流体中3D3C流场检测方案(CCD相机)

智能文字提取功能测试中

Research ArticleOptics EXPRESsVol.26， No. 13 | 25 Jun 2018|OPTICS EXPRESS 16708 Research Article https：//doi.org/10.1364/OE.26.016708#326765Journal C 2018Received 23 Mar 2018； revised 24 May 2018； accepted 5 Jun 2018； published 14 Jun 2018 Self-calibrated microscopic dual-viewtomographic holography for 3D flowmeasurements JIAN GAO AND JOSEPH KATZ Department of Mechanical Engineering， Johns Hopkins University， 3400 N. Charles St.， Baltimore， MD21218， USA "katz@jhu.edu Abstract： This paper introduces the application of microscopic dual-view tomographic holog-raphy (M-DTH) to measure the 3D position and motion of micro-particles located in densesuspensions. Pairing of elongated traces of the same particle in the two inclined reconstructedfields requires precise matching of the entire sample volume that accounts for the inherent distor-tions in each view. It is achieved by an iterative volumetric self-calibration method， consisting ofmapping one view onto the next， dividing the sample volume into slabs， and cross-correlatingthe two views. Testing of the procedures using synthetic particle fields with imposed distortionand realistic errors in particle locations shows that the self-calibration method achieves a 3Duncertainty of about 1um， a third of the particle diameter. Multiplying the corrected intensityfields is used for truncating the elongated traces， whose centers are located within 1um of theexact value. Without correction， only a small fraction of the traces even overlap. The distortioncorrection also increases the number of intersecting traces in experimental data along with theirintensity. Application of this method for 3D velocity measurements is based on the centroids ofthe truncated/shortened particle traces. Matching of these traces in successive fields is guidedby several criteria， including results of volumetric cross-correlation of the multiplied intensityfields. The resulting 3D velocity distribution is substantially more divergence-free， i.e.， satisfiesconservation of mass， compared to analysis performed using single-view data. Sample applica-tion of the new method shows the 3D flow structure around a pair of cubic roughness elementsembedded in the inner part of a high Reynolds number turbulent boundary layer. C 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes： (090.1995) Digital holography； (110.0180) Microscopy； (110.6960) Tomography； (100.6890) Three-dimensional image processing； (280.2490) Flow diagnostics. References and links 1. JJ. Sheng， E. Malkiel， and J. Katz， “Using digital holographic microscopy for simultaneous measurements of 3D nearwall velocity and wall shear stress in a turbulent boundary layer，"Exp. 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Fluids 54， 1557 ( 2013). ) 1..lIIntroduction： applying tomography to alleviate the depth-of-focus problem in1digital holography Digital in-line holography is an effective tool for quantifying the 3D location， size， and shapeof particles distributed in a sample volume. Sequential exposures in time can be used fordetermining the velocity of these particles as well. Relevant applications include e.g. trackingof micro-particles for 3D flow measurements [1-3]， as well as characterization of droplets andbubbles in multiphase flows [4-7]， organisms and cells in biological studies [8-10]， and fuelparticles in combustion research [11，12]. However， the elongated reconstructed image in theaxial direction of the illuminating laser beam， the so-called extended depth-of-focus (DOF)problem， has been a primary shortcoming of in-line holography [13]. Consequently， the resultinguncertainty of several particle diameters [14-16] in the axial (depth) location of the particlecenter is significantly larger than those in directions aligned perpendicularly to the beam axialdirection. For applications involving a single beam， numerous studies have already introducedmethods for improving the depth uncertainty， e.g.， [17-20]， too many to summarize in a singlepaper. In some cases， methods like the inverse problem approach [17] have been successful inalleviating most of the adverse effects for sample volumes containing sparse particle distributions，where the interference pattern of spherical particles can be precisely resolved without significantdistortions caused by neighboring particles. For dense micro-particle suspensions needed for 3Dvelocity measurements， approaches involving e.g. spatial correlations of the 3D traces [19] havealso improved the depth accuracy， but not to levels comparable to the perpendicular (in-plane)resolution. Recording at least one additional inclined view to probe the sample volume is perhaps themost straightforward way to overcome the DOF problem since information in the depth directionof one view can be extracted from the in-plane data of the others. Knowledge of how to mapeach point in the inclined reconstructed 3D fields， the so-called mapping function， is essential formatching images of the same particle in the multiple views and for determining its location anddisplacement between exposures based on in-plane data [21-23]. Alternatively， the elongatedtraces can be truncated by multiplying the 3D intensity fields， which retains only the volumeswhere they overlap [24，25]. Subsequently， the particle displacement between exposures can bedetermined using particle-tracking or 3D cross-correlation of the truncated field. This approachis referred to as dual-view tomographic holography (DTH) in the present paper. Determining andcalibrating the mapping function which relates the two fields is the critical step in implementingthis procedure since errors in the order of a few microns in applications involving microscopicholography are sufficient for traces of small particles not to overlap. Prior applications have beenbased on geometric mapping， which accounts for relative angle， magnification， and translationbetween the two coordinate systems， without corrections for distortions [21-26]. The calibrationshave been based on recording holograms of targets translated within the sample volume. The precision of the mapping function is crucial for effective matching of the 3D intensityfields associated with individual particles. For the microscopic DTH implemented in the presentstudy for 3D velocity measurements， the mapping precision has to be in the order of micronsconsidering that the nominal diameter of the tracer particles is 2 