).
[CrossRef]

Dupont, L.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gianino, P. D.

Goodman, J. W.

Horner, J. L.

Hsu, Y. N.

Ichioka, Y.

Jennison, B. K.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

Juday, R. D.

Juvells, I.

I. Juvells, A. Carnicer, S. Vallmitjana, J. Campos, “Implementation of real filters in a joint transform correlator using a positive-only display,” J. Opt. 25, 33–40 (1994).
[CrossRef]

Keryer, G.

Y. Pétillot, G. Keryer, J. L. de Bougrenet de la Tocnaye, “Real-time distortion-invariant joint transform correlator using ferroelectric liquid crystal spatial light modulators,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, Ph. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Wash., 1994), pp. 267–274.

Laude, V.

Lee, W.-H.

Legeard, L.

LoCicero, J. L.

Mahalanobis, A.

B. V. K. Vijaya Kumar, D. W. Carlson, A. Mahalanobis, “Efficient determination of the optimum gain and angle in the design of optical correlation filters,” Opt. Eng. 37, 132–137 (1998).
[CrossRef]

B. V. K. Vijaya Kumar, D. W. Carlson, A. Mahalanobis, “Optimal trade-off synthetic discriminant function filters for arbitrary devices,” Opt. Lett. 19, 1556–1558 (1994).
[CrossRef]

Mahlab, U.

Maloney, W. T.

Männer, R.

Maréchal, A.

A. Maréchal, P. Croce, “Un filtre de fréquences spatiales pour l’amélioration du contraste des images optiques,” C. R. Acad. Sci. 237, 607–609 (1953).

Matsuoka, K.

Mazé, S.

Neil, M. A. A.

Noehte, S.

Paige, E. G. S.

Pétillot, Y.

Y. Pétillot, G. Keryer, J. L. de Bougrenet de la Tocnaye, “Real-time distortion-invariant joint transform correlator using ferroelectric liquid crystal spatial light modulators,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, Ph. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Wash., 1994), pp. 267–274.

Rajan, K. P.

Réfrégier, P.

Réfrégier, Ph.

Roe, M. G.

M. G. Roe, K. L. Schehrer, R. Dobson, L. Schirber, “Distortion-invariant optical pattern recognition using composite binary filters,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE1959, 203–213 (1993).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schehrer, K. L.

M. G. Roe, K. L. Schehrer, R. Dobson, L. Schirber, “Distortion-invariant optical pattern recognition using composite binary filters,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE1959, 203–213 (1993).
[CrossRef]

Schirber, L.

M. G. Roe, K. L. Schehrer, R. Dobson, L. Schirber, “Distortion-invariant optical pattern recognition using composite binary filters,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE1959, 203–213 (1993).
[CrossRef]

Seldowitz, M. A.

Sezan, M. I.

H. Stark, M. I. Sezan, “Image processing using projection methods,” in Real-Time Optical Information Processing, B. Javidi, J. L. Horner, eds. (Academic, San Diego, Calif., 1994), pp. 185–232.

Shamir, J.

Stark, H.

H. Stark, W. C. Catino, J. L. LoCicero, “Design of phase gratings by generalized projections,” J. Opt. Soc. Am. A 8, 566–571 (1991).
[CrossRef]

H. Stark, M. I. Sezan, “Image processing using projection methods,” in Real-Time Optical Information Processing, B. Javidi, J. L. Horner, eds. (Academic, San Diego, Calif., 1994), pp. 185–232.

Sweeney, D. W.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
[CrossRef] [PubMed]

J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. SPIE884, 2–9 (1988).
[CrossRef]

Taniguchi, M.

Vallmitjana, S.

I. Juvells, A. Carnicer, S. Vallmitjana, J. Campos, “Implementation of real filters in a joint transform correlator using a positive-only display,” J. Opt. 25, 33–40 (1994).
[CrossRef]

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

Vijaya Kumar, B. V. K.

Weaver, C. S.

Wu, S.-T.

S.-T. Wu, “Nematic liquid crystal,” in Spatial Light Modulator Technology: Materials, Devices, and Applications, U. Efron, ed. (Marcel Dekker, New York, 1995), pp. 1–31.

Wyrowski, F.

F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3865–3870 (1989).
[CrossRef]

Yaroslavsky, L. P.

Zhang, E.

