Abstract

We study a cascade of linear shift-invariant processing modules (correlators), each augmented with a nonlinear threshold as a means to increase the performance of high-speed optical pattern recognition. This configuration is a special class of multilayer, feed-forward neural networks and has been proposed in the literature as a relatively fast best-guess classifier. However, it seems that, although cascaded correlation has been proposed in a number of specific pattern recognition problems, the importance of the configuration has been largely overlooked. We prove that the cascaded architecture is the exact structure that must be adopted if a multilayer feed-forward neural network is trained to produce a shift-invariant output. In contrast with more generalized multilayer networks, the approach is easily implemented in practice with optical techniques and is therefore ideally suited to the high-speed analysis of large images. We have trained a digital model of the system using a modified backpropagation algorithm with optimization using simulated annealing techniques. The resulting cascade has been applied to a defect recognition problem in the canning industry as a benchmark for comparison against a standard linear correlation filter, the minimum average correlation energy (MACE) filter. We show that the nonlinear performance of the cascade is a significant improvement over that of the linear MACE filter in this case.

© 2001 Optical Society of America

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References

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  1. D. Casasent, L. M. Nieberg, “Classifier and shift-invariant automatic target recognition,” Neural Networks 8, 1117–1129 (1995).
    [CrossRef]
  2. X.-Y. Su, G.-S. Zhang, L.-R. Guo, “Phase-only composite filter,” Opt. Eng. 26, 520–523 (1987).
    [CrossRef]
  3. S.-C. B. Lo, S.-L. A. Lou, J.-S. Lin, M. T. Freedman, M. V. Chien, S. K. Mun, “Artificial convolution neural network techniques and applications for lung nodule detection,” IEEE Trans. Med. Imaging 14, 711–718 (1995).
    [CrossRef] [PubMed]
  4. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
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    [CrossRef] [PubMed]
  6. S. J. Perantonis, P. J. G. Lisboa, “Translation, rotation and scale invariant pattern recognition by high-order neural networks and moment classifiers,” IEEE Trans. Neural Networks 3, 241–251 (1992).
    [CrossRef]
  7. M. D. Richard, R. P. Lippmann, “Neural network classifiers estimate Bayesian a posteriori probabilities,” Neural Comput. 3, 461–483 (1991).
    [CrossRef]
  8. F. Kanaya, S. Miyake, “Bayes statistical behavior and valid generalization of pattern classifying neural networks,” IEEE Trans. Neural Networks 2, 471–475 (1991).
    [CrossRef]
  9. J. Shamir, H. J. Caulfield, R. B. Johnson, “Massive holographic interconnection networks and their limitations,” Appl. Opt. 28, 311–324 (1989).
    [CrossRef] [PubMed]
  10. M. A. Neifeld, “Optical dual-scale architecture for neural image recognition,” Appl. Opt. 34, 5920–5927 (1995).
    [CrossRef] [PubMed]
  11. D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [CrossRef]
  12. H. J. Caulfield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
    [CrossRef] [PubMed]
  13. B. V. K. Vijaya Kumar, “Minimum variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
  14. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [CrossRef] [PubMed]
  15. K. Fukushima, “Cognitron: a self-organizing multilayered neural network model,” Biol. Cybern. 20, 121–136 (1975).
    [CrossRef] [PubMed]
  16. K. Fukushima, “Neocognitron: a self-organizing multilayered neural network model for a mechanism of pattern recognition unaffected by shift in position,” Biol. Cybern. 36, 193–202 (1980).
    [CrossRef]
  17. B. Sahiner, H.-P. Chan, N. Petrick, D. Wei, M. A. Helvie, D. D. Adler, M. M. Goodsitt, “Classification of mass and normal breast tissue: a convolution neural network classifier with spatial domain and texture images,” IEEE. Trans. Med. Imaging 15, 598–610 (1996).
    [CrossRef] [PubMed]
  18. P. D. Gader, J. R. Miramonti, Y. Won, P. Coffield, “Segmentation free shared weight networks for automatic vehicle detection,” Neural Networks 8, 1457–1473 (1995).
    [CrossRef]
  19. F. Dubois, “Nonlinear cascaded correlation processes to improve the performances of automatic spatial-frequency-selective filters in pattern recognition,” Appl. Opt. 35, 4589–4597 (1996).
    [CrossRef] [PubMed]
  20. D. Psaltis, J. Hong, “Shift-invariant optical associative memories,” Opt. Eng. 26, 10–15 (1987).
    [CrossRef]
  21. F. T. S. Yu, S. Jutamulia, “Implementation of symbolic substitution logic using optical associative memories,” Appl. Opt. 26, 2293–2294 (1987).
    [CrossRef] [PubMed]
  22. S. D. Goodman, W. T. Rhodes, “Symbolic substitution applications to image processing,” Appl. Opt. 27, 1708–1714 (1988).
    [CrossRef] [PubMed]
  23. S. Reed, J. M. Coupland, “Statistical performance of cascaded shift invariant processing,” Appl. Opt. 39, 5949–5955 (2000).
    [CrossRef]
  24. S. Reed, J. M. Coupland, “Rotation invariance using cascaded optical processing architectures,” Asian J. Phys. 8, 421–429 (1999).
  25. W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).
  26. R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1985).
    [CrossRef]
  27. E. Barnard, E. C. Botha, “Back-propagation uses prior information efficiently,” IEEE Trans. Neural Networks 4, 794–802 (1993).
    [CrossRef]
  28. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimisation by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  29. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).

