Abstract

Level-crossing statistics of aperture-integrated laser speckle are studied. In particular, expressions are obtained for the level-crossing rate and for the above-level dwell distance.

© 1988 Optical Society of America

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References

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  1. K. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1521 (1979).
    [CrossRef]
  2. R. Barakat, “The level-crossing rate and above-level duration time of the intensity of a Gaussian random process,” Inf. Sci. (NY) 20, 83–87 (1980).
    [CrossRef]
  3. R. D. Bahuguna, K. K. Gupta, K. Singh, “Expected number of intensity level crossing in a normal speckle pattern,”J. Opt. Soc. Am. 70, 874–876 (1980).
    [CrossRef]
  4. K. Ebeling, “Experimental investigation of some statistical properties of monochromatic speckle patterns,” Opt. Acta 26, 1345–1349 (1979).
    [CrossRef]
  5. N. Takai, T. Iwai, T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossing of laser speckle,”J. Opt. Soc. Am. 70, 450–455 (1980).
    [CrossRef]
  6. N. Takai, T. Iwai, T. Asakura, “Laser speckle velocimeters using a zero-crossing technique for spatially integrated intensity fluctuation,” Opt. Eng. 20, 320–324 (1981).
    [CrossRef]
  7. N. Takai, T. Asakura, “Displacement measurement of speckles using a 2-D level-crossing technique,” Appl. Opt. 22, 3514–3519 (1983).
    [CrossRef] [PubMed]
  8. N. Takai, T. Iwai, T. Ushizaka, T. Asakura, “Zero-crossing study on dynamic properties of speckles,”J. Opt. (Paris) 11, 93–101 (1980).
    [CrossRef]
  9. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975).
    [CrossRef]
  10. J. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), Chap. 10.
  11. R. Barakat, “Second-order statistics of integrated intensities and of detected photoelectrons,” J. Mod. Opt. 34, 91–102 (1987).
    [CrossRef]
  12. G. Szegö, Orthogonal Polynomials (American Mathematical Society, Providence, R.I., 1939).
  13. R. A. Silverman, “The fluctuation rate of the chi process,” Trans. IRE IT-4, 30–34 (1958).
  14. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  15. K. Ebeling, “K-distributed spatial intensity derivatives in monochromatic speckle patterns,” Opt. Commun. 35, 323–326 (1980).
    [CrossRef]
  16. I. F. Blake, W. C. Lindsey, “Level-crossing problems for random processes,”IEEE Trans. Inf. Theory IT-19, 295–315 (1973).
    [CrossRef]
  17. J. Abrahams, “A survey of recent progress on level-crossing problems for random processes,” in Communications and Networks, A Survey of Recent Advances, I. F. Blake, H. V. Poor, eds. (Springer-Verlag, New York, 1986).
  18. C. Helstrom, Statistical Theory of Signal Detection (Pergamon, New York, 1968), pp. 305–306.
  19. A. R. Pratt, “Some theoretical considerations concerning time statistics in signal detection,” in Signal Processing, J. Griffiths, P. Stocklin, C. van Schooneveld, eds. (Academic, New York, 1973).
  20. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1968).
  21. H. Gray, R. Thompson, G. McWilliams, “A new approximation for the chi-square integral,” Math. Computat. 23, 85–89 (1969).
    [CrossRef]

1987 (1)

R. Barakat, “Second-order statistics of integrated intensities and of detected photoelectrons,” J. Mod. Opt. 34, 91–102 (1987).
[CrossRef]

1983 (1)

1981 (1)

N. Takai, T. Iwai, T. Asakura, “Laser speckle velocimeters using a zero-crossing technique for spatially integrated intensity fluctuation,” Opt. Eng. 20, 320–324 (1981).
[CrossRef]

1980 (5)

K. Ebeling, “K-distributed spatial intensity derivatives in monochromatic speckle patterns,” Opt. Commun. 35, 323–326 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, “Zero-crossing study on dynamic properties of speckles,”J. Opt. (Paris) 11, 93–101 (1980).
[CrossRef]

