Abstract

Some general properties of the circular harmonic expansion relevant to their use for pattern recognition are derived. Both circular harmonic filters and Fourier-Mellin descriptors, which are used as the moments of circular harmonic functions, are considered. Expressions for the asymptotic energy in terms of the circular harmonic orders are derived and experimentally verified.

© 1986 Optical Society of America

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References

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  1. Y.-N. Hsu, H. H. Arsenault, G. April, “Rotation-Invariant Digital Pattern Recognition Using Circular Harmonic Expansion,” Appl. Opt. 21, 4012 (1982).
    [CrossRef] [PubMed]
  2. H. H. Arsenault, Y.-N. Hsu, Chalasinska-Macukow, “Rotation-Invariant Pattern Recognition,” Opt. Eng. 23, 705 (1984).
    [CrossRef]
  3. Y. Sheng, J. Duvernoy, “Circular Fourier-Radial Mellin Transform Descriptors (FMDs) for Pattern Recognition,” J. Opt. Soc. Am. 6, 885 (1986).
    [CrossRef]
  4. Y. Sheng, H. H. Arsenault, “Experiments on Pattern Recognition Using Fourier-Mellin Descriptors,” J. Opt. Soc. Am. 6, 771 (1986).
    [CrossRef]
  5. M. Hu, “Visual Pattern Recognition by Moment Invariants,” IRE Trans. Inf. Theory IT-8, 179 (1962).
  6. Y. S. Abu-Mostafa, D. Psaltis, “Recognition Aspects of Moment Invariants,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-6, 698 (1984).
    [CrossRef]
  7. Y.-N. Hsu, H. H. Arsenault, “Pattern Discrimination by Multiple Circular Harmonic Components,” Appl. Opt. 23, 841 (1984).
    [CrossRef] [PubMed]
  8. R. Wu, H. Stark, “Rotation-Invariant Pattern Recognition Using a Vector Reference,” Appl. Opt. 23, 838 (1984).
    [CrossRef] [PubMed]
  9. H. H. Arsenault, C. Belisle, “Contrast-Invariant Pattern Recognition Using Circular Harmonic Components,” Appl. Opt. 24, 2072 (1985).
    [CrossRef] [PubMed]
  10. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), Chaps. 8 and 9.
  11. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 362 and 363.
  12. E. W. Hansen, J. G. Verly, E. B. Keirstead, “Rotation-Invariant Optical Processing,” J. Opt. Soc. Am. 72, 1670 (1982).
    [CrossRef]
  13. H. S. Carlslaw, An Introduction to the Theory of Fourier Series and Integrals (Dover, New York, 1950), Chap. 8.
  14. T. Pavlidis, Structural Pattern Recognition (Springer-Verlag, Berlin, 1977), Chap. 7.

1986 (2)

Y. Sheng, J. Duvernoy, “Circular Fourier-Radial Mellin Transform Descriptors (FMDs) for Pattern Recognition,” J. Opt. Soc. Am. 6, 885 (1986).
[CrossRef]

Y. Sheng, H. H. Arsenault, “Experiments on Pattern Recognition Using Fourier-Mellin Descriptors,” J. Opt. Soc. Am. 6, 771 (1986).
[CrossRef]

1985 (1)

1984 (4)

Y. S. Abu-Mostafa, D. Psaltis, “Recognition Aspects of Moment Invariants,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-6, 698 (1984).
[CrossRef]

Y.-N. Hsu, H. H. Arsenault, “Pattern Discrimination by Multiple Circular Harmonic Components,” Appl. Opt. 23, 841 (1984).
[CrossRef] [PubMed]

R. Wu, H. Stark, “Rotation-Invariant Pattern Recognition Using a Vector Reference,” Appl. Opt. 23, 838 (1984).
[CrossRef] [PubMed]

H. H. Arsenault, Y.-N. Hsu, Chalasinska-Macukow, “Rotation-Invariant Pattern Recognition,” Opt. Eng. 23, 705 (1984).
[CrossRef]

1982 (2)

1962 (1)

M. Hu, “Visual Pattern Recognition by Moment Invariants,” IRE Trans. Inf. Theory IT-8, 179 (1962).

Abu-Mostafa, Y. S.

Y. S. Abu-Mostafa, D. Psaltis, “Recognition Aspects of Moment Invariants,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-6, 698 (1984).
[CrossRef]

April, G.

Arsenault, H. H.

Belisle, C.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 362 and 363.

Carlslaw, H. S.

H. S. Carlslaw, An Introduction to the Theory of Fourier Series and Integrals (Dover, New York, 1950), Chap. 8.

Chalasinska-Macukow,

H. H. Arsenault, Y.-N. Hsu, Chalasinska-Macukow, “Rotation-Invariant Pattern Recognition,” Opt. Eng. 23, 705 (1984).
[CrossRef]

Duvernoy, J.

Y. Sheng, J. Duvernoy, “Circular Fourier-Radial Mellin Transform Descriptors (FMDs) for Pattern Recognition,” J. Opt. Soc. Am. 6, 885 (1986).
[CrossRef]

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), Chaps. 8 and 9.

Hansen, E. W.

Hsu, Y.-N.

Hu, M.

M. Hu, “Visual Pattern Recognition by Moment Invariants,” IRE Trans. Inf. Theory IT-8, 179 (1962).

Keirstead, E. B.

Pavlidis, T.

T. Pavlidis, Structural Pattern Recognition (Springer-Verlag, Berlin, 1977), Chap. 7.

Psaltis, D.

Y. S. Abu-Mostafa, D. Psaltis, “Recognition Aspects of Moment Invariants,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-6, 698 (1984).
[CrossRef]

Sheng, Y.

