Abstract

We introduce a modification of the nonlinear morphological correlation for optical rotation-invariant pattern recognition. The high selectivity of the morphological correlation is conserved compared with standard linear correlation. The operation performs the common morphological correlation by extraction of the information by means of a circular-harmonic component of a reference. In spite of some loss of information good discrimination is obtained, especially for detecting images with a high degree of resemblance. Computer simulations are presented, as well as optical experiments implemented with a joint transform correlator.

© 2000 Optical Society of America

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References

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  1. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973), Chap. 8.
  2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  3. Y. N. Hsu, H. H. Arsenault, G. April, “Rotation-invariant digital pattern recognition using circular-harmonic expansion,” Appl. Opt. 21, 4012–4015 (1982).
    [Crossref] [PubMed]
  4. Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular-harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [Crossref] [PubMed]
  5. D. Casasent, D. Psaltis, “Scale-invariant optical transform,” Opt. Eng. 15, 258–261 (1976).
    [Crossref]
  6. D. Mendlovic, N. Konforti, E. Marom, “Shift- and projection-invariant pattern recognition,” Appl. Opt. 28, 4784–4789 (1990).
    [Crossref]
  7. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  8. C. S. Weaver, J. W. Goodman, “Technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [Crossref] [PubMed]
  9. P. Garcia-Martinez, D. Mas, J. Garcia, C. Ferreira, “Nonlinear morphological correlation: optoelectronic implementation,” Appl. Opt. 37, 2112–2118 (1998).
    [Crossref]
  10. A. Shemer, D. Mendlovic, G. Shabtay, P. Garcia-Martinez, J. Garcia, “Modified morphological correlation based on bit-map representation,” Appl. Opt. 38, 781–787 (1999).
    [Crossref]
  11. P. Maragos, R. W. Shafer, “Morphological systems for multidimensional signal processing,” Proc. IEEE 78, 894–902 (1982).
  12. Y. Sheng, H. H. Arsenault, “Method of determining expansion centers and predicting circular-harmonic filters,” J. Opt. Soc. Am. A 4, 1793–1797 (1987).
    [Crossref]
  13. G. Premont, Y. Sheng, “Fast design of circular-harmonic filters using simulated annealing,” Appl. Opt. 32, 3116–3121 (1993).
    [Crossref] [PubMed]
  14. P. Garcia-Martinez, J. Garcia, C. Ferreira, “A new criterion for determining the expansion center for circular-harmonic filters,” Opt. Commun. 117, 399–405 (1995).
    [Crossref]
  15. F. T. S. Yu, X. Li, E. Tam, S. Jutamulia, D. A. Gregory, “Rotation-invariant pattern recognition with a programmable joint transform correlator,” Appl. Opt. 28, 4725–4727 (1989).
    [Crossref] [PubMed]
  16. D. Mendlovic, E. Marom, N. Konforti, “Complex-reference joint transform correlator,” Opt. Lett. 15, 1224–1226 (1990).
    [Crossref] [PubMed]
  17. R. Pierstun, J. Rosen, J. Shamir, “Generation of continuous complex-valued functions of a joint transform correlator,” Appl. Opt. 33, 4398–4405 (1994).
    [Crossref]
  18. H. H. Arsenault, C. Ferreira, M. P. Levesque, T. Szoplik, “Simpler filter with limited rotation invariance,” Appl. Opt. 25, 3230–3234 (1986).
    [Crossref]
  19. Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum circular symmetrical filters and their uses in pattern recognition,” Opt. Acta 29, 627–644 (1982).
    [Crossref]

1999 (1)

1998 (1)

1995 (1)

P. Garcia-Martinez, J. Garcia, C. Ferreira, “A new criterion for determining the expansion center for circular-harmonic filters,” Opt. Commun. 117, 399–405 (1995).
[Crossref]

1994 (1)

1993 (1)

1990 (2)

D. Mendlovic, E. Marom, N. Konforti, “Complex-reference joint transform correlator,” Opt. Lett. 15, 1224–1226 (1990).
[Crossref] [PubMed]

D. Mendlovic, N. Konforti, E. Marom, “Shift- and projection-invariant pattern recognition,” Appl. Opt. 28, 4784–4789 (1990).
[Crossref]

1989 (1)

1987 (1)

1986 (1)

1982 (4)

Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum circular symmetrical filters and their uses in pattern recognition,” Opt. Acta 29, 627–644 (1982).
[Crossref]

Y. N. Hsu, H. H. Arsenault, G. April, “Rotation-invariant digital pattern recognition using circular-harmonic expansion,” Appl. Opt. 21, 4012–4015 (1982).
[Crossref] [PubMed]

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

P. Maragos, R. W. Shafer, “Morphological systems for multidimensional signal processing,” Proc. IEEE 78, 894–902 (1982).

1976 (1)

D. Casasent, D. Psaltis, “Scale-invariant optical transform,” Opt. Eng. 15, 258–261 (1976).
[Crossref]

1966 (1)

1964 (1)

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

April, G.

Arsenault, H. H.

Casasent, D.

D. Casasent, D. Psaltis, “Scale-invariant optical transform,” Opt. Eng. 15, 258–261 (1976).
[Crossref]

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973), Chap. 8.

Ferreira, C.

Garcia, J.

Garcia-Martinez, P.

Goodman, J. W.

Gregory, D. A.

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973), Chap. 8.

Hsu, Y. N.

Jutamulia, S.

Konforti, N.

D. Mendlovic, E. Marom, N. Konforti, “Complex-reference joint transform correlator,” Opt. Lett. 15, 1224–1226 (1990).
[Crossref] [PubMed]

D. Mendlovic, N. Konforti, E. Marom, “Shift- and projection-invariant pattern recognition,” Appl. Opt. 28, 4784–4789 (1990).
[Crossref]

Levesque, M. P.

Li, X.

Maragos, P.

P. Maragos, R. W. Shafer, “Morphological systems for multidimensional signal processing,” Proc. IEEE 78, 894–902 (1982).

Marom, E.

D. Mendlovic, N. Konforti, E. Marom, “Shift- and projection-invariant pattern recognition,” Appl. Opt. 28, 4784–4789 (1990).
[Crossref]

D. Mendlovic, E. Marom, N. Konforti, “Complex-reference joint transform correlator,” Opt. Lett. 15, 1224–1226 (1990).
[Crossref] [PubMed]

Mas, D.

Mendlovic, D.

Pierstun, R.

Premont, G.

Psaltis, D.

D. Casasent, D. Psaltis, “Scale-invariant optical transform,” Opt. Eng. 15, 258–261 (1976).
[Crossref]

Rosen, J.

Shabtay, G.

Shafer, R. W.

P. Maragos, R. W. Shafer, “Morphological systems for multidimensional signal processing,” Proc. IEEE 78, 894–902 (1982).

Shamir, J.

Shemer, A.

Sheng, Y.

Szoplik, T.

Tam, E.

VanderLugt, A.

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

Weaver, C. S.

Yang, Y.

Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum circular symmetrical filters and their uses in pattern recognition,” Opt. Acta 29, 627–644 (1982).
[Crossref]

Yu, F. T. S.

Appl. Opt. (10)

Y. N. Hsu, H. H. Arsenault, G. April, “Rotation-invariant digital pattern recognition using circular-harmonic expansion,” Appl. Opt. 21, 4012–4015 (1982).
[Crossref] [PubMed]

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

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

P. Garcia-Martinez, D. Mas, J. Garcia, C. Ferreira, “Nonlinear morphological correlation: optoelectronic implementation,” Appl. Opt. 37, 2112–2118 (1998).
[Crossref]

A. Shemer, D. Mendlovic, G. Shabtay, P. Garcia-Martinez, J. Garcia, “Modified morphological correlation based on bit-map representation,” Appl. Opt. 38, 781–787 (1999).
[Crossref]

D. Mendlovic, N. Konforti, E. Marom, “Shift- and projection-invariant pattern recognition,” Appl. Opt. 28, 4784–4789 (1990).
[Crossref]

G. Premont, Y. Sheng, “Fast design of circular-harmonic filters using simulated annealing,” Appl. Opt. 32, 3116–3121 (1993).
[Crossref] [PubMed]

F. T. S. Yu, X. Li, E. Tam, S. Jutamulia, D. A. Gregory, “Rotation-invariant pattern recognition with a programmable joint transform correlator,” Appl. Opt. 28, 4725–4727 (1989).
[Crossref] [PubMed]

R. Pierstun, J. Rosen, J. Shamir, “Generation of continuous complex-valued functions of a joint transform correlator,” Appl. Opt. 33, 4398–4405 (1994).
[Crossref]

H. H. Arsenault, C. Ferreira, M. P. Levesque, T. Szoplik, “Simpler filter with limited rotation invariance,” Appl. Opt. 25, 3230–3234 (1986).
[Crossref]

IEEE Trans. Inf. Theory (1)

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

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

Opt. Acta (1)

Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum circular symmetrical filters and their uses in pattern recognition,” Opt. Acta 29, 627–644 (1982).
[Crossref]

Opt. Commun. (1)

P. Garcia-Martinez, J. Garcia, C. Ferreira, “A new criterion for determining the expansion center for circular-harmonic filters,” Opt. Commun. 117, 399–405 (1995).
[Crossref]

Opt. Eng. (1)

D. Casasent, D. Psaltis, “Scale-invariant optical transform,” Opt. Eng. 15, 258–261 (1976).
[Crossref]

Opt. Lett. (1)

Proc. IEEE (1)

P. Maragos, R. W. Shafer, “Morphological systems for multidimensional signal processing,” Proc. IEEE 78, 894–902 (1982).

Other (2)

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973), Chap. 8.

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

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Figures (7)

Fig. 1
Fig. 1

JTC setup. SLM, spatial light modulator.

Fig. 2
Fig. 2

Input scene with the reference object rotated by 90°. The object in the lower right-hand corner is to be rejected.

Fig. 3
Fig. 3

Rotation-invariant correlation by use of a third-order CHF.

Fig. 4
Fig. 4

Rotation MC by use of a third-order CH.

Fig. 5
Fig. 5

(a) Joint input scene for the optical linear correlation and (b) a q-sliced joint input scene for the optical RIMC.

Fig. 6
Fig. 6

Three-dimensional plots of the optical correlation output for linear correlation: a false alarm occurs. The normalized correlation-peak values are shown.

Fig. 7
Fig. 7

Three-dimensional plots of the optical correlation output for the RIMC: no false alarm occurs. The normalized correlation-peak values are shown.

Equations (10)

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

MAEx, y=ζ,ηW |gζ+x, η+y-fζ, η|.
μgfx, y=ζ,ηWmingζ+x, η+y, fζ, η,
μgfx, y=q=1Q-1 μgqfqx, y=q=1Q-1 γgqfqx, y=q=1Q-1gq  fqx, y,
JPSΣu, ν=q=1Q-1JPSqu, ν=q=1Q-1 |Fqu, ν|2+q=1Q-1 |Gqu, ν|2+q=1Q-1 Fq*u, νGqu, νexp-i2ϕqu, ν+q=1Q-1 Fqu, νGq*u, νexpi2ϕqu, ν.
fr, θ=m=- fmrexpimθ,
fmr=02π fr, θexp-imθdθ,
γmr, θ=gr, θ  fmr, θ,
μr, θ=q=1Q-1 gqr, θ  fqr, θ×q=1Q-1gqr, θ  m=- fqmr, θ×m=-q=1Q-1gqr, θ  fqmr, θ.
μ˜gfmr, θ=q=1Q-1 gqr, θ  fqmr, θ.
JPSΣCHu, ν=q=1Q-1JPSqCHu, ν=q=1Q-1 |Fqm=0u, ν|2+q=1Q-1 |Gqu, ν|2+q=1Q-1 Fqm=0*u, νGqu, ν×exp-i2ϕqm=0u, ν+q=1Q-1 Fqm=0u, νGq*u, ν×expi2ϕqm=0u, ν.

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