## Abstract

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 1*µm*, a third of the particle diameter. Multiplying the corrected intensity fields is used for truncating the elongated traces, whose centers are located within 1*µm* 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.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction: applying tomography to alleviate the depth-of-focus problem in digital holography

Digital in-line holography is an effective tool for quantifying the 3D location, size, and shape of particles distributed in a sample volume. Sequential exposures in time can be used for determining the velocity of these particles as well. Relevant applications include e.g. tracking of micro-particles for 3D flow measurements [1–3], as well as characterization of droplets and bubbles in multiphase flows [4–7], organisms and cells in biological studies [8–10], and fuel particles in combustion research [11, 12]. However, the elongated reconstructed image in the axial 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 resulting uncertainty of several particle diameters [14–16] in the axial (depth) location of the particle center is significantly larger than those in directions aligned perpendicularly to the beam axial direction. For applications involving a single beam, numerous studies have already introduced methods for improving the depth uncertainty, e.g., [17–20], too many to summarize in a single paper. In some cases, methods like the inverse problem approach [17] have been successful in alleviating most of the adverse effects for sample volumes containing sparse particle distributions, where the interference pattern of spherical particles can be precisely resolved without significant distortions caused by neighboring particles. For dense micro-particle suspensions needed for 3D velocity measurements, approaches involving e.g. spatial correlations of the 3D traces [19] have also 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 the most straightforward way to overcome the DOF problem since information in the depth direction of one view can be extracted from the in-plane data of the others. Knowledge of how to map each point in the inclined reconstructed 3D fields, the so-called mapping function, is essential for matching images of the same particle in the multiple views and for determining its location and displacement between exposures based on in-plane data [21–23]. Alternatively, the elongated traces can be truncated by multiplying the 3D intensity fields, which retains only the volumes where they overlap [24,25]. Subsequently, the particle displacement between exposures can be determined using particle-tracking or 3D cross-correlation of the truncated field. This approach is referred to as dual-view tomographic holography (DTH) in the present paper. Determining and calibrating the mapping function which relates the two fields is the critical step in implementing this procedure since errors in the order of a few microns in applications involving microscopic holography are sufficient for traces of small particles not to overlap. Prior applications have been based on geometric mapping, which accounts for relative angle, magnification, and translation between the two coordinate systems, without corrections for distortions [21–26]. The calibrations have 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 intensity fields associated with individual particles. For the microscopic DTH implemented in the present study for 3D velocity measurements, the mapping precision has to be in the order of microns considering that the nominal diameter of the tracer particles is 2 *µm*. Geometric mapping is insufficient 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.). For example, a misalignment angle of 1 degree could lead to tens of microns of mapping error as the reconstruction depth increases [25]. Distortions are probably the reason for the limited matching of 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 holography to quantify a distortion function, namely the 3D deviation of the actual mapping function from 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, the self-calibration procedure provides precise matching of the lines of sight of cameras observing the sample volume from different angles [28]. The present method is also based on matching the particle fields based on their spatial distributions in the reconstructed sample volume. This method takes advantage of the known, although less accurate, depth location of the particle in the sample volume. The optical setup is discussed in Section 2. The new self-calibration procedure is introduced in Section 3, where synthetic data is used for demonstrating that an uncertainty of about 1 *µm* is achievable. Implementation using experimental data and the following particle tracking and 3D velocity measurements are discussed in Section 4.

## 2. Optical setup for microscopic dual-view tomographic holography (M-DTH)

The motivation for the present development is an experimental study aimed at measuring the flow around a pair of cubic roughness elements immersed in the inner part of a turbulent channel flow. The schematic of the experimental facility is shown in Fig. 1. These cubes have a height of *a* = 1 *mm*, and are aligned in the spanwise direction (*y*) and separated by 1.6*a*. To gain unobstructed optical access to the sample volume, the walls of the channel as well as the cubes are made of acrylic whose refractive index (~1.49) is matched with that of the working fluid, a concentrated aqueous solution of sodium iodide [29,30]. A dense particle suspension is required to resolve the flow structure at scales much small than the cube, e.g. 50 *µm*. Hence, the flow field is seeded “locally” by injecting 2-*µm* tracer particles from 52 ports located sufficiently far (330 port 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 are similar to those used in previous studies [1, 2, 31]. Furthermore, the present fully-developed smooth-wall turbulent channel flow is similar and of the same scale as in [1]. As argued in the latter based on prior publications about jets-in-cross-flow, the effect of the injectors on the flow in the sample volume is negligible. Support for this claim is provided in [31], albeit for a rough-wall channel flow, by showing an agreement between the mean velocity profiles obtained using digital holographic microscopy with local seeding and 2D PIV with global seeding. The number of ports 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 30000 particles broadly distributed in the sample volume.

