Abstract

Use of a copula for generating a sequence of correlated speckle patterns is introduced. The chief characteristic of this algorithm is that it generates a continuous speckle sequence with a specified evolution of the correlation and does so with just two arrays of random numbers. Thus, physically realistic temporally varying speckle patterns with proper first- and second-order statistics are easily realized. We illustrate use of the algorithm for generating sequences with prescribed Gaussian, exponential, and equal-interval correlations and demonstrate how correlation times can be specified independently. This approach to generating sequences of random realizations with prescribed correlations should prove useful in modeling such phenomena as dynamic light scatter, flow-dependent laser speckle contrast, and propagation of spatial coherence.

© 2008 Optical Society of America

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References

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  1. H. J. Rabal, R. Arizaga, N. L. Cap, E. Grumel, and M. Trivi, "Numerical model for dynamic speckle: an approach using the movement of the scatterers," J. Opt. A, Pure Appl. Opt. 5, S381-S385 (2003).
    [CrossRef]
  2. A. Federico, G. H. Kaufmann, G. E. Galizzi, H. Rabal, M. Trivi, and R. Arizaga, "Simulation of dynamic speckle sequences and its application to the analysis of transient processes," Opt. Commun. 260, 493-499 (2006).
    [CrossRef]
  3. H. J. Rabal, N. L. Cap, M. Trivi, R. Arizaga, A. Federico, G. E. Galizzi, and G. H. Kaufmann, "Speckle activity images based on the spatial variance of the phase," Appl. Opt. 45, 8733-8738 (2006).
    [CrossRef] [PubMed]
  4. J. W. Goodman, Statistical Optics (Wiley, 1985).
  5. R. B. Nelson, An Introduction to Copulas (Springer-Verlag, 1999).
  6. R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, "Speckle-visibility spectroscopy: a tool to study time-varying dynamics," Rev. Sci. Instrum. 76, 093110 (2005).
    [CrossRef]
  7. A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).
  8. J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975), pp. 9-75.
    [CrossRef]
  9. A. Oulamara, G. Tribillon, and J. Duvernoy, "Biological activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle," J. Mod. Opt. 36, 165-179 (1989).
    [CrossRef]
  10. D. D. Duncan, F. F. Mark, and L. W. Hunter, "A new speckle technique for noncontact measurement of small creep rates," Opt. Eng. 31, 1583-1589 (1992).
    [CrossRef]
  11. L. Ji and G. Danuser, "Tracking quasi-stationary flow of weak fluorescent signals by adaptive multi-frame correlation," J. Microsc. 220, 150-167 (2005).
    [CrossRef] [PubMed]
  12. M. E. Thomas and D. D. Duncan, "Atmospheric transmission," in Atmospheric Propagation of Radiation, Vol. 2 of The Infrared and Electro-Optical Systems Handbook, F.G.Smith, ed. (ERIM Infrared Information Analysis Center and SPIE, 1993).
  13. S. R. Sadda, S. Joeres, Z. Wu, P. Updike, P. Romano, A. T. Collins, and A. C. Walsh, "Error correction and quantitative subanalysis of optical coherence tomography data using computer-assisted grading," Invest. Ophthalmol. Visual Sci. 48, 839-848 (2007).
    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).
  15. U. Cherubini, E. Luciano, and W. Vecchiato, Copula Methods in Finance (Wiley, 2004).

2007

S. R. Sadda, S. Joeres, Z. Wu, P. Updike, P. Romano, A. T. Collins, and A. C. Walsh, "Error correction and quantitative subanalysis of optical coherence tomography data using computer-assisted grading," Invest. Ophthalmol. Visual Sci. 48, 839-848 (2007).
[CrossRef]

2006

A. Federico, G. H. Kaufmann, G. E. Galizzi, H. Rabal, M. Trivi, and R. Arizaga, "Simulation of dynamic speckle sequences and its application to the analysis of transient processes," Opt. Commun. 260, 493-499 (2006).
[CrossRef]

H. J. Rabal, N. L. Cap, M. Trivi, R. Arizaga, A. Federico, G. E. Galizzi, and G. H. Kaufmann, "Speckle activity images based on the spatial variance of the phase," Appl. Opt. 45, 8733-8738 (2006).
[CrossRef] [PubMed]

