Abstract

We propose the application of a method based on the discrete wavelet transform to detect, identify, and measure scaling behavior in dynamic speckle. The multiscale phenomena presented by a sample and displayed by its speckle activity are analyzed by processing the time series of dynamic speckle patterns. The scaling analysis is applied to the temporal fluctuation of the speckle intensity and also to the two derived data sets generated by its magnitude and sign. The application of the method is illustrated by analyzing paint-drying processes and bruising in apples. The results are discussed taking into account the different time organizations obtained for the scaling behavior of the magnitude and the sign of the intensity fluctuation.

© 2007 Optical Society of America

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  1. Y. Aizu and T. Asakura, "Biospeckle," in Trends in Optics, A. Consortini, ed. (Academic, 1996), Chap. 2.
  2. A. Oulamara, G. Tribillon, and J. Dovernoy, "Biological activity measurements on botanical specimen surfaces using a temporal decorrelation effect of laser speckle," J. Mod. Opt. 36, 165-179 (1989).
    [CrossRef]
  3. M. F. Limia, A. M. Nuñez, H. Rabal, and M. Trivi, "Wavelet transform analysis of dynamic speckle patterns texture," Appl. Opt. 41, 6745-6750 (2002).
    [CrossRef] [PubMed]
  4. M. Pajuelo, G. Baldwin, H. Rabal, N. Cap, R. Arizaga, and M. Trivi, "Biospeckle assessment of bruising in fruits," Opt. Lasers Eng. 40, 13-24 (2003).
    [CrossRef]
  5. H. 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 5, 5381-5385 (2003).
    [CrossRef]
  6. B. Ruiz, N. Cap, and H. Rabal, "Local correlation in dynamic speckle," Opt. Commun. 245, 103-111 (2005).
    [CrossRef]
  7. G. J. Tearney and B. E. Bouma, "Atherosclerotic plaque characterization by spatial and temporal speckle pattern analysis," Opt. Lett. 27, 533-535 (2002).
    [CrossRef]
  8. P. Yu, I. Peng, M. Mustata, J. J. Turek, M. R. Melloch, and D. D. Nolte, "Time-dependent speckle in holographic optical coherence imaging and the health of tumor tissue," Opt. Lett. 29, 68-70 (2004).
    [CrossRef] [PubMed]
  9. G. H. Sendra, R. Arizaga, H. Rabal, and M. Trivi, "Decomposition of biospeckle images in temporary spectral bands," Opt. Lett. 30, 1641-1643 (2005).
    [CrossRef] [PubMed]
  10. L. T. Passoni, H. Rabal, and C. M. Arizmendi, "Characterizing dynamic speckle time data set with the Hurst coefficient concept," Fractals 12, 319-328 (2004).
    [CrossRef]
  11. G. M. Zaslavsky, "Chaos, fractional kinetics, and anomalous transport," Phys. Rep. 371, 461-580 (2002).
    [CrossRef]
  12. P. Abry, P. Flandrin, M. Taqqu, and D. Veitch, "Wavelets for the analysis, estimation and synthesis of scaling data," in Self-Similar Network Traffic and Performance Evaluation, K. Park and W. Willinger, eds. (Wiley, 2000), pp. 39-87.
  13. H. Rabal, N. 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]
  14. A. Federico and G. H. Kaufmann, "Evaluation of dynamic speckle activity using the empirical mode decomposition method," Opt. Commun. 267, 287-294 (2006).
    [CrossRef]
  15. E. Bacry, J. F. Muzy, and A. Arnéodo, "Singularity spectrum of fractal signals from wavelet analysis: exact results," J. Stat. Phys. 70, 635-675 (1993).
    [CrossRef]
  16. S. Jaffard, "Multifractal formalism for functions part I: results valid for all functions," SIAM J. Math. Anal. 28, 944-970 (1997).
    [CrossRef]
  17. S. Jaffard, "Multifractal formalism for functions part II: self-similar functions," SIAM J. Math. Anal. 28, 971-998 (1997).
    [CrossRef]
  18. A. Federico and G. H. Kaufmann, "Multifractals and dynamic speckle," Proc. SPIE 6341, 63412J (2006).
  19. J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, "Testing for nonlinearity in time series: the method of surrogate data," Physica D 58, 77-94 (1992).
    [CrossRef]

2006 (3)

H. Rabal, N. 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 and G. H. Kaufmann, "Evaluation of dynamic speckle activity using the empirical mode decomposition method," Opt. Commun. 267, 287-294 (2006).
[CrossRef]

A. Federico and G. H. Kaufmann, "Multifractals and dynamic speckle," Proc. SPIE 6341, 63412J (2006).

2005 (2)

2004 (2)

L. T. Passoni, H. Rabal, and C. M. Arizmendi, "Characterizing dynamic speckle time data set with the Hurst coefficient concept," Fractals 12, 319-328 (2004).
[CrossRef]

P. Yu, I. Peng, M. Mustata, J. J. Turek, M. R. Melloch, and D. D. Nolte, "Time-dependent speckle in holographic optical coherence imaging and the health of tumor tissue," Opt. Lett. 29, 68-70 (2004).
[CrossRef] [PubMed]

2003 (2)

M. Pajuelo, G. Baldwin, H. Rabal, N. Cap, R. Arizaga, and M. Trivi, "Biospeckle assessment of bruising in fruits," Opt. Lasers Eng. 40, 13-24 (2003).
[CrossRef]

H. 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 5, 5381-5385 (2003).
[CrossRef]

2002 (3)

1997 (2)

S. Jaffard, "Multifractal formalism for functions part I: results valid for all functions," SIAM J. Math. Anal. 28, 944-970 (1997).
[CrossRef]

S. Jaffard, "Multifractal formalism for functions part II: self-similar functions," SIAM J. Math. Anal. 28, 971-998 (1997).
[CrossRef]

1993 (1)

E. Bacry, J. F. Muzy, and A. Arnéodo, "Singularity spectrum of fractal signals from wavelet analysis: exact results," J. Stat. Phys. 70, 635-675 (1993).
[CrossRef]

1992 (1)

J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, "Testing for nonlinearity in time series: the method of surrogate data," Physica D 58, 77-94 (1992).
[CrossRef]

1989 (1)

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

Abry, P.

P. Abry, P. Flandrin, M. Taqqu, and D. Veitch, "Wavelets for the analysis, estimation and synthesis of scaling data," in Self-Similar Network Traffic and Performance Evaluation, K. Park and W. Willinger, eds. (Wiley, 2000), pp. 39-87.

Aizu, Y.

Y. Aizu and T. Asakura, "Biospeckle," in Trends in Optics, A. Consortini, ed. (Academic, 1996), Chap. 2.

Arizaga, R.

H. Rabal, N. 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]

G. H. Sendra, R. Arizaga, H. Rabal, and M. Trivi, "Decomposition of biospeckle images in temporary spectral bands," Opt. Lett. 30, 1641-1643 (2005).
[CrossRef] [PubMed]

M. Pajuelo, G. Baldwin, H. Rabal, N. Cap, R. Arizaga, and M. Trivi, "Biospeckle assessment of bruising in fruits," Opt. Lasers Eng. 40, 13-24 (2003).
[CrossRef]

H. 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 5, 5381-5385 (2003).
[CrossRef]

Arizmendi, C. M.

L. T. Passoni, H. Rabal, and C. M. Arizmendi, "Characterizing dynamic speckle time data set with the Hurst coefficient concept," Fractals 12, 319-328 (2004).
[CrossRef]

Arnéodo, A.

E. Bacry, J. F. Muzy, and A. Arnéodo, "Singularity spectrum of fractal signals from wavelet analysis: exact results," J. Stat. Phys. 70, 635-675 (1993).
[CrossRef]

Asakura, T.

Y. Aizu and T. Asakura, "Biospeckle," in Trends in Optics, A. Consortini, ed. (Academic, 1996), Chap. 2.

Bacry, E.

E. Bacry, J. F. Muzy, and A. Arnéodo, "Singularity spectrum of fractal signals from wavelet analysis: exact results," J. Stat. Phys. 70, 635-675 (1993).
[CrossRef]

Baldwin, G.

M. Pajuelo, G. Baldwin, H. Rabal, N. Cap, R. Arizaga, and M. Trivi, "Biospeckle assessment of bruising in fruits," Opt. Lasers Eng. 40, 13-24 (2003).
[CrossRef]

Bouma, B. E.

Cap, N.

H. Rabal, N. 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]

B. Ruiz, N. Cap, and H. Rabal, "Local correlation in dynamic speckle," Opt. Commun. 245, 103-111 (2005).
[CrossRef]

M. Pajuelo, G. Baldwin, H. Rabal, N. Cap, R. Arizaga, and M. Trivi, "Biospeckle assessment of bruising in fruits," Opt. Lasers Eng. 40, 13-24 (2003).
[CrossRef]

Cap, N. L.

H. 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 5, 5381-5385 (2003).
[CrossRef]

Dovernoy, J.

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

Eubank, S.

J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, "Testing for nonlinearity in time series: the method of surrogate data," Physica D 58, 77-94 (1992).
[CrossRef]

Farmer, J. D.

J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, "Testing for nonlinearity in time series: the method of surrogate data," Physica D 58, 77-94 (1992).
[CrossRef]

Federico, A.

