Abstract

A crowded photographic emulsion is viewed as a sandwich of stacked, crowded monolayers. An earlier renewal model of granularity in a crowded monolayer, combined with a new analysis of the general way in which granularity propagates through layers, leads to predictions of the granularity of the multilayer sandwich as a function of the number of layers. For a fixed concentration of grains per unit projected area in the sandwich, rms density fluctuations increase as the number of layers decreases because rms transmittance fluctuations decrease at a slower rate than mean transmittance. These changes are similar to the entropy decrease of grain configurations in the emulsion. For sandwiches consisting of at least 15 layers having a maximum density not greater than 2, the change of rms density fluctuations vs mean density for an exposure series is accurately predicted by the honest random-dot model. Any discrepancy between the theoretical predictions of the honest random-dot model and experimental data for normal emulsions cannot be attributed solely to the neglect of crowding constraints by that model.

© 1972 Optical Society of America

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References

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  1. We are indebted to Dr. George C. Higgins of the Kodak Research Laboratories for this query.
  2. For a review of the literature on photographic granularity, see the paper cited in Ref. 4.
  3. E. A. Trabka, J. Opt. Soc. Am. 61, 800 (1971).
    [Crossref]
  4. J. F. Hamilton, W. H. Lawton, and E. A. Trabka, in Stochastic Point Processes: Statistical Analysis, Theory and Applications, edited by P. A. W. Lewis (Wiley, New York, 1972), Sec. 3.1.2.
  5. All theoretical spectra P†(ν) depicted in this paper are single sided in the sense that the area above the positive frequency axis yields the total variance. Moreover, they have been folded (see Ref. 6) about the Nyquist frequency νN= 2/l, corresponding to a sampling interval ΔX= l/4 with grain length l= 4. Consequently, the spectra shown are what we would expect to get by sampling realizations of the process at this rate.
  6. R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959), p. 120.
  7. E. A. Trabka, J. Opt. Soc. Am. 59, 662 (1969).
    [Crossref]
  8. J. H. Altman, Appl. Opt. 3, 35 (1964).
    [Crossref]
  9. Reference 4, Sec. 2.2.
  10. The sticking probability p0 of Ref. 3 is here and in Ref. 4 referred to as a packing factor, since the former term has connotations we wish to avoid.
  11. P. G. Nutting, Phil. Mag. No. 6,  26, 423 (1913).
  12. See E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 117.
  13. E. L. O’Neill, J. Opt. Soc. Am. 48, 945 (1958).
    [Crossref]
  14. Reference 4, Sec. 3.1.1.

1971 (1)

1969 (1)

1964 (1)

1958 (1)

1913 (1)

P. G. Nutting, Phil. Mag. No. 6,  26, 423 (1913).

Altman, J. H.

Blackman, R. B.

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959), p. 120.

Hamilton, J. F.

J. F. Hamilton, W. H. Lawton, and E. A. Trabka, in Stochastic Point Processes: Statistical Analysis, Theory and Applications, edited by P. A. W. Lewis (Wiley, New York, 1972), Sec. 3.1.2.

Lawton, W. H.

J. F. Hamilton, W. H. Lawton, and E. A. Trabka, in Stochastic Point Processes: Statistical Analysis, Theory and Applications, edited by P. A. W. Lewis (Wiley, New York, 1972), Sec. 3.1.2.

Nutting, P. G.

P. G. Nutting, Phil. Mag. No. 6,  26, 423 (1913).

O’Neill, E. L.

E. L. O’Neill, J. Opt. Soc. Am. 48, 945 (1958).
[Crossref]

See E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 117.

Trabka, E. A.

E. A. Trabka, J. Opt. Soc. Am. 61, 800 (1971).
[Crossref]

E. A. Trabka, J. Opt. Soc. Am. 59, 662 (1969).
[Crossref]

J. F. Hamilton, W. H. Lawton, and E. A. Trabka, in Stochastic Point Processes: Statistical Analysis, Theory and Applications, edited by P. A. W. Lewis (Wiley, New York, 1972), Sec. 3.1.2.

Tukey, J. W.

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959), p. 120.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

Phil. Mag. No. 6 (1)

P. G. Nutting, Phil. Mag. No. 6,  26, 423 (1913).

Other (9)

See E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 117.