The microscopic dual-view tomographic holography (M-DTH) setup is illustrated in Fig. 2. 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 the channel wall, and the second is nominally inclined by 36*°*, with the overlapping volume centered on the space between the cubes (Fig. 2, top right). The inline holograms are magnified by identical 8X microscope objectives (MO1 and MO2), and then recorded by 6600 × 4400-pixel interline-transfer, CCD cameras (Imperx B6640), resulting in an effective pixel size of ~0.68 *µm*. The sample volume size is 4.2 × 2.8 × 1.5 *mm*^{3} in the streamwise, spanwise, and wall-normal directions. The origin of the corresponding physical/experimental coordinate system, (*x, y, z*), is located on the wall, between front surfaces of the cubes, as indicated in the lower-right inset of Fig. 2. The coordinate systems of the two holograms are denoted by (*x*_{1}, *y*_{1}, *z*_{1}) and (*x*_{2}, *y*_{2}, *z*_{2}), as indicated in the zoomed-in part of Fig. 2. Sequential hologram pairs are acquired by both cameras at a rate of 1.5 *Hz*, with 25 *µs* delay between exposures. Thousands of hologram pairs are recorded to obtain converged statistics of the mean flow field.

## 3. Self-calibration procedures

#### 3.1. Imposing and modeling of the distortion function

Using index notation, the coordinates in view 1 are represented by *p _{i}*, corresponding to

*x*

_{1},

*y*

_{1}, and

*z*

_{1}. Similarly, the coordinates in view 2 are represented by

*q*, corresponding to

_{i}*x*

_{2},

*y*

_{2}, and

*z*

_{2}. The 3D mapping function relating the

*p*field to the

_{i}*q*field includes a geometric displacement,

_{i}*C*, geometric mapping,

_{i}*a*, and distortion,

_{ij}*e*(

_{i}*p*), resulting in the following expression,

_{i}*o*= −

_{i}*a*. For clarification,

_{ij}e_{j}*a*accounts for magnification and roation, and has the form

_{ij}*M*

_{1},

*M*

_{2}, and

*M*

_{3}are the magnifications in the

*x*

_{1},

*y*

_{1}, and

*z*

_{1}directions, respectively, and

*ψ*,

*φ*, and

*θ*are angles of rotation about the

*x*

_{1},

*y*

_{1}, and

*z*

_{1}axes, respectively. The goal of the self-calibration is to determine

*e*(

_{i}*p*). In the present analysis, view 1, which has less interfaces and thus a simpler relationship with the physical coordinates, is used as a reference, and the distortion is associated with mapping from view 2 to this reference.

_{i}The self-calibration method is illustrated and tested by generating a virtual field containing synthetic particles with (*M*_{1}, *M*_{2}, *M*_{3}) = (1.005, 1.015, 1.045), (*C*_{1}, *C*_{2}, *C*_{3}) = (7.4, 473.9, 7.6) *µm*, and (*ψ, φ, θ*) = (37.1*°*, −0.3*°*, −0.55*°*). The rotation angles are chosen to resemble those used in the experimental configuration (*φ, θ* ≈ 0, *ψ* ≈ 36*°*). Subsequently, view 2 is distorted by a known function, *f _{i}*(

*p*), where

_{i}*G*,

_{i}*B*,

_{i}*µ*, and

_{ij}*σ*are provided in Table 1. To examine the effects of varying the amplitudes of distortion, the values of

_{ij}*G*are selected such that the distortion in the

_{i}*x*

_{1}direction is the lowest, and that in the

*z*

_{1}direction is the highest. These choices are consistent with the experimental conditions. As an example, the distribution of the prescribed distortion field in the plane

*z*

_{1}= 350

*µm*is shown in Fig. 3.

