A comparative analysis of the uncertainty of astigmatismμPTV, stereoμPIV, and μPIV
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Abstract
Astigmatism or wavefront deformation, microscopic particle tracking velocimetry (AμPTV) (Chen et al. in Exp Fluids 47:849–863, 2009; Cierpka et al. in Meas Sci Technol 21:045401, 2010b) is a method to determine the complete 3D3C velocity field in microfluidic devices with a single camera. By using an intrinsic calibration procedure that enables a robust and precise calibration on the basis of the measured data itself (Cierpka et al. in Meas Sci Technol 22:015401, doi: 10.1088/09570233/22/1/015401, 2011), accurate results without errors due to spatial averaging or bias due to the depth of correlation can be obtained. This method takes all image aberrations into account, allows for the use of the whole CCD sensor, and is easy to apply without expert knowledge. In this paper, a comparative study is presented to assess the uncertainties of two stateoftheart methods for 3C3D velocity field measurements in microscopic flows: stereoscopic microparticle image velocimetry (SμPIV) and astigmatism microparticle tracking velocimetry (AμPTV). First, the main parameters affecting all methods’ measurement uncertainty are identified, described, and quantified. Second, the test case of the flow over a backwardfacing step is analyzed using all methods. For comparison, standard 2D2C μPIV measurements and numerical flow simulations are shown as well. Advantages and disadvantages of both methods are discussed.
Keywords
Particle Image Velocimetry Focal Plane Particle Image Astigmatism Cylindrical Lens1 Introduction

the spatial resolution of the depth direction is determined by the imaging optics’ depth of focus and thus limited to several μm

outoffocus particles also contribute to the crosscorrelation (depth of correlation) and, hence, introduce a bias in the measurements

only 2C2D velocity fields can be measured.
The improvement and adaptation of the macroscopic μPIV technique are still ongoing processes. Reviews of the state of the art of μPIV and of its relevant applications were published by Lindken et al. (2009) and Wereley and Meinhart (2010). Several methods have been proposed to extend the velocity reconstruction to the third component. Reviews about advanced 3D methods can be found by Lee and Kim (2009); Chen et al. (2009) and Cierpka et al. (2010b). One method consists of using different viewing perspectives. Stereoscopic μPIV (SμPIV), derived from μPIV, takes advantage of a stereoscopic microscope to observe the flow field in the measurement region from two slightly different viewing angles. The inplane particle image displacement observed by two cameras under different angles can be used to estimate the inplane velocity by 2D crosscorrelation. The third component is then reconstructed by the inplane velocities as will be explained in detail in Sect. 2 Another approach is the tomographic reconstruction of the particle distribution in the volume, after which 3D crosscorrelation is applied to obtain the 3D velocity field. An inherent problem for multicamera approaches is the need for a very precise calibration and the small viewing angles applied (Lindken et al. 2006). Thus, their applicability, especially for the tomographic methods, to microfluidic devices seems to be quite limited, and alternative imaging approaches are necessary. Recently, inline holography (Lee and Kim 2009; Ooms et al. 2009) was applied to 3D velocity field measurements in microscopic channels. However, the numerical reconstruction process is rather time consuming, and the optical setup has to be built with great care to have an acceptable accuracy of the outofplane velocity. To overcome the difficulties of the complex calibration procedure in holography and multicamera techniques, a method using just one camera is favorable. The depth coding via three pinholes in the imaging system is a smart technique, estimating the particle’s depth position via twodimensional images (Pereira and Gharib 2002; Willert and Gharib 1992). With the three pinholes, a particle is imaged as a triplet. The distance between the edges of the triplet is related to the depth position. This concept is more robust than holography and was successfully applied to microfluidics by Yoon and Kim (2006). Aside from masking the optics, there are other methods that rely on breaking the axis symmetry of an optical system. This allows for the coding of the depth position of particles in a 2D image. By later reconstruction of the particles’ position in real space, the velocity field can be evaluated by correlation algorithms or tracking methods. So far, a bent dichroic mirror (Ragan et al. 2006), diffraction gratings for multiplanar imaging (AngaritaJaimes et al. 2006), an optical filter plate at an angle (van Hinsberg et al. 2008), and the observation under an angle (Hain and Kähler 2006) was used. For microfluidic applications, cylindrical lenses were successfully used by Chen et al. (2009) and Cierpka et al. (2010b). The approach based on cylindrical lenses, especially, is a very powerful and simple method, which allows for the extension of existing 2D measurement systems to fully 3D measurements. Kao and Verkman (1994) applied this technique to the measurement of the position of fluorescent particles in living cells. Today astigmatic imaging is commercially used in nearly every CD or DVD player to precisely determine the distance between CD and laser head. A big advantage of this approach is the possibility to adjust the measurement depth and resolution by changing the focal length of the cylindrical lens. The use of the recently presented intrinsic calibration procedure (Cierpka et al. 2011) makes the technique easy applicable without special expert knowledge.
