Abstract

A logarithmic harmonic filter can detect objects at different projection angles. The Mexican-hat wavelet function can extract edges of equal width for objects, regardless of their sizes. Hence incorporating wavelet filtering in the logarithmic harmonic filter can improve its performance. The theory is presented together with computer simulation. Finally, an experiment using a joint transform correlator is presented to verify the capability of the proposed filter.

© 2001 Optical Society of America

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References

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1998 (1)

1997 (1)

Y. S. Cheng, “Real-time shift-invariant optical pattern recognition,” Int. J. High Speed Electron. Syst. 8, 733–748 (1997).
[CrossRef]

1996 (2)

1995 (2)

D. Mendlovic, Z. Zalevsky, I. Kiruschev, G. Lebreton, “Composite harmonic filter for scale-, projection-, and shift-invariant pattern recognition,” Appl. Opt. 34, 310–316 (1995).
[CrossRef] [PubMed]

H. F. Yau, Y. OuYang, S. W. Wang, “Shift, rotation and limited scale invariant pattern recognition using synthetic discriminant functions,” Opt. Rev. 2, 266–269 (1995).
[CrossRef]

1992 (2)

Y. Sheng, D. Roberge, H. H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

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

1990 (1)

1988 (1)

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

1984 (1)

1982 (1)

1967 (1)

A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1737–1748 (1967).
[CrossRef]

1966 (1)

1964 (1)

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

Arsenault, H. H.

Casasent, D.

Cheng, Y. S.

Y. S. Cheng, “Real-time shift-invariant optical pattern recognition,” Int. J. High Speed Electron. Syst. 8, 733–748 (1997).
[CrossRef]

Ferreira, C.

Garcia, J.

Goodman, J. W.

Gregory, D. A.

F. T. S. Yu, D. A. Gregory, “Optical pattern recognition: architectures and techniques,” Proc. IEEE 84, 733–752 (1996).
[CrossRef]

Hsu, Y. N.

Kiruschev, I.

Konforti, N.

D. Mendlovic, N. Konforti, E. Maron, “Shift and projection invariant pattern recognition using logarithmic harmonics,” Appl. Opt. 29, 4784–4789 (1990).
[CrossRef] [PubMed]

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

Lebreton, G.

Lohmann, A. W.

A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1737–1748 (1967).
[CrossRef]

Maron, E.

D. Mendlovic, N. Konforti, E. Maron, “Shift and projection invariant pattern recognition using logarithmic harmonics,” Appl. Opt. 29, 4784–4789 (1990).
[CrossRef] [PubMed]

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

Mendlovic, D.

Moya, A.

OuYang, Y.

H. F. Yau, Y. OuYang, S. W. Wang, “Shift, rotation and limited scale invariant pattern recognition using synthetic discriminant functions,” Opt. Rev. 2, 266–269 (1995).
[CrossRef]

Paris, D. P.

A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1737–1748 (1967).
[CrossRef]

Roberge, D.

Y. Sheng, D. Roberge, H. H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

Sheng, Y.

Y. Sheng, D. Roberge, H. H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

Szu, H. H.

Y. Sheng, D. Roberge, H. H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

VanderLugt, A.

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

VijayaKumar, B. V. K.

Wang, S. W.

H. F. Yau, Y. OuYang, S. W. Wang, “Shift, rotation and limited scale invariant pattern recognition using synthetic discriminant functions,” Opt. Rev. 2, 266–269 (1995).
[CrossRef]

Weaver, C. S.

Yau, H. F.

H. F. Yau, Y. OuYang, S. W. Wang, “Shift, rotation and limited scale invariant pattern recognition using synthetic discriminant functions,” Opt. Rev. 2, 266–269 (1995).
[CrossRef]

Yau, J.

Yu, F. T. S.

F. T. S. Yu, D. A. Gregory, “Optical pattern recognition: architectures and techniques,” Proc. IEEE 84, 733–752 (1996).
[CrossRef]

Zalevsky, Z.