Appl. Opt. (18)

C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
[CrossRef] [PubMed]

H. J. Caulfield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
[CrossRef] [PubMed]

Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
[CrossRef]

R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
[CrossRef] [PubMed]

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

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

L. P. Yaroslavsky, “Is the phase-only filter and its modifications optimal in terms of the discrimination capability in pattern recognition?” Appl. Opt. 31, 1677–1679 (1992).
[CrossRef] [PubMed]

J. L. de Bougrenet de la Tocnaye, L. Dupont, “Complex amplitude modulation by use of liquid-crystal spatial light modulators,” Appl. Opt. 36, 1730–1741 (1997).
[CrossRef]

U. Mahlab, J. Shamir, “Iterative optimization algorithms for filter generation in optical correlators: a comparison,” Appl. Opt. 31, 1117–1125 (1992).
[CrossRef] [PubMed]

F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3865–3870 (1989).
[CrossRef]

E. Zhang, S. Noehte, C. H. Dietrich, R. Männer, “Gradual and random binarization of gray-scale holograms,” Appl. Opt. 34, 5987–5995 (1995).
[CrossRef] [PubMed]

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
[CrossRef] [PubMed]

M. Taniguchi, K. Matsuoka, Y. Ichioka, “Computer-generated multiple-object discriminant correlation filters: design by simulated annealing,” Appl. Opt. 34, 1379–1385 (1995).
[CrossRef] [PubMed]

W.-H. Lee, “Sampled Fourier transform hologram generated by computer,” Appl. Opt. 9, 639–643 (1970).
[CrossRef] [PubMed]

S. Mazé, P. Réfrégier, “Optical correlation: influence of the coding of the input image,” Appl. Opt. 33, 6788–6796 (1994).
[CrossRef] [PubMed]

L. Legeard, Ph. Réfrégier, P. Ambs, “Multicriteria optimality for iterative encoding of computer generated holograms,” Appl. Opt. 36, 7444–7449 (1997).
[CrossRef]

S. E. Broomfield, M. A. A. Neil, E. G. S. Paige, “Programmable multiple-level phase modulation using ferroelectric liquid crystal spatial light modulators,” Appl. Opt. 34, 6652–6665 (1995).
[CrossRef] [PubMed]

C. R. Acad. Sci. (1)

A. Maréchal, P. Croce, “Un filtre de fréquences spatiales pour l’amélioration du contraste des images optiques,” C. R. Acad. Sci. 237, 607–609 (1953).

Dokl. Akad. Nauk SSSR (1)

L. M. Bregman, “Finding the common point of convex sets by the method of successive projections,” Dokl. Akad. Nauk SSSR 162, 487–490 (1965).

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

J. Opt. (1)

I. Juvells, A. Carnicer, S. Vallmitjana, J. Campos, “Implementation of real filters in a joint transform correlator using a positive-only display,” J. Opt. 25, 33–40 (1994).
[CrossRef]

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

Opt. Eng. (2)

B. V. K. Vijaya Kumar, D. W. Carlson, A. Mahalanobis, “Efficient determination of the optimum gain and angle in the design of optical correlation filters,” Opt. Eng. 37, 132–137 (1998).
[CrossRef]

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

Opt. Lett. (3)

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. IEEE (1)

J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filters,” Proc. IEEE 55, 599–601 (1967).
[CrossRef]

Other (5)

H. Stark, M. I. Sezan, “Image processing using projection methods,” in Real-Time Optical Information Processing, B. Javidi, J. L. Horner, eds. (Academic, San Diego, Calif., 1994), pp. 185–232.

J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer generated holography,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. SPIE884, 2–9 (1988).
[CrossRef]

S.-T. Wu, “Nematic liquid crystal,” in Spatial Light Modulator Technology: Materials, Devices, and Applications, U. Efron, ed. (Marcel Dekker, New York, 1995), pp. 1–31.

M. G. Roe, K. L. Schehrer, R. Dobson, L. Schirber, “Distortion-invariant optical pattern recognition using composite binary filters,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE1959, 203–213 (1993).
[CrossRef]

Y. Pétillot, G. Keryer, J. L. de Bougrenet de la Tocnaye, “Real-time distortion-invariant joint transform correlator using ferroelectric liquid crystal spatial light modulators,” in Euro-American Workshop on Optical Pattern Recognition, B. Javidi, Ph. Réfrégier, eds. (SPIE Optical Engineering Press, Bellingham, Wash., 1994), pp. 267–274.

