We have analyzed the image formation and dynamic properties in laser speckle
imaging (LSI) both experimentally and with Monte-Carlo simulation. We show for
the case of a liquid inclusion that the spatial resolution and the signal
itself are both significantly affected by scattering from the turbid
environment. Multiple scattering leads to blurring of the dynamic inhomogeneity
as detected by LSI. The presence of a non-fluctuating component of scattered
light results in the significant increase in the measured image contrast and
complicates the estimation of the relaxation time. We present a refined
processing scheme that allows a correct estimation of the relaxation time from
LSI data.Comment: submitted to Optics Letter

Alfred Buck

Andreas Völker

Bandyopadhyay

Boas

Briers

Bruno Weber

Cheng

Dunn

Durduran

Durian

Frank Scheffold

Heckmeier

Pavel Zakharov

Prahl

Schätzel

Völker

Weber

Zimnyakov

English

arXiv

We have analyzed the image formation and dynamic properties in laser speckle  imaging (LSI) both experimentally and with Monte Carlo simulation. We show for the  case of a liquid inclusion that the spatial resolution and the signal itself are both  significantly affected by scattering from the turbid environment. Multiple scattering  leads to blurring of the dynamic inhomogeneity as detected by LSI. The presence of a  nonfluctuating component of scattered light results in the significant increase in the  measured image contrast and complicates the estimation of the relaxation time. We  present a refined processing scheme that allows a correct estimation of the relaxation  time from LSI data

