Abstract

Extreme-value statistics of the maximum intensity of a laser speckle pattern observed through a finite-size scanning aperture are evaluated. It is shown that, in most circumstances, a recording medium able to record intensities some 3–5 times the average extreme-value intensity should be adequate for recording all measurements.

© 1978 Optical Society of America

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References

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  1. R. Barakat, “Some extreme value statistics of laser speckle patterns,” Opt. Commun. 10, 107 (1974).
    [Crossref]
  2. M. A. Condie, “An Experimental Investigation of the Statistics of Diffusely Reflected Coherent Light,” Thesis, (Stanford University, 1966) (unpublished).
  3. J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327 (1971).
    [Crossref]
  4. R. Barakat, “First order probability densities of laser speckle patterns observed through finite size scanning apertures,” Opt. Acta 20, 729 (1973).
    [Crossref]
  5. A. A. Scribot, “First order probability density functions of speckle measured with a finite aperture,” Opt. Commun. 11, 238 (1974).
    [Crossref]
  6. R. Barakat and J. Blake, “Second-order statistics of speckle patterns observed through finite size scanning apertures,” J. Opt. Soc. Am.,  68, 614–622 (1978).
    [Crossref]
  7. H. A. David, Order Statistics (Wiley, New York, 1970), p. 207.
  8. E. J. Gumbel, Statistics of Extremes (Columbia U. P., New York, 1958), p. 171.
  9. D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I.,” Bell Syst. Tech. J. 40, 43 (1961).
    [Crossref]
  10. B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals based on use of the prolate functions,” in Progress in Optics, Vol. IX, edited by E. Wolf (North-Holland, Amsterdam, 1971).
    [Crossref]
  11. D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of order zero,” Bell Syst. Tech. J. 44, 1745 (1966).
  12. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York, 1975).
    [Crossref]
  13. S. Gupta, “Order statistics from the gamma distribution,” Technometrics 2, 243 (1960).
    [Crossref]
  14. P. Krishnaiah and M. Rizvi, “A note on moments of gamma order statistics,” Technometrics 9, 315 (1967).
    [Crossref]

1978 (1)

1974 (2)

R. Barakat, “Some extreme value statistics of laser speckle patterns,” Opt. Commun. 10, 107 (1974).
[Crossref]

A. A. Scribot, “First order probability density functions of speckle measured with a finite aperture,” Opt. Commun. 11, 238 (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]

1971 (1)

J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327 (1971).
[Crossref]

1967 (1)

P. Krishnaiah and M. Rizvi, “A note on moments of gamma order statistics,” Technometrics 9, 315 (1967).
[Crossref]

1966 (1)

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of order zero,” Bell Syst. Tech. J. 44, 1745 (1966).

1961 (1)

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I.,” Bell Syst. Tech. J. 40, 43 (1961).
[Crossref]

1960 (1)

S. Gupta, “Order statistics from the gamma distribution,” Technometrics 2, 243 (1960).
[Crossref]

Barakat, R.

R. Barakat and J. Blake, “Second-order statistics of speckle patterns observed through finite size scanning apertures,” J. Opt. Soc. Am.,  68, 614–622 (1978).
[Crossref]

R. Barakat, “Some extreme value statistics of laser speckle patterns,” Opt. Commun. 10, 107 (1974).
[Crossref]

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

Blake, J.

Condie, M. A.

M. A. Condie, “An Experimental Investigation of the Statistics of Diffusely Reflected Coherent Light,” Thesis, (Stanford University, 1966) (unpublished).

Dainty, J. C.

J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327 (1971).
[Crossref]

David, H. A.

H. A. David, Order Statistics (Wiley, New York, 1970), p. 207.

Frieden, B. R.

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals based on use of the prolate functions,” in Progress in Optics, Vol. IX, edited by E. Wolf (North-Holland, Amsterdam, 1971).
[Crossref]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York, 1975).
[Crossref]

Gumbel, E. J.

E. J. Gumbel, Statistics of Extremes (Columbia U. P., New York, 1958), p. 171.

Gupta, S.

S. Gupta, “Order statistics from the gamma distribution,” Technometrics 2, 243 (1960).
[Crossref]

Krishnaiah, P.

P. Krishnaiah and M. Rizvi, “A note on moments of gamma order statistics,” Technometrics 9, 315 (1967).
[Crossref]

Pollak, H. O.

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I.,” Bell Syst. Tech. J. 40, 43 (1961).
[Crossref]

Rizvi, M.

P. Krishnaiah and M. Rizvi, “A note on moments of gamma order statistics,” Technometrics 9, 315 (1967).
[Crossref]

Scribot, A. A.

A. A. Scribot, “First order probability density functions of speckle measured with a finite aperture,” Opt. Commun. 11, 238 (1974).
[Crossref]

Slepian, D.

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of order zero,” Bell Syst. Tech. J. 44, 1745 (1966).

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I.,” Bell Syst. Tech. J. 40, 43 (1961).
[Crossref]

Sonnenblick, E.

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of order zero,” Bell Syst. Tech. J. 44, 1745 (1966).

Bell Syst. Tech. J. (2)

D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I.,” Bell Syst. Tech. J. 40, 43 (1961).
[Crossref]

D. Slepian and E. Sonnenblick, “Eigenvalues associated with prolate spheroidal wave functions of order zero,” Bell Syst. Tech. J. 44, 1745 (1966).