R. Barakat, “The level-crossing rate and above-level duration time of the intensity of a Gaussian random process,” Inf. Sci. (NY) 20, 83–87 (1980).
[CrossRef]

R. D. Bahuguna, K. K. Gupta, K. Singh, “Expected number of intensity level crossing in a normal speckle pattern,”J. Opt. Soc. Am. 70, 874–876 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossing of laser speckle,”J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

1979 (2)

K. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1521 (1979).
[CrossRef]

K. Ebeling, “Experimental investigation of some statistical properties of monochromatic speckle patterns,” Opt. Acta 26, 1345–1349 (1979).
[CrossRef]

1973 (1)

I. F. Blake, W. C. Lindsey, “Level-crossing problems for random processes,”IEEE Trans. Inf. Theory IT-19, 295–315 (1973).
[CrossRef]

1969 (1)

H. Gray, R. Thompson, G. McWilliams, “A new approximation for the chi-square integral,” Math. Computat. 23, 85–89 (1969).
[CrossRef]

1958 (1)

R. A. Silverman, “The fluctuation rate of the chi process,” Trans. IRE IT-4, 30–34 (1958).

Abrahams, J.

J. Abrahams, “A survey of recent progress on level-crossing problems for random processes,” in Communications and Networks, A Survey of Recent Advances, I. F. Blake, H. V. Poor, eds. (Springer-Verlag, New York, 1986).

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1968).

Asakura, T.

N. Takai, T. Asakura, “Displacement measurement of speckles using a 2-D level-crossing technique,” Appl. Opt. 22, 3514–3519 (1983).
[CrossRef] [PubMed]

N. Takai, T. Iwai, T. Asakura, “Laser speckle velocimeters using a zero-crossing technique for spatially integrated intensity fluctuation,” Opt. Eng. 20, 320–324 (1981).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossing of laser speckle,”J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, “Zero-crossing study on dynamic properties of speckles,”J. Opt. (Paris) 11, 93–101 (1980).
[CrossRef]

Bahuguna, R. D.

Barakat, R.

R. Barakat, “Second-order statistics of integrated intensities and of detected photoelectrons,” J. Mod. Opt. 34, 91–102 (1987).
[CrossRef]

R. Barakat, “The level-crossing rate and above-level duration time of the intensity of a Gaussian random process,” Inf. Sci. (NY) 20, 83–87 (1980).
[CrossRef]

Bendat, J.

J. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), Chap. 10.

Blake, I. F.

I. F. Blake, W. C. Lindsey, “Level-crossing problems for random processes,”IEEE Trans. Inf. Theory IT-19, 295–315 (1973).
[CrossRef]

Ebeling, K.

K. Ebeling, “K-distributed spatial intensity derivatives in monochromatic speckle patterns,” Opt. Commun. 35, 323–326 (1980).
[CrossRef]

K. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1521 (1979).
[CrossRef]

K. Ebeling, “Experimental investigation of some statistical properties of monochromatic speckle patterns,” Opt. Acta 26, 1345–1349 (1979).
[CrossRef]

Goodman, J. W.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Gray, H.

H. Gray, R. Thompson, G. McWilliams, “A new approximation for the chi-square integral,” Math. Computat. 23, 85–89 (1969).
[CrossRef]

Gupta, K. K.

Helstrom, C.

C. Helstrom, Statistical Theory of Signal Detection (Pergamon, New York, 1968), pp. 305–306.

Iwai, T.

N. Takai, T. Iwai, T. Asakura, “Laser speckle velocimeters using a zero-crossing technique for spatially integrated intensity fluctuation,” Opt. Eng. 20, 320–324 (1981).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossing of laser speckle,”J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, “Zero-crossing study on dynamic properties of speckles,”J. Opt. (Paris) 11, 93–101 (1980).
[CrossRef]

Lindsey, W. C.

I. F. Blake, W. C. Lindsey, “Level-crossing problems for random processes,”IEEE Trans. Inf. Theory IT-19, 295–315 (1973).
[CrossRef]

McWilliams, G.