2000 (1)

1999 (1)

S. Reed, J. M. Coupland, “Rotation invariance using cascaded optical processing architectures,” Asian J. Phys. 8, 421–429 (1999).

1996 (2)

F. Dubois, “Nonlinear cascaded correlation processes to improve the performances of automatic spatial-frequency-selective filters in pattern recognition,” Appl. Opt. 35, 4589–4597 (1996).
[CrossRef] [PubMed]

B. Sahiner, H.-P. Chan, N. Petrick, D. Wei, M. A. Helvie, D. D. Adler, M. M. Goodsitt, “Classification of mass and normal breast tissue: a convolution neural network classifier with spatial domain and texture images,” IEEE. Trans. Med. Imaging 15, 598–610 (1996).
[CrossRef] [PubMed]

1995 (4)

P. D. Gader, J. R. Miramonti, Y. Won, P. Coffield, “Segmentation free shared weight networks for automatic vehicle detection,” Neural Networks 8, 1457–1473 (1995).
[CrossRef]

M. A. Neifeld, “Optical dual-scale architecture for neural image recognition,” Appl. Opt. 34, 5920–5927 (1995).
[CrossRef] [PubMed]

D. Casasent, L. M. Nieberg, “Classifier and shift-invariant automatic target recognition,” Neural Networks 8, 1117–1129 (1995).
[CrossRef]

S.-C. B. Lo, S.-L. A. Lou, J.-S. Lin, M. T. Freedman, M. V. Chien, S. K. Mun, “Artificial convolution neural network techniques and applications for lung nodule detection,” IEEE Trans. Med. Imaging 14, 711–718 (1995).
[CrossRef] [PubMed]

1993 (1)

E. Barnard, E. C. Botha, “Back-propagation uses prior information efficiently,” IEEE Trans. Neural Networks 4, 794–802 (1993).
[CrossRef]

1992 (1)

S. J. Perantonis, P. J. G. Lisboa, “Translation, rotation and scale invariant pattern recognition by high-order neural networks and moment classifiers,” IEEE Trans. Neural Networks 3, 241–251 (1992).
[CrossRef]

1991 (2)

M. D. Richard, R. P. Lippmann, “Neural network classifiers estimate Bayesian a posteriori probabilities,” Neural Comput. 3, 461–483 (1991).
[CrossRef]

F. Kanaya, S. Miyake, “Bayes statistical behavior and valid generalization of pattern classifying neural networks,” IEEE Trans. Neural Networks 2, 471–475 (1991).
[CrossRef]

1989 (2)

1988 (1)

1987 (5)

1986 (1)

1984 (1)

1983 (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimisation by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

1980 (1)

K. Fukushima, “Neocognitron: a self-organizing multilayered neural network model for a mechanism of pattern recognition unaffected by shift in position,” Biol. Cybern. 36, 193–202 (1980).
[CrossRef]

1975 (1)

K. Fukushima, “Cognitron: a self-organizing multilayered neural network model,” Biol. Cybern. 20, 121–136 (1975).
[CrossRef] [PubMed]

1969 (1)

Adler, D. D.