J. Opt. Soc. Am. (1)

Opt. Acta (2)

J. C. Dainty, “Detection of images immersed in speckle noise,” Opt. Acta 18, 327 (1971).
[Crossref]

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

Opt. Commun. (2)

A. A. Scribot, “First order probability density functions of speckle measured with a finite aperture,” Opt. Commun. 11, 238 (1974).
[Crossref]

R. Barakat, “Some extreme value statistics of laser speckle patterns,” Opt. Commun. 10, 107 (1974).
[Crossref]

Technometrics (2)

S. Gupta, “Order statistics from the gamma distribution,” Technometrics 2, 243 (1960).
[Crossref]

P. Krishnaiah and M. Rizvi, “A note on moments of gamma order statistics,” Technometrics 9, 315 (1967).
[Crossref]

Other (5)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer-Verlag, New York, 1975).
[Crossref]

M. A. Condie, “An Experimental Investigation of the Statistics of Diffusely Reflected Coherent Light,” Thesis, (Stanford University, 1966) (unpublished).

H. A. David, Order Statistics (Wiley, New York, 1970), p. 207.

E. J. Gumbel, Statistics of Extremes (Columbia U. P., New York, 1958), p. 171.

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals based on use of the prolate functions,” in Progress in Optics, Vol. IX, edited by E. Wolf (North-Holland, Amsterdam, 1971).
[Crossref]

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Figures (2)

FIG. 1
FIG. 1

PDF of intensity at a fixed point in speckle pattern: - - -, Eq. (1); —, Eq. (3); 〈h〉 = 1.893 for both cases.

FIG. 2
FIG. 2

Mode ĥN and selected % points: - - -, 90%; —·—, 95%; —· ·—, 99% as functions of N for m = 5 for gamma density approximation.

Tables (2)

Tables Icon

TABLE I Eigenvalues μn and Cn coefficients for C = 3 in Eq. (33).

Tables Icon

TABLE II Characteristics of distribution of extreme values for data listed in Table I (〈h〉 = 1.893).

Equations (38)

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f h ( h ) = μ h - 1 e - h / μ h ,             h 0 ,
h A ( x , y ) = A h ( x , y ) × B ( x - x , y - y ) d x d y ,
f h A ( h A ) = n = 0 μ n - 1 C n e - h A / μ n ,             h A 0 ,
C n l = 1 ( 1 - μ l μ n ) - 1 ,             ( l n ) .
μ n ϕ n ( x , y ) = A r ( x , y ; x , y ) × B ( x , y ) ϕ n ( x , y ) d x d y ,
n = 0 C n = 1 ,
n = 0 μ n C n = n = 0 μ n = E ( h A ) .
F h A ( h A ) = n = 0 1 μ n C n 0 h A e - h / μ n d h = n = 0 C n - n = 0 C n e - h A / μ n = 1 - n = 0 C n e - h A / μ n
G ( h N ) = [ F h ( h N ) ] N
g ( h N ) = N f h ( h N ) [ F h ( h N ) ] N - 1 .
f h ( h N ) F H ( h N ) + ( N - 1 ) [ f h ( h N ) ] 2 = 0.
h N = μ 0 log C 0 ( N - 1 ) + μ 0 log ( d 3 2 / d 1 d 2 ) ,
d 1 1 - j = 0 C j e - h N / μ j ,
d 2 1 + j = 1 μ 0 2 C j μ j 2 C 0 e - h N ( μ j - 1 - μ 0 - 1 ) ,
d 3 1 + j = 1 μ 0 C j μ j C 0 e - h N ( μ j - 1 - μ 0 - 1 ) .
G ( h ) = p .
h = μ 0 log [ C 0 / ( 1 - p 1 / N ) ] + μ 0 log ( 1 + j = 1 C j C 0 - 1 e - h ( μ j - 1 - μ 0 - 1 ) ) .
lim h d d g [ 1 - F h ( h ) f h ( h ) ] = 0.
lim N P [ α N ( h N - l N ) < u ] = exp [ - exp ( - u ) ] ,
N - 1 = 1 - F h ( l N ) ,
α N = N f h ( l N ) .
lim h d d h [ 1 - F h ( h ) f h ( h ) ] = lim h d d h [ ( j = 0 C j e - h / μ j ) / ( j = 0 μ j - 1 C j e - h / μ j ) ] = 0.
G ( h N ) = exp { - exp [ - α N ( h N - l N ) ] } .
N - 1 = j = 0 C j e - l N / μ j
α N = N j = 0 μ j - 1 C j e - l N / μ j .
mode h N ĥ N = l N ,
E ( h N ) = l N + γ / α N ,
var h N = π 2 / 6 α N ,
l N = μ 0 log C 0 N + μ 0 log ( 1 + j = 1 C j C 0 - 1 e - l N ( μ j - 1 - μ 0 - 1 ) ) .
l N ( 1 ) = μ 0 log C 0 N ,
h = l N - ( 1 / α N ) log ( - log p ) .
h ( 1 ) = μ 0 log C 0 N - μ 0 log ( - log p ) .
μ n ϕ n ( x ) = - 1 + 1 ϕ n ( x ) sin c ( x - x ) π ( x - x ) d x .
f h ( h ) = 1 h Γ ( M ) ( M m h h ) e - M h / h ,
ĥ N = h x / M ,
1 Γ ( M ) x t M - 1 e - t d t = 1 + N - 1 ( M - 1 - x ) x M e - x .
ĥ p = h x / M ,
x t M - 1 e - t d t = Γ ( M ) ( 1 - p 1 / N ) .