Abstract

A method is presented for designing translation-invariant optical correlation filters that have a specified rotational response for each of several input images. The correlation filter is postulated to have the form of an infinite linear combination of the angular Fourier harmonics of the input images. The corresponding response of the optical system to rotations of the multiple input images is described by a vector–matrix convolution equation. The solution of this equation for the unknown correlation filter is presented in terms of Fourier series. Use of one term in the Fourier series gives the multiple circular-harmonic filter that provides a specified rotationally invariant response for each of the multiple input images. Applications to rotationally invariant discrimination are described, and examples are given.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. F. Schils, D. W. Sweeney, “Rotationally invariant correlation filtering,” J. Opt. Soc. Am. A 2, 1411–1418 (1985).
    [Crossref]
  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]
  3. Y.-N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [Crossref] [PubMed]
  4. H. H. Arsenault, Y.-N. Hsu, K. Chalasinska-Macukow, “Rotation-invariant pattern recognition,” Opt. Eng. 23, 705–709 (1984).
    [Crossref]
  5. A. Vander Lugt, “Signal detection by complex spatial filtering,” IRE Trans. Inf. Theory IT-10, 139–145 (1964).
    [Crossref]
  6. H. H. Arsenault, Y.-N. Hsu, “Rotation-invariant discrimination between almost similar objects,” Appl. Opt. 22, 130–132 (1983).
    [Crossref] [PubMed]
  7. E. G. Paek, S. S. Lee, “Discrimination enhancement in optical pattern recognition by using a modified matched filter,” Can. J. Phys. 57, 1335–1339 (1979).
    [Crossref]
  8. Y.-N. Hsu, H. H. Arsenault, “Pattern discrimination by multiple circular harmonic components,” Appl. Opt. 23, 841–844 (1984).
    [Crossref] [PubMed]
  9. R. Wu, H. Stark, “Rotation-invariant pattern recognition using a vector reference,” Appl. Opt. 23, 838–840 (1984).
    [Crossref]
  10. H. J. Caulfield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
    [Crossref] [PubMed]
  11. B. Braunecker, R. Hauck, A. W. Lohmann, “Optical character recognition based on nonredundant correlation measurements,” Appl. Opt. 18, 2746–2753 (1979).
    [Crossref] [PubMed]
  12. H. J. Caulfield, “Linear combinations of filters for character recognition: a unified treatment,” Appl. Opt. 19, 3877–3878 (1980).
    [Crossref] [PubMed]
  13. C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [Crossref] [PubMed]
  14. C. F. Hester, D. Casasent, “Intra-class infrared (IR) tank pattern recognition using synthetic discriminant functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 292, 25–33 (1981).
  15. C. F. Hester, D. Casasent, “Inter-class discrimination using synthetic discriminant functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108–116 (1981).
  16. D. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [Crossref] [PubMed]
  17. Actually a large number of images exist that give the specified rotational responses at the origin of the correlation plane. However, in general there is only one image that lies in the range spanned by linear combinations of rotations of the input images. This image formed by linear combinations is a desirable solution in the sense that it is the solution with least energy, or least norm.11
  18. If any of the Fourier-series representations in the analysis here does not converge, then it is implicitly assumed to be the Fourier-series representation of a periodic distribution. The use of the term function should be interpreted to mean generalized function.

1985 (1)

1984 (4)

1983 (1)

1982 (2)

1981 (2)

C. F. Hester, D. Casasent, “Intra-class infrared (IR) tank pattern recognition using synthetic discriminant functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 292, 25–33 (1981).

C. F. Hester, D. Casasent, “Inter-class discrimination using synthetic discriminant functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108–116 (1981).

1980 (2)

1979 (2)

B. Braunecker, R. Hauck, A. W. Lohmann, “Optical character recognition based on nonredundant correlation measurements,” Appl. Opt. 18, 2746–2753 (1979).
[Crossref] [PubMed]

E. G. Paek, S. S. Lee, “Discrimination enhancement in optical pattern recognition by using a modified matched filter,” Can. J. Phys. 57, 1335–1339 (1979).
[Crossref]

1969 (1)

1964 (1)

A. Vander Lugt, “Signal detection by complex spatial filtering,” IRE Trans. Inf. Theory IT-10, 139–145 (1964).
[Crossref]

April, G.

Arsenault, H. H.

Braunecker, B.

Casasent, D.

D. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
[Crossref] [PubMed]

C. F. Hester, D. Casasent, “Inter-class discrimination using synthetic discriminant functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108–116 (1981).

C. F. Hester, D. Casasent, “Intra-class infrared (IR) tank pattern recognition using synthetic discriminant functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 292, 25–33 (1981).

C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
[Crossref] [PubMed]

Caulfield, H. J.

Chalasinska-Macukow, K.

H. H. Arsenault, Y.-N. Hsu, K. Chalasinska-Macukow, “Rotation-invariant pattern recognition,” Opt. Eng. 23, 705–709 (1984).
[Crossref]

Hauck, R.

Hester, C. F.

