Abstract

We study the noise of the intensity variance and of the intensity correlation and structure functions measured in light scattering from a random medium in the case when these quantities are obtained by averaging over a finite number N of pixels of a digital camera. We show that the noise scales as 1/N in all cases and that it is sensitive to correlations of signals corresponding to adjacent pixels as well as to the effective time averaging (due to the finite integration time) and spatial averaging (due to the finite pixel size). Our results provide a guide to estimation of noise levels in such applications as multi-speckle dynamic light scattering, time-resolved correlation spectroscopy, speckle visibility spectroscopy, laser speckle imaging etc.

© 2010 Optical Society of America

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  4. C. Zhou, G. Yu, F. Daisuke, J. H. Greenberg, A. G. Yodh, and T. Durduran, ”Diffuse optical correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 14, 1125–1144 (2006).
    [CrossRef] [PubMed]
  5. F. Scheffold and R. Cerbino, ”New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12, 50–57 (2007).
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  6. S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys. 104, 1758–1761 (1996).
    [CrossRef]
  7. L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, “Universal aging features in the restructuring of fractal colloidal gels,” Phys. Rev. Lett. 84, 2275–2278 (2000).
    [CrossRef] [PubMed]
  8. A. Knaebel, M. Bellour, J. P. Munch, V. Viasnoff, F. Lequeux, and J. L. Harden, “Aging behavior of Laponite clay particle suspensions,” Europhys. Lett. 52, 73–79 (2000).
    [CrossRef]
  9. V. Viasnoff, F. Lequeux, and D. J. Pine, “Multispeckle diffusing-wave spectroscopy: a tool to study slow relaxation and time-dependent dynamics,” Rev. Sci. Instrum. 73, 2336–2344 (2002).
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    [CrossRef]
  12. P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. 90, 184302 (2003).
    [CrossRef] [PubMed]
  13. 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]
  14. R. Cerbino and A. Vailati, “Near-field scattering techniques: novel instrumentation and results from time and spatially resolved investigations of soft matter systems,” Curr. Opin. Colloid Interface Sci. 14, 416–425 (2009).
    [CrossRef]
  15. J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): a nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1, 174–179 (1996).
    [CrossRef]
  16. P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009).
    [CrossRef]
  17. . C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Steady light transport under flow: characterization of evolving dense random media,” Phys. Rev. E 71, 066603 (2005).
    [CrossRef]
  18. D. D. Duncan, S. J. Kirkpatrick, and R. K. Wang, ”Statistics of local speckle contrast,” J. Opt. Soc. Am. A 25(1), 9–15 (2008).
    [CrossRef]
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  20. P. Zakharov and F. Scheffold, “Monitoring spatially heterogeneous dynamics in a drying colloidal thin film,” Soft Materials, to appear (2010).
  21. . J. D. Briers, “Laser doppler, speckle and related techniques for blood perfusion mapping and imaging,” Physiol. Meas. 22, R35–R66 (2001).
    [CrossRef]
  22. A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21, 195–201 (2001).
    [CrossRef] [PubMed]
  23. B. Weber, C. Burger, M. T. Wyss, G. K. von Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20(10), 2664–2670 (2004).
    [CrossRef] [PubMed]
  24. T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518–525 (2004).
    [CrossRef] [PubMed]
  25. . A. K. Dunn, A. Devor, A. M. Dale, and D. A. Boas, “Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex,” Neuroimage 27(15), 279–290 (2005).
    [CrossRef] [PubMed]
  26. T. Yoshimara, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am A 3, 1032–1054 (1986).
    [CrossRef]
  27. K. Sch¨atzel, “Noise on photon correlation data. i. autocorrelation functions,” Quantum Opt.: J. Eur. Opt. Soc. B 2, 287–305 (1990).
    [CrossRef]
  28. . M. Erpelding, A. Amon, and J. E. Crassous, “Diffusive wave spectroscopy applied to the spatially resolved deformation of a solid,” Phys. Rev. E 78,046104 (2008).
    [CrossRef]
  29. . N. Willenbacher, C. Oelschlaeger, M. Schspferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. 99, 068302 (2007).
    [CrossRef] [PubMed]

2010 (1)

P. Zakharov and F. Scheffold, “Monitoring spatially heterogeneous dynamics in a drying colloidal thin film,” Soft Materials, to appear (2010).

