Abstract

The second-order statistics for speckle patterns are derived through the use of an integral equation which determines the moment generating function. A specific geometry of apertures is treated as an example of the techniques developed. The inversion problem of determining the field correlation function from measurements of the integrated intensities is examined in the context of singular-value decomposition. The joint probability density function for the integrated intensities is evaluated.

© 1978 Optical Society of America

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References

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  1. Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer, Berlin, 1975).
  2. J. C. Dainty, “The Statistics of Speckle Patterns,” in Progress in Optics, Vol. 14, edited by E. Wolf (North-Holland, Amsterdam, 1977).
    [Crossref]
  3. Special issue on “Speckle Phenomena,” J. Opt. Soc. Am. 66,Nov.1976.
  4. M. A. Condie, “An Experimental Investigation of the Statistics of Diffusely Reflected Coherent Light,” thesis (Standford University, 1966).
  5. J. C. Dainty, “Detection of Images Immersed in Speckle Noise,” Opt. Acta 18, 327 (1971).
    [Crossref]
  6. R. Barakat, “First-Order Probability Densities of Laser Speckle Patterns Observed Through Finite Size Scanning Apertures,” Opt. Acta 20, 729 (1973).
    [Crossref]
  7. G. Parry, “The Scattering of Polychromatic Light from Rough Surfaces: First Order Statistics,” Opt. Quant. Elec. 7, 311 (1975).
    [Crossref]
  8. T. S. McKechnie, “Reduction of Speckle by a Moving Aperture—First Order Statistics,” Opt. Comm. 13, 35 (1975).
    [Crossref]
  9. A. Scribot, “First Order Probability Density Functions of Speckle Measured with a Finite Aperture,” Opt. Comm. 11, 238 (1974).
    [Crossref]
  10. F. Gori and C. Palma, “On the Eigenvalues of the Sinc2Kernel,” J. Phys. A 8, 1709 (1975).
    [Crossref]
  11. R. Barakat and P. Fortini, “Extreme Value Statistics of Speckle Patterns Observed Through Finite Size Scanning Apertures,” J. Opt. Soc. Am. (to be published).
  12. R. Barakat and R. Glauber, Quantum Theory of Photoelectron Counting Statistics (Unpublished report, Physics Dept., Harvard R. Barakat and J. Blake 621 University, 1966).
  13. E. Jakeman and E. R. Pike, “The Intensity Distribution of Gaussian Light,” J. Phys. A. 1, 128 (1968).
    [Crossref]
  14. E. Jakeman, “Theory of Optical Spectroscopy by Digital Autocorrelation of Photon-Counting Fluctuations,” J. Phys. A 3, 201 (1970).
    [Crossref]
  15. J. Blake and R. Barakat, Twofold Photoelectron Counting Statistics: the Clipped Correlation Function, J. Phys. A 6, 1196 (1973).
    [Crossref]
  16. R. Barakat, “First Order Statistics of Random Sinusoidal Waves with Applications to Laser Speckle Patterns,” Opt. Acta 21, 903 (1974).
    [Crossref]
  17. R. Barakat and J. Blake, “Onefold Photoelectron Counting Statistics for Non-Gaussian Light: Scattering from Polydispersive Suspensions,” Phys. Rev. A 13, 1122 (1976). See Appendix B.
    [Crossref]
  18. M. Kac and A. Siegert, “On the Theory of Noise in Radio Receivers with Square Law Detectors,” J. Appl. Phys. 18, 383 (1947).
    [Crossref]
  19. J. B. Thomas, Statistical Communication Theory (Wiley, New York, 1969).
  20. R. Glauber, “Photon Statistics,” in Fundamental Problems in Statistical Mechanics, edited by E. D. G. Cohen (North-Holland, Amsterdam, 1969).
  21. W. Martienssen and E. Spiller, “Coherence and Fluctuations in Light Beams,” Am. J. Phys. 32, 919 (1964).
    [Crossref]
  22. J. Blake and R. Barakat, “Threefold Photoelectron Counting Statistics for Gaussian Light,” Opt. Commun. 16, 303 (1976).
    [Crossref]
  23. J. Blake and R. Barakat, “The Inversion Problem for Twofold Photoelectron Counting Statistics,” Can. J. Phys. 53, 1215 (1975).
    [Crossref]
  24. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1974), Chaps. 4 and 5.
  25. G. H. Golub and C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numer. Math. 14, 403 (1970).
    [Crossref]
  26. I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Oliver and Boyd, Edinburgh, 1956).
  27. G. Szegö, Orthogonal Polynomials, 3rd ed. (American Mathematical Society, Providence, 1967).