B. Sahiner, H.-P. Chan, N. Petrick, D. Wei, M. A. Helvie, D. D. Adler, M. M. Goodsitt, “Classification of mass and normal breast tissue: a convolution neural network classifier with spatial domain and texture images,” IEEE. Trans. Med. Imaging 15, 598–610 (1996).
[CrossRef] [PubMed]

Barnard, E.

E. Barnard, E. C. Botha, “Back-propagation uses prior information efficiently,” IEEE Trans. Neural Networks 4, 794–802 (1993).
[CrossRef]

Botha, E. C.

E. Barnard, E. C. Botha, “Back-propagation uses prior information efficiently,” IEEE Trans. Neural Networks 4, 794–802 (1993).
[CrossRef]

Casasent, D.

Caulfield, H. J.

Chan, H.-P.

B. Sahiner, H.-P. Chan, N. Petrick, D. Wei, M. A. Helvie, D. D. Adler, M. M. Goodsitt, “Classification of mass and normal breast tissue: a convolution neural network classifier with spatial domain and texture images,” IEEE. Trans. Med. Imaging 15, 598–610 (1996).
[CrossRef] [PubMed]

Chien, M. V.

S.-C. B. Lo, S.-L. A. Lou, J.-S. Lin, M. T. Freedman, M. V. Chien, S. K. Mun, “Artificial convolution neural network techniques and applications for lung nodule detection,” IEEE Trans. Med. Imaging 14, 711–718 (1995).
[CrossRef] [PubMed]

Coffield, P.

P. D. Gader, J. R. Miramonti, Y. Won, P. Coffield, “Segmentation free shared weight networks for automatic vehicle detection,” Neural Networks 8, 1457–1473 (1995).
[CrossRef]

Coupland, J. M.

S. Reed, J. M. Coupland, “Statistical performance of cascaded shift invariant processing,” Appl. Opt. 39, 5949–5955 (2000).
[CrossRef]

S. Reed, J. M. Coupland, “Rotation invariance using cascaded optical processing architectures,” Asian J. Phys. 8, 421–429 (1999).

Dubois, F.

Freedman, M. T.

S.-C. B. Lo, S.-L. A. Lou, J.-S. Lin, M. T. Freedman, M. V. Chien, S. K. Mun, “Artificial convolution neural network techniques and applications for lung nodule detection,” IEEE Trans. Med. Imaging 14, 711–718 (1995).
[CrossRef] [PubMed]

Fukushima, K.

K. Fukushima, “Neocognitron: a self-organizing multilayered neural network model for a mechanism of pattern recognition unaffected by shift in position,” Biol. Cybern. 36, 193–202 (1980).
[CrossRef]

K. Fukushima, “Cognitron: a self-organizing multilayered neural network model,” Biol. Cybern. 20, 121–136 (1975).
[CrossRef] [PubMed]

Gader, P. D.

P. D. Gader, J. R. Miramonti, Y. Won, P. Coffield, “Segmentation free shared weight networks for automatic vehicle detection,” Neural Networks 8, 1457–1473 (1995).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimisation by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Giles, C. L.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Goodman, S. D.

Goodsitt, M. M.

B. Sahiner, H.-P. Chan, N. Petrick, D. Wei, M. A. Helvie, D. D. Adler, M. M. Goodsitt, “Classification of mass and normal breast tissue: a convolution neural network classifier with spatial domain and texture images,” IEEE. Trans. Med. Imaging 15, 598–610 (1996).
[CrossRef] [PubMed]

Guo, L.-R.

X.-Y. Su, G.-S. Zhang, L.-R. Guo, “Phase-only composite filter,” Opt. Eng. 26, 520–523 (1987).
[CrossRef]

Helvie, M. A.

B. Sahiner, H.-P. Chan, N. Petrick, D. Wei, M. A. Helvie, D. D. Adler, M. M. Goodsitt, “Classification of mass and normal breast tissue: a convolution neural network classifier with spatial domain and texture images,” IEEE. Trans. Med. Imaging 15, 598–610 (1996).
[CrossRef] [PubMed]

Hong, J.

