Abstract

An optical processor that realizes a generalized chord transformation is described. The wedge-ring detector samples of an autocorrelation are shown to be the histograms of the chord distributions. This dimensionality reduced set of features is used as the feature vector inputs for a Fisher linear classifier to determine the class of the input object independent of geometrical distortions. Initial discussions on the use of different classifiers, the polarity of the classifier’s output, and selection of the image training set are also advanced.

© 1983 Optical Society of America

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References

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  1. H. J. Caulfield, R. Haimes, D. Casasent, Opt. Eng. 19, 152 (1980).
  2. D. Casasent, A. Furman, Appl. Opt. 16, 1662 (1977).
    [CrossRef] [PubMed]
  3. D. Casasent, D. Psaltis, Proc. IEEE 65, 77 (1977).
    [CrossRef]
  4. H. J. Caulfield, R. Haimes, Appl. Opt. 19, 181 (1980).
    [CrossRef] [PubMed]
  5. C. F. Hester, D. Casasent, Appl. Opt. 19, 1758 (1980).
    [CrossRef] [PubMed]
  6. D. Casasent, J. Pauly, D. Fetterly, Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126 (1981).
  7. H. Kasdan, D. Meade, Proc. Electron. Opt. Syst. Des. 248 (1975).
  8. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973); see also Y. T. Chien, Interactive Pattern Recognition (Dekker, New York, 1978).
  9. G. Tenery, IEEE Trans. Mil. Electron. ME-7, 196 (1963).
    [CrossRef]
  10. D. J. H. Moore, D. J. Parker, Pattern Recognition 6, 149 (1974).
    [CrossRef]
  11. D. G. Nichols, Opt. Commun. 43, 168 (1982).
    [CrossRef]
  12. J. Rau, J. Opt. Soc. Am. 56, 1490 (1966).
    [CrossRef]
  13. H. N. Kannapell, “Image Classification Based on a Three-Dimensional Angle Histogram,” M. Eng. Thesis, U. Louisville, Louisville (1981).
  14. S. P. Smith, A. K. Jain, Comput. Graphics Image Process. 20, 1699 (1982).
    [CrossRef]
  15. H. J. Caulfield, M. H. Weinberg, Appl. Opt. 21, 1699 (1982).
    [CrossRef] [PubMed]
  16. J. R. Leger, S. H. Lee, J. Opt. Soc. Am. 72, 556 (1982).
    [CrossRef]
  17. Z. H. Gu, J. R. Leger, S. H. Lee, J. Opt. Soc. Am. 72, 787 (1982).
    [CrossRef]
  18. T. Marill, D. M. Green, IEEE Trans. Inf. Theory IT-9, 11 (1963).
    [CrossRef]

1982 (5)

1981 (1)

D. Casasent, J. Pauly, D. Fetterly, Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126 (1981).

1980 (3)

1977 (2)

1975 (1)

H. Kasdan, D. Meade, Proc. Electron. Opt. Syst. Des. 248 (1975).

1974 (1)

D. J. H. Moore, D. J. Parker, Pattern Recognition 6, 149 (1974).
[CrossRef]

1966 (1)

1963 (2)

G. Tenery, IEEE Trans. Mil. Electron. ME-7, 196 (1963).
[CrossRef]

T. Marill, D. M. Green, IEEE Trans. Inf. Theory IT-9, 11 (1963).
[CrossRef]

Casasent, D.

D. Casasent, J. Pauly, D. Fetterly, Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126 (1981).

H. J. Caulfield, R. Haimes, D. Casasent, Opt. Eng. 19, 152 (1980).

C. F. Hester, D. Casasent, Appl. Opt. 19, 1758 (1980).
[CrossRef] [PubMed]

D. Casasent, A. Furman, Appl. Opt. 16, 1662 (1977).
[CrossRef] [PubMed]

D. Casasent, D. Psaltis, Proc. IEEE 65, 77 (1977).
[CrossRef]

Caulfield, H. J.

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973); see also Y. T. Chien, Interactive Pattern Recognition (Dekker, New York, 1978).

Fetterly, D.

D. Casasent, J. Pauly, D. Fetterly, Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126 (1981).

Furman, A.

Green, D. M.

T. Marill, D. M. Green, IEEE Trans. Inf. Theory IT-9, 11 (1963).
[CrossRef]

Gu, Z. H.

Haimes, R.

H. J. Caulfield, R. Haimes, Appl. Opt. 19, 181 (1980).
[CrossRef] [PubMed]

H. J. Caulfield, R. Haimes, D. Casasent, Opt. Eng. 19, 152 (1980).

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973); see also Y. T. Chien, Interactive Pattern Recognition (Dekker, New York, 1978).

Hester, C. F.

Jain, A. K.

S. P. Smith, A. K. Jain, Comput. Graphics Image Process. 20, 1699 (1982).
[CrossRef]

Kannapell, H. N.

H. N. Kannapell, “Image Classification Based on a Three-Dimensional Angle Histogram,” M. Eng. Thesis, U. Louisville, Louisville (1981).

Kasdan, H.

H. Kasdan, D. Meade, Proc. Electron. Opt. Syst. Des. 248 (1975).

Lee, S. H.

