Abstract

The zero-crossing rate of differentiated speckle whose intensity is governed by a negative exponential probability-density function (i.e., fully developed speckle) is evaluated in closed form with the use of an exact expression for the joint probability-density function of the intensity and its first two derivatives. The conditional zero-crossing rate of differentiated speckle, given that the intensity is specified, is also obtained in closed form.

© 1994 Optical Society of America

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References

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  1. N. Takai, T. Iwai, T. Asakura, “Real-time velocity measurements for a diffuse object using zero-crossing of laser speckle,”J. Opt. Soc. Am. 70, 450–455 (1980).
    [CrossRef]
  2. N. Takai, T. Iwai, T. Asakura, “Laser speckle velocimeters using a zero-crossing technique for spatially integrated intensity fluctuations,” Opt. Eng. 20, 320–324 (1981).
    [CrossRef]
  3. N. Takai, T. Asakura, “Displacement measurement of speckles using a 2-D level-crossing technique,” Appl. Opt. 22, 3514–3519 (1983).
    [CrossRef] [PubMed]
  4. N. Takai, T. Iwai, T. Ushizaka, T. Asakura, “Zero-crossing study on dynamic properties of speckles,”J. Opt. (Paris) 11, 93–101 (1980).
    [CrossRef]
  5. J. Ohtsubo, “Exact solution of the zero-crossing rate of a differentiated speckle pattern,” Opt. Commun. 42, 13–18 (1982).
    [CrossRef]
  6. J. S. Bendat, Principles and Applications of Random Noise Theory (Wiley, New York, 1958), Chap. 10.
  7. L. F. Blake, W. C. Lindsey, “Level-crossing problems for random processes,”IEEE Trans. Inf. Theory IT-19, 295–315 (1973).
    [CrossRef]
  8. M. Leadbetter, G. Lindgren, H. Rootzen, Extremes and Related Properties of Random Sequences and Processes (Springer-Verlag, New York, 1983), p. 115.
  9. K. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1527 (1979).
    [CrossRef]
  10. R. Barakat, “The level-crossing rate and above-levelduration time of the intensity of a Gaussian random process,” Inf. Sci. (New York) 20, 83–87 (1980).
  11. R. D. Bahuguna, K. K. Gupta, K. Singh, “Expected number of intensity level crossings in a normal speckle pattern,”J. Opt. Soc. Am. 70, 874–876 (1980).
    [CrossRef]
  12. R. Barakat, “Level-crossing statistics of aperture integrated isotropic speckle,” J. Opt. Soc. Am. A 5, 1244–1247 (1988).
    [CrossRef]
  13. R. I. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1967), Chap. 3.
  14. T. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

1988 (1)

1983 (1)

1982 (1)

J. Ohtsubo, “Exact solution of the zero-crossing rate of a differentiated speckle pattern,” Opt. Commun. 42, 13–18 (1982).
[CrossRef]

1981 (1)

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

1980 (4)

R. Barakat, “The level-crossing rate and above-levelduration time of the intensity of a Gaussian random process,” Inf. Sci. (New York) 20, 83–87 (1980).

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

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

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

1979 (1)

K. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1527 (1979).
[CrossRef]

1973 (1)

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

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 fluctuations,” Opt. Eng. 20, 320–324 (1981).
[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]

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

Bahuguna, R. D.

Barakat, R.

R. Barakat, “Level-crossing statistics of aperture integrated isotropic speckle,” J. Opt. Soc. Am. A 5, 1244–1247 (1988).
[CrossRef]

R. Barakat, “The level-crossing rate and above-levelduration time of the intensity of a Gaussian random process,” Inf. Sci. (New York) 20, 83–87 (1980).

Bendat, J. S.

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

Blake, L. F.

L. 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, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1527 (1979).
[CrossRef]

Gradshteyn, T. S.

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

Gupta, K. K.

Iwai, T.

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

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

Leadbetter, M.

M. Leadbetter, G. Lindgren, H. Rootzen, Extremes and Related Properties of Random Sequences and Processes (Springer-Verlag, New York, 1983), p. 115.

Lindgren, G.

M. Leadbetter, G. Lindgren, H. Rootzen, Extremes and Related Properties of Random Sequences and Processes (Springer-Verlag, New York, 1983), p. 115.

Lindsey, W. C.

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

Ohtsubo, J.

J. Ohtsubo, “Exact solution of the zero-crossing rate of a differentiated speckle pattern,” Opt. Commun. 42, 13–18 (1982).
[CrossRef]

Rootzen, H.

M. Leadbetter, G. Lindgren, H. Rootzen, Extremes and Related Properties of Random Sequences and Processes (Springer-Verlag, New York, 1983), p. 115.

Ryzhik, I. M.

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

Singh, K.

Stratonovich, R. I.

R. I. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1967), Chap. 3.

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 fluctuations,” Opt. Eng. 20, 320–324 (1981).
[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]

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

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

Inf. Sci. (New York) (1)

R. Barakat, “The level-crossing rate and above-levelduration time of the intensity of a Gaussian random process,” Inf. Sci. (New York) 20, 83–87 (1980).

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)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

K. Ebeling, “Statistical properties of spatial derivatives of the amplitude and intensity of monochromatic speckle patterns,” Opt. Acta 26, 1505–1527 (1979).
[CrossRef]

Opt. Commun. (1)

J. Ohtsubo, “Exact solution of the zero-crossing rate of a differentiated speckle pattern,” Opt. Commun. 42, 13–18 (1982).
[CrossRef]

Opt. Eng. (1)

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

Other (4)

R. I. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1967), Chap. 3.

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

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

M. Leadbetter, G. Lindgren, H. Rootzen, Extremes and Related Properties of Random Sequences and Processes (Springer-Verlag, New York, 1983), p. 115.