D. Psaltis, J. Hong, “Shift-invariant optical associative memories,” Opt. Eng. 26, 10–15 (1987).
[CrossRef]

Horn, R. A.

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1985).
[CrossRef]

Javidi, B.

Johnson, C. R.

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1985).
[CrossRef]

Johnson, R. B.

Jutamulia, S.

Kanaya, F.

F. Kanaya, S. Miyake, “Bayes statistical behavior and valid generalization of pattern classifying neural networks,” IEEE Trans. Neural Networks 2, 471–475 (1991).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimisation by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Lin, J.-S.

S.-C. B. Lo, S.-L. A. Lou, J.-S. Lin, M. T. Freedman, M. V. Chien, S. K. Mun, “Artificial convolution neural network techniques and applications for lung nodule detection,” IEEE Trans. Med. Imaging 14, 711–718 (1995).
[CrossRef] [PubMed]

Lippmann, R. P.

M. D. Richard, R. P. Lippmann, “Neural network classifiers estimate Bayesian a posteriori probabilities,” Neural Comput. 3, 461–483 (1991).
[CrossRef]

Lisboa, P. J. G.

S. J. Perantonis, P. J. G. Lisboa, “Translation, rotation and scale invariant pattern recognition by high-order neural networks and moment classifiers,” IEEE Trans. Neural Networks 3, 241–251 (1992).
[CrossRef]

Lo, S.-C. B.

S.-C. B. Lo, S.-L. A. Lou, J.-S. Lin, M. T. Freedman, M. V. Chien, S. K. Mun, “Artificial convolution neural network techniques and applications for lung nodule detection,” IEEE Trans. Med. Imaging 14, 711–718 (1995).
[CrossRef] [PubMed]

Lou, S.-L. A.

S.-C. B. Lo, S.-L. A. Lou, J.-S. Lin, M. T. Freedman, M. V. Chien, S. K. Mun, “Artificial convolution neural network techniques and applications for lung nodule detection,” IEEE Trans. Med. Imaging 14, 711–718 (1995).
[CrossRef] [PubMed]

Mahalanobis, A.

Maloney, W. T.

Maxwell, T.

Miramonti, J. R.

P. D. Gader, J. R. Miramonti, Y. Won, P. Coffield, “Segmentation free shared weight networks for automatic vehicle detection,” Neural Networks 8, 1457–1473 (1995).
[CrossRef]

Miyake, S.

F. Kanaya, S. Miyake, “Bayes statistical behavior and valid generalization of pattern classifying neural networks,” IEEE Trans. Neural Networks 2, 471–475 (1991).
[CrossRef]

Mun, S. K.

S.-C. B. Lo, S.-L. A. Lou, J.-S. Lin, M. T. Freedman, M. V. Chien, S. K. Mun, “Artificial convolution neural network techniques and applications for lung nodule detection,” IEEE Trans. Med. Imaging 14, 711–718 (1995).
[CrossRef] [PubMed]

Neifeld, M. A.

Nieberg, L. M.

D. Casasent, L. M. Nieberg, “Classifier and shift-invariant automatic target recognition,” Neural Networks 8, 1117–1129 (1995).
[CrossRef]

Perantonis, S. J.

S. J. Perantonis, P. J. G. Lisboa, “Translation, rotation and scale invariant pattern recognition by high-order neural networks and moment classifiers,” IEEE Trans. Neural Networks 3, 241–251 (1992).
[CrossRef]

Petrick, N.

B. Sahiner, H.-P. Chan, N. Petrick, D. Wei, M. A. Helvie, D. D. Adler, M. M. Goodsitt, “Classification of mass and normal breast tissue: a convolution neural network classifier with spatial domain and texture images,” IEEE. Trans. Med. Imaging 15, 598–610 (1996).
[CrossRef] [PubMed]

Pratt, W. K.

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).

Psaltis, D.

D. Psaltis, J. Hong, “Shift-invariant optical associative memories,” Opt. Eng. 26, 10–15 (1987).
[CrossRef]

Reed, S.

