Abstract

We describe an approach to compute filters that automatically performs a spatial frequency selection to improve interclass discrimination and to reduce intraclass sensitivity. This approach is achieved by using as input to the filter synthesis a set of reference images to compute the filters and a set of distorted images to introduce the distortion or noise model of the reference images. Simulation results of correlation examples are provided for two pattern-recognition problems and are compared with the ones obtained with the standard minimum average correlation energy filters.

© 1993 Optical Society of America

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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. D. P. Casasent, D. Psaltis, “Position, rotation and scale-invariant optical correlation, “ Appl. Opt. 15, 1795–1799 (1976).
    [Crossref] [PubMed]
  3. Yuan-neng Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [Crossref] [PubMed]
  4. G. F. Schils, D. W. Sweeney, “Rotationally invariant correlation filtering,” J. Opt. Soc. Am. A 2, 1411–1418 (1985).
    [Crossref]
  5. T. Szoplik, H. H. Arsenault, “Shift and scale-invariant anamorphic Fourier correlator using multiple circular filters,” Appl. Opt. 24, 3179–3183 (1985).
    [Crossref] [PubMed]
  6. A. S. Jensen, L. Lindvold, E. Rasmussen, “Transformation of image positions, rotations and sizes into shift parameters,” Appl. Opt. 26, 1775–1781 (1987).
    [Crossref] [PubMed]
  7. K. Mersereau, G. M. Morris, “Scale, rotation and shift invariant image recognition,” Appl. Opt. 25, 2338–2342 (1986).
    [Crossref] [PubMed]
  8. J. Rosen, J. Shamir, “Circular harmonic phase filters for efficient rotation-invariant pattern recognition,” Appl. Opt. 27, 2895–2899 (1988).
    [Crossref] [PubMed]
  9. J. Rosen, J. Shamir, “Scale invariant pattern recognition with logarithmic radial harmonic filters,” Appl. Opt. 28, 240–244(1989).
    [Crossref] [PubMed]
  10. D. Casasent, A. Iyer, G. Ravichandran, “Circular harmonic function, minimum correlation energy filters,” Appl. Opt. 35, 5169–5175 (1991).
    [Crossref]
  11. H. J. Caufield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
    [Crossref]
  12. D. P. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [Crossref] [PubMed]
  13. D. P. Casasent, Wen-Thong Chang, “Correlation synthetic discriminant functions,” Appl. Opt. 25, 2343–2350 (1986).
    [Crossref] [PubMed]
  14. J. R. Leger, S. H. Lee, “Image classification by an optical implementation of the Fukunaga-Koontz transform,” J. Opt. Soc. Am. 72, 556–564 (1982).
    [Crossref]
  15. B. Braunecker, R. Hauck, A. W. Lohmann, “Optical character recognition based on the nonredundant correlation measurements,” Appl. Opt. 18, 2746–2753 (1979).
    [Crossref] [PubMed]
  16. G. F. Schils, D. W. Sweeney, “Rotationally correlation filtering for multiple images,” J. Opt. Soc. Am. 3, 902–908 (1986).
    [Crossref]
  17. R. R. Kallman, “Construction of low noise optical corelation filters,” Appl. Opt. 25, 1032–1033 (1986).
    [Crossref] [PubMed]
  18. D. P. Casasent, W. A. Rozzi, “Modified MSF synthesis by Fisher and mean-square-error techniques,” Appl. Opt. 25, 184–186 (1986).
    [Crossref] [PubMed]
  19. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average corelation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [Crossref] [PubMed]
  20. M. Fleisher, U. Mahlab, J. Shamir, “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
    [Crossref] [PubMed]
  21. A. Mahalanobis, D. Casasent, “Performance evaluation of minimum average correlation filters,” Appl. Opt. 30, 561–572 (1991).
    [Crossref] [PubMed]
  22. S. I. Sudharsanan, A. Mahalanobis, M. K. Sundareshan, “Unified framework for synthesis of discriminant functions with reduced noise variance and sharp correlation structure,” Opt. Eng. 29, 1021–1028 (1990).
    [Crossref]
  23. B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. 3, 1579–1584 (1986).
    [Crossref]
  24. D. Casasent, G. Ravichandran, S. Bollapragada, “Gaussian-minimum correlation energy filters,” Appl. Opt. 30, 5176–5181 (1991).
    [Crossref] [PubMed]
  25. Ph. Refregier, J. P. Huignard, “Phase selection of synthetic discriminant function filters,” Appl. Opt. 29, 4772–4778 (1990).
    [Crossref] [PubMed]
  26. Ph. Refregier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [Crossref] [PubMed]
  27. B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [Crossref]
  28. J. L. Horner, “Clarification of Horner efficiency,” Appl. Opt. 31, 4629 (1992).
    [Crossref] [PubMed]

1992 (1)

1991 (4)

1990 (4)

1989 (1)

1988 (1)

1987 (2)

1986 (6)

1985 (2)

1984 (1)

1982 (2)

1979 (1)

1976 (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.

