Abstract

Spatial filtering is shown to apply not only to the identification of deterministic signals but also to the classification of images. The spectral content of images is put into a classifier that extracts the dominant eigenvectors responsible for statistical features. Principal images that carry most of the information are obtained by using optical representations of eigenvectors as spatial filters. The statistical stability and the intrinsic dimensionality of Fourier spectra are related to the fast estimation of useful eigenvectors.

© 1978 Optical Society of America

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References

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  1. B. J. Turner, “Cluster Analysis of Multispectral Scanner Remote Sensor Data,” in Remote Sensing of Earth Resources, Vol. 1, F. Shahrokhi, Ed., University of Tennessee, Tullahoma (1972).
  2. G. E. Lowitz, Automatisme 21 (314), 83 (1976).
  3. J. W. Goodman, H. Kato, Opt. Commun. 8, 378 (1973).
    [CrossRef]
  4. D. Casasent, D. Psaltis, Appl. Opt. 15, 1795 (1976).
    [CrossRef] [PubMed]
  5. J. Duvernoy, Appl. Opt. 15, 1584 (1976).
    [CrossRef] [PubMed]
  6. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), Chap. 10, pp. 288–301.
  7. B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).
    [CrossRef] [PubMed]
  8. S. Watanabe, “Karhunen-Loève Expansion and Factor Analysis,” in Proceedings 4th Prague Conference on Information Theory (1965), pp. 635–660, in Pattern Recognition: Introduction and Foundations, ed. J. Sklansky (Dowden, Hutchinson, Ross Inc., Stroudsburg, Pennsylvania, 1973).
  9. J. A. McLaughlin, J. Raviv, Inf. Control 12, 121 (1969).
    [CrossRef]
  10. Ref. 6, Chap. 8, pp. 249–250.
  11. S. Watanabe, Knowing and Guessing (Wiley, New York, 1969), Sec. 7.6.
  12. B. R. Frieden, J. Opt. Soc. Am. 57, 1013 (1967).
    [CrossRef]
  13. J. Duvernoy, D. Charraut, P. Y. Baures, Opt. Acta 24, 795 (1977).
    [CrossRef]

1977 (1)

J. Duvernoy, D. Charraut, P. Y. Baures, Opt. Acta 24, 795 (1977).
[CrossRef]

1976 (3)

1973 (1)

J. W. Goodman, H. Kato, Opt. Commun. 8, 378 (1973).
[CrossRef]

1972 (1)

1969 (1)

J. A. McLaughlin, J. Raviv, Inf. Control 12, 121 (1969).
[CrossRef]

1967 (1)

Baures, P. Y.

J. Duvernoy, D. Charraut, P. Y. Baures, Opt. Acta 24, 795 (1977).
[CrossRef]

Casasent, D.

Charraut, D.

J. Duvernoy, D. Charraut, P. Y. Baures, Opt. Acta 24, 795 (1977).
[CrossRef]

Duvernoy, J.

J. Duvernoy, D. Charraut, P. Y. Baures, Opt. Acta 24, 795 (1977).
[CrossRef]

J. Duvernoy, Appl. Opt. 15, 1584 (1976).
[CrossRef] [PubMed]

Frieden, B. R.

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), Chap. 10, pp. 288–301.

Goodman, J. W.

J. W. Goodman, H. Kato, Opt. Commun. 8, 378 (1973).
[CrossRef]

Kato, H.

J. W. Goodman, H. Kato, Opt. Commun. 8, 378 (1973).
[CrossRef]

Lowitz, G. E.

G. E. Lowitz, Automatisme 21 (314), 83 (1976).

McLaughlin, J. A.

J. A. McLaughlin, J. Raviv, Inf. Control 12, 121 (1969).
[CrossRef]

Psaltis, D.

Raviv, J.

J. A. McLaughlin, J. Raviv, Inf. Control 12, 121 (1969).
[CrossRef]

Turner, B. J.

B. J. Turner, “Cluster Analysis of Multispectral Scanner Remote Sensor Data,” in Remote Sensing of Earth Resources, Vol. 1, F. Shahrokhi, Ed., University of Tennessee, Tullahoma (1972).

Watanabe, S.

S. Watanabe, Knowing and Guessing (Wiley, New York, 1969), Sec. 7.6.

S. Watanabe, “Karhunen-Loève Expansion and Factor Analysis,” in Proceedings 4th Prague Conference on Information Theory (1965), pp. 635–660, in Pattern Recognition: Introduction and Foundations, ed. J. Sklansky (Dowden, Hutchinson, Ross Inc., Stroudsburg, Pennsylvania, 1973).

