Abstract

Properties of the entropy function encountered in physics and information theory are employed in the generation of highly selective spatial filters for pattern recognition. Computer simulations and laboratory demonstrate efficient recognition of single patterns or classes even when these are submerged in experiments high level random noise.

© 1990 Optical Society of America

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References

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  1. A. B. VanderLugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  2. H. J. Caulfield, W. T. Maloney, “Improved Discrimination in Optical Character Recognition,” Appl. Opt. 8, 2354–2356 (1969).
    [CrossRef] [PubMed]
  3. B. Braunecker, R. Hauck, A. W. Lohmann, “Optical Character Recognition Based on Nonredundant Correlation Measurements,” Appl. Opt. 18, 2746–2753 (1979).
    [CrossRef] [PubMed]
  4. H. H. Arsenault, Y. Sheng, J. Bulabois, “Modified Composite Filter for Pattern Recognition in the Presence of Noise with a Non-Zero Mean,” Opt. Commun. 63, 15–20 (1987).
    [CrossRef]
  5. G. F. Schils, D. W. Sweeney, “Rotation Invariant Correlation Filtering,” J. Opt. Soc. Am. A 2, 1411–1418 (1985).
    [CrossRef]
  6. 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]
  7. D. P. Casasent, W. A. Rozzi, “Modified MSF Synthesis by Fisher and Mean-Square-Error Techniques,” Appl. Opt. 25, 184–186 (1986).
    [CrossRef] [PubMed]
  8. Y. Sheng, J. Duvernoy, “Circular-Fourier-Radial-Mellin Transform Descriptors for Pattern Recognition,” J. Opt. Soc. Am. A 3, 885–887 (1986).
    [CrossRef] [PubMed]
  9. H. H. Arsenault, Y. Sheng, “Properties of Circular Harmonic Expansion for Rotation-Invariant Pattern Recognition,” Appl. Opt. 25, 3225–3229 (1986).
    [CrossRef] [PubMed]
  10. G. F. Schils, D. W. Sweeney, “Rotationally Invariant Correlation Filters for Multiple Images,” J. Opt. Soc. Am. A 3, 902–908 (1986).
    [CrossRef]
  11. J. Rosen, J. Shamir, “Distortion Invariant Pattern Recognition with Phase Filters,” Appl. Opt. 26, 2315–2319 (1987).
    [CrossRef] [PubMed]
  12. G. F. Schils, D. W. Sweeney, “Iterative Technique for the Synthesis of Distortion-Invariant Optical Correlation Filters,” Opt. Lett. 12, 307–309 (1987).
    [CrossRef] [PubMed]
  13. J. Rosen, J. Shamir, “Circular Harmonic Phase Filters for Efficient Rotation-Invariant Pattern Recognition,” Appl. Opt. 27, 2895–2899 (1988).
    [CrossRef] [PubMed]
  14. J. Rosen, J. Shamir, “Scale Invariant Pattern Recognition with Logarithmic Radial Harmonic Filters,” Appl. Opt. 28, 240–244 (1989).
    [CrossRef] [PubMed]
  15. D. Casasent, W.-T. Chang, “Correlation Synthetic Discriminant Functions,” Appl. Opt. 25, 2343–2350 (1986).
    [CrossRef] [PubMed]
  16. J. L. Horner, P. D. Gianino, “Applying the Phase-Only Filter Concept to the Synthetic Discriminant Function Correlation Filter,” Appl. Opt. 24, 851–855 (1985).
    [CrossRef] [PubMed]
  17. J. Shamir, H. J. Caulfield, J. Rosen, “Pattern Recognition Using Reduced Information Content Filters,” Appl. Opt. 26, 2311–2314 (1987).
    [CrossRef] [PubMed]
  18. A. Mahalanobis, B. V. K. V. Kumar, D. P. Casasent, “Minimum Average Correlation Energy Filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  19. R. D. Juday, B. J. Daiuto, “Relaxation Method of Compensation in an Optical Correlator,” Opt. Eng. 26, 1094–1101 (1987).
    [CrossRef]
  20. See, for example, A. Ilobson, Concepts in Statistical Mechanics (Gordon & Breach, New York, 1971).
  21. C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, 1949).
  22. R. Kikuchi, B. H. Soffer, “Maximum Entropy Image Restoration. I. The Entropy Expression,” J. Opt. Soc. Am. 67, 1656–1665 (1977).
    [CrossRef]
  23. R. K. Bryan, J. Skilling, “Maximum Entropy Image Reconstruction From Phaseless Fourier Data,” Opt. Acta 33, 287–299 (1986).
    [CrossRef]
  24. E. S. Meinel, “Maximum-Entropy Image Restoration: Lagrange and Recursive Techniques,” J. Opt. Soc. Am. A 5, 25–29 (1988).
    [CrossRef]
  25. J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  26. J. L. Horner, J. R. Leger, “Pattern Recognition with Binary Phase-Only Filter,” Appl. Opt. 24, 609–611 (1985).
    [CrossRef] [PubMed]
  27. H. J. Caulfield, “Role of the Horner Efficiency in the Optimization of Spatial Filters for Optical Pattern Recognition,” Appl. Opt. 21, 4391–4392 (1982).
    [CrossRef] [PubMed]
  28. U. Mahlab, M. Fleisher, J. Shamir, “Error Probability in Optical Pattern Recognition,” Opt. Commun. (to be published).

