Abstract

Moment invariants of the Fourier transform of an image are introduced. It is found that a feature set composed of moment invariants from both the space domain and the Fourier domain gives better performance for a wide range of classification tasks than does the same number of moment invariants from either domain alone. Redundancy among moments of the two domains is examined by using the correlation coefficient between the feature kernels as a measure. Examples are used to compare the feature sets and to assess their performance in classification tasks. Moment invariants of the magnitude of the Fourier transform and, by inference, some popular features, such as the spectral ring–wedge detector, are found to fall far short in performance compared with those in which the phase of the Fourier transform is also utilized. Coherent optical systems to compute the dual-domain moment invariants are proposed.

© 1988 Optical Society of America

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References

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  1. D. Casasent, L. Cheatham, D. Fetterly, “Optical system to compute intensity moments: design,” Appl. Opt. 21, 3292–3298 (1982).
    [Crossref] [PubMed]
  2. Y. S. Abu-Mostafa, D. Psaltis, “Recognitive aspects of moment invariants,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
    [Crossref]
  3. Y. S. Abu-Mostafa, D. Psaltis, “Image normalization by complex moments,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46–55 (1985).
    [Crossref]
  4. H. Kasdan, “Industrial applications of diffraction pattern sampling,” Opt. Eng. 18, 496–503 (1979).
    [Crossref]
  5. J. Duvernoy, “Statistical signatures of scale and amplitude variations in Fourier spectra,” Opt. Commun. 42, 386–390 (1982).
    [Crossref]
  6. T. Minemoto, I. Tsuchimoto, S. Imi, “Hybrid pattern recognition using the Fraunhofer diffraction pattern,” Opt. Commun. 51, 221–226 (1984).
    [Crossref]
  7. T. Minemoto, J. Narano, “Hybrid pattern recognition by features extracted from object patterns and Fraunhofer diffraction patterns,” Appl. Opt. 24, 2914–2920 (1985).
    [Crossref] [PubMed]
  8. M. R. Teague, “Optical calculation of irradiance moments,” Appl. Opt. 19, 1353–1356 (1980).
    [Crossref] [PubMed]
  9. B. V. K. Vijaya Kumar, C. A. Rahenkamp, “Calculation of geometric moments using Fourier plane intensities,” Appl. Opt. 25, 997–1007 (1986):
    [Crossref]
  10. G. R. Gindi, A. F. Gmitro, “Optical feature extraction via the Radon transform,” Opt. Eng. 23, 499–506 (1984).
    [Crossref]
  11. B. V. K. Vijaya Kumar, “Geometric moments computed from the Hartley transform,” Opt. Eng. 25, 1327–1356 (1986).
  12. M. K. Hu, “Visual pattern recognition by moment invariants,”IRE Trans. Inf. Theory IT-8, 179–187 (1962).
  13. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 757.
  14. T. Y. Young, T. W. Calvert, Classification, Estimation and Pattern Recognition (Elsevier, New York, 1974), p. 244.
  15. J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, Reading, Mass., 1974), p. 293.
  16. M. O. Freeman, B. E. A. Saleh, “Optical location of centroids of nonoverlapping objects,” Appl. Opt. 26, 2752–2759 (1987).
    [Crossref] [PubMed]
  17. B. E. A. Saleh, M. O. Freeman, “Optical transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987), pp. 281–333.
  18. J. R. Leger, S. H. Lee, “Hybrid optical processor for pattern recognition and classification using a generalized set of pattern functions,” Appl. Opt. 21, 274–287 (1982).
    [Crossref] [PubMed]
  19. M. R. Teague, “Image analysis via the general theory of moments,”J. Opt. Soc. Am. 70, 920–930 (1980).
    [Crossref]

1987 (1)

1986 (2)

B. V. K. Vijaya Kumar, C. A. Rahenkamp, “Calculation of geometric moments using Fourier plane intensities,” Appl. Opt. 25, 997–1007 (1986):
[Crossref]

B. V. K. Vijaya Kumar, “Geometric moments computed from the Hartley transform,” Opt. Eng. 25, 1327–1356 (1986).

1985 (2)

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

T. Minemoto, J. Narano, “Hybrid pattern recognition by features extracted from object patterns and Fraunhofer diffraction patterns,” Appl. Opt. 24, 2914–2920 (1985).
[Crossref] [PubMed]

1984 (3)

T. Minemoto, I. Tsuchimoto, S. Imi, “Hybrid pattern recognition using the Fraunhofer diffraction pattern,” Opt. Commun. 51, 221–226 (1984).
[Crossref]

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

G. R. Gindi, A. F. Gmitro, “Optical feature extraction via the Radon transform,” Opt. Eng. 23, 499–506 (1984).
[Crossref]

1982 (3)

1980 (2)

1979 (1)

H. Kasdan, “Industrial applications of diffraction pattern sampling,” Opt. Eng. 18, 496–503 (1979).
[Crossref]

1962 (1)

M. K. 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, “Recognitive aspects of moment invariants,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 757.