1976 (3)

Special issue on “Speckle Phenomena,” J. Opt. Soc. Am. 66,Nov.1976.

R. Barakat and J. Blake, “Onefold Photoelectron Counting Statistics for Non-Gaussian Light: Scattering from Polydispersive Suspensions,” Phys. Rev. A 13, 1122 (1976). See Appendix B.
[Crossref]

J. Blake and R. Barakat, “Threefold Photoelectron Counting Statistics for Gaussian Light,” Opt. Commun. 16, 303 (1976).
[Crossref]

1975 (4)

J. Blake and R. Barakat, “The Inversion Problem for Twofold Photoelectron Counting Statistics,” Can. J. Phys. 53, 1215 (1975).
[Crossref]

G. Parry, “The Scattering of Polychromatic Light from Rough Surfaces: First Order Statistics,” Opt. Quant. Elec. 7, 311 (1975).
[Crossref]

T. S. McKechnie, “Reduction of Speckle by a Moving Aperture—First Order Statistics,” Opt. Comm. 13, 35 (1975).
[Crossref]

F. Gori and C. Palma, “On the Eigenvalues of the Sinc2Kernel,” J. Phys. A 8, 1709 (1975).
[Crossref]

1974 (2)

A. Scribot, “First Order Probability Density Functions of Speckle Measured with a Finite Aperture,” Opt. Comm. 11, 238 (1974).
[Crossref]

R. Barakat, “First Order Statistics of Random Sinusoidal Waves with Applications to Laser Speckle Patterns,” Opt. Acta 21, 903 (1974).
[Crossref]

1973 (2)

J. Blake and R. Barakat, Twofold Photoelectron Counting Statistics: the Clipped Correlation Function, J. Phys. A 6, 1196 (1973).
[Crossref]

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]

1970 (2)

E. Jakeman, “Theory of Optical Spectroscopy by Digital Autocorrelation of Photon-Counting Fluctuations,” J. Phys. A 3, 201 (1970).
[Crossref]

G. H. Golub and C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numer. Math. 14, 403 (1970).
[Crossref]

1968 (1)

E. Jakeman and E. R. Pike, “The Intensity Distribution of Gaussian Light,” J. Phys. A. 1, 128 (1968).
[Crossref]

1964 (1)

W. Martienssen and E. Spiller, “Coherence and Fluctuations in Light Beams,” Am. J. Phys. 32, 919 (1964).
[Crossref]

1947 (1)

M. Kac and A. Siegert, “On the Theory of Noise in Radio Receivers with Square Law Detectors,” J. Appl. Phys. 18, 383 (1947).
[Crossref]

Barakat, R.

R. Barakat and J. Blake, “Onefold Photoelectron Counting Statistics for Non-Gaussian Light: Scattering from Polydispersive Suspensions,” Phys. Rev. A 13, 1122 (1976). See Appendix B.
[Crossref]

J. Blake and R. Barakat, “Threefold Photoelectron Counting Statistics for Gaussian Light,” Opt. Commun. 16, 303 (1976).
[Crossref]

J. Blake and R. Barakat, “The Inversion Problem for Twofold Photoelectron Counting Statistics,” Can. J. Phys. 53, 1215 (1975).
[Crossref]

R. Barakat, “First Order Statistics of Random Sinusoidal Waves with Applications to Laser Speckle Patterns,” Opt. Acta 21, 903 (1974).
[Crossref]

J. Blake and R. Barakat, Twofold Photoelectron Counting Statistics: the Clipped Correlation Function, J. Phys. A 6, 1196 (1973).
[Crossref]

R. Barakat, “First-Order Probability Densities of Laser Speckle Patterns Observed Through Finite Size Scanning Apertures,” Opt. Acta 20, 729 (1973).
[Crossref]

R. Barakat and R. Glauber, Quantum Theory of Photoelectron Counting Statistics (Unpublished report, Physics Dept., Harvard R. Barakat and J. Blake 621 University, 1966).

R. Barakat and P. Fortini, “Extreme Value Statistics of Speckle Patterns Observed Through Finite Size Scanning Apertures,” J. Opt. Soc. Am. (to be published).

