Abstract

Recent work by Jones and Higgins has led to an objective method for evaluating the granular structure of the developed photographic image by which materials of widely differing granularity can be ordered in correspondence with subjective graininess measurements. The present paper gives the theoretical calculations of statistical fluctuations of density and syzygetic density differences (SΔD) for simplified density patterns when scanned with varied-size apertures. Average syzygetic density values, 〈SΔDAv, versus density curves are derived first for square, checkerboard density patterns and second for circular patterns containing uniform-size grains with random arrangement. By combining the experimental, threshold, visual, gradient sensitivity curve (〈SΔDAvversus density for critical aperture), obtained by Jones and Higgins, with theoretical curves of 〈SΔDAvversus density calculated here, it is shown how the graininess-density curve can be obtained.

© 1955 Optical Society of America

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References

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  1. L. A. Jones and N. Deisch, J. Franklin Inst. 190, 657 (1920).
    [CrossRef]
  2. J. Eggert and A. Kuster, Kinotechnik 16, 127, 291, 308 (1934).
  3. A. Van Kreveld, Phot. J. 74, 590 (1934); J. Opt. Soc. Am. 26, 170 (1936); A. Van Kreveld and J. C. Schaffer, J. Opt. Soc. Am. 27, 100 (1937).
    [CrossRef]
  4. A. Goetz and W. O. Gould, J. Soc. Motion Picture Engrs. 29, 510 (1937); Goetz, Gould, and Dember, ibid. 34, 279 (1940); A. Goetz and F. W. Brown, ibid. 39, 375 (1942).
  5. E. W. H. Selwyn, Phot. J. 75, 571 (1935); Phot. J. 79, 513 (1939).
  6. W. Raemer and E. W. H. Selwyn, Phot. J. 83, 17 (1943).
  7. E. W. H. Selwyn, Phot. J. 83, 227 (1943).
  8. L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 35, 435 (1945).
    [CrossRef]
  9. L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 36, 203 (1946).
    [CrossRef]
  10. L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 37, 217 (1947).
    [CrossRef] [PubMed]
  11. L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 38, 398 (1948).
    [CrossRef] [PubMed]
  12. L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 41, 41 (1951).
    [CrossRef]
  13. L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 41, 64 (1951).
    [CrossRef]
  14. L. A. Jones and G. C. Higgins, J. Opt. Soc. Am. 41, 192 (1951).
    [CrossRef]
  15. Jones, Higgins, and Stultz, “Photographic Granularity and Graininess. VIII. A Method of Measuring Granularity in Terms of the Scanning Area Giving a Threshold Luminance Gradient,” J. Opt. Soc. Am. 45, 107 (1955).
    [CrossRef]
  16. L. Silberstein and A. P. H. Trivelli, J. Opt. Soc. Am. 28, 441 (1938).
    [CrossRef]

1955 (1)

1951 (3)

1948 (1)

1947 (1)

1946 (1)

1945 (1)

1943 (2)

W. Raemer and E. W. H. Selwyn, Phot. J. 83, 17 (1943).

E. W. H. Selwyn, Phot. J. 83, 227 (1943).

1938 (1)

1937 (1)

A. Goetz and W. O. Gould, J. Soc. Motion Picture Engrs. 29, 510 (1937); Goetz, Gould, and Dember, ibid. 34, 279 (1940); A. Goetz and F. W. Brown, ibid. 39, 375 (1942).

1935 (1)

E. W. H. Selwyn, Phot. J. 75, 571 (1935); Phot. J. 79, 513 (1939).

1934 (2)

J. Eggert and A. Kuster, Kinotechnik 16, 127, 291, 308 (1934).

A. Van Kreveld, Phot. J. 74, 590 (1934); J. Opt. Soc. Am. 26, 170 (1936); A. Van Kreveld and J. C. Schaffer, J. Opt. Soc. Am. 27, 100 (1937).
[CrossRef]

1920 (1)

L. A. Jones and N. Deisch, J. Franklin Inst. 190, 657 (1920).
[CrossRef]

Deisch, N.

L. A. Jones and N. Deisch, J. Franklin Inst. 190, 657 (1920).
[CrossRef]

Eggert, J.

J. Eggert and A. Kuster, Kinotechnik 16, 127, 291, 308 (1934).

Goetz, A.

A. Goetz and W. O. Gould, J. Soc. Motion Picture Engrs. 29, 510 (1937); Goetz, Gould, and Dember, ibid. 34, 279 (1940); A. Goetz and F. W. Brown, ibid. 39, 375 (1942).

Gould, W. O.

A. Goetz and W. O. Gould, J. Soc. Motion Picture Engrs. 29, 510 (1937); Goetz, Gould, and Dember, ibid. 34, 279 (1940); A. Goetz and F. W. Brown, ibid. 39, 375 (1942).

Higgins,

Higgins, G. C.

Jones,

Jones, L. A.

Kuster, A.

J. Eggert and A. Kuster, Kinotechnik 16, 127, 291, 308 (1934).

Raemer, W.

W. Raemer and E. W. H. Selwyn, Phot. J. 83, 17 (1943).

Selwyn, E. W. H.

W. Raemer and E. W. H. Selwyn, Phot. J. 83, 17 (1943).

E. W. H. Selwyn, Phot. J. 83, 227 (1943).

E. W. H. Selwyn, Phot. J. 75, 571 (1935); Phot. J. 79, 513 (1939).

Silberstein, L.

Stultz,

Trivelli, A. P. H.

Van Kreveld, A.

A. Van Kreveld, Phot. J. 74, 590 (1934); J. Opt. Soc. Am. 26, 170 (1936); A. Van Kreveld and J. C. Schaffer, J. Opt. Soc. Am. 27, 100 (1937).
[CrossRef]

J. Franklin Inst. (1)

L. A. Jones and N. Deisch, J. Franklin Inst. 190, 657 (1920).
[CrossRef]

J. Opt. Soc. Am. (9)

J. Soc. Motion Picture Engrs. (1)

A. Goetz and W. O. Gould, J. Soc. Motion Picture Engrs. 29, 510 (1937); Goetz, Gould, and Dember, ibid. 34, 279 (1940); A. Goetz and F. W. Brown, ibid. 39, 375 (1942).

Kinotechnik (1)

J. Eggert and A. Kuster, Kinotechnik 16, 127, 291, 308 (1934).

Phot. J. (4)

A. Van Kreveld, Phot. J. 74, 590 (1934); J. Opt. Soc. Am. 26, 170 (1936); A. Van Kreveld and J. C. Schaffer, J. Opt. Soc. Am. 27, 100 (1937).
[CrossRef]

E. W. H. Selwyn, Phot. J. 75, 571 (1935); Phot. J. 79, 513 (1939).

W. Raemer and E. W. H. Selwyn, Phot. J. 83, 17 (1943).

E. W. H. Selwyn, Phot. J. 83, 227 (1943).

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Figures (11)

Fig. 1
Fig. 1

Threshold gradient sensitivity curve of the eye (〈SΔDAvversus density with critical aperture φc) in two forms: (A) Log 〈SΔDAvversus density, and (B) 〈SΔDAvversus density.

Fig. 2
Fig. 2

Illustrative drawings to show equality of granular structure of checkerboard density patterns when magnified in inverse ratio to grain size. Schematic ΔD-frequency and SΔD-frequency curves at right.

Fig. 3
Fig. 3

Checkerboard density pattern of mean density 0.2558 and elementary units of 1μ2 area, enclosed by square scanning aperture of linear dimensions 3μ.

Fig. 4
Fig. 4

Transmission-frequency curve for checkerboard density pattern of mean density 0.2558 scanned by square aperture of area 9μ2 shown in Fig. 3.

Fig. 5
Fig. 5

Density-frequency curve for checkerboard density pattern of mean density 0.2558 derived from transmission-frequency curve of Fig. 4.

Fig. 6
Fig. 6

SΔD-frequency curve for checkerboard density pattern of mean density 0.2558 derived from density-frequency curve of Fig. 5.

Fig. 7
Fig. 7

Curves of 〈SΔDAvversus density for checkerboard density patterns with varying-size scanning apertures.

Fig. 8
Fig. 8

Density pattern formed of opaque circular grains of uniform size and random distribution.

Fig. 9
Fig. 9

Curves of 〈SΔDAvversus density for randomly distributed grain pattern of type shown in Fig. 8, calculated for varying-size scanning apertures.

Fig. 10
Fig. 10

Method of combining theoretical curves of 〈SΔDAvversus density for varying aperture size, φ, and the threshold visual gradient sensitivity curve (A) to obtain critical φ values in graininess determination.

Fig. 11
Fig. 11

Theoretical graininess versus density curve obtained by combining 〈SΔDAvversus density curves of varying aperture size and the threshold visual gradient sensitivity curve as illustrated in Fig. 10.

Tables (5)

Tables Icon

Table I Transmission-frequency data calculated by binomial theorem for checkerboard density pattern of Fig. 3.

Tables Icon

Table II Transformed density-frequency data calculated for checkerboard density pattern of Fig. 3.

Tables Icon

Table III SΔD-frequency data calculated for checkerboard density pattern of Fig. 3 for average density 0.2558.

Tables Icon

Table IV Theoretical values of syzygetic density difference 〈SΔDAv for the density pattern of Fig. 3, calculated for different-sized scanning apertures φ and at different density levels.

