Abstract

Speckle produced by strongly-scattering media contains information about its optical properties. Statistical speckle study allows discrimination between media and enables one to characterize any change. Two approaches of the speckle phenomenon are used in the measurement of speckle produced by monodisperse-polystyrene microspheres in solution and mixtures of them: a stochastic approach based on the fractional Brownian motion and a classical frequential approach based on speckle size measurement. In this paper, we introduce an approach that contains the multi-scale aspect of the speckle; therefore it provides more information on the medium than the speckle dimension. The obtained results show that the stochastic approach allows a better samples discrimination than the classical frequential approach.

© 2007 Optical Society of America

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References

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  5. J. D. Briers, G. Richard, and X. W. He, “Capillary blood flow monitoring using laser speckle contrast analysis (LASCA),” J. Biomed. Opt. 4, 164–175 (1999).
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    [Crossref] [PubMed]
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    [Crossref]
  15. L. othuaud, et al., “Fractal analysis of trabecular bone texture on radiographs: discriminant value in post menopausal osteoporosis,” Osteoporos. Int. 8, 618–625 (1998).
    [Crossref]
  16. G. M. Tosoni, A. G. Lurie, A. E. Cowan, and J.A. Burleson, “Pixel intensity and fractal analyses: detecting osteoporosis in perimenopausal and postmenopausal women by using digital panoramic images,” Oral Surgery, Oral Medicine, Oral Pathology, Oral Radiology, and Endodontology, 102, 235–241 (2006).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  22. T. D. Frank, A. Daffertshofer, and PJ. Beek, “Multivariate Ornstein-Uhlenberg processes with mean field-dependent coefficients-application to postural sway,” Phys. Rev. E 63, (2001).
  23. Pentland A. et al. “Fractal-based description of natural scenes,” IEEE Trans. Patt. Mach. Int. 6, No 6, 661–674 (1984).
    [Crossref]
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    [Crossref] [PubMed]
  25. K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. 114, 203–210 (1995).
    [Crossref]
  26. J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. 156, 350–358 (1998).
    [Crossref]
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    [Crossref]
  30. R. Simpson, et al., “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2007 (1)

2006 (2)

L. Zhifand, L. Hui, and Y. Qiu, “Fractal analysis of laser speckle for measuring roughness,” Proc. SPIE 6027, 470–476 (2006).

G. M. Tosoni, A. G. Lurie, A. E. Cowan, and J.A. Burleson, “Pixel intensity and fractal analyses: detecting osteoporosis in perimenopausal and postmenopausal women by using digital panoramic images,” Oral Surgery, Oral Medicine, Oral Pathology, Oral Radiology, and Endodontology, 102, 235–241 (2006).
[Crossref]

2005 (2)

2004 (2)

2001 (3)

T. D. Frank, A. Daffertshofer, and PJ. Beek, “Multivariate Ornstein-Uhlenberg processes with mean field-dependent coefficients-application to postural sway,” Phys. Rev. E 63, (2001).

C. L. Benhamou, et al., “Fractal Analysis of radiographic Trabecular Bone Texture and Bone Mineral Density: Two Complementary Parameters Related to Osteoporotic Fractures,” Journal Bone Miner. Res. 16, 697–704 (2001).
[Crossref]

I. V. Fedosov and V. V. Tuchin, “The use of dynamic speckle field space-time correlation function estimates for the direction and velocity determination of blood flow,” Proc. SPIE 4434, 192–196 (2001).
[Crossref]

2000 (1)

1999 (2)

P. Lehmann, “Surface-roughness measurement based on the intensity correlation function of scattered light under speckle-pattern illumination,” Appl. Opt. 38, 1144–1152 (1999).
[Crossref]

J. D. Briers, G. Richard, and X. W. He, “Capillary blood flow monitoring using laser speckle contrast analysis (LASCA),” J. Biomed. Opt. 4, 164–175 (1999).
[Crossref]

1998 (3)

L. othuaud, et al., “Fractal analysis of trabecular bone texture on radiographs: discriminant value in post menopausal osteoporosis,” Osteoporos. Int. 8, 618–625 (1998).
[Crossref]

J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. 156, 350–358 (1998).
[Crossref]

R. Simpson, et al., “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

1997 (4)

1995 (1)

K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. 114, 203–210 (1995).
[Crossref]

1994 (1)

1992 (1)

1984 (1)

Pentland A. et al. “Fractal-based description of natural scenes,” IEEE Trans. Patt. Mach. Int. 6, No 6, 661–674 (1984).
[Crossref]

1964 (1)

L. I. Goldfisher, “Autocorrelation function and power spectral density of last-produced speckle pattern,” J. Opt. Soc. Am. A 55, 247–253 (1964).

