Abstract

Optical correlation schemes based on a matched filter containing a single logarithmic harmonic of an object are described. This correlator can provide shift and projection (tilt) invariant pattern recognition. The logarithmic harmonics, their orthogonality, and their completeness are presented, as well as experimental results using computer simulations and real optical targets. The projection invariance and the discrimination ability of this filter are successfully demonstrated.

© 1990 Optical Society of America

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References

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  1. A. VanderLugt, “Signal Detection by Complex Spatial Filter,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  2. D. Casasent, S. F. Xia, A. J. Lee, J. Z. Song, “Real-Time Deformation Invariant Optical Pattern Recognition Using Coordinate Transformations,” Appl. Opt. 26, 938–942 (1987).
    [CrossRef] [PubMed]
  3. Y. Saito, S. Komatsu, H. Ohzu, “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8–11 (1983).
    [CrossRef]
  4. D. Mendlovic, N. Konforti, E. Marom, “Scale and Projection Invariant Pattern Recognition,” Appl. Opt. 28, 4982–4986 (1989).
    [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. D. Mendlovic, E. Marom, N. Konforti, “Shift and Scale Invariant Pattern Recognition Using Mellin Radial Harmonics,” Opt. Commun. 67, 172–176 (1988).
    [CrossRef]
  7. J. Rosen, J. Shamir, “Scale Invariant Pattern Recognition with Logarithmic Radial Harmonic Filters,” Appl. Opt. 28, 240–244 (1989).
    [CrossRef] [PubMed]
  8. T. Szoplik, H. H. Arsenault, “Rotation-Invariant Optical Data Processing Using the 2-D Nonsymmetric Fourier Transform,” Appl. Opt. 24, 168–172 (1985).
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  9. 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]
  10. G. F. Schils, D. W. Sweeney, “Optical Processor for Recognition of Three-Dimensional Targets Viewed from Any Direction,” J. Opt. Soc. Am. A 5, 1309–1321 (1988).
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  11. D. Casasent, J. H. Song, “Optical Projection Correlation,” Appl. Opt. 27, 4977–4984 (1988).
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  12. IMSL Library, fortran Subroutines for Mathematics and Statistics (IMSL, Houston, TX, 1984).
  13. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer Holograms,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]

1989 (2)

1988 (3)

1987 (1)

1985 (2)

1983 (1)

Y. Saito, S. Komatsu, H. Ohzu, “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

1982 (1)

1967 (1)

1964 (1)

A. VanderLugt, “Signal Detection by Complex Spatial Filter,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Arsenault, H. H.

Casasent, D.

Hsu, Y. N.

Komatsu, S.

Y. Saito, S. Komatsu, H. Ohzu, “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

Konforti, N.

D. Mendlovic, N. Konforti, E. Marom, “Scale and Projection Invariant Pattern Recognition,” Appl. Opt. 28, 4982–4986 (1989).
[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]

Lee, A. J.

Lohmann, A. W.

Marom, E.

D. Mendlovic, N. Konforti, E. Marom, “Scale and Projection Invariant Pattern Recognition,” Appl. Opt. 28, 4982–4986 (1989).
[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]

Mendlovic, D.

D. Mendlovic, N. Konforti, E. Marom, “Scale and Projection Invariant Pattern Recognition,” Appl. Opt. 28, 4982–4986 (1989).
[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]

Ohzu, H.

Y. Saito, S. Komatsu, H. Ohzu, “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

Paris, D. P.

Rosen, J.

Saito, Y.

Y. Saito, S. Komatsu, H. Ohzu, “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

Schils, G. F.

Shamir, J.

Song, J. H.

Song, J. Z.

Sweeney, D. W.

Szoplik, T.

VanderLugt, A.

A. VanderLugt, “Signal Detection by Complex Spatial Filter,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Xia, S. F.

Appl. Opt. (8)

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal Detection by Complex Spatial Filter,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

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

Opt. Commun. (2)

Y. Saito, S. Komatsu, H. Ohzu, “Scale and Rotation Invariant Real Time Optical Correlator Using Computer Generated Hologram,” Opt. Commun. 47, 8–11 (1983).
[CrossRef]

D. Mendlovic, E. Marom, N. Konforti, “Shift and Scale Invariant Pattern Recognition Using Mellin Radial Harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Other (1)

IMSL Library, fortran Subroutines for Mathematics and Statistics (IMSL, Houston, TX, 1984).

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Figures (10)

Fig. 1
Fig. 1

Outline of the object (F-18 aircraft). Note its dynamic range [exp(−L) < |x| < 1].