The virtual cameras in views 1 and 2 have 1037 × 1037 and 1054 × 1555 pixels, respectively. With 8X magnification, the 5.5 *µm* pixel size corresponds to 0.68 *µm*. The synthetic particle field containing randomly distributed particles has dimensions of 700 × 700 × 750 *µm*^{3} in the (*x*_{1}, *y*_{1}, *z*_{1}) directions, respectively, centered in the overlapping region of the two views. The particle size is normally distributed with a mean of 3 *µm* and a standard deviation of 0.5 *µm*. 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*^{3}. The coordinates of the particles in view 1, *P _{i}* = (

*X*

_{1},

*Y*

_{1},

*Z*

_{1}), are transformed to the corresponding coordinates in view 2,

*Q*= (

_{i}*X*

_{2},

*Y*

_{2},

*Z*

_{2}), using Eq. (1). These locations represent the exact/true coordinates 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 reconstructed images, especially in the depth direction. Hence, the detected coordinates of individual particles in view 1, ${P}_{i}^{\prime}=({X}_{1}^{\prime},{Y}_{1}^{\prime},{Z}_{1}^{\prime})$, and view 2, ${Q}_{i}^{\prime}=({X}_{2}^{\prime},{Y}_{2}^{\prime},{Z}_{2}^{\prime})$, deviate from the true coordinates. We assume that the positional errors, ${\delta}_{1i}={P}_{i}^{\prime}-{P}_{i}$ and ${\delta}_{2i}={Q}_{i}^{\prime}-{Q}_{i}$, are normally distributed with zero mean in their own fields. Therefore, the corresponding probability density functions (PDF) of positional errors are

*σ*

_{1}and

*σ*

_{2}denote the standard deviations of in-plane errors, and

*σ*

_{3}denotes that of the out-of-plane error. Hence,

*σ*

_{1}and

*σ*

_{2}are in the order of one pixel, and

*σ*

_{3}is significantly larger, 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 is denoted 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-calibration procedure, as listed in Table 2.

_{r}#### 3.2. Self-calibration Procedures

Initially the distortion field is determined in a series of (*x*_{1}, *y*_{1}) planes established by dividing the particle field into *N _{c}* thin slabs, as illustrated in Fig. 4. The division to slabs is readily achievable in the reconstructed 3D domain based on the intensity distribution. Each slab has a prescribed thickness in the depth direction, denoted as

*w*, and it extends over the entire (

_{z}*x*

_{1},

*y*

_{1}) plane. The center of the

*n*slab, where the calibration is performed, is denoted by ${z}_{1}^{(n)}$. After calibrating the

^{th}*N*planes, the 3D distortion field is determined by interpolation in the depth direction. In the virtual experiment,

_{c}*N*= 14,

_{c}*w*= 100

_{z}*µm*, and there is 50% overlap between adjacent slabs, i.e., ${z}_{1}^{(n+1)}-{z}_{1}^{(n)}=50\mu m$.

The in-plane distortions of each slab (*e*_{1} and *e*_{2}) are determined first. The images of all the particles located within the slab in view 1, which are illustrated as elongated red traces in Fig. 4, are projected onto the ${z}_{1}^{(n)}$-plane. Here, the projection utilizes the true particle in-plane positions, (*X*_{1}, *Y*_{1}), because in experiments, it corresponds to the reconstructed intensity distribution within the particle in experimental data, without calculating its location. From view 2, we use the corresponding particle centroids with position errors added, and then map them into view 1 using only the geometric mapping. Due to the out-of-plane distortion (*e*_{3}) and the errors, the mapped centers may fall outside of the slab, as illustrated by the solid blue dots in Fig. 4. Hence, an extended slab, $[{z}_{1}^{(n)}-0.5{w}_{z}-{z}_{1}^{ex},{z}_{1}^{(n)}+0.5{w}_{z}+{z}_{1}^{ex}]$, is used for the mapped view-2 particle centroids. In the present analysis, ${z}_{1}^{ex}=50\mu m$, consistent with the magnitude of *e*_{3}. The intensity distribution within each projected 2D image is assumed to be Gaussian, using the intensity integration method described in [33] for generating the particle images. Extending the slab 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 on the results is very small since they are not correlated with the view-1 particles located within the original narrower slab. Denoting the coordinates of the mapped particles from view 2 as ${P}_{mi}^{\prime}=({X}_{1m}^{\prime},{Y}_{1m}^{\prime},{Z}_{1m}^{\prime})$ and using only geometric mapping,