A., Pentland

Pentland A. et al. “Fractal-based description of natural scenes,” IEEE Trans. Patt. Mach. Int. 6, No 6, 661–674 (1984).
[Crossref]

Abgrall, J. F.

Abry, P.

P. Abry, P. Gonçalves, and P. Flandrin, Spectrum analysis and 1/f processes (Springer, Berlin, 1995).

Alexander, T. L.

Asakura, T.

J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. 156, 350–358 (1998).
[Crossref]

K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. 114, 203–210 (1995).
[Crossref]

B., Gelebart

Gelebart B., et al., “Time- and space-resolved reflectance applied to the analysis of multi-layered turbid media,” J. Opt. 28, 234–244 (1997).
[Crossref]

Barnsley, M. F.

M. F. Barnsley, R.L. Devaney, B.B. Mandelbrot, H.-O. Peitgen, D. Saupe, and R. F. Voss, The science of fractal images (Springer, New-York, 1988).
[Crossref]

Beek, PJ.

T. D. Frank, A. Daffertshofer, and PJ. Beek, “Multivariate Ornstein-Uhlenberg processes with mean field-dependent coefficients-application to postural sway,” Phys. Rev. E 63, (2001).

Benhamou, C. L.

C. L. Benhamou, et al., “Fractal Analysis of radiographic Trabecular Bone Texture and Bone Mineral Density: Two Complementary Parameters Related to Osteoporotic Fractures,” Journal Bone Miner. Res. 16, 697–704 (2001).
[Crossref]

Berlasso, R.

Bigio, I. J.

Blouch, M. T.

Boas, D. A.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley, New York, 1983).

Boulvert, F.

Briers, J. D.

J. D. Briers, G. Richard, and X. W. He, “Capillary blood flow monitoring using laser speckle contrast analysis (LASCA),” J. Biomed. Opt. 4, 164–175 (1999).
[Crossref]

D. A. Zimnyakov, J. D. Briers, and V. V. Tuchin, “Speckle technologies for monitoring and imaging of tissues and tissue like phantoms,” Chap.18 in Handbook of biomedical diagnostics, Valery V. Tuchin, Ed. (SPIE press, Bellingham2002).

Brun, G. Le

Burleson, J.A.

G. M. Tosoni, A. G. Lurie, A. E. Cowan, and J.A. Burleson, “Pixel intensity and fractal analyses: detecting osteoporosis in perimenopausal and postmenopausal women by using digital panoramic images,” Oral Surgery, Oral Medicine, Oral Pathology, Oral Radiology, and Endodontology, 102, 235–241 (2006).
[Crossref]

Cariou, J.

Chiang, F. P.

Cowan, A. E.

G. M. Tosoni, A. G. Lurie, A. E. Cowan, and J.A. Burleson, “Pixel intensity and fractal analyses: detecting osteoporosis in perimenopausal and postmenopausal women by using digital panoramic images,” Oral Surgery, Oral Medicine, Oral Pathology, Oral Radiology, and Endodontology, 102, 235–241 (2006).
[Crossref]

Daffertshofer, A.

T. D. Frank, A. Daffertshofer, and PJ. Beek, “Multivariate Ornstein-Uhlenberg processes with mean field-dependent coefficients-application to postural sway,” Phys. Rev. E 63, (2001).

Deléchelle, E.

S. Guyot, M. C. Péron, and E. Deléchelle, “Spatial Speckle Characterization by Brownian Motion analysis,” Phys. Rev. E 70, 046618 (2004).
[Crossref]

Devaney, R.L.

M. F. Barnsley, R.L. Devaney, B.B. Mandelbrot, H.-O. Peitgen, D. Saupe, and R. F. Voss, The science of fractal images (Springer, New-York, 1988).
[Crossref]

Fedosov, I. V.