Therefore, the SμPIV and AμPTV approaches will be studied and discussed in the following. In order to determine the accuracy and uncertainty of both techniques, measurements of the flow over a backwardfacing step will be compared with standard μPIV as well as numerical simulations. The backwardfacing step flow was chosen, since it offers a velocity field that is well known and mainly one directional prior to the step. Furthermore, it has a very pronounced outofplane component shortly downstream of the step. Other groups have also verified their 3D measurement methods with backwardfacing step flows (Chen et al. 2009; Yoon and Kim 2006; Bown et al. 2006). A combined stereo PIV/PTV approach was used by Bown et al. (2006). They measured the flow over a 232μm step in a 466μm high channel. Glycerol was used as working fluid, resulting in a Re _{ h } = 0.004. The flow was investigated with stereoscopic μPIV at 23 different planes in the zdirection. The accuracy of the correlationbased results was found to be limited by the misalignment or nonoverlapping of the two focal planes of the stereo microscope. To improve the accuracy, a super resolution PTV approach was applied. Using a PTV, algorithm allows to restrict valid measurements only to strongly focused particles, which decreases the effect of the depth of correlation. The authors reported uncertainties for the averaged vector map in the order of 0.35 μm/s (3% of the mean velocity) for the inplane components, and 0.82 μm/s (7% of the mean velocity) for the outofplane component of the correlationbased velocity estimation. The uncertainty was decreased to 2 and 3% for the inplane and outofplane velocity, respectively, with the PTV algorithm. Unfortunately, the way the uncertainties were determined was not reported, and a comparison is therefore difficult. Chen et al. (2009) used a cylindrical lens with \(f_{\hbox{cyl}} = 500\,\hbox{mm}\) to measure a \(600\,\upmu\hbox{m}\) range at a \(170\,\upmu \hbox{m}\) backwardfacing step, inside a 500 μm high channel. The uncertainty for the depth position was reported to be \(2.8\,\upmu\hbox{m}\) for the calibration images. Unfortunately, no uncertainty of the single measurements was given. The measured RMS value of the velocity was \(3.3\,\upmu\hbox{m/s}\), even though \(2.8\,\upmu\hbox{m/s}\) was expected from the measurement uncertainty. This is above one third of \(u_{\infty}.\) The investigated Reynolds number was Re _{ h } = 0.0015. The images were taken in single frame mode, probably with continuous laser light illumination and are of higher quality than double frame images with very short laser light pulses. The separation time between successive images in the study was \(\Updelta t = 2\,\hbox{s}\). The authors stated that 3,000 images were acquired, which takes 100 min. This and the very low Reynolds number are far beyond realistic ‘Labonachip’ applications, which range in the order of \(Re_h = 1,\ldots,100\). For these devices the acquisition of double frame images in a short time, which suffer from large noise levels, is necessary.
