Abstract

When image plane speckle intensity integrated over a finite aperture is submitted to a logarithmic transformation, the noise becomes additive and signal independent. The first- and second-order moments of the probability distribution are derived. It is found that the logarithm of speckle noise approaches a normal distribution much faster than speckle intensity. The properties of speckle noise are different from those of film-grain noise; for example, neither Nutting’s law nor Selwyn’s law is satisfied by speckle.

© 1976 Optical Society of America

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References

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  1. H. H. Arsenault and N. Brousseau, “Digital Image Processing and Preprocessing,” J. Opt. Soc. Am. 65, 1203 (1975).
  2. J. F. Walkup and R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258 (1974).
    [Crossref]
  3. L. S. Barbanel, “Signal discrimination and dection in the presence of nonadditive noise,” Opt. Spectrosc. (USSR) 33, 1145 (1972).
  4. J. W. Goodman and H. Kato, “Nonlinear filtering in coherent optical systems through halftone screen processes,” Appl. Opt. 14, 1813 (1975).
    [Crossref] [PubMed]
  5. T. G. Stockham, “Image processing in the context of a visual model,” Proc. IEEE 60, 828 (1972).
    [Crossref]
  6. S. Lowenthal and H. H. Arsenault, “Image formation for coherent diffuse objects: statistical properties,” J. Opt. Soc. Am. 60, 1478 (1970).
    [Crossref]
  7. J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688 (1965).
    [Crossref]
  8. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York), pp. 9–75.
  9. R. Barakat, “First-order probability densities of laser speckle patterns observed through finite size scanning apertures,” Opt. Acta 20, 729 (1973).
    [Crossref]
  10. M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions,” (Dover, New York, 1965), p. 930.
  11. Reference (10), p. 258.
  12. H. J. Gerritsen, W. J. Hannan, and E. G. Ramberg, “Elimination of speckle noise in holograms with redundancy,” Appl. Opt. 7, 2301 (1968).
    [Crossref] [PubMed]
  13. F. C. Billingsley, Noise Considerations in Digital Image Processing Hardware, in Picture Processing and Digital Filtering, edited by T. S. Huang (Springer-Verlag, New York, 1975), p. 259.
  14. H. H. Arsenault and G. April, “Speckle removal by optical and digital processing,” J. Opt. Soc. Am. 66, 177 (1976).

1976 (1)

H. H. Arsenault and G. April, “Speckle removal by optical and digital processing,” J. Opt. Soc. Am. 66, 177 (1976).

1975 (2)

J. W. Goodman and H. Kato, “Nonlinear filtering in coherent optical systems through halftone screen processes,” Appl. Opt. 14, 1813 (1975).
[Crossref] [PubMed]

H. H. Arsenault and N. Brousseau, “Digital Image Processing and Preprocessing,” J. Opt. Soc. Am. 65, 1203 (1975).

1974 (1)

J. F. Walkup and R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258 (1974).
[Crossref]

1973 (1)

R. Barakat, “First-order probability densities of laser speckle patterns observed through finite size scanning apertures,” Opt. Acta 20, 729 (1973).
[Crossref]

1972 (2)

L. S. Barbanel, “Signal discrimination and dection in the presence of nonadditive noise,” Opt. Spectrosc. (USSR) 33, 1145 (1972).

T. G. Stockham, “Image processing in the context of a visual model,” Proc. IEEE 60, 828 (1972).
[Crossref]

1970 (1)

1968 (1)

1965 (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688 (1965).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions,” (Dover, New York, 1965), p. 930.

April, G.

H. H. Arsenault and G. April, “Speckle removal by optical and digital processing,” J. Opt. Soc. Am. 66, 177 (1976).

Arsenault, H. H.

H. H. Arsenault and G. April, “Speckle removal by optical and digital processing,” J. Opt. Soc. Am. 66, 177 (1976).

H. H. Arsenault and N. Brousseau, “Digital Image Processing and Preprocessing,” J. Opt. Soc. Am. 65, 1203 (1975).

S. Lowenthal and H. H. Arsenault, “Image formation for coherent diffuse objects: statistical properties,” J. Opt. Soc. Am. 60, 1478 (1970).
[Crossref]

Barakat, R.

R. Barakat, “First-order probability densities of laser speckle patterns observed through finite size scanning apertures,” Opt. Acta 20, 729 (1973).
[Crossref]

Barbanel, L. S.

L. S. Barbanel, “Signal discrimination and dection in the presence of nonadditive noise,” Opt. Spectrosc. (USSR) 33, 1145 (1972).

Billingsley, F. C.

F. C. Billingsley, Noise Considerations in Digital Image Processing Hardware, in Picture Processing and Digital Filtering, edited by T. S. Huang (Springer-Verlag, New York, 1975), p. 259.

Brousseau, N.

H. H. Arsenault and N. Brousseau, “Digital Image Processing and Preprocessing,” J. Opt. Soc. Am. 65, 1203 (1975).

Choens, R. C.

J. F. Walkup and R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258 (1974).
[Crossref]

Gerritsen, H. J.

Goodman, J. W.

J. W. Goodman and H. Kato, “Nonlinear filtering in coherent optical systems through halftone screen processes,” Appl. Opt. 14, 1813 (1975).
[Crossref] [PubMed]

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688 (1965).
[Crossref]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York), pp. 9–75.