um. Geometric mapping isinsufficient since it does not account for effects of alignment errors， presence of windows，refractive index changes， and distortion caused by the imaging optics (aberrations， etc.). Forexample， a misalignment angle of 1 degree could lead to tens of microns of mapping error as thereconstruction depth increases [25]. Distortions are probably the reason for the limited matchingof droplets traces between the two views in [23]， and between flow tracer particles in [22]. This paper introduces and implements a“self-calibration”procedure for tomographic hologra-phy to quantify a distortion function， namely the 3D deviation of the actual mapping functionfrom the geometric mapping. In stereoscopic particle image velocimetry (PIV)， a 2D self-calibration procedure， which is based on matching the particle field in the sample area， follows an initial/coarse calibration obtained by translating a target in this area [27]. In tomographic PIV， theself-calibration procedure provides precise matching of the lines of sight of cameras observingthe sample volume from different angles [28]. The present method is also based on matchingthe particle fields based on their spatial distributions in the reconstructed sample volume. Thismethod takes advantage of the known， although less accurate， depth location of the particle in thesample volume. The optical setup is discussed in Section 2. The new self-calibration procedureis introduced in Section 3， where synthetic data is used for demonstrating that an uncertainty ofabout 1 pm is achievable. Implementation using experimental data and the following particletracking and 3D velocity measurements are discussed in Section 4. Fig. 1. Schematic of the experimental facility showing the channel， the pair of cubes on thebottom wall， and the local particle injection system. 2.. Optical setup for microscopic dual-view tomographic holography (M-DTH) The motivation for the present development is an experimental study aimed at measuring theflow around a pair of cubic roughness elements immersed in the inner part of a turbulent channelflow. The schematic of the experimental facility is shown in Fig. 1. These cubes have a heightof a =1 mm， and are aligned in the spanwise direction (y) and separated by 1.6a. To gainunobstructed optical access to the sample volume， the walls of the channel as well as the cubesare made of acrylic whose refractive index (~1.49) is matched with that of the working fluid， aconcentrated aqueous solution of sodium iodide [29，30]. A dense particle suspension is requiredto resolve the flow structure at scales much small than the cube， e.g. 50 um. Hence， the flow fieldis seeded“locally”by injecting 2-um tracer particles from 52 ports located sufficiently far (330port diameters) upstream of the cubes， at a very low velocity (3% of the centerline velocity)，as shown in the upper-left inset of Fig. 1. The location， port size， and speed of injection aresimilar to those used in previous studies [1，2， 31]. Furthermore， the present fully-developedsmooth-wall turbulent channel flow is similar and of the same scale as in [1]. As argued in thelatter based on prior publications about jets-in-cross-flow， the effect of the injectors on the flow inthe sample volume is negligible. Support for this claim is provided in [31]， albeit for a rough-wallchannel flow， by showing an agreement between the mean velocity profiles obtained using digitalholographic microscopy with local seeding and 2D PIV with global seeding. The number ofports is aimed at improving the homogeneity of the particle distributions in the sample volume.While， their distributions vary， a substantial fraction of the holograms contain well above 30000particles broadly distributed in the sample volume. The microscopic dual-view tomographic holography (M-DTH) setup is illustrated in Fig. 2. Research Article Vol.26，No. 13| 25 Jun 2018|OPTICS EXPRESS 16712 The beam of a dual-head Nd： YAG laser (New Wave Solo PIV) is spatially filtered， collimated，and split to illuminate the sample volume from two angles. One beam is perpendicular to thechannel wall， and the second is nominally inclined by 36°， with the overlapping volume centeredon the space between the cubes (Fig. 2， top right). The inline holograms are magnified byidentical 8X microscope objectives (MO1 and MO2)， and then recorded by 6600× 4400-pixelinterline-transfer， CCD cameras (Imperx B6640)， resulting in an effective pixel size of ~0.68 um.The sample volume size is 4.2×2.8 ×1.5 mm’ in the streamwise， spanwise， and wall-normaldirections. The origin of the corresponding physical/experimental coordinate system， (x，y，z)， islocated on the wall， between front surfaces of the cubes， as indicated in the lower-right inset ofFig. 2. The coordinate systems of the two holograms are denoted by (x1，y1，zi) and (x2，y2，Z2)，as indicated in the zoomed-in part of Fig. 2. Sequential hologram pairs are acquired by bothcameras at a rate of 1.5 Hz， with 25 us delay between exposures. Thousands of hologram pairsare recorded to obtain converged statistics of the mean flow field. Fig. 2. The microscopic dual-view tomographic holography (M-DTH) setup. 3. Self-calibration procedures 3.1. Imposing and modeling of the distortion function Using index notation， the coordinates in view 1 are represented by pi，corresponding to xi， y1，and z1. Similarly， the coordinates in view 2 are represented by qi， corresponding to x2，y2， and z2.The 3D mapping function relating the pi field to the qi field includes a geometric displacement，Ci， geometric mapping， aij， and distortion，ei(pi)，resulting in the following expression， Here， repeated indices indicate summation. The inverse mapping is denoted as where Aij=a，C=-AijCj，and oi=-aijej. For clarification， aij accounts for magnifica-tion and roation， and has the form (3) where M1， M2， and M3 are the magnifications in the x1， y1， and z directions， respectively， andw，y， and 0 are angles of rotation about the x1，y1， and z1 axes， respectively. The goal of theself-calibration is to determine ei(pi). In the present analysis， view 1， which has less interfacesand thus a simpler relationship with the physical coordinates， is used as a reference， and thedistortion is associated with mapping from view 2 to this reference. The self-calibration method is illustrated and tested by generating a virtual field containingsynthetic particles with (M1， M2，M3)=(1.005，1.015，1.045)，(C1，C2，C3)=(7.4，473.9，7.6)um， and(w，p，0)=(37.1，-0.3°，-0.55°). The