2005

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, "Speckle-visibility spectroscopy: a tool to study time-varying dynamics," Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

L. Ji and G. Danuser, "Tracking quasi-stationary flow of weak fluorescent signals by adaptive multi-frame correlation," J. Microsc. 220, 150-167 (2005).
[CrossRef] [PubMed]

2004

U. Cherubini, E. Luciano, and W. Vecchiato, Copula Methods in Finance (Wiley, 2004).

2003

H. J. Rabal, R. Arizaga, N. L. Cap, E. Grumel, and M. Trivi, "Numerical model for dynamic speckle: an approach using the movement of the scatterers," J. Opt. A, Pure Appl. Opt. 5, S381-S385 (2003).
[CrossRef]

2002

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

1999

R. B. Nelson, An Introduction to Copulas (Springer-Verlag, 1999).

1993

M. E. Thomas and D. D. Duncan, "Atmospheric transmission," in Atmospheric Propagation of Radiation, Vol. 2 of The Infrared and Electro-Optical Systems Handbook, F.G.Smith, ed. (ERIM Infrared Information Analysis Center and SPIE, 1993).

1992

D. D. Duncan, F. F. Mark, and L. W. Hunter, "A new speckle technique for noncontact measurement of small creep rates," Opt. Eng. 31, 1583-1589 (1992).
[CrossRef]

1989

A. Oulamara, G. Tribillon, and J. Duvernoy, "Biological activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle," J. Mod. Opt. 36, 165-179 (1989).
[CrossRef]

1985

J. W. Goodman, Statistical Optics (Wiley, 1985).

1975

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975), pp. 9-75.
[CrossRef]

Arizaga, R.

A. Federico, G. H. Kaufmann, G. E. Galizzi, H. Rabal, M. Trivi, and R. Arizaga, "Simulation of dynamic speckle sequences and its application to the analysis of transient processes," Opt. Commun. 260, 493-499 (2006).
[CrossRef]

H. J. Rabal, N. L. Cap, M. Trivi, R. Arizaga, A. Federico, G. E. Galizzi, and G. H. Kaufmann, "Speckle activity images based on the spatial variance of the phase," Appl. Opt. 45, 8733-8738 (2006).
[CrossRef] [PubMed]

H. J. Rabal, R. Arizaga, N. L. Cap, E. Grumel, and M. Trivi, "Numerical model for dynamic speckle: an approach using the movement of the scatterers," J. Opt. A, Pure Appl. Opt. 5, S381-S385 (2003).
[CrossRef]

Bandyopadhyay, R.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, "Speckle-visibility spectroscopy: a tool to study time-varying dynamics," Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Cap, N. L.

H. J. Rabal, N. L. Cap, M. Trivi, R. Arizaga, A. Federico, G. E. Galizzi, and G. H. Kaufmann, "Speckle activity images based on the spatial variance of the phase," Appl. Opt. 45, 8733-8738 (2006).
[CrossRef] [PubMed]

H. J. Rabal, R. Arizaga, N. L. Cap, E. Grumel, and M. Trivi, "Numerical model for dynamic speckle: an approach using the movement of the scatterers," J. Opt. A, Pure Appl. Opt. 5, S381-S385 (2003).
[CrossRef]

Cherubini, U.

U. Cherubini, E. Luciano, and W. Vecchiato, Copula Methods in Finance (Wiley, 2004).

Collins, A. T.

S. R. Sadda, S. Joeres, Z. Wu, P. Updike, P. Romano, A. T. Collins, and A. C. Walsh, "Error correction and quantitative subanalysis of optical coherence tomography data using computer-assisted grading," Invest. Ophthalmol. Visual Sci. 48, 839-848 (2007).
[CrossRef]

Danuser, G.

L. Ji and G. Danuser, "Tracking quasi-stationary flow of weak fluorescent signals by adaptive multi-frame correlation," J. Microsc. 220, 150-167 (2005).
[CrossRef] [PubMed]

Dixon, P. K.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, "Speckle-visibility spectroscopy: a tool to study time-varying dynamics," Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Duncan, D. D.