Calvert, T. W.

T. Y. Young, T. W. Calvert, Classification, Estimation and Pattern Recognition (Elsevier, New York, 1974), p. 244.

Casasent, D.

Cheatham, L.

Duvernoy, J.

J. Duvernoy, “Statistical signatures of scale and amplitude variations in Fourier spectra,” Opt. Commun. 42, 386–390 (1982).
[Crossref]

Fetterly, D.

Freeman, M. O.

M. O. Freeman, B. E. A. Saleh, “Optical location of centroids of nonoverlapping objects,” Appl. Opt. 26, 2752–2759 (1987).
[Crossref] [PubMed]

B. E. A. Saleh, M. O. Freeman, “Optical transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987), pp. 281–333.

Gindi, G. R.

G. R. Gindi, A. F. Gmitro, “Optical feature extraction via the Radon transform,” Opt. Eng. 23, 499–506 (1984).
[Crossref]

Gmitro, A. F.

G. R. Gindi, A. F. Gmitro, “Optical feature extraction via the Radon transform,” Opt. Eng. 23, 499–506 (1984).
[Crossref]

Gonzalez, R. C.

J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, Reading, Mass., 1974), p. 293.

Hu, M. K.

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

Imi, S.

T. Minemoto, I. Tsuchimoto, S. Imi, “Hybrid pattern recognition using the Fraunhofer diffraction pattern,” Opt. Commun. 51, 221–226 (1984).
[Crossref]

Kasdan, H.

H. Kasdan, “Industrial applications of diffraction pattern sampling,” Opt. Eng. 18, 496–503 (1979).
[Crossref]

Lee, S. H.

Leger, J. R.

Minemoto, T.

T. Minemoto, J. Narano, “Hybrid pattern recognition by features extracted from object patterns and Fraunhofer diffraction patterns,” Appl. Opt. 24, 2914–2920 (1985).
[Crossref] [PubMed]

T. Minemoto, I. Tsuchimoto, S. Imi, “Hybrid pattern recognition using the Fraunhofer diffraction pattern,” Opt. Commun. 51, 221–226 (1984).
[Crossref]

Narano, J.

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, “Recognitive aspects of moment invariants,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[Crossref]

Rahenkamp, C. A.

Saleh, B. E. A.

M. O. Freeman, B. E. A. Saleh, “Optical location of centroids of nonoverlapping objects,” Appl. Opt. 26, 2752–2759 (1987).
[Crossref] [PubMed]

B. E. A. Saleh, M. O. Freeman, “Optical transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987), pp. 281–333.

Teague, M. R.

Tou, J. T.

J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, Reading, Mass., 1974), p. 293.

Tsuchimoto, I.

T. Minemoto, I. Tsuchimoto, S. Imi, “Hybrid pattern recognition using the Fraunhofer diffraction pattern,” Opt. Commun. 51, 221–226 (1984).
[Crossref]

Vijaya Kumar, B. V. K.

B. V. K. Vijaya Kumar, “Geometric moments computed from the Hartley transform,” Opt. Eng. 25, 1327–1356 (1986).

B. V. K. Vijaya Kumar, C. A. Rahenkamp, “Calculation of geometric moments using Fourier plane intensities,” Appl. Opt. 25, 997–1007 (1986):
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 757.

Young, T. Y.

T. Y. Young, T. W. Calvert, Classification, Estimation and Pattern Recognition (Elsevier, New York, 1974), p. 244.

Appl. Opt. (6)

IEEE Trans. Pattern Anal. Mach. Intell. (2)

Y. S. Abu-Mostafa, D. Psaltis, “Recognitive 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. K. Hu, “Visual pattern recognition by moment invariants,”IRE Trans. Inf. Theory IT-8, 179–187 (1962).

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

J. Duvernoy, “Statistical signatures of scale and amplitude variations in Fourier spectra,” Opt. Commun. 42, 386–390 (1982).
[Crossref]

T. Minemoto, I. Tsuchimoto, S. Imi, “Hybrid pattern recognition using the Fraunhofer diffraction pattern,” Opt. Commun. 51, 221–226 (1984).
[Crossref]

Opt. Eng. (3)

H. Kasdan, “Industrial applications of diffraction pattern sampling,” Opt. Eng. 18, 496–503 (1979).
[Crossref]

G. R. Gindi, A. F. Gmitro, “Optical feature extraction via the Radon transform,” Opt. Eng. 23, 499–506 (1984).
[Crossref]

B. V. K. Vijaya Kumar, “Geometric moments computed from the Hartley transform,” Opt. Eng. 25, 1327–1356 (1986).

Other (4)

B. E. A. Saleh, M. O. Freeman, “Optical transformations,” in Optical Signal Processing, J. L. Horner, ed. (Academic, New York, 1987), pp. 281–333.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), p. 757.