Blake, J.

R. Barakat and J. Blake, “Onefold Photoelectron Counting Statistics for Non-Gaussian Light: Scattering from Polydispersive Suspensions,” Phys. Rev. A 13, 1122 (1976). See Appendix B.
[Crossref]

J. Blake and R. Barakat, “Threefold Photoelectron Counting Statistics for Gaussian Light,” Opt. Commun. 16, 303 (1976).
[Crossref]

J. Blake and R. Barakat, “The Inversion Problem for Twofold Photoelectron Counting Statistics,” Can. J. Phys. 53, 1215 (1975).
[Crossref]

J. Blake and R. Barakat, Twofold Photoelectron Counting Statistics: the Clipped Correlation Function, J. Phys. A 6, 1196 (1973).
[Crossref]

Condie, M. A.

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

Dainty, J. C.

J. C. Dainty, “Detection of Images Immersed in Speckle Noise,” Opt. Acta 18, 327 (1971).
[Crossref]

J. C. Dainty, “The Statistics of Speckle Patterns,” in Progress in Optics, Vol. 14, edited by E. Wolf (North-Holland, Amsterdam, 1977).
[Crossref]

Fortini, P.

R. Barakat and P. Fortini, “Extreme Value Statistics of Speckle Patterns Observed Through Finite Size Scanning Apertures,” J. Opt. Soc. Am. (to be published).

Glauber, R.

R. Barakat and R. Glauber, Quantum Theory of Photoelectron Counting Statistics (Unpublished report, Physics Dept., Harvard R. Barakat and J. Blake 621 University, 1966).

R. Glauber, “Photon Statistics,” in Fundamental Problems in Statistical Mechanics, edited by E. D. G. Cohen (North-Holland, Amsterdam, 1969).

Golub, G. H.

G. H. Golub and C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numer. Math. 14, 403 (1970).
[Crossref]

Gori, F.

F. Gori and C. Palma, “On the Eigenvalues of the Sinc2Kernel,” J. Phys. A 8, 1709 (1975).
[Crossref]

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1974), Chaps. 4 and 5.

Jakeman, E.

E. Jakeman, “Theory of Optical Spectroscopy by Digital Autocorrelation of Photon-Counting Fluctuations,” J. Phys. A 3, 201 (1970).
[Crossref]

E. Jakeman and E. R. Pike, “The Intensity Distribution of Gaussian Light,” J. Phys. A. 1, 128 (1968).
[Crossref]

Kac, M.

M. Kac and A. Siegert, “On the Theory of Noise in Radio Receivers with Square Law Detectors,” J. Appl. Phys. 18, 383 (1947).
[Crossref]

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1974), Chaps. 4 and 5.

Martienssen, W.

W. Martienssen and E. Spiller, “Coherence and Fluctuations in Light Beams,” Am. J. Phys. 32, 919 (1964).
[Crossref]

McKechnie, T. S.

T. S. McKechnie, “Reduction of Speckle by a Moving Aperture—First Order Statistics,” Opt. Comm. 13, 35 (1975).
[Crossref]

Palma, C.

F. Gori and C. Palma, “On the Eigenvalues of the Sinc2Kernel,” J. Phys. A 8, 1709 (1975).
[Crossref]

Parry, G.

G. Parry, “The Scattering of Polychromatic Light from Rough Surfaces: First Order Statistics,” Opt. Quant. Elec. 7, 311 (1975).
[Crossref]

Pike, E. R.

E. Jakeman and E. R. Pike, “The Intensity Distribution of Gaussian Light,” J. Phys. A. 1, 128 (1968).
[Crossref]

Reinsch, C.

G. H. Golub and C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numer. Math. 14, 403 (1970).
[Crossref]

Scribot, A.

A. Scribot, “First Order Probability Density Functions of Speckle Measured with a Finite Aperture,” Opt. Comm. 11, 238 (1974).
[Crossref]

Siegert, A.

M. Kac and A. Siegert, “On the Theory of Noise in Radio Receivers with Square Law Detectors,” J. Appl. Phys. 18, 383 (1947).
[Crossref]

Sneddon, I. N.

I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Oliver and Boyd, Edinburgh, 1956).

Spiller, E.

W. Martienssen and E. Spiller, “Coherence and Fluctuations in Light Beams,” Am. J. Phys. 32, 919 (1964).
[Crossref]

Szegö, G.

