Abstract

The phase-only logarithmic radial harmonic (LRH) filter has been shown to be suitable for scale-invariant block object recognition. However, an important set of objects is the collection of contour functions that results from a digital edge extraction of the original block objects. These contour functions have a constant width that is independent of the scale of the original object. Therefore, since the energy of the contour objects decreases more slowly with the scale factor than does the energy of the block objects, the phase-only LRH filter has difficulties in the recognition tasks when these contour objects are used. We propose a modified LRH filter that permits the realization of a shift- and scale-invariant optical recognition of contour objects. The modified LRH filter is a complex filter that compensates the energy variation resulting from the scaling of contour objects. Optical results validate the theory and show the utility of the newly proposed method.

© 2000 Optical Society of America

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References

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  1. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  2. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  3. D. Casasent, D. P. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  4. D. Casasent, D. P. Psaltis, “Scale invariant optical correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
    [CrossRef]
  5. J. Duvernoy, “Optical-digital processing of directional terrain textures invariant under translation, rotation, and change of scale,” Appl. Opt. 23, 828–837 (1984).
    [CrossRef] [PubMed]
  6. Y. Sheng, J. Duvernoy, “Circular-Fourier-radial-Mellin transform descriptors for pattern recognition,” J. Opt. Soc. Am. A 3, 885–888 (1986).
    [CrossRef] [PubMed]
  7. Y. Sheng, C. Lejeune, “Invariant pattern recognition using Fourier–Mellin transforms and neural networks,” J. Opt. (Paris) 22, 223–228 (1991).
    [CrossRef]
  8. Y. Sheng, H. H. Arsenault, “Experiments on pattern recognition using invariant Fourier–Mellin descriptors,” J. Opt. Soc. Am. A 3, 771–776 (1986).
    [CrossRef] [PubMed]
  9. M. Fang, G. Häusler, “Class of transforms invariant under shift, rotation, and scaling,” Appl. Opt. 29, 704–708 (1990).
    [CrossRef] [PubMed]
  10. E. W. Hansen, J. W. Goodman, “Optical reconstruction from projections via circular harmonic expansion,” Opt. Commun. 24, 268–272 (1978).
    [CrossRef]
  11. Y. N. Hsu, H. H. Arsenault, G. April, “Rotational-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4012–4015 (1982).
    [CrossRef] [PubMed]
  12. Y. N. Hsu, H. H. Arsenault, “Optical recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  13. T. Szoplik, “Shift and scale-invariant anamorphic Fourier correlator,” J. Opt. Soc. Am. A 2, 1419–1423 (1985).
    [CrossRef]
  14. T. Szoplik, H. H. Arsenault, “Shift and scale-invariant anamorphic Fourier correlator using multiple circular harmonic filters,” Appl. Opt. 24, 3179–3183 (1985).
    [CrossRef] [PubMed]
  15. J. García, T. Szoplik, C. Ferreira, “Shift-and-scale-invariant pattern recognition using an elliptic coordinate-transformed phase-only filter,” Appl. Opt. 31, 4823–4828 (1992).
    [CrossRef] [PubMed]
  16. D. Cojoc, M. T. Molina, J. García, C. Ferreira, “Coordinate-transformed filter for shift-invariant and scale-invariant pattern recognition,” Appl. Opt. 36, 4812–4815 (1997).
    [CrossRef] [PubMed]
  17. K. Mersereau, G. M. Morris, “Scale, rotation, and shift invariant image recognition,” Appl. Opt. 25, 2338–2342 (1986).
    [CrossRef] [PubMed]
  18. J. J. Esteve-Taboada, J. García, C. Ferreira, “Extended scale-invariant pattern recognition with white-light illumination,” Appl. Opt. 39, 1268–1271 (2000).
    [CrossRef]
  19. D. Mendlovic, E. Marom, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
    [CrossRef]
  20. J. Rosen, J. Shamir, “Scale invariant pattern recognition with logarithmic radial harmonic filters,” Appl. Opt. 28, 240–244 (1989).
    [CrossRef] [PubMed]
  21. A. Moya, D. Mendlovic, J. García, C. Ferreira, “Projection-invariant pattern recognition with a phase-only logarithmic-harmonic-derived filter,” Appl. Opt. 35, 3862–3867 (1996).
    [CrossRef] [PubMed]
  22. J. W. Lee, I. S. Kweon, “Extraction of line features in a noisy image,” Pattern Recogn. 30, 1651–1660 (1997).
    [CrossRef]
  23. A. Elmabrouk, A. Aggoun, “Edge detection using local histogram analysis,” Electron. Lett. 34, 1216–1217 (1998).
    [CrossRef]
  24. A. Barducci, I. Pippi, “Object recognition by edge analysis: a case study,” Opt. Eng. 38, 284–294 (1999).
    [CrossRef]
  25. A. Bandera, C. Urdiales, F. Arrebola, F. Sandoval, “2D object recognition based on curvature functions obtained from local histograms of the contour chain code,” Pattern Recogn. Lett. 20, 49–55 (1999).
    [CrossRef]
  26. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  27. I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
    [CrossRef]