T. Y. Young, T. W. Calvert, Classification, Estimation and Pattern Recognition (Elsevier, New York, 1974), p. 244.

J. T. Tou, R. C. Gonzalez, Pattern Recognition Principles (Addison-Wesley, Reading, Mass., 1974), p. 293.

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Figures (5)

Fig. 1
Fig. 1

Correlation coefficients between complex moments. (a) Within one domain any given complex moment kernel indexed by (p, q) is correlated with all other complex moment kernels indexed by (s, t) such that (pq) = (st). (b) Correlation coefficient between adjacent complex-moment kernels within one domain as a function of the order of the moment. (c) Correlation between a moment kernel in the space domain and a moment kernel in the Fourier domain; this is limited to moments that have the same indices. (d) Upper bound on the magnitude of the correlation coefficient between complex moment kernels in the space and Fourier domains with matched indices as a function of the order of the moment.

Fig. 2
Fig. 2

Sample images from the four classes tested.

Fig. 3
Fig. 3

Two complex quantities, C1 and C2, can be determined in magnitude and relative phase difference by optically creating the quantities C1, C2, C1 + C2, and C1 + C2* and measuring their magnitudes. Only four possible solutions, each with the same phase difference, exist for C1 and C2.

Fig. 4
Fig. 4

An optical system to compute a set of complex moments in both the space domain and the Fourier domain simultaneously on orthogonal polarizations, Fourier-domain moments along the path passing directly through the beam splitters, and space-domain moments along the path that is reflected at the beam splitters. The polarizations are arranged so that no light is lost at the polarizing beam splitter. Hologram CGH1 multiplies the appropriate field by a random phase, while the kernels for the moment computations are encoded along with the random phase on hologram CGH2.

Fig. 5
Fig. 5

A three-channel system whereby a small number of Fourier-domain moments can be computed without requiring critical positioning of the input. The lowest channel is a random-phase-coded processor for features that do not require input alignment, e.g., space-domain moments. The middle channel correlates the input with Fourier-domain complex-moment kernels. The top channel locates the centroid of the input, thereby determining the correct location at which to sample the middle channel to obtain the Fourier-domain complex moments. CGH, Computer-generated hologram.

Tables (2)

Tables Icon

Table 1 Set of Moment Invariants through Degree 5a

Tables Icon

Table 2 Divergences for Various Feature Sets and Class Combinationsa

Equations (48)