*P*and ${P}_{mi}^{\prime}$ can be determined using spatial cross-correlation analysis, widely used in PIV [34]. Due to the imposed positional errors, non-uniform particle distributions, and finite number of particles per image, cross-correlating one image pair might be insufficient for obtaining accurate data on the in-plane distortion. This problem can be readily solved by application of the sum-of-cross-correlation method [35] to combine the results from multiple realizations. This method consists of adding the cross-correlation results of all the images to obtain an ensemble-averaged correlation, which is inherently much less sensitive to the randomly distributed positional errors and sparse/non-uniform particle distributions. In the present virtual experiment, the interrogation window size 128 by 128 pixels, and the spacing between windows is 32 pixels (75% overlap). The multi-pass calculations are performed using a commercial code (DaVis 8.2, LaVision GmbH). The displacement of the peak location from the window center indicates the local in-plane distortion. Sample sum-of-correlation maps for one interrogation window based on 500 realizations are shown in Figs. 5(a) and 5(b). Without any positional errors, Fig. 5(a) shows a sharp peak, which spreads and becomes elongated upon addition of the positional errors [Fig. 5(b)]. The elongation is primarily in the

_{i}*y*

_{1}-direction due to the imposed larger errors in the

*z*

_{2}direction, as illustrated in Fig. 5(c). Nevertheless, the elongated sum-of-correlation map still has a distinct peak that can be readily detected. The correlation map provides the calculated mean in-plane distortion components, ${E}_{1}^{(0)}({p}_{i})$ and ${E}_{2}^{(0)}({p}_{i})$, spatially averaged over the window and slab depth. The superscripts indicate the iteration number. As demonstrated later, the positional errors introduce uncertainty, which diminishes with increasing number of samples and particle concentrations. This procedure is repeated for all the slabs and then smoothed by low pass filtering in all directions. The low pass filtering is based on first-order weighted interpolation using single value decomposition (SVD) with distortion values in a 3 × 3 × 3 volume surrounding each point. The values of ${E}_{1}^{(0)}({p}_{i})$ and ${E}_{2}^{(0)}({p}_{i})$ are our first estimation for the true

*e*

_{1}(

*p*) and

_{i}*e*

_{2}(

*p*).

_{i}The out-of-plane distortion (*e*_{3}) is determined by applying the sum-of-correlation method in the view 2 domain. Figure 6 shows a slab aligned in the *z*_{1} direction cut through the view-2 volume. It contains the mapped particles from view 1 (red circles), and the elongated traces of the view-2 particles. As demonstrated in the insert, the latter are projected along the *z*_{2} direction onto a (*x*_{2}, *y*_{2}) plane as long as they are located within the extended slab. Their projected images represent their true in-plane positions with respect to view 2, *Q _{i}*. In the actual experiment, this operation is equivalent to calculating the peak intensity distribution in the

*z*

_{2}direction through the region defined by the extended slab. The coordinates of the view-1 particles within the ${z}_{1}^{(n)}$ slab, including their true location and positional errors, are mapped into view 2 according to both the geometric mapping and the already-determined in-plane distortions. Their mapped locations are then

*e*and ${E}_{i}^{(0)}$, and the unknown

_{i}*e*

_{3}. After fitting a Gaussian intensity distribution in the

*x*

_{2}−

*y*

_{2}plane to each mapped particle, as 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 of

*δ*

_{1}

*, the measured average displacement in the*

_{i}*y*

_{2}direction is

*a*

_{23}

*e*

_{3}. Hence,

*e*

_{3}can be calculated directly from ${D}_{{y}_{2}}$. The analysis gives an estimated depth distortion Alternately, the same term can be calculated by ${D}_{{x}_{2}}\mathrm{/}{a}_{13}$, where ${D}_{{x}_{2}}$ is the