Appl. Opt. (8)

IEEE Trans. Inf. Theory (1)

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

Int. J. High Speed Electron. Syst. (1)

Y. S. Cheng, “Real-time shift-invariant optical pattern recognition,” Int. J. High Speed Electron. Syst. 8, 733–748 (1997).
[CrossRef]

Opt. Commun. (1)

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

Opt. Eng. (1)

Y. Sheng, D. Roberge, H. H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

Opt. Lett. (1)

Opt. Rev. (1)

H. F. Yau, Y. OuYang, S. W. Wang, “Shift, rotation and limited scale invariant pattern recognition using synthetic discriminant functions,” Opt. Rev. 2, 266–269 (1995).
[CrossRef]

Proc. IEEE (1)

F. T. S. Yu, D. A. Gregory, “Optical pattern recognition: architectures and techniques,” Proc. IEEE 84, 733–752 (1996).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Images of two Chinese words, “chung” and “yang,” at various projection angles. The original words are in the left column while those in the middle and in the right columns are for θ = 60° (α = 0.5) and θ = 75° (α = 0.26), respectively.

Fig. 2
Fig. 2

(a) The Chinese word for “yang” at the lower middle of Fig. 1 filtered by the Mexican-hat wavelet with parameters s x = s y = 0.5. (b) The intensity distribution of the LHW filter (M = 1.5, L = 3) obtained from the object in (a). (c) The intensity distribution of the LH filter (M = 1.5, L = 3) obtained from the lower middle object in Fig. 1.

Fig. 3
Fig. 3

(a) Normalized correlation peak intensity as a function of the projection angle of the object for the case with the LHW filter (M = 1.5, L = 3, s x = s y = 0.5). (b) The normalized correlation peak intensity as a function of the projection angle of the object for the case with the LH filter (M = 1.5, L = 3).

Fig. 4
Fig. 4

(a) The objects of Fig. 1, which corresponds to α = 1, α = 0.5, and α = 0.26. (b) The intensity distribution of the correlation output when the word “yang” with α = 0.5 is used directly as the filter. (c) The intensity distribution of the correlation output when the LH filter (M = 1.5, L = 3) is used. (d) The intensity distribution of the correlation output when the LHW filter (M = 1.5, L = 3) is used.

Fig. 5
Fig. 5

Cross-sectional intensity distribution across (in the x direction) the correlation peaks of Figs. 4(b)4(d), respectively.

Fig. 6
Fig. 6

Typical joint transform correlator.

Fig. 7
Fig. 7

(a) Two words at different projection angles θ = 0° and θ = 60° are utilized as the input objects in the experiment. (b) The spatial domain LHW filter that is placed in the input object plane and is at a proper distance from the two input objects.

Fig. 8
Fig. 8

(a) Correlation peaks, after detection with a threshold, at the output plane of the joint transform correlator. (b) Cross-sectional light distribution across the correlation peaks in (a).

Equations (11)

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fx, y=12Lm=- fmy; x0, y0|x|i2πm-1/2,
fmy; x0, y0=-X-X exp-L fx, y; x0, y0-x-i2πm-1/2dx+X exp-LX fx, y; x0, y0×x-i2πm-1/2dx.
fFx, y; x0, y0=12L fMy; x0, y0|x|i2πM-1/2.
gx, y=12Ln=- gny; x, y|x|i2πn-1/2,
cgfx, y; x0, y0=-- gx, yfF*x, y; x0, y0dxdy=12L- gMy; x, yfM*y; x0, y0dy.
gx/α, y=12Ln=- gny; x, yxαi2πn-1/2,
|cgx/α,yfx, y; x0, y0|2=α|cgfx, y; x0, y0|2,
hx, y=1-x2+y2exp-x2+y2/2.
Wfsx, sy; u, v=1sxsy-- hsx,sy*x-u, y-v×fx, ydxdy,
hsx,syx, y=1sxsy hxsx, ysy,
WsLWf=shLWf=-1sh*LWf=-1SH**Wf=s-1HWf,

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