Abstract

A Fourier-domain-based recognition technique is proposed for periodic and quasiperiodic pattern recognition. It is based on the angular correlation of the moduli of the sample and the reference Fourier spectra centered at the maximum central point. As in other correlation techniques, recognition is achieved when a high correlation peak is obtained, and this result occurs when the two spectra coincide. The angular correlation is a one-dimensional function of the rotation angle. The position of the correlation peak indicates the rotation angle between two similar patterns in the original images. Some optimizations for the discrete calculation of the Fourier-domain-based angular correlation are also proposed. Some applications of this technique to web inspection tasks, such as pattern recognition and classification, damaged web evaluation, and detection of defects, are presented and discussed.

© 1996 Optical Society of America

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References

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  1. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), p. 112.
  2. J. B. DeVelis, G. B. Parrent, G. O. Reynolds, B. J. Thompson, eds., The New Physical Optics Notebook: Tutorials in Fourier Optics Vol. PM01 of SPIE Press Monographs Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989), Chap. 3.
  3. E. G. Steward, “Fourier Optics”: An Introduction (Ellis Horwood, New York, 1983), p. 95.
  4. T. Matsuyama, S.-I. Miura, M. Nagao, “Structural analysis of natural textures by Fourier transformation,” Comput. Vision, Graphics, Image Process. 24, 347–362 (1983).
    [CrossRef]
  5. Ref. 3, pp. 96–101.
  6. R. Furter, “Evenness testing in yarn production,” in Manual of Textile Technology, Part I, (Textile Institute, Manchester, England, 1982), Chap. 4.
  7. E. J. Wood, “Applying Fourier and associated transforms to pattern characterization in textiles,” Text. Res. J. 60, 212–220 (1990).
    [CrossRef]
  8. N. George, J. T. Thomasson, A. Spindel, “Photodetector light-pattern detector,” U.S. patent3,689,772 (5September1972).
  9. Y. Wu, B. Pourdeyhimi, S. M. Spivak, “Texture evaluation of carpets using image analysis,” Text. Res. J. 61, 407–419 (1991).
    [CrossRef]
  10. Ref. 1, pp. 297–298.
  11. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973), Chap. 2.

1991 (1)

Y. Wu, B. Pourdeyhimi, S. M. Spivak, “Texture evaluation of carpets using image analysis,” Text. Res. J. 61, 407–419 (1991).
[CrossRef]

1990 (1)

E. J. Wood, “Applying Fourier and associated transforms to pattern characterization in textiles,” Text. Res. J. 60, 212–220 (1990).
[CrossRef]

1983 (1)

T. Matsuyama, S.-I. Miura, M. Nagao, “Structural analysis of natural textures by Fourier transformation,” Comput. Vision, Graphics, Image Process. 24, 347–362 (1983).
[CrossRef]

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973), Chap. 2.

Furter, R.

R. Furter, “Evenness testing in yarn production,” in Manual of Textile Technology, Part I, (Textile Institute, Manchester, England, 1982), Chap. 4.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), p. 112.

George, N.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light-pattern detector,” U.S. patent3,689,772 (5September1972).

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973), Chap. 2.

Matsuyama, T.

T. Matsuyama, S.-I. Miura, M. Nagao, “Structural analysis of natural textures by Fourier transformation,” Comput. Vision, Graphics, Image Process. 24, 347–362 (1983).
[CrossRef]

Miura, S.-I.

T. Matsuyama, S.-I. Miura, M. Nagao, “Structural analysis of natural textures by Fourier transformation,” Comput. Vision, Graphics, Image Process. 24, 347–362 (1983).
[CrossRef]

Nagao, M.

T. Matsuyama, S.-I. Miura, M. Nagao, “Structural analysis of natural textures by Fourier transformation,” Comput. Vision, Graphics, Image Process. 24, 347–362 (1983).
[CrossRef]

Pourdeyhimi, B.

Y. Wu, B. Pourdeyhimi, S. M. Spivak, “Texture evaluation of carpets using image analysis,” Text. Res. J. 61, 407–419 (1991).
[CrossRef]

Spindel, A.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light-pattern detector,” U.S. patent3,689,772 (5September1972).

Spivak, S. M.

Y. Wu, B. Pourdeyhimi, S. M. Spivak, “Texture evaluation of carpets using image analysis,” Text. Res. J. 61, 407–419 (1991).
[CrossRef]

Steward, E. G.

E. G. Steward, “Fourier Optics”: An Introduction (Ellis Horwood, New York, 1983), p. 95.

Thomasson, J. T.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light-pattern detector,” U.S. patent3,689,772 (5September1972).

