Abstract

The cross-correlation matrix of a set of signals is obtained by multiple Fourier holography. Various sources of error make this kind of matrix not directly interpretable. Global methods that correct deterministic fluctuations as well as random perturbations are proposed. They do not take into account any individual element of the matrix, but they minimize the over-all variance. The efficiency of these methods is illustrated by the example of the classification of sonagrams.

© 1978 Optical Society of America

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References

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  1. J. C. Viénot, J. Bulabois, G. Perrin, C. R. Acad. Sci. Ser. B 263, 1300 (1966).
  2. J. M. Fournier, J. C. Viénot, Isr. J. Technol. 9, 281 (1971).
  3. F. M. Schafner, R. O. Webb, Appl. Opt. 11, 1991 (1972).
    [CrossRef]
  4. R. O. Harger, Appl. Opt. 4, 383 (1965).
    [CrossRef]
  5. J. E. Wasielewski, Appl. Opt. 10, 2439 (1971).
    [CrossRef] [PubMed]
  6. D. Casasent, A. Furman, Appl. Opt. 16, 1652 (1977).
    [CrossRef] [PubMed]
  7. J. W. Goodman, J. Opt. Soc. Am. 57, 493 (1967).
    [CrossRef] [PubMed]
  8. A. Van der Lugt, F. B. Rotz, A. Klooster, Optical and Electro-Optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), p. 135.
  9. B. J. Pernick, S. Levinson, C. Bartolotta, Appl. Opt. 9, 1902 (1970).
    [CrossRef] [PubMed]
  10. H. S. Caulfield, W. T. Maloney, Appl. Opt. 8, 2354 (1969).
    [CrossRef] [PubMed]
  11. W. T. Maloney, Appl. Opt. 10, 2127 (1971).
    [CrossRef] [PubMed]
  12. P. S. Naidu, Opt. Commun. 13, 28 (1974).
  13. J. C. Viénot, J. Bulabois, L. R. Guy, Opt. Commun. 2, 431 (1971).
    [CrossRef]
  14. G. W. Stroke, R. Restrick, A. Funkhauser, Phys. Lett. 18, 274 (1965).
    [CrossRef]
  15. J. E. Rau, J. Opt. Soc. Am. 56, 1490 (1966).
    [CrossRef]
  16. J. C. Viénot, J. Duvernoy, G. Tribillon, Nouv. Rev. Opt. 2, 269 (1971).
    [CrossRef]
  17. N. Aebischer, Opt. Technol. 1, 89 (1969).
    [CrossRef]

1977 (1)

1974 (1)

P. S. Naidu, Opt. Commun. 13, 28 (1974).

1972 (1)

1971 (5)

J. C. Viénot, J. Duvernoy, G. Tribillon, Nouv. Rev. Opt. 2, 269 (1971).
[CrossRef]

J. C. Viénot, J. Bulabois, L. R. Guy, Opt. Commun. 2, 431 (1971).
[CrossRef]

J. M. Fournier, J. C. Viénot, Isr. J. Technol. 9, 281 (1971).

J. E. Wasielewski, Appl. Opt. 10, 2439 (1971).
[CrossRef] [PubMed]

W. T. Maloney, Appl. Opt. 10, 2127 (1971).
[CrossRef] [PubMed]

1970 (1)

1969 (2)

1967 (1)

1966 (2)

J. E. Rau, J. Opt. Soc. Am. 56, 1490 (1966).
[CrossRef]

J. C. Viénot, J. Bulabois, G. Perrin, C. R. Acad. Sci. Ser. B 263, 1300 (1966).

1965 (2)

G. W. Stroke, R. Restrick, A. Funkhauser, Phys. Lett. 18, 274 (1965).
[CrossRef]

R. O. Harger, Appl. Opt. 4, 383 (1965).
[CrossRef]

Aebischer, N.

N. Aebischer, Opt. Technol. 1, 89 (1969).
[CrossRef]

Bartolotta, C.