I. V. Fedosov and V. V. Tuchin, “The use of dynamic speckle field space-time correlation function estimates for the direction and velocity determination of blood flow,” Proc. SPIE 4434, 192–196 (2001).
[Crossref]

Flandrin, P.

P. Abry, P. Gonçalves, and P. Flandrin, Spectrum analysis and 1/f processes (Springer, Berlin, 1995).

Françon, M.

M. Françon, Granularite Laser, speckle, application en optique, Masson (Paris, 1978).

Frank, T. D.

T. D. Frank, A. Daffertshofer, and PJ. Beek, “Multivariate Ornstein-Uhlenberg processes with mean field-dependent coefficients-application to postural sway,” Phys. Rev. E 63, (2001).

Funamizu, H.

Gaggioli, N. G.

Goldfisher, L. I.

L. I. Goldfisher, “Autocorrelation function and power spectral density of last-produced speckle pattern,” J. Opt. Soc. Am. A 55, 247–253 (1964).

Gonçalves, P.

P. Abry, P. Gonçalves, and P. Flandrin, Spectrum analysis and 1/f processes (Springer, Berlin, 1995).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 4, pp. 124–127; Chap. 7, 340–350.

J. W. Goodman, “Statistical Properties of Laser Speckle Pattern,” in Laser Speckle and Related Phenomena, Vol. 9 in series Topics in Applied Physics, J.C. Dainty, ed., (Springer-Verlag, Berlin, Heidelberg New York Tokyo, 1984)

Guern, Y.

Guyot, S.

S. Guyot, M. C. Péron, and E. Deléchelle, “Spatial Speckle Characterization by Brownian Motion analysis,” Phys. Rev. E 70, 046618 (2004).
[Crossref]

Ha, T. Hyon

T. Hyon Ha, et al. “Fractal dimension of cerebral cortical surface in schizophrenia and obsessive- compulsive disorder,” Neurosci. Lett. 384, 172–176 (2005).
[Crossref] [PubMed]

Harvey, J. E.

He, X. W.

J. D. Briers, G. Richard, and X. W. He, “Capillary blood flow monitoring using laser speckle contrast analysis (LASCA),” J. Biomed. Opt. 4, 164–175 (1999).
[Crossref]

Hielscher, A. H.

Hogg, R.V.

R.V. Hogg and J. Ledolter, Engineering statistics, (Macmillan Publishing Company, New-York, 1987).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley, New York, 1983).

Hui, L.

L. Zhifand, L. Hui, and Y. Qiu, “Fractal analysis of laser speckle for measuring roughness,” Proc. SPIE 6027, 470–476 (2006).

Ibrahim, M.

J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. 156, 350–358 (1998).
[Crossref]

Jeune, B. Le

Ledolter, J.

R.V. Hogg and J. Ledolter, Engineering statistics, (Macmillan Publishing Company, New-York, 1987).

Lehmann, P.

Li, Q. B.

Lurie, A. G.

G. M. Tosoni, A. G. Lurie, A. E. Cowan, and J.A. Burleson, “Pixel intensity and fractal analyses: detecting osteoporosis in perimenopausal and postmenopausal women by using digital panoramic images,” Oral Surgery, Oral Medicine, Oral Pathology, Oral Radiology, and Endodontology, 102, 235–241 (2006).
[Crossref]

Mandelbrot, B.B.

M. F. Barnsley, R.L. Devaney, B.B. Mandelbrot, H.-O. Peitgen, D. Saupe, and R. F. Voss, The science of fractal images (Springer, New-York, 1988).
[Crossref]

Meur, J. Le

Mishin, A. A.

Mourant, J. R.

othuaud, L.

L. othuaud, et al., “Fractal analysis of trabecular bone texture on radiographs: discriminant value in post menopausal osteoporosis,” Osteoporos. Int. 8, 618–625 (1998).
[Crossref]

Peitgen, H.-O.

M. F. Barnsley, R.L. Devaney, B.B. Mandelbrot, H.-O. Peitgen, D. Saupe, and R. F. Voss, The science of fractal images (Springer, New-York, 1988).
[Crossref]

Péron, M. C.

S. Guyot, M. C. Péron, and E. Deléchelle, “Spatial Speckle Characterization by Brownian Motion analysis,” Phys. Rev. E 70, 046618 (2004).
[Crossref]

Piederrière, Y.