M. E. Thomas and D. D. Duncan, "Atmospheric transmission," in Atmospheric Propagation of Radiation, Vol. 2 of The Infrared and Electro-Optical Systems Handbook, F.G.Smith, ed. (ERIM Infrared Information Analysis Center and SPIE, 1993).

D. D. Duncan, F. F. Mark, and L. W. Hunter, "A new speckle technique for noncontact measurement of small creep rates," Opt. Eng. 31, 1583-1589 (1992).
[CrossRef]

Durian, D. J.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, "Speckle-visibility spectroscopy: a tool to study time-varying dynamics," Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Duvernoy, J.

A. Oulamara, G. Tribillon, and J. Duvernoy, "Biological activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle," J. Mod. Opt. 36, 165-179 (1989).
[CrossRef]

Federico, A.

A. Federico, G. H. Kaufmann, G. E. Galizzi, H. Rabal, M. Trivi, and R. Arizaga, "Simulation of dynamic speckle sequences and its application to the analysis of transient processes," Opt. Commun. 260, 493-499 (2006).
[CrossRef]

H. J. Rabal, N. L. Cap, M. Trivi, R. Arizaga, A. Federico, G. E. Galizzi, and G. H. Kaufmann, "Speckle activity images based on the spatial variance of the phase," Appl. Opt. 45, 8733-8738 (2006).
[CrossRef] [PubMed]

Galizzi, G. E.

H. J. Rabal, N. L. Cap, M. Trivi, R. Arizaga, A. Federico, G. E. Galizzi, and G. H. Kaufmann, "Speckle activity images based on the spatial variance of the phase," Appl. Opt. 45, 8733-8738 (2006).
[CrossRef] [PubMed]

A. Federico, G. H. Kaufmann, G. E. Galizzi, H. Rabal, M. Trivi, and R. Arizaga, "Simulation of dynamic speckle sequences and its application to the analysis of transient processes," Opt. Commun. 260, 493-499 (2006).
[CrossRef]

Gittings, A. S.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, "Speckle-visibility spectroscopy: a tool to study time-varying dynamics," Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

J. W. Goodman, Statistical Optics (Wiley, 1985).

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975), pp. 9-75.
[CrossRef]

Grumel, E.

H. J. Rabal, R. Arizaga, N. L. Cap, E. Grumel, and M. Trivi, "Numerical model for dynamic speckle: an approach using the movement of the scatterers," J. Opt. A, Pure Appl. Opt. 5, S381-S385 (2003).
[CrossRef]

Hunter, L. W.

D. D. Duncan, F. F. Mark, and L. W. Hunter, "A new speckle technique for noncontact measurement of small creep rates," Opt. Eng. 31, 1583-1589 (1992).
[CrossRef]

Ji, L.

L. Ji and G. Danuser, "Tracking quasi-stationary flow of weak fluorescent signals by adaptive multi-frame correlation," J. Microsc. 220, 150-167 (2005).
[CrossRef] [PubMed]

Joeres, S.

S. R. Sadda, S. Joeres, Z. Wu, P. Updike, P. Romano, A. T. Collins, and A. C. Walsh, "Error correction and quantitative subanalysis of optical coherence tomography data using computer-assisted grading," Invest. Ophthalmol. Visual Sci. 48, 839-848 (2007).
[CrossRef]

Kaufmann, G. H.

A. Federico, G. H. Kaufmann, G. E. Galizzi, H. Rabal, M. Trivi, and R. Arizaga, "Simulation of dynamic speckle sequences and its application to the analysis of transient processes," Opt. Commun. 260, 493-499 (2006).
[CrossRef]

H. J. Rabal, N. L. Cap, M. Trivi, R. Arizaga, A. Federico, G. E. Galizzi, and G. H. Kaufmann, "Speckle activity images based on the spatial variance of the phase," Appl. Opt. 45, 8733-8738 (2006).
[CrossRef] [PubMed]

Luciano, E.

U. Cherubini, E. Luciano, and W. Vecchiato, Copula Methods in Finance (Wiley, 2004).

Mark, F. F.

D. D. Duncan, F. F. Mark, and L. W. Hunter, "A new speckle technique for noncontact measurement of small creep rates," Opt. Eng. 31, 1583-1589 (1992).
[CrossRef]

Nelson, R. B.

R. B. Nelson, An Introduction to Copulas (Springer-Verlag, 1999).

Oulamara, A.