Bollapragada, S.

Braunecker, B.

Casasent, D.

Casasent, D. P.

Caufield, H. J.

Chang, Wen-Thong

Fleisher, M.

Hassebrook, L.

Hauck, R.

Horner, J. L.

Hsu, Yuan-neng

Huignard, J. P.

Iyer, A.

D. Casasent, A. Iyer, G. Ravichandran, “Circular harmonic function, minimum correlation energy filters,” Appl. Opt. 35, 5169–5175 (1991).
[Crossref]

Jensen, A. S.

Kallman, R. R.

Lee, S. H.

Leger, J. R.

Lindvold, L.

Lohmann, A. W.

Mahalanobis, A.

A. Mahalanobis, D. Casasent, “Performance evaluation of minimum average correlation filters,” Appl. Opt. 30, 561–572 (1991).
[Crossref] [PubMed]

S. I. Sudharsanan, A. Mahalanobis, M. K. Sundareshan, “Unified framework for synthesis of discriminant functions with reduced noise variance and sharp correlation structure,” Opt. Eng. 29, 1021–1028 (1990).
[Crossref]

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average corelation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[Crossref] [PubMed]

Mahlab, U.

Maloney, W. T.

Mersereau, K.

Morris, G. M.

Psaltis, D.

Rasmussen, E.

Ravichandran, G.

D. Casasent, A. Iyer, G. Ravichandran, “Circular harmonic function, minimum correlation energy filters,” Appl. Opt. 35, 5169–5175 (1991).
[Crossref]

D. Casasent, G. Ravichandran, S. Bollapragada, “Gaussian-minimum correlation energy filters,” Appl. Opt. 30, 5176–5181 (1991).
[Crossref] [PubMed]

Refregier, Ph.

Rosen, J.

Rozzi, W. A.

Schils, G. F.

G. F. Schils, D. W. Sweeney, “Rotationally correlation filtering for multiple images,” J. Opt. Soc. Am. 3, 902–908 (1986).
[Crossref]

G. F. Schils, D. W. Sweeney, “Rotationally invariant correlation filtering,” J. Opt. Soc. Am. A 2, 1411–1418 (1985).
[Crossref]

Shamir, J.

Sudharsanan, S. I.

S. I. Sudharsanan, A. Mahalanobis, M. K. Sundareshan, “Unified framework for synthesis of discriminant functions with reduced noise variance and sharp correlation structure,” Opt. Eng. 29, 1021–1028 (1990).
[Crossref]

Sundareshan, M. K.

S. I. Sudharsanan, A. Mahalanobis, M. K. Sundareshan, “Unified framework for synthesis of discriminant functions with reduced noise variance and sharp correlation structure,” Opt. Eng. 29, 1021–1028 (1990).
[Crossref]

Sweeney, D. W.

G. F. Schils, D. W. Sweeney, “Rotationally correlation filtering for multiple images,” J. Opt. Soc. Am. 3, 902–908 (1986).
[Crossref]

G. F. Schils, D. W. Sweeney, “Rotationally invariant correlation filtering,” J. Opt. Soc. Am. A 2, 1411–1418 (1985).
[Crossref]

Szoplik, T.

VanderLugt, A. B.

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

Vijaya Kumar, B. V. K.

Appl. Opt. (21)

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

A. S. Jensen, L. Lindvold, E. Rasmussen, “Transformation of image positions, rotations and sizes into shift parameters,” Appl. Opt. 26, 1775–1781 (1987).
[Crossref] [PubMed]

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

D. Casasent, A. Iyer, G. Ravichandran, “Circular harmonic function, minimum correlation energy filters,” Appl. Opt. 35, 5169–5175 (1991).
[Crossref]

H. J. Caufield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
[Crossref]

D. P. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
[Crossref] [PubMed]

D. P. Casasent, Wen-Thong Chang, “Correlation synthetic discriminant functions,” Appl. Opt. 25, 2343–2350 (1986).
[Crossref] [PubMed]

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

Yuan-neng Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
[Crossref] [PubMed]

B. Braunecker, R. Hauck, A. W. Lohmann, “Optical character recognition based on the nonredundant correlation measurements,” Appl. Opt. 18, 2746–2753 (1979).
[Crossref] [PubMed]

R. R. Kallman, “Construction of low noise optical corelation filters,” Appl. Opt. 25, 1032–1033 (1986).
[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]

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average corelation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[Crossref] [PubMed]

M. Fleisher, U. Mahlab, J. Shamir, “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
[Crossref] [PubMed]

A. Mahalanobis, D. Casasent, “Performance evaluation of minimum average correlation filters,” Appl. Opt. 30, 561–572 (1991).
[Crossref] [PubMed]

D. Casasent, G. Ravichandran, S. Bollapragada, “Gaussian-minimum correlation energy filters,” Appl. Opt. 30, 5176–5181 (1991).
[Crossref] [PubMed]