H. Gray, R. Thompson, G. McWilliams, “A new approximation for the chi-square integral,” Math. Computat. 23, 85–89 (1969).
[CrossRef]

Pratt, A. R.

A. R. Pratt, “Some theoretical considerations concerning time statistics in signal detection,” in Signal Processing, J. Griffiths, P. Stocklin, C. van Schooneveld, eds. (Academic, New York, 1973).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Silverman, R. A.

R. A. Silverman, “The fluctuation rate of the chi process,” Trans. IRE IT-4, 30–34 (1958).

Singh, K.

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1968).

Szegö, G.

G. Szegö, Orthogonal Polynomials (American Mathematical Society, Providence, R.I., 1939).

Takai, N.

N. Takai, T. Asakura, “Displacement measurement of speckles using a 2-D level-crossing technique,” Appl. Opt. 22, 3514–3519 (1983).
[CrossRef] [PubMed]

N. Takai, T. Iwai, T. Asakura, “Laser speckle velocimeters using a zero-crossing technique for spatially integrated intensity fluctuation,” Opt. Eng. 20, 320–324 (1981).
[CrossRef]

N. Takai, T. Iwai, T. Asakura, “Real-time velocity measurement for a diffuse object using zero-crossing of laser speckle,”J. Opt. Soc. Am. 70, 450–455 (1980).
[CrossRef]

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, “Zero-crossing study on dynamic properties of speckles,”J. Opt. (Paris) 11, 93–101 (1980).
[CrossRef]

Thompson, R.

H. Gray, R. Thompson, G. McWilliams, “A new approximation for the chi-square integral,” Math. Computat. 23, 85–89 (1969).
[CrossRef]

Ushizaka, T.

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, “Zero-crossing study on dynamic properties of speckles,”J. Opt. (Paris) 11, 93–101 (1980).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Inf. Theory (1)

I. F. Blake, W. C. Lindsey, “Level-crossing problems for random processes,”IEEE Trans. Inf. Theory IT-19, 295–315 (1973).
[CrossRef]

Inf. Sci. (NY) (1)

R. Barakat, “The level-crossing rate and above-level duration time of the intensity of a Gaussian random process,” Inf. Sci. (NY) 20, 83–87 (1980).
[CrossRef]

J. Mod. Opt. (1)

R. Barakat, “Second-order statistics of integrated intensities and of detected photoelectrons,” J. Mod. Opt. 34, 91–102 (1987).
[CrossRef]

J. Opt. (Paris) (1)

N. Takai, T. Iwai, T. Ushizaka, T. Asakura, “Zero-crossing study on dynamic properties of speckles,”J. Opt. (Paris) 11, 93–101 (1980).
[CrossRef]

J. Opt. Soc. Am. (2)

Math. Computat. (1)

H. Gray, R. Thompson, G. McWilliams, “A new approximation for the chi-square integral,” Math. Computat. 23, 85–89 (1969).
[CrossRef]

Opt. Acta (2)

K. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1521 (1979).
[CrossRef]

K. Ebeling, “Experimental investigation of some statistical properties of monochromatic speckle patterns,” Opt. Acta 26, 1345–1349 (1979).
[CrossRef]

Opt. Commun. (1)

K. Ebeling, “K-distributed spatial intensity derivatives in monochromatic speckle patterns,” Opt. Commun. 35, 323–326 (1980).
[CrossRef]

Opt. Eng. (1)

N. Takai, T. Iwai, T. Asakura, “Laser speckle velocimeters using a zero-crossing technique for spatially integrated intensity fluctuation,” Opt. Eng. 20, 320–324 (1981).
[CrossRef]

Trans. IRE (1)

R. A. Silverman, “The fluctuation rate of the chi process,” Trans. IRE IT-4, 30–34 (1958).

Other (8)

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

G. Szegö, Orthogonal Polynomials (American Mathematical Society, Providence, R.I., 1939).

J. Abrahams, “A survey of recent progress on level-crossing problems for random processes,” in Communications and Networks, A Survey of Recent Advances, I. F. Blake, H. V. Poor, eds. (Springer-Verlag, New York, 1986).