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Equations (37)

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E ( t ) = U ( t ) + i V ( t ) ,
V ( t ) = 1 π - U ( t ) d t t - t .
U ( t ) = V ( t ) = 0 , U ( t ) U ( t + τ ) = V ( t ) V ( t + τ ) = R ( τ ) .
U ( t ) V ( t ) 0 ,
I ( t ) E ( t ) 2 = U 2 ( t ) + V 2 ( t )
W ( I ) = ( 1 / μ ) exp ( - I / μ ) ,             I 0 ,
N ( Z = Z 0 ) = - Z t W ( Z 0 , Z t ) d Z t ,
N ( Z t = Z t 0 ) = - Z t t W ( Z t 0 , Z t t ) d Z t t .
N ( I = I 0 ) = ( 2 r 2 π ) 1 / 2 ( I 0 μ ) 1 / 2 exp ( - I 0 / μ ) ,
W ( U , U t , U t t , V , V t , V t t ) = W ( U , U t , U t t ) W ( V , V t , V t t ) .
W ( U , U t , U t t , V , V t , V t t ) = W ( U , U t t ) W ( U t ) W ( V , V t t ) W ( V t ) ,
W ( U , U t t ) = 1 2 π σ ( r 4 - r 2 2 ) 1 / 2 × exp [ - ( r 4 U 2 + 2 r 2 U U t t + U t t 2 2 σ 2 ( r 4 - r 2 2 ) ] ,
W ( U t ) = 1 ( 2 π σ 2 r 2 ) 1 / 2 exp ( - U t 2 2 σ 2 r 2 ) .
r 2 l = ( - 1 ) l d 2 l d τ 2 l r ( τ ) τ = 0 ,
δ r 4 - r 2 2 > 0.
I = U 2 + V 2 , I t = 2 U U t + 2 V V t , I t t = 2 U t 2 + 2 V t 2 + 2 U U t t + 2 V V t t .
W ( I , I t , I t t ) = 1 π μ ( 2 r 2 δ ) 1 / 2 I exp [ - Q ( I , I t , I t t ) ] , Q ( I , I t , I t t ) = r 4 I δ μ + I t 2 2 r 2 μ I + ( I t t - I t 2 / 2 I ) 2 δ μ I + 2 r 2 ( I t t - I t 2 / 2 I ) δ μ .
W ( I , I t ) = 1 ( 2 π r 2 μ I ) 1 / 2 exp ( - 1 μ - I t 2 2 r 2 μ I ) .
N ( I t = 0 ) = - I t t W ( 0 , I t t ) d ( I t t ) .
W ( 0 , I t t ) 0 W ( I , 0 , I t t ) d I ,
W ( 0 , I t t ) = 2 1 / 2 π μ ( r 2 δ ) 1 / 2 exp ( - 2 r 2 I t t δ μ ) K 0 ( 2 r 4 1 / 2 δ μ I t t ) ,
0 I - exp { - 1 2 ( 2 r 4 δ μ ) [ I + ( I t t 2 r 4 ) I - 1 ] } d I = 2 K 0 ( 2 r 4 I t t δ μ ) .
N ( I t = 0 ) = 2 3 / 2 π μ ( r 2 δ ) 1 / 2 0 I t t exp ( - 2 r 2 I t t δ μ ) × K 0 ( 2 r 4 1 / 2 I t t δ μ ) d I t t .
N ( I t = 0 ) = ( r 4 - r 2 2 ) 3 / 2 Γ ( 5 / 2 ) ( 2 π r 4 ) 1 / 2 ( r 4 1 / 2 + r 2 ) 2 × F 2 1 ( 2 , 1 2 , 5 2 , r 2 - r 4 1 / 2 r 2 + r 4 1 / 2 ) .
N ( I t = 0 ) = A ( r 2 , r 4 ) τ 0 .
r ( τ ) = exp ( - τ 2 / τ G 2 ) ,
r ( τ ) = τ L 2 τ 2 + τ L 2 ,
r 2 = 2 / τ G 2 ,             r 4 = 12 / τ G 4 ,
r 2 = 2 / τ L 2 ,             r 4 = 24 / τ L 4 .
r 4 1 / 2 - r 2 r 4 1 / 2 + r 2 = 0.2679 ( Gauss ) = 0.4202 ( Lorentz ) ,
N ( I t = 0 ) 0.144 / τ G 0.363 / τ L .
N ( I t = 0 I ) = - I t t W ( I t 0 , I t t I ) d I t t .
W ( I t , I t t I ) = W ( I , I t t , I t t ) W ( I ) = 1 π ( 2 r 2 δ ) 1 / 2 I exp ( - Q 2 ) ,
Q 2 = r 2 2 I δ μ + I t 2 2 τ 2 μ I + ( I t t - I t 2 / 2 I ) 2 δ μ I + 2 r 2 ( I t t - I t 2 / 2 I ) δ μ .
N ( I t 0 = 0 I ) = 1 π ( 2 r 2 δ ) 1 / 2 I exp ( - r 2 2 I / δ μ ) × - exp [ - ( p I t t 2 + 2 q I t t ) ] I t t d I t t ,
N ( I t = 0 I ) = ( r 2 I 2 π μ ) 1 / 2 .
N ( I t = 0 I ) = ( const . ) ( I / μ ) 1 / 2 τ 0 .

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