Abstract

Image descriptors based on the circular-Fourier–radial-Mellin transform are used for position-, rotation-, scale-, and intensity-invariant multiclass pattern recognition. The orders of the radial moments and of the circular harmonics are chosen to obtain an efficient image description. The first-order radial moments of three circular harmonics are sufficient to obtain a satisfactory recognition performance. The influence of additive noise is investigated. Experimental results are shown.

© 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, 4012–4015 (1982).
    [CrossRef] [PubMed]
  2. R. Wu, H. Stark, “Rotation-invariant pattern recognition using optimum feature extraction,” Appl. Opt. 24, 179–184 (1985).
    [CrossRef] [PubMed]
  3. Y. Sheng, J. Duvernoy, “Circular-Fourier–radial-Mellin transform descriptors for pattern recognition,” J. Opt. Soc. Am. A 3, 885–888 (1986).
    [CrossRef] [PubMed]
  4. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), Chap. 12.
  5. D. Casasent, D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef] [PubMed]
  6. J. Altmann, H. J. P. Reitbock, “A fast correlation method for scale- and translation-invariant pattern recognition,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 46–57 (1984).
    [CrossRef]
  7. M. Hu, “Visual pattern recognition by moment invariants,”IRE Trans. Inf. Theory IT-8, 179–187 (1962).
  8. S. Maitra, “Moment invariants,” Proc. IEEE 67, 697–699 (1979).
    [CrossRef]
  9. D. Casasent, J. Pauly, “Infrared ship classification using a new moment pattern recognition concept,” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126–133 (1981).
  10. S. Reddi, “Radial and angular moment invariants for image identification,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-3, 240–242 (1982).
    [CrossRef]
  11. A. Goshtasby, “Template matching in rotated images,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 338–344 (1985).
    [CrossRef]
  12. Y. S. Abu-Mostafa, D. Psaltis, “Recognition aspects of moment invariants,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
    [CrossRef]
  13. Y. S. Abu-Mostafa, D. Psaltis, “Image normalization by complex moments,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46–55 (1985).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics, 3rd. ed. (Pergamon, Oxford, 1975), App. VII.
  15. J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, London, 1974), Chaps. 3 and 7.

1986 (1)

1985 (3)

R. Wu, H. Stark, “Rotation-invariant pattern recognition using optimum feature extraction,” Appl. Opt. 24, 179–184 (1985).
[CrossRef] [PubMed]

A. Goshtasby, “Template matching in rotated images,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 338–344 (1985).
[CrossRef]

Y. S. Abu-Mostafa, D. Psaltis, “Image normalization by complex moments,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46–55 (1985).
[CrossRef]

1984 (2)

Y. S. Abu-Mostafa, D. Psaltis, “Recognition aspects of moment invariants,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[CrossRef]

J. Altmann, H. J. P. Reitbock, “A fast correlation method for scale- and translation-invariant pattern recognition,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 46–57 (1984).
[CrossRef]

1982 (2)

Y. N. Hsu, H. H. Arsenault, G. April, “Rotation-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4012–4015 (1982).
[CrossRef] [PubMed]

S. Reddi, “Radial and angular moment invariants for image identification,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-3, 240–242 (1982).
[CrossRef]

1981 (1)

D. Casasent, J. Pauly, “Infrared ship classification using a new moment pattern recognition concept,” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126–133 (1981).

1979 (1)

S. Maitra, “Moment invariants,” Proc. IEEE 67, 697–699 (1979).
[CrossRef]

1976 (1)

1962 (1)

M. Hu, “Visual pattern recognition by moment invariants,”IRE Trans. Inf. Theory IT-8, 179–187 (1962).

Abu-Mostafa, Y. S.

Y. S. Abu-Mostafa, D. Psaltis, “Image normalization by complex moments,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46–55 (1985).
[CrossRef]

Y. S. Abu-Mostafa, D. Psaltis, “Recognition aspects of moment invariants,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[CrossRef]

Altmann, J.

J. Altmann, H. J. P. Reitbock, “A fast correlation method for scale- and translation-invariant pattern recognition,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 46–57 (1984).
[CrossRef]

April, G.

Arsenault, H. H.

Born, M.

M. Born, E. Wolf, Principles of Optics, 3rd. ed. (Pergamon, Oxford, 1975), App. VII.

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), Chap. 12.

Casasent, D.

D. Casasent, J. Pauly, “Infrared ship classification using a new moment pattern recognition concept,” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126–133 (1981).

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

Duvernoy, J.

Gonzalez, R. C.

J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, London, 1974), Chaps. 3 and 7.

Goshtasby, A.

A. Goshtasby, “Template matching in rotated images,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 338–344 (1985).
[CrossRef]

Hsu, Y. N.

Hu, M.

M. Hu, “Visual pattern recognition by moment invariants,”IRE Trans. Inf. Theory IT-8, 179–187 (1962).

Maitra, S.

S. Maitra, “Moment invariants,” Proc. IEEE 67, 697–699 (1979).
[CrossRef]

Pauly, J.

D. Casasent, J. Pauly, “Infrared ship classification using a new moment pattern recognition concept,” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126–133 (1981).

Psaltis, D.

Y. S. Abu-Mostafa, D. Psaltis, “Image normalization by complex moments,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46–55 (1985).
[CrossRef]

Y. S. Abu-Mostafa, D. Psaltis, “Recognition aspects of moment invariants,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[CrossRef]

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

Reddi, S.

