Abstract

A general equation is derived which gives the root-mean-square transmission (and hence the granularity) as an integral of the product of the autocorrelation and an “aperture weighting function.” A method using expansion into series is described which inverts this equation and gives the autocorrelation in terms of the root-mean-square transmission. Solutions in closed form have been obtained for two special cases of physical importance to which the series solutions are shown to be equivalent. In view of this interconvertibility, the choice of optical systems for making measurements which will correlate well with visual graininess is a matter of practical convenience in obtaining the relevant parameters.

© 1957 Optical Society of America

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References

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  1. A. Marriage and E. Pitts, J. Opt. Soc. Am. 46, 1019 (1956).
    [CrossRef]
  2. E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, London, 1927).
  3. R. Clark Jones, J. Opt. Soc. Am. 45, 799 (1955).
    [CrossRef]

1956 (1)

1955 (1)

Clark Jones, R.

Marriage, A.

Pitts, E.

Watson, G. N.

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, London, 1927).

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, London, 1927).

J. Opt. Soc. Am. (2)

Other (1)

E. T. Whittaker and G. N. Watson, Modern Analysis (Cambridge University Press, London, 1927).

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Figures (3)

Fig. 1
Fig. 1

The function W(z) for a circular aperture defined by Eq. (2.4).

Fig. 2
Fig. 2

The correlation r(x), together with the derived function g(x) (smooth curves) and values of r calculated using Eq. (5.2) (circles).

Fig. 3
Fig. 3

The first four terms of Eq. (5.2): Curve 1: g; Curve 2: −1.209xg1; Curve 3: 0.291x2g2; Curve 4: −0.0116x3g3.

Equations (45)