Bulabois, J.

J. C. Viénot, J. Bulabois, L. R. Guy, Opt. Commun. 2, 431 (1971).
[CrossRef]

J. C. Viénot, J. Bulabois, G. Perrin, C. R. Acad. Sci. Ser. B 263, 1300 (1966).

Casasent, D.

Caulfield, H. S.

Duvernoy, J.

J. C. Viénot, J. Duvernoy, G. Tribillon, Nouv. Rev. Opt. 2, 269 (1971).
[CrossRef]

Fournier, J. M.

J. M. Fournier, J. C. Viénot, Isr. J. Technol. 9, 281 (1971).

Funkhauser, A.

G. W. Stroke, R. Restrick, A. Funkhauser, Phys. Lett. 18, 274 (1965).
[CrossRef]

Furman, A.

Goodman, J. W.

Guy, L. R.

J. C. Viénot, J. Bulabois, L. R. Guy, Opt. Commun. 2, 431 (1971).
[CrossRef]

Harger, R. O.

Klooster, A.

A. Van der Lugt, F. B. Rotz, A. Klooster, Optical and Electro-Optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), p. 135.

Levinson, S.

Maloney, W. T.

Naidu, P. S.

P. S. Naidu, Opt. Commun. 13, 28 (1974).

Pernick, B. J.

Perrin, G.

J. C. Viénot, J. Bulabois, G. Perrin, C. R. Acad. Sci. Ser. B 263, 1300 (1966).

Rau, J. E.

Restrick, R.

G. W. Stroke, R. Restrick, A. Funkhauser, Phys. Lett. 18, 274 (1965).
[CrossRef]

Rotz, F. B.

A. Van der Lugt, F. B. Rotz, A. Klooster, Optical and Electro-Optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), p. 135.

Schafner, F. M.

Stroke, G. W.

G. W. Stroke, R. Restrick, A. Funkhauser, Phys. Lett. 18, 274 (1965).
[CrossRef]

Tribillon, G.

J. C. Viénot, J. Duvernoy, G. Tribillon, Nouv. Rev. Opt. 2, 269 (1971).
[CrossRef]

Van der Lugt, A.

A. Van der Lugt, F. B. Rotz, A. Klooster, Optical and Electro-Optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), p. 135.

Viénot, J. C.

J. C. Viénot, J. Bulabois, L. R. Guy, Opt. Commun. 2, 431 (1971).
[CrossRef]

J. M. Fournier, J. C. Viénot, Isr. J. Technol. 9, 281 (1971).

J. C. Viénot, J. Duvernoy, G. Tribillon, Nouv. Rev. Opt. 2, 269 (1971).
[CrossRef]

J. C. Viénot, J. Bulabois, G. Perrin, C. R. Acad. Sci. Ser. B 263, 1300 (1966).

Wasielewski, J. E.

Webb, R. O.

Appl. Opt. (7)

C. R. Acad. Sci. Ser. B (1)

J. C. Viénot, J. Bulabois, G. Perrin, C. R. Acad. Sci. Ser. B 263, 1300 (1966).

Isr. J. Technol. (1)

J. M. Fournier, J. C. Viénot, Isr. J. Technol. 9, 281 (1971).

J. Opt. Soc. Am. (2)

Nouv. Rev. Opt. (1)

J. C. Viénot, J. Duvernoy, G. Tribillon, Nouv. Rev. Opt. 2, 269 (1971).
[CrossRef]

Opt. Commun. (2)

P. S. Naidu, Opt. Commun. 13, 28 (1974).

J. C. Viénot, J. Bulabois, L. R. Guy, Opt. Commun. 2, 431 (1971).
[CrossRef]

Opt. Technol. (1)

N. Aebischer, Opt. Technol. 1, 89 (1969).
[CrossRef]

Phys. Lett. (1)

G. W. Stroke, R. Restrick, A. Funkhauser, Phys. Lett. 18, 274 (1965).
[CrossRef]

Other (1)

A. Van der Lugt, F. B. Rotz, A. Klooster, Optical and Electro-Optical Information Processing, J. T. Tippett et al., Eds. (MIT Press, Cambridge, 1965), p. 135.