Qiu, Y.

L. Zhifand, L. Hui, and Y. Qiu, “Fractal analysis of laser speckle for measuring roughness,” Proc. SPIE 6027, 470–476 (2006).

Quintian, F. Perz

Raffo, C. A.

Rebollo, M. A.

Richard, G.

J. D. Briers, G. Richard, and X. W. He, “Capillary blood flow monitoring using laser speckle contrast analysis (LASCA),” J. Biomed. Opt. 4, 164–175 (1999).
[Crossref]

Saupe, D.

M. F. Barnsley, R.L. Devaney, B.B. Mandelbrot, H.-O. Peitgen, D. Saupe, and R. F. Voss, The science of fractal images (Springer, New-York, 1988).
[Crossref]

Simpson, R.

R. Simpson, et al., “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

Tosoni, G. M.

G. M. Tosoni, A. G. Lurie, A. E. Cowan, and J.A. Burleson, “Pixel intensity and fractal analyses: detecting osteoporosis in perimenopausal and postmenopausal women by using digital panoramic images,” Oral Surgery, Oral Medicine, Oral Pathology, Oral Radiology, and Endodontology, 102, 235–241 (2006).
[Crossref]

Tuchin, V. V.

I. V. Fedosov and V. V. Tuchin, “The use of dynamic speckle field space-time correlation function estimates for the direction and velocity determination of blood flow,” Proc. SPIE 4434, 192–196 (2001).
[Crossref]

D. A. Zimnyakov, V. V. Tuchin, and A. A. Mishin, “Spatial speckle correlometry in applications to speckle structure monitoring,” Appl. Opt. 36, 5594–5607 (1997).
[Crossref] [PubMed]

D. A. Zimnyakov, J. D. Briers, and V. V. Tuchin, “Speckle technologies for monitoring and imaging of tissues and tissue like phantoms,” Chap.18 in Handbook of biomedical diagnostics, Valery V. Tuchin, Ed. (SPIE press, Bellingham2002).

Uno, K.

K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. 114, 203–210 (1995).
[Crossref]

Uozumi, J.

H. Funamizu and J. Uozumi, “Generation of fractal speckles by means of a spatial light modulator,” Opt. Express 15, 7415–7422 (2007).
[Crossref] [PubMed]

J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. 156, 350–358 (1998).
[Crossref]

K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. 114, 203–210 (1995).
[Crossref]

Voss, R. F.

M. F. Barnsley, R.L. Devaney, B.B. Mandelbrot, H.-O. Peitgen, D. Saupe, and R. F. Voss, The science of fractal images (Springer, New-York, 1988).
[Crossref]

Weeks, A. R.

Yodh, A. G.

Zhifand, L.

L. Zhifand, L. Hui, and Y. Qiu, “Fractal analysis of laser speckle for measuring roughness,” Proc. SPIE 6027, 470–476 (2006).

Zimnyakov, D. A.

D. A. Zimnyakov, V. V. Tuchin, and A. A. Mishin, “Spatial speckle correlometry in applications to speckle structure monitoring,” Appl. Opt. 36, 5594–5607 (1997).
[Crossref] [PubMed]

D. A. Zimnyakov, J. D. Briers, and V. V. Tuchin, “Speckle technologies for monitoring and imaging of tissues and tissue like phantoms,” Chap.18 in Handbook of biomedical diagnostics, Valery V. Tuchin, Ed. (SPIE press, Bellingham2002).

Appl. Opt. (6)

IEEE Trans. Patt. Mach. Int. (1)

Pentland A. et al. “Fractal-based description of natural scenes,” IEEE Trans. Patt. Mach. Int. 6, No 6, 661–674 (1984).
[Crossref]

J. Biomed. Opt. (1)

J. D. Briers, G. Richard, and X. W. He, “Capillary blood flow monitoring using laser speckle contrast analysis (LASCA),” J. Biomed. Opt. 4, 164–175 (1999).
[Crossref]

J. Opt. (1)

Gelebart B., et al., “Time- and space-resolved reflectance applied to the analysis of multi-layered turbid media,” J. Opt. 28, 234–244 (1997).
[Crossref]

J. Opt. Soc. Am. A (2)

D. A. Boas and A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A 14, 192–215 (1997).
[Crossref]

L. I. Goldfisher, “Autocorrelation function and power spectral density of last-produced speckle pattern,” J. Opt. Soc. Am. A 55, 247–253 (1964).