*x*

_{2}-component of the measured displacement. However, since

*x*

_{1}- and

*x*

_{2}-axes are nearly parallel in the present setup,

*a*

_{13}≈ 0, hence the

*y*

_{2}-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 distribution of 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, ${E}_{i}^{(0)}-{e}_{i}$, for case 3 (see Table 2), at *z*_{1} = 350 *µm*, are shown in Figs. 7(a)–7(c). For this case, which represents larger number of realizations, and moderate elongation of 2 particle diameters, ${E}_{1}^{(0)}-{e}_{1}$ is negligible, ${E}_{2}^{(0)}-{e}_{2}$ is about 10% of the diameter, and ${E}_{3}^{(0)}-{e}_{3}$ is about a third of the diameter over a broad area. Summary of results for all the cases are summarized in Table 3. For each case and component, it provides the volume-averaged error and its spatial standard deviation. In case 1, the positional errors are zero, representing an ideal condition where the particle coordinates are determined accurately in each view. In cases 2 and 3, the standard deviation of the depth positional errors is twice the particle diameter, which represents the accurate end of previously claimed particle detection uncertainty in digital holography [14,15,19]. In cases 4–6, the depth uncertainty 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 analysis are also evaluated by varying *N _{r}* and

*SD*for the two depth uncertainties. Without positional errors (case 1), the initially determined ${E}_{1}^{(0)}-{e}_{1}$ and ${E}_{2}^{(0)}-{e}_{2}$ are essentially zero, and values of ${E}_{3}^{(0)}-{e}_{3}$ 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 2–6, introduction of positional errors not only increases the standard deviation of distortion error but also causes a mean bias error in the out-of-plane component, both of which increase with positional uncertainty. Furthermore, the means and standard deviations of ${E}_{2}^{(0)}-{e}_{2}$ and ${E}_{3}^{(0)}-{e}_{3}$ increase with decreasing total number of particles involved in the sum-of-correlation analysis, as shown by comparing the results for cases 4 to 6. The Differences between errors in the

*x*

_{1}and

*y*

_{1}directions are associated with the present particular geometric configuration (

*φ, θ*≈ 0) and the higher magnitude of

*e*

_{2}compared to

*e*

_{1}.

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 the previous iteration for mapping the particles between the two views. For example, in the *k ^{th}* iteration, Eq. (7) is replaced with

*N*= 500), determining the in-plane distortion components for one slab in view 1 (1037 × 1037 pixels) takes 12 minutes for the sum of correlations and additional 1 minute for the subsequent analysis using an 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 to complete the analysis. Accordingly, each iteration of case 1 takes about 7 hours for the 14 slabs.

_{r}Sample results obtained after the 1* ^{st}* iteration (following the initial analysis) are presented in 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 in error for all three components, including ${E}_{3}^{(1)}-{e}_{3}$, to the 0.1–0.2

*µm*range. Statistics are provided in Table 3, in rows corresponding to ${E}_{1}^{(1)}-{e}_{1}$, ${E}_{2}^{(1)}-{e}_{2}$ and ${E}_{3}^{(1)}-{e}_{3}$. Without positional errors (case 1), the mean and standard deviation of the error fall below 0.1

*µm*. For cases 2–3 with mild depth uncertainty, the mean error decreases by three times 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. For cases 4–6, the first iteration results for ${E}_{2}^{(1)}-{e}_{2}$ are not significantly different from the initial results, but the mean values of ${E}_{3}^{(1)}-{e}_{3}$ are 2–4 times smaller, with minimal influence on the standard deviation. Increasing the number of particles and realizations decreases the error. For the best case (case 6), the mean error is less than 1% and its standard deviation is less than 4% of the imposed standard deviation of the positional error. Additional iterations, as we have tried for case 4, reduce the mean error further, but increase the standard deviations slightly. Further reduction in the standard deviation can be achieved by increasing the number of particles/images involved in the self-calibration. For example, for case 4 (worst case), performing a 2

*iteration using a different set of 500 synthetic particle fields reduces ${E}_{3}^{(1)}-{e}_{3}$ to 0.22 ± 1.44*

^{nd}*µm*.

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 the latter 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 of magnitude smaller than the error in the depth direction. Such precision ensures that when the particles of view 2 are mapped into view 1, they are highly likely to intersect with the view-1 traces. Quantitative results follow.