2000 (1)

1999 (2)

A. Barducci, I. Pippi, “Object recognition by edge analysis: a case study,” Opt. Eng. 38, 284–294 (1999).
[CrossRef]

A. Bandera, C. Urdiales, F. Arrebola, F. Sandoval, “2D object recognition based on curvature functions obtained from local histograms of the contour chain code,” Pattern Recogn. Lett. 20, 49–55 (1999).
[CrossRef]

1998 (1)

A. Elmabrouk, A. Aggoun, “Edge detection using local histogram analysis,” Electron. Lett. 34, 1216–1217 (1998).
[CrossRef]

1997 (2)

1996 (1)

1995 (1)

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
[CrossRef]

1992 (1)

1991 (1)

Y. Sheng, C. Lejeune, “Invariant pattern recognition using Fourier–Mellin transforms and neural networks,” J. Opt. (Paris) 22, 223–228 (1991).
[CrossRef]

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]

1986 (3)

1985 (2)

1984 (1)

1982 (2)

1978 (1)

E. W. Hansen, J. W. Goodman, “Optical reconstruction from projections via circular harmonic expansion,” Opt. Commun. 24, 268–272 (1978).
[CrossRef]

1976 (2)

D. Casasent, D. P. Psaltis, “Scale invariant optical correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

D. Casasent, D. P. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[CrossRef] [PubMed]

1967 (1)

1966 (1)

1964 (1)

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

Aggoun, A.

A. Elmabrouk, A. Aggoun, “Edge detection using local histogram analysis,” Electron. Lett. 34, 1216–1217 (1998).
[CrossRef]

April, G.

Arrebola, F.

A. Bandera, C. Urdiales, F. Arrebola, F. Sandoval, “2D object recognition based on curvature functions obtained from local histograms of the contour chain code,” Pattern Recogn. Lett. 20, 49–55 (1999).
[CrossRef]

Arsenault, H. H.

Bandera, A.

A. Bandera, C. Urdiales, F. Arrebola, F. Sandoval, “2D object recognition based on curvature functions obtained from local histograms of the contour chain code,” Pattern Recogn. Lett. 20, 49–55 (1999).
[CrossRef]

Barducci, A.

A. Barducci, I. Pippi, “Object recognition by edge analysis: a case study,” Opt. Eng. 38, 284–294 (1999).
[CrossRef]

Campos, J.

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
[CrossRef]

Casasent, D.

D. Casasent, D. P. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[CrossRef] [PubMed]

D. Casasent, D. P. Psaltis, “Scale invariant optical correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

Cojoc, D.

Duvernoy, J.

Elmabrouk, A.

A. Elmabrouk, A. Aggoun, “Edge detection using local histogram analysis,” Electron. Lett. 34, 1216–1217 (1998).
[CrossRef]

Esteve-Taboada, J. J.

Fang, M.

Ferreira, C.

García, J.

Goodman, J. W.

E. W. Hansen, J. W. Goodman, “Optical reconstruction from projections via circular harmonic expansion,” Opt. Commun. 24, 268–272 (1978).
[CrossRef]

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

Gorecki, C.

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
[CrossRef]

Hansen, E. W.

E. W. Hansen, J. W. Goodman, “Optical reconstruction from projections via circular harmonic expansion,” Opt. Commun. 24, 268–272 (1978).
[CrossRef]

Häusler, G.

Hsu, Y. N.

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]

Kweon, I. S.