Ph. Refregier, J. P. Huignard, “Phase selection of synthetic discriminant function filters,” Appl. Opt. 29, 4772–4778 (1990).
[Crossref] [PubMed]

B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[Crossref]

J. L. Horner, “Clarification of Horner efficiency,” Appl. Opt. 31, 4629 (1992).
[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. (3)

B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. 3, 1579–1584 (1986).
[Crossref]

G. F. Schils, D. W. Sweeney, “Rotationally correlation filtering for multiple images,” J. Opt. Soc. Am. 3, 902–908 (1986).
[Crossref]

J. R. Leger, S. H. Lee, “Image classification by an optical implementation of the Fukunaga-Koontz transform,” J. Opt. Soc. Am. 72, 556–564 (1982).
[Crossref]

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

Opt. Eng. (1)

S. I. Sudharsanan, A. Mahalanobis, M. K. Sundareshan, “Unified framework for synthesis of discriminant functions with reduced noise variance and sharp correlation structure,” Opt. Eng. 29, 1021–1028 (1990).
[Crossref]

Opt. Lett. (1)

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Figures (10)

Fig. 1
Fig. 1

Image of object 1.

Fig. 2
Fig. 2

Image of object 2.

Fig. 3
Fig. 3

Responses of the hk filters as functions of the object images.

Fig. 4
Fig. 4

Example of response 1 correlation intensity in the rotation invariant problem. This is the central region of 64 × 64 pixels taken from the correlation intensity of the 128 × 128 pixels (correlation intensities of an image of object 1 with filter h1).

Fig. 5
Fig. 5

Example of response 0 correlation intensity in the rotation invariant problem. This is the central region of 64 × 64 pixels taken from the correlation intensity of the 128 × 128 pixels (correlation intensities of an image of object 2 with filter h1).

Fig. 6
Fig. 6

Maximum detected correlation intensities (arbitrary units) as functions of the input image orientation of object 1 for (a) filter 1 and (b) filter 2.

Fig. 7
Fig. 7

Summary of the results obtained in the correlation simulations in the case of rotation invariance where I1, I0, SD1, and SD0 are the averages of the maximum detected intensities for responses 1 and 0 and the square roots of the corresponding variances. Here G is a dimensionless performance estimator and PCE is the average value of the peak to correlation energy.

Fig. 8
Fig. 8

Responses of the hk filters as functions of the object images in the case of the character-recognition problem.

Fig. 9
Fig. 9

Distorted characters used to compute the filters.

Fig. 10
Fig. 10

Summary of the results obtained in the correlation simulations in the case of character recognition where I1, I0, SD1, and SD0 are the averages of the maximum detected intensities for responses 1 and 0 and the square roots of the corresponding variances. Here G is a dimensionless performance estimator and PCE is the average value of the peak to correlation energy.

Equations (17)

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r k q m = s , t = 0 n - 1 h m ( s , t ) x k q ( s , t ) .
r k q m = U , V = - n / 2 ( n / 2 ) - 1 H m ( U , V ) X k q * ( U , V ) .
r k q m = x k q · h m ,
r k q m = X k q * · H m ,
F m = s , t = - n / 2 ( n / 2 ) - 1 k q i H m ( U , V ) [ X k q ( U , V ) - Y k q i ( U , V ) ] * 2 .
X k q ( U , V ) = 0 if 1 2 L K k , k , q , q X k q ( U , V ) X k q * ( U , V ) X k q ( U , V ) X k q ( U , V ) η ,
E ( Z , Z ) = k , q , i X k q ( Z ) - Y k q i ( Z ) 2 ,
F m = H m + E H m ,
H m = E - 1 X ( X + E - 1 X ) r m .
Y k q i ( U , V ) = X k q ( U , V ) + N i ( U , V ) ,
F m = s , t = - n / 2 ( n / 2 ) - 1 k q i H m ( U , V ) N i * ( u , V ) 2 .
FR = E [ y 1 ( 0 , 0 ) ] - E [ y 2 ( 0 , 0 ) ] 2 { VAR [ y 1 ( 0 , 0 ) ] + VAR [ y 2 ( 0 , 0 ) ] } / 2 ,
G = E [ y 1 ( 0 , 0 ) 2 ] - E [ max ( y 2 2 ) ] { VAR [ y 1 ( 0 , 0 ) 2 ] } 1 / 2 + { VAR [ max ( y 2 2 ) ] } 1 / 2 ,
G I 1 - I 0 SD 1 + SD 0 ,
PCE = y 1 ( 0 , 0 ) 2 s , t = 0 n - 1 y 1 ( s , t ) 2 .
SNR = E [ y 1 ( 0 , 0 ) ] 2 VAR [ y 1 ( 0 , 0 ) ] .
HE = y 1 ( 0 , 0 ) 2 U , V X ( U , V ) 2 ,

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