Fig. 2
Fig. 2

(a) Object used for the computer simulations and (b) real part of its second-order logarithmic harmonic.

Fig. 3
Fig. 3

Relative energy distribution among the different harmonics. Order N is not necessarily an integer.

Fig. 4
Fig. 4

Energy distribution in the correlation plane for the case of a matched filter generated from the second-order logarithmic harmonic: (a) correlation with original input and (b) correlation with a projected version of the original input with an aspect ratio of 0.6.

Fig. 5
Fig. 5

Cross correlation with different objects: (a) input targets (1) F-19 (Stealth), (2) Tornado with closed wings, and (3) Tornado with open wings; (b) cross correlation of (a1) with the logarithmic harmonic of order 2; (c) same as (b) but with (a2) as the input; (d) same as (b) but with (a3) as the input.

Fig. 6
Fig. 6

Classic VanderLugt matched filter correlation: (a) autocorrelation distribution and (b) cross correlation with a projected version (aspect ratio of 0.6) around the y-axis. The same vertical scale was used in both figures.

Fig. 7
Fig. 7

Computer-generated matched filters of the F-18 model used in our experiment: (a) single logarithmic harmonic of order 2 and (b) the classical matched filter.

Fig. 8
Fig. 8

Correlation output patterns generated with the matched filter (LHF) of Fig. 7(a). Input patterns are with four different aspect ratios: (a) 1.0, (b) 0.9, (c) 0.6, and (d) 0.45.

Fig. 9
Fig. 9

Cross-correlation output pattern generated with the same filter [Fig. 7(a)] for three different inputs: (a) F-19 (Stealth), (b) Tornado with open wings, and (c) Tornado with closed wings.

Fig. 10
Fig. 10

Correlation output patterns generated with the classical matched filter of Fig. 7(b). The input patterns have three different aspect ratios: (a) 1.0, (b) 0.9, and (c) 0.6.

Equations (15)

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Φ N ( x ) = x i 2 π N - 1 / 2 = exp [ i ( 2 π N - 1 2 ) ln ( x ) ]             N = 0 , ± 1 , ± 2 , .
Φ N ( x ) = x i 2 π N - 1 / 2 = exp [ i ( 2 π N - 1 2 ) ln x ] .
{ Φ N ( x ) , Φ M ( x ) } = i . i . Φ N ( x ) Φ M * ( x ) d x = - 1 - exp ( - L ) ( - x ) i 2 π N - 1 / 2 ( - x ) - i 2 π M - 1 / 2 d x + exp ( - L ) 1 x i 2 π N - 1 / 2 x - i 2 π M - 1 / 2 d x ,
{ Φ N ( x ) , Φ M ( x ) } = 2 - L 0 exp [ i ( N - M ) 2 π ξ ] d ξ = 2 L N = M = 0 N M ,
x 2 Φ N ( x ) ± 2 x Φ N ( x ) + ( 1 4 + 4 π 2 N 2 ) Φ N ( x ) = 0 ,
f ( x , y ) = 1 2 L N = - f N ( y ) x i 2 π N - 1 / 2 ,
f N ( y ) = - 1 - exp ( - L ) f ( x , y ) ( - x ) - i 2 π N - 1 / 2 d x + exp ( - L ) 1 f ( x , y ) x - 2 π N - 1 / 2 d x .
h ( x , y ) = 1 2 L f N ( y ) x i 2 π N - 1 / 2 .
c f g ( x , y ) = - - g ( x + x , y + y ) h * ( x , y ) d x d y ,
g ( x + x , y + y ) = 1 2 L M = - g M ( x ) ( y + y ) x i 2 π M - 1 / 2 ,
c f g ( x , y ) = 1 2 L - i . i . M = - × [ g M ( x ) ( y + y ) ] x i 2 π M - 1 / 2 h * ( x , y ) d x d y .
c f g ( x , y ) = 1 2 L - [ g N ( x ) ( y + y ) ] f N * ( y ) d y .
c f g [ α ] ( x , y ) = - i . i . g ( α x + x , y + y ) h * ( x , y ) d x d y .
Φ N ( α x ) = α i 2 π N - 1 / 2 x i 2 π N - 1 / 2 = α - 1 / 2 exp ( i 2 π N ln α ) x i 2 π N - 1 / 2 .
c f g [ α ] ( x , y ) = α - 1 / 2 exp ( i 2 π N ln α ) c f g [ 1 ] ( x , y ) .

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