Appl. Opt. (2)

Automatisme (1)

G. E. Lowitz, Automatisme 21 (314), 83 (1976).

Inf. Control (1)

J. A. McLaughlin, J. Raviv, Inf. Control 12, 121 (1969).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Acta (1)

J. Duvernoy, D. Charraut, P. Y. Baures, Opt. Acta 24, 795 (1977).
[CrossRef]

Opt. Commun. (1)

J. W. Goodman, H. Kato, Opt. Commun. 8, 378 (1973).
[CrossRef]

Other (5)

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972), Chap. 10, pp. 288–301.

S. Watanabe, “Karhunen-Loève Expansion and Factor Analysis,” in Proceedings 4th Prague Conference on Information Theory (1965), pp. 635–660, in Pattern Recognition: Introduction and Foundations, ed. J. Sklansky (Dowden, Hutchinson, Ross Inc., Stroudsburg, Pennsylvania, 1973).

Ref. 6, Chap. 8, pp. 249–250.

S. Watanabe, Knowing and Guessing (Wiley, New York, 1969), Sec. 7.6.

B. J. Turner, “Cluster Analysis of Multispectral Scanner Remote Sensor Data,” in Remote Sensing of Earth Resources, Vol. 1, F. Shahrokhi, Ed., University of Tennessee, Tullahoma (1972).

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Figures (11)

Fig. 1
Fig. 1

Typical remains of a Roman cadastration (Yugoslavia).

Fig. 2
Fig. 2

Direction basis and spatial frequency basis used for the statistical analysis of Fourier spectra.

Fig. 3
Fig. 3

Filtering of a Roman cadastration along the direction with (and without) selection of a characteristic frequency that includes noise.

Fig. 4
Fig. 4

Statistical stability of the Fourier spectrum considered as the super imposition of different realizations of the same sampling (1 sfu = 1/710 m−1).

Fig. 5
Fig. 5

Statistical stability of the eigenvalues corresponding to a sampling of 2 sfu.

Fig. 6
Fig. 6

First twenty-three eigenvalues of the 44 × 44 covariance matrix and their estimates from the 36 × 36 reduced covariance matrix. The rotation of the corresponding eigenvectors from their actual values to their estimated values is indicated.

Fig. 7
Fig. 7

Statistical stability of the estimates of the first six eigenvalues, obtained from 6 × 6 covariance matrices.

Fig. 8
Fig. 8

Projection of the Fourier spectrum of an aerial photograph, considered as a set of thirty-six directions in the spatial frequency basis, in the space of the first three eigenvectors of this basis.

Fig. 9
Fig. 9

Projection of the Fourier spectrum of an aerial photograph, considered as a set of forty-four spatial frequencies in the direction basis, in the space of the first three eigenvectors of this basis.

Fig. 10
Fig. 10

Optical setup for statistical spatial filtering: the object is filtered by an optical representatiion of the eigenvector being considered. In the direction basis (left) the K-L image is obtained by incoherent superimposition of elementary images transmitted sector by sector. In the spatial frequency basis the K-L image is given by rings the transmittance of which is proportional to the components of the eigenvector.

Fig. 11
Fig. 11

The principal K-L image of the previous aerial photograph defined as its projection on the first eigenvector of the direction basis. In this basis two classes of spatial frequencies are separated (Fig. 8). Here these classes appear as high frequencies on the right side of the river, and as medium frequencies on the left side of the river.

Equations (15)

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N k j = k / 710 m 1 , with k = 1 , , 22 and j = 1,2 ,
σ k j 2 = | N k j N k j | 2 ,
σ k 2 = σ k j 2 ¯
( Δ σ k 2 ) 2 = | σ k j 2 σ k 2 | 2 ¯ ,
p k = σ k 2 / Δ σ k 2 .
p k = ( λ k ) / ( Δ λ k ) ,
λ k = λ k j ¯ , j = 1,2 ,
λ k 2 = ( λ k j λ k ) 2 . ¯
C m m = ( 1 / n ) D m n · D n m t ,
C n n = ( 1 / n ) D n m t · D m n ,
λ m = λ n , m n ,
φ m = D m n · φ n , m n .
C d d i = ( 1 / d ) D d m i t · D m d i ,
λ m = d n i = 1 n / d λ d i , m d ,
φ m = d n i = 1 n / d D m d i · φ d i , m d .

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