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2 i < j a i a j t i t j δ 2 + a i 2 t i 2 δ 2 ,
g = w ( x ) r ( x ) d x ,
w ( x ) d x = 1.
g ( y ) = k W ( x / y ) r ( x ) d x ,
k W ( x / y ) d x = 1
g ( y ) = 1 y W ( x / y ) r ( x ) d x ,
g ( y ) = 0 α W ( z ) r ( y z ) d z ,
0 α W ( z ) d z = 1.
W ( z ) = 2 ( 1 - z ) ,             0 z 1 ;
W ( z ) = 32 z π 0 cos - 1 z sin 2 θ d θ = 16 z π [ cos - 1 z - z ( 1 - z 2 ) 1 2 ]             0 z 1.
r ( x ) = A 0 g ( x ) + A 1 x g 1 ( x ) + A 2 x 2 g 2 ( x ) / 2 ! + + A n x n g n ( x ) / n ! +
g ( b ) = [ A 0 g ( b y ) + + A n b n y n g n ( b y ) / n ! + ] W ( y ) d y .
g 1 ( b ) = [ ( A 0 + A 1 ) y g 1 ( b y ) + ( A 1 + A 2 ) b y 2 g 2 ( b y ) + ] W ( y ) d y , g n ( b ) = { [ A 0 + ( n 1 ) A 1 + + ( n r ) A r + + A n ] y n g n ( b y ) + O ( b ) } W ( y ) d y .
M n = y n W ( y ) d y ,
1 = A 0 M 0 , 1 = ( A 0 + A 1 ) M 1 , 1 = [ A 0 + ( n 1 ) A 1 + ( n 2 ) A 2 + + ( n r ) A r + + A n ] M n ,
A 0 = M 0 - 1 = 1 , A 1 = ( M 1 - 1 - M 0 - 1 ) , A n = M n - 1 - ( n 1 ) M n - 1 - 1 + + ( - 1 ) r ( n r ) M n - r - 1 + + ( - 1 ) n M 0 - 1 .
A ( x ) = 0 A n x n .
A ( x ) = n = 0 r = 0 n x n ( - 1 ) r n ! ( n - r ) ! r ! M n - r .
A ( x ) = m = 0 r = 0 x m + r ( - 1 ) r ( m + r ) ! m ! r ! M m = m = 0 x m ( 1 + x ) - ( m + 1 ) M m - 1 .
W ( z ) = 2 ( 1 - z ) ,
M n = 2 0 1 z n ( 1 - z ) d z = 2 / ( n + 1 ) ( n + 2 ) ,
A ( x ) = m = 0 x m ( 1 + x ) - ( m + 1 ) ( m + 1 ) ( m + 2 ) / 2 = ( 1 + x ) - 1 [ 1 - x ( 1 + x ) - 1 ] - 3 = ( 1 + x ) 2 .
r ( x ) = g ( x ) + 2 x g 1 ( x ) + x 2 g 2 ( x ) / 2 = 1 2 2 x 2 [ x 2 g ( x ) ] ,
M n = 32 π 0 1 y n + 1 0 cos - 1 y sin 2 θ d θ d y .
M n = 32 π ( n + 2 ) 0 1 y n + 2 ( 1 - y 2 ) 1 2 d y = 8 Γ [ ( n + 3 ) / 2 ] ( π ) 1 2 ( n + 2 ) Γ [ ( n + 6 ) / 2 ] .
A ( x ) = π 8 0 x m ( 1 + x ) - ( m + 1 ) × ( m + 2 ) Γ [ ( m + 6 ) / 2 ] / Γ [ ( m + 3 ) / 2 ] .
r ( x ) = g ( x ) + 1.209 x g 1 ( x ) + 0.291 x 2 g 2 ( x ) + 0.0116 x 3 g 3 ( x ) + .
g ( x ) = 32 π 0 1 r ( x y ) y 0 cos - 1 y sin 2 θ d θ d y ,
x 2 g ( x ) = 32 π 0 x r ( ξ ) ξ ξ / x 1 ( 1 - z 2 ) 1 2 d z d ξ ,
x ( x 2 g ) = 32 π 0 x r ( ξ ) ξ 2 ( x 2 - ξ 2 ) 1 2 d ξ / x 3 , 2 x 2 ( x 2 g ) = - 32 π 0 x r ( ξ ) ξ 2 ( 2 x 2 - 3 ξ 2 ) d ξ x 4 ( x 2 - ξ 2 ) 1 2 .
2 x 2 ( x 2 g ) + 3 x x ( x 2 g ) = 32 π 0 z r ( ξ ) ξ 2 d ξ x 2 ( x 2 - ξ 2 ) 1 2 .
x 2 2 x 2 ( x 2 g ) + 3 x x ( x 2 g ) = 32 π 0 π / 2 x 2 sin 2 ϕ r ( x sin ϕ ) d ϕ = 2 π 0 π / 2 f ( x sin ϕ ) d ϕ ,
f ( y ) = 16 y 2 r ( y ) .
16 x 2 r ( x ) = x 0 π / 2 { ξ [ ξ 2 2 ξ 2 ( ξ 2 g ) + 3 ξ ξ ( ξ 2 g ) ] } ξ = x sin ϕ d ϕ .
ξ = x y y = sin ϕ ,
16 r ( x ) = 0 1 y { y 2 2 y 2 [ y 2 g ( x y ) ] + 3 y y [ y 2 g ( x , y ) ] } d y / ( 1 - y 2 ) 1 2
16 x r ( x ) = 0 π / 2 [ x 4 sin 4 ϕ g 3 ( x sin ϕ ) + 11 x 3 sin 3 ϕ g 2 ( x sin ϕ ) + 29 x 2 sin 2 ϕ × g 1 ( x sin ϕ ) + 16 x sin ϕ g ( x sin ϕ ) ] d ϕ .
g ( x sin ϕ ) = n = 0 ( x sin ϕ - x ) n g n ( x ) / n ! = n = 0 ( - x ) n ( 1 - sin ϕ ) n g n ( x ) / n ! .
16 ( - x ) n n ! 0 π / 2 x sin ϕ ( 1 - sin ϕ ) n d ϕ + 29 ( - x ) n - 1 ( n - 1 ) ! 0 π / 2 x 2 sin 2 ϕ ( 1 - sin ϕ ) n - 1 d ϕ + 11 ( - x ) n - 2 ( n - 2 ) ! 0 π / 2 x 3 sin 3 ϕ ( 1 - sin ϕ ) n - 2 d ϕ + ( - x ) n - 3 ( n - 3 ) ! 0 π / 2 x 4 sin 4 ϕ ( 1 - sin ϕ ) n - 3 d ϕ .
( - 1 ) n 16 A n = 16 0 π / 2 sin ϕ ( 1 - sin ϕ ) n d ϕ - 29 n × 0 π / 2 sin 2 ϕ ( 1 - sin ϕ ) n - 1 d ϕ + 11 n ( n - 1 ) × 0 π / 2 sin 3 ϕ ( 1 - sin ϕ ) n - 2 d ϕ - n ( n - 1 ) × ( n - 2 ) 0 π / 2 sin 4 ϕ ( 1 - sin ϕ ) n - 3 d ϕ .
M n - 1 = ( π ) 1 2 8 ( n + 2 ) Γ [ ( n + 6 ) / 2 ] Γ [ ( n + 3 ) / 2 ] = ( π ) 1 2 32 ( n + 2 ) 2 ( n + 4 ) Γ [ ( n + 2 ) / 2 ] Γ [ ( n + 3 ) / 2 ] = ( n + 2 ) 2 ( n + 4 ) 16 0 π / 2 sin n + 1 ϕ d ϕ .
16 ( - 1 ) n A n = 0 π / 2 0 n ( - 1 ) r ( n r ) ( r + 4 ) ( r + 2 ) 2 sin r + 1 ϕ d ϕ .
x 4 ( 1 - x sin ϕ ) n sin ϕ d ϕ = 0 n ( - 1 ) r ( n r ) x 4 + r sin r + 1 ϕ d ϕ ,
( d d x ) ( x d d x ) ( 1 x d d x ) [ x 4 ( 1 - x sin ϕ ) n sin ϕ ] = 0 n ( - 1 ) r ( n r ) ( 4 + r ) ( r + 2 ) 2 x r + 1 sin r + 1 ϕ ,
( - 1 ) n 16 A n = { ( d d x ) ( x d d x ) ( 1 x d d x ) × 0 π / 2 x 4 ( 1 - x sin ϕ ) n sin ϕ d ϕ } x = 1 .