2009 (2)

P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009).
[CrossRef]

R. Cerbino and A. Vailati, “Near-field scattering techniques: novel instrumentation and results from time and spatially resolved investigations of soft matter systems,” Curr. Opin. Colloid Interface Sci. 14, 416–425 (2009).
[CrossRef]

2008 (2)

D. D. Duncan, S. J. Kirkpatrick, and R. K. Wang, ”Statistics of local speckle contrast,” J. Opt. Soc. Am. A 25(1), 9–15 (2008).
[CrossRef]

. M. Erpelding, A. Amon, and J. E. Crassous, “Diffusive wave spectroscopy applied to the spatially resolved deformation of a solid,” Phys. Rev. E 78,046104 (2008).
[CrossRef]

2007 (2)

. N. Willenbacher, C. Oelschlaeger, M. Schspferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. 99, 068302 (2007).
[CrossRef] [PubMed]

F. Scheffold and R. Cerbino, ”New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12, 50–57 (2007).
[CrossRef]

2006 (2)

2005 (3)

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]

. C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Steady light transport under flow: characterization of evolving dense random media,” Phys. Rev. E 71, 066603 (2005).
[CrossRef]

. A. K. Dunn, A. Devor, A. M. Dale, and D. A. Boas, “Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex,” Neuroimage 27(15), 279–290 (2005).
[CrossRef] [PubMed]

2004 (2)

B. Weber, C. Burger, M. T. Wyss, G. K. von Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20(10), 2664–2670 (2004).
[CrossRef] [PubMed]

T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518–525 (2004).
[CrossRef] [PubMed]

2003 (2)

L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, and S. Mazoyer, “Time-resolved correlation: a new tool for studying temporally heterogeneous dynamics,” J. Phys. Cond. Mat. 15, 257–262 (2003).
[CrossRef]

P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. 90, 184302 (2003).
[CrossRef] [PubMed]

2002 (1)

V. Viasnoff, F. Lequeux, and D. J. Pine, “Multispeckle diffusing-wave spectroscopy: a tool to study slow relaxation and time-dependent dynamics,” Rev. Sci. Instrum. 73, 2336–2344 (2002).
[CrossRef]

2001 (2)

. J. D. Briers, “Laser doppler, speckle and related techniques for blood perfusion mapping and imaging,” Physiol. Meas. 22, R35–R66 (2001).
[CrossRef]

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21, 195–201 (2001).
[CrossRef] [PubMed]

2000 (2)

L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, “Universal aging features in the restructuring of fractal colloidal gels,” Phys. Rev. Lett. 84, 2275–2278 (2000).
[CrossRef] [PubMed]

A. Knaebel, M. Bellour, J. P. Munch, V. Viasnoff, F. Lequeux, and J. L. Harden, “Aging behavior of Laponite clay particle suspensions,” Europhys. Lett. 52, 73–79 (2000).
[CrossRef]

1996 (2)

S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys. 104, 1758–1761 (1996).
[CrossRef]

J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): a nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1, 174–179 (1996).
[CrossRef]

1990 (1)

K. Sch¨atzel, “Noise on photon correlation data. i. autocorrelation functions,” Quantum Opt.: J. Eur. Opt. Soc. B 2, 287–305 (1990).
[CrossRef]

1986 (1)

T. Yoshimara, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am A 3, 1032–1054 (1986).
[CrossRef]

1983 (1)

K. Schatzel, “Noise in photon correlation and photon structure functions,” J. Mod. Opt. 30, 155–166 (1983).

Amon, A.

. M. Erpelding, A. Amon, and J. E. Crassous, “Diffusive wave spectroscopy applied to the spatially resolved deformation of a solid,” Phys. Rev. E 78,046104 (2008).
[CrossRef]

Ball, R. C.

L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, “Universal aging features in the restructuring of fractal colloidal gels,” Phys. Rev. Lett. 84, 2275–2278 (2000).
[CrossRef] [PubMed]

Ballesta, P.

L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, and S. Mazoyer, “Time-resolved correlation: a new tool for studying temporally heterogeneous dynamics,” J. Phys. Cond. Mat. 15, 257–262 (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]

Baravian, C.

. C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Steady light transport under flow: characterization of evolving dense random media,” Phys. Rev. E 71, 066603 (2005).
[CrossRef]

Bartsch, E.

S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys. 104, 1758–1761 (1996).
[CrossRef]

Bellour, M.

A. Knaebel, M. Bellour, J. P. Munch, V. Viasnoff, F. Lequeux, and J. L. Harden, “Aging behavior of Laponite clay particle suspensions,” Europhys. Lett. 52, 73–79 (2000).
[CrossRef]

Bhat, S.