Tables Icon

Table V Theoretical values of syzygetic density difference 〈SΔDAv for the density pattern of Fig. 8, calculated for different-sized scanning apertures φ and at different density levels.

Equations (23)

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G = 1000 / B . M .
φ c = 1.5 B . M . ( 2000 17.8 - 1 ) ,
T = m m + n ,             m + n = constant ,
D = Log 1 T = Log m + n m .
P ( m , n ) = ( m + n ) ! m ! n ! p m ( 1 - p ) n ,
0 1.0 f s d T = 0 f s d T d D d D .
d T / d D = 0.4343 T .
S Δ D f ( frequency ) 0.00 2 ( a 2 + b 2 + c 2 + d 2 + e 2 + f 2 + g 2 + h 2 + i 2     ·     ·     ·     ·     ·     ·     +     q 2 ) 0.05 2 ( a b + b c + c d + d e + e f + f g + g h + h i     ·     ·     ·     ·     ·     ·     +     p q ) 0.10 2 ( a c + b d + c e + d f + e g + f h + g i     ·     ·     ·     ·     ·     ·     ·     +     o q ) 0.15 2 ( a d + b e + c f + d g + e h + f i     ·     ·     ·     ·     ·     ·     ·     +     n q ) 0.20 2 ( a e + b f + c g + d h + e i     ·     ·     ·     ·     ·     ·     ·     ·     +     m q ) 0.25 2 ( a f + b g + c h + d i     ·     ·     ·     ·     ·     ·     ·     ·     ·     +     l q ) 0.30 2 ( a g + b h + c i     ·     ·     ·     ·     ·     ·     ·     ·     ·     +     k q ) 0.35 2 ( a h + b i + c j     ·     ·     ·     ·     ·     ·     ·     ·     +     j q ) 0.40 2 ( a i + b j + c k     ·     ·     ·     ·     ·     ·     ·     +     i q ) 0.45 2 ( a j + b k + c l     ·     ·     ·     ·     ·     +     h q ) 0.50 2 ( a k + b l + c m     ·     ·     ·     ·     +     g q ) 0.55 2 ( a l + b m + c n     ·     ·     ·     +     f q ) 0.60 2 ( a m + b n + c o     ·     ·     +     e q ) 0.65 2 ( a n + b o + c p + d q ) 0.70 2 ( a o + b p + c q ) 0.75 2 ( a p + b q ) 0.80 2 ( a q )
S Δ D Av = i f s × S Δ D i i f s = 0.1371.
density = M n a ,
a = d 2 / φ 2 .
D Av = M n ¯ a ,
P ( n ) = exp ( - n ¯ ) n ¯ n n !         Poisson law ,
P ( n ) = 1 ( 2 π n ¯ ) 1 2 exp [ - ( n - n ¯ ) 2 2 n ¯ ]             Normal law .
G = 1000 / B . M . ,
φ c = 1.5 B . M . ( 2000 17.8 - 1 ) .
y = f ( x ) ,
y = f ( x ) ,
P { S Δ x = k } = 0 1 - k f ( x ) f ( x + k ) d x + k 1.0 f ( x - k ) f ( x ) d x ,
S Δ x Av = 0 1.0 k P { S Δ x = k } d k = 0 1.0 k d k 0 1 - k f ( x ) f ( x + k ) d x + 0 1.0 k d k k 1.0 f ( x - k ) f ( x ) d x .
P { S Δ x = k } = n = 0 N - k f ( x n ) f ( x n + k ) + n = k N f ( x n - k ) f ( x n ) ,
P k = [ f ( x 0 ) f ( x 0 + k ) + f ( x 0 ) f ( x k ) ] + [ f ( x 1 ) f ( x 1 + k ) + f ( x 1 ) f ( x 1 + k ) ] + + [ f ( x N - k ) f ( x N ) + f ( x N - k ) f ( x N ) ] .
P 0 = 2 [ f ( x 0 ) f ( x 0 ) + f ( x 1 ) f ( x 1 ) + f ( x 2 ) f ( x 2 ) + f ( x 3 ) f ( x 3 ) + f ( x 4 ) f ( x 4 ) ] P 1 = 2 [ f ( x 0 ) f ( x 1 ) + f ( x 1 ) f ( x 2 ) + f ( x 2 ) f ( x 3 ) + f ( x 3 ) f ( x 4 ) ] P 2 = 2 [ f ( x 0 ) f ( x 2 ) + f ( x 1 ) f ( x 3 ) + f ( x 2 ) f ( x 4 ) ] P 3 = 2 [ f ( x 0 ) f ( x 3 ) + f ( x 1 ) f ( x 4 ) ] P 4 = 2 [ f ( x 0 ) f ( x 4 ) ] .