1989 (1)

1988 (2)

1987 (6)

1986 (6)

1985 (4)

1984 (1)

1982 (1)

1979 (1)

1977 (1)

1969 (1)

1964 (1)

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

Arsenault, H. H.

Braunecker, B.

Bryan, R. K.

R. K. Bryan, J. Skilling, “Maximum Entropy Image Reconstruction From Phaseless Fourier Data,” Opt. Acta 33, 287–299 (1986).
[CrossRef]

Bulabois, J.

H. H. Arsenault, Y. Sheng, J. Bulabois, “Modified Composite Filter for Pattern Recognition in the Presence of Noise with a Non-Zero Mean,” Opt. Commun. 63, 15–20 (1987).
[CrossRef]

Casasent, D.

Casasent, D. P.

Caulfield, H. J.

Chang, W.-T.

Daiuto, B. J.

R. D. Juday, B. J. Daiuto, “Relaxation Method of Compensation in an Optical Correlator,” Opt. Eng. 26, 1094–1101 (1987).
[CrossRef]

Duvernoy, J.

Fleisher, M.

U. Mahlab, M. Fleisher, J. Shamir, “Error Probability in Optical Pattern Recognition,” Opt. Commun. (to be published).

Gianino, P. D.

Hauck, R.

Horner, J. L.

Ilobson, A.

See, for example, A. Ilobson, Concepts in Statistical Mechanics (Gordon & Breach, New York, 1971).

Juday, R. D.

R. D. Juday, B. J. Daiuto, “Relaxation Method of Compensation in an Optical Correlator,” Opt. Eng. 26, 1094–1101 (1987).
[CrossRef]

Kikuchi, R.

Kumar, B. V. K. V.

Leger, J. R.

Lohmann, A. W.

Mahalanobis, A.

Mahlab, U.

U. Mahlab, M. Fleisher, J. Shamir, “Error Probability in Optical Pattern Recognition,” Opt. Commun. (to be published).

Maloney, W. T.

Meinel, E. S.

Rosen, J.

Rozzi, W. A.

Schils, G. F.

Shamir, J.

Shannon, C. E.

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, 1949).

Sheng, Y.

Skilling, J.

R. K. Bryan, J. Skilling, “Maximum Entropy Image Reconstruction From Phaseless Fourier Data,” Opt. Acta 33, 287–299 (1986).
[CrossRef]

Soffer, B. H.

Sweeney, D. W.

Szoplik, T.

VanderLugt, A. B.

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

Weaver, W.

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, 1949).

Appl. Opt. (15)

B. Braunecker, R. Hauck, A. W. Lohmann, “Optical Character Recognition Based on Nonredundant Correlation Measurements,” Appl. Opt. 18, 2746–2753 (1979).
[CrossRef] [PubMed]

H. J. Caulfield, “Role of the Horner Efficiency in the Optimization of Spatial Filters for Optical Pattern Recognition,” Appl. Opt. 21, 4391–4392 (1982).
[CrossRef] [PubMed]

J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812–816 (1984).
[CrossRef] [PubMed]

J. L. Horner, P. D. Gianino, “Applying the Phase-Only Filter Concept to the Synthetic Discriminant Function Correlation Filter,” Appl. Opt. 24, 851–855 (1985).
[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]

D. P. Casasent, W. A. Rozzi, “Modified MSF Synthesis by Fisher and Mean-Square-Error Techniques,” Appl. Opt. 25, 184–186 (1986).
[CrossRef] [PubMed]

D. Casasent, W.-T. Chang, “Correlation Synthetic Discriminant Functions,” Appl. Opt. 25, 2343–2350 (1986).
[CrossRef] [PubMed]

H. H. Arsenault, Y. Sheng, “Properties of Circular Harmonic Expansion for Rotation-Invariant Pattern Recognition,” Appl. Opt. 25, 3225–3229 (1986).
[CrossRef] [PubMed]

J. Shamir, H. J. Caulfield, J. Rosen, “Pattern Recognition Using Reduced Information Content Filters,” Appl. Opt. 26, 2311–2314 (1987).
[CrossRef] [PubMed]

J. Rosen, J. Shamir, “Distortion Invariant Pattern Recognition with Phase Filters,” Appl. Opt. 26, 2315–2319 (1987).
[CrossRef] [PubMed]

A. Mahalanobis, B. V. K. V. Kumar, D. P. Casasent, “Minimum Average Correlation Energy Filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

J. Rosen, J. Shamir, “Circular Harmonic Phase Filters for Efficient Rotation-Invariant Pattern Recognition,” Appl. Opt. 27, 2895–2899 (1988).
[CrossRef] [PubMed]