S. Reed, J. M. Coupland, “Statistical performance of cascaded shift invariant processing,” Appl. Opt. 39, 5949–5955 (2000).
[CrossRef]

S. Reed, J. M. Coupland, “Rotation invariance using cascaded optical processing architectures,” Asian J. Phys. 8, 421–429 (1999).

Rhodes, W. T.

Richard, M. D.

M. D. Richard, R. P. Lippmann, “Neural network classifiers estimate Bayesian a posteriori probabilities,” Neural Comput. 3, 461–483 (1991).
[CrossRef]

Sahiner, B.

B. Sahiner, H.-P. Chan, N. Petrick, D. Wei, M. A. Helvie, D. D. Adler, M. M. Goodsitt, “Classification of mass and normal breast tissue: a convolution neural network classifier with spatial domain and texture images,” IEEE. Trans. Med. Imaging 15, 598–610 (1996).
[CrossRef] [PubMed]

Shamir, J.

Su, X.-Y.

X.-Y. Su, G.-S. Zhang, L.-R. Guo, “Phase-only composite filter,” Opt. Eng. 26, 520–523 (1987).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimisation by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Vijaya Kumar, B. V. K.

Wei, D.

B. Sahiner, H.-P. Chan, N. Petrick, D. Wei, M. A. Helvie, D. D. Adler, M. M. Goodsitt, “Classification of mass and normal breast tissue: a convolution neural network classifier with spatial domain and texture images,” IEEE. Trans. Med. Imaging 15, 598–610 (1996).
[CrossRef] [PubMed]

Won, Y.

P. D. Gader, J. R. Miramonti, Y. Won, P. Coffield, “Segmentation free shared weight networks for automatic vehicle detection,” Neural Networks 8, 1457–1473 (1995).
[CrossRef]

Yu, F. T. S.

Zhang, G.-S.

X.-Y. Su, G.-S. Zhang, L.-R. Guo, “Phase-only composite filter,” Opt. Eng. 26, 520–523 (1987).
[CrossRef]

Appl. Opt. (11)

J. Shamir, H. J. Caulfield, R. B. Johnson, “Massive holographic interconnection networks and their limitations,” Appl. Opt. 28, 311–324 (1989).
[CrossRef] [PubMed]

M. A. Neifeld, “Optical dual-scale architecture for neural image recognition,” Appl. Opt. 34, 5920–5927 (1995).
[CrossRef] [PubMed]

D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
[CrossRef]

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

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[CrossRef] [PubMed]

C. L. Giles, T. Maxwell, “Learning, invariance, and generalization in high-order neural networks,” Appl. Opt. 26, 4972–4978 (1987).
[CrossRef] [PubMed]

F. Dubois, “Nonlinear cascaded correlation processes to improve the performances of automatic spatial-frequency-selective filters in pattern recognition,” Appl. Opt. 35, 4589–4597 (1996).
[CrossRef] [PubMed]

F. T. S. Yu, S. Jutamulia, “Implementation of symbolic substitution logic using optical associative memories,” Appl. Opt. 26, 2293–2294 (1987).
[CrossRef] [PubMed]

S. D. Goodman, W. T. Rhodes, “Symbolic substitution applications to image processing,” Appl. Opt. 27, 1708–1714 (1988).
[CrossRef] [PubMed]

S. Reed, J. M. Coupland, “Statistical performance of cascaded shift invariant processing,” Appl. Opt. 39, 5949–5955 (2000).
[CrossRef]

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

Asian J. Phys. (1)

S. Reed, J. M. Coupland, “Rotation invariance using cascaded optical processing architectures,” Asian J. Phys. 8, 421–429 (1999).