C. Helstrom, Statistical Theory of Signal Detection (Pergamon, New York, 1968), pp. 305–306.

A. R. Pratt, “Some theoretical considerations concerning time statistics in signal detection,” in Signal Processing, J. Griffiths, P. Stocklin, C. van Schooneveld, eds. (Academic, New York, 1973).

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1968).

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

J. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), Chap. 10.

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Figures (1)

Fig. 1
Fig. 1

Normalized level-crossing rate L ( Ω 0 ) L ( Ω 0 ) / | r ¨ ( 0 ) | for an aperture corresponding to ——, α = 1; - · -, α = 4; and - - -,α = 7.

Equations (45)

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Ω = aperture | E ( x ) | 2 d x ,
W ( Ω ) = 1 μ exp ( Ω / μ ) ,
W g ( Ω ) = 1 Γ ( α ) ( α μ ) α Ω α 1 exp ( α Ω / μ ) .
α = Ω 2 var ( Ω ) .
W ( Ω 1 , Ω 2 ) = W g ( Ω 1 ) W g ( Ω 2 ) n = 0 r Ω n ( n + α 1 n ) L n ( α 1 ) ( α Ω 1 μ ) × L n ( α 1 ) ( α Ω 2 μ ) ,
r Ω Ω 1 Ω 2 Ω 2 Ω 2 Ω 2 , 0 r Ω 1.
W ( Ω 1 , Ω 2 ) = W g ( Ω 2 ) δ ( Ω 1 Ω 2 ) .
W ( Ω 1 , Ω 2 ) = 1 ( 1 r Ω ) Γ ( α ) ( α μ ) 2 ( α 2 Ω 2 Ω 1 r Ω μ 2 ) ( α 1 ) / 2 × I α 1 [ 2 ( 1 r Ω ) ( α 2 r Ω Ω 2 Ω 1 μ 2 ) 1 / 2 ] × exp [ ( 1 r Ω ) 1 ( α μ ) ( Ω 2 + Ω 1 ) ] .
Ω ˙ ( x ) lim η 0 Ω ( x + η ) Ω ( x ) η ,
Ω 1 = Ω ( x ) , Ω 2 = Ω ( x + η ) ,
Ω 1 = Ω , Ω 2 = Ω + Ω ˙ η ,
| ( Ω 1 , Ω 2 ) ( Ω , Ω ˙ ) | = η ,
W ( Ω , Ω ˙ ) = lim η 0 η W ( Ω , Ω + Ω ˙ η ) .
I ν ( z ) 1 ( 2 π z ) 1 / 2 e z , | z | 1.
( α / μ ) ( 2 α + 1 ) / 2 Ω ( 2 α 3 ) / 2 2 Γ ( α ) π lim η 0 A ( η ) exp [ ( α μ ) B ( η ) ] ,
A ( η ) η [ 1 r ( η ) ] 1 / 2 [ r ( η ) ] 1 / 4 ,
B ( η ) ( 2 Ω + Ω ˙ η ) + 2 [ r ( η ) ] 1 / 2 ( Ω ˙ 2 + Ω Ω ˙ η ) 1 / 2 [ 1 r ( η ) ] 1 / 2 .
r ( η ) = 1 + r ¨ ( 0 ) 2 η 2 + ,
W ( Ω , Ω ˙ ) = ( α / μ ) α + ( 1 / 2 ) Ω α ( 3 / 2 ) Γ ( α ) ( 2 π | r ¨ ( 0 ) | ) 1 / 2 exp ( α Ω μ α Ω ˙ 2 2 | r ¨ ( 0 ) | μ Ω )