G. Szegö, Orthogonal Polynomials, 3rd ed. (American Mathematical Society, Providence, 1967).

Thomas, J. B.

J. B. Thomas, Statistical Communication Theory (Wiley, New York, 1969).

Am. J. Phys. (1)

W. Martienssen and E. Spiller, “Coherence and Fluctuations in Light Beams,” Am. J. Phys. 32, 919 (1964).
[Crossref]

Can. J. Phys. (1)

J. Blake and R. Barakat, “The Inversion Problem for Twofold Photoelectron Counting Statistics,” Can. J. Phys. 53, 1215 (1975).
[Crossref]

J. Appl. Phys. (1)

M. Kac and A. Siegert, “On the Theory of Noise in Radio Receivers with Square Law Detectors,” J. Appl. Phys. 18, 383 (1947).
[Crossref]

J. Opt. Soc. Am. (1)

Special issue on “Speckle Phenomena,” J. Opt. Soc. Am. 66,Nov.1976.

J. Phys. A (3)

F. Gori and C. Palma, “On the Eigenvalues of the Sinc2Kernel,” J. Phys. A 8, 1709 (1975).
[Crossref]

E. Jakeman, “Theory of Optical Spectroscopy by Digital Autocorrelation of Photon-Counting Fluctuations,” J. Phys. A 3, 201 (1970).
[Crossref]

J. Blake and R. Barakat, Twofold Photoelectron Counting Statistics: the Clipped Correlation Function, J. Phys. A 6, 1196 (1973).
[Crossref]

J. Phys. A. (1)

E. Jakeman and E. R. Pike, “The Intensity Distribution of Gaussian Light,” J. Phys. A. 1, 128 (1968).
[Crossref]

Numer. Math. (1)

G. H. Golub and C. Reinsch, “Singular Value Decomposition and Least Squares Solutions,” Numer. Math. 14, 403 (1970).
[Crossref]

Opt. Acta (3)

R. Barakat, “First Order Statistics of Random Sinusoidal Waves with Applications to Laser Speckle Patterns,” Opt. Acta 21, 903 (1974).
[Crossref]

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. Comm. (2)

T. S. McKechnie, “Reduction of Speckle by a Moving Aperture—First Order Statistics,” Opt. Comm. 13, 35 (1975).
[Crossref]

A. Scribot, “First Order Probability Density Functions of Speckle Measured with a Finite Aperture,” Opt. Comm. 11, 238 (1974).
[Crossref]

Opt. Commun. (1)

J. Blake and R. Barakat, “Threefold Photoelectron Counting Statistics for Gaussian Light,” Opt. Commun. 16, 303 (1976).
[Crossref]

Opt. Quant. Elec. (1)

G. Parry, “The Scattering of Polychromatic Light from Rough Surfaces: First Order Statistics,” Opt. Quant. Elec. 7, 311 (1975).
[Crossref]

Phys. Rev. A (1)

R. Barakat and J. Blake, “Onefold Photoelectron Counting Statistics for Non-Gaussian Light: Scattering from Polydispersive Suspensions,” Phys. Rev. A 13, 1122 (1976). See Appendix B.
[Crossref]

Other (10)

R. Barakat and P. Fortini, “Extreme Value Statistics of Speckle Patterns Observed Through Finite Size Scanning Apertures,” J. Opt. Soc. Am. (to be published).

R. Barakat and R. Glauber, Quantum Theory of Photoelectron Counting Statistics (Unpublished report, Physics Dept., Harvard R. Barakat and J. Blake 621 University, 1966).

J. B. Thomas, Statistical Communication Theory (Wiley, New York, 1969).

R. Glauber, “Photon Statistics,” in Fundamental Problems in Statistical Mechanics, edited by E. D. G. Cohen (North-Holland, Amsterdam, 1969).

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

Laser Speckle and Related Phenomena, edited by J. C. Dainty (Springer, Berlin, 1975).

J. C. Dainty, “The Statistics of Speckle Patterns,” in Progress in Optics, Vol. 14, edited by E. Wolf (North-Holland, Amsterdam, 1977).
[Crossref]

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1974), Chaps. 4 and 5.

I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Oliver and Boyd, Edinburgh, 1956).

G. Szegö, Orthogonal Polynomials, 3rd ed. (American Mathematical Society, Providence, 1967).

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Figures (5)

FIG. 1
FIG. 1

Geometry of receiving apertures.