Bissig, H.

L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, and S. Mazoyer, “Time-resolved correlation: a new tool for studying temporally heterogeneous dynamics,” J. Phys. Cond. Mat. 15, 257–262 (2003).
[CrossRef]

Boas, D. A.

. A. K. Dunn, A. Devor, A. M. Dale, and D. A. Boas, “Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex,” Neuroimage 27(15), 279–290 (2005).
[CrossRef] [PubMed]

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21, 195–201 (2001).
[CrossRef] [PubMed]

Bolay, H.

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21, 195–201 (2001).
[CrossRef] [PubMed]

Briers, J. D.

. J. D. Briers, “Laser doppler, speckle and related techniques for blood perfusion mapping and imaging,” Physiol. Meas. 22, R35–R66 (2001).
[CrossRef]

Briers, J.D.

J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): a nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1, 174–179 (1996).
[CrossRef]

Buck, A.

P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009).
[CrossRef]

B. Weber, C. Burger, M. T. Wyss, G. K. von Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20(10), 2664–2670 (2004).
[CrossRef] [PubMed]

Burger, C.

B. Weber, C. Burger, M. T. Wyss, G. K. von Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20(10), 2664–2670 (2004).
[CrossRef] [PubMed]

Burnett, M. G.

T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518–525 (2004).
[CrossRef] [PubMed]

Calcinaghi, N.

P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009).
[CrossRef]

Cardinaux, F.

. N. Willenbacher, C. Oelschlaeger, M. Schspferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. 99, 068302 (2007).
[CrossRef] [PubMed]

Caton, F.

. C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Steady light transport under flow: characterization of evolving dense random media,” Phys. Rev. E 71, 066603 (2005).
[CrossRef]

Cerbino, R.

R. Cerbino and A. Vailati, “Near-field scattering techniques: novel instrumentation and results from time and spatially resolved investigations of soft matter systems,” Curr. Opin. Colloid Interface Sci. 14, 416–425 (2009).
[CrossRef]

F. Scheffold and R. Cerbino, ”New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12, 50–57 (2007).
[CrossRef]

Cipelletti, L.

L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, and S. Mazoyer, “Time-resolved correlation: a new tool for studying temporally heterogeneous dynamics,” J. Phys. Cond. Mat. 15, 257–262 (2003).
[CrossRef]

L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, “Universal aging features in the restructuring of fractal colloidal gels,” Phys. Rev. Lett. 84, 2275–2278 (2000).
[CrossRef] [PubMed]

Crassous, J. E.

. M. Erpelding, A. Amon, and J. E. Crassous, “Diffusive wave spectroscopy applied to the spatially resolved deformation of a solid,” Phys. Rev. E 78,046104 (2008).
[CrossRef]

Daisuke, F.

Dale, A. M.

. A. K. Dunn, A. Devor, A. M. Dale, and D. A. Boas, “Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex,” Neuroimage 27(15), 279–290 (2005).
[CrossRef] [PubMed]

Detre, J. A.

T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518–525 (2004).
[CrossRef] [PubMed]

Devor, A.

. A. K. Dunn, A. Devor, A. M. Dale, and D. A. Boas, “Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex,” Neuroimage 27(15), 279–290 (2005).
[CrossRef] [PubMed]

Dillet, J.

. C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Steady light transport under flow: characterization of evolving dense random media,” Phys. Rev. E 71, 066603 (2005).
[CrossRef]

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]

P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. 90, 184302 (2003).
[CrossRef] [PubMed]

Duncan, D. D.

Dunn, A. K.

. A. K. Dunn, A. Devor, A. M. Dale, and D. A. Boas, “Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex,” Neuroimage 27(15), 279–290 (2005).
[CrossRef] [PubMed]

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21, 195–201 (2001).
[CrossRef] [PubMed]

Durduran, T.

C. Zhou, G. Yu, F. Daisuke, J. H. Greenberg, A. G. Yodh, and T. Durduran, ”Diffuse optical correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 14, 1125–1144 (2006).
[CrossRef] [PubMed]

T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518–525 (2004).
[CrossRef] [PubMed]

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]

P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. 90, 184302 (2003).
[CrossRef] [PubMed]

Erpelding, M.

. M. Erpelding, A. Amon, and J. E. Crassous, “Diffusive wave spectroscopy applied to the spatially resolved deformation of a solid,” Phys. Rev. E 78,046104 (2008).
[CrossRef]

Fischer, P.