rotation angles are chosen to resemble thoseused in the experimental configuration (9，0~0，≈36°). Subsequently， view2 is distorted bya known function， fi(pi)， where The selected function is The magnitudes ofGi， Bi，pij， and o ij are provided in Table 1. To examine the effects of varyingthe amplitudes of distortion， the values of Gi are selected such that the distortion in the xidirection is the lowest， and that in the z direction is the highest. These choices are consistentwith the experimental conditions. As an example， the distribution of the prescribed distortionfield in the plane z1=350 um is shown in Fig. 3. Table 1. Parameters for the analytical functions fi (uint： um) i G； Bi di1 di2 Ji3 0i1 0i2 0i3 1 10 -5 173 173 173 346 346 700 2 50 -20 519 519 519 350 350 700 3 -80 50 345 345 345 500 500 1000 The virtual cameras in views 1 and 2 have 1037 ×1037 and 1054×1555 pixels， respectively.With 8X magnification， the 5.5 um pixel size corresponds to 0.68 um. The synthetic particlefield containing randomly distributed particles has dimensions of 700 × 700 × 750 um inthe (x1，y1，Z1) directions， respectively， centered in the overlapping region of the two views.The particle size is normally distributed with a mean of 3 um and a standard deviation of 0.5um. The particle field density is characterized by its shadow density (SD) [32]. For example，at SD= 4.6%， each field contains 3200 particles， corresponding to 8707 particles/mm. Thecoordinates of the particles in view 1，Pi=(Xi，Y1，Zi)，are transformed to the correspondingcoordinates in view 2，Qi =(X2，Y2，Z2)， using Eq. (1). These locations represent the exact/truecoordinates of the particle centroids in the two views. Errors are then added to each location，reflecting uncertainties in our ability to determine the particle center from the reconstructedimages，especially in the depth direction. Hence， the detected coordinates of individual particlesin view 1，P?=(X’Y’，Z)， and view 2，Q=(X，Y，Z)， deviate from the true coordinates.We assume that the positional errors， 61i=P’- Pi and82i=Q-Qi， are normally distributed Fig. 3. Sample cross-section of the imposed distortion，(a) e1， (b) e2， and (c) e3 in planez1=350 um. with zero mean in their own fields. Therefore， the corresponding probability density functions(PDF) of positional errors are Here， o and 02 denote the standard deviations of in-plane errors， and o3 denotes that of theout-of-plane error.Hence， o and o2 are in the order of one pixel， and 03 is significantlylarger， extending to several particle diameters. Five diameters are used as the present worst case.The number of uncorrelated particle fields used to implement the self-calibration procedures isdenoted by N，. Different conditions have been investigated to study the effects of positional errors，particle number density， and number of particle fields on the accuracy of the self-calibrationprocedure， as listed in Table 2. Table 2. Conditions of the different cases used to test the self-calibration procedure Case 1 2 3 4 5 6 oi(um) 0 1.4 1.4 1.4 1.4 1.4 02 (um) 0 1.4 1.4 1.4 1.4 1.4 03 (um) 0 6 6 15 15 15 SD 4.6% 4.6% 4.6% 4.6% 18.4% 18.4% Nr 500 500 2000 500 2000 4000 3.2. Self-calibration Procedures Initially the distortion field is determined in a series of (x1，yi) planes established by dividing theparticle field into N thin slabs， as illustrated in Fig. 4. The division to slabs is readily achievablein the reconstructed 3D domain based on the intensity distribution. Each slab has a prescribedthickness in the depth direction， denoted as wz， and it extends over the entire (x1，yi) plane. Thecenter of the nth slab， where the calibration is performed， is denoted by z(n). After calibratingthe Ne planes， the 3D distortion field is determined by interpolation in the depth direction. In thevirtual experiment， Nc=14，Wz= 100 um， and there is 50% overlap between adjacent slabs，i.e.，z(n+1)_z(n)=50 pm. The in-plane distortions of each slab (e and e2) are determined first. The images of all theparticles located within the slab in view 1， which are illustrated as elongated red traces in Fig. 4，are projected onto the z"-plane. Here， the projection utilizes the true particle in-plane positions，(X1，Y1)， because in experiments， it corresponds to the reconstructed intensity distribution within Vol. 26， No. 13| 25 Jun 2018|OPTICS EXPRESS 16715 Fig. 4. A sample slab illustrating the elongated traces from view 1 along with the corre-sponding mapped particle centers from view 2. It is used for determining ejand e2. the particle in experimental data， without calculating its location. From view 2， we use thecorresponding particle centroids with position errors added， and then map them into view 1using only the geometric mapping. Due to the out-of-plane distortion (e3) and the errors， themapped centers may fall outside of the slab， as illustrated by the solid blue dots in Fig. 4. Hence，an extended slab， [z(n)-0.5wz-zex，z(n)+0.5wz + z*]， is used for the mapped view-2particle centroids. In the present analysis， zex= 50 um， consistent with the magnitude of e3.The intensity distribution within each projected 2D image is assumed to be Gaussian， using theintensity integration method described in [33] for generating the particle images. Extending theslab also projects additional mapped view-2 particles， which do not have view-1 counterparts，and are illustrated as hollow dots in Fig. 4. The following analysis shows that their effect onthe results is very small since they are not correlated with the view-1 particles located withinthe original narrower slab. Denoting the coordinates of the mapped particles from view 2 asPmi=(X{m，’imZim) and using only geometric mapping， Due to distortion and detection error， P+ Pi. Based on Eqs. (1) and(7)， For example， as illustrated in the zoomed part of Fig. 4， The mapping along with projection generates two images of each instantaneous slab. Thedisplacement of in-plane components of Pi and P. can be determined using spatial cross-correlation analysis， widely used in PIV [34]. Due to the imposed positional errors， non-uniformparticle distributions， and finite number of particles per image， cross-correlating one image pairmight be insufficient for obtaining accurate data on the in-plane distortion. This problem can bereadily solved by application of the sum-of-cross-correlation method [35] to combine the resultsfrom multiple realizations. This method consists of adding the