S. Reddi, “Radial and angular moment invariants for image identification,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-3, 240–242 (1982).
[CrossRef]

Reitbock, H. J. P.

J. Altmann, H. J. P. Reitbock, “A fast correlation method for scale- and translation-invariant pattern recognition,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 46–57 (1984).
[CrossRef]

Sheng, Y.

Stark, H.

Tou, J. T.

J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, London, 1974), Chaps. 3 and 7.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 3rd. ed. (Pergamon, Oxford, 1975), App. VII.

Wu, R.

Appl. Opt. (3)

IEEE Trans. Pattern Anal. Mach. Intell. (5)

J. Altmann, H. J. P. Reitbock, “A fast correlation method for scale- and translation-invariant pattern recognition,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 46–57 (1984).
[CrossRef]

S. Reddi, “Radial and angular moment invariants for image identification,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-3, 240–242 (1982).
[CrossRef]

A. Goshtasby, “Template matching in rotated images,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 338–344 (1985).
[CrossRef]

Y. S. Abu-Mostafa, D. Psaltis, “Recognition aspects of moment invariants,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[CrossRef]

Y. S. Abu-Mostafa, D. Psaltis, “Image normalization by complex moments,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46–55 (1985).
[CrossRef]

IRE Trans. Inf. Theory (1)

M. Hu, “Visual pattern recognition by moment invariants,”IRE Trans. Inf. Theory IT-8, 179–187 (1962).

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

Proc. IEEE (1)

S. Maitra, “Moment invariants,” Proc. IEEE 67, 697–699 (1979).
[CrossRef]

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

D. Casasent, J. Pauly, “Infrared ship classification using a new moment pattern recognition concept,” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 126–133 (1981).

Other (3)

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), Chap. 12.

M. Born, E. Wolf, Principles of Optics, 3rd. ed. (Pergamon, Oxford, 1975), App. VII.

J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, London, 1974), Chaps. 3 and 7.

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Figures (4)

Fig. 1
Fig. 1

Three pattern groups: 1, with prototypes; 2, scaled by the factor 0.77 and with different gray levels; 3, scaled by the factor 0.5 and with different gray levels; 4, rotated by 30°; 5, rotated by 90°.

Fig. 2
Fig. 2

Noisy images En, Fn, and Hn, and deformed images Ed, Fd, and Hd.

Fig. 3
Fig. 3

Pattern recognition in 3-D feature space.

Fig. 4
Fig. 4

Pattern dispersion σ as function of order s.

Tables (8)

Tables Icon

Table 1 Interclass Distances among E, F, and H

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Table 2 Intraclass Distances for the Image Group E, F, and H

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Table 3 Distances from Noisy Images En, Fn and Hn, as well as Deformed Images Ed, Fd, and Hd, to the Prototypes E, F, and H

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Table 4 Contributions ( D 2 ¯ ) m of each 14 Φs,m to Mean-Square Distance D 2 ¯ among the Prototypes

Tables Icon

Table 5 Interclass Distances among E, F, and H Calculated in Three-Dimensional Feature Space

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Table 6 Intraclass Distances for the Image Group E, F, and H Calculated in Three-Dimensional Feature Space

Tables Icon

Table 7 Distances from Noisy Images En, Fn, and Hn, as well as Deformed Images Ed, Fd, and Hd, to the Prototypes E, F, and H, Calculated in Three-Dimensional Feature Space

Tables Icon

Table 8 Misclassification Rate as a Function of s (m = 2, 4, 5)

Equations (14)

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f m ( r ) = 1 2 π 0 2 π f ( r , θ ) exp ( - j m θ ) d θ ,
M s , m = 0 1 r s - 1 f m ( r ) d r ,
Φ s , m = M s , m 2 / m = 0 N M s , m 2 .
M s , m 2 = α 2 s k 2 M s , m 2 .
0 1 Q n ( r ) Q n ( r ) d r = δ n n ,
Q n ( r ) = 1 ( 2 n + 1 ) 1 / 2 s = 0 n ( n + s ) ! ( n - s ) ! s ! r s .
Q 0 ( r ) = 1 Q 1 ( r ) = ( 1 - 2 r ) / 3 , Q 2 ( r ) = ( 1 - 6 r + 6 r 2 ) / 5 , Q 3 ( r ) = ( 1 - 12 r + 30 r 2 - 20 r 3 ) / 7 , . . . .
r s = A r s - 1 + r s - 1 ( r - A )
0 1 A r s - 1 r s - 1 ( r - A ) d r = 0 ,
ρ ( s ) = { 0 1 [ r s - 1 ( r - A ) ] 2 d r 0 1 ( A r s - 1 ) d r } 1 / 2 = [ 1 ( 2 s - 1 ) ( 2 s + 1 ) ] 1 / 2 .
E [ | 0 1 r s - 1 n ( r , θ ) exp ( - j m θ ) d r d θ | 2 ] .
σ = 1 N i = 2 N ( Φ 2 , m ) i - ( Φ 2 , m ) 1 2 ,
Φ s , m = M s , m 2 M s , 0 2 ,
D 2 ¯ = m = 1 N [ ( Φ 2 , m ) 2 ¯ - ( Φ ¯ 2 , m ) 2 ] ,

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