## Noise in laser speckle correlation and imaging techniques |

Optics Express, Vol. 18, Issue 14, pp. 14519-14534 (2010)

http://dx.doi.org/10.1364/OE.18.014519

Acrobat PDF (1598 KB)

### 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

## 1. Introduction

*dynamic light scattering*(DLS), also known as photon correlation spectroscopy (PCS), where the fluctuations of the far-field speckle are analyzed [2]. The method is widely used to study polymers in solution or for the sizing of sub-micron particles. The same method applied to turbid media is known as

*diffusing-wave spectroscopy*(DWS). In both DLS and DWS, one usually makes use of the ergodicity of intensity fluctuations and replace ensemble averaging by time averaging. Both are typically applied to study fluctuations in the (sub-) millisecond range. Thus, for a total measurement time of a few minutes, millions of fluctuations are sampled which provides an excellent signal to noise ratio. Using a single photon counter and a digital correlator, nanosecond time resolution is commonly achieved. Even though at the shortest times the signal to noise ratio is limited by photon shot noise [3, 4

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). [CrossRef]

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]

11. 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]

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]

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]

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

*in vivo*with excellent temporal and spatial resolution. LSI is also becoming increasingly popular in soft material sciences as a probe of heterogenous dynamic properties [20, 28

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]

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]

## 2. Properties of stationary speckle patterns

*N*values of integrated (over the area of a pixel) intensity

*I*,

_{α}*α*= 1, . . .

*N*, corresponding to

*N*distinct pixels.

*N*could be a (typically, large) total number of available pixels, but it could also be just a small subset of pixels carrying information about only a part of the optical field, scattered by a small part of the sample (see Fig. 2). Our purpose is to characterize the statistical properties of the optical field based on this information. Note that quite generally the statistical properties of the fully developed speckle patterns considered here are the same for measurements in the image plane or in the far-field [1]. Therefore our results should find applications in many research areas where speckle based techniques are employed.

### 2.1. Negative exponential distribution of integrated intensities

*I*of light scattered from such a random medium follows the negative exponential distribution:

*P*(

*I*) = (1/〈

*I*〉)exp(−

*I*/〈

*I*〉) [1]. In the experiment we only have access to

*i*and

*c*found by averaging over an ensemble of realizations are equal to the actual values of the average intensity and of its variance, respectively:

*c*〉/〈

*I*〉

^{2}is equal to one in this case. The

*speckle contrast K*can be defined as

### 2.2. Gamma distribution of integrated intensities

*b*≪

*a*) the integrated intensities

*I*corresponding to different pixels can be considered independent, whereas the distribution of each

_{α}*I*, according to Goodman [1], can be modeled with the so-called gamma distribution:

_{α}*μ*≥ 1 depends on the pixel shape and size, as well as on the spatial correlation function

*g*

_{2}(Δ

**r**) of intensity

*I*(

**r**):

*μ*provides a measure of how the ratio of speckle size

*b*to the pixel size

*a*influences the statistical properties of the measured intensity. It is worthwhile to note that in the photon correlation spectroscopy, 1/

*μ*is equal to the intercept (usually denoted by

*β*) of the autocorrelation function:

*g*

_{2}(0) = 1+1/

*μ*(see also Section 3.1).

*b*quantifies the extent of spatial correlations of

*I*(

**r**) (i.e., the size of a single speckle spot). An analytic expression for

*μ*can only be obtained for very small speckles

*b*≪

*a*:

*μ*⋍ (

*a*/

*b*)

^{2}/(4

*π*). Figure 3 (left) shows a speckle image of light scattered from solid Teflon taken for the case of large speckles that can be resolved in space (

*f*/# = 32). Be

*Ĩ*(

**q**) the Fourier transform of the image

*I*(

**r**). From the inverse Fourier transform of the power spectrum ∣

*I*(

**q**)∣

^{2}we directly obtain the normalized intensity correlation function

*g*

_{2}(Δ

*r*) [1] [see Fig. 3 (right)]. In our example, for speckles of finite size

*b*/

*a*= 1.24, the intercept of the correlation function is given by 1+1/

*μ*= 1.92. In Fig. 4 we compare Eq. (11) with experimental results and find an excellent agreement.

*i*and

*c*defined by Eqs. (2) and (3) can be calculated for the gamma distribution using the known expression for its statistical moments:

*I*are independent random variables obeying the gamma distribution [Eq. (10)]. As expected, they coincide with Eqs. (4), (5), (7), and (8) for

_{α}*μ*= 1, when Eq. (10) becomes identical to the negative exponential distribution. When

*μ*> 1, the variance of intensities

*c*of the speckle pattern is reduced due to the effective spatial averaging over the area of a pixel that becomes comparable to the typical size of the speckle spot

*b*[see Eq. (16)]. At the same time, fluctuations of

*c*from one meta-pixel to another are suppressed [see Eq. (17)]. In the limit of large

*N*we have

### 2.3. Role of correlations between neighboring pixels

*a*larger than the correlation range

*b*of the speckle field, it appears reasonable to take into account only correlations between integrated intensities

*I*corresponding to neighboring pixels and to neglect correlations between intensities of more distant pixels. The impact of correlations on the average values and variances of

_{α}*i*and

*c*is described by two additional parameters

*i*. The average 〈

*i*〉=〈

*I*〉 is the same as in the absence of correlations between pixels. To compute the variance of

*i*,

*σ*

^{2}

*= 〈*

_{i}*i*

^{2}〉−〈

*i*〉

^{2}, we need to know 〈

*i*

^{2}〉 which is the ensemble average of the square of Eq. (2):

*I*and the parameters

_{α}*μ*,

*μ*

_{2}and

*μ*

_{3}that we introduced in Eqs. (11), (19), and (20), the correlation function 〈

*I*

_{α}*I*

_{α′}〉 that appears in Eq. (21) is equal to

*α*=

*α*′,

*α*and

*α*′ correspond to two neighboring pixels situated side by side as the pixels 1 and 2 in Fig. 2,

*α*and

*α*′ correspond to two neighboring pixels situated next to each other on a diagonal as the pixels 1 and 3 in Fig. 2, or simply 〈

*I*〉

^{2}for pixels that are not neighbors. The sum in Eq. (21) contains

*N*terms of type Eq. (22) (one term per pixel), 4(

*N*− √

*N*) terms of type Eq. (23), 4(√

*N*−1)

^{2}terms of type Eq. (24) and

*N*

^{2}−(3√

*N*−2)

^{2}terms 〈

*I*〉

^{2}. These expressions follow from accurately counting the number of neighbors of types 2 and 3 for all pixels with account for the fact that the pixels situated at the boundary of a square matrix of pixels have less neighbors. Combining all these results we have

*c*can be found by noting that the definition of

*c*, Eq. (3), can be rewritten as

*I*

^{2}

*〉 ≡ 〈*

_{α}*I*

^{2}〉 and Eq. (25) for 〈

*i*

^{2}〉:

*c*becomes

*N*-dependent. Because we saw in the previous sections that when different pixels are considered independent, 〈

*c*〉 does not depend on

*N*[cf. Eqs. (5) and (16)], we conclude that the dependence of 〈

*c*〉 on

*N*is a signature of correlation between signals corresponding to different pixels. The estimator [Eq. (3)] acquires a bias. Interestingly, the spatial range of correlations

*b*or, more precisely, the ratio

*b*/

*a*can be estimated by observing the

*N*-dependence of 〈

*c*〉.

*c*,

*σ*

^{2}

*, can be calculated along the same lines as 〈*

_{c}*c*〉. The calculation, however, is much more involved and rapidly leads to cumbersome equations. Instead of dealing with this lengthy and, anyway, approximate analytic calculation, we estimate

*σ*

^{2}

*by a numerical simulation. We use a computer to generate a set of 4096×4096 speckle patterns with correlation lengths*

_{c}*b*= 2, 4, and 8, respectively. These speckle patterns are then used to generate integrated speckle patterns with pixel sizes

*a*varying from 2 to 64. The integrated speckle patterns model outputs of a typical CCD or CMOS camera. Finally, we numerically compute the dependencies of 〈

*c*〉 and

*σ*

^{2}

*on the number*

_{c}*N*of pixels, for fixed ratios

*b*/

*a*. For

*b*/

*a*≪ 1 the parameter

*μ*characterizing integrated speckle patterns [Eq. (11)] is a function of

*b*/

*a*only. Hence, by inverting this function we can express

*b*/

*a*and hence

*σ*

^{2}

*/ 〈*

_{c}*c*〉

^{2}as a function of

*μ*instead of

*b*/

*a*. We know that Eq. (17) is valid in the limit of

*μ*→ ∞ (or, equivalently,

*b*/

*a*→0) which suggests that

*σ*

^{2}

*/〈*

_{c}*c*〉

^{2}can be expressed as a series in 1/

*μ*of which the terms of order (1/

*μ*)

^{0}and (1/

*μ*)

^{1}are already known from Eq. (17). In the limit of large

*N*, the results of our simulations can be fit by adding a term quadratic in 1/

*μ*:

### 2.4. Properties of time-integrated speckle patterns

*I*of scattered light fluctuates in time. In such an experiment the intensity

*I*of scattered light changes not only as a function of position

**r**, but also as a function of time

*t*:

*I*=

*I*(

**r**,

*t*). Fluctuations of

*I*in time can be characterized by its autocorrelation function,

**r**from the arguments of

*I*, for clarity. The analysis presented in the previous section applies to this situation too, provided that the integration time

*T*(i.e. the time during which the signal is accumulated to determine

*I*) is much shorter than the characteristic correlation time

_{α}*τ*

_{c}of variations of

*I*. In the opposite case, i.e. when

*T*≳

*τ*

_{c}, the definition of

*I*should include not only the spatial integration over the pixel area, but the temporal integration as well.

_{α}*a*≪

*b*and statistically independent

*I*following the negative exponential distribution, the effect of finite integration time can be described by a parameter

_{α}*ν*[13

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]

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

*μ*in Eq. (11), with the integrations over space (i.e. over a pixel) being replaced by integrations over time. Both integrations represent (partial) averaging over position or time due to either the finite area of the pixel [in Eq. (11)] or non-zero integration time

*T*[in Eq. (31)]. Because the integrand of Eq. (31) is a function only of the difference

*t*−

*t*′, we can change the variables of integration to

*τ*=

*t*−

*t*′ and perform integration over

*t̄*. The expression for

*ν*then becomes

*I*follows the gamma distribution [Eq. (10)] with

_{α}*ν*substituted for

*μ*[1]. The average value and the normalized variance of

*c*are given by Eqs. (16) and (17), respectively.

*a*that becomes comparable to the correlation length

*b*of the speckle pattern, we can take into account correlations between integrated intensities corresponding to neighboring pixels following the approach of Sec. 2.3. The outcome of our analysis is quite simple: all results of Sec. 2.3—and, in particular, Eqs. (26), (28), and (29)—still apply but with

*μ*,

*μ*

_{2}and

*μ*

_{3}replaced by

*μ*×

*ν*,

*μ*

_{2}×

*ν*and

*μ*

_{3}×

*ν*, respectively. Here

*μ*,

*μ*

_{2}and

*μ*

_{3}describe the impact of spatial correlations, whereas ν accounts for time integration. The effects of spatial and time correlations thus fully decouple. This is due to the fact that the spatio-temporal correlation function

*g*

_{2}(Δ

**r**,

*τ*)−1 = 〈

*I*(

**r**,

*t*)

*I*(

**r**+Δ

**r**,

*t*+

*τ*)〉/〈

*I*〉

^{2}− 1 decouples, in its turn, into a product of position- and time-dependent parts.

### 2.5. Comparison with experiment

*λ*= 532 nm) is expanded and collimated to a beam of several centimeters in waist to create an approximately homogeneous illumination spot on the sample surface (see Fig. 1). Liquid samples are contained in large glass containers whereas solid samples are measured in air. The diffuse reflected light is monitored in the image plane in the crossed polarization channel with a CCD camera PCO Pixelfly (640×480 pixels of 9.9×9.9 µm

^{2}size, 12 bit) at magnification one. The estimated depth-of-focus of the imaging system is close to 0.1 mm. The actual size of speckle spots on the CCD chip can be adjusted by varying the aperture of the camera objective. For the simple configuration of a single lens placed at a distance

*l*from the screen and a circular aperture with radius

*q*the speckle correlation length is [26

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

*b*= 2

*l*/

*q*= [4

*l*/

*kf*] ×

*f*/#.

*f*= 50 mm and is placed at a distance

*l*~ 100 mm from the CCD sensor. The

*f*-number of the objective can be varied from

*f*/#=2.8 to 32. The accessible values of

*b*are thus comparable or smaller than the pixel size

*b*≤

*a*. The camera exposure time is adjustable with the shortest time used of 0.1 ms. The maximum speed of data acquisition at full resolution is 50 frames per second. The recorded images are corrected for dark counts and slight spatial variations of the average intensity. Namely, an average over a large number of statistically independent images it computed. Next the reference image is smoothed by taking a sliding window average (10×10 pixel) in order to eliminate residual signatures of the speckle. Subsequently, each recorded image is divided by this mean and the statistical properties are calculated. We have carefully checked that the choice of the sliding window average does not alter the experimental results. Throughout this article we compare two different types of experiments with theory. In an idealized experiment we analyze only a

*subset*of pixels where neighboring pixels are omitted. The procedure effectively eliminates the effect of spatial correlations between the pixels for all situations considered here. The second set of data is obtained by analyzing all available pixels, a situation typically encountered in actual applications.

*l** ≃ 0.25 mm. In Fig. 5, we compare our experimental results for 〈

*c*〉 (left panel) and its normalized variance

*σ*

^{2}

*/〈*

_{c}*c*〉

^{2}(right panel) with the predictions of the theoretical model developed above. The parameter

*μ*is determined from the best fit to the data via the relation

*μ*= 〈

*I*〉

^{2}/〈

*c*〉 for

*N*→ ∞. The values obtained for

*μ*are found in excellent agreement with the theoretical predictions as shown in Fig. 4. Here the parameter

*b*for all settings has been obtained from the relation

*b*∝

*f*/# using the known value of

*b*/

*a*= 1.24 for

*f*/# = 32 (Fig. 3). For consistency, in the following discussion, for dynamic media, we use the fitted values of

*μ*for each setting.

## 3. Noise in photon correlation spectroscopy

### 3.1. Fluctuations of the intensity correlation function

*T*≪

*τ*and correlating speckle images separated by a time lag

_{c}*τ*larger than

*T*. When intensities

*I*(

_{α}*t*) are measured for

*N*pixels, the degree of intensity correlation can be estimated as

*I*〉; it is equivalent to

*i*of Sec. 2. However, it will be important for us to keep track of the difference between

*i*(

*t*) and

*i*(

*t*+

*τ*) in the definition [Eq. (33)] of

*c*(

*τ*).

*c*(

*τ*) for independent pixels with

*I*(

_{α}*t*) distributed according to the negative exponential distribution are:

*N*we have

*g*

_{2}(Δ

**r**,

*τ*)−1 = 〈

*I*(

**r**,

*t*)

*I*(

**r**+Δ

**r**,

*t*+

*τ*)〉/〈

*I*〉

^{2}− 1 decouples into a product of position- and time-dependent parts, we readily obtain

*c*(

*τ*). Here

*μ*,

*μ*

_{2}and

*μ*

_{3}account for spatial correlations of the instantaneous speckle pattern and are given by Eqs. (11), (19) and (20), respectively.

*c*(

*τ*) with account for correlations between neighboring pixels appears to be much more involved. However, an approximate result can still be obtained by an

*ad hoc*combination of Eqs. (29) and (36):

*l** ≃ 73

*µ*m at 532 nm. A detailed characterization of the system is given in Ref. [29

29. N. Willenbacher, C. Oelschlaeger, M. Schšpferer, 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]

*T*≃ 22° C which sets the relaxation time to several tens of seconds. We first determine the full autocorrelation function from a combined photon PCS and CCD-camera based experiment [9

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). [CrossRef]

*g*

_{2}(

*τ*) of scattered light (see the inset of Fig. 6). We then perform measurements of the noise of the speckle correlation coefficient

*c*(

*τ*) for different sizes of the camera objective aperture corresponding to the values of

*μ*ranging from 1.29 to 5.15, and as a function of the number of pixels

*N*. In all cases we observe that the normalized variance of

*c*(

*τ*) scales with 1/

*N*as expected from the theory. In Fig. 6 we show

*Nσ*

^{2}

*/〈*

_{c}*c*〉

^{2}as a function of the time lag

*τ*. The general rule is that the normalized variance of the correlation coefficient

*c*(

*τ*) increases with

*τ*. Experiment and theory are in fairly good agreement if the analysis is performed on a subset of pixels (right panel of Fig. 6), especially at small

*μ*, when the statistics of intensity fluctuations is close to negative exponential. The agreement is worse when all pixels are analyzed (left panel of Fig. 6), even though the theory describes the general trend of our data quite well. This suggests that either we were unable to eliminate residual experimental errors or that additional factors, not taken into account in our theoretical model, enter into play when important correlations between signals corresponding to different pixels are present.

### 3.2. Fluctuations of the intensity structure function

*g*

_{2}(

*τ*), fluctuations of intensity can be characterized by a

*structure*function

*I*(

_{α}*t*), assuming no correlation between intensities at different pixels, we obtain

*a*that can become comparable to the correlations length

*b*of the speckle pattern, we obtain

*s*(

*τ*) taking into account correlations between neighboring pixels appears to be much more involved. However, an approximate result can still be obtained by a (though

*ad-hoc*) combination of Eqs. (29) and (44):

*s*(

*τ*) scales with 1/

*N*and that it is independent of

*τ*. However, theory and experiment do not show quantitative agreement: theoretical lines lie either higher (blue and red lines in the left panel of Fig. 7) or lower (all other lines in Fig. 7) than the data. The reasons behind this disagreement are the same as in the case of intensity correlation coefficient

*c*(

*τ*) in Sec. 3.1.

## 4. Conclusions

*N*of pixels of a digital camera. In particular, we studied the noise of the speckle intensity variance

*c*, and of the intensity correlation and structure coefficients

*c*(

*τ*) and

*s*(

*τ*), respectively. The variances of all these quantities decrease as 1/

*N*when the number of pixels is increased and depend in complex ways on the spatial (due to the finite pixel size) and temporal (due to the finite integration time) correlations of intensity in the speckle pattern. For stationary speckle patterns, we obtained quantitative agreement between measurements and the theoretical model that we developed in this paper. For dynamic speckle patterns, theoretical predictions reproduce general trends of our data but fail to provide a fully quantitative description. We believe that this is due to the approximate character of our theoretical model (neglecting correlations between distant pixels, gamma distribution of intensity at a single pixel) as well as to imperfections of our experiment (impossibility to achieve a perfectly uniform illumination of the sample, necessity to work with weak signals, etc.). Despite the absence of quantitative agreement between theory and experiment in the latter case, the results presented in the paper provide an important starting point for estimation of the noise level in such applications as the multi-speckle dynamic light scattering, time-resolved correlation spectroscopy, speckle visibility spectroscopy, near-field scattering, laser speckle imaging and echo speckle imaging.

## Acknowledgements

## References and links

1. | J.W. Goodman, |

2. | B.J. Berne and R. Pecora, |

3. | K. Schätzel, “Noise in photon correlation and photon structure functions,” J. Mod. Opt. |

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 |

5. | F. Scheffold and R. Cerbino, “New trends in light scattering,” Curr. Opin. Colloid Interface Sci. |

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. |

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. |

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. |

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. |

10. | P. Zakharov, S. Bhat, P. Schurtenberger, and F. Scheffold, “Multiple scattering suppression in dynamic light scattering based on a digital camera detection scheme,” Appl. Opt. |

11. | 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. |

12. | P. K. Dixon and D. J. Durian, “Speckle visibility spectroscopy and variable granular fluidization,” Phys. Rev. Lett. |

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. |

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. |

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. |

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 |

17. | C. Baravian, F. Caton, J. Dillet, and J. Mougel, “Steady light transport under flow: characterization of evolving dense random media,” Phys. Rev. E |

18. | D. D. Duncan, S. J. Kirkpatrick, and R. K. Wang, “Statistics of local speckle contrast,” J. Opt. Soc. Am. A |

19. | P. Zakharov and F. Scheffold, “Advances in dynamic light scattering techniques,” in: A. A. Kokhanovsky, ed. |

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. |

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. |

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. |

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 |

26. | T. Yoshimara, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am A |

27. | K. Schätzel, “Noise on photon correlation data. i. autocorrelation functions,” Quantum Opt.: J. Eur. Opt. Soc. B |

28. | M. Erpelding, A. Amon, and J. E. Crassous, “Diffusive wave spectroscopy applied to the spatially resolved deformation of a solid,” Phys. Rev. E |

29. | N. Willenbacher, C. Oelschlaeger, M. Schšpferer, P. Fischer, F. Cardinaux, and F. Scheffold, “Broad bandwidth optical and mechanical rheometry of wormlike micelle solutions,” Phys. Rev. Lett. |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(100.4550) Image processing : Correlators

(110.6150) Imaging systems : Speckle imaging

(170.3880) Medical optics and biotechnology : Medical and biological imaging

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: May 17, 2010

Revised Manuscript: June 15, 2010

Manuscript Accepted: June 20, 2010

Published: June 22, 2010

**Virtual Issues**

Vol. 5, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

S. E. Skipetrov, J. Peuser, R. Cerbino, P. Zakharov, B. Weber, and F. Scheffold, "Noise in laser speckle correlation and imaging techniques," Opt. Express **18**, 14519-14534 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14519

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### References

- 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).
- K. Schatzel, “Noise in photon correlation and photon structure functions,” J. Mod. Opt. 30, 155–166 (1983).
- 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]
- F. Scheffold and R. Cerbino, ”New trends in light scattering,” Curr. Opin. Colloid Interface Sci. 12, 50–57 (2007). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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. 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]
- 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]
- P. Zakharov and F. Scheffold, “Advances in dynamic light scattering techniques,” in: A. A. Kokhanovsky, ed,. Light Scattering Reviews 4 (Springer, Heidelberg, 2009).
- P. Zakharov and F. Scheffold, “Monitoring spatially heterogeneous dynamics in a drying colloidal thin film,” Soft Materials, to appear (2010).
- . 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]
- 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]
- . 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]
- T. Yoshimara, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am A 3, 1032–1054 (1986). [CrossRef]
- K. Sch¨atzel, “Noise on photon correlation data. i. autocorrelation functions,” Quantum Opt.: J. Eur. Opt. Soc. B 2, 287–305 (1990). [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]
- . 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]

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