W ( Ω , Ω ˙ ) = 1 ( 2 π | r ¨ ( 0 ) | μ Ω ) 1 / 2 exp ( Ω μ Ω ˙ 2 2 | r ¨ ( 0 ) | μ Ω ) ,
W ( Ω ˙ ) = α π Γ ( α ) 2 α 1 σ ˙ ( α 2 | Ω ˙ | σ ˙ ) α ( 1 / 2 ) K α ( 1 / 2 ) [ α 2 | Ω ˙ | σ ˙ ] ,
σ ˙ [ μ 2 | r ¨ ( 0 ) | ] 1 / 2 .
0 t ν 1 exp ( β t γ t ) d t = 2 ( β γ ) ν / 2 K ν [ 2 ( β γ ) 1 / 2 ] .
W ( Ω ˙ ) = 1 σ ˙ 2 exp ( 2 | Ω ˙ | / σ ˙ ) .
x μ K ν ( b x ) d x = 2 μ 1 b ( μ + 1 ) Γ ( 1 + μ + ν 2 ) Γ ( 1 + μ ν 2 ) .
Ω ˙ 2 = ( σ ˙ 2 α ) , Ω ˙ 4 = 3 Ω ˙ 2 2 ( 1 + 1 α ) , Ω ˙ 6 = 15 Ω ˙ 2 3 ( 1 + 1 α ) ( 1 + 2 α ) .
L ( Ω 0 ) = | Ω ˙ | W ( Ω 0 , Ω ˙ ) d Ω ˙ ,
L ( Ω 0 ) = 1 Γ ( α ) ( 2 | r ¨ ( 0 ) | π ) 1 / 2 ( α Ω 0 μ ) α ( 1 / 2 ) exp ( α Ω 0 / μ ) .
Ω 0 | max = μ ( 1 1 2 α ) .
Ω 0 | max = μ / 2 .
L ( Ω 0 | max ) = 1 Γ ( α ) ( 2 | r ¨ ( 0 ) | π ) 1 / 2 ( α 1 2 ) α ( 1 / 2 ) × exp [ ( α 1 2 ) ] .
L ( Ω 0 | max ) | r ¨ ( 0 ) | 1 / 2 = 0.3422 ( α = 1 ) = 0.3263 ( α = 3 ) = 0.3213 ( α = 5 ) = 0.3204 ( α = 7 ) .
L ( Ω 0 | max ) | r ¨ ( 0 ) | 1 / 2 1 π ( 1 + 1 6 α ) 0.3183 ( 1 + 1 6 α ) .
L ( Ω 0 ) | r ¨ ( 0 ) | 1 / 2 π ( Ω 0 μ ) α ( 1 / 2 ) exp [ ( α / μ ) ( Ω 0 μ ) ] .
W ( Ω ˙ | Ω ) = W ( Ω , Ω ˙ ) / W ( Ω ) .
W ( Ω ˙ | Ω ) = [ α 2 π | r ¨ ( 0 ) | μ Ω ] 1 / 2 exp [ α Ω ˙ 2 2 | r ¨ ( 0 ) | μ Ω ] .
Ω ˙ | Ω = 0 ,
Ω ˙ 2 | Ω = | r ¨ ( 0 ) | μ Ω α .
x + | Ω Ω 0 W ( Ω ) d Ω 1 2 L ( Ω 0 ) .
Ω 0 W g ( Ω ) d Ω = 1 Γ ( α ) ( α Ω 0 / μ ) t α 1 e t d t = 1 Γ ( α ) Γ ( α , α Ω 0 μ ) ,
x + | Ω 0 = ( 2 π μ | r ¨ ( 0 ) | Ω 0 ) 1 / 2 ,
x + | Ω 0 = Γ ( α ) ( 2 π | r ¨ ( 0 ) | ) 1 / 2 ( μ α Ω 0 ) α ( 1 / 2 ) .
x t α 1 e t d t x e x ( x + α 1 ) [ 1 + ( α 1 ) ( x α + 1 ) 2 + 2 x ]
x + | Ω 0 ( 2 π α Ω 0 | r ¨ ( 0 ) | μ ) 1 / 2 ( α Ω 0 μ + α 1 ) 1 × [ 1 + ( α 1 ) ( α Ω 0 μ α + 1 ) 2 + ( 2 α Ω 0 μ ) ] .
x + | Ω 0 ~ ( 2 π μ | r ¨ ( 0 ) | α Ω 0 ) 1 / 2

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