cross-correlation results of all theimages to obtain an ensemble-averaged correlation， which is inherently much less sensitive tothe randomly distributed positional errors and sparse/non-uniform particle distributions. In thepresent virtual experiment， the interrogation window size 128 by 128 pixels， and the spacingbetween windows is 32 pixels (75% overlap). The multi-pass calculations are performed usinga commercial code (DaVis 8.2， LaVision GmbH). The displacement of the peak location fromthe window center indicates the local in-plane distortion. Sample sum-of-correlation maps for Fig.5. Sample sum-of-cross-correlation maps for one interrogation window based on 500synthetic realizations： (a) without imposed positional errors； and (b) with imposed positionalerrors. (c) an illustration clarifying the causes for the elongated correlation profile withpositional errors. one interrogation window based on 500 realizations are shown in Figs.5(a) and 5(b). Withoutany positional errors， Fig. 5(a) shows a sharp peak， which spreads and becomes elongated uponaddition of the positional errors [Fig. 5(b)]. The elongation is primarily in the yi-direction due tothe imposed larger errors in the z2 direction， as illustrated in Fig. 5(c). Nevertheless， the elongatedsum-of-correlation map still has a distinct peak that can be readily detected. The correlation mculated mean in-planFO(niland Fapprovides the calOnie distortion components， E(pi) and E(pi)， spatiallyaveraged over the window and slab depth. The superscripts indicate the iteration number. Asdemonstrated later， the positional errors introduce uncertainty， which diminishes with increasingnumber of samples and particle concentrations. This procedure is repeated for all the slabsand then smoothed by low pass filtering in all directions. The low pass filtering is based onfirst-order weighted interpolation using single value decomposition (SVD) with distortion valuesin a 3×3×3 volume surrounding each point. The values ofE(pi) and E(pi) are our firstestimation for the true ei(pi) and e2(pi). The out-of-plane distortion (e3) is determined by applying the sum-of-correlation method inthe view 2 domain. Figure 6 shows a slab aligned in the zi direction cut through the view-2volume. It contains the mapped particles from view 1 (red circles)， and the elongated traces ofthe view-2 particles. As demonstrated in the insert， the latter are projected along the Z2 directiononto a (x2，y2) plane as long as they are located within the extended slab. Their projected imagesrepresent their true in-plane positions with respect to view 2， Qi. In the actual experiment， thisoperation is equivalent to calculating the peak intensity distribution in the z2 direction throughthe region defined by the extended slab. The coordinates of the view-1 particles within the zn)slab， including their true location and positional errors， are mapped into view 2 according to boththe geometric mapping and the already-determined in-plane distortions. Their mapped locations are then These coordinates differ from the corresponding exact locations in view 2 by reflecting effects of positional errors， differences between ei and E， and the unknown e3.After fitting a Gaussian intensity distribution in the x2-y2 plane to each mapped particle， as 22 ： particle traces reconstructed in view 2 centered at (Y，Z2)corresponding to the red particles： particle traces reconstructed in view 2 not correlated to any particles in the slab 宁2 Fig. 6. Mapping of synthetic particle centers from view 1 into view 2 along with thecorresponding elongated traces of view 2 used for determining e3. discussed before， they are cross-correlated with the view-2 projections， and then ensemble-averaged using the sum-of-correlation method. Assuming that averaging minimizes the effect of81i， the measured average displacement in the y2 direction is If e-E ande2-E are small， Dy， is dominated by a23e3. Hence， e3 can be calculateddirectly from Dyz. The analysis gives an estimated depth distortion Alternately， the same term can be calculated by Dx，/a13， where Dx， is the x2-component of themeasured displacement. However， since x1-and x2-axes are nearly parallel in the present setup，a13 ~0， hence the y2-displacement is a better choice. Repeating this procedure for all the slabs，followed by 3D interpolations and filtering， provides the initial estimation of the 3D distributionof the out-of-plane component of the distortion. 3.3. Iterative improvements to the error Sample spatial distributions of the components of the error in the distortion function， E0)-ei， forcase 3 (see Table 2)， at zi=350 um， are shown in Figs. 7(a)-7(c). For this case， which representslarger number of realizations， and moderate elongation of 2 particle diameters，E)-ej isnegligible，E 0)- e2 is about 10% of the diameter， and E0) -e3 is about a third of the diameterover a broad area. Summary of results for all the cases are summarized in Table 3. For each caseand component， it provides the volume-averaged error and its spatial standard deviation. In case1， the positional errors are zero， representing an ideal condition where the particle coordinatesare determined accurately in each view. In cases 2 and 3， the standard deviation of the depthpositional errors is twice the particle diameter， which represents the accurate end of previouslyclaimed particle detection uncertainty in digital holography [14，15，19]. In cases 4-6， the depthuncertainty is 5 times the particle diameter， representing more realistic uncertainties in single-view holographic microscopy. The effects of the number of particles involved in the analysisare also evaluated by varying N， and SD for the two depth uncertainties. Without positionalerrors (case 1)， the initially determined E)-eand E0)-e2are essentially zero， and values ofE0”-e3 are still well in the sub-micron range. The latter is caused mostly by spatial averaging inherent to PIV cross-correlation analysis， especially in high gradient regions. Propagation of the errors associated with in-plane components also effects the out-of-plane distortion. As for cases Fig. 7. (a)-(C) The initial three components of the error (in um) in the calculated distortionfield at z1=350 um for case 3； and (d)-(f) the corresponding errors after the first iteration.Note the difference in scales between columns. Table 3. Mean and standard deviation of distortion error (unit： um) Case 1 2 3 4 5 6 EO)) -e1 -0.01±0.04 -0.02±0.06 -0.02±0.04 -0.02±0.07 -0.02±0.05 -0.02±0.04 E：(1) e1 0.01±0.01 -0.01±0.04 -0.01±0.02 0.00±0.06 0.00±0.02 -0.01±0.02 E0) -e2 0.01±0.10 0.04±0.23 0.02±0.17 0.08±0.86 0.09±0.44 0.02±0.29 E，2 -e2 0.00±0.02 0.02±0.22 0.01±0.12 0.07±0.98 0.07±0.47 0.00±0.30 EO -e3 0.21±0.44 0.45±0.52 0.45±0.46 0.92±1.45 0.62±0.77 0.55±0.61 E，) -e3 -0.04±0.07 0.16±0.38 0.14±0.20 0.44±1.63 0.15±0.80 0.13±0.53 2-6， introduction of positional errors not only increases the standard deviation of distortion errorbut also causes a mean bias error in the out-of-plane component， both of which increase withpositional uncertainty. Furthermore， the means and standf? afard deviations of E) -e2 and E )- e(30)increase with decreasing total number of particles involved in the sum-of-correlation analysis， asshown by comparing the results for cases 4 to6. The Differences between errors in the x and yidirections are associated with the present particular geometric configuration (p，0~0) and thehigher magnitude of e2 compared to e1. An iterative procedure can be used for reducing the initial errors， by repeating the sum-of-correlation analysis using the geometric mapping and the calculated distortion field of theprevious iteration for mapping the particles between the two views. For example， in the kiniteration， Eq. (7) is replaced with where E(k-1) is the distortion function obtained in the previous iteration. This analysis calculatesthe residual distortion field， which then has to be added to the result of the previous iteration Research Article Vol. 26， No. 13| 25 Jun 2018|OPTICS EXPRESS 16719 to obtain the distribution of E(). The present version of the code is written in MATLAB，which utilizes the previously-mentioned DaVis software (LaVision GmbH) for calculatingthe correlations. As an example for the computation time involved， for case 1 (N，= 500)，determining the in-plane distortion components for one slab in view 1 (1037×1037 pixels) takes12 minutes for the sum of correlations and additional 1 minute for the subsequent analysis usingan Intel i9-7920X CPU. Determining the out-of-plane component of the distortion in view 2(1054×1555 pixels) takes about 18 minutes for sum of correlations， and additional 2 minutes tocomplete the analysis. Accordingly， each iteration of case 1 takes about 7 hours for the 14 slabs. Sample results obtained after the 1iteration (following the initial analysis) are presentedin Figs. 7(d)-7(f)， using the same scale as those of the initial values a row above. As is evident，there is a broad reduction (1)in error for all three components， including E)-e3， to the 0.1-0.2um range. Statistics are provided in Table 3， in rows corresponding to E(l)-e1，E，)-e2， andE)-e3. Without positional errors (case 1)， the mean and standard deviation of the error fallbelow 0.1 um. For cases 2-3 with mild depth uncertainty， the mean error decreases by threetimes and the standard deviation by about two times compared to those of the previous iteration.As before， increasing the number of particles reduces the standard deviation of the error. Forcases 4-6， the first iteration results for E ) - e2 are not significantly different from the initialresults， but the mean values of E l)-e3 are 2-4 times smaller， with minimal influence on thestandard deviation. Increasing the number of particles and realizations decreases the error. Forthe best case (case 6)， the mean error is less than 1% and its standard deviation is less than 4% ofthe imposed standard deviation of the positional error. Additional iterations， as we have triedfor case 4， reduce the mean error further， but increase the standard deviations slightly. Furtherreduction in the standard deviation can be achieved by increasing the number of particles/imagesinvolved in the self-calibration. For example， for case 4 (worst case)， performing a 2nd iterationusing a different set of 500 synthetic particle fields reduces E)-e3 to 0.22±1.44 um. In summary， the performance of the self-calibration procedure depends on several parameters，including the accuracy of the depth detection， and the total number of particle involved， with thelatter compensating in part inaccuracies introduced by the former. Yet， even in the worst case，the error in the calculated distortion field is only a fraction of the particle size， and an order ofmagnitude smaller than the error in the depth direction. Such precision ensures that when theparticles of view 2 are mapped into view 1， they are highly likely to intersect with the view-1traces. Quantitative results follow. 3.4. Truncating the particle traces The intersection of two inclined views can be used for truncating the elongated particle traces. Asillustrated in Fig. 8(a)， when the elongated 3D particle trace reconstructed in view 2 is mapped toview 1， if the mapping is sufficiently accurate， the oblique traces intersect， and the overlappingregion can be adopted as the truncated particle trace. This operation could be performed， e.g.， bymultiplying the two intensity fields [24]. For cases involving micro-particles， precise mapping toachieve sufficient overlap between traces is essential for successful application of this procedure. This truncation procedure has been implemented using the previously described syntheticparticle fields to determine the influence of uncertainties in the calculated distortion functions onthe accuracy of particle detection. For this analysis， each trace is assumed to have a cylindricalshape with a diameter equal to that of the particle (3 um) with Gaussian intensity distribution，all with the same peak， and length of ten times the diameter. In each view， the centroid of thecylinder is located at the true particle position in its field. After mapping view-2 into 1， andmultiplying the intensity fields， a level of 64% of the original peak intensity squared is usedas a threshold for truncating the traces. The fraction of intersecting particles is over 97% forall 6 cases， while without distortion correction， only less than 5% of the particle traces overlap.Presumably， the intensity of a reconstructed particle is maximized at its center. Therefore， the Fig. 8. (a) An illustration of the truncation of an elongated particle trace based on theirintersection in the two distortion-corrected views； (b， c) distribution of depth error ofthe particle location detected using synthetic truncated traces with (b) intensity-weightedcentroids and (c) geometric centroids. intensity-weighted centroid of the truncated