Abstract

We present a simple method of constructing synthetic discriminant function filters optimized to take into account the modulation of liquid-crystal devices. This relaxation algorithm, a generalization of the Jared and Ennis method, is an iterative method that includes arbitrary modulations for both scene and filter, extending the problem to the complex plane. Simulated and experimental results obtained in a VanderLugt correlator are presented for a two-class recognition problem. The optimal number of images needed to describe an object in a filter generated in this way is discussed, and the influence of the spatial light modulation resolution on the correlation is studied.

© 2004 Optical Society of America

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  1. C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [CrossRef] [PubMed]
  2. Y. N. Hsu, H. H. Arsenault, “Optical character recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  3. D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
    [CrossRef]
  4. S. Jutamulia, T. Asakura, “Rotation-invariant joint transform correlator,” Appl. Opt. 33, 5440–5442 (1994).
    [CrossRef] [PubMed]
  5. R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
    [CrossRef] [PubMed]
  6. J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to the synthetic discriminant function correlation filter,” Appl. Opt. 24, 851–855 (1985).
    [CrossRef] [PubMed]
  7. D. Casasent, W. A. Rozzi, “Computer-generated and phase-only synthetic discriminant function filters,” Appl. Opt. 25, 3767–3772 (1986).
    [CrossRef] [PubMed]
  8. R. R. Kallman, “Direct construction of phase-only filters,” Appl. Opt. 26, 5200–5201 (1987).
    [CrossRef] [PubMed]
  9. D. A. Jared, D. J. Ennis, “Inclusion of filter modulation in synthetic-discriminant-function construction,” Appl. Opt. 28, 232–239 (1989).
    [CrossRef] [PubMed]
  10. M. Montes-Usategui, J. Campos, I. Juvells, “Computation of arbitrarily constrained synthetic discriminant functions,” Appl. Opt. 34, 3904–3914 (1995).
    [CrossRef] [PubMed]
  11. R. D. Juday, “Generality of matched filtering and minimum Euclidean distance projection for optical pattern recognition,” J. Opt. Soc. Am. A 18, 1882–1896 (2001).
    [CrossRef]
  12. M. B. Reid, P. W. Ma, J. D. Downie, E. Ochoa, “Experimental verification of modified synthetic discriminant function filters for rotation invariance,” Appl. Opt. 29, 1209–1212 (1990).
    [CrossRef] [PubMed]
  13. P. C. Miller, R. S. Caprari, “Demonstration of improved automatic target-recognition performance by moment analysis of correlation peaks,” Appl. Opt. 38, 1325–1331 (1999).
    [CrossRef]
  14. D. W. Carlson, B. V. K. Vijaya Kumar, “Synthetic discriminant functions for implementations on arbitrarily constrained devices,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. SPIE1772, 10–20 (1992).
    [CrossRef]
  15. C. Zeile, E. Lüder, “Complex transmission of liquid crystal light modulators in optical signal processing applications,” in Liquid Crystal Materials, Devices, and Applications II, U. Efron, M. D. Wand, eds., Proc. SPIE1911, 195–206 (1993).
    [CrossRef]
  16. E. Martín-Badosa, A. Carnicer, I. Juvells, S. Vallmitjana, “Complex modulation characterization of liquid crystal devices by interferometric data correlation,” Meas. Sci. Technol. 8, 764–772 (1997).
    [CrossRef]
  17. M. Montes-Usategui, S. E. Monroe, R. D. Juday, “Automated self-alignment procedure for optical correlators,” Opt. Eng. 36, 1782–1791 (1997).
    [CrossRef]

2001 (1)

1999 (1)

1997 (2)

E. Martín-Badosa, A. Carnicer, I. Juvells, S. Vallmitjana, “Complex modulation characterization of liquid crystal devices by interferometric data correlation,” Meas. Sci. Technol. 8, 764–772 (1997).
[CrossRef]

M. Montes-Usategui, S. E. Monroe, R. D. Juday, “Automated self-alignment procedure for optical correlators,” Opt. Eng. 36, 1782–1791 (1997).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

1990 (1)

1989 (1)

1988 (1)

D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

1987 (1)

1986 (1)

1985 (1)

1982 (1)

1980 (1)

Arsenault, H. H.

Asakura, T.

Campos, J.

Caprari, R. S.

Carlson, D. W.

D. W. Carlson, B. V. K. Vijaya Kumar, “Synthetic discriminant functions for implementations on arbitrarily constrained devices,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. SPIE1772, 10–20 (1992).
[CrossRef]

Carnicer, A.

E. Martín-Badosa, A. Carnicer, I. Juvells, S. Vallmitjana, “Complex modulation characterization of liquid crystal devices by interferometric data correlation,” Meas. Sci. Technol. 8, 764–772 (1997).
[CrossRef]

Casasent, D.

Downie, J. D.

Ennis, D. J.

Gianino, P. D.

Hester, C. F.

Horner, J. L.

Hsu, Y. N.

Jared, D. A.

Juday, R. D.

Jutamulia, S.

Juvells, I.

E. Martín-Badosa, A. Carnicer, I. Juvells, S. Vallmitjana, “Complex modulation characterization of liquid crystal devices by interferometric data correlation,” Meas. Sci. Technol. 8, 764–772 (1997).
[CrossRef]

M. Montes-Usategui, J. Campos, I. Juvells, “Computation of arbitrarily constrained synthetic discriminant functions,” Appl. Opt. 34, 3904–3914 (1995).
[CrossRef] [PubMed]

Kallman, R. R.

Konforti, N.

D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Lüder, E.

C. Zeile, E. Lüder, “Complex transmission of liquid crystal light modulators in optical signal processing applications,” in Liquid Crystal Materials, Devices, and Applications II, U. Efron, M. D. Wand, eds., Proc. SPIE1911, 195–206 (1993).
[CrossRef]

Ma, P. W.

Marom, E.

D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Martín-Badosa, E.

E. Martín-Badosa, A. Carnicer, I. Juvells, S. Vallmitjana, “Complex modulation characterization of liquid crystal devices by interferometric data correlation,” Meas. Sci. Technol. 8, 764–772 (1997).
[CrossRef]

Mendlovic, D.

D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Miller, P. C.

Monroe, S. E.

M. Montes-Usategui, S. E. Monroe, R. D. Juday, “Automated self-alignment procedure for optical correlators,” Opt. Eng. 36, 1782–1791 (1997).
[CrossRef]

Montes-Usategui, M.

M. Montes-Usategui, S. E. Monroe, R. D. Juday, “Automated self-alignment procedure for optical correlators,” Opt. Eng. 36, 1782–1791 (1997).
[CrossRef]

M. Montes-Usategui, J. Campos, I. Juvells, “Computation of arbitrarily constrained synthetic discriminant functions,” Appl. Opt. 34, 3904–3914 (1995).
[CrossRef] [PubMed]

Ochoa, E.

Reid, M. B.

Rozzi, W. A.

Vallmitjana, S.

E. Martín-Badosa, A. Carnicer, I. Juvells, S. Vallmitjana, “Complex modulation characterization of liquid crystal devices by interferometric data correlation,” Meas. Sci. Technol. 8, 764–772 (1997).
[CrossRef]

Vijaya Kumar, B. V. K.

D. W. Carlson, B. V. K. Vijaya Kumar, “Synthetic discriminant functions for implementations on arbitrarily constrained devices,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. SPIE1772, 10–20 (1992).
[CrossRef]

Zeile, C.

C. Zeile, E. Lüder, “Complex transmission of liquid crystal light modulators in optical signal processing applications,” in Liquid Crystal Materials, Devices, and Applications II, U. Efron, M. D. Wand, eds., Proc. SPIE1911, 195–206 (1993).
[CrossRef]

Appl. Opt. (11)

C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
[CrossRef] [PubMed]

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

J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to the synthetic discriminant function correlation filter,” Appl. Opt. 24, 851–855 (1985).
[CrossRef] [PubMed]

D. Casasent, W. A. Rozzi, “Computer-generated and phase-only synthetic discriminant function filters,” Appl. Opt. 25, 3767–3772 (1986).
[CrossRef] [PubMed]

R. R. Kallman, “Direct construction of phase-only filters,” Appl. Opt. 26, 5200–5201 (1987).
[CrossRef] [PubMed]

D. A. Jared, D. J. Ennis, “Inclusion of filter modulation in synthetic-discriminant-function construction,” Appl. Opt. 28, 232–239 (1989).
[CrossRef] [PubMed]

M. B. Reid, P. W. Ma, J. D. Downie, E. Ochoa, “Experimental verification of modified synthetic discriminant function filters for rotation invariance,” Appl. Opt. 29, 1209–1212 (1990).
[CrossRef] [PubMed]

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

S. Jutamulia, T. Asakura, “Rotation-invariant joint transform correlator,” Appl. Opt. 33, 5440–5442 (1994).
[CrossRef] [PubMed]

P. C. Miller, R. S. Caprari, “Demonstration of improved automatic target-recognition performance by moment analysis of correlation peaks,” Appl. Opt. 38, 1325–1331 (1999).
[CrossRef]