. N. Willenbacher, C. Oelschlaeger, M. Schspferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. 99, 068302 (2007).
[CrossRef] [PubMed]

Frenz, V.

S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys. 104, 1758–1761 (1996).
[CrossRef]

Furuya, D.

T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518–525 (2004).
[CrossRef] [PubMed]

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]

Greenberg, J. H.

C. Zhou, G. Yu, F. Daisuke, J. H. Greenberg, A. G. Yodh, and T. Durduran, ”Diffuse optical correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 14, 1125–1144 (2006).
[CrossRef] [PubMed]

T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518–525 (2004).
[CrossRef] [PubMed]

Haiss, F.

P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009).
[CrossRef]

Harden, J. L.

A. Knaebel, M. Bellour, J. P. Munch, V. Viasnoff, F. Lequeux, and J. L. Harden, “Aging behavior of Laponite clay particle suspensions,” Europhys. Lett. 52, 73–79 (2000).
[CrossRef]

Kirkpatrick, S. J.

Kirsch, S.

S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys. 104, 1758–1761 (1996).
[CrossRef]

Knaebel, A.

A. Knaebel, M. Bellour, J. P. Munch, V. Viasnoff, F. Lequeux, and J. L. Harden, “Aging behavior of Laponite clay particle suspensions,” Europhys. Lett. 52, 73–79 (2000).
[CrossRef]

Lequeux, F.

V. Viasnoff, F. Lequeux, and D. J. Pine, “Multispeckle diffusing-wave spectroscopy: a tool to study slow relaxation and time-dependent dynamics,” Rev. Sci. Instrum. 73, 2336–2344 (2002).
[CrossRef]

A. Knaebel, M. Bellour, J. P. Munch, V. Viasnoff, F. Lequeux, and J. L. Harden, “Aging behavior of Laponite clay particle suspensions,” Europhys. Lett. 52, 73–79 (2000).
[CrossRef]

Manley, S.

L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, “Universal aging features in the restructuring of fractal colloidal gels,” Phys. Rev. Lett. 84, 2275–2278 (2000).
[CrossRef] [PubMed]

Mazoyer, S.

L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, and S. Mazoyer, “Time-resolved correlation: a new tool for studying temporally heterogeneous dynamics,” J. Phys. Cond. Mat. 15, 257–262 (2003).
[CrossRef]

Moskowitz, M. A.

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21, 195–201 (2001).
[CrossRef] [PubMed]

Mougel, J.

. C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Steady light transport under flow: characterization of evolving dense random media,” Phys. Rev. E 71, 066603 (2005).
[CrossRef]

Munch, J. P.

A. Knaebel, M. Bellour, J. P. Munch, V. Viasnoff, F. Lequeux, and J. L. Harden, “Aging behavior of Laponite clay particle suspensions,” Europhys. Lett. 52, 73–79 (2000).
[CrossRef]

Oelschlaeger, C.

. N. Willenbacher, C. Oelschlaeger, M. Schspferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. 99, 068302 (2007).
[CrossRef] [PubMed]

Pine, D. J.

V. Viasnoff, F. Lequeux, and D. J. Pine, “Multispeckle diffusing-wave spectroscopy: a tool to study slow relaxation and time-dependent dynamics,” Rev. Sci. Instrum. 73, 2336–2344 (2002).
[CrossRef]

Sch?spferer, M.

. N. Willenbacher, C. Oelschlaeger, M. Schspferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. 99, 068302 (2007).
[CrossRef] [PubMed]

Sch¨atzel, K.

K. Sch¨atzel, “Noise on photon correlation data. i. autocorrelation functions,” Quantum Opt.: J. Eur. Opt. Soc. B 2, 287–305 (1990).
[CrossRef]

Schartl, W.

S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys. 104, 1758–1761 (1996).
[CrossRef]

Schatzel, K.

K. Schatzel, “Noise in photon correlation and photon structure functions,” J. Mod. Opt. 30, 155–166 (1983).

Scheffold, F.

P. Zakharov and F. Scheffold, “Monitoring spatially heterogeneous dynamics in a drying colloidal thin film,” Soft Materials, to appear (2010).

P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009).
[CrossRef]

F. Scheffold and R. Cerbino, ”New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12, 50–57 (2007).
[CrossRef]

. N. Willenbacher, C. Oelschlaeger, M. Schspferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. 99, 068302 (2007).
[CrossRef] [PubMed]

P. Zakharov, S. Bhat, P. Schurtenberger, F. Scheffold, “Multiple scattering suppression in dynamic light scattering based on a digital camera detection scheme,” Appl. Opt. 45(8), 1756–1764 (2006)
[CrossRef] [PubMed]

B. Weber, C. Burger, M. T. Wyss, G. K. von Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20(10), 2664–2670 (2004).
[CrossRef] [PubMed]

Schurtenberger, P.