2 Experimental setup
2.1 The backwardfacing step flow, numerical simulation, and conventional μPIV
For the sake of a proper comparison, all experiments were performed in the same microchannel to avoid variations in the boundary conditions. The microchannels are fabricated out of elastomeric polydimethylsiloxane (PDMS) on a 0.6mm thick glass plate by the Institute for Microtechnology of the Technical University Braunschweig. They possess inlet and outlet crosssectional areas of \(500 \times 150\,\upmu\hbox{m}^{2}\) and \(500 \times 200\,\upmu\hbox{m}^{2}\), respectively. The channel was approximately 30 mm in length, with the backwardfacing step at about 15 mm from the inlet to assure fully developed flow conditions upstream of the step. The flow in the channel was seeded with polystyrene latex particles, fabricated by Microparticles GmbH. The particle material was premixed with a fluorescent dye, and the surface of the latex microspheres was later PEG modified to make them hydrophilic. Agglomeration of particles at the channel walls can be avoided by this procedure, allowing for long duration measurements without cleaning the channels or even clogging. The particles showed very high fluorescence signals that allowed for the extension of the measurement depth for the astigmatic measurements (Cierpka et al. 2011). To investigate the downward flow close to the step in greater detail with standard μPIV, an additional measurement was performed with a channel allowing optical access from the side. The data were evaluated using the single pixel procedure outlined by Scharnowski et al. (2010).
The mean diameter of the monodisperse particle distribution was \(d_P=2\,\upmu\hbox{m}\) (standard deviation = \(0.04\,\upmu\hbox{m}\)) for the AμPTV measurements and \(d_P = 1\,\upmu \hbox{m}\) for all PIV measurements. The fluid was distilled water, which was pushed by a high precision Nexus 3000 syringe pump (manufactured by Chemix) with constant flow rate through the channel. The Reynolds number based on the step height was Re _{ h } = 3.75 and based on the hydraulic diameter of the inlet, \(Re_{\hbox{HD}}=17.3\). For the illumination of the particles, a two cavity frequencydoubled Litron Nano S Nd:YAG laser system was used. The image recording was performed with the DaVis 7.4 software package from LaVision. The images were acquired in double exposure mode, where the camera shutter is activated two times. The time delay between the two successive frames was set to \(\Updelta t = 200\,\upmu\hbox{s}.\) 1,000 images were recorded at each zposition for all three techniques. The AμPIV measurements, as well as the 2D2C conventional μPIV measurements, were performed using an Axio Observer Z.1 inverted microscope by Carl Zeiss AG with a LDPlan Neofluar objective with a numerical aperture of NA = 0.4 and a magnification of M = 20×. To reconstruct the velocity field in the volume from conventional PIV, the raw image pairs were preprocessed and crosscorrelated. Preprocessing consisted of subtracting the sliding minimum over time, followed by the same substraction in space to decrease nonuniformities and backreflections. These steps are followed by a bandwidth filter and constant background subtraction, used to sort out particle agglomerations and eliminate the remaining background noise. 2D velocity fields were measured for seven equidistant planes inside the channel, starting from \(z = 37\,\upmu\hbox{m}\) and ending at \(z=177\,\upmu\hbox{m}\). The image pairs were crosscorrelated with the DaVis 7.2 software package from LaVision. A normalized multipass algorithm with a final interrogation window size of 32 × 32 pixels was used with 50% overlapping of the interrogation windows with an average of 3–5 particle images per window. Since the flow was laminar and stationary, the vector fields were averaged to get the final vector fields.
For the numerical flow simulation, the microchannel was modeled with a solid modeler to extract the microchannel boundaries; the boundaries were meshed in CDadapco STARCCM+ 4, and a finite volume model was set for a laminar and viscid fluid with a constant density (water). The computational domain exceeded 600,000 hexahedral cells. In the step region, four times the channel width, the mesh size was equal to \(6.25\,\upmu\hbox{m}\) (1/80 of step width) to ensure an optimal velocity resolution. The noslip condition was set at the boundaries of the computational domain. At the inlet, the velocity was set to match the Reynolds number of the experiment. At the outlet, the pressure was set to a reference value. To avoid entrance effects, two flow extensions were located at the inlet and the outlet; uniform boundary conditions were set at a distance of twenty times the channel width. The steady solution converged using the implicit solver in 500 steps; the relative errors of residuals of continuity and momentum were less than 10^{−6}.