FIG. 2
FIG. 2

Correlation function c(l) for Gauss (upper curves) and negative exponential (lower curves) speckle correlation, rh(l), with σ2L = 0.3: —, L = 0.05; –·–, L = 1.0, - - -, L = 2.0.

FIG. 3
FIG. 3

Reconstruction of rh(l) = exp(−2|l|) from c(l) with L = 0.1 and 4% noise: —, theoretical value; open circles, all 20 singular values included; closed circles, 17 singular values included.

FIG. 4
FIG. 4

Reconstruction of rh(l) = exp(−2l2) from c(l) with L = 0.1 and 4% noise: —, theoretical value; open circles, all 20 singular values included; closed circles, 17 singular values included.

FIG. 5
FIG. 5

f(Ω,Ω) as a function of L: –·–, L = 0.2; —, L = 0.3; - - -, L = 0.5 with σ2 = 1 and rh(l) = 0.64.

Equations (88)

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U ( x ˆ ) = V ( x ˆ ) + i W ( x ˆ ) .
E [ V ( x ˆ ) ] = E [ W ( x ˆ ) ] = 0 ,
E [ V ( x ˆ ) W ( x ˆ ) ] = E [ V ( x ˆ ) ] E [ W ( x ˆ ) ] = 0.
E [ V ( x ˆ ) V ( x ˆ ) ] = E [ W ( x ˆ ) W ( x ˆ ) ] = ( σ 2 / 2 ) r ( x ˆ , x ˆ ) ,
h ( x ˆ ) U ( x ˆ ) 2 = V 2 ( x ˆ ) + W 2 ( x ˆ ) .
f h ( h ) = ( 1 / h 0 ) e - h / h 0 ,             h 0
h A ( x ˆ ) = A h ( x ˆ ) B ( x ˆ - x ˆ ) d x ˆ .
U ( x ˆ ) = k = 0 U k ψ k ( x ˆ ) ,             x ˆ A
A ψ j ( x ˆ ) ψ k * ( x ˆ ) d x ˆ = δ j k .
E [ U k U l * ] = σ k 2 δ k l ,
E [ U ( x ˆ ) U * ( x ˆ ) ] = k = 0 l = 0 U k U l * ψ k ( x ˆ ) ψ l * ( x ˆ ) = k = 0 σ k ψ k ( x ˆ ) ψ k * ( x ˆ )
A r ( x ˆ - x ˆ ) ψ l ( x ˆ ) d x ˆ = ( σ l σ ) 2 ψ l ( x ˆ ) ,
f ( U 1 , U 2 , ) f { U k } = k = 0 1 π σ k 2 e - U k 2 / σ κ 2 .
U k = V k + i W k ,
Q ( λ ) 0 f { U k } e - λ Ω ( x ˆ ) k = 0 d 2 U k = E [ e - λ Ω ] ,
d 2 U k d V k d W k .
E ( Ω m ) = ( - 1 ) m m λ m Q ( λ ) | λ = 0 .
Ω = k = 0 l = 0 U k * U l A ψ k * ( x ˆ ) ψ l ( x ˆ ) d x ˆ = k = 0 U k 2 0 .
Q ( λ ) = 0 exp { - k = 0 σ k - 2 U k 2 - λ k = 0 U k 2 } k = 0 d 2 U k π σ k = 0 exp ( - k = 0 ( σ k - 2 + λ ) U k 2 ) k = 0 d 2 U k π σ k = k = 0 ( π σ k 2 ) - 1 0 e - ( σ κ - 2 + λ ) U k 2 d 2 U k = k = 0 ( 1 + λ σ k 2 ) - 1 .
