Abstract

A logarithmic–logarithmic coordinate transformation is used to perform successfully scale and projection (tilt) invariant optical pattern recognition. The coordinate transformation is achieved optically using a computer generated hologram divided into four different quadrants since the logarithmic function does not exist for negative values of the argument. The transformed pattern is introduced into a VanderLugt correlator setup through a liquid crystal light valve. Initial experimental results are presented.

© 1989 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, 139–146 (1964).
    [CrossRef]
  2. Y. Y. Hsu, H. H. Arsenault, G. April, “Rotation-Invariant Digital Pattern Recognition Using Circular Harmonics Expansion,” Appl. Opt. 21, 4012–4015 (1982).
    [CrossRef] [PubMed]
  3. Y. N. Hsu, H. H. Arsenault, “Optical Pattern Recognition Using Circular Harmonics Expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [CrossRef] [PubMed]
  4. D. Mendlovic, E. Marom, N. Konforti, “Shift and Scale Invariant Pattern Recognition Using Mellin Radial Harmonics,” Opt. Commun. 67, 172–176 (1988).
    [CrossRef]
  5. J. Rosen, J. Shamir, “Scale Invariant Pattern Recognition with Logarithmic Radial Harmonic Filters,” Appl. Opt. 28, 240–244 (1989).
    [CrossRef] [PubMed]
  6. 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]
  7. D. Casasent, S.-F. Xia, A. J. Lee, J.-Z. Shong, “Real-Time Deformation Invariant Optical Pattern Recognition Using Coordinate Transformations,” Appl. Opt. 26, 938–942 (1987).
    [CrossRef] [PubMed]
  8. T. Szoplik, H. H. Arsenault, “Rotation-Invariant Optical Data Processing Using the 2-D Nonsymmetrical Fourier Transform,” Appl. Opt. 24, 168–172 (1985).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  11. W.-H. Lee, “Binary Synthetic Holograms,” Appl. Opt. 13, 1677–1682 (1974).
    [CrossRef] [PubMed]
  12. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, NJ, 1973), Chap. 4, p. 421.
  13. O. Bryngdahl, “Geometrical Transformations in Optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
    [CrossRef]

1989 (1)

1988 (2)

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

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).
[CrossRef]

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 (2)

1974 (2)

1964 (1)

A. VanderLugt, “Signal Detection by Complex Spatial Filter,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).
[CrossRef]

April, G.

Arsenault, H. H.

Bryngdahl, O.

Casasent, D.

Felsen, L. B.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, NJ, 1973), Chap. 4, p. 421.

Hsu, Y. N.

Hsu, Y. Y.

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

Lee, A. J.

Lee, W.-H.

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, NJ, 1973), Chap. 4, p. 421.

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.

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]

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.

Shong, J.-Z.

Sweeney, D. W.

Szoplik, T.

VanderLugt, A.

A. VanderLugt, “Signal Detection by Complex Spatial Filter,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).
[CrossRef]

Xia, S.-F.

Appl. Opt. (7)

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal Detection by Complex Spatial Filter,” IEEE Trans. Inf. Theory IT-10, 139–146 (1964).
[CrossRef]

J. Opt. Soc. Am. (1)

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)

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, NJ, 1973), Chap. 4, p. 421.

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Figures (11)

Fig. 1
Fig. 1

Boundaries of the processed area in the input plane.

Fig. 2
Fig. 2

Basic system used for the coordinate transformations of the input f(x,y). The new coordinates are displayed at the transform plane F(u,υ).

Fig. 3
Fig. 3

Computer generated hologram for 2-D logarithmic coordinate transformations. Lee's method was used in generating this filter.

Fig. 4
Fig. 4

Setup for scale and projection invariant pattern recognition.

Fig. 5
Fig. 5

(a) Object chosen for experimental testing. The axes are chosen so that the first moments vanish; (b) its logarithmic coordinate transformation; and (c) its matched filter.

Fig. 6
Fig. 6

Experimental results for the original input: (a) input pattern; (b) output plane distribution. Left: transformed image of the input pattern after passing through the LCLV; right: the correlation plane distribution.

Fig. 7
Fig. 7

(a) Object scaled along the x-axis only; (b) correlation (spot at right side) with the original object shown in Fig. 5(a).

Fig. 8
Fig. 8

(a) Object scaled along the y-axis only; (b) correlation (spot at right) with the original object shown in Fig. 5(a).

Fig. 9
Fig. 9

(a) Object scaled differently along the x- and y-axes; (b) correlation (spot at right) with the original object shown in Fig. 5(a).

Fig. 10
Fig. 10

System addressed by a different input, +: (a) input pattern; (b) cross-correlation (at right).

Fig. 11
Fig. 11

Same as Fig. 10, but the input is the letter Y.

Equations (7)

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F ( u , υ ) = F [ ln ( x ) , ln ( y ) ] = f ( x , y ) .
F ( u , υ ) = F [ ln ( a x ) , ln ( y ) ] = F [ ln ( x ) + ln ( a ) , ln ( y ) ] .
1 st quadrant u = ln ( x ) ln ( x min ) , υ = ln ( y ) ln ( y min ) , 2 nd quadrant u = ln ( x ) ln ( x max ) , υ = ln ( y ) ln ( y min ) , 3 rd quadrant u = ln ( x ) ln ( x max ) , υ = ln ( y ) ln ( y max ) , 4 th quadrant u = ln ( x ) ln ( x min ) , υ = ln ( y ) ln ( y max ) ,
F ( u , υ ) = f ( x , y ) exp [ i ϕ ( x , y ) exp [ i 2 π λ f ( x u + y υ ) ] dxdy ,
ϕ x = 2 π u λ f , ϕ y = 2 π υ λ f ,
ϕ 1 = 2 π λ f { x [ ln ( x ) 1 ln ( x min ) ] + y [ ln ( y ) 1 ln ( y min ) ] } , ϕ 2 = 2 π λ f { x [ ln ( x ) 1 ln ( x max ) ] + y [ ln ( y ) 1 ln ( y min ) ] } , ϕ 3 = 2 π λ f { x [ ln ( x ) 1 ln ( x max ) ] y [ ln ( y ) 1 ln ( y max ) ] } , ϕ 4 = 2 π λ f { x [ ln ( x ) 1 ln ( x min ) ] y [ ln ( y ) 1 ln ( y max ) ] } .
ϕ ( x , y ) 2 π α x = 2 π N ,

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