A. Federico and G. H. Kaufmann, "Multifractals and dynamic speckle," Proc. SPIE 6341, 63412J (2006).

A. Federico and G. H. Kaufmann, "Evaluation of dynamic speckle activity using the empirical mode decomposition method," Opt. Commun. 267, 287-294 (2006).
[CrossRef]

H. Rabal, N. 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]

Flandrin, P.

P. Abry, P. Flandrin, M. Taqqu, and D. Veitch, "Wavelets for the analysis, estimation and synthesis of scaling data," in Self-Similar Network Traffic and Performance Evaluation, K. Park and W. Willinger, eds. (Wiley, 2000), pp. 39-87.

Galdrikian, B.

J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, "Testing for nonlinearity in time series: the method of surrogate data," Physica D 58, 77-94 (1992).
[CrossRef]

Galizzi, G. E.

Grumel, E.

H. 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 5, 5381-5385 (2003).
[CrossRef]

Jaffard, S.

S. Jaffard, "Multifractal formalism for functions part I: results valid for all functions," SIAM J. Math. Anal. 28, 944-970 (1997).
[CrossRef]

S. Jaffard, "Multifractal formalism for functions part II: self-similar functions," SIAM J. Math. Anal. 28, 971-998 (1997).
[CrossRef]

Kaufmann, G. H.

A. Federico and G. H. Kaufmann, "Multifractals and dynamic speckle," Proc. SPIE 6341, 63412J (2006).

H. Rabal, N. 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 and G. H. Kaufmann, "Evaluation of dynamic speckle activity using the empirical mode decomposition method," Opt. Commun. 267, 287-294 (2006).
[CrossRef]

Limia, M. F.

Longtin, A.

J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, "Testing for nonlinearity in time series: the method of surrogate data," Physica D 58, 77-94 (1992).
[CrossRef]

Melloch, M. R.

Mustata, M.

Muzy, J. F.

E. Bacry, J. F. Muzy, and A. Arnéodo, "Singularity spectrum of fractal signals from wavelet analysis: exact results," J. Stat. Phys. 70, 635-675 (1993).
[CrossRef]

Nolte, D. D.

Nuñez, A. M.

Oulamara, A.

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

Pajuelo, M.

M. Pajuelo, G. Baldwin, H. Rabal, N. Cap, R. Arizaga, and M. Trivi, "Biospeckle assessment of bruising in fruits," Opt. Lasers Eng. 40, 13-24 (2003).
[CrossRef]

Passoni, L. T.

L. T. Passoni, H. Rabal, and C. M. Arizmendi, "Characterizing dynamic speckle time data set with the Hurst coefficient concept," Fractals 12, 319-328 (2004).
[CrossRef]

Peng, I.

Rabal, H.

H. Rabal, N. 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]

G. H. Sendra, R. Arizaga, H. Rabal, and M. Trivi, "Decomposition of biospeckle images in temporary spectral bands," Opt. Lett. 30, 1641-1643 (2005).
[CrossRef] [PubMed]

B. Ruiz, N. Cap, and H. Rabal, "Local correlation in dynamic speckle," Opt. Commun. 245, 103-111 (2005).
[CrossRef]

L. T. Passoni, H. Rabal, and C. M. Arizmendi, "Characterizing dynamic speckle time data set with the Hurst coefficient concept," Fractals 12, 319-328 (2004).
[CrossRef]

H. 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 5, 5381-5385 (2003).
[CrossRef]

M. Pajuelo, G. Baldwin, H. Rabal, N. Cap, R. Arizaga, and M. Trivi, "Biospeckle assessment of bruising in fruits," Opt. Lasers Eng. 40, 13-24 (2003).
[CrossRef]

M. F. Limia, A. M. Nuñez, H. Rabal, and M. Trivi, "Wavelet transform analysis of dynamic speckle patterns texture," Appl. Opt. 41, 6745-6750 (2002).
[CrossRef] [PubMed]

Ruiz, B.

B. Ruiz, N. Cap, and H. Rabal, "Local correlation in dynamic speckle," Opt. Commun. 245, 103-111 (2005).
[CrossRef]

Sendra, G. H.

Taqqu, M.

P. Abry, P. Flandrin, M. Taqqu, and D. Veitch, "Wavelets for the analysis, estimation and synthesis of scaling data," in Self-Similar Network Traffic and Performance Evaluation, K. Park and W. Willinger, eds. (Wiley, 2000), pp. 39-87.

Tearney, G. J.

Theiler, J.

J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, "Testing for nonlinearity in time series: the method of surrogate data," Physica D 58, 77-94 (1992).
[CrossRef]

Tribillon, G.

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

Trivi, M.

Turek, J. J.

Veitch, D.

P. Abry, P. Flandrin, M. Taqqu, and D. Veitch, "Wavelets for the analysis, estimation and synthesis of scaling data," in Self-Similar Network Traffic and Performance Evaluation, K. Park and W. Willinger, eds. (Wiley, 2000), pp. 39-87.

Yu, P.

Zaslavsky, G. M.

G. M. Zaslavsky, "Chaos, fractional kinetics, and anomalous transport," Phys. Rep. 371, 461-580 (2002).
[CrossRef]

Appl. Opt. (2)

Fractals (1)

L. T. Passoni, H. Rabal, and C. M. Arizmendi, "Characterizing dynamic speckle time data set with the Hurst coefficient concept," Fractals 12, 319-328 (2004).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. A (1)

H. 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 5, 5381-5385 (2003).
[CrossRef]

J. Stat. Phys. (1)

E. Bacry, J. F. Muzy, and A. Arnéodo, "Singularity spectrum of fractal signals from wavelet analysis: exact results," J. Stat. Phys. 70, 635-675 (1993).
[CrossRef]

Opt. Commun. (2)

A. Federico and G. H. Kaufmann, "Evaluation of dynamic speckle activity using the empirical mode decomposition method," Opt. Commun. 267, 287-294 (2006).
[CrossRef]

B. Ruiz, N. Cap, and H. Rabal, "Local correlation in dynamic speckle," Opt. Commun. 245, 103-111 (2005).
[CrossRef]

Opt. Lasers Eng. (1)

M. Pajuelo, G. Baldwin, H. Rabal, N. Cap, R. Arizaga, and M. Trivi, "Biospeckle assessment of bruising in fruits," Opt. Lasers Eng. 40, 13-24 (2003).
[CrossRef]

Opt. Lett. (3)

Phys. Rep. (1)

G. M. Zaslavsky, "Chaos, fractional kinetics, and anomalous transport," Phys. Rep. 371, 461-580 (2002).
[CrossRef]

Physica D (1)

J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, "Testing for nonlinearity in time series: the method of surrogate data," Physica D 58, 77-94 (1992).
[CrossRef]

SIAM J. Math. Anal. (2)

S. Jaffard, "Multifractal formalism for functions part I: results valid for all functions," SIAM J. Math. Anal. 28, 944-970 (1997).
[CrossRef]

S. Jaffard, "Multifractal formalism for functions part II: self-similar functions," SIAM J. Math. Anal. 28, 971-998 (1997).
[CrossRef]

Other (3)

A. Federico and G. H. Kaufmann, "Multifractals and dynamic speckle," Proc. SPIE 6341, 63412J (2006).

P. Abry, P. Flandrin, M. Taqqu, and D. Veitch, "Wavelets for the analysis, estimation and synthesis of scaling data," in Self-Similar Network Traffic and Performance Evaluation, K. Park and W. Willinger, eds. (Wiley, 2000), pp. 39-87.

Y. Aizu and T. Asakura, "Biospeckle," in Trends in Optics, A. Consortini, ed. (Academic, 1996), Chap. 2.

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Figures (5)

Fig. 1
Fig. 1

Logscale Diagram of the data sets (a) Δ I , (b) | Δ I | , and (c) sgn ( Δ I ) , corresponding to a paint-drying process showing the different temporal sequences ai.

Fig. 2
Fig. 2

Logscale Diagram of the data sets (a) Δ I , (b) | Δ I | , and (c) sgn ( Δ I ) , corresponding to the temporal sequence i; original data (dashed curve), randomized Fourier phase (solid curve).

Fig. 3
Fig. 3

(a) The qth-order scaling exponent α q and (b) the scaling exponent h q , corresponding to the magnitude data set of the sequence i with the Fourier phase randomized.

Fig. 4
Fig. 4

The Logscale Diagram of the data sets (a) Δ I , (b) | Δ I | , and (c) sgn ( Δ I ) , corresponding to a bruised apple for the healthy (dashed curve) and bruised (solid curve) regions.

Fig. 5
Fig. 5

(a) The qth order scaling exponent α q and (b) the scaling exponent h q , corresponding to the magnitude data set in the healthy region.

Equations (8)

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

E [ d X 2 ( j , k ) ] = 2 j α c f C ( α , ψ 0 ) ,
c f C ( α , ψ 0 ) = σ 2 d t | t | 2 H [ d u ψ 0 ( u ) ψ 0 ( t u ) ] ,
C ( α , ψ 0 ) = d v | v | α | F [ ψ 0 ( v ) ] | 2 ,
μ j = 1 n j k = 1 n j | d X ( j , k ) | 2 , j = 1 J ,
μ j ( q ) = 1 n j k = 1 n j | d X ( j , k ) | q
E [ | d X ( j , k ) | q ] = E [ | d X ( 0 , k ) | q ] 2 j ( q H + q / 2 ) .
E [ μ j ( q ) ] = C q 2 j [ ζ ( q ) + q / 2 ] ,
d t | T X ( a , t ) | q a ζ ( q ) + q / 2 , a 0 + .

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