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Figures (10)

Fig. 1
Fig. 1

Examples of coding domains: (a) unity disk, (b) real axis ([-1;1]), (c) amplitude only ([0;1]), (d) phase only, (e) binary, (f) ternary, (g) spiral.

Fig. 2
Fig. 2

Reference image used for synthesizing OT filters.

Fig. 3
Fig. 3

Locus of points (SNR, PBCE). The filters are not constrained to a particular coding domain.

Fig. 4
Fig. 4

Locus of points (PBCE, η) for α = 0 in the case in which the filter is constrained onto a binary domain. Various methods were investigated.

Fig. 5
Fig. 5

Locus of points (SNR, η) for β = 0 in the case in which the filter is constrained onto a binary domain. Various methods were investigated.

Fig. 6
Fig. 6

Locus of points (SNR, PBCE) in the case in which the filter is constrained onto a binary domain (γ has been optimized). Various methods were investigated.

Fig. 7
Fig. 7

Implementation of OT filters onto various coding domains. The curves describe the compromise between peak sharpness and optical efficiency.

Fig. 8
Fig. 8

Implementation of OT filters onto various coding domains. The curves describe the compromise between noise resistance and optical efficiency.

Fig. 9
Fig. 9

Implementation of OT filters onto various coding domains. The curves describe the compromise between noise resistance and peak sharpness. As explained in Section 5, the results for [0;1] are not explicitly given.

Fig. 10
Fig. 10

Complex histograms of OT filters implemented onto a spiral coding domain. The original filter (a) is supported by the real axis (thick line). When one implements a filter onto a spiral, the angle φ evolves in the function of the gain parameter γ [in (c) γ is 40% smaller than in (a); the histogram is therefore more compact] to make the projection onto the histogram minimize the Euclidean distance between the original filter and its projection onto the spiral. In (b) the implementation is performed with φ = 0; in (d), the filter has to be dephased by 100° before projection to make the projection onto the spiral (e) minimize the Euclidean distance.

Equations (29)

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

hˆMk=rˆk*Sˆk,
MSE=k Sˆk|hˆk|2.
CPE=k Dˆk|hˆk|2,
hˆIk=rˆk*Dˆk=rˆk*|rˆk|2.
η=|hˆ+·rˆ|2=khˆk*rˆk2,
hˆPOk=rˆk*|rˆk|.
Ehˆ=αMSE+βCPE-2γη,
hˆOTk=γ rˆk*Bˆk,
Bˆu=αSˆ+βDˆ.
SNRh=ηhMSEh=khˆk*rˆk2k Sˆk|hˆk|2,
PBCEh=ηhCPEh=khˆk*rˆk2k Dˆk|hˆk|2.
Eφhˆ=kBˆuk|hˆk-hˆu0k expiφ|2,
Eφhˆ=k |hˆk-hˆu0k expiφ|2.
Eφh=Bu+·h-hu0 exp-iφ** h-hu0 exp-iφ=kkBuk·hk-hu0k exp-iφ·hk-k-hu0k-k exp-iφ*,
Eφh=h-hu0 exp-iφ*·h-hu0 exp-iφ=k |hˆk-hˆu0k expiφ|2=k |hk-hu0k expiφ|2,
h0;1k=0.5·h-1;1k+1.
hˆ0;1k=0.5·hˆ-1;1k,  k0,
hpk=Phk.
hˆOK=hˆpk,hˆk+1n=hˆOKn+hˆu0n-hˆOKnspeedBˆun;
hˆk+1n=hˆOKn+hˆu0n-hˆOKn×speedBˆun.
hpk=Phk expiφ.
Ek+1=nBˆun·|hˆu0n-hˆpkn|2; if Ek+1<Ebest,perform φbest=φ,  Ebest=Ek+1.
Ek+1=Ebest,  φ=φbest,  hpk=Phk exp(iφ]
hˆOK=hˆpk,hˆk+1n=hˆOKn+hˆu0n-hˆOKn speed Bˆun;
hˆk+1n=hˆOKn+hˆu0n-hˆOKn speed Bˆun.
level1,, levelN=0, 1N-1,, KN-1,, 1-KN-1,, 1-1N-1, 1.
hpkn=hknifKN-1<hkn<1-KN-1Phknelse,
hˆOK=hˆpk,hˆk+1n=hˆOKn+hˆu0n-hˆOKn speed Bˆun;
hˆk+1n=hˆOKn+hˆu0n-hˆOKn×speedBˆun.

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