f Ω ( Ω ) = 1 2 π i λ 0 - i λ 0 + i Q ( λ ) e λ Ω d λ = 1 2 π i λ 0 - i λ 0 + i k = 0 ( 1 + λ σ k 2 ) - 1 e λ Ω d λ .
f Ω ( Ω ) = k = 0 σ k - 2 C k e - Ω / σ κ 2 ,             Ω 0
C k = l = 1 ( 1 - σ l 2 σ k 2 ) - 1 .
k = 0 σ k - 2 C k = 0 ,
f Ω ( Ω ) = [ 1 / Ω Γ ( ) ] ( Ω / Ω ) e - M Ω / Ω ,
Ω j = A j h ( x ˆ ) d x ˆ ,             j = 1 , 2
U ( x ˆ ) = k = 0 U k ψ k ( x ˆ ) ,             x ˆ A 1 and A 2
E [ U k U l * ] = σ k 2 δ k l .
h ( x ˆ ) = k = 0 l = 0 U k U l * ψ k ( x ˆ ) ψ l * ( x ˆ )
λ 1 Ω 1 + λ 2 Ω 2 = k = 0 l = 0 U k U l * ( λ 1 A 1 ψ k ( x ˆ ) ψ l * ( x ˆ ) d x ˆ + λ 2 A 2 ψ k ( x ˆ ) ψ l * ( x ˆ ) d x ˆ ) .
( λ 1 A 1 + λ 2 A 2 ) ψ k ( x ˆ ) ψ l * ( x ˆ ) d x ˆ = δ k l .
( λ 1 A 1 + λ 2 A 2 ) r ( x ˆ - x ˆ ) ψ l ( x ˆ ) d x ˆ = k = 0 ( σ k σ ) 2 ψ k ( x ˆ ) ( λ 1 A 1 + λ 2 A 2 ) × ψ k * ( x ˆ ) ψ l ( x ˆ ) d x ˆ = k = 0 ( σ k σ ) 2 ψ k ( x ˆ ) δ k l = ( σ l / σ ) 2 ψ l ( x ˆ ) .
Q ( λ 1 , λ 2 ) = 0 f ( { U k } ) e - λ 1 Ω 1 - λ Ω 2 k = 0 d 2 U k = E [ e - λ 1 Ω 1 e - λ 2 Ω 2 ] .
E [ Ω 1 k Ω 2 l ] = ( - 1 ) k + l k + l λ 1 k λ 2 l Q ( λ 1 , λ 2 ) | λ 1 = λ 2 = 0 .
E [ Ω 1 Ω 2 ] = 2 λ 1 λ 2 Q ( λ 1 , λ 2 ) | λ 1 = λ 2 = 0 .
Q ( λ 1 , λ 2 ) = 0 exp ( - k = 0 σ k - 2 U k 2 - k = 0 U k 2 ) k = 0 d 2 U k π σ k 2 .
Q ( λ 1 , λ 2 ) = k = 0 ( 1 + σ k 2 ) - 1 .
λ 1 - L / 2 L / 2 r ( x - x ) ψ l ( x ) d x + λ 2 l - L / 2 l + L / 2 r ( x - x ) ψ l ( x ) d x = m 1 ψ l ( x ) .
λ 1 L r ( x ) ψ k ( 0 ) + λ 2 L r ( x - l ) ψ k ( l ) = m ψ k ( x ) .
| λ 1 L r ( 0 ) λ 2 L r ( l ) λ 1 L r ( l ) λ 2 L r ( 0 ) | | ψ k ( l ) ψ k ( l ) | = ( σ k σ ) 2 | ψ k ( 0 ) ψ k ( l ) |
| λ 1 L r ( 0 ) σ + 1             λ 2 L r ( l ) σ λ 1 L r ( l ) σ             λ 2 L r ( 0 ) σ + 1 | | ψ k ( 0 ) ψ k ( l ) | = ( 1 + σ k 2 ) | ψ k ( 0 ) ψ k ( l ) |
 ψ ˆ k = ( 1 + σ k 2 ) ψ ˆ k .
Q ( λ 1 , λ 2 ) - 1 = ( 1 + σ 1 2 ) ( 1 + σ 2 2 ) = det Â
Q ( λ 1 , λ 2 ) = { 1 + ( λ 1 + λ 2 ) σ 2 L + λ 1 λ 2 ( σ 2 L ) 2 [ 1 - r 2 ( l ) ] } - 1 .
E [ Ω 1 Ω 2 ] = ( σ 2 L ) 2 [ 1 + r 2 ( l ) ] .
c ( l ) E [ Ω 1 Ω 2 ] / ( σ 2 L ) 2 = 1 + r 2 ( l ) .