Leger, J. R.

Marill, T.

T. Marill, D. M. Green, IEEE Trans. Inf. Theory IT-9, 11 (1963).
[CrossRef]

Meade, D.

H. Kasdan, D. Meade, Proc. Electron. Opt. Syst. Des. 248 (1975).

Moore, D. J. H.

D. J. H. Moore, D. J. Parker, Pattern Recognition 6, 149 (1974).
[CrossRef]

Nichols, D. G.

D. G. Nichols, Opt. Commun. 43, 168 (1982).
[CrossRef]

Parker, D. J.

D. J. H. Moore, D. J. Parker, Pattern Recognition 6, 149 (1974).
[CrossRef]

Pauly, J.

D. Casasent, J. Pauly, D. Fetterly, Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126 (1981).

Psaltis, D.

D. Casasent, D. Psaltis, Proc. IEEE 65, 77 (1977).
[CrossRef]

Rau, J.

Smith, S. P.

S. P. Smith, A. K. Jain, Comput. Graphics Image Process. 20, 1699 (1982).
[CrossRef]

Tenery, G.

G. Tenery, IEEE Trans. Mil. Electron. ME-7, 196 (1963).
[CrossRef]

Weinberg, M. H.

Appl. Opt. (4)

Comput. Graphics Image Process. (1)

S. P. Smith, A. K. Jain, Comput. Graphics Image Process. 20, 1699 (1982).
[CrossRef]

IEEE Trans. Inf. Theory (1)

T. Marill, D. M. Green, IEEE Trans. Inf. Theory IT-9, 11 (1963).
[CrossRef]

IEEE Trans. Mil. Electron. (1)

G. Tenery, IEEE Trans. Mil. Electron. ME-7, 196 (1963).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Commun. (1)

D. G. Nichols, Opt. Commun. 43, 168 (1982).
[CrossRef]

Opt. Eng. (1)

H. J. Caulfield, R. Haimes, D. Casasent, Opt. Eng. 19, 152 (1980).

Pattern Recognition (1)

D. J. H. Moore, D. J. Parker, Pattern Recognition 6, 149 (1974).
[CrossRef]

Proc. Electron. Opt. Syst. Des. (1)

H. Kasdan, D. Meade, Proc. Electron. Opt. Syst. Des. 248 (1975).

Proc. IEEE (1)

D. Casasent, D. Psaltis, Proc. IEEE 65, 77 (1977).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. Casasent, J. Pauly, D. Fetterly, Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126 (1981).

Other (2)

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973); see also Y. T. Chien, Interactive Pattern Recognition (Dekker, New York, 1978).

H. N. Kannapell, “Image Classification Based on a Three-Dimensional Angle Histogram,” M. Eng. Thesis, U. Louisville, Louisville (1981).

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Figures (4)

Fig. 1
Fig. 1

Chord transformation.

Fig. 2
Fig. 2

Simplified block diagram of a hybrid optical/digital distortion-invariant pattern recognition system using a generalized chord transformation.

Fig. 3
Fig. 3

Representative images for the class 1 ship imagery: (top right) 0, (top left) 45, (bottom) 90°.

Fig. 4
Fig. 4

Representative images for the class 2 ship imagery: (top right) 0, (top left) 45, (bottom 90°.

Tables (3)

Tables Icon

Table I Image Training Sets Used

Tables Icon

Table II Simulated Test Data (Binary Imagery)

Tables Icon

Table III Simulated Test Data (Gray-Scale Imagery)

Equations (20)

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

g ( x , y , r , θ ) = b ( x , y ) b ( x + r cos θ , y + r sin θ ) = 1 ,
h ( r , θ ) = g ( x , y , r , θ ) d x d y = b ( x , y ) b ( x + r cos θ , y + r sin θ ) d x d y .
h ( ξ , η ) = b ( x , y ) b ( x + ξ , y + η ) d x d y = b ( x , y ) b ( x , y ) ,
h G ( ξ , η ) = f ( x , y ) f ( x + ξ , y + η ) d x d y = f f .
h ( θ ) = h ( r cos θ , r sin θ ) r d r ,
h ( r ) = h ( r cos θ , r sin θ ) r d θ .
m 1 = ( 1 / N 1 ) i f 1 i .
S 1 = i ( f 1 i - m 1 ) ( f 1 i - m 1 ) T .
J = m 1 - m 2 2 S 1 2 + S 2 2 .
m 1 = ( 1 / N 1 ) i w T f 1 i = w T m 1 ,
S 1 2 = ( Projections 1 i - m 1 ) 2 = ( w T f 1 i - w T m 1 ) 2 = w T S 1 w .
( m 1 - m 2 ) 2 = ( w T m 1 - w T m 2 ) 2 = w T S B w ,
S B = ( m 1 - m 2 ) ( m 1 - m 2 ) T .
S 1 2 + S 2 2 = w T S 1 w + w T S 2 w = w T S W w ,
S W = S 1 + S 2 .
J ( w ) = w T S B w / w T S W w .
S B w = λ S W w .
S W - 1 S B w = λ w .
w = S W - 1 ( m 1 - m 2 ) .
T = ( m 1 + m 2 ) / 2.

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