Sillescu, H.

S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys. 104, 1758–1761 (1996).
[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]

Trappe, V.

L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, and S. Mazoyer, “Time-resolved correlation: a new tool for studying temporally heterogeneous dynamics,” J. Phys. Cond. Mat. 15, 257–262 (2003).
[CrossRef]

Vailati, A.

R. Cerbino and A. Vailati, “Near-field scattering techniques: novel instrumentation and results from time and spatially resolved investigations of soft matter systems,” Curr. Opin. Colloid Interface Sci. 14, 416–425 (2009).
[CrossRef]

Viasnoff, V.

V. Viasnoff, F. Lequeux, and D. J. Pine, “Multispeckle diffusing-wave spectroscopy: a tool to study slow relaxation and time-dependent dynamics,” Rev. Sci. Instrum. 73, 2336–2344 (2002).
[CrossRef]

A. Knaebel, M. Bellour, J. P. Munch, V. Viasnoff, F. Lequeux, and J. L. Harden, “Aging behavior of Laponite clay particle suspensions,” Europhys. Lett. 52, 73–79 (2000).
[CrossRef]

Volker, A. C.

P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009).
[CrossRef]

von Schulthess, G. K.

B. Weber, C. Burger, M. T. Wyss, G. K. von Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20(10), 2664–2670 (2004).
[CrossRef] [PubMed]

Wang, R. K.

Weber, B.

P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009).
[CrossRef]

B. Weber, C. Burger, M. T. Wyss, G. K. von Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20(10), 2664–2670 (2004).
[CrossRef] [PubMed]

Webster, S.

J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): a nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1, 174–179 (1996).
[CrossRef]

Weitz, D. A.

L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, “Universal aging features in the restructuring of fractal colloidal gels,” Phys. Rev. Lett. 84, 2275–2278 (2000).
[CrossRef] [PubMed]

Willenbacher, N.

. N. Willenbacher, C. Oelschlaeger, M. Schspferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. 99, 068302 (2007).
[CrossRef] [PubMed]

Wyss, M. T.

P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009).
[CrossRef]

B. Weber, C. Burger, M. T. Wyss, G. K. von Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20(10), 2664–2670 (2004).
[CrossRef] [PubMed]

Yodh, A. G.

C. Zhou, G. Yu, F. Daisuke, J. H. Greenberg, A. G. Yodh, and T. Durduran, ”Diffuse optical correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 14, 1125–1144 (2006).
[CrossRef] [PubMed]

T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518–525 (2004).
[CrossRef] [PubMed]

Yoshimara, T.

T. Yoshimara, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am A 3, 1032–1054 (1986).
[CrossRef]

Yu, G.

C. Zhou, G. Yu, F. Daisuke, J. H. Greenberg, A. G. Yodh, and T. Durduran, ”Diffuse optical correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 14, 1125–1144 (2006).
[CrossRef] [PubMed]

T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518–525 (2004).
[CrossRef] [PubMed]

Zakharov, P.

P. Zakharov and F. Scheffold, “Monitoring spatially heterogeneous dynamics in a drying colloidal thin film,” Soft Materials, to appear (2010).

P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009).
[CrossRef]

P. Zakharov, S. Bhat, P. Schurtenberger, F. Scheffold, “Multiple scattering suppression in dynamic light scattering based on a digital camera detection scheme,” Appl. Opt. 45(8), 1756–1764 (2006)
[CrossRef] [PubMed]

Zhou, C.

C. Zhou, G. Yu, F. Daisuke, J. H. Greenberg, A. G. Yodh, and T. Durduran, ”Diffuse optical correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 14, 1125–1144 (2006).
[CrossRef] [PubMed]

T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518–525 (2004).
[CrossRef] [PubMed]

Zunzunegui, C.