Journal Bone Miner. Res. (1)

C. L. Benhamou, et al., “Fractal Analysis of radiographic Trabecular Bone Texture and Bone Mineral Density: Two Complementary Parameters Related to Osteoporotic Fractures,” Journal Bone Miner. Res. 16, 697–704 (2001).
[Crossref]

Neurosci. Lett. (1)

T. Hyon Ha, et al. “Fractal dimension of cerebral cortical surface in schizophrenia and obsessive- compulsive disorder,” Neurosci. Lett. 384, 172–176 (2005).
[Crossref] [PubMed]

Opt. Commun. (2)

K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. 114, 203–210 (1995).
[Crossref]

J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. 156, 350–358 (1998).
[Crossref]

Opt. Express (3)

Oral Surgery, Oral Medicine, Oral Pathology, Oral Radiology, and Endodontology, (1)

G. M. Tosoni, A. G. Lurie, A. E. Cowan, and J.A. Burleson, “Pixel intensity and fractal analyses: detecting osteoporosis in perimenopausal and postmenopausal women by using digital panoramic images,” Oral Surgery, Oral Medicine, Oral Pathology, Oral Radiology, and Endodontology, 102, 235–241 (2006).
[Crossref]

Osteoporos. Int. (1)

L. othuaud, et al., “Fractal analysis of trabecular bone texture on radiographs: discriminant value in post menopausal osteoporosis,” Osteoporos. Int. 8, 618–625 (1998).
[Crossref]

Phys. Med. Biol. (1)

R. Simpson, et al., “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

Phys. Rev. E (2)

T. D. Frank, A. Daffertshofer, and PJ. Beek, “Multivariate Ornstein-Uhlenberg processes with mean field-dependent coefficients-application to postural sway,” Phys. Rev. E 63, (2001).

S. Guyot, M. C. Péron, and E. Deléchelle, “Spatial Speckle Characterization by Brownian Motion analysis,” Phys. Rev. E 70, 046618 (2004).
[Crossref]

Proc. SPIE (2)

L. Zhifand, L. Hui, and Y. Qiu, “Fractal analysis of laser speckle for measuring roughness,” Proc. SPIE 6027, 470–476 (2006).

I. V. Fedosov and V. V. Tuchin, “The use of dynamic speckle field space-time correlation function estimates for the direction and velocity determination of blood flow,” Proc. SPIE 4434, 192–196 (2001).
[Crossref]

Other (8)

D. A. Zimnyakov, J. D. Briers, and V. V. Tuchin, “Speckle technologies for monitoring and imaging of tissues and tissue like phantoms,” Chap.18 in Handbook of biomedical diagnostics, Valery V. Tuchin, Ed. (SPIE press, Bellingham2002).

J. W. Goodman, “Statistical Properties of Laser Speckle Pattern,” in Laser Speckle and Related Phenomena, Vol. 9 in series Topics in Applied Physics, J.C. Dainty, ed., (Springer-Verlag, Berlin, Heidelberg New York Tokyo, 1984)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 4, pp. 124–127; Chap. 7, 340–350.

P. Abry, P. Gonçalves, and P. Flandrin, Spectrum analysis and 1/f processes (Springer, Berlin, 1995).

M. F. Barnsley, R.L. Devaney, B.B. Mandelbrot, H.-O. Peitgen, D. Saupe, and R. F. Voss, The science of fractal images (Springer, New-York, 1988).
[Crossref]

M. Françon, Granularite Laser, speckle, application en optique, Masson (Paris, 1978).

R.V. Hogg and J. Ledolter, Engineering statistics, (Macmillan Publishing Company, New-York, 1987).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (Wiley, New York, 1983).

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Figures (8)

Fig. 1.
Fig. 1.

Normalized autocovariance function cI (x,0) , from speckle pattern of Fig. 4(b). dx is the width at half maximum.

Fig. 2.
Fig. 2.

Power Spectral Density of a) experimental speckle patterns and b) of speckle pattern of Fig. 4(b) (log-log scale).