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Figures (8)

Fig. 1
Fig. 1

Experimental setup that gives the cross-correlation matrix of the two series of signals S.

Fig. 2
Fig. 2

Cross-correlation matrices corresponding to (a) 2 series of 11 identical holes; (b) 2 series of 11 replicas of the same photographic signal; (c) 2 series of 11 different photographic signals.

Fig. 3
Fig. 3

Correction of the rough matrix A. The variations of the average cross-correlation level along rows (Mj) and columns (Ni) are found to follow a deterministic law. Experimental correction factors m(j) and n(i) are optimized by the Newton method.

Fig. 4
Fig. 4

Definition of the correction factor f(j) of the rows, starting from experimental values 1/Mj; m(j) is derived from them only by reordering.

Fig. 5
Fig. 5

Assessment of global random fluctuations that affect an element Bij of the previously corrected matrix. Variations of Mj (or Ni) represent the average error introduced by sj (or si).

Fig. 6
Fig. 6

Efficiency of the successive corrections evaluated on the three matrices of Fig. 2.

Fig. 7
Fig. 7

Variations of the degree of resemblance of the fourth signal with all 11, measured along the fourth row (r) and the fourth column (c) of the matrix in Fig. 2(c), without and with successive corrections.

Fig. 8
Fig. 8

Variations of the average degree of resemblance between the 11 signals.

Equations (30)

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D i = y [ log ( E i ) ]
log ( E i ) = z ( D i ) ,
log ( E i ) = k = 1 5 a ( k ) · P k ( D i ) , i = 1 , , N ,
R i j = ρ i j / ( ρ i i ρ j j ) 0.5 .
R i j = R j i ,
Q = μ / σ ,
μ = 1 121 i = 1 11 j = 1 11 ρ i j
σ = [ 1 120 i = 1 11 j = 1 11 ( ρ i j μ ) 2 ] 0.5
s ij = 1 2 | ρ i j + ρ j i ρ i j ρ j i | , i = 1 , , 11 and j = i + 1 , , 10 ,
S = 1 55 i = 1 11 j = i + 1 10 s i j .
M j = 1 11 i = 1 11 A i j
N i = 1 11 j = 1 11 A i j ,
T = j = 1 11 [ 1 / M j m ( j ) ] 2 m ( j )
K Q [ m ( j ) ]
K Q [ m 1 ( j ) ] = 0.
K Q [ m 1 ( j ) ] = 0 = K Q [ m ( j ) ] [ m 1 ( j ) m ( j ) ] · d Q [ m ( j ) ] d m ( j )
m 1 ( j ) = m ( j ) + { K Q [ m ( j ) ] } · { d Q [ m ( j ) ] d m ( j ) } 1 .
B i j = A i j · f ( j ) · g ( i ) .
ρ i j = ρ 0 i j · k i j ,
r i j = k i j / ( k i i · k j j ) 0.5
ρ i j = ρ 0 · ( k i i · k j j ) 0.5 · r i j ,
ρ i j = ρ 0 · ( E i · E j ) 0.5 · r i j ,
r i j = r 0 · ( 1 + i j ) ,
M j = B ¯ j i i = ρ 0 r 0 E j 0.5 · ( E i 0.5 ) ¯ i
M j / M 0 = E j 0.5 / ( E j 0.5 ) ¯ j ,
( ρ i j · M 0 M j ρ i j · M 0 M j ¯ j ) 2 j
1 + [ ( M 0 M j ) / M 0 ]
B i j = M 0 N 0 ,
C i j = B i j · ( 1 + M 0 M j M 0 ) · ( 1 + N 0 N i N 0 ) ;
C i j M 0 N 0

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