J. W. Lee, I. S. Kweon, “Extraction of line features in a noisy image,” Pattern Recogn. 30, 1651–1660 (1997).
[CrossRef]

Lee, J. W.

J. W. Lee, I. S. Kweon, “Extraction of line features in a noisy image,” Pattern Recogn. 30, 1651–1660 (1997).
[CrossRef]

Lejeune, C.

Y. Sheng, C. Lejeune, “Invariant pattern recognition using Fourier–Mellin transforms and neural networks,” J. Opt. (Paris) 22, 223–228 (1991).
[CrossRef]

Lohmann, A. 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]

Mendlovic, D.

A. Moya, D. Mendlovic, J. García, C. Ferreira, “Projection-invariant pattern recognition with a phase-only logarithmic-harmonic-derived filter,” Appl. Opt. 35, 3862–3867 (1996).
[CrossRef] [PubMed]

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

Mersereau, K.

Molina, M. T.

Moreno, I.

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
[CrossRef]

Morris, G. M.

Moya, A.

Paris, D. P.

Pippi, I.

A. Barducci, I. Pippi, “Object recognition by edge analysis: a case study,” Opt. Eng. 38, 284–294 (1999).
[CrossRef]

Psaltis, D. P.

D. Casasent, D. P. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[CrossRef] [PubMed]

D. Casasent, D. P. Psaltis, “Scale invariant optical correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

Rosen, J.

Sandoval, F.

A. Bandera, C. Urdiales, F. Arrebola, F. Sandoval, “2D object recognition based on curvature functions obtained from local histograms of the contour chain code,” Pattern Recogn. Lett. 20, 49–55 (1999).
[CrossRef]

Shamir, J.

Sheng, Y.

Szoplik, T.

Urdiales, C.

A. Bandera, C. Urdiales, F. Arrebola, F. Sandoval, “2D object recognition based on curvature functions obtained from local histograms of the contour chain code,” Pattern Recogn. Lett. 20, 49–55 (1999).
[CrossRef]

VanderLugt, A.

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

Weaver, C. S.

Yzuel, M. J.

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
[CrossRef]

Appl. Opt. (14)

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

D. Casasent, D. P. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[CrossRef] [PubMed]

J. Duvernoy, “Optical-digital processing of directional terrain textures invariant under translation, rotation, and change of scale,” Appl. Opt. 23, 828–837 (1984).
[CrossRef] [PubMed]

M. Fang, G. Häusler, “Class of transforms invariant under shift, rotation, and scaling,” Appl. Opt. 29, 704–708 (1990).
[CrossRef] [PubMed]

T. Szoplik, H. H. Arsenault, “Shift and scale-invariant anamorphic Fourier correlator using multiple circular harmonic filters,” Appl. Opt. 24, 3179–3183 (1985).
[CrossRef] [PubMed]

J. García, T. Szoplik, C. Ferreira, “Shift-and-scale-invariant pattern recognition using an elliptic coordinate-transformed phase-only filter,” Appl. Opt. 31, 4823–4828 (1992).
[CrossRef] [PubMed]

D. Cojoc, M. T. Molina, J. García, C. Ferreira, “Coordinate-transformed filter for shift-invariant and scale-invariant pattern recognition,” Appl. Opt. 36, 4812–4815 (1997).
[CrossRef] [PubMed]

K. Mersereau, G. M. Morris, “Scale, rotation, and shift invariant image recognition,” Appl. Opt. 25, 2338–2342 (1986).
[CrossRef] [PubMed]

J. J. Esteve-Taboada, J. García, C. Ferreira, “Extended scale-invariant pattern recognition with white-light illumination,” Appl. Opt. 39, 1268–1271 (2000).
[CrossRef]

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

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

J. Rosen, J. Shamir, “Scale invariant pattern recognition with logarithmic radial harmonic filters,” Appl. Opt. 28, 240–244 (1989).
[CrossRef] [PubMed]

A. Moya, D. Mendlovic, J. García, C. Ferreira, “Projection-invariant pattern recognition with a phase-only logarithmic-harmonic-derived filter,” Appl. Opt. 35, 3862–3867 (1996).
[CrossRef] [PubMed]

A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
[CrossRef] [PubMed]

Electron. Lett. (1)

A. Elmabrouk, A. Aggoun, “Edge detection using local histogram analysis,” Electron. Lett. 34, 1216–1217 (1998).
[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. (Paris) (1)