A. Oulamara, G. Tribillon, and J. Duvernoy, "Biological activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle," J. Mod. Opt. 36, 165-179 (1989).
[CrossRef]

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Rabal, H.

A. Federico, G. H. Kaufmann, G. E. Galizzi, H. Rabal, M. Trivi, and R. Arizaga, "Simulation of dynamic speckle sequences and its application to the analysis of transient processes," Opt. Commun. 260, 493-499 (2006).
[CrossRef]

Rabal, H. J.

H. J. Rabal, N. L. Cap, M. Trivi, R. Arizaga, A. Federico, G. E. Galizzi, and G. H. Kaufmann, "Speckle activity images based on the spatial variance of the phase," Appl. Opt. 45, 8733-8738 (2006).
[CrossRef] [PubMed]

H. J. Rabal, R. Arizaga, N. L. Cap, E. Grumel, and M. Trivi, "Numerical model for dynamic speckle: an approach using the movement of the scatterers," J. Opt. A, Pure Appl. Opt. 5, S381-S385 (2003).
[CrossRef]

Romano, P.

S. R. Sadda, S. Joeres, Z. Wu, P. Updike, P. Romano, A. T. Collins, and A. C. Walsh, "Error correction and quantitative subanalysis of optical coherence tomography data using computer-assisted grading," Invest. Ophthalmol. Visual Sci. 48, 839-848 (2007).
[CrossRef]

Sadda, S. R.

S. R. Sadda, S. Joeres, Z. Wu, P. Updike, P. Romano, A. T. Collins, and A. C. Walsh, "Error correction and quantitative subanalysis of optical coherence tomography data using computer-assisted grading," Invest. Ophthalmol. Visual Sci. 48, 839-848 (2007).
[CrossRef]

Suh, S. S.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, "Speckle-visibility spectroscopy: a tool to study time-varying dynamics," Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Thomas, M. E.

M. E. Thomas and D. D. Duncan, "Atmospheric transmission," in Atmospheric Propagation of Radiation, Vol. 2 of The Infrared and Electro-Optical Systems Handbook, F.G.Smith, ed. (ERIM Infrared Information Analysis Center and SPIE, 1993).

Tribillon, G.

A. Oulamara, G. Tribillon, and J. Duvernoy, "Biological activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle," J. Mod. Opt. 36, 165-179 (1989).
[CrossRef]

Trivi, M.

H. J. Rabal, N. L. Cap, M. Trivi, R. Arizaga, A. Federico, G. E. Galizzi, and G. H. Kaufmann, "Speckle activity images based on the spatial variance of the phase," Appl. Opt. 45, 8733-8738 (2006).
[CrossRef] [PubMed]

A. Federico, G. H. Kaufmann, G. E. Galizzi, H. Rabal, M. Trivi, and R. Arizaga, "Simulation of dynamic speckle sequences and its application to the analysis of transient processes," Opt. Commun. 260, 493-499 (2006).
[CrossRef]

H. J. Rabal, R. Arizaga, N. L. Cap, E. Grumel, and M. Trivi, "Numerical model for dynamic speckle: an approach using the movement of the scatterers," J. Opt. A, Pure Appl. Opt. 5, S381-S385 (2003).
[CrossRef]

Updike, P.

S. R. Sadda, S. Joeres, Z. Wu, P. Updike, P. Romano, A. T. Collins, and A. C. Walsh, "Error correction and quantitative subanalysis of optical coherence tomography data using computer-assisted grading," Invest. Ophthalmol. Visual Sci. 48, 839-848 (2007).
[CrossRef]

Vecchiato, W.

U. Cherubini, E. Luciano, and W. Vecchiato, Copula Methods in Finance (Wiley, 2004).

Walsh, A. C.

S. R. Sadda, S. Joeres, Z. Wu, P. Updike, P. Romano, A. T. Collins, and A. C. Walsh, "Error correction and quantitative subanalysis of optical coherence tomography data using computer-assisted grading," Invest. Ophthalmol. Visual Sci. 48, 839-848 (2007).
[CrossRef]

Wu, Z.