Y. Sheng, H. H. Arsenault, “Experiments on Pattern Recognition Using Fourier-Mellin Descriptors,” J. Opt. Soc. Am. 6, 771 (1986).
[CrossRef]

Y. Sheng, J. Duvernoy, “Circular Fourier-Radial Mellin Transform Descriptors (FMDs) for Pattern Recognition,” J. Opt. Soc. Am. 6, 885 (1986).
[CrossRef]

Stark, H.

Verly, J. G.

Wu, R.

Appl. Opt. (4)

IEEE Trans. Pattern Anal. Machine Intell. (1)

Y. S. Abu-Mostafa, D. Psaltis, “Recognition Aspects of Moment Invariants,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-6, 698 (1984).
[CrossRef]

IRE Trans. Inf. Theory (1)

M. Hu, “Visual Pattern Recognition by Moment Invariants,” IRE Trans. Inf. Theory IT-8, 179 (1962).

J. Opt. Soc. Am. (3)

E. W. Hansen, J. G. Verly, E. B. Keirstead, “Rotation-Invariant Optical Processing,” J. Opt. Soc. Am. 72, 1670 (1982).
[CrossRef]

Y. Sheng, J. Duvernoy, “Circular Fourier-Radial Mellin Transform Descriptors (FMDs) for Pattern Recognition,” J. Opt. Soc. Am. 6, 885 (1986).
[CrossRef]

Y. Sheng, H. H. Arsenault, “Experiments on Pattern Recognition Using Fourier-Mellin Descriptors,” J. Opt. Soc. Am. 6, 771 (1986).
[CrossRef]

Opt. Eng. (1)

H. H. Arsenault, Y.-N. Hsu, Chalasinska-Macukow, “Rotation-Invariant Pattern Recognition,” Opt. Eng. 23, 705 (1984).
[CrossRef]

Other (4)

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), Chaps. 8 and 9.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 362 and 363.

H. S. Carlslaw, An Introduction to the Theory of Fourier Series and Integrals (Dover, New York, 1950), Chap. 8.

T. Pavlidis, Structural Pattern Recognition (Springer-Verlag, Berlin, 1977), Chap. 7.

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Figures (3)

Fig. 1
Fig. 1

Reconstruction of the image of the space shuttle with its circular harmonic components (CHCs). From top left to bottom right: original image, the zero-order CHC of the image, reconstructed images adding successive orders up to m = 14.

Fig. 2
Fig. 2

Output correlation peak |Cm| of a rotation-invariant circular harmonic filter of the letter A converges with the circular harmonic order m. Dots represent experimental data; the straight line is the function |Cm|/|C0| = α(m + β)γ, where the values of the constants α, β, and γ are given in Table I.

Fig. 3
Fig. 3

Radial moments of the circular harmonic functions |Ms,m|/|Ms,o| of the letter A converge with the circular harmonic order m. Dots represent experimental data; the straight line is the function |Ms,m|/|Ms,o| = α(m + β)γ, where the values of the constants α, β, and γ are given in Table II.

Tables (2)

Tables Icon

Table I Values of the Constants in the Experimental Convergence Equations |Cm|/|C0| = α(m + β)γ of the Output Correlation Peak |Cm| in the Circular Harmonic Filter for Six Letters and for the image of the Space Shuttle

Tables Icon

Table II Values of the Constants in the Experimental Convergence Equations |Ms,m|/|Ms,o| = α(m + β)γ of the Radial Moments of the Circular Harmonic Functions for Six Letters and for the image of the Space Shuttle

Equations (16)

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f ( r , θ ) = m = - f m ( r ) exp ( j m θ ) ,
f m ( r ) = 1 2 π 0 2 π f ( r , θ ) exp ( - j m θ ) d θ .
f r ( r , θ ) = f m ( r ) exp ( j m θ ) ,
C m ( x , y ) = - f ( ξ , η ) f r * ( ξ - x , η - y ) d ξ d η .
C m = C m ( 0 , 0 ) = 2 π 0 f m ( r ) 2 r d r exp ( j m α ) ,
C m = 1 2 π 0 0 2 π f ( r , θ ) exp ( - j m θ ) d θ × 0 2 π f * ( r , θ ) exp ( j m θ ) d θ r d r = 1 2 π 0 0 2 π [ - θ 2 π - θ f ( r , θ + ϕ ) exp ( - j m ϕ ) d ϕ ] × f * ( r , θ ) d θ r d r ,
C m = 2 π 0 f m ( r ) 2 r d r = 1 2 π 0 0 2 π 0 2 π f * ( r , θ ) f ( r , θ + ϕ ) exp ( - j m ϕ ) d θ d ϕ r d r .
Φ ( ϕ ) = 0 0 2 π f * ( r , θ ) f ( r , θ + ϕ ) d θ r d r ,
C m = 1 2 π 0 2 π Φ ( ϕ ) exp ( - j m ϕ ) d ϕ .
M s , m = 0 2 π 0 r s - 1 f ( r , θ ) exp ( - j m θ ) d r d θ ,
Φ s ( θ ) = 0 r s - 1 f ( r , θ ) d r ,
M s , m = 0 2 π Φ s ( θ ) exp ( - j m θ ) d θ .
M s , m < 2 π [ 1 2 S - 1 0 f m ( r ) 2 d r ] 1 / 2 .
C 0 = 1 2 π 0 2 π 0 [ 0 2 π f * ( r , θ ) f ( r , θ + ϕ ) d θ ] r d r d ϕ ,
M s , o = 0 0 2 π r s - 1 f ( r , θ ) d θ d r ,
log 10 C m = α - γ log 10 ( m + β )

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