Abstract

Optical correlation schemes based on circular or radial harmonics have been recently suggested for performing pattern recognition in the presence of rotated or scaled versions of the object. Those schemes, however, are based on using a single harmonic at one time. In this paper we present two approaches that overcome this restriction.

© 1989 Optical Society of America

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References

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  1. Y. N. Hsu, H. H. Arsenault, “Optical Pattern Recognition Using Circular Harmonics Expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [Crossref] [PubMed]
  2. D. Mendlovic, E. Marom, N. Konforti, “Shift and Scale Invariant Pattern Recognition Using Mellin Radial Harmonics,” Opt. Commun. 67, 172–176 (1988).
    [Crossref]
  3. G. F. Schils, D. W. Sweeney, “Iterative Technique for the Synthesis of Optical-Correlation Filters,” J. Opt. Soc. Am. A 3, 1433–1442 (1986).
    [Crossref]
  4. R. Wu, H. Stark, “Rotation-Invariant Pattern Recognition Using a Vector Reference,” Appl. Opt. 23, 838–840 (1984).
    [Crossref] [PubMed]
  5. Y. N. Hsu, H. H. Arsenault, “Pattern Discrimination by Multiple Circular Harmonic Components,” Appl. Opt. 23, 841–844 (1984).
    [Crossref] [PubMed]
  6. H. H. Arsenault, Y. N. Hsu, “Rotation-Invariant Discrimination Between Almost Similar Objects,” Appl. Opt. 22, 130–132 (1983).
    [Crossref] [PubMed]
  7. J. Rosen, J. Shamir, “Circular Harmonic Phase Filters for Efficient Rotation Invariant Pattern Recognition,” Appl. Opt. 27, 2895–2899 (1988).
    [Crossref] [PubMed]
  8. J. L. Horner, J. R. Leger, “Pattern Recognition with Binary Phase-Only Filter,” Appl. Opt. 24, 609–611 (1985).
    [Crossref] [PubMed]
  9. Y. Sheng, H. H. Arsenault, “Method for Determining Expansion Centers and Predicting Sidelobe Levels for Circular-Harmonic Filters,” J. Opt. Soc. Am. A 4, 1793–1797 (1987).
    [Crossref]

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]

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

1987 (1)

1986 (1)

1985 (1)

1984 (2)

1983 (1)

1982 (1)

Arsenault, H. H.

Horner, J. L.

Hsu, Y. N.

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]

Leger, J. R.

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]

Rosen, J.

Schils, G. F.

Shamir, J.

Sheng, Y.

Stark, H.

Sweeney, D. W.

Wu, R.

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Figures (7)

Fig. 1
Fig. 1

Composite secondary correlation for single circular harmonic M. First filter matched to the Mth harmonic of the input pattern, is positioned in plane A. The correlation pattern is displayed in plane B. A second filter matched to the (−M)th harmonic of plane B is placed in plane C. Second correlation is displayed in plane D.

Fig. 2
Fig. 2

Computer simulation of the autocorrelation using the composite secondary circular harmonics of orders 1, 2, and 3, as well as the zero-order circular harmonic. a) The input objects. b) The intensity distribution at the correlation plane.

Fig. 3
Fig. 3

Computer simulation of the cross correlation (recognition capability). a) The input objects E, C, F, and X. b) The intensity at the correlation plane using the second circular harmonic only. c) The intensity at the correlation plane using the composite secondary circular harmonics of orders 1, 2, and 3, as well as the zero-order circular harmonic. Peak correlation intensities are indicated in the square box.

Fig. 4
Fig. 4

(a) Same as Fig. 3(b) but with POF. b) Same as Fig. 3(c) but with POF.

Fig. 5
Fig. 5

Geometric representation showing that the effect of the choice of the expansion center results only in a lateral shift. a) For scale changes. b) For different angular orientations.

Fig. 6
Fig. 6

Block diagram outlining the steps taken for generation of the multiple expansion centers filter.

Fig. 7
Fig. 7

Input patterns for the results presented in Table I. (a) The original input pattern. (b) Test pattern.

Tables (1)

Tables Icon

Table I Recognition Capability of the Multiple Expansion Centers Filter Used to Detect Object of Fig. 7(a)a

Equations (16)

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f ( r , θ ) = M = f M ( r ) exp ( i M θ ) ,
f M ( r ) = 1 2 π 0 2 π f ( r , θ ) exp ( i M θ ) d θ .
h M ( r , θ ) = f M ( r ) exp ( i M θ ) .
C [ f ( r , θ + α ) , h M ] = exp ( i M α ) C [ f ( r , θ ) , h M ] .
f ( ρ , θ ) = N = f N ( θ ) ρ i N 1 ,
f N ( θ ) = 1 2 π r 0 R 1 f ( ρ , θ ) ρ i N 1 ρ d ρ ,
h N ( ρ , θ ) f N ( θ ) ρ i N 1 .
C [ f ( β ρ , θ ) h N ] = 1 β exp [ i N ln ( β ) ] C [ f ( ρ , θ ) , h N ] .
h ( r , θ ) = f M ( r ) exp ( i M θ ) + f M ( r ) exp ( i M θ ) ,
C [ f ( r , θ + α ) , h ( r , θ ) ] = C [ f ( r , θ ) , h ( r , θ ) ] cos ( M α ) .
F 1 ( r , θ ) = F . T . [ f M ( r ) exp ( i M θ ) ] .
C B [ f ( r , θ + α ) , f M ( r ) exp ( i M θ ) ] = C B , α [ f ( r , θ ) , f M ( r ) exp ( i M θ ) ] exp ( i M α ) ,
F 2 ( r , θ ) = F . T . [ C M B ( r ) exp ( i M θ ) ] ,
C B = M = C M B ( r ) exp ( i M θ ) .
C D [ C B , α exp ( i M α ) , C M B ( r ) exp ( i M θ ) ] = C D [ C B , α , C M B ( r ) exp ( i M θ ) ] exp ( i M α ) = { C D , α [ C B , C M B ( r ) exp ( i M θ ) ] exp ( i M α ) } exp ( i M α ) = C D , α [ C B , C M B ( r ) exp ( i M θ ) ] .
W ( r i ) = 0.2 + ( r i r 0 ) 2 ,

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