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m ( p , q ) = x p y q f ( x , y ) d x d y , 0 , 1 , 2 , .
C s ( p , q ) = ( x + i y ) p ( x i y ) q f ( x , y ) d x d y , p , q = 0 , 1 , 2 , ,
f ( x , y ) = n = c n ( r ) e i n θ ,
C s ( p , q ) = 2 π 0 r p + q + 1 C q p ( r ) d r .
C 0 ( p , q ) = m = 0 p n = 0 q ( p m ) ( q n ) ( 1 ) m + n [ C ( 1 , 0 ) C ( 0 , 0 ) ] m × [ C ( 1 , 0 ) C ( 0 , 0 ) ] n C ( p m , q n ) .
C N ( p , q ) = C 0 ( 0 , 0 ) ( p + q 2 ) / 2 C 0 ( 1 , 1 ) ( p + q ) / 2 C 0 ( p , q ) .
[ C N ( p , q ) C N ( q , p ) ] [ = | C N ( p , q ) | 2 ]
{ C N ( r , s ) [ C N ( t , u ) ] k } + { C N ( s , r ) [ C N ( u , t ) k ] } , where ( r s ) + k ( t u ) = 0 [ = 2 | C N ( r , s ) | | C N ( u , t ) | k cos ( θ r s k θ u t ) ] .
L j = f ( x , y ) g j ( x , y ) d x d y ,
ξ j k = g j ( x , y ) g k * ( x , y ) d x d y [ | g j ( x , y ) | 2 d x d y ] 1 / 2 [ | g k ( x , y ) | 2 d x d y ] 1 / 2 .
ξ s s ( p , q , p + 1 , q + 1 ) = 2 π 0 1 r p + q r p + q + 2 r d r [ 2 π 0 1 | r p + q | 2 r d r ] 1 / 2 [ 2 π 0 1 | r p + q + 2 | 2 r d r ] 1 / 2 = [ 1 1 ( p + q + 2 ) 2 ] 1 / 2 .
C F ( p , q ) = ( u + i υ ) p ( u i υ ) q F ( u , υ ) d u d υ ,
F ( u , υ ) = n = b n ( ρ ) e i n ϕ ,
C F ( p , q ) = 2 π 0 ρ p + q + 1 b q p ( ρ ) d ρ .
g ( p , q , x , y ) = x p y q ,
G ( s , t , x , y ) = u s υ t exp [ i 2 π ( u x + υ y ) ] d u d υ
R m M ( p , q , s , t ) = g ( p , q , x , y ) G ( s , t , x , y ) d x d y = x p y q u s υ t × exp [ i 2 π ( u x + υ y ) ] d u d υ d x d y .
x n f ( x ) i n d n F ( u ) d u n ,
R m M ( p , q , s , t ) = i s + t x p y q d s δ ( x ) d x s d t δ ( y ) d y t d x d y .
f ( x ) d n δ ( x ) d x n d x = ( 1 ) n d n f ( x ) d x n | x = 0 ,
R m M ( p , q , s , t ) = ( i ) p + q p ! q ! δ p s δ q t ,
C s ( p , q ) = r s α r s m ( r , s ; r + s = p + q ) ,
C F ( a , b ) = d e α d e M ( d , e ; d + e = a + b ) ,
a = p + n , b = q + n ,
ξ s s ( p , q , p + 1 , q + 1 ) = [ 1 1 ( p + q + 2 ) 2 ] 1 / 2 .
ξ s F 2 ( p , q , p , q , ) 1 max ξ s s 2 ( p , q , p + n , q + n ) 1 ( p + q + 2 ) 2 .
| ξ s F ( p , q , s , t ) | 1 p + q + 2 δ p s δ q t .
F ( u , υ ) = n = b n ( ρ ) e i n ϕ
| F ( u , υ ) | 2 = n = γ 2 n ( ρ ) exp ( i 2 n ϕ ) ,
γ 2 n ( ρ ) = n = b m * ( ρ ) b m + 2 n ( ρ ) .
C ( p , q ) = m = 63 64 m = 63 64 ( m + i n ) p ( m i n ) q f ( m , n ) ,
J = i K i K E { L i j ( x ) | x class i } ,
L i j ( x ) = ln [ p i ( x ) p j ( x ) ]
J = i K j K l F 1 2 σ j l 2 [ σ i l 2 + σ j l 2 + ( m i l m j l ) 2 ] ,
C ( p , q ) = C * ( q , p ) for a real function
= C * ( q , p ) for a complex-conjucate symmetric function if q p is even
= C * ( q , p ) for a complex-conjucate symmetric function if q p is odd .
M = FT [ f ( x , y ) g ( x , y ) ] | u = υ = 0 ,
M = [ f ( x , y ) g ( x , y ) ] | x = y = 0 ,
T ( u , υ ) = FT 1 { p + q 5 α p q ( x i y ) p ( x + i y ) q × exp [ i θ r ( x Δ p , y Δ q ) ] + r , s p , q α p q r s [ ( x i y ) p ( x + i y ) q + ( x i y ) r ( x + i y ) s ] × exp [ i θ r ( x Δ p , r , y Δ q , s ) ] } ,
α p q | C s ( p , q ) | α p q [ | ( x + i y ) p ( x i y ) q | 2 d x d y ] 1 / 2 × [ | f ( x , y ) | 2 d x d y ] 1 / 2
α r s | C F ( r , s ) | α r s [ | ( u + i υ ) r ( u i υ ) s | 2 d u d υ ] 1 / 2 × [ | F ( u , υ ) | 2 d u d υ ] 1 / 2 ,
[ | f ( x , y ) | 2 d x d y ] 1 / 2 = [ | F ( u , υ ) | 2 d u d υ ] 1 / 2 .
α p q = [ | ( x + i y ) p ( x i y ) q | 2 d x d y ] 1 / 2
α p q r s = [ | ( x + i y ) p ( x i y ) q ( x + i y ) r ( x i y ) s | 2 d x d y ] 1 / 2 ,
T ( u , υ ) = α p q ( u + i υ ) p ( u i υ ) q × exp [ i 2 π ( u Δ x , p + υ Δ y , q ) ] ,
o ( x , y ) = FT { FT [ f ( x x 0 , y y 0 ) T ( u , υ ) ] } = F ( u , υ ) exp [ j 2 π ( u x 0 + υ y 0 ) ] T ( u , υ ) × exp [ j 2 π ( u x + υ y ) ] d u d υ .
C F ( p , q ) = o ( x , y ) | x = x 0 Δ x , p ; y = y 0 Δ y , q .

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