volume， assumed to be the detected particle location，is compared to the prescribed center to estimate the uncertainty in the present procedures. ThePDFs of depth error are plotted in Fig.8(b)， with the corresponding mean and standard deviationsprovided in the inserted table. As a comparison， one could also estimate the particle locationusing the geometric centroid of the truncated volume， resulting in the PDF presented in Fig. 8(c).As is evident， for case 1， while the peak PDF in Fig. 8(b) is higher， the tables indicate that theapproach used for determining the centroids has marginal influence on the uncertainty in particlelocation. The sub-micron shifts in the PDF peaks and the standard deviations have magnitudethat are similar to that of the distortion errors (Table 3). Most notable， however， the results forcase 6， which imposes large particle location errors， but involves the largest number of particles，indicate that the uncertainty in particle location can be maintained well below 1 um， a quarter ofthe particle diameter， and less than 5% of the imposed error. Before concluding this section， it should be noted that the presently selected angle betweenviews (~ 36°) is based on geometric constraints of the experimental setup， including the thicknessof the channel window and the working distance of the microscope objective. While we do notquantify the effect of this angle on the uncertainty in mapping and velocity measurements， afew comments should be made. In terms of mapping one view into the other， for each slab，the uncertainty in the in-plane location of the particle increases and that of the depth locationdecreases with increasing angle between views. As the presently observed trends indicate， thein-plane uncertainties can be remedied by increasing the number of realizations used for thesum of correlations. In subsequent steps， the length of the truncated traces should inherentlydecrease with increasing angle between views， presumably improving the accuracy of the 3Dvelocity measurements. Consistent with the latter observation， several previous studies involvingmulti-view holography have used an angle of 90°[21，22，24，25]. In this case， the impact of thedepth positional error (823) in the yi direction is maximized， resulting in a longer correlationprofile in Fig. 5. The same applies to the projection of view 1 into 2 to determine the out-of-planedistortion. However， the magnitude of a23≈ -1 is maximized， reducing the impact of Dy onE)[Eq.(13)]. These effects will be quantified in future studies. 4.Experimental implementation of self-calibrated tomographic holography The self-calibration is implemented on experimental holograms of tracer particles acquiredusing the M-DTH setup described in Section 2. Prior to reconstruction， the non-uniform Vol.26， No. 13| 25 Jun 2018|OPTICS EXPRESS 16721 background is homogenized by normalizing each pixel by the time-averaged intensity. Subse-quently， a spatial high-pass filter is used for removing low-frequency noise. The geometric map-ping parameters are： (M1，M2，M3)=(1.005，1.015，1.045)， (C1，C2，C3)=(43.8，934.3，-88.4)um， and(y，p，0)=(37.40，-0.4°，-0.55). Eleven slabs separated by 150 um are used in theself-calibration procedure，each with a thickness of 400 um. The analysis used in the sum-of-correlation procedure is based on recording 478 hologram pairs. In the sample volume， eachhologram contains at least 30000 particles， corresponding to SD ~ 1%. The interrogation win-dow size for cross-correlation is 256×256 pixels (174×174 um-) with 50% overlap. The largerinterrogation window size along with the thicker slabs are selected to accommodate the lowerparticle density and non-uniform distribution. The calculated distortion field at z=500 um afterthe first iteration is shown in Fig. 9. Similar to the virtual experiments， due to the particularorientation of the coordinate system， the smallest distortion is in the x direction， and the othertwo components are similar in magnitude. Fig. 9. Experimentally determined values of (a) E，(b)E，and(c) Ein a plane located500-um away from the wall. To illustrate the improvements in particle centroid detection， we provide both sample datafor illustration， as well as quantitative data evaluating the fraction of particle with overlappingvolumes after mapping view 2 into view 1. In single-view based detection described in [14]， alocal signal-to-noise ratio (SNR)， defined as I(x，y，z)-I /o，， is used as a criterion for distin-guishing between a particle trace and the background noise. Here， I(x，y，z) is the reconstructedpixel intensity， and I， and or， are， respectively， the average and standard deviation of the intensityin a local volume surrounding the particle of interest. Based on manual evaluation of many tracesin the present data， a reasonable level for SNR is 3.4. Pixels of SNR higher than this level areconsidered as being part of a particle trace. In the tomographic analysis， we initially use a lowerthreshold of1.5. Accordingly， before multiplication， the reconstructed intensity fields of pixelswith SNR lower than 1.5 is set to zero. Since the SNR effectively normalizes and homogenizesthe intensity field， we use it instead of the particle intensity when the traces are multiplied. Aftermultiplication， an SNR’ level of 8 is used as the threshold for defining the truncated particletraces. To evaluate the results， we determine the number of particles detected using these criteriain 29 realizations (~10° particles) and compare it to that obtained from an intensity-based single-view procedure for SNR of 3.4 [14，31]. With distortion correction， the number of truncatedtraces is 79.7% of that obtained from single-view traces. Without distortion correction， only59.6% of the traces remain after multiplication. Furthermore， as demonstrated by the probabilitydensity distributions shown in Fig. 10(g)， the truncated traces with distortion correction havehigher levels of SNR， indicating that the traces intersect closer to the particle centers comparedto those obtained without correction. Clearly， the distortion correction is critical for applicationof tomographic holography. Figure 10 provides an illustration of the improvement in particledetection. Figures 10(a) and 10(b) show two views of the reconstructed particle traces in a smallfraction (260×274×960 m’) of view 1. In Fig. 10(a) 162 reconstructed planes are compressed Fig. 10. (a，c， e) Particle images detected over a depth of ▲z1 =960 um compressed intoan (x1，)i)plane， and (b， d， f) the corresponding compressed (x1，Z1) planes. (a， b) Originalview-1 data； (c， d) results after multiplying the projected views with distortion correction，and (e， f) multiplied results without distortion correction. (g) distributions of SNRwithintruncated traces in the experimental data.