S. R. Sadda, S. Joeres, Z. Wu, P. Updike, P. Romano, A. T. Collins, and A. C. Walsh, "Error correction and quantitative subanalysis of optical coherence tomography data using computer-assisted grading," Invest. Ophthalmol. Visual Sci. 48, 839-848 (2007).
[CrossRef]

Appl. Opt.

Invest. Ophthalmol. Visual Sci.

S. R. Sadda, S. Joeres, Z. Wu, P. Updike, P. Romano, A. T. Collins, and A. C. Walsh, "Error correction and quantitative subanalysis of optical coherence tomography data using computer-assisted grading," Invest. Ophthalmol. Visual Sci. 48, 839-848 (2007).
[CrossRef]

J. Microsc.

L. Ji and G. Danuser, "Tracking quasi-stationary flow of weak fluorescent signals by adaptive multi-frame correlation," J. Microsc. 220, 150-167 (2005).
[CrossRef] [PubMed]

J. Mod. Opt.

A. Oulamara, G. Tribillon, and J. Duvernoy, "Biological activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle," J. Mod. Opt. 36, 165-179 (1989).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

H. J. Rabal, R. Arizaga, N. L. Cap, E. Grumel, and M. Trivi, "Numerical model for dynamic speckle: an approach using the movement of the scatterers," J. Opt. A, Pure Appl. Opt. 5, S381-S385 (2003).
[CrossRef]

Opt. Commun.

A. Federico, G. H. Kaufmann, G. E. Galizzi, H. Rabal, M. Trivi, and R. Arizaga, "Simulation of dynamic speckle sequences and its application to the analysis of transient processes," Opt. Commun. 260, 493-499 (2006).
[CrossRef]

Opt. Eng.

D. D. Duncan, F. F. Mark, and L. W. Hunter, "A new speckle technique for noncontact measurement of small creep rates," Opt. Eng. 31, 1583-1589 (1992).
[CrossRef]

Rev. Sci. Instrum.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, "Speckle-visibility spectroscopy: a tool to study time-varying dynamics," Rev. Sci. Instrum. 76, 093110 (2005).
[CrossRef]

Other

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

J. W. Goodman, "Statistical properties of laser speckle patterns," in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975), pp. 9-75.
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, 1985).

R. B. Nelson, An Introduction to Copulas (Springer-Verlag, 1999).

M. E. Thomas and D. D. Duncan, "Atmospheric transmission," in Atmospheric Propagation of Radiation, Vol. 2 of The Infrared and Electro-Optical Systems Handbook, F.G.Smith, ed. (ERIM Infrared Information Analysis Center and SPIE, 1993).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

U. Cherubini, E. Luciano, and W. Vecchiato, Copula Methods in Finance (Wiley, 2004).

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Figures (14)

Fig. 1
Fig. 1

Illustration of the synthetic speckle algorithm. Shaded region of the matrix is filled with complex numbers of unity amplitude and phases uniformly distributed over ( 0 , 2 π ) .

Fig. 2
Fig. 2

Illustration of evolution of a correlated sequence of N speckle patterns. Phase arrays T 1 and T 2 with degree of correlation r are generated from bivariate Gaussian arrays Z 1 and Z 2 , which in turn are generated from S.I. arrays Y 1 and Y 2 .

Fig. 3
Fig. 3

Calculated speckle correlation coefficient versus the model based on variance of the phase difference, for the specified Gaussian sequential correlation function.

Fig. 4
Fig. 4

Speckle correlation coefficient as a function of sequence number, for the specified Gaussian sequential correlation function.

Fig. 5
Fig. 5

Sequential speckle correlation coefficient, for the specified Gaussian sequential correlation function.

Fig. 6
Fig. 6

(a) First frame from the speckle realization cube. (b) Slice through the central row in the speckle cube illustrating the temporal continuity of the speckle pattern. The spatial dimension is along the horizontal axis, and time is along the vertical axis.

Fig. 7
Fig. 7

Calculated speckle correlation coefficient versus the model based on variance of the phase difference, for the specified exponential sequential correlation function.

Fig. 8
Fig. 8

Speckle correlation coefficient as a function of sequence number, for the specified exponential sequential correlation function.

Fig. 9
Fig. 9

Sequential correlation coefficient for the exponential model, for the specified exponential sequential correlation function.