P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009).
[CrossRef]

Appl. Opt. (1)

Curr. Opin. Colloid Interface Sci. (2)

R. Cerbino and A. Vailati, “Near-field scattering techniques: novel instrumentation and results from time and spatially resolved investigations of soft matter systems,” Curr. Opin. Colloid Interface Sci. 14, 416–425 (2009).
[CrossRef]

F. Scheffold and R. Cerbino, ”New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12, 50–57 (2007).
[CrossRef]

Eur. J. Neurosci. (1)

B. Weber, C. Burger, M. T. Wyss, G. K. von Schulthess, F. Scheffold, and A. Buck, “Optical imaging of the spatiotemporal dynamics of cerebral blood flow and oxidative metabolism in the rat barrel cortex,” Eur. J. Neurosci. 20(10), 2664–2670 (2004).
[CrossRef] [PubMed]

Europhys. Lett. (1)

A. Knaebel, M. Bellour, J. P. Munch, V. Viasnoff, F. Lequeux, and J. L. Harden, “Aging behavior of Laponite clay particle suspensions,” Europhys. Lett. 52, 73–79 (2000).
[CrossRef]

J. Biomed. Opt. (1)

J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): a nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1, 174–179 (1996).
[CrossRef]

J. Cereb. Blood Flow Metab. (2)

T. Durduran, M. G. Burnett, C. Zhou, G. Yu, D. Furuya, A. G. Yodh, J. A. Detre, and J. H. Greenberg, “Spatiotemporal quantification of cerebral blood flow during functional activation in rat somatosensory cortex using laser-speckle flowmetry,” J. Cereb. Blood Flow Metab. 24, 518–525 (2004).
[CrossRef] [PubMed]

A. K. Dunn, H. Bolay, M. A. Moskowitz, and D. A. Boas, “Dynamic imaging of cerebral blood flow using laser speckle,” J. Cereb. Blood Flow Metab. 21, 195–201 (2001).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys. 104, 1758–1761 (1996).
[CrossRef]

J. Mod. Opt. (1)

K. Schatzel, “Noise in photon correlation and photon structure functions,” J. Mod. Opt. 30, 155–166 (1983).

J. Opt. Soc. Am A (1)

T. Yoshimara, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am A 3, 1032–1054 (1986).
[CrossRef]

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

J. Phys. Cond. Mat. (1)

L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, and S. Mazoyer, “Time-resolved correlation: a new tool for studying temporally heterogeneous dynamics,” J. Phys. Cond. Mat. 15, 257–262 (2003).
[CrossRef]

Neuroimage (1)

. A. K. Dunn, A. Devor, A. M. Dale, and D. A. Boas, “Spatial extent of oxygen metabolism and hemodynamic changes during functional activation of the rat somatosensory cortex,” Neuroimage 27(15), 279–290 (2005).
[CrossRef] [PubMed]

Opt. Express (2)

P. Zakharov, A. C. Volker, M. T. Wyss, F. Haiss, N. Calcinaghi, C. Zunzunegui, A. Buck, F. Scheffold, and B. Weber, “Dynamic laser speckle imaging of cerebral blood flow,” Opt. Express 16, 13904–13917 (2009).
[CrossRef]

C. Zhou, G. Yu, F. Daisuke, J. H. Greenberg, A. G. Yodh, and T. Durduran, ”Diffuse optical correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 14, 1125–1144 (2006).
[CrossRef] [PubMed]

Phys. Rev. E (2)

. C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Steady light transport under flow: characterization of evolving dense random media,” Phys. Rev. E 71, 066603 (2005).
[CrossRef]

. M. Erpelding, A. Amon, and J. E. Crassous, “Diffusive wave spectroscopy applied to the spatially resolved deformation of a solid,” Phys. Rev. E 78,046104 (2008).
[CrossRef]

Phys. Rev. Lett. (3)

. N. Willenbacher, C. Oelschlaeger, M. Schspferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. 99, 068302 (2007).
[CrossRef] [PubMed]

P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. 90, 184302 (2003).
[CrossRef] [PubMed]

L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, “Universal aging features in the restructuring of fractal colloidal gels,” Phys. Rev. Lett. 84, 2275–2278 (2000).
[CrossRef] [PubMed]

Physiol. Meas. (1)

. J. D. Briers, “Laser doppler, speckle and related techniques for blood perfusion mapping and imaging,” Physiol. Meas. 22, R35–R66 (2001).
[CrossRef]

Quantum Opt.: J. Eur. Opt. Soc. B (1)

K. Sch¨atzel, “Noise on photon correlation data. i. autocorrelation functions,” Quantum Opt.: J. Eur. Opt. Soc. B 2, 287–305 (1990).
[CrossRef]

Rev. Sci. Instrum. (2)

V. Viasnoff, F. Lequeux, and D. J. Pine, “Multispeckle diffusing-wave spectroscopy: a tool to study slow relaxation and time-dependent dynamics,” Rev. Sci. Instrum. 73, 2336–2344 (2002).
[CrossRef]

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]

Soft Materials (1)

P. Zakharov and F. Scheffold, “Monitoring spatially heterogeneous dynamics in a drying colloidal thin film,” Soft Materials, to appear (2010).