C. F. Hester, D. Casasent, “Intra-class infrared (IR) tank pattern recognition using synthetic discriminant functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 292, 25–33 (1981).

C. F. Hester, D. Casasent, “Inter-class discrimination using synthetic discriminant functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108–116 (1981).

C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
[Crossref] [PubMed]

Hsu, Y.-N.

Lee, S. S.

E. G. Paek, S. S. Lee, “Discrimination enhancement in optical pattern recognition by using a modified matched filter,” Can. J. Phys. 57, 1335–1339 (1979).
[Crossref]

Lohmann, A. W.

Maloney, W. T.

Paek, E. G.

E. G. Paek, S. S. Lee, “Discrimination enhancement in optical pattern recognition by using a modified matched filter,” Can. J. Phys. 57, 1335–1339 (1979).
[Crossref]

Schils, G. F.

Stark, H.

Sweeney, D. W.

Vander Lugt, A.

A. Vander Lugt, “Signal detection by complex spatial filtering,” IRE Trans. Inf. Theory IT-10, 139–145 (1964).
[Crossref]

Wu, R.

Appl. Opt. (10)

Y.-N. Hsu, H. H. Arsenault, “Pattern discrimination by multiple circular harmonic components,” Appl. Opt. 23, 841–844 (1984).
[Crossref] [PubMed]

R. Wu, H. Stark, “Rotation-invariant pattern recognition using a vector reference,” Appl. Opt. 23, 838–840 (1984).
[Crossref]

H. J. Caulfield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2356 (1969).
[Crossref] [PubMed]

B. Braunecker, R. Hauck, A. W. Lohmann, “Optical character recognition based on nonredundant correlation measurements,” Appl. Opt. 18, 2746–2753 (1979).
[Crossref] [PubMed]

H. J. Caulfield, “Linear combinations of filters for character recognition: a unified treatment,” Appl. Opt. 19, 3877–3878 (1980).
[Crossref] [PubMed]

C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
[Crossref] [PubMed]

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]

Y.-N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
[Crossref] [PubMed]

H. H. Arsenault, Y.-N. Hsu, “Rotation-invariant discrimination between almost similar objects,” Appl. Opt. 22, 130–132 (1983).
[Crossref] [PubMed]

D. Casasent, “Unified synthetic discriminant function computational formulation,” Appl. Opt. 23, 1620–1627 (1984).
[Crossref] [PubMed]

Can. J. Phys. (1)

E. G. Paek, S. S. Lee, “Discrimination enhancement in optical pattern recognition by using a modified matched filter,” Can. J. Phys. 57, 1335–1339 (1979).
[Crossref]

IRE Trans. Inf. Theory (1)

A. Vander Lugt, “Signal detection by complex spatial filtering,” IRE Trans. Inf. Theory IT-10, 139–145 (1964).
[Crossref]

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

Opt. Eng. (1)

H. H. Arsenault, Y.-N. Hsu, K. Chalasinska-Macukow, “Rotation-invariant pattern recognition,” Opt. Eng. 23, 705–709 (1984).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

C. F. Hester, D. Casasent, “Intra-class infrared (IR) tank pattern recognition using synthetic discriminant functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 292, 25–33 (1981).

C. F. Hester, D. Casasent, “Inter-class discrimination using synthetic discriminant functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108–116 (1981).

Other (2)

Actually a large number of images exist that give the specified rotational responses at the origin of the correlation plane. However, in general there is only one image that lies in the range spanned by linear combinations of rotations of the input images. This image formed by linear combinations is a desirable solution in the sense that it is the solution with least energy, or least norm.11

If any of the Fourier-series representations in the analysis here does not converge, then it is implicitly assumed to be the Fourier-series representation of a periodic distribution. The use of the term function should be interpreted to mean generalized function.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

(a) Triangle (128 × 128) and square (128 × 128) (superimposed onto a 256 × 256 input image) used as inputs to both a circular-harmonic filter and a multiple circular-harmonic filter. (b) Digitally calculated response of the k = 0 circular-harmonic filter matched to the triangle of (a). Shown is the modulus in the correlation plane. (c) Slices of the intensity distribution in the correlation plane passing through the image locations. Both the triangle and the square produce strong correlation intensities where the filter is the circular-harmonic filter.

Fig. 2
Fig. 2

(a) The k = 0 order multiple circular-harmonic filter g (256 × 256) that is designed so that the triangle and the square produce the intensities 1 and 0, respectively, regardless of image orientation. The darkest shade corresponds to a negative value for the real image g. (b) Plot of the multiple circular-harmonic filter g as a function of the radial distance r. (c) Modulus of the correlation plane image when the input image of Fig. 1(a) is digitally correlated with the k = 0 order multiple circular-harmonic filter shown in (a). (d) Slices of the intensity distribution in the correlation plane passing through the image locations. At the known locations corresponding to the image centroids, the triangle is seen to produce a rotationally invariant intensity of 1, while the square produces 0. These widely separated intensity values allow the triangle to be discriminated from the square.