Zakharov, Pavel

Völker, Andreas

Buck, Alfred

Weber, Bruno

Scheffold, Frank

RERO DOC Digital Library

http://doc.rero.chQuantitative modeling of laser speckle imagingPavel Zakharov and Andreas Vo¨lkerDepartment of Physics, University of Fribourg, 1700 Fribourg, SwitzerlandAlfred Buck and Bruno WeberDivision of Nuclear Medicine, University Hospital Zurich, 8091 Zurich, SwitzerlandFrank ScheﬀoldDepartment of Physics, University of Fribourg, 1700 Fribourg, SwitzerlandWe have analyzed the image formation and dynamic properties in laser speckle imaging (LSI) both experi-mentally and with Monte-Carlo simulation. We show for the case of a liquid inclusion that the spatial resolutionand the signal itself are both signiﬁcantly aﬀected by scattering from the turbid environment. Multiple scat-tering leads to blurring of the dynamic inhomogeneity as detected by LSI. The presence of a non-ﬂuctuatingcomponent of scattered light results in the signiﬁcant increase in the measured image contrast and complicatesthe estimation of the relaxation time. We present a reﬁned processing scheme that allows a correct estimationof the relaxation time from LSI data.Laser speckle imaging (LSI) is an eﬃcient and simplemethod for full-ﬁeld monitoring of dynamics in hetero-geneous media.1 It is widely used in biomedical imagingof blood ﬂow1–5 since it provides access to physiologi-cal processes in vivo with excellent temporal and spatialresolution.In a more general context LSI can be considered asimpliﬁed version of the dynamic light scattering (DLS)approach, which analyzes the temporal intensity ﬂuctu-ations of scattered laser light in order to derive the mi-croscopic properties of the scatterers position and mo-tion.6,7 In LSI an image of dynamic heterogeneities isobtained by analysing the local speckle contrast K inthe image plane. K is deﬁned by the variance of theintensity ﬂuctuations for a given integration time T :8K2(T ) = 〈I2〉/〈I〉2 − 1. In the absence of scatterers mo-tion the contrast takes a maximum value, while motiondecreases the contrast. Therefore the value K can beused to construct an image of local dynamic properties.1However the quantitative interpretation of the LSI datais not straightforward. Multiple scattering of light caninﬂuence the apparent size of the object due to diﬀuseblurring. Access to the local dynamic properties, such asblood ﬂow or Brownian motion, is complicated by thecomplex interplay between the measured contrast andthe full ﬂuctuation spectrum of scattered light. The lat-ter is usually not accessible if technical simplicity of LSIis to be preserved.In this letter we present a quantitative approach toanalyze LSI images. We address the problem of spatialresolution and blurring due to multiple scattering viamodel experiments and Monte-Carlo simulations. Fur-ther we will demonstrate that previous attempts to re-late quantitatively LSI images to the microscopic motionhave been hampered by an incorrect data analysis.Wepresent a reﬁned processing scheme to access this infor-mation from a standard LSI experiment. The main newelement of our analysis is to take into account the contri-bution of the non-ﬂuctuating part of scattered intensity.We furthermore suggest a simple experimental procedurethat allows to access this important quantity.Model experiments have been carried out using a ho-mogeneous block of solid Teﬂon and a home-made het-erogeneous sample. This medical phantom mimics a liq-uid inclusion in solid tissue. It is obtained by milling acylindrical hole of diameter D = 3 mm in a block of solidTeﬂon. A layer of variable thickness 0.1− 2.1 mm sepa-rates the cylindrical inclusion from the interface that isimaged. The void is ﬁlled with a dispersion of 710 nmpolystyrene particles in water. The particle concentra-tion is adjusted to match the optical properties of theliquid dispersion to the solid such that no static scatte-ring diﬀerences could be detected anymore (volume frac-tion ca. 1.3%; scattering coeﬃcient μs = 36 mm−1, meanfree path l∗  277 μm and anisotropy factor g ≈ 0.9).The imaging setup has been described in detail in Ref. 5.Brieﬂy, the cylindric inclusion with ﬂat base of 3 mm di-ameter oriented to the surface is imaged with a CCDcamera (PCO Pixelﬂy, Germany, exposure time 130 ms)with a standard camera objective lens (f = 50 mm).The sample is illuminated with a diode laser (wavelength785 nm, max. 50 mW). Two crossed polarizers, posi-tioned before and after the sample, ensure that we de-tect only depolarized light back reﬂected from the sam-ple. The incident beam is expanded by a slow-rotatingground glass in order to reduce statistical noise.5 Foreach depth the contrast as a function of radial distancer from the inclusion center is determined (Fig. 1).For the Monte-Carlo simulation we followed the pho-ton packet approach of light propagation in a turbid1Published in "Optics Letters 31(23): 3465-3467, 2006"which should be cited to refer to this work. http://doc.rero.ch 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0  1  2  3  4  5r, mmKd = 0.10 mmd = 0.45 mmd = 1.20 mmd = 2.10 mm0.40.50.60.70.0 0.5 1.0 1.5 2.0d, mmσ, mmFig. 1. Contrast K as a function of distance r from thecenter of the inclusion for diﬀerent depths d. Lines - sim-ulation results, Symbols - experimental results. Dottedline - inclusion boundary. Inset: Width of the liquid solid-boundary as a function of depth (• - simulation data, - experiment, line is a guide for the eye).medium.9 The Henyey-Greenstein phase function wasused based on an average scattering angle 〈cos θ〉.9 Thedegree of polarization was assumed to decay exponen-tially.10 The reﬂections and refractions on the bound-aries were treated according to Fresnel formulas. Depo-larized photon packets back-reﬂected from the samplewhere registered within a numerical aperture 0.17. Theﬁeld auto-correlation function (ACF) g1(τ) of the scat-tered light was determined as explained in Ref. 7.In order to quantify the eﬀect of image blurring causedby multiple scattering we determine the standard devi-ation σ of the contrast gradient dK/dr which is a mea-sure of the apparent interfacial width. As shown in Fig. 1,starting from the smallest depths, the apparent widthis increased. This increase is of the order of a few l∗,which is the relevant length scale for an incident pho-ton to propagate laterally. The smearing increases lin-earily with depth until for large depths 2σ becomes ofthe order of the depth d, as expected for diﬀuse lightpropagation.11 Our results show the importance of dif-fuse blurring in the image formation already for smalland moderate depths and can thus provide importantguidelines for experiments in biomedical imaging.2,3A typical set of simulated correlation functions is pre-sented in the Fig.2. If g1(τ) is known, then the specklecontrast K of the time-integrated speckle ﬂuctuationscan be obtained via the following relation12,13K2(T ) =〈I2〉〈I〉2 − 1 =2T∫ T0β|g1(τ)|2 (1− τ/T ) dτ, (1)where T is the integration time of the detector andβ ≤ 1 the coherence factor of the detection optics. Notethat this equation diﬀers from the traditionally usedexpression of Fercher and Briers.1 As pointed out byDurian and coworkers13 the expression of Fercher and 0 0.2 0.4 0.6 0.8 110-6 10-5 10-4 10-3 10-2 10-1 100τ, sg 1 (τ)r = 0.0 mmr = 1.5 mmr = 3.0 mmr = 4.5 mm3 mmrFig. 2. Simulated correlation functions for the inclusiondepth d = 100 μm (Symbols). Solid lines: Fits with DWStheory including a baseline; τ0 values for the ﬁts wereobtained by the contrast K inversion (see Fig. 3).Briers is incorrect and it is important to take into ac-count triangular averaging of the correlation functionvia the factor (1− τ/T ).12 As shown in Fig. 1 the re-sulting values of the contrast are found in quantitativeagreement with the experimental data.A comparison of Fig. 1 and Fig. 2 immediately revealsthe main limitation of LSI. A single contrast value K isrecorded in practice as compared to the complex correla-tion function characterizing the full spectrum of intensityﬂuctuations. To address this problem we ﬁrst analyzethe characteristic features of the correlation functionspresented in Fig. 2. Overall the functions are well de-scribed by the usual stretched-exponential form derivedfor diﬀusing-wave spectroscopy (DWS) of a colloid sus-pension: g1(τ) = exp(−γ√6τ/τ0), where in our case therelaxation time τ0 = 1/Dk20 characterizes Brownian mo-tion (diﬀusion coeﬃcient D) and γ  2 is a constant.7It is worthwhile to note that for the case of ﬂuid ﬂowthe relaxation time τ0 is inversely proportional to thecharacteristic velocity of the scatterers.1,14The presence of a non-ﬂuctuating static scatteringpart, however, leads to a non-zero baseline for the ACF.Since the relative amount of the static light scattering in-creases with the distance r from the center, the plateauof the ACF increases as well. It has been noted pre-viously that a non-zero baseline signiﬁcantly inﬂuencesthe resulting values of the contrast K.1 Nevertheless it israther common in the biomedical community to convertthe obtained contrast values directly to relaxation timesneglecting contributions of static scattering.3,4,15If the detected light is composed of dynamic and staticcomponents: E(τ) = Ed(τ)+Es the ﬁeld ACF is deﬁnedas following:11g1(τ) = 〈E(τ)E(0)〉/〈E〉2 = (1− ρ) |g1 d (τ)|+ ρ, (2)where ρ = 〈Is〉/(〈Id〉+ 〈Is〉) characterizes the static partof the detected light intensity and g1 d (τ) is the ﬁeld2http://doc.rero.ch10-210-1100101102 0  0.5  1  1.5  2d, mm1 / τ0,  1 / sFig. 3. Inverse of the relaxation time 1/τ0 as a functionof depth obtained by directly converting the contrastK(r ≈ 0) based on Eq.1 neglecting the static part: • -simulation data,  - experiment and using the correctprocedure based on Eq. 3:  - simulation and ♦ - ex-periment. Horizontal dashed line - actual value of therelaxation time 1/τ0  72s−1.correlation function of the dynamic component. The con-trast value can be calculated by substituting eq. (2) intoeq. (1):K2 =2βT∫ T0[(1− ρ) |g1d (τ)|+ ρ]2 (1− τ/T ) dτ=[(1− ρ)2 K22 d + 2ρ (1− ρ)K21 d + βρ2], (3)where K22 d = 2β/T∫ T0|g1 d (τ)|2 (1− τ/T ) dτ andK21 d = 2β/T∫ T0|g1 d (τ)| (1− τ/T ) dτ are the normal-ized variances of the intensity and ﬁeld ﬂuctuations ofthe dynamic part of the detected signal with K2 d <K1 d.A correct analysis of the laser speckle image can onlybe performed if the value of ρ is known.16 This can beachieved by an additional processing step . The cameraexposure time T is usually larger compared to the re-laxation time τ0 related to blood ﬂow and subsequentframes are separated by a period Δt larger or equal T .Since Δt ≥ T > τ0 two sequential frames are free of thedynamic component of interest and thus the space de-pendent static component can be easily found by cross-correlating a sub-set of N pixels i, j around a given po-sition: βρ2 + 1 = 〈I0IΔt〉N /〈I〉2NFig. 3 shows results of the contrast inversion appliedto our experiments. While the correct procedure yieldsvalues close to the relaxation time of Brownian motionτ0 the direct conversion, neglecting static scattering, candeviate by several orders of magnitude.In conclusion, we studied the image blurring of anobject buried in a turbid medium and found that theresolution of the obtained images can be aﬀected sig-niﬁcantly by multiple scattering. We furthermore intro-duced a model that reﬂects the impact of the staticscattering on the interpretation of LSI images. A simpleprocedure has been suggested to perform a quantitativeanalysis in actual LSI experiments.Financial support from the Swiss National Sciencefoundation (Project No. 111824) is gratefully acknowl-edged. Correspondence should be addressed to FrankScheﬀold (e-mail, Frank.Scheﬀold@unifr.ch) or Pavel Za-kharov (e-mail, Pavel.Zakharov@unifr.ch)References1. J. D. Briers. Laser doppler, speckle and related tech-niques for blood perfusion mapping and imaging. Phys-iological Measurement, 22(4):R35–R66, 2001.2. B. Weber, C. Burger, M. T. Wyss, G. K. von Schulthess,F. Scheﬀold, and A. Buck. Optical imaging of the spa-tiotemporal dynamics of cerebral blood ﬂow and oxida-tive metabolism in the rat barrel cortex. Eur J Neurosci,20(10):2664, 2004.3. T. Durduran, M. G. Burnett, C. Zhou G. Yu, D. Furuya,A. G. Yodh, J. A. Detre, and J. H. Greenberg. Spa-tiotemporal quantiﬁcation of cerebral blood ﬂow duringfunctional activation in rat somatosensory cortex usinglaser-speckle ﬂowmetry. Journal of Cerebral Blood Flow& Metabolism, 24:518–525, 2004.4. A. Dunn, A. Devor, M. Andermann, H. Bolay,M. Moskowitz, A. Dale, and D. Boas. Simultaneousimaging of total cerebral hemoglobin concentration, oxy-genation and blood ﬂow during functional activation.Optics Letters, 28:28 – 30, 2003.5. A.C. Vo¨lker, P. Zakharov, B. Weber, F. Buck, andF. Scheﬀold. Laser speckle imaging with active noisereduction scheme. Opt. Exp., 13(24):9782 – 9787, 2005.6. B.J. Berne and R. Pecora. Dynamic Light Scattering.With Applications to Chemistry, Biology, and Physics.Dover Publications, Inc., New York, 2000.7. D.J. Durian. Accuracy of diﬀusing-wave spectroscopytheories. Phys. Rev. E , 51:3350 – 3358, 1995.8. Brackets 〈〉 denote the ensemble average. In practice thisaverage can be obtained by a spatial or temporal analysisof the intensity ﬂuctuations, for details see Refs. 1–59. S. A. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch.A monte carlo model of light propagation in tissue. InG. J. Mu¨ller and D. H. Sliney, editors, SPIE Proceed-ings of Dosimetry of Laser Radiation in Medicine andBiology, volume IS 5, pages 102 – 111, 1989.10. D. A. Zimnyakov. On some manifestations of similarityin multiple scattering of coherent light. Waves RandomMedia, 10:417 – 434, 2000.11. D. A. Boas and A. G. Yodh. Spatially varying dynamicalproperties of turbid media probed with diﬀusing tempo-ral light correlation. JOSA, 14(1):192 – 215, 1997.12. K Scha¨tzel. Noise on photon correlation data. i. auto-correlation functions. Quantum Optics: Journal of theEuropean Optical Society Part B, 2(4):287–305, 1990.13. R. Bandyopadhyay, A.S. Gittings, S.S. Suh, P.K. Dixon,and D.J. Durian. Speckle-visibility spectroscopy: A toolto study time-varying dynamics. Rev. Sci. Instrum.,76:093110, 2005.14. M. Heckmeier and G. Maret Visualization of Flow inMultiple Scattering Liquids, Europhys. Lett. 34, 257(1996)3http://doc.rero.ch15. Haiying Cheng, Qingming Luo, Qian Liu, Qiang Lu,Hui Gong, and Shaoqun Zeng. Laser speckle imagingof blood ﬂow in microcirculation. Physics in Medicineand Biology, 49(7):1347–1357, 2004.16. We note that our results are in contradiction to previ-ous claims that simply analyzing dynamic speckles elim-inates the contamination due to stationary speckles asreported by P. Li, S. Ni, L. Zhang, S. Zeng, and Q . LuoImaging cerebral blood ﬂow through the intact rat skullwith temporal laser speckle imaging. Optics Letters,31:1824 – 1827, 2006.4

Quantitative modeling of laser speckle imaging

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Quantitative modeling of laser speckle imaging

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