Reference 4, Sec. 3.1.1.

J. F. Hamilton, W. H. Lawton, and E. A. Trabka, in Stochastic Point Processes: Statistical Analysis, Theory and Applications, edited by P. A. W. Lewis (Wiley, New York, 1972), Sec. 3.1.2.

All theoretical spectra P†(ν) depicted in this paper are single sided in the sense that the area above the positive frequency axis yields the total variance. Moreover, they have been folded (see Ref. 6) about the Nyquist frequency νN= 2/l, corresponding to a sampling interval ΔX= l/4 with grain length l= 4. Consequently, the spectra shown are what we would expect to get by sampling realizations of the process at this rate.

R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra (Dover, New York, 1959), p. 120.

Reference 4, Sec. 2.2.

The sticking probability p0 of Ref. 3 is here and in Ref. 4 referred to as a packing factor, since the former term has connotations we wish to avoid.

We are indebted to Dr. George C. Higgins of the Kodak Research Laboratories for this query.

For a review of the literature on photographic granularity, see the paper cited in Ref. 4.

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Figures (11)

Fig. 1
Fig. 1

Limiting cases of point transmittance spectra P(ν) vs ν in cycles per unit length for μt = 0.4. The dashed curve applies when all grains are in one layer, whereas the solid curve is the prediction of the one-dimensional random-dot model for an effectively infinite number of layers. The spatial frequency ν = 1 4 is the reciprocal of the grain length l = 4.

Fig. 2
Fig. 2

Point transmittance spectrum P(ν) vs ν in cycles per unit length for a completely exposed monolayer with μt = 0.4. The continuous curve is theoretical, whereas the curve with jumps depicts spectral estimates from Monte Carlo simulation whose resolution is given by the length of the line segment between jumps.

Fig. 3
Fig. 3

Monolayer covariance Ct(r) vs displacement r for grains of length l = 4 corresponding to the spectrum shown in Fig. 2.

Fig. 4
Fig. 4

Point transmittance spectrum P(ν) vs ν in cycles per unit length for a two-layer sandwich with μπ = 0.16. The continuous curve is theoretical, whereas the curve with jumps depicts spectral estimates from Monte Carlo simulation whose resolution is given by the length of the line segment between jumps.

Fig. 5
Fig. 5

Point transmittance spectra P(ν) vs ν in cycles per unit length for sandwiches of m = 1, 2, 4, 8, ∞ layers with μπ = 0.16 showing the conversion of high-frequency fluctuations to low-frequency fluctuations as the number of layers increases.

Fig. 6
Fig. 6

Mean transmittance μΠ of a multilayer sandwich vs the number m of layers for a fixed concentration n ¯ such that μΠ = 0.01 when m = ∞ as indicated by the dashed horizontal asymptote. The points denoted by + are from Monte Carlo simulation.

Fig. 7
Fig. 7

Mean density μΔ (solid curve) of a multilayer sandwich vs the number m of layers for a fixed concentration n ¯ such that μΔ = 2.0 when m = ∞. The points denoted by + are from Monte Carlo simulation. The dashed curve is the normalized change in entropy (l/L) H ˜ of grain configurations in the emulsion.

Fig. 8
Fig. 8

Normalized rms aperture transmittance N 1 2 σ Π of a multilayer sandwich vs the number m of layers for a fixed concentration n ¯ such that μΔ = 2.0 when m = ∞. The dashed horizontal asymptote is the prediction of the honest random-dot model. The points denoted by + are from Monte Carlo simulation.

Fig. 9
Fig. 9

Normalized rms aperture density ( N 1 2 K - 1 ) σ Δ of a multilayer sandwich vs the number m of layers for a fixed concentration n ¯ such that μΔ = 2.0 when m = ∞. The dashed horizontal asymptote is the prediction of the honest random-dot model. The points denoted by + are from Monte Carlo simulation.

Fig. 10
Fig. 10

Relative error using the honest random-dot model vs the number m of layers for concentration n ¯ yielding various values of D = K n ¯ a. The solid curves show predictions of mean density μΔ, whereas the points denoted by circles are predictions of rms aperture density.