Biol. Cybern. (2)

K. Fukushima, “Cognitron: a self-organizing multilayered neural network model,” Biol. Cybern. 20, 121–136 (1975).
[CrossRef] [PubMed]

K. Fukushima, “Neocognitron: a self-organizing multilayered neural network model for a mechanism of pattern recognition unaffected by shift in position,” Biol. Cybern. 36, 193–202 (1980).
[CrossRef]

IEEE Trans. Med. Imaging (1)

S.-C. B. Lo, S.-L. A. Lou, J.-S. Lin, M. T. Freedman, M. V. Chien, S. K. Mun, “Artificial convolution neural network techniques and applications for lung nodule detection,” IEEE Trans. Med. Imaging 14, 711–718 (1995).
[CrossRef] [PubMed]

IEEE Trans. Neural Networks (3)

S. J. Perantonis, P. J. G. Lisboa, “Translation, rotation and scale invariant pattern recognition by high-order neural networks and moment classifiers,” IEEE Trans. Neural Networks 3, 241–251 (1992).
[CrossRef]

F. Kanaya, S. Miyake, “Bayes statistical behavior and valid generalization of pattern classifying neural networks,” IEEE Trans. Neural Networks 2, 471–475 (1991).
[CrossRef]

E. Barnard, E. C. Botha, “Back-propagation uses prior information efficiently,” IEEE Trans. Neural Networks 4, 794–802 (1993).
[CrossRef]

IEEE. Trans. Med. Imaging (1)

B. Sahiner, H.-P. Chan, N. Petrick, D. Wei, M. A. Helvie, D. D. Adler, M. M. Goodsitt, “Classification of mass and normal breast tissue: a convolution neural network classifier with spatial domain and texture images,” IEEE. Trans. Med. Imaging 15, 598–610 (1996).
[CrossRef] [PubMed]

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

Neural Comput. (1)

M. D. Richard, R. P. Lippmann, “Neural network classifiers estimate Bayesian a posteriori probabilities,” Neural Comput. 3, 461–483 (1991).
[CrossRef]

Neural Networks (2)

P. D. Gader, J. R. Miramonti, Y. Won, P. Coffield, “Segmentation free shared weight networks for automatic vehicle detection,” Neural Networks 8, 1457–1473 (1995).
[CrossRef]

D. Casasent, L. M. Nieberg, “Classifier and shift-invariant automatic target recognition,” Neural Networks 8, 1117–1129 (1995).
[CrossRef]

Opt. Eng. (2)

X.-Y. Su, G.-S. Zhang, L.-R. Guo, “Phase-only composite filter,” Opt. Eng. 26, 520–523 (1987).
[CrossRef]

D. Psaltis, J. Hong, “Shift-invariant optical associative memories,” Opt. Eng. 26, 10–15 (1987).
[CrossRef]

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimisation by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other (3)

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).

R. A. Horn, C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, UK, 1985).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

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Figures (6)

Fig. 1
Fig. 1

Multilayer feed-forward NN.

Fig. 2
Fig. 2

Optical configuration for can defect image acquisition.

Fig. 3
Fig. 3

Selection of in-class training images with the corresponding desired output fields.

Fig. 4
Fig. 4

Selection of sample out-of-class training images.

Fig. 5
Fig. 5

(a) Sample input image with (b) MACE defect recognition and (c) cascaded correlator defect recognition output.

Fig. 6
Fig. 6

(a) Sample input image intermediate to training set can rotations with (b) MACE defect recognition and (c) cascaded correlator defect recognition output.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

Ok,j=mWk,mfiWm,iAi,j,
Ai+1,j+1=Ai,j,
Ok+1,j+1=Ok,j.
Ri=mHi,mbm,
Ok+1,j+1=mWk+1,mfiWm,iAi,j+1.
Ok+1,j+1=mWk+1,m+nfiWm+n,i+1Ai+1,j+1.
mWk,mfiWm,iAi,j=mWk+1,m+nfiWm+n,i+1Ai+1,j+1.
mWk,ma+b iWm,iAi,j+c ipWm,iWm,pAi,jAp,j+=mWk+1,m+na+b iWm+n,i+1Ai+1,j+1+c ipWm+n,i+1Wm+n,p+1Ai+1,j+1Ap+1,j+1+.
mWk,m=mWk+1,m+n,
mWk,mWm,i=mWk+1,m+nWm+n,i+1,
mWk,mWm,iWm,p=mWk+1,m+nWm+n,i+1Wm+n,p+1,
Wm,i=Wm+1,i+1,
Wk,m=Wk+1,m+1.
fx=|x|2,  |x|2t,fx=0,|x|2<t,
B=D-1XX+D-1X-1u,
Dk,k=i=1M |Xi,k|2.

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