## Statistical speckle study to characterize scattering media: use of two complementary approaches

Optics Express, Vol. 15, Issue 21, pp. 13817-13831 (2007)

http://dx.doi.org/10.1364/OE.15.013817

Acrobat PDF (365 KB)

### 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

## 1. Introduction

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

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

## 2. Speckle statistics

### 2.1 Statistic of 1^{st} order

*N*is the scatterers number,

*a*and

_{k}*φ*the amplitude and the phase of the

_{k}*k*

^{th}contribution, respectively.

- The amplitude
*a*and phase_{k}*φ*of the_{k}*k*^{th}contribution are independent between them and of other contribution. - The phases
*φ*are uniformly distributed on [0;2_{k}*π*].

*I*can be expressed by:

*I*is [19]:

_{d}*M*= 〈

*I*〉

_{d}^{2}/σ

_{I}

^{2}and σ

_{I}

^{2}the variance of the intensity.

*M*can be interpreted as the number of speckle grains detected. For

*M*= 1, this probability density corresponds to be a function in exponential decrease and tends to be a Gaussian distribution when

*M*→∞. Experimentally, it tends to a Gaussian distribution when

*M*≫ 1 [19].

### 2.2 Statistic of 2^{nd} order

*FT*). Then, the PSD of the intensity is:

*x*

_{1},

*y*

_{1}) and (

*x*

_{2},

*y*

_{2}) in the observation plane (

*x*,

*y*) is defined as follow:

*x*=

*x*

_{1}-

*x*

_{2}and ∆

*y*=

*y*

_{1}-

*y*

_{2}. If

*x*

_{2}= 0,

*y*

_{2}= 0,

*x*

^{1}=

*x*and

*y*

_{1}=

*y*we can write:

*FT*

^{-1}) of the

*PSD*of the intensity:

*c*(

_{I}*x*,

*y*), corresponds to the normalized autocorrelation function with a zero base and provides a reasonable measurement of the “average width” of a speckle [7]:

*c*(

_{I}*x*,0) and

*c*(0,

_{I}*y*) correspond to the horizontal and the vertical profiles of

*c*(

_{I}*x*,

*y*) , respectively. Their width at half maximum are denoted

*dx*and

*dy*, respectively. Figure 1 shows the horizontal profile

*c*(

_{I}*x*,0) of the normalized autocovariance function

*c*(

_{I}*x*,

*y*) .

*dx*and

*dy*give the “average width” of speckle grains in the two dimensions of the image and they constitute the “speckle size”. The well known classical frequential approach of the phenomenon speckle is based on this statistical property. This approach allows to characterize spatially the speckle pattern through the characteristic of these grains.

## 3. Fractional Brownian motion (fBm) applied to the speckle phenomenon: diffusion equation of the speckle pattern

*X*such as a Brownian motion is described by:

*t*is the time, ∆

*t*is the time increment and ∝ is the proportional sign.

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

*H*= 0.5. Indeed, the increment process of the fractional Brownian motion is expressed as follow:

*H*∈ [0,1] is the Hurst coefficient that reflects the Hölderian regularity in each point of the trajectory [21

21. 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]

*X*. In fact, the fractional Brownian motion is the generalization of the Brownian motion. For the Brownian motion, there is no correlation between the increment because

*H*= 0.5 [20].

*y*of the speckle pattern, with the hypothesis of stationarity for the second order, we obtain:

*C*

_{ff}is the autocorrelation function of the speckle pattern intensity

*I*and〈

*I*(

*x*,

*y*)

^{2}〉 = σ

_{I}

^{2}+ 〈

*I*(

*x*,

*y*)〉

^{2}.

*X*which presents local regularity is described for its autocorrelation function by:

*y*, is [18

**70**, 046618 (2004). [CrossRef]

*H*, the Hurst coefficient, is related to the image fractal dimension*D*according to the expression_{f}*D*=_{f}*d*+1-*H*, where*d*is the topologic dimension. When*H*→ 1, then*D*→_{f}*d*; therefore we tend towards a perfect regular image, in the “Hölderian” sense.*H*characterizes the image at a local scale. Then, it is a characteristic of speckle grains. Indeed, it can be easily shown that, if ∆*x*≪*λ**i.e*. at a local scale in the speckle pattern and for*H*> 0,5, the Eq. (17) can be written like: log(〈[l(*x*+∆*x*,*y*)-*I*(*x*,*y*)]^{2}〉)∝ 2*H*log(∆*x*). We can note the analogy with the increment process of the fractional Brownian motion [Eq. (12)].*S*, the Self-similar element, given by*π*/*λ*[22], allows the quantization of the dimension in the image that separates the classic from the self-similarity properties of the speckle. In this dimension, the process is scale invariant.*G*, the Saturation of the variance equal to 2σ_{I}^{2}, characterizes the image at a global scale. For a speckle pattern and for every horizontal dimension*y*, the intensity increments*f*(*x*) = log(〈[*I*(*x*+ ∆*x*,*y*)-*I*(*x*,*y*)]^{2}〉) are calculated and approximated according to Eq. (17) by:

*ε*= 10

^{-5}is a mean square error. We can identify then the three coefficients

*a,b,c*according to Eq. (17):

21. 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]

*H*can be considered as a regularity parameter.

*H*value found is lower than 1 (≈ 0,77, see Section 5) that means that this speckle pattern is more rough than that of Fig. 3(b) in the Hölderian sense, this can be appreciate through the comparison of Fig. 3(b) and Fig. 4(b).

## 4. Experimental setup and methods

^{-4}s, which is shorter than the correlation times measured for our samples. For each sample under study, 200 speckle patterns were recorded with 25 Hz frequency and were digitized with an 8-bit precision by the analog-to-digital converter.

*L*between the surface of the medium and the CCD camera [10

10. Q. B. Li and F. P. Chiang, “Three-dimensional of laser speckle,” Appl. Opt. **31**, 6287–6291 (1992). [CrossRef] [PubMed]

27. T. L. Alexander, J. E. Harvey, and A. R. Weeks, “Average speckle size as a function of intensity threshold level: comparison of experimental measurements with theory,” Appl. Opt. **33**, 8240–8250 (1994). [CrossRef] [PubMed]

*L*= 20 cm from the surface of the medium.

*N*

_{0}is the initial concentration defined as the number per cubed meter. For polystyrene microspheres and at the wavelength used here, the absorption can be omitted regarding the scattering effect. By dilution in deionized water, we can adjust their concentration

*c*to obtain the scattering coefficient

*μ*required, according to Mie theory [28]. Table 1 presents the initial number of particles per cubed meter

_{s}*N*

_{0}and scattering parameters issued from Mie calculation for each microsphere diameter

*d*: the scattering coefficient

*μ*with respect to

_{s}*c*, the scattering efficiency factor

*Qs*and the anisotropy factor

*g*. The reduced scattering coefficient

*μ*’ introduced as the equivalent isotropic scattering coefficient of an otherwise anisotropically-scattering medium is also presented.

_{s}- mixture
*m*: all size microspheres with a volume fraction*f*of 1/6 for each diameter. Two mixtures_{v}*m*were carried out for each microspheres addition type. - mixture
*m*(3:1): composed of mixture*m*supplemented with 0.20 μm (or 2.00 μm) in volume ratio (3:1). - mixture
*m*(3:2): composed of mixture*m*supplemented with 0.20 μm (or 2.00 μm) in volume ratio (3:2). - mixture
*m*(1:1): composed of mixture*m*supplemented with 0.20 μm (or 2.00 μm) in volume ratio (1:1).

## 5. Results

*p*, the p-value for the null hypothesis H

_{0}, corresponding to the variability between groups. Nevertheless, the ANOVA tests the hypothesis that all groups are statically the same against the general alternative that they are not all the same. If the global null hypothesis H

_{0}is rejected and in order to know which pairs of groups are significantly different, and which are not, we used a “Multiple Comparison test”. From each sample under study as described in Section 4, and for each of 200 speckle patterns recorded by the CCD camera, we calculated, the speckle size from the spatial autocorrelation function described in Subsection 2.2, and the model parameters according to the procedure described in Section 3. We deduced then the corresponding mean and standard deviation values.

### 4.1 Influence of microspheres sizes on speckle statistic: classical frequential and fractal approach.

*length dx*and with a p-value less than 0.01, we have discrimination between all miscrosphere diameters except between 1.00, 1.50 and 2.00 μm. For Hurst coefficient

*H*, and Self-similarity

*S*, we have a very good discrimination between all diameters with a p-value

*p*≪ 0.01. The Saturation of the variance

*G*, discriminates all microsphere diameters except for 0.75 and 1.50 μm.

### 4.2 Influence of the size distribution of smaller and larger microspheres on speckle statistic: classical frequential and fractal approach.

*dx*and

*S. H*discriminates all mixtures except mixtures

*m*(3:2) and

*m*(1:1).

*G*does not discriminate mixtures

*m*(3:1) and

*m*(1:1). In the case of addition of larger microspheres (2.00 μm), all mixtures have been discriminated by

*S*and

*H*. Nevertheless,

*G*does not discriminate mixtures

*m*and

*m*(3:1), and

*dx*does not discriminate

*m*(3:2) and

*m*(1:1), as shown the same statistic tests used.

## 6. Discussions

### 6.1 Samples discrimination

*S*, and the Hurst coefficient

*H*, discriminate all microsphere sizes studied with a p-value

*p*≪ 0.01, contrary to speckle size which not discriminate 1.00, 1.50 and 2.00 μm.

*H*and

*S*allow discrimination for all mixtures samples for larger microspheres addition, contrary to speckle size which does not allow discrimination for mixtures

*m*(3:2) and

*m*(1:1). For addition of smaller microspheres, we have discrimination between all mixtures for the Self-similarity

*S*, and speckle size

*dx*. Therefore, at least one of model parameters discriminate all mixtures in the two types of them, that is not the case of the speckle size which discriminates all mixtures only in case of addition of smaller microspheres.

### 6.2 Parameters evolution

*G*increases with microsphere diameters until one reaches values about the wavelength used,

*i.e*. 0.75 μm. Beyond that, it is difficult to exploit.

*c*(

_{I}*x*,0) , that allow extraction of more parameters than the speckle size. Consequently, integrating the multi-scale aspect of the speckle pattern, model parameters would provide complementary information about the image and thus, about the medium.

## 7. Conclusions

**70**, 046618 (2004). [CrossRef]

*H*, the Self-similar element

*S*, and the Saturation of the variance

*G*. These parameters characterize the image at three different scales.

*in vivo*several cutaneous pathologies. We hope to understand the optical signification of these model parameters by continuing to evaluate the relationship between their evolution and modifications in scattering media. We wish that speckle study becomes thus a non invasive tool aimed at helping diagnosis, which will be of great interest for dermatologists.

## Acknowledgments

## References and links

1. | P. Lehmann, “Surface-roughness measurement based on the intensity correlation function of scattered light under speckle-pattern illumination,” Appl. Opt. |

2. | R. Berlasso, F. Perz Quintian, M. A. Rebollo, C. A. Raffo, and N. G. Gaggioli, “Study of speckle size of light scattered from cylindrical rough surfaces,” Appl. Opt. |

3. | 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 |

4. | 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 |

5. | J. D. Briers, G. Richard, and X. W. He, “Capillary blood flow monitoring using laser speckle contrast analysis (LASCA),” J. Biomed. Opt. |

6. | 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 |

7. | J. W. Goodman, “Statistical Properties of Laser Speckle Pattern,” in |

8. | L. I. Goldfisher, “Autocorrelation function and power spectral density of last-produced speckle pattern,” J. Opt. Soc. Am. A |

9. | M. Françon, |

10. | Q. B. Li and F. P. Chiang, “Three-dimensional of laser speckle,” Appl. Opt. |

11. | Y. Piederrière, F. Boulvert, J. Cariou, B. Le Jeune, Y. Guern, and G. Le Brun, “Backscattered speckle size as a function of polarization : influence of particle-size and -concentration,” Opt. Express |

12. | Y. Piederrière, J. Le Meur, J. Cariou, J. F. Abgrall, and M. T. Blouch, “Particle aggregation monitoring by speckle size measurement; application to blood platelets aggregation,” Opt. Express |

13. | L. Zhifand, L. Hui, and Y. Qiu, “Fractal analysis of laser speckle for measuring roughness,” Proc. SPIE |

14. | C. L. Benhamou, |

15. | L. othuaud, |

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, |

17. | T. Hyon Ha, |

18. | S. Guyot, M. C. Péron, and E. Deléchelle, “Spatial Speckle Characterization by Brownian Motion analysis,” Phys. Rev. E |

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

20. | P. Abry, P. Gonçalves, and P. Flandrin, |

21. | M. F. Barnsley, R.L. Devaney, B.B. Mandelbrot, H.-O. Peitgen, D. Saupe, and R. F. Voss, |

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 |

23. | Pentland A. et al. “Fractal-based description of natural scenes,” |

24. | H. Funamizu and J. Uozumi, “Generation of fractal speckles by means of a spatial light modulator,” Opt. Express |

25. | K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. |

26. | J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. |

27. | T. L. Alexander, J. E. Harvey, and A. R. Weeks, “Average speckle size as a function of intensity threshold level: comparison of experimental measurements with theory,” Appl. Opt. |

28. | C. F. Bohren and D. R. Huffman, |

29. | Gelebart B., |

30. | R. Simpson, |

31. | R.V. Hogg and J. Ledolter, |

32. | D. A. Zimnyakov, V. V. Tuchin, and A. A. Mishin, “Spatial speckle correlometry in applications to speckle structure monitoring,” Appl. Opt. |

33. | A. H. Hielscher, J. R. Mourant, and I. J. Bigio, “Influence of particle size and concentration on the diffuse backscattering of polarized light from tissue phantoms and biological cell suspensions,” Appl. Opt. |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: June 18, 2007

Revised Manuscript: August 1, 2007

Manuscript Accepted: August 29, 2007

Published: October 5, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

O. Carvalho, B. Clairac, M. Benderitter, and L. Roy, "Statistical speckle study to characterize scattering media: use of two complementary approaches," Opt. Express **15**, 13817-13831 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-21-13817

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### References

- 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]
- R. Berlasso, F. Perz Quintian, M. A. Rebollo, C. A. Raffo and N. G. Gaggioli, "Study of speckle size of light scattered from cylindrical rough surfaces," Appl. Opt. 39, 5811-5819 (2000). [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]
- 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]
- 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," in Handbook of Biomedical Diagnostics, Valery V. Tuchin, ed., (SPIE press, Bellingham 2002) Chap. 18.
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- Q. B. Li and F. P. Chiang, "Three-dimensional of laser speckle," Appl. Opt. 31, 6287-6291 (1992). [CrossRef] [PubMed]
- Y. Piederrière, F. Boulvert, J. Cariou, B. Le Jeune, Y. Guern, G. Le Brun, "Backscattered speckle size as a function of polarization : influence of particle-size and -concentration," Opt. Express 13, 5030-5039 (2005). [CrossRef] [PubMed]
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