Q ( λ 1 , λ 2 ) = det { Î N + ½ σ 2 L ( λ 1 + λ 2 ) B ˆ N ( 0 ) + ( ½ σ 2 L ) 2 B ˆ N ( 0 ) B ˆ N ( 0 ) - ( ½ σ 2 L ) 2 B ˆ N ( l ) B ˆ N ( - l ) } - 1 ,
{ B N ( l ) } i j = ( H i H j ) 1 / 2 r [ ( L / 2 ) ( x j - x i ) + l ] .
c ( l ) = 1 + ( 1 / 4 ) tr B ˆ N ( l ) B ˆ N ( - l ) .
B ˆ 2 ( l ) = | r ( l )             r ( l - δ ) r ( l + δ )             r ( l ) | ,
c ( l ) = 1 + ¼ [ r 2 ( l - δ ) + 2 r 2 ( l ) + r 2 ( l + δ ) ] = 1 + ¼ [ r h ( l - δ ) + 2 r h ( l ) + r h ( l + δ ) ] .
c ( l ) = 1 + 1 L 2 - L / 2 L / 2 r h ( l - l + l ) d l d l
r ( l ) = e - l ,             r h ( l ) = e - 2 l ,
c ( l ) = 1 + e - 2 l [ ( sinh L ) / L ] 2 ,             0 L l = 1 + L - 2 [ ¼ e - 2 ( L + l ) + ¼ e - 2 ( L - l ) - ½ e - 2 L + L - l ] ,             0 l L .
r ( l ) = e - l 2 ,             r h ( l ) = e - 2 l 2
 r ˆ = b ˆ ,             b ˆ 4 ( x ˆ - î ) ,
 = û Ŝ V ˆ + ,
Ŝ diag ( σ 1 + , σ 2 + , , σ N + , 0 , , 0 ) ,
σ n + = 1 / σ n             if σ n 0 = 0             if σ n = 0
 + = V ˆ Ŝ û + .
r ˆ = Â b ˆ = V ˆ Ŝ û + b ˆ .
r ˆ = n = 1 k ( Û n b ˆ / σ n ) v ˆ n ,
σ n + = 1 / σ n             if σ n > ɛ , = 0             if σ n < ɛ .
ɛ / σ 0 noise .
{ c } j c ( j δ ) - ¼ δ j 1 ,
{ r } j r 2 ( j δ ) ,
{ A } i j = 2 δ i j + δ i , j + 1 + δ i , j - 1 .
ĉ = î + ¼ Â r ˆ ,
r ˆ = 4 Â - 1 ( ĉ - î ) .
{ A - 1 } i j = j ( N - j + 1 ) ,             i = j = ( - 1 ) i + j ( N - j + 1 ) ,             i < j < N = { A - 1 } j i ,             i > j .
f ( Ω 1 , Ω 2 ) = 1 ( 2 π i ) 2 B 1 B 2 Q ( λ 1 , λ 2 ) × e λ 1 Ω 1 + λ 2 Ω 2 d λ 1 d λ 2 λ 1 - 1 λ 2 - 1 Q ( λ 1 , λ 2 ) ,
Q ( λ 1 , λ 2 ) = [ ( 1 + σ 2 L λ 1 ) ( 1 + σ 2 L λ 2 ) - ( σ 2 L ) 2 r 2 ( l ) λ 1 λ 2 ] - 1 = n = 0 [ r 2 ( l ) ] n ( σ 2 L λ 1 ) n ( σ 2 L λ 2 ) n [ ( 1 + σ 2 L λ 1 ) ( 1 + σ 2 L λ 2 ) ] n + 1 .
f ( Ω 1 , Ω 2 ) = n = 0 [ r 2 ( l ) ] n λ 1 - 1 [ ( σ 2 L λ 1 ) n / ( 1 + σ 2 L λ 1 ) n + 1 ] × λ 2 - 1 [ ( σ 2 L λ 2 ) n / ( 1 + σ 2 L λ 2 ) n + 1 ]
λ - 1 [ ( σ 2 L λ ) n / ( 1 + σ 2 L λ ) n + 1 ] = ( σ 2 L ) - 1 e - Ω / σ 2 L L n ( Ω / σ 2 L ) ,