Fig. 3
Fig. 3

(a) More complex images to distinguish using the multiple circular-harmonic filter. Each subimage is 128 × 128 so that the input image is 256 × 256. (b) The k = 0 order multiple circular-harmonic filter g that is designed using Eqs. (19), (21), and (22) to produce the intensity values 0, 1, and 2 corresponding to the images in the upper right, lower right, and left of (a). The darkest shade corresponds to a negative value for the real image g. (c) Plot of the multiple circular-harmonic filter g as a function of the radial distance r. (d) Modulus of the correlation plane image when the input image (a) is digitally correlated with the k = 0 order multiple circular-harmonic filter shown in (b) and (c). (e) Slices of the correlation plane intensity passing through the image centroids. The desired intensity values are seen to be produced at these locations.

Equations (29)

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

f ( r , θ ) = m = f m ( r ) e i m θ ,
f m ( r ) = 1 2 π 0 2 π f ( r , θ ) e i m θ d θ .
E m = 2 π 0 | f m ( r ) | 2 r d r .
C f g ( x ) = R 2 f ( ξ ) g * ( ξ x ) d ξ , x , ξ R 2 .
C f g ( 0 ) = f , g = 0 0 2 π f ( r , θ ) g * ( r , θ ) d θ r d r .
C f g ( α ) = f ( r , θ + α ) , g ( r , θ ) = m c m e i m α .
c m = 2 π 0 f m ( r ) g m * ( r ) r d r .
C f f ( α ) = f ( r , θ + α ) , f ( r , θ ) = m E m e i m α ,
g ( r , θ ) = 0 2 π [ a 1 * ( γ ) f 1 ( r , θ + γ ) + + a n * ( γ ) f n ( r , θ + γ ) ] d γ .
a i ( γ ) = m a i m e i m γ ,
a i m = 1 2 π 0 2 π a i ( γ ) e i m γ d γ .
g ( r , θ ) = m 2 π [ a 1 m * f 1 m ( r ) + + a n m * f n m ( r ) ] e i m θ .
f i ( r , θ + α ) , 0 2 π [ a 1 * ( γ ) f 1 ( r , θ + γ ) + + a n * ( γ ) f n ( r , θ + γ ) ] d γ = C f i g ( α ) , 0 2 π [ f i ( r , θ + α ) , f 1 ( r , θ + γ ) a 1 ( γ ) + + f i ( r , θ + α ) , f n ( r , θ + γ ) a n ( γ ) ] d γ = C f i g ( α ) , 0 2 π [ C f i f 1 ( α γ ) a 1 ( γ ) + + C f i f n ( α γ ) a n ( γ ) ] d γ = C f i g ( α )
0 2 π [ C f 1 f 1 ( α γ ) C f 1 f n ( α γ ) C f n f 1 ( α γ ) C f n f n ( α γ ) ] × [ a 1 ( γ ) a n ( γ ) ] d γ = [ C f 1 g ( α ) C f n g ( α ) ] .
0 2 π C ff ( α γ ) a ( γ ) d γ = C f g ( α ) .
C ff ( α ) = m E m e i m α , a ( α ) = m a m e i m α , C f g ( α ) = m c m e i m α
m 2 π E m a m e i m α = m c m e i m α .
2 π E m a m = c m
C f g ( α ) = [ c 1 e i ϕ 1 ( α ) c n e i ϕ n ( α ) ] .
c m = c k δ m k = [ c 1 c n ] δ m k .
a m = 1 2 π E k 1 c k δ m k .
g ( r , θ ) = 2 π [ a 1 k * f 1 k ( r ) + + a n k * f n k ( r ) ] e i k θ ,
E m = [ 2 π 0 | f 1 m ( r ) | 2 r d r 2 π 0 f 1 m ( r ) f n m * ( r ) r d r 2 π 0 f n m ( r ) f 1 m * ( r ) r d r 2 π 0 | f n m ( r ) | 2 r d r ] = 2 π 0 [ f 1 m ( r ) f n m ( r ) ] [ f 1 m ( r ) f n m ( r ) ] * T r d r ,
E m = 2 π [ 0 | f 1 m ( r ) | 2 r d r 0 f 1 m ( r ) f 2 m * ( r ) r d r 0 f 2 m ( r ) f 1 m * ( r ) r d r 0 | f 2 m ( r ) | 2 r d r ] = [ E 1 m c 12 m c 12 m * E 2 m ] .
ρ m 2 = | c 12 m | 2 E 1 m E 2 m
g ( r , θ ) = 1 ( 1 ρ k 2 ) E 1 k [ f 1 k ( r ) c 12 k E 2 k f 2 k ( r ) ] e i k θ .
ρ 2 = | R 2 f 2 ( x ) f 1 * ( x ) d x | 2 R 2 | f 1 ( x ) | 2 d x R 2 | f 2 ( x ) | 2 d x .
ρ m 2 = | 2 π 0 f 2 m ( r ) f 1 m * ( r ) r d r | 2 2 π 0 | f 1 m ( r ) | 2 r d r 2 π 0 | f 2 m ( r ) | 2 r d r ,
ρ m 2 = | c 21 m | 2 E 1 m E 2 m .

Metrics