J. Rosen, J. Shamir, “Scale Invariant Pattern Recognition with Logarithmic Radial Harmonic Filters,” Appl. Opt. 28, 240–244 (1989).
[CrossRef] [PubMed]

H. J. Caulfield, W. T. Maloney, “Improved Discrimination in Optical Character Recognition,” Appl. Opt. 8, 2354–2356 (1969).
[CrossRef] [PubMed]

J. L. Horner, J. R. Leger, “Pattern Recognition with Binary Phase-Only Filter,” Appl. Opt. 24, 609–611 (1985).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

R. K. Bryan, J. Skilling, “Maximum Entropy Image Reconstruction From Phaseless Fourier Data,” Opt. Acta 33, 287–299 (1986).
[CrossRef]

Opt. Commun. (1)

H. H. Arsenault, Y. Sheng, J. Bulabois, “Modified Composite Filter for Pattern Recognition in the Presence of Noise with a Non-Zero Mean,” Opt. Commun. 63, 15–20 (1987).
[CrossRef]

Opt. Eng. (1)

R. D. Juday, B. J. Daiuto, “Relaxation Method of Compensation in an Optical Correlator,” Opt. Eng. 26, 1094–1101 (1987).
[CrossRef]

Opt. Lett. (1)

Other (3)

See, for example, A. Ilobson, Concepts in Statistical Mechanics (Gordon & Breach, New York, 1971).

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. Illinois Press, Urbana, 1949).

U. Mahlab, M. Fleisher, J. Shamir, “Error Probability in Optical Pattern Recognition,” Opt. Commun. (to be published).

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Figures (13)

Fig. 1
Fig. 1

4-f optical correlator.

Fig. 2
Fig. 2

Input pattern for the first set of experiments.

Fig. 3
Fig. 3

Correlation of Fig. 2 with (a) a matched filter for the letter P and (b) a phase-only matched filter for the letter P.

Fig. 4
Fig. 4

Amplitude distribution in an entropy optimized filter for the letter P of Fig. 2.

Fig. 5
Fig. 5

Correlation of Fig. 2 with the EOF of Fig. 4.

Fig. 6
Fig. 6

Comparison between the correlation function obtained for the letter P using an MF (a) and EOF (b).

Fig. 7
Fig. 7

(a) Experimental result with a holographic EOF corresponding to the computer simulation of Fig. 5. (b) Performance of a MF. The output plane is rotated so that the intensity profile (along the line) crosses the two correlation peaks.

Fig. 8
Fig. 8

Input of four patterns only one of which was a member of the training set.

Fig. 9
Fig. 9

Correlation of Fig. 8 with the EOF of Fig. 4: (a) computer simulation; (b) laboratory result. (The intensity profile is taken along a diagonal line through the correlation peaks of the two P letters.)

Fig. 10
Fig. 10

Training set for detecting G and P and rejecting F and O (a) and correlation with the resulting EOF (b).

Fig. 11
Fig. 11

Input pattern immersed in clutter having a peak amplitude of 60% of the signal level (a) and correlation output with the EOF (b).

Fig. 12
Fig. 12

Same letter P deteriorated by the noise of various levels immersed in the background of 60% noise level (a) and the correlation output with the EOF (b).

Fig. 13
Fig. 13

(a) Speckle pattern multiplying the input of Fig. 2, (b) deteriorated input to laboratory correlator, and (c) output pattern with intensity profile through the two correlation peaks.

Equations (18)

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C ( x 0 , y 0 ) = - - f ( x , y ) h * ( x + x 0 , y + y 0 ) d x d y
ϕ ( x 0 , y 0 ) = C ( x 0 , y 0 ) 2 - - C ( x 0 , y 0 ) 2 d x 0 d y 0 .
S = - - - ϕ ( x 0 , y 0 ) log ϕ ( x 0 , y 0 ) d x 0 d y 0 .
f ( x , y ) f ( i , j )             i , j = 1 , 2 , , N .
h ( x , y ) h ( i , j )             i , j = 1 , 2 , , N .
C ( m , n ) = i , j N f ( i , j ) h * ( i + m , j + n )             m , n = 1 , 2 , , ( 2 N - 1 ) ,
ϕ ( m , n ) = C ( m , n ) 2 m , n C ( m , n ) 2 .
S = - m , n ϕ ( m , n ) log ϕ ( m , n ) ,
ϕ D ( m , n ) = { 1 m = k and n = l , 0 otherwise ,
ϕ R ( m , n ) = 1 ( 2 N - 1 ) 2             ( m , n ) .
S max R = - log - 1 ( 2 N - 1 ) 2 ,
S min D = 0.
M = S R - S D ,
M ideal = S max R - S min D = - log 1 ( 2 N - 1 ) 2 .
M max = M [ h EOF ( i , j ) ] .
M = f n R S n - f n D S n .
h ( t + 1 ) ( i , j ) = h ( t ) ( i , j ) + η [ M ( t ) ] i , j ,
h ( 0 ) ( i , j ) = 1.

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