Fig. 3.
Fig. 3.

Speckle diffusion function a) from synthetic speckle pattern b).

Fig. 4.
Fig. 4.

Speckle diffusion function a) from speckle pattern b) obtained by monodisperse-polystyrene microspheres in solution of deionised water (0, 20-μm microspheres).

Fig. 5.
Fig. 5.

Experimental setup (Face view).

Fig. 6.
Fig. 6.

Evolution of speckle size dx (a) and model parameters (Self-similarity S (b), Saturation of the variance G (c) and Hurst coefficient H (d)) versus the microsphere diameter d.

Fig. 7.
Fig. 7.

Evolution of speckle size dx (a) and model parameters (Self-similarity S (b), Saturation of the variance G (c) and Hurst coefficient H (d)), versus microspheres size distribution. Mixture m corresponds to a mixture with a volume fraction fv =1/6 for each microsphere size, mixture m(3:1), m(3:2) and m(1:1) correspond to the mixture m supplemented with smaller (0.20 μm) or larger microspheres (2.00 μm) with a ratio of (3:1), (3:2) and (1:1), respectively.

Fig. 8.
Fig. 8.

Comparison of the evolution aspect of scattering efficiency factor, Qs and Hurst coefficient H, according to the microsphere size d.

Tables (3)

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Table 1. Scattering coefficient with respect to c, scattering efficiency factor and anisotropy factor of polystyrene microspheres (Mie Calculation). c can be adjusted to obtain μs required.

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Table 2. Speckle size dx and model parameters (Saturation of the variance G, Self-similarity S, and Hurst coefficient H) for each microsphere diameter d.

Tables Icon

Table 3. Speckle size dx and model parameters (Saturation of the variance G, Self-similarity S, and Hurst coefficient H) for each mixture. Mixture m corresponds to a mixture with a volume fraction fv =1/6 for each microsphere size, mixture m(3:1), m(3:2) and m(1:1) correspond to the mixture m supplemented with smaller (0.20 μm) or larger microspheres (2.00 μm) with a ratio of (3:1), (3:2) and (1:1), respectively.

Equations (19)

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A ( x , y , z ) = 1 N Σ a k exp ( j φ k )
P ( A ( r ) , A ( i ) ) = 1 2 πσ 2 exp { [ A ( r ) ] 2 + [ A ( i ) ] 2 2 σ 2 }
P ( I ) = 1 2 σ 2 exp ( I 2 σ 2 )
P ( I d ) = ( M I d ) M I d M 1 Γ ( M ) exp ( M . I d I d )
PSD ( I ( x , y ) ) = FT ( I ( x , y ) ) 2
R I ( Δ x , Δ y ) = I ( x y , y 1 ) I ( x 2 , y 2 )
R I ( Δ x , Δ y ) = R I ( x , y )
R I ( x , y ) = FT 1 [ PSD ( I ( x , y ) ) ]
c I ( x , y ) = R I ( x , y ) I ( x , y ) 2 I ( x , y ) 2 I ( x , y ) 2
c I ( x , y ) = FT 1 ( FT ( I ( x , y ) ) 2 ) I ( x , y ) 2 I ( x , y ) 2 I ( x , y ) 2
[ X ( t + Δ t ) X ( t ) ] 2 Δ t
[ X ( t + Δ t ) X ( t ) ] 2 Δ t 2 H
[ I ( x + Δ x , y ) I ( x , y ) ] 2 = 2 ( I ( x , y ) 2 I ( x + Δ x , y ) I ( x , y ) )
[ I ( x + Δ x , y ) I ( x , y ) ] 2 = 2 ( I ( x , y ) 2 C ff )
C ff = X ( t ) X ( t + Δ t ) = σ 2 exp ( λ Δ t 2 H )
[ I ( x + Δ x , y ) I ( x , y ) ] 2 = 2 σ I 2 ( 1 exp ( λ Δ x 2 H ) )
log ( [ I ( x + Δ x , y ) I ( x , y ) ] 2 ) = log ( 2 σ I 2 ) + log ( 1 exp ( λ Δ x 2 H ) )
f ( x ) [ a + log ( 1 exp ( b Δ x c ) ) ] 2 2 ε
{ a = log ( 2 σ I 2 ) b = λ c = 2 H

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