Abstract

The application of coherent optical techniques to the study of correlations in the detailed structure of electron (or light) micrographs was considered for the case when the structure appears to be random. A statistical autocorrelation analysis was made for a random array of transparent disks on an opaque background, relating the radii of the central and first autocorrelation maxima to the radius and mean separation, respectively, of the disks. Experimental determinations of the two-dimensional autocorrelation function were made both for arrays of disks and ellipses. In each case, the shape of the central maximum was similar to the shape of the objects (when all had the same shape and orientation). In addition, the shape of the first autocorrelation maximum was dependent only upon the distribution of centroids of the objects in the array and exhibited the proper asymmetry when the mean separation of centroids differed for the two dimensions. In this latter case, a rectangular first autocorrelation maximum was observed, implying separability of the two-dimensional autocorrelation function and providing a technique for direct quantitative determinations of the mean separations.

© 1971 Optical Society of America

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References

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  1. L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
    [CrossRef]
  2. R. Hosemann, S. N. Bagehi, Direct Analysis of Diffraction by Matter (North-Holland, Amsterdam, 1962), Chap 16
  3. R. E. Beissner, R. L. Bond, W. W. Bradshaw, J. Opt. Soc. Am. 60, 1551 (1970).
  4. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), p. 115.

1970

R. E. Beissner, R. L. Bond, W. W. Bradshaw, J. Opt. Soc. Am. 60, 1551 (1970).

1960

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
[CrossRef]

Bagehi, S. N.

R. Hosemann, S. N. Bagehi, Direct Analysis of Diffraction by Matter (North-Holland, Amsterdam, 1962), Chap 16

Beissner, R. E.

R. E. Beissner, R. L. Bond, W. W. Bradshaw, J. Opt. Soc. Am. 60, 1551 (1970).

Bond, R. L.

R. E. Beissner, R. L. Bond, W. W. Bradshaw, J. Opt. Soc. Am. 60, 1551 (1970).

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), p. 115.

Bradshaw, W. W.

R. E. Beissner, R. L. Bond, W. W. Bradshaw, J. Opt. Soc. Am. 60, 1551 (1970).

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
[CrossRef]

Hosemann, R.

R. Hosemann, S. N. Bagehi, Direct Analysis of Diffraction by Matter (North-Holland, Amsterdam, 1962), Chap 16

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
[CrossRef]

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
[CrossRef]

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
[CrossRef]

IRE Trans. Information Theory

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, IRE Trans. Information Theory IT-6, 386 (1960).
[CrossRef]

J. Opt. Soc. Am.

R. E. Beissner, R. L. Bond, W. W. Bradshaw, J. Opt. Soc. Am. 60, 1551 (1970).

Other

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), p. 115.

R. Hosemann, S. N. Bagehi, Direct Analysis of Diffraction by Matter (North-Holland, Amsterdam, 1962), Chap 16

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Figures (8)

Fig. 1
Fig. 1

Schematic representation of t(x) (solid lines) and t(x + α) (dashed lines) for the one-dimensional array of transparent circular disks shown at the top, in the interval |α| ≤ d.

Fig. 2
Fig. 2

Schematic representation of t(x) (solid lines) and t(x + α′), t(x + a″) (dashed lines) indicating the various degrees of overlap in intervals α′, α″ > x0/2.

Fig. 3
Fig. 3

Plot of reduced one-dimensional autocorrelation function f′(α) = (x0/d)f(α) for lightly packed array of circular disks.

Fig. 4
Fig. 4

Schematic representation of decomposition (for N = 3) of t(x) for a densely packed array.

Fig. 5
Fig. 5

Schematic representation of experimental arrangement for optical determination of the autocorrelation function.

Fig. 6
Fig. 6

Object and corresponding autocorrelation for a random array of circular disks.

Fig. 7
Fig. 7

Object and corresponding autocorrelation for a quasi-random array of circular disks for which mean separations differed in the two dimensions.

Fig. 8
Fig. 8

Object and corresponding autocorrelation for a random array of ellipsoidal disks.

Equations (21)

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F ( α , β ) = lim S [ 1 S S T ( x , y ) T ( x + α , y + β ) d x d y ] ,
T ( x , y ) = 1 , within a disk , = 0 , elsewhere .
T ( x , y ) = t ( x ) u ( y ) , F ( α , β ) = f ( α ) g ( β ) = lim L x , L y [ 1 L x L x t ( x ) t ( x + α ) d x 1 L y × L y u ( y ) u ( y + β ) d y ] .
f ( α ) = ξ - α / x 0 ,             for α d ,
f ( α ) = f ( α ) = 1 x 0 α d ( ξ - α ) P ( ξ ) d ξ = ( d - α ) 2 2 x 0 d .
f ( α ) = E ( α ) / x 0 = [ E c ( α ) + E p ( α ) ] / x 0 ,
E c ( α ) = ξ η , ξ ξ = 0 d - α n = ξ + α d ξ P ( ξ , η ; α ) d η d ξ ,
P ( ξ , η ; α ) = P ( ξ ) P ( η ; α ¯ ) .
E c ( α ¯ ) = ( 1 / 6 d 2 ) ( d - α ¯ ) 3 ; 0 α ¯ d .
E p ( α ¯ ) = η - α ¯ η , ξ ξ = η - α ¯ d η = α ¯ d ( η - α ¯ ) P ( ξ ) P ( η ; α ¯ ) d η d ξ = ( 1 / 6 d 2 ) [ 3 d ( d - α ¯ ) 2 - 2 ( d - α ¯ ) 3 ] ;             0 α ¯ d .
f n ( α ) = ( 1 / 6 x 0 d 2 ) [ 3 d ( d + n x 0 - α ) 2 - ( d + n x 0 - α ) 3 ] ,
t ( x ) = i = 1 N t i ( x ) .
f ( α ) = i , j = 1 N lim L x [ 1 L x L x t i ( x ) t j ( x + α ) d x ] i , j = 1 N f i j ( α ) = i = 1 N f i i ( α ) + i j N f i j ( α ) .
i = 1 N f i i ( α ) = N f i i ( α ) .
i = 1 N f i i ( α ) = N 2 x 0 d ( d - α ) 2 = 1 2 x 0 d ( d - α ) 2 .
i = 1 N f i i ( α ) = 1 6 x 0 d 2 [ 3 d ( d + n N x 0 - α ) 2 - ( d + n N x 0 - α ) 3 ] .
i j N f i j ( α ) = 2 i < j N f i j ( α ) = 2 i < j N 1 6 ( j - i ) x 0 d 2 [ 3 d ( d + n ( j - i ) x 0 - α ) 2 - ( d + n ( j - i ) x 0 - α ) 3 ] .
I ( x , y ) = K 2 F ( q , p ) 2 ,
F ( q , p ) = K S t ( x , y ) e - i ( x q + y p ) d x d y ,
g ( α , β ) = K S F ( q , p ) 2 e - i ( α q + β p ) d q d p ,
g ( α , β ) = K S t ( x , y ) t ( x + α , y + β ) d x d y ,

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