(h) Iso-surface plot of the 3D intensity fields ofthe particle marked in (a). Yellow circles point at the same particles in the two views. Thepixel aspect ratio in (b)， (d)， and (f) is not 1. in the z direction， and in Fig. 10(b)， 403 planes are compressed in the yi direction. For theselected three particles， the corresponding views are highlighted. As expected， all the tracesare elongated in the z direction. The truncated intensity distributions with distortion correctionare presented in Figs. 10(c) and 10(d)， and results obtained without correction are shown inFigs. 10(e) and 10(f). As is evident， without the distortion correction， after multiplication alarge fraction of the particle images are lost， including the highlighted ones. Conversely， withdistortion correction， the traces intersect， causing a substantial reduction in the length of thetraces. A sample 3D depiction of the impact of distortion correction for the particle marked byan arrow [Fig. 10(a)] is presented in Fig. 10(h). Evidently， without correction， the traces fail tointersect， while with correction， they intersect near the centers of the elongated traces. After performing the self-calibration and truncating the particle traces following the proceduredescribed above， the SNR’ distributions are used for measuring the 3D velocity field. Thisapproach has two advantages： First， with the much shorter traces in the z direction， our abilityto determine the 3D particle coordinates is greatly improved compared to the single-view data.Second， in the single-view particle tracking procedures described in [1] and [31]， matchingof traces of the same particle in successive exposures is guided， among other parameters，by calculating the planar (x，y) distributions of velocity using 2D PIV cross-correlations ofcompressed images in a series of slabs. This analysis does not account for the depth-displacement.Conversely， using the present truncated traces， the 3D tracking is guided by 3D cross-correlations.Given that the flow around the cubes is highly 3D， the volumetric cross-correlations shouldpresumably provide better guidance for matching corresponding successive traces. The rest of (a) Fig. 11. Sample data showing legs of horseshoe vortices between the cubes visualizedusing： (a) iso-surfaces of streamwise vorticity， and (b) vectors showing the spanwise andwall-normal velocity components superimposed on the streamwise vorticity at (y，z) planeslocated x/a =0.48，1.62，and 2.58. the data analysis procedures follow [1，31]， including： (i) the use multi-dimensional criteria formatching traces， (ii) calculating the displacement of individual particles based on the locationof their SNR2-weighted centroids， and (iii) projecting the unstructured data onto regular gridsusing first-order SVD. This analysis provides the local velocity gradients as well. Sample ensemble-averaged flow structures based on analyzing 193 instantaneous realizationsare presented in Fig. 11. Figure 11(a) visualizes the pair of counter-rotating quasi-streamwisevortices developing between the cubes by iso-surfaces of streamwise vorticity，wx=0uz/0y-Ouy/Oz， where (ux，uy，uz) are the velocity component in the (x，y，z) directions. Each of thesestructures is part of a horseshoe vortex that wraps around the base of the cube [36]. Vectorscomposed of uy anduz along with contour plots of wx in three selected (y， z) planes are presentedin Fig. 11(b). The vector spacing is 60 um. Interestingly， interactions of the vortex legs with thewall generates counter-rotating structures in the space between the cubes. A complete descriptionof this flow and effects of the cube spacing on the structure will be the subject of future papers.However， before concluding， the accuracy of the 3D velocity field is quantified by examining howwell the instantaneous velocity distributions conserve mass， namely satisfy the 3D continuityequation，oux/0x +0uy/oy +Ouz/Oz = 0. Following [1， 37]， we calculate the probabilitydistribution of the normalized divergence， When the continuity equation is satisfied， n=0. For random data， the mean value of n is 1.Figure 12 shows the cumulative distribution function of the normalized divergence， comparing theresults obtained using a single view， to those obtained using M-DTH. Two results are presented.The first is based on using 3D cross-correlations with interrogation volume of 174×174×166um for calculating the velocity. The second is based on particle tracking (and SVD) results，which uses the correlations for guidance. The solid lines show results based on local gradients，and the dashed and dash-dotted lines are based on repeating the analysis using spatially averagedgradients over volumes of 5 ×5×5(300×300×300 um)and 8×8×8(480×480 ×480um’) vector spacings. As is evident， the M-DTH results are significantly more divergent-free Vol. 26， No. 13|25 Jun 2018|OPTICS EXPRESS 16724 than the single-view data. Furthermore， enlarging the volume reduces the divergence errorsignificantly， consistent with [1，37].The divergence may be further reduced by post-processingthe velocity distributions using divergence correction schemes introduced in recent tomographicPIV applications [38，39]. Fig. 12. Cumulative distribution of the experimental normalized divergence，n. 5. Summary and Conclusions A microscopic dual-view tomographic holography (M-DTH) system is introduced and imple-mented for high-resolution 3D flow measurements. The critical step of M-DTH application mustinvolve precise matching of the two reconstructed volumes. Otherwise， substantial fraction ofthe elongated traces of microscopic particles in the two views do not overlap， and even whenthey do， their intersection is not necessarily located at the correct center of the particle. Geo-metric mapping of the two fields is insufficient due to the effects of even slight image distortion.Hence， this paper introduces an iterative volumetric self-calibration procedure that determinesthe 3D distortion function by mapping one view onto the next， dividing the sample volume intoslabs， and cross-correlating corresponding intensity fields in the two views. Sum-of-correlationsachieved