Y. Sheng, C. Lejeune, “Invariant pattern recognition using Fourier–Mellin transforms and neural networks,” J. Opt. (Paris) 22, 223–228 (1991).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (3)

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

D. Casasent, D. P. Psaltis, “Scale invariant optical correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

E. W. Hansen, J. W. Goodman, “Optical reconstruction from projections via circular harmonic expansion,” Opt. Commun. 24, 268–272 (1978).
[CrossRef]

Opt. Eng. (2)

A. Barducci, I. Pippi, “Object recognition by edge analysis: a case study,” Opt. Eng. 38, 284–294 (1999).
[CrossRef]

I. Moreno, C. Gorecki, J. Campos, M. J. Yzuel, “Comparison of computer-generated holograms produced by laser printers and lithography: application to pattern recognition,” Opt. Eng. 34, 3520–3525 (1995).
[CrossRef]

Pattern Recogn. (1)

J. W. Lee, I. S. Kweon, “Extraction of line features in a noisy image,” Pattern Recogn. 30, 1651–1660 (1997).
[CrossRef]

Pattern Recogn. Lett. (1)

A. Bandera, C. Urdiales, F. Arrebola, F. Sandoval, “2D object recognition based on curvature functions obtained from local histograms of the contour chain code,” Pattern Recogn. Lett. 20, 49–55 (1999).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Circular crown of average radius r 0 and width Δ: (a) without scaling, (b) with scale factor β.

Fig. 2
Fig. 2

Contour functions used as input objects in experiments. Objects A2 and B2 are two scaled versions with scale factor β = 2 of objects A1 and B1, respectively.

Fig. 3
Fig. 3

Horizontal profile through a curve that contains the central maximum of the modulus of the Fourier transform for objects A1 and A2.

Fig. 4
Fig. 4

Intensity of correlation centers in terms of frequency p of phase-only LRH filter matched to object A2 during correlation with object A1 (A2–A1) and object A2 (A2–A2).

Fig. 5
Fig. 5

Intensity of the correlation centers in terms of frequency p of MLRH filter matched to object A2 during correlation with object A1 (A2–A1) and object A2 (A2–A2).

Fig. 6
Fig. 6

Simulated correlation between input scene shown in Fig. 2 and phase-only LRH filter matched to object A2 with frequency p = 2.3.

Fig. 7
Fig. 7

Simulated correlation between input scene shown in Fig. 2 and MLRH filter matched to object A2 with frequency p = 2.3.

Fig. 8
Fig. 8

Optical correlation between input scene shown in Fig. 2 and phase-only LRH filter matched to object A2 with frequency p = 2.3.

Fig. 9
Fig. 9

Optical correlation between input scene shown in Fig. 2 and MLRH filter matched to object A2 with frequency p = 2.3.

Equations (16)

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

Gρ, ϕ=1β2 Fρβ, ϕ,
Cf,h=dD02π Fρ, ϕR*ρS*ϕρdρdϕ,
Cf,hβ=d/βD/β02π Fτ, ϕR*βτS*ϕτdτdϕ,
Cf,hβ=Cf,h expiσβ,
H*ρ, ϕ=expiΩϕρ/dip/w,
w=12πlnDd,
Ωϕ=-argdD Fρ, ϕρdip/wρdρ.
Cf,h;pβ=βdip/w02πexpiΩϕ×d/βD/β Fτ, ϕτip/wτdτdϕ,
|Cf,h;p1|=02πdD Fτ, ϕτip/wτdτdϕ.
Fρ, ϕ=A r0-Δ/2r0+Δ/202πexp-i2πrρ cosϕ-θrdrdθ.
Gρ, ϕ=A r0/β-Δ/2r0/β+Δ/202πexp-i2πrρ×cosϕ-θrdrdθ.
Gρ, ϕ=1β Fρβ, ϕ.
Cf,hβ=β D/βD/β02π Fτ, ϕR*βτS*ϕτdτdϕ.
H*ρ, ϕ=expiΩϕρ/dip/w-1,
Cf,h;pβ=βdip/wd 02πexpiΩϕ×d/βD/β Fτ, ϕτip/wdτdϕ,
|Cf,h;p1|=d 02πdD Fτ, ϕτip/wdτdϕ.

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