Abstract

Photographic noise is studied from the point of view of continuous-parameter Markov chains. A general, conditional-probability distribution of the undeveloped grains in a cell is used for a pure-death process. The corresponding mean and variance are obtained. A new theoretical expression relating rms density to average density is also determined. The application of this probability distribution to a typical photographic film is demonstrated and the informational sensitivity, defined for the effectiveness of information transmission, is also derived. Finally, it is concluded that the photographic process is nonhomogeneous in the sense that the probability of grain development (death) is a function of exposure time.

© 1969 Optical Society of America

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References

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  1. R. C. Jones, J. Opt. Soc. Am. 45, 799 (1955).
    [CrossRef]
  2. H. J. Zweig, J. Opt. Soc. Am. 46, 805 (1956).
    [CrossRef]
  3. H. J. Zweig, J. Opt. Soc. Am. 46, 812 (1956).
    [CrossRef]
  4. H. J. Zweig, J. Opt. Soc. Am. 49, 238 (1959).
    [CrossRef]
  5. B. Picinbono, Compt. Rend. 240, 2206 (1955).
  6. M. Savelli, Compt. Rend. 246, 3605 (1958).
  7. E. Parzen, Stochastic Processes (Holden-Day Publ. Co., San Francisco, 1962), p. 288 ff.
  8. J. H. Altman and H. J. Zweig, Phot. Sci. Eng. 7, 173 (1963).
  9. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw–Hill Book Co., New York, 1965), prob. 15–7, p. 552 ff.
  10. C. E. K. Mees, The Theory of the Photographic Process (The Macmillan Book Co., New York, 1954), rev. ed., p. 162 ff.
  11. Reference 9, p. 300 ff.
  12. H. J. Zweig, G. C. Higgins, and D. L. MacAdam, J. Opt. Soc. Am. 48, 926 (1958).
    [CrossRef]
  13. C. E. K. Mees and T. H. James, The Theory of the Photographic Process (The Macmillan Book Co., New York, 1966), 3rd ed., p. 540 ff.

1963 (1)

J. H. Altman and H. J. Zweig, Phot. Sci. Eng. 7, 173 (1963).

1959 (1)

1958 (2)

1956 (2)

1955 (2)

B. Picinbono, Compt. Rend. 240, 2206 (1955).

R. C. Jones, J. Opt. Soc. Am. 45, 799 (1955).
[CrossRef]

Altman, J. H.

J. H. Altman and H. J. Zweig, Phot. Sci. Eng. 7, 173 (1963).

Higgins, G. C.

James, T. H.

C. E. K. Mees and T. H. James, The Theory of the Photographic Process (The Macmillan Book Co., New York, 1966), 3rd ed., p. 540 ff.

Jones, R. C.

MacAdam, D. L.

Mees, C. E. K.

C. E. K. Mees, The Theory of the Photographic Process (The Macmillan Book Co., New York, 1954), rev. ed., p. 162 ff.

C. E. K. Mees and T. H. James, The Theory of the Photographic Process (The Macmillan Book Co., New York, 1966), 3rd ed., p. 540 ff.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw–Hill Book Co., New York, 1965), prob. 15–7, p. 552 ff.

Parzen, E.

E. Parzen, Stochastic Processes (Holden-Day Publ. Co., San Francisco, 1962), p. 288 ff.

Picinbono, B.

B. Picinbono, Compt. Rend. 240, 2206 (1955).

Savelli, M.

M. Savelli, Compt. Rend. 246, 3605 (1958).

Zweig, H. J.

Compt. Rend. (2)

B. Picinbono, Compt. Rend. 240, 2206 (1955).

M. Savelli, Compt. Rend. 246, 3605 (1958).

J. Opt. Soc. Am. (5)

Phot. Sci. Eng. (1)

J. H. Altman and H. J. Zweig, Phot. Sci. Eng. 7, 173 (1963).

Other (5)

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw–Hill Book Co., New York, 1965), prob. 15–7, p. 552 ff.

C. E. K. Mees, The Theory of the Photographic Process (The Macmillan Book Co., New York, 1954), rev. ed., p. 162 ff.

Reference 9, p. 300 ff.

C. E. K. Mees and T. H. James, The Theory of the Photographic Process (The Macmillan Book Co., New York, 1966), 3rd ed., p. 540 ff.

E. Parzen, Stochastic Processes (Holden-Day Publ. Co., San Francisco, 1962), p. 288 ff.

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Figures (4)

Fig. 1
Fig. 1

Solid curves: Density of the developed grains as a function of exposure time t, for various values of irradiance I. Dashed curves: Slope of the solid curves against the exposure time t.

Fig. 2
Fig. 2

Mean of the developed and the undeveloped grains as a function of exposure time t.

Fig. 3
Fig. 3

Mean-square fluctuation of the grains as a function of exposure time t.

Fig. 4
Fig. 4

Informational sensitivity as a function of average density D.

Equations (18)

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P m ( t ) = P [ x ( t ) = m x ( 0 ) = M ] ,             x ( t ) = 0 , 1 , 2 , , M ,
d P M ( t ) / d t = - M q ( t ) P M ( t ) ,             for m = M
d P m ( t ) / d t = - m q ( t ) P m ( t ) + ( m + 1 ) q ( t ) P m + 1 ( t ) ,             for m < M ,
P M ( 0 ) = 1
P m ( 0 ) = 0 ,             for m < M ,
P m ( t ) = M ! ( M - m ) ! m ! [ 1 - e - ρ ( t ) ] M - m e - m ρ ( t ) ,             for m M ,
ρ ( t ) = 0 t q ( t ) d t .
m ¯ = M e - ρ ( t )
σ 2 = M e - ρ ( t ) [ 1 - e - ρ ( t ) ] .
σ = [ m ¯ ( 1 - m ¯ M ) ] 1 2 = [ n ¯ ( 1 - n ¯ M ) ] 1 2
d D d t = { K M ( t / α 2 ) exp [ - 1 2 ( t / α ) 2 ] , for t 0 0 , otherwise ,
D = K ( M - m ¯ ) .
d D / d t = K M q ( t ) e - ρ ( t ) .
q ( t ) = t / α 2 ,
ρ ( t ) = 1 2 ( t / α ) 2 .
g = d D / d E ,
g = ( K M / 1 ) ( t / α 2 ) exp [ - 1 2 ( t / α ) 2 ]
g σ = K ( M 1 2 ) I t α 2 [ 1 e ρ ( t ) - 1 ] 1 2 .