M. Montes-Usategui, J. Campos, I. Juvells, “Computation of arbitrarily constrained synthetic discriminant functions,” Appl. Opt. 34, 3904–3914 (1995).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

E. Martín-Badosa, A. Carnicer, I. Juvells, S. Vallmitjana, “Complex modulation characterization of liquid crystal devices by interferometric data correlation,” Meas. Sci. Technol. 8, 764–772 (1997).
[CrossRef]

Opt. Commun. (1)

D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Opt. Eng. (1)

M. Montes-Usategui, S. E. Monroe, R. D. Juday, “Automated self-alignment procedure for optical correlators,” Opt. Eng. 36, 1782–1791 (1997).
[CrossRef]

Other (2)

D. W. Carlson, B. V. K. Vijaya Kumar, “Synthetic discriminant functions for implementations on arbitrarily constrained devices,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. SPIE1772, 10–20 (1992).
[CrossRef]

C. Zeile, E. Lüder, “Complex transmission of liquid crystal light modulators in optical signal processing applications,” in Liquid Crystal Materials, Devices, and Applications II, U. Efron, M. D. Wand, eds., Proc. SPIE1911, 195–206 (1993).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Operative curves considered in the filter generation: (a) modulation of the scene (high-contrast configuration), (b) modulation of the filter (phase-mostly configuration).

Fig. 2
Fig. 2

Images of the training set: (a) class A, (b) class B.

Fig. 3
Fig. 3

Evolution of the central correlation amplitude along the iteration in the generation with the generalized method of a filter that includes four images, two of each class, with the detection of class A.

Fig. 4
Fig. 4

Central correlation amplitude with each of the 36 images belonging to class A and a filter generated from all of them.

Fig. 5
Fig. 5

Influence of the number of images per filter in the correlation. The filled squares show the mean value of the maxima of correlation between filters, which were generated from a number of images each rotated 5°, and images of the detected class, those belonging to the training set, and intermediate images. The open circles indicate the mean value of the maximum with images of the rejected class and thus should have a low value. The detected class is sharks, and the number of images in the filter is equal for each class. The correlation values are normalized to the autocorrelation, as in Figs. 4, 5, 6, 7, 8, 9 and 11.

Fig. 6
Fig. 6

Maximum correlation amplitude of a single matched filter against the rotation of the input object. At a rotation of 2° the amplitude has decreased to half of the autocorrelation value.

Fig. 7
Fig. 7

Maximum correlation amplitude with a filter generated from 20 images, each rotated 3° and both belonging to the training set and intermediate images.

Fig. 8
Fig. 8

Influence in the correlation of the sampling interval of the images in the training set. The filled squares show the mean value of the correlation maxima between filters and images of the detected class, both belonging to the training set and intermediate images. The open circles indicate the mean value of the maximum with images of the rejected class. The detected class is sharks.

Fig. 9
Fig. 9

Influence in the correlation of the number of gray levels in which the filter is adapted. The filled squares show the mean value of the maxima of correlation between filters, which were generated from 20 images of each class separated 3°, and images of the detected class, both belonging to the training set and intermediate images. The open circles indicate the mean value of the maximum with images of the rejected class. The detected class is sharks.

Fig. 10
Fig. 10

Picture of the VanderLugt convergent correlator. 1, laser; 2, mirror; 3, 8, 10, and 13, polarizing elements; 4, attenuation filter; 5, pinhole; 6 and 12, convergent lenses; 7 and 11, LCD; 9, divergent lens; 14, CCD camera; 15, video projector.

Fig. 11
Fig. 11

Maximum correlation amplitude captured experimentally with each of the 20 images included in the training set; the sharks had a rotation of 3° each.

Fig. 12
Fig. 12

(a) Input scene with two images of class A and (b) experimental correlation plane captured with a filter generated from 20 images belonging to class A.

Fig. 13
Fig. 13

(a) Input scene with an image of class A and another of class B and (b) experimental correlation plane captured with a filter generated from 20 images belonging to class A.

Tables (3)

Tables Icon

Table 1 Influence of the Number of Images per Filter in the Fisher Ratio

Tables Icon

Table 2 Influence of the Angular Separation among Images in the Filter in the Fisher Ratio

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Table 3 Influence of the Number of Gray Levels in the Filter in the Fisher Ratio

Equations (8)

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tnx, yhx, y0,0=cn,
hx, y=n=0N antnx, y,
ani+1=ani+βcn-c0mnim0i,
mni=tnx, yhix, y0,0,
Hu, v=fn=0N-1 anitnx, y*,
ani+1=ani+βcn-c0mnim0i,
ani+1=ani+βcn-c0mnim0i.
J=IA-IB2σA2+σB2,

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