Other (3)

P. Zakharov and F. Scheffold, “Advances in dynamic light scattering techniques,” in: A. A. Kokhanovsky, ed,. Light Scattering Reviews 4 (Springer, Heidelberg, 2009).

J. W. Goodman, Speckle Phenomena in Optics (Roberts and Company, Englewood, Colorado, 2007).

.B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, New York, 1976).

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Figures (7)

Fig. 1.
Fig. 1.

Experimental setup. A flat sample is illuminated with a polarized expanded laser beam. The diffuse reflected light is detected in the cross polarization channel by imaging the surface of the sample with a digital camera. The size of individual speckle spots in the image plane can be adjusted by changing the aperture of the camera objective.

Fig. 2.
Fig. 2.

Schematic representation of the matrix of pixels of a digital camera. The full matrix is divided in a set of meta-pixels Nx × Ny = N pixels each. Spatially-varying statistical properties of the speckle pattern imaged by the camera (i.e., the variance of intensities c) are estimated by averaging over all pixels within the same meta-pixel. In our calculation (Sec. 2.3), we are taking into account correlations between intensities at neighboring pixels in all directions. A pixel which is not at the boundary of the square matrix (pixel 1 in the figure) has 8 neighbors: 4 neighbors of type 2 and 4 neighbors of type 3.

Fig. 3.
Fig. 3.

Left: False-color image of light intensities in a speckle pattern (60 × 60 pixels) obtained by using the smallest available aperture setting (f/# = 32). Intensity scale from 0 to 104 in arbitrary units. Right: Intensity correlation function obtained by the inverse Fourier transform of the speckle power spectrum (full frame 640×480 pixels). The data is quantitatively described by Eq. (13) with b = 1.24a and μ = 1.09.

Fig. 4.
Fig. 4.

Speckle parameter μ as a function of the speckle size b divided by the size of the camera pixel a. Symbols: experimental results for the case of scattering from a solid sample (Teflon). Solid line: Eq. (11). Dashed line: the approximate result in the small-speckle limit ba.

Fig. 5.
Fig. 5.

Average variance of the intensity fluctuations 〈c〉 and its noise σ 2 c as functions of the number of pixels N. Speckle pattern is recorded in the image plane for light reflected from a solid piece of Teflon. Exposure time is 1 ms, transport mean free path in Teflon l* ≃ 0.25 mm, camera pixel size a = 9.9 µm, magnification one. Left panel: analysis using a subset of pixels (all neighboring pixels omitted, full symbols) leads to a constant value of 〈c〉 [Eq. (16), dotted lines]. Analysis using all pixels (open symbols) is compared to the prediction of Eq. (28) (solid lines). Right panel: thick black and thin blue lines show predictions of Eqs. (18) and (29), respectively. The inset shows a comparison of the experimental values (symbols) and theoretical predictions for H(μ) = 2 c /〈c2: Eq. (18) (blue line) and Eq. (29) (red line).

Fig. 6.
Fig. 6.

Noise of the speckle correlation coefficient as a function of the time lag τ (symbols). Lines show predictions by Eqs. (39) (left) and (36) (right). Inset of the left panel shows the intensity correlation function g 2(τ).

Fig. 7.
Fig. 7.

Noise of the speckle structure coefficient as a function of time lag τ (symbols). As predicted by the theory, the noise is independent of τ. Lines show predictions of Eqs. (46) (left) and (44) (right). The inset of the right panel shows the intensity structure function d(τ).

Equations (50)

