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 d<sub>max</sub> 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.

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Altman, J. H.

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

Bayer, B. E.

B. E. Bayer, J. Opt. Soc. Am. 54, 1485 (1964).

Cox, D. R.

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

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

Davenport, Jr., W. B.

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

Dillon, P. L. P.

J. C. Marchant and P. L. P. Dillon, J. Opt. Soc. Am. 51, 641 (1961).

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.

J. C. Marchant and P. L. P. Dillon, J. Opt. Soc. Am. 51, 641 (1961).

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).

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, Jr. and W. L. Root, An itntroduction to the Theory of Random Signals and Noise (McGraw–Hill, New York, 1958), p. 102.

Savelli, M.

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

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

Silberstein, L.

L. Silberstein and A. P. H. Trivelli, J. Opt. Soc. Am. 28, 441 1938).

Smith, W. L.

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

Trabka, E. A.

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

Trivelli, A. P. H.

L. Silberstein and A. P. H. Trivelli, J. Opt. Soc. Am. 28, 441 1938).

Watts, D. G.

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

Other (23)

L. Silberstein and A. P. H. Trivelli, J. Opt. Soc. Am. 28, 441 1938).

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

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

J. C. Marchant and P. L. P. Dillon, J. Opt. Soc. Am. 51, 641 (1961).

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

B. E. Bayer, J. Opt. Soc. Am. 54, 1485 (1964).

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

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

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

Reference 12, p. 32.

Reference 12, p. 48.

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

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

E. Parzen, Technometrics 3, 171 (1961).

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

Reference 12, pp. 46, 55.

Reference 12, p. 54.

W. B. Davenport, Jr. and W. L. Root, An itntroduction 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.

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