#### 3.4. Truncating the particle traces

The intersection of two inclined views can be used for truncating the elongated particle traces. As illustrated in Fig. 8(a), when the elongated 3D particle trace reconstructed in view 2 is mapped to view 1, if the mapping is sufficiently accurate, the oblique traces intersect, and the overlapping region can be adopted as the truncated particle trace. This operation could be performed, e.g., by multiplying the two intensity fields [24]. For cases involving micro-particles, precise mapping to achieve sufficient overlap between traces is essential for successful application of this procedure.

This truncation procedure has been implemented using the previously described synthetic particle fields to determine the influence of uncertainties in the calculated distortion functions on the accuracy of particle detection. For this analysis, each trace is assumed to have a cylindrical shape with a diameter equal to that of the particle (3 *µm*) with Gaussian intensity distribution, all with the same peak, and length of ten times the diameter. In each view, the centroid of the cylinder is located at the true particle position in its field. After mapping view-2 into 1, and multiplying the intensity fields, a level of 64% of the original peak intensity squared is used as a threshold for truncating the traces. The fraction of intersecting particles is over 97% for all 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 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. The PDFs of depth error are plotted in Fig. 8(b), with the corresponding mean and standard deviations provided in the inserted table. As a comparison, one could also estimate the particle location using 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 the approach used for determining the centroids has marginal influence on the uncertainty in particle location. The sub-micron shifts in the PDF peaks and the standard deviations have magnitude that are similar to that of the distortion errors (Table 3). Most notable, however, the results for case 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 *µm*, a quarter of the particle diameter, and less than 5% of the imposed error.

Before concluding this section, it should be noted that the presently selected angle between views (∼ 36*°*) is based on geometric constraints of the experimental setup, including the thickness of the channel window and the working distance of the microscope objective. While we do not quantify the effect of this angle on the uncertainty in mapping and velocity measurements, a few 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 location decreases with increasing angle between views. As the presently observed trends indicate, the in-plane uncertainties can be remedied by increasing the number of realizations used for the sum of correlations. In subsequent steps, the length of the truncated traces should inherently decrease with increasing angle between views, presumably improving the accuracy of the 3D velocity measurements. Consistent with the latter observation, several previous studies involving multi-view holography have used an angle of 90*°* [21,22,24,25]. In this case, the impact of the depth positional error (*δ*_{23}) in the *y*_{1} direction is maximized, resulting in a longer correlation profile in Fig. 5. The same applies to the projection of view 1 into 2 to determine the out-of-plane distortion. However, the magnitude of *a*_{23} ≈ −1 is maximized, reducing the impact of ${D}_{{y}_{2}}$ on ${E}_{3}^{(0)}$ [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 acquired using the M-DTH setup described in Section 2. Prior to reconstruction, the non-uniform background is homogenized by normalizing each pixel by the time-averaged intensity. Subsequently, a spatial high-pass filter is used for removing low-frequency noise. The geometric mapping parameters are: (*M*_{1}, *M*_{2}, *M*_{3})=(1.005, 1.015, 1.045), (*C*_{1}, *C*_{2}, *C*_{3})=(43.8, 934.3, −88.4) *µm*, and (*ψ, φ, θ*)=(37.4*°*, −0.4*°*, −0.55*°*). Eleven slabs separated by 150 *µm* are used in the self-calibration procedure, each with a thickness of 400 *µm*. The analysis used in the sum-of-correlation procedure is based on recording 478 hologram pairs. In the sample volume, each hologram contains at least 30000 particles, corresponding to *SD* ≈ 1%. The interrogation window size for cross-correlation is 256×256 pixels (174×174 *µm*^{2}) with 50% overlap. The larger interrogation window size along with the thicker slabs are selected to accommodate the lower particle density and non-uniform distribution. The calculated distortion field at *z* = 500 *µm* after the first iteration is shown in Fig. 9. Similar to the virtual experiments, due to the particular orientation of the coordinate system, the smallest distortion is in the *x* direction, and the other two components are similar in magnitude.

To illustrate the improvements in particle centroid detection, we provide both sample data for illustration, as well as quantitative data evaluating the fraction of particle with overlapping volumes after mapping view 2 into view 1. In single-view based detection described in [14], a local signal-to-noise ratio (*SNR*), defined as $[I(x,y,z)-{\overline{I}}_{v}]\mathrm{/}{\sigma}_{v}$, is used as a criterion for distinguishing between a particle trace and the background noise. Here, *I*(*x, y, z*) is the reconstructed pixel intensity, and ${\overline{I}}_{v}$ and *σ _{v}* are, respectively, the average and standard deviation of the intensity in a local volume surrounding the particle of interest. Based on manual evaluation of many traces in the present data, a reasonable level for

*SNR*is 3.4. Pixels of

*SNR*higher than this level are considered as being part of a particle trace. In the tomographic analysis, we initially use a lower threshold of 1.5. Accordingly, before multiplication, the reconstructed intensity fields of pixels with

*SNR*lower than 1.5 is set to zero. Since the

*SNR*effectively normalizes and homogenizes the intensity field, we use it instead of the particle intensity when the traces are multiplied. After multiplication, an

*SNR*

^{2}level of 8 is used as the threshold for defining the truncated particle traces. To evaluate the results, we determine the number of particles detected using these criteria in 29 realizations (~10

^{6}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 truncated traces is 79.7% of that obtained from single-view traces. Without distortion correction, only 59.6% of the traces remain after multiplication. Furthermore, as demonstrated by the probability density distributions shown in Fig. 10(g), the truncated traces with distortion correction have higher levels of

*SNR*

^{2}, indicating that the traces intersect closer to the particle centers compared to those obtained without correction. Clearly, the distortion correction is critical for application of tomographic holography. Figure 10 provides an illustration of the improvement in particle detection. Figures 10(a) and 10(b) show two views of the reconstructed particle traces in a small fraction (260 × 274 × 960

*µm*

^{3}) of view 1. In Fig. 10(a) 162 reconstructed planes are compressed in the

*z*

_{1}direction, and in Fig. 10(b), 403 planes are compressed in the

*y*

_{1}direction. For the selected three particles, the corresponding views are highlighted. As expected, all the traces are elongated in the

*z*

_{1}direction. The truncated intensity distributions with distortion correction are presented in Figs. 10(c) and 10(d), and results obtained without correction are shown in Figs. 10(e) and 10(f). As is evident, without the distortion correction, after multiplication a large fraction of the particle images are lost, including the highlighted ones. Conversely, with distortion correction, the traces intersect, causing a substantial reduction in the length of the traces. A sample 3D depiction of the impact of distortion correction for the particle marked by an arrow [Fig. 10(a)] is presented in Fig. 10(h). Evidently, without correction, the traces fail to intersect, while with correction, they intersect near the centers of the elongated traces.

After performing the self-calibration and truncating the particle traces following the procedure described above, the *SNR*^{2} distributions are used for measuring the 3D velocity field. This approach has two advantages: First, with the much shorter traces in the *z*_{1} direction, our ability to 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], matching of 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 of compressed 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 should presumably provide better guidance for matching corresponding successive traces. The rest of the data analysis procedures follow [1,31], including: (i) the use multi-dimensional criteria for matching traces, (ii) calculating the displacement of individual particles based on the location of their *SNR*^{2}-weighted centroids, and (iii) projecting the unstructured data onto regular grids using first-order SVD. This analysis provides the local velocity gradients as well.

Sample ensemble-averaged flow structures based on analyzing 193 instantaneous realizations are presented in Fig. 11. Figure 11(a) visualizes the pair of counter-rotating quasi-streamwise vortices developing between the cubes by iso-surfaces of streamwise vorticity, *ω _{x}* =

*∂u*−

_{z}/∂y*∂u*, where (

_{y}/∂z*u*) are the velocity component in the (

_{x}, u_{y}, u_{z}*x, y, z*) directions. Each of these structures is part of a horseshoe vortex that wraps around the base of the cube [36]. Vectors composed of

*u*and

_{y}*u*along with contour plots of

_{z}*ω*in three selected (

_{x}*y, z*) planes are presented in Fig. 11(b). The vector spacing is 60

*µm*. Interestingly, interactions of the vortex legs with the wall generates counter-rotating structures in the space between the cubes. A complete description of 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 how well the instantaneous velocity distributions conserve mass, namely satisfy the 3D continuity equation,

*∂u*+

_{x}/∂x*∂u*+

_{y}/∂y*∂u*= 0. Following [1, 37], we calculate the probability distribution of the normalized divergence,