Fig. 11
Fig. 11

Normalized rms aperture density ( N 1 2 K - 1 ) σ Δ vs mean density μΔ for sandwiches of opaque grains in m = 1, 5, ∞ layers, each with (μΔ)max = 2.0 at complete exposure. The dashed curve corresponds to the dishonest random-dot model, whereas the curve for m = ∞ corresponds to the honest random-dot model.

Tables (1)

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Table I Notation.

Equations (55)

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σ t 2 = μ t ( 1 - μ t ) ,
T ( P ) = 1 A A ( P ) t ( Q ) d Q .
T = lim A T ( P )
μ T = μ t = T .
π ( P ) = i = 1 m t i ( P ) .
Π ( P ) i = 1 m T i ( P ) ,
Π = E [ i = 1 m t i ( P ) ] = i = i m E [ t i ( P ) ] = T m .
Π = T m
μ π = μ t m , μ Π = μ T m .
C π ( r ) = E [ π ( P ) π ( P + Q ) ] - E [ π ( P ) ] E [ π ( P + Q ) ] ,
C π ( r ) = [ C t ( r ) + T 2 ] m - T 2 m .
C π ( r ) = j = 1 m ( m j ) [ C t ( r ) ] j T 2 ( m - j ) .
P π ( ω ) = - + C π ( r ) exp ( i ω · r ) d r ,
P π ( ω ) = j = 1 m ( m j ) T 2 ( m - j ) F j ( ω ) ,
F j ( ω ) = - + [ C t ( r ) ] j exp ( i ω · r ) d r .
P π ( ω ) Π 2 = j = 1 m ( m j ) ( P t ( ω ) T 2 ) * j ,
P Π ( ω ) = H ( ω ) 2 P π ( ω ) ,
σ Π 2 B P π ( 0 ) ,
A σ Π 2 - + C π ( r ) d r ,
D = - log 10 T
Δ = m D .
d ( P ) = - log 10 μ t - K μ t - 1 [ t ( P ) - μ t ]
D ( P ) = - log 10 T - K T - 1 [ T ( P ) - T ] .
μ d = μ D = D ,
C d ( r ) = K 2 C t ( r ) T - 2
C D ( r ) = K 2 C T ( r ) T - 2 .
σ d 2 = K 2 σ t 2 T - 2 ,
σ D 2 = K 2 σ T 2 T - 2 .
P d ( ω ) = K 2 P t ( ω ) T - 2 ,
P D ( ω ) = K 2 P T ( ω ) T - 2 .
μ δ = m μ d , μ Δ = m μ D .
δ ( P ) m d ( P ) , Δ ( P ) m D ( P ) .
D ( P ) = - log T ( P ) ,
C δ ( r ) = K - 2 ( m - 1 ) [ C d ( r ) + K 2 ] m - K 2
C δ ( r ) = j = 1 m ( m j ) K - 2 ( j - 1 ) [ C d ( r ) ] j .
P δ ( ω ) = j = 1 m ( m j ) K - 2 ( j - 1 ) [ P d ( ω ) ] * j .
σ Δ 2 = K 2 σ Π 2 T - 2
σ Δ 2 m σ D 2 + A - 1 j = 2 m ( m j ) K - 2 ( j - 1 ) × - + [ C d ( r ) ] j d r .
C t ( r ) = μ t 2 [ μ t - Γ ( r ) - 1 ] ,
T = μ T = μ t = 1 - ρ a ,
μ Π = [ 1 - ( n ¯ a / m ) ] m ,
μ Δ = - m log 10 [ 1 - ( n ¯ a ) / m ] .
lim m μ Δ = D = K n ¯ a .
( l / L ) H ˜ = - n ¯ l log [ 1 - ( n ¯ l / m ) ] ,
μ Δ = - m log 10 [ 1 - p ( n ¯ a ) / m ] ;
x = ( x 1 , x 2 , , x N )
H ( P ) = - R P ( x ) log P ( x ) d x .
e i ( c ) = x i + ( i - 1 ) l
e i ( u ) = x i
H ˜ = m ( H c - H u ) .
P I ( x ) = N ! I - N
H ( P I ) = - log ( N ! I - N ) .
H c = H ( P L - N l ) ,
H u = H ( P L ) .
H ˜ = - n ¯ L log [ 1 - ( n ¯ l / m ) ] ,