Abstract

Photographic grains do not always behave like Poisson-distributed disks, as the random-dot model assumes. Because real grains cannot occupy the same space at the same time, they crowd one another out at high densities, imposing constraints on their locations, which, consequently, cannot be completely random. The theory presented here takes this into account for nonscattering grains illuminated by collimated light. It predicts rms density fluctuations that can sometimes reach a peak at a mean density less than dmax and then decrease beyond this point. This is explained by granularity spectra that, at high densities, have their greatest power per unit bandwidth at high frequencies that are strongly attenuated by the aperture. These results, markedly different from those predicted by the familiar random-dot model of grains overlapping in multilayers, are by no means universally true; for crowded monolayers, in which grains tend to stick to one another, can have granularity curves so similar to those for uncrowded multilayers as to be virtually indistinguishable in practice.

© 1971 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Silberstein and A. P. H. Trivelli, J. Opt. Soc. Am. 28, 441 (1938).
    [Crossref]
  2. M. Picinbono, Compt. Rend. 240, 23 (1955).
  3. M. Savelli, Compt. Rend. 244, 1710 (1957).
  4. J. C. Marchant and P. L. P. Dillon, J. Opt. Soc. Am. 51, 641 (1961).
    [Crossref]
  5. E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 115.
  6. B. E. Bayer, J. Opt. Soc. Am. 54, 1485 (1964).
    [Crossref]
  7. E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962), p. 32.
  8. Reference 7, p. 56.
  9. E. C. Pielou, An Introduction to Mathematical Ecology (Wiley–Interscience, New York, 1969).
  10. E. A. Trabka, J. Opt. Soc. Am. 59, 662 (1969).
    [Crossref]
  11. J. H. Altman, Appl. Opt. 3, 35 (1964).
    [Crossref]
  12. D. R. Cox, Renewal Theory (Wiley, New York, 1962), p. 40.
  13. Reference 12, p. 32.
  14. Reference 12, p. 48.
  15. W. L. Smith, J. Roy. Statist. Soc. 20B, 267 (1958).
  16. G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, 1968), p. 285.
  17. E. Parzen, Technometrics 3, 171 (1961).
    [Crossref]
  18. M. Savelli, Compt. Rend. 246, 229 (1958).
  19. Reference 12, pp. 46, 55.
  20. Reference 12, p. 54.
  21. W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 102.
  22. D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events (Wiley, New York, 1966), p. 74.
  23. Reference 12, p. 63.

1969 (1)

1964 (2)

1961 (2)

1958 (2)

M. Savelli, Compt. Rend. 246, 229 (1958).

W. L. Smith, J. Roy. Statist. Soc. 20B, 267 (1958).

1957 (1)

M. Savelli, Compt. Rend. 244, 1710 (1957).

1955 (1)

M. Picinbono, Compt. Rend. 240, 23 (1955).

1938 (1)

Altman, J. H.

Bayer, B. E.

Cox, D. R.

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events (Wiley, New York, 1966), p. 74.

D. R. Cox, Renewal Theory (Wiley, New York, 1962), p. 40.

Davenport, W. B.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 102.

Dillon, P. L. P.

Jenkins, G. M.

G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, 1968), p. 285.

Lewis, P. A. W.

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events (Wiley, New York, 1966), p. 74.

Marchant, J. C.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 115.

Parzen, E.

E. Parzen, Technometrics 3, 171 (1961).
[Crossref]

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962), p. 32.

Picinbono, M.

M. Picinbono, Compt. Rend. 240, 23 (1955).

Pielou, E. C.

E. C. Pielou, An Introduction to Mathematical Ecology (Wiley–Interscience, New York, 1969).

Root, W. L.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 102.

Savelli, M.

M. Savelli, Compt. Rend. 246, 229 (1958).

M. Savelli, Compt. Rend. 244, 1710 (1957).

Silberstein, L.

Smith, W. L.

W. L. Smith, J. Roy. Statist. Soc. 20B, 267 (1958).

Trabka, E. A.

Trivelli, A. P. H.

Watts, D. G.

G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, 1968), p. 285.

Appl. Opt. (1)

Compt. Rend. (3)

M. Picinbono, Compt. Rend. 240, 23 (1955).

M. Savelli, Compt. Rend. 244, 1710 (1957).

M. Savelli, Compt. Rend. 246, 229 (1958).

J. Opt. Soc. Am. (4)

J. Roy. Statist. Soc. (1)

W. L. Smith, J. Roy. Statist. Soc. 20B, 267 (1958).

Technometrics (1)

E. Parzen, Technometrics 3, 171 (1961).
[Crossref]

Other (13)

Reference 12, pp. 46, 55.

Reference 12, p. 54.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958), p. 102.

D. R. Cox and P. A. W. Lewis, The Statistical Analysis of Series of Events (Wiley, New York, 1966), p. 74.

Reference 12, p. 63.

G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, 1968), p. 285.

D. R. Cox, Renewal Theory (Wiley, New York, 1962), p. 40.

Reference 12, p. 32.

Reference 12, p. 48.

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962), p. 32.

Reference 7, p. 56.

E. C. Pielou, An Introduction to Mathematical Ecology (Wiley–Interscience, New York, 1969).

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 115.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

F. 1
F. 1

Normalized rms density fluctuations N 1 2 σ d / k vs mean density μd of monolayer for various values of the Poisson excess when tmin = 0.3 (dmax = 0.52). The dashed curve labeled = 0 corresponds to a completely random or Poisson distribution of grains while those for < 0 correspond to constrained randomness due to grains of finite size crowding one another.

F. 2
F. 2

Log of normalized rms density fluctuations N 1 2 σ d / k vs log of mean density μd predicted by the random-dot model (points marked ○) and the best-fitting crowded monolayer (solid curve) with dmax = 0.5.

F. 3
F. 3

Absorptance profiles for crowded monolayers with 0.32 average transmittance or 0.5 average density. (a) Sticking probability p0 = 0. (b) Sticking probability p0 = 0.4.

F. 4
F. 4

Normalized variance σ2L of the number of events in an interval of length L at the output of a type-I event counter with dead-time l vs the average number λL at its input. The parameter λl denotes the average number of input events in a dead-time interval. The vertical bars on each curve mark the left-hand end of the asymptotic region as estimated by Cox. Points denoted by ○ were obtained by Monte Carlo simulation.

F. 5
F. 5

Wiener spectrum P(ν) of transmittance fluctuations in a completely exposed crowded monolayer with no sticking vs ν in cycles per unit length for tmin = 0.3,p0 = 0, C2 = 0.09, and = −0.91. 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.

F. 6
F. 6

Wiener spectrum P(ν) of transmittance fluctuations in a completely exposed crowded monolayer with moderate sticking vs ν in cycles per unit length for tmin = 0.3, p0 = 0.38, C2 = 0.2, and = − 0.8. 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.

F. 7
F. 7

Wiener spectrum P(ν) of transmittance fluctuations in a completely exposed crowded monolayer with intense sticking vs ν in cycles per unit length for tmin = 0.3, p0 = 0.63, C2 = 0.4, and = − 0.6. 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.

F. 8
F. 8

Normalized Wiener spectrum P(υ)/l of transmittance fluctuations in completely exposed crowded monolayers with no sticking vs υ = πνl, where ν is in cycles per unit length and l is the grain length. Curves show how the shape of the spectrum depends upon the minimum transmittance.

F. 9
F. 9

Normalized Wiener spectrum P(υ)/l of transmittance fluctuations in partially exposed crowded monolayers with no sticking vs υ = πνl, where v is in cycles per unit length and l is the grain length. Curves show how the shape of the spectrum depends upon the average transmittance when the minimum possible transmittance for complete exposure is tmin = 0.15.

F. 10
F. 10

Normalized rms density fluctuations N 1 2 σ d / k vs mean density μd of crowded monolayers with no sticking. The parameter tmin denotes the minimum transmittance possible when all grains develop.

Equations (89)

Equations on this page are rendered with MathJax. Learn more.

p ( r ) = λ 4 π r 2 e λ ( 4 / 3 ) π r 3 .
P ( r ) = 1 e λ ( 4 / 3 ) π r 3 .
β = P ( 2 r 0 ) .
n ¯ = λ δ 2 r 0 .
β = 1 e ( 16 / 3 ) ( n ¯ a / δ ) ,
δ = 16 D / [ 3 log ( 1 β ) ] .
t = 1 ( s / N ) .
μ t = 1 ( μ s / N ) ,
σ t 2 = σ s 2 / N 2 .
z i = 0 , if the i th grain does not get a developable exposure ; = 1 , otherwise .
s = z 1 + z 2 + + z h ,
μ s = μ h μ z ,
σ s 2 = μ h σ z 2 + μ z 2 σ h 2 .
μ z = p ,
σ z 2 = p ( 1 p ) .
μ t = 1 p n ¯ a ,
N σ t 2 = ( 1 μ t ) ( 1 + p ) ,
n ¯ = μ h / A
= ( σ h 2 / μ h ) 1
d = log t ,
μ d = log μ t ,
σ d 2 = k 2 σ t 2 / μ t 2 ,
μ d = log ( 1 p n ¯ a ) ,
N σ d 2 / k 2 = [ ( 1 μ t ) / μ t 2 ] ( 1 + p ) .
N σ d 2 k 2 = 10 2 μ d ( 1 10 μ d ) { 1 + 1 10 μ d 1 10 d max } ,
d max = log ( 1 n ¯ a )
N 1 2 σ d / k = c ( μ d / k ) 1 2 ,
μ h ( 1 / μ x ) L ,
σ h 2 ( σ x 2 / μ x 3 ) L ,
σ h 2 / μ h = C 2 ,
C 2 = σ x 2 / μ x 2 .
= C 2 1
f ( x ) = p 0 δ ( x l ) + ( 1 p 0 ) U ( x l ) λ e λ ( x l ) ,
μ x = ( 1 + λ l p 0 ) / λ ,
σ x 2 = ( 1 p 0 2 ) / λ 2 ,
C 2 = ( 1 p 0 2 ) / ( 1 + λ l p 0 ) 2 .
μ t = [ 1 + ( 1 p ) λ l p 0 ] / ( 1 + λ l p 0 ) ,
t min = ( 1 p 0 ) / ( 1 + λ l p 0 )
C 2 = [ ( 1 + p 0 ) / ( 1 p 0 ) ] ( t min ) 2 .
d max = log t min .
L μ x 3 / σ x 2 .
λ L ( 1 + λ l ) 3 .
σ h 2 / λ L ( 1 + λ L ) 3 .
P ( 0 ) = 2 l ( 1 μ t ) ( 1 + p ) .
σ T 2 B P ( 0 ) .
σ t 2 = μ t ( 1 μ t ) ,
a = b 1 / Δ ( lags ) ,
ρ = 2 b 1 ( points ) / ( lags ) .
μ h = L / μ x
σ h 2 = ψ ( L ) + μ h μ h 2 .
ψ * ( s ) = 2 / { s 2 μ x [ 1 f * ( s ) ] } ,
f * ( s ) = λ e s l / ( S + λ ) .
ψ * ( s ) = 2 s 2 μ x [ 1 + n = 1 λ n e s n l ( s + λ ) n ] ,
σ h 2 λ L = 1 1 + w { 2 ν + 1 z 1 + w 1 z [ ν ( ν + 1 ) + w ν ( ν + 1 ) 2 n = 1 ν e ( z n w ) j = 0 n 1 ( n j ) ( z n w ) j j ! ] } ,
ν = [ L / l ] = greatest interger L / l
w = λ l , z = λ L .
P ( w ) = 2 + c ( τ ) e i w τ d τ
c ( τ ) = cov { t ( u ) , t ( u + τ ) } = cov { a ( u ) , a ( u + τ ) } .
S n = i = 1 n x i .
a ( u ) = n = 1 A ( u S n ; y n )
a 1 ( u ) = Ê [ a ( u ) ] ,
a 2 ( u , υ ) = Ê [ a ( u , y n ) a ( υ , y n ) ] ,
E [ a ( u ) ] μ x 1 0 a 1 ( θ ) d θ ,
μ x = E [ x i ]
P ( ω ) = ( 2 / μ x ) [ γ 2 ( ω , ω ) + 2 | γ 1 ( ω ) | 2 Re ρ * ( i ω ) ] ,
γ 1 ( ζ ) = 0 e i ζ u a 1 ( u ) d u ,
γ 2 ( ξ , η ) = 0 0 e i ( ξ u + η υ ) a 2 ( u , υ ) dud υ ,
ρ * ( s ) = 0 e s u ρ ( u ) d u
ρ * ( s ) = [ f * ( s ) ] / [ 1 f * ( s ) ] ,
A ( u ) = A 0 ( 0 u l ) = 0 ( u < 0 , u > l ) ,
a 1 ( u ) = p ( 0 u l ) = 0 ( u < 0 , u > l ) ,
a 2 ( u , υ ) = ( 1 / p ) a 1 ( u ) a 1 ( υ ) ,
γ 1 ( ζ ) = p l e ζ l / 2 [ sin ( ζ l / 2 ) ] / ( ζ l / 2 )
γ 2 ( ξ , η ) = ( 1 / p ) γ 1 ( ξ ) γ 2 ( η ) ,
E [ a ( u ) ] = ( p l ) / μ x ,
P ( ω ) = ( 2 / p μ x ) | γ 1 ( ω ) | 2 [ 1 + 2 p Re ρ * ( i ω ) ] .
| γ 1 ( ω ) | 2 = ( p 2 l 2 sin 2 X ) / X 2
X = ω l / 2
E [ t ( u ) ] = t min ( 1 + 1 p 1 p 0 λ l ) ,
ρ * ( s ) = [ ( s p 0 + λ ) / ( s + λ ) ] e τ s
2 Re ρ * ( i ω ) = Z 2 X 2 p 0 ( 1 p 0 ) X 2 ( 1 p 0 ) 2 + Z ,
Z = sin X [ ( 4 X 2 p 0 + λ 2 l 2 ) sin X + 2 X λ l ( 1 p 0 ) cos X ] .
α = ( 1 t min ) / t min , q 0 = 1 p 0 ,
λ l = q 0 α ,
q 0 = 2 / [ C 2 ( 1 + α ) 2 + 1 ] ,
x = l ( υ < p 0 ) = l λ 1 ln [ ( 1 υ ) / ( 1 p 0 ) ] ( υ p 0 ) .
g ( x 0 ) = μ x 1 x f ( u ) d u .
x 0 = [ ( 1 + λ l p 0 ) / λ ] υ
x 0 = l 1 λ ln { 1 υ [ 1 + λ l p 0 ] λ l 1 p 0 }