Wood, E. J.

E. J. Wood, “Applying Fourier and associated transforms to pattern characterization in textiles,” Text. Res. J. 60, 212–220 (1990).
[CrossRef]

Wu, Y.

Y. Wu, B. Pourdeyhimi, S. M. Spivak, “Texture evaluation of carpets using image analysis,” Text. Res. J. 61, 407–419 (1991).
[CrossRef]

Comput. Vision, Graphics, Image Process. (1)

T. Matsuyama, S.-I. Miura, M. Nagao, “Structural analysis of natural textures by Fourier transformation,” Comput. Vision, Graphics, Image Process. 24, 347–362 (1983).
[CrossRef]

Text. Res. J. (2)

E. J. Wood, “Applying Fourier and associated transforms to pattern characterization in textiles,” Text. Res. J. 60, 212–220 (1990).
[CrossRef]

Y. Wu, B. Pourdeyhimi, S. M. Spivak, “Texture evaluation of carpets using image analysis,” Text. Res. J. 61, 407–419 (1991).
[CrossRef]

Other (8)

Ref. 1, pp. 297–298.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973), Chap. 2.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light-pattern detector,” U.S. patent3,689,772 (5September1972).

Ref. 3, pp. 96–101.

R. Furter, “Evenness testing in yarn production,” in Manual of Textile Technology, Part I, (Textile Institute, Manchester, England, 1982), Chap. 4.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), p. 112.

J. B. DeVelis, G. B. Parrent, G. O. Reynolds, B. J. Thompson, eds., The New Physical Optics Notebook: Tutorials in Fourier Optics Vol. PM01 of SPIE Press Monographs Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989), Chap. 3.

E. G. Steward, “Fourier Optics”: An Introduction (Ellis Horwood, New York, 1983), p. 95.

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Figures (9)

Fig. 1
Fig. 1

Transformation of the modulus of a Fourier-spectrum matrix, R or S, from rectangular (u, υ) into polar (r, θ) coordinates.

Fig. 2
Fig. 2

Plain webs: (a) reference r(x, y) with warp in the vertical direction, (b) sample s(x, y).

Fig. 3
Fig. 3

Fourier spectra of the plain webs (Fig. 2): (a) |R(u, υ)|, (b) |S(u, υ)|.

Fig. 4
Fig. 4

When the pixels with low values (In the interval [0, 25]) in the web spectra of Fig. 3 are taken and summed up along radial directions, two maxima appear that correspond to warp and weft directions: (a) for the reference web, (b) for the sample web.

Fig. 5
Fig. 5

Angular correlation of web spectra: C 1(α) S(olid curve) corresponds to the webs in Fig. 2; in the case of C 2(α) (dashed curve), the sample web is simply the reference scaled and rotated.

Fig. 6
Fig. 6

Class-conditional density functions based on the use of the maximum of the angular correlation as a feature for the classification.

Fig. 7
Fig. 7

(a) Cotton web: original (upper part) and after washing and ironing (bottom part), (b) Fourier-domain-based angular correlations: autocorrelation for the original (reference) and the washed and ironed (sample) cotton webs, (c) angular correlation obtained by the rescaling of the FT's of the original and altered cotton webs.

Fig. 8
Fig. 8

(a) Original web (upper part) and after abrasion (bottom part), (b) angular correlation between the original and the abraded web spectra. The curve is normalized to the angular autocorrelation value of the original web.

Fig. 9
Fig. 9

(a) Knitted web: original (upper part) and defective, knitted with a thinner yarn (bottom part), (b) angular autocorrelation of the reference knitted web spectrum (dashed curve) and angular correlation between the original and the defective web spectra (solid curve), (c) angular correlation obtained by the scaling of the FT's of the original and the defective knitted webs.

Equations (6)

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C ( α ) = 0 0 π | R ( r , θ ) | | S ( r , θ α ) | d r d θ ,
C ( α ) = C ( n π M ) = 1 ( i j R i j 2 i j S i j 2 ) 1 / 2 × ( i = 1 , j = n L , M | R i , j | | S i , j n + 1 | + i = 1 , j = 1 L , n 1 | R i , j | × | S i , M n + 1 + j | ) , 1 n M .
decide w 1 = plain if p ( V w 1 ) p ( V w 2 ) > λ 12 λ 22 P ( w 2 ) λ 21 λ 11 P ( w 1 ) ,
λ i j = { 0 , i = j 1 , i j ,
decide w 1 = plain if p ( V w 1 ) > p ( V w 2 ) .
g i ( V ) = p ( V / w i ) ,

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