Hannan, W. J.

Kato, H.

Lowenthal, S.

Ramberg, E. G.

Stegun, I.

M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions,” (Dover, New York, 1965), p. 930.

Stockham, T. G.

T. G. Stockham, “Image processing in the context of a visual model,” Proc. IEEE 60, 828 (1972).
[Crossref]

Walkup, J. F.

J. F. Walkup and R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258 (1974).
[Crossref]

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

H. H. Arsenault and G. April, “Speckle removal by optical and digital processing,” J. Opt. Soc. Am. 66, 177 (1976).

H. H. Arsenault and N. Brousseau, “Digital Image Processing and Preprocessing,” J. Opt. Soc. Am. 65, 1203 (1975).

S. Lowenthal and H. H. Arsenault, “Image formation for coherent diffuse objects: statistical properties,” J. Opt. Soc. Am. 60, 1478 (1970).
[Crossref]

Opt. Acta (1)

R. Barakat, “First-order probability densities of laser speckle patterns observed through finite size scanning apertures,” Opt. Acta 20, 729 (1973).
[Crossref]

Opt. Eng. (1)

J. F. Walkup and R. C. Choens, “Image processing in signal-dependent noise,” Opt. Eng. 13, 258 (1974).
[Crossref]

Opt. Spectrosc. (USSR) (1)

L. S. Barbanel, “Signal discrimination and dection in the presence of nonadditive noise,” Opt. Spectrosc. (USSR) 33, 1145 (1972).

Proc. IEEE (2)

T. G. Stockham, “Image processing in the context of a visual model,” Proc. IEEE 60, 828 (1972).
[Crossref]

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688 (1965).
[Crossref]

Other (4)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York), pp. 9–75.

M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions,” (Dover, New York, 1965), p. 930.

Reference (10), p. 258.

F. C. Billingsley, Noise Considerations in Digital Image Processing Hardware, in Picture Processing and Digital Filtering, edited by T. S. Huang (Springer-Verlag, New York, 1975), p. 259.

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Figures (6)

FIG. 1
FIG. 1

The probability density of the relative intensity of speckle noise. I0 is the mean intensity. (See Ref. 8, p. 54).

FIG. 2
FIG. 2

The probability density of the logarithm of the intensity. When speckle noise intensity is transformed in this way, it becomes additive.

FIG. 3
FIG. 3

Distortion of the speckle mean intensity as a function of the number M of speckles in the aperture. The full curve is the difference between the mean density and the log of the mean intensity, whereas the dotted curve is the approximation given by Eq. (11).

FIG. 4
FIG. 4

Speckle density rms noise versus the number M of speckles in the aperture. The full curve corresponds to values calculated from Eq. (13), and the dotted curve is the approximation σ D = 1 / M.

FIG. 5
FIG. 5

The rms signal-to-noise ratio for speckle density, vs the number of speckles in the aperture. The dotted curve is the approximation 2 M corresponding to D0 = 2.

FIG. 6
FIG. 6

A comparison of the granularity of the noise vs density for speckle and film-grain noise, for an ideal emulsion following Selwyn’s law. Film-grain noise increases with density, whereas speckle noise is constant. The full curve corresponds to speckle and the dotted curve to film.

Equations (24)

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g ( D ) = - log f ( I ) .
p I ( I ) = M M Γ ( M ) I 0 ( I I 0 ) M - 1 exp ( - M I I 0 ) ,
R = ( a D / λ d ) 2 ,
M = ( 2 0 1 ( 1 - τ ) sinc 2 ( R τ ) d τ ) - 2 ,
M R for R 1 , 1 for R 1.
f D ( D ) = [ M M / Γ ( M ) ] exp [ - M ( D - D 0 ) ] exp { - M exp [ - ( D - D 0 ) ] }
f D ( D ) = [ M M / Γ ( M ) ] exp { - M [ D + exp ( - D ) ] } .
f D ( D ) M M Γ ( M ) exp ( - M ) exp M D 2 2             1 - D 3 .
f D ( D ) [ M M exp ( - M ) / Γ ( M ) exp ( - M D 2 / 2 ) .
f D ( D ) ( M / 2 Π ) 1 / 2 exp ( - M D 2 / 2 ) .
D ¯ = [ M M / Γ ( M ) ] - D exp { - M [ D + exp ( - D ) ] } d D .
D ¯ = D 0 - Γ ( M ) / Γ ( M ) + ln ( M ) ,
D ¯ = D 0 + γ ,
D ¯ D 0 + 1 / 2 M .
σ D 2 = [ M M / Γ ( M ) ] - D 2 exp { - M [ D + exp ( - D ) ] } d D .
σ D 2 = π 2 6 - ( γ + Γ ( M ) Γ ( M ) ) 2 + 2 K = 1 M - 2 1 M - K j = 1 M - K - 1 1 j .
σ D 2 1 / M ,
( S / N ) rms = D ¯ / σ D .
( S / N ) rms D ¯ M .
J 1 + D m / 3 σ D 1 + ( D m / 3 ) M ,
a ( J - 1 ) λ d / D ,
a 6 ( J - 1 ) μ m ,
C / speckle = ( 1 / M ) log 2 J .
C / speckle = ( 1 / M ) log 2 ( 1 + ( D m / 3 ) M ) .