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

I α = 1 a 2 pixel α d 2 r I ( r ) ,
i = 1 N Σ α = 1 N I α ,
c = 1 N 1 Σ α = 1 N ( I α i ) 2 .
i = I ,
c = ( I I ) 2 = I 2 .
K = c I .
σ i 2 i 2 = i 2 i 2 i 2 = 1 N ,
σ c 2 c 2 = c 2 c 2 c 2 = 8 N × N 3 4 N 1 .
σ c 2 c 2 N = 8 N .
P ( I α ) = 1 Γ ( μ ) ( μ I ) μ I α μ 1 exp ( μ I α I ) .
1 μ = 1 a 4 pixel d 2 r pixel d 2 r ' [ g 2 ( r r ' ) 1 ] ,
g 2 ( Δ r ) = I ( r ) I ( r + Δ r ) I 2 = 1 + exp ( Δ r 2 b 2 ) ,
g 2 ( Δ r ) = 1 + [ 2 J 1 ( Δ r b ) ( Δ r b ) ] 2 .
I α n = Γ ( μ + n ) Γ ( μ ) ( I μ ) n .
i = I , σ i 2 i 2 = 1 μ N ,
c = I 2 μ ,
σ c 2 c 2 = 2 N 1 [ 1 + 3 μ ( 1 1 N ) ] .
σ c 2 c 2 N = 2 N ( 1 + 3 μ ) .
1 μ 2 = 1 a 4 pixel 1 d 2 r pixel 2 d 2 r [ g 2 ( r r ) 1 ] ,
1 μ 3 = 1 a 4 pixel 1 d 2 r pixel 3 d 2 r [ g 2 ( r r ) 1 ] ,
i 2 = 1 N 2 Σ α = 1 N Σ α = 1 N I α I α .
I 2 = 1 a 4 pixel d 2 r pixel d 2 r ' I ( r ) I ( r ' )
= I 2 ( 1 + 1 μ ) ,
I 1 I 2 = 1 a 4 pixel 1 d 2 r pixel 2 d 2 r ' I ( r ) I ( r ' )
= I 2 ( 1 + 1 μ 2 ) ,
I 1 I 3 = 1 a 4 pixel 1 d 2 r pixel 3 d 2 r ' I ( r ) I ( r ' )
= I 2 ( 1 + 1 μ 3 ) ,
i 2 = 1 N 2 { N I 2 + 4 ( N N ) I 1 I 2
+ 4 ( N 1 ) 2 I 1 I 3 + [ N 2 ( 3 N 2 ) 2 ] I 2 } .
σ i 2 i 2 = 1 N [ 1 μ + 4 μ 2 ( 1 1 N ) + 4 μ 3 ( 1 2 N + 1 N ) ] .
c = 1 N 1 Σ α = 1 N I α 2 i 2 1 1 N .
c = I 2 [ 1 μ 4 μ 2 N ( N + 1 ) 4 ( N 1 ) μ 3 N ( N + 1 ) ] .
σ c 2 c 2 2 N [ 1 + 3 μ + 12 μ 2 ] = H ( μ ) N ,
g 2 ( τ ) = I ( t ) I ( t + τ ) I 2 ,
1 v = 1 T 2 0 T d t 0 T d t ' [ g 2 ( t t ' ) 1 ] .
1 v = 2 T 0 T [ g 2 ( τ ) t ] ( 1 τ / T ) d τ .
c ( τ ) = 1 N 1 Σ α = 1 N [ I α ( t ) i ( t ) ] [ I α ( t + τ ) i ( t + τ ) ] ,
i ( t ) = 1 N Σ α = 1 N I α ( t )
c ( τ ) = I 2 [ g 2 ( τ ) 1 ] ,
σ c ( τ ) 2 c ( τ ) 2 = g 2 ( τ ) 2 ( 3 2 N ) 2 [ g 2 ( τ ) 1 N ] ( N 1 ) [ g 2 ( τ ) 1 ] 2 .
σ c ( τ ) 2 c ( τ ) 2 N = 1 N × g 2 ( τ ) [ 3 g 2 ( τ ) 2 ] [ g 2 ( τ ) 1 ] 2 .
c ( τ ) = I 2 [ 1 μ 4 μ 2 N ( N + 1 ) 4 ( N 1 ) μ 3 N ( N + 1 ) ] [ g 2 ( τ ) 1 ]
σ c ( τ ) 2 c ( τ ) 2 H ( μ ) 8 N × g 2 ( τ ) [ 3 g 2 ( τ ) 2 ] [ g 2 ( τ ) 1 ] 2 .
D ( τ ) = [ I ( t ) I ( t + τ ) ] 2 = I 2 d ( τ ) ,
d ( τ ) = [ I ( t ) I ( t + τ ) ] 2 I 2 = 2 [ g 2 ( 0 ) g 2 ( τ ) ] .
s ( τ ) = 1 N Σ α = 1 N [ I α ( t ) I α ( t + τ ) ] 2 .
s ( τ ) = D ( τ ) ,
σ s ( τ ) 2 s ( τ ) 2 = 5 N .
s ( τ ) = D ( τ ) μ .
σ s ( τ ) 2 s ( τ ) 2 5 8 N H ( μ ) .

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