by combining data obtained from numerous instantaneous realizations provide robuststatistics on the 3D distortion field. The accuracy of the 3D self-calibration method is tested using distorted synthetic (3-umdiameter) particles fields with varying density， number of holograms， and imposed particlepositional errors. The latter have magnitudes consistent with realistic particle detection errorin holography， including the reduced accuracy in the depth direction. Without any positionalerror， the self-calibration determines the distortion field with negligible errors of less than 0.1um. The error increases with increasing particle position error， but decreases， with increasingnumber of particles used for calculating the sum-of-correlation. The worst presently testedscenario involves the least number of particles and position error as high as 5 times the particlediameter. For this case， the mean and standard deviation of the distortion error are about 15%and 54% of the particle diameter， respectively， the latter being about 10% of the imposed particlepositional errors. For the same position error， increasing the particle density and/or number ofholograms， reduces the mean and standard deviation of error to 4% and 18% of the particlediameter. Subsequently， the intensity multiplication method is adopted to truncate the particletraces for particle centroid detection. After correction for distortion and mapping one view into Research Article Vol.26， No. 13|25 Jun 2018|OPTICS EXPRESS 16725 the next， the elongated particle traces are truncated by multiplying the two intensity fields. Whenthe intensity-weighted centroids of the truncated traces are compared to the original particlelocations， the mean and standard deviations of the errors are 11% and 55% of the diameter，respectively for the worst case， and 3% and 24% for the denser fields. In contrast， withoutdistortion correction， only 5% of the traces overlap. The self-calibration and intensity multiplication procedures are implemented to analyzeexperimental data obtained in a turbulent boundary layer. The flow is seeded by 2-um particles，the two views are inclined by approximately 37°， and the sample volume is 4.2×2.8×1.5 mm'.Distortion correction substantially increases the number of intersecting inclined traces close tothe particle center， as demonstrated by an increase in both the number of detected particles andthe normalized intensity in the truncated traces. For velocity measurements， the displacement ofindividual particles is determined based on the 3D location of the intensity-weighted centroid ofthe truncated traces. Matching of traces is guided by a series of criteria， including volumetriccross-correlations of the truncated intensity fields. The unstructured data is then interpolated ontoa regular grid with 60-um spacing using SVD， which also determines the spatial distributions ofvelocity gradients. The improvement in the data quality is demonstrated by a reduction in thedivergence of the velocity field， i.e.， compliance with conservation of mass， in comparison toresults obtained by analyzing data obtained from a single hologram. Increasing the volume overwhich conservation of mass is evaluated reduces the error further. As noted in the introduction， truncation of particle traces based on dual-view holographyhas already been implemented in previous applications [24-26]. However， in all of these cases，matching of the two views is based on geometric mapping， without correction for the spatiallyvarying distortion. Furthermore， their particles are larger and their concentrations are lower thanthe present ones， e.g. one 5-um cell in [25]， a few 110-um particles in [24]， as well as a layer of11-um particles sandwiched between microscope slides and five 600-um glass spheres in [26].While the likelihood of overlap between the two views is expected to improve with increasingparticle size and decreasing concentration， in the present dense suspension of 2-um particles，very few(5%) overlap without the 3D distortion correction， and many of them， in the wrong place.For experimental data， applying this self-calibration method leads to significant improvements inthe accuracy of the 3D velocity field. While the present implementation of the self-calibrationprocedure and tomographic truncation of traces focuses on 3D velocity measurements at highmagnification， the same approach could be readily utilized in other applications of digitalholography at other magnifications. Funding National Science Foundation (NSF) (CBET-1438203)； Office of Naval Research (ONR) (N00014-15-1-2404， N00014-17-1-2955)； Gulf of Mexico Research Initiative (GoMRI) (DROPPS II).      This paper introduces the application of microscopic dual-view tomographic holography(M-DTH) to measure the 3D position and motion of micro-particles located in dense suspensions. Pairing of elongated traces of the same particle in the two inclined reconstructed fields requires precise matching of the entire sample volume that accounts for the inherent distortions in each view. It is achieved by an iterative volumetric self-calibration method, consisting of mapping one view onto the next, dividing the sample volume into slabs, and cross-correlating the two views. Testing of the procedures using synthetic particle fields with imposed distortion and realistic errors in particle locations shows that the self-calibration method achieves a 3D uncertainty of about 1um, a third of the particle diameter. Multiplying the corrected intensity fields is used for truncating the elongated traces, whose centers are located within 1um of the exact value. Without correction, only a small fraction of the traces even overlap. The distortion correction also increases the number of intersecting traces in experimental data along with their intensity. Application of this method for 3D velocity measurements is based on the centroids of the truncated/shortened particle traces. Matching of these traces in successive fields is guided by several criteria, including results of volumetric cross-correlation of the multiplied intensity fields. The resulting 3D velocity distribution is substantially more divergence-free, i.e., satisfies conservation of mass, compared to analysis performed using single-view data. Sample application of the new method shows the 3D flow structure around a pair of cubic roughness elements embedded in the inner part of a high Reynolds number turbulent boundary layer.