_{z}/∂z*η*= 0. For random data, the mean value of

*η*is 1. Figure 12 shows the cumulative distribution function of the normalized divergence, comparing the results 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 × 166

*µm*

^{3}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 averaged gradients over volumes of 5 × 5 × 5 (300 × 300 × 300

*µm*

^{3}) and 8 × 8 × 8 (480 × 480 × 480

*µm*

^{3}) vector spacings. As is evident, more divergent-free the M-DTH results are significantly than the single-view data. Furthermore, enlarging the volume reduces the divergence error significantly, consistent with [1,37]. The divergence may be further reduced by post-processing the velocity distributions using divergence correction schemes introduced in recent tomographic PIV applications [38,39].

## 5. Summary and Conclusions

A microscopic dual-view tomographic holography (M-DTH) system is introduced and implemented for high-resolution 3D flow measurements. The critical step of M-DTH application must involve precise matching of the two reconstructed volumes. Otherwise, substantial fraction of the elongated traces of microscopic particles in the two views do not overlap, and even when they do, their intersection is not necessarily located at the correct center of the particle. Geometric 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 determines the 3D distortion function by mapping one view onto the next, dividing the sample volume into slabs, and cross-correlating corresponding intensity fields in the two views. Sum-of-correlations achieved by combining data obtained from numerous instantaneous realizations provide robust statistics on the 3D distortion field.

The accuracy of the 3D self-calibration method is tested using distorted synthetic (3-*µm* diameter) particles fields with varying density, number of holograms, and imposed particle positional errors. The latter have magnitudes consistent with realistic particle detection error in holography, including the reduced accuracy in the depth direction. Without any positional error, the self-calibration determines the distortion field with negligible errors of less than 0.1 *µm*. The error increases with increasing particle position error, but decreases, with increasing number of particles used for calculating the sum-of-correlation. The worst presently tested scenario involves the least number of particles and position error as high as 5 times the particle diameter. 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 particle positional errors. For the same position error, increasing the particle density and/or number of holograms, reduces the mean and standard deviation of error to 4% and 18% of the particle diameter. Subsequently, the intensity multiplication method is adopted to truncate the particle traces for particle centroid detection. After correction for distortion and mapping one view into the next, the elongated particle traces are truncated by multiplying the two intensity fields. When the intensity-weighted centroids of the truncated traces are compared to the original particle locations, 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, without distortion correction, only 5% of the traces overlap.

The self-calibration and intensity multiplication procedures are implemented to analyze experimental data obtained in a turbulent boundary layer. The flow is seeded by 2-*µm* particles, the two views are inclined by approximately 37*°*, and the sample volume is 4.2 × 2.8 × 1.5 *mm*^{3}. Distortion correction substantially increases the number of intersecting inclined traces close to the particle center, as demonstrated by an increase in both the number of detected particles and the normalized intensity in the truncated traces. For velocity measurements, the displacement of individual particles is determined based on the 3D location of the intensity-weighted centroid of the truncated traces. Matching of traces is guided by a series of criteria, including volumetric cross-correlations of the truncated intensity fields. The unstructured data is then interpolated onto a regular grid with 60-*µm* spacing using SVD, which also determines the spatial distributions of velocity gradients. The improvement in the data quality is demonstrated by a reduction in the divergence of the velocity field, i.e., compliance with conservation of mass, in comparison to results obtained by analyzing data obtained from a single hologram. Increasing the volume over which conservation of mass is evaluated reduces the error further.

As noted in the introduction, truncation of particle traces based on dual-view holography has 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 spatially varying distortion. Furthermore, their particles are larger and their concentrations are lower than the present ones, e.g. one 5-*µm* cell in [25], a few 110-*µm* particles in [24], as well as a layer of 11-*µm* particles sandwiched between microscope slides and five 600-*µm* glass spheres in [26]. While the likelihood of overlap between the two views is expected to improve with increasing particle size and decreasing concentration, in the present dense suspension of 2-*µm* 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 in the accuracy of the 3D velocity field. While the present implementation of the self-calibration procedure and tomographic truncation of traces focuses on 3D velocity measurements at high magnification, the same approach could be readily utilized in other applications of digital holography 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).

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