Abstract

A well-known, simple theory predicts that, for a given grain size, the granularity of a developed photographic image is proportional to the square root of the density. In this paper, it is shown that, in the case of a two-dimensional grain model consisting of randomly placed opaque disks (dots), this simple relation progressively underestimates granularity as density is increased. The disparity exists because, with the random-dot model, density is not solely proportional to the number of dots in the aperture, but may vary at random according to the amount of overlap among the dots. It is suggested that the effect of random overlap may be relatively small in a developed photographic image, which differs from a random-dot model in that the grains are distributed in depth.

© 1964 Optical Society of America

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References

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  1. J. C. Marchant and P. L. P. Dillon, J. Opt. Soc. Am. 51, 641 (1961).
    [CrossRef]
  2. E. W. H. Selwyn, Phot. J. 75, 571 (1935).
  3. B. Picinbono, Compt. Rend. 240, 2206 (1955).
  4. A. Marriage and E. Pitts, J. Opt. Soc. Am. 46, 1019 (1956).
    [CrossRef]
  5. E. Pitts and A. Marriage, J. Opt. Soc. Am. 47, 321 (1957).
    [CrossRef]
  6. L. Silberstein and A. P. H. Trivelli, J. Opt. Soc. Am. 28, 441 (1938).
    [CrossRef]
  7. J. H. Webb, J. Opt. Soc. Am. 45, 379 (1955).
    [CrossRef]
  8. Throughout this paper, the notation E(x) will be used to denote the mean value of a random variable, x.
  9. J. H. Altman, Appl. Opt. 3, 35 (1964).
    [CrossRef]
  10. P. G. Nutting, Phil. Mag. 26, 423 (1913).

1964 (1)

1961 (1)

1957 (1)

1956 (1)

1955 (2)

J. H. Webb, J. Opt. Soc. Am. 45, 379 (1955).
[CrossRef]

B. Picinbono, Compt. Rend. 240, 2206 (1955).

1938 (1)

1935 (1)

E. W. H. Selwyn, Phot. J. 75, 571 (1935).

1913 (1)

P. G. Nutting, Phil. Mag. 26, 423 (1913).

Appl. Opt. (1)

Compt. Rend. (1)

B. Picinbono, Compt. Rend. 240, 2206 (1955).

J. Opt. Soc. Am. (5)

Phil. Mag. (1)

P. G. Nutting, Phil. Mag. 26, 423 (1913).

Phot. J. (1)

E. W. H. Selwyn, Phot. J. 75, 571 (1935).

Other (1)

Throughout this paper, the notation E(x) will be used to denote the mean value of a random variable, x.

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Figures (4)

F. 1
F. 1

Random-dot model for photographic grain. The scale is arbitrary.

F. 2
F. 2

Illustration of the problem of finding the probability that two photons, incident upon a random-dot pattern, will both be transmitted.

F. 3
F. 3

Factor by which simple granularity formula must be multiplied to obtain granularity of random-dot pattern. Curves are shown for infinitely large aperture (top curve), aperture four times the area of the grain (middle curve), and aperture equal in area to the grain (bottom curve).

F. 4
F. 4

Three-dimensional model for photographic grain. Opaque spheres are suspended in a thick transparent layer.

Equations (51)

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p 0 = e n ¯ α .
D = log T .
D T = log E ( T ) .
( D D T ) [ log e / E ( T ) ] [ T E ( T ) ] .
H T = [ log e / E ( T ) ] σ T ,
D T = ( log 10 e ) n ¯ a .
z 1 + z 2 + + z N .
lim N [ z 1 + z 2 + z N N ] = T .
E ( z 1 ) = E ( z 2 ) = = E ( z N ) .
lim N [ E ( z 1 ) + E ( z 2 ) + + E ( z N ) N ] = E ( T ) .
E ( z 1 ) = E ( T ) .
e n ¯ a .
E ( T ) = e n ¯ a .
D T = ( log e ) n ¯ a .
H T = [ log e / E ( T ) ] σ T .
σ T = ( E { [ T E ( T ) ] 2 } ) 1 2
σ T = { E ( T 2 ) [ E ( T ) ] 2 } 1 2 .
lim N [ z 1 + z 2 + + z N N ] = T .
lim N [ z 1 2 + z 1 z 2 + + z N 2 N 2 ] = T 2 .
lim N [ E ( z 1 2 ) + E ( z 1 z 2 ) + + E ( z N 2 ) N 2 ] = E ( T 2 ) .
lim N [ N E ( z 1 2 ) + N ( N 1 ) E ( z 1 z 2 ) N 2 ] = E ( T 2 ) .
E ( z 1 z 2 ) = E ( T 2 ) .
a · g ( u ) ,
g ( u ) = ( 2 / π ) [ cos 1 u u ( 1 u 2 ) 1 2 ] , 0 u 1 . g ( u ) = 0 , u 1 .
2 a a g ( u ) .
e n ¯ [ 2 a a g ( u ) ] .
E ( T ) = e n ¯ a ,
[ E ( T ) ] 2 e n ¯ a g ( u ) .
f ( u ) d u ,
0 [ E ( T ) ] 2 e n ¯ a g ( u ) f ( u ) d u .
σ T = ( { 0 [ E ( T ) ] 2 e n ¯ a g ( u ) f ( u ) d u } [ E ( T ) ] 2 ) 1 2 .
0 f ( u ) d u = 1 .
σ T = E ( T ) { 0 1 [ e n ¯ a g ( u ) 1 ] f ( u ) d u } 1 2 .
8 ρ g ( ρ ) d ρ ,
ρ = u ( a / A ) 1 2 .
f ( u ) d u = 8 ( a / A ) u g [ u ( a / A ) 1 2 ] d u .
σ T = E ( T ) { ( a A ) 0 1 [ e n ¯ a g ( u ) 1 ] 8 u g [ u ( a A ) 1 2 ] d u } 1 2 .
H T = log 10 e { ( a A ) 0 1 [ e n ¯ a g ( u ) 1 ] 8 u g [ u ( a A ) 1 2 ] d u } 1 2 .
H M = log e [ ( n ¯ a ) ( a A ) ] 1 2 ,
( log e ) ( a / A ) M .
D M = ( log e ) ( a / A ) ( n ¯ A ) = ( log e ) n ¯ a .
H M = ( log e ) ( a / A ) ( n ¯ A ) 1 2 = ( log e ) [ ( a / A ) ( n ¯ a ) ] 1 2 .
H T H M = ( 1 ( n ¯ a ) 0 1 [ e n ¯ a g ( u ) 1 ] { 8 u g [ u ( a / A ) 1 2 ] } d u ) 1 2 .
lim n ¯ a 0 [ e n ¯ a g ( u ) 1 n ¯ a g ( u ) ] = 1 lim a / A 0 { g [ u ( a / A ) 1 2 ] } = 1 ,
lim n ¯ a 0 a / A 0 [ H T H M ] = { 1 ( n ¯ a ) 0 1 [ n ¯ a g ( u ) ] [ 8 u ] d u } 1 2 = 1 .
{ [ e n ¯ a g ( u ) 1 ] = [ n ¯ a g ( u ) ] + 1 2 [ n ¯ a g ( u ) ] 2 + , g [ u ( a / A ) 1 2 ] = 1 4 / π [ u ( a / A ) 1 2 ] + ,
H T / H M = [ 1 + b 1 ( n ¯ a ) b 2 ( a A ) 1 2 + ] 1 2 ,
b 1 = ( 3 π 2 16 ) / 6 π 2 0.23 , b 2 = 256 / 45 π 2 0.58 .
8 ρ g ( ρ ) d ρ ,
g ( ρ ) = ( 2 / π ) [ cos 1 ρ ρ ( 1 ρ 2 ) 1 2 ] , 0 ρ 1 . g ( ρ ) = 0 , ρ 1 .
0 1 K g ( ρ ) ρ d ρ = 1 .