L n ( x ) = j = 0 n n ! ( - x ) j ( j ! ) ( n - j ) ! .
f ( Ω 1 , Ω 2 ) = ( σ 2 L ) - 2 e - ( Ω 1 + Ω 2 ) / σ 2 L × n = 0 r 2 n ( l ) L n ( Ω 1 σ 2 L ) L n ( Ω 2 σ 2 L ) .
n = 0 ρ n ( n + m ) ! L n ( m ) ( x ) L n ( m ) ( y ) = 1 ( 1 - ρ ) ( ρ x y ) m / 2 e - ρ ( x + y ) / 1 - ρ × I m [ 2 ( ρ x y ) 1 / 2 / ( 1 - ρ ) ] ,
f ( Ω 1 , Ω 2 ) = ( 1 - r h ) - 1 ( σ 2 L ) - 2 e - ( Ω 1 + Ω 2 ) / ( 1 - r h ) σ 2 L × I 0 [ 2 ( r h Ω 1 Ω 2 ) 1 / 2 / ( 1 - r h ) σ 2 L ] .
Q ( λ 1 , λ 2 ) - 1 = det | 1 + s 1 + s 2 + s 1 s 2 ( 1 + Y 2 - X 2 - W 2 ) ( s 1 + s 2 ) Y + s 1 s 2 ( 2 Y - X U - X W ) ( s 1 + s 2 ) Y + s 1 s 2 ( 2 Y - X U - X W ) 1 + s 1 + s 2 + s 1 s 2 ( 1 + Y 2 - X 2 - U 2 ) | ,
Q ( λ 1 , λ 2 ) = Q 1 2 ( λ 1 , λ 2 ) [ 1 + A s 1 s 2 Q 1 ( λ 1 , λ 2 ) + Y 2 ( s 1 2 + s 2 2 ) Q 1 2 ( λ 1 , λ 2 ) + 2 Y 2 s 1 s 2 Q 1 2 ( λ 1 , λ 2 ) + B s 1 2 s 2 2 Q 1 2 ( λ 1 , λ 2 ) ] - 1 = Q 2 ( λ 1 , λ 2 ) - A s 1 s 2 Q 3 ( λ 1 , λ 2 ) - Y 2 [ s 1 2 + 2 s 1 s 2 + s 2 2 + Y - 2 ( B - A 2 ) s 1 2 s 2 2 ] Q 4 ( λ 1 , λ 2 ) + [ 2 A B s 1 3 s 2 3 + 2 Y 2 s 1 s 2 ( s 1 2 + 2 s 1 s 2 + s 2 2 ) ] Q 5 ( λ 1 , λ 2 ) + ,
A 2 Y 2 - W 2 - U 2 B ( 2 Y - U X - W X ) 2 - Y 4 - Y 2 ( W 2 + U 2 ) + W 2 U 2 .
Q l ( λ 1 , λ 2 ) [ 1 + ( s 1 + s 2 ) + s 1 s 2 ( 1 - r 2 ) ] - l ,
Q l k ( λ 1 , λ 2 ) = Q l + k ( λ 1 , λ 2 ) .
f l ( Ω 1 , Ω 2 ) = ( Ω 1 Ω 2 / r h ) ( l - 1 ) / 2 ( l - 1 ) ! ( 1 - r h ) ( σ 2 L ) l + 1 e - ( Ω 1 + Ω 2 ) / ( 1 - r h ) σ 2 L × I l - 1 [ 2 ( r h Ω 1 Ω 2 ) 1 / 2 / ( 1 - r h ) σ 2 L ] .
0 f l ( Ω 1 , Ω 2 ) d Ω 1 d Ω 2 = 1.
f ( Ω 1 , Ω 2 ) = f 2 ( Ω 1 , Ω 2 ) - A 2 Ω 1 Ω 2 f 3 ( Ω 1 , Ω 2 ) - Y 2 ( 2 Ω 1 2 + 2 Ω 2 2 + 2 2 Ω 1 Ω 2 + Y - 2 ( B - A 2 ) 4 Ω 1 2 d Ω 2 2 ) f 4 ( Ω 1 , Ω 2 ) + [ 2 Y 2 4 Ω 1 3 Ω 2 + 2 Y 2 4 Ω 1 Ω 2 3 + 4 Y 2 4 Ω 1 2 Ω 2 2 + 2 A B 6 Ω 1 3 Ω 2 3 ] × f 5 ( Ω 1 , Ω 2 ) + .
0 G ( Ω 1 , Ω 2 ) d Ω 1 d Ω 2 = 0.
f l ( Ω 1 , Ω 2 ) ~ ( Ω 1 Ω 2 ) l - 1
f ( Ω , Ω ) = f 2 ( Ω , Ω ) - A d 2 d Ω 2 f 3 ( Ω , Ω ) - Y 2 ( 4 d 2 d Ω 2 + Y - 2 ( B - A 2 ) d 4 d Ω 4 ) × f 4 ( Ω , Ω ) + .