Fig. 10
Fig. 10

Speckle correlation coefficient versus the model based on variance of the phase difference, for the specified constant sequential correlation function.

Fig. 11
Fig. 11

Speckle correlation coefficient as a function of sequence number, for the specified constant sequential correlation function.

Fig. 12
Fig. 12

Sequential speckle polarization coefficient (mean is 0.9962), for the specified constant sequential correlation function. Note the constant observed value for this case.

Fig. 13
Fig. 13

Speckle correlation coefficient as a function of sequence number for the multiplicative factor of m = 3 . Results are shown for frames 1 and 25 chosen as the reference, for the specified constant sequential correlation function.

Fig. 14
Fig. 14

Sequential speckle polarization coefficient (mean is 0.9673), for the specified constant sequential correlation function and multiplicative factor of m = 3 .

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

H ( x , y ) = C [ F ( x ) , G ( y ) ] .
Y 1 = μ + σ 2 ln X 1 cos ( 2 π X 2 ) ,
Y 2 = μ + σ 2 ln X 1 sin ( 2 π X 2 ) ,
E { Y i } = μ , E { ( Y i E { Y i } ) 2 } = σ 2 ,
f Y 1 , Y 2 = N ( μ 1 , μ 2 ; σ 1 , σ 2 ; r ) = N ( μ , μ ; σ , σ ; 0 ) ,
Z 1 Z 2 = 1 2 1 1 1 1 1 + r 0 0 1 r Y 1 Y 2 ,
T 1 = F Z ( Z 1 ) , T 2 = F Z ( Z 2 ) ,
ρ = exp { σ Δ ϕ 2 } ,
r 1 k E { ( T 11 μ 11 ) ( T 1 k μ 1 k ) } σ 11 σ 1 k = 1 + r 2 .
var { ϕ 1 ϕ 2 } = ( 2 π ) 2 E { [ ( T 11 μ 11 ) ( T 1 k μ 1 k ) ] 2 } = ( 2 π ) 2 { σ 11 2 2 σ 11 σ 1 k r 1 k + σ 1 k 2 } = ( 2 π ) 2 2 σ T 2 ( 1 r 1 k ) = ( 2 π ) 2 ( 1 r 1 k ) 6 ,
ρ 1 k = exp { ( 2 π ) 2 6 ( 1 r 1 k ) } ,
1 r 1 k = ( k 1 N 1 ) ν ,
r = 2 [ 1 ( k 1 N 1 ) ν ] 2 1 .
ρ 1 k = exp { 2 ( k 1 c ) ν } ,
c = ( N 1 ) ( 3 π 2 ) 1 ν .
Z 1 = 2 ln X 1 cos ( 2 π X 2 + ϕ ) ,
ϕ = tan 1 ( 1 r 1 + r ) .
Z 1 = 2 ln X 1 cos ( 2 π X 2 ) ,
Z 1 = 2 ln X 1 cos ( 2 π X 2 + π 2 ) .
r = cos ( π k 1 N 1 ) ,
ϕ = π 2 k 1 N 1 .
Z 1 ( k ) = 2 ln X 1 cos ( 2 π X 2 + π 2 k 1 N 1 ) .
r 1 k = cos ( π 2 k 1 N 1 ) .
ρ 1 k = exp { 2 ( k 1 c ) 2 } ,
c = ( N 1 ) ( 3 π 2 ) 1 2 .
E { ρ k , k + 1 } = 1 10 N 2 .
r = cos [ π k 1 N 1 ] ,
ρ 1 k = exp { ( 2 π ) 2 3 sin 2 [ π 4 ( k 1 N 1 ) ] } ,
exp { i 2 π T } ,
exp { i 2 π m T } ,
ρ 1 k = exp { ( 2 π m ) 2 6 ( 1 r 1 k ) } .
m = 3 ,
r 1 k = cos ( π 2 k 1 N 1 ) .
E { ρ k , k + 1 } = 1 10 ( m N ) 2 .
cube 1 : Z 1 = 2 ln X 1 cos ( 2 π X 2 + ϕ ) ,
cube 2 : Z 1 = 2 ln X 1 cos ( 2 π X 3 + ϕ ) ,
ϕ = tan 1 ( 1 r 1 + r ) ,
I = F 1 { H F { exp ( i ϕ ) } } 2 ,

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