2.2 Stereoscopic μPIV
A major problem in SμPIV measurements is given by the possible mismatch of the two focal planes caused by optical aberrations and imperfections in the construction of the microscope. In SμPIV, as well as in μPIV, volume illumination is used and the measurement volume observed by one camera corresponds to its focal plane. The evaluated 3D velocity vectors result from the recombination of the 2D velocity fields observed by cameras 1 and 2, under the assumption that their measurement volumes are exactly cospatial. The thickness of the measurement volume can be estimated using the depth of correlation (Olsen and Adrian 2000). A misalignment of the two cameras’ measurement planes with respect to each other introduces an additional bias error, especially when velocity gradients are present (Rossi et al. 2010). What is more important, this error cannot be corrected since it inherently depends on the design and construction of the microscope.
It can be observed that the focal planes are curved and overlap only partially, even when the finite thickness of the measurement volume is considered. Particularly for the case in water, in the region where the velocity measurements on the backwardfacing step were taken, a mean difference of \(4.1\,\upmu\hbox{m}\) was estimated between the two focal planes, with a maximum of \(11.2\,\upmu\hbox{m}\). This error is only negligible when the depth of the measurement volume is large compared to the mismatch. However, an additional error is introduced by averaging the velocity measurement through the depth of correlation in this case. For this setup, using \(1\,\upmu\hbox{m}\) diameter particles, the depth of correlation was estimated to be equal to \(30\,\upmu\hbox{m}\), which means that in the worst case onethird of the measurement thickness was not correlated. This can already lead to substantial systematic errors (Kähler 2004). With regards to the PIV analysis, the images were first preprocessed using a sliding minimum filter for background removal and a smoothing median filter for image random noise reduction. Subsequently, an ensemble correlation over 1,000 images per plane was calculated, using a multipass algorithm with final interrogation window of 64 × 64 pixels and 50% overlap. The vector fields were recombined using the empirical calibration to reconstruct the third velocity component. The results were later organized on a Cartesian grid with the same grid size as the results of the conventional μPIV, with \(\Updelta x = \Updelta y = 15\,\upmu\hbox{m}\) and \(\Updelta z = 29\,\upmu\hbox{m}\).
2.3 Astigmatism μPTV
The depth coding of the particle position on the images is achieved by a cylindrical lens in the imaging system. Similar setups were used in previous studies (Chen et al. 2009; Cierpka et al. 2010b). The cylindrical lens for the current investigation had a focal length of \(f_{\rm cyl} = 100\,\hbox{mm}\) and was directly placed in front of the CCD chip (Cierpka et al. 2010a). The curvature of the cylindrical lens only acts in one direction and causes two focal planes in the x and yaxis to be formed. For the setup used here, these planes are separated by \(\Updelta z \approx 45.2\,\upmu \hbox{m}\) in the measurement volume. Particles that are close to one focal plane, e.g. the xaxis focal plane, appear as small and sharp images in that axis. They are now far from the infocus plane in the ydirection and result in defocused, larger images in the yaxis. Thus, an elliptical image is formed on the CCD sensor, with a small horizontal axis, denoted as a _{ x }, and a large vertical axis, denoted as a _{ y }. By evaluating the particle image’s width and height, the depth position can be found using a calibration procedure. The position in the xyplane is determined by a waveletbased algorithm, which gives reliable results with subpixel accuracy up to high background noise levels (Cierpka et al. 2010b). The ratio between background and signal intensity was below 0.1 for the measurements presented, which results in an error of 0.05 pixels for the inplane position. This relates to an absolute error of \(0.031\,\upmu\hbox{m}\) in the xdirection and \(0.038\,\upmu\hbox{m}\) in the ydirection. For the final procedure, image preprocessing is applied to the images. First, a sliding minimum over time is subtracted to remove background noise. Smoothing and segmentation filters are then used to highlight regions of possible particle candidates. Based on this initial guess for particle positions, the algorithm determines a _{ x }, a _{ y }, x, and y in the originally backgroundsubtracted images. For the details of the particle image detection algorithm, the interested reader is referred to Cierpka et al. