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 n 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.

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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).
  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 (v) 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 vN=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).
  8. J. H. Altman, App1. Opt. 3, 35 (1964).
  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).
  14. Reference 4, Sec. 3.1.1.

1971 (1)

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

1969 (1)

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

1964 (1)

J. H. Altman, App1. Opt. 3, 35 (1964).

1958 (1)

E. L. O'Neill, J. Opt. Soc. Am. 48, 945 (1958).

1913 (1)

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

Altman, J. H.

J. H. Altman, App1. Opt. 3, 35 (1964).

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).

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).

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

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.

Other (14)

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.

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

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 (v) 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 vN=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.

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

J. H. Altman, App1. Opt. 3, 35 (1964).

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.

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

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

E. L. O'Neill, J. Opt. Soc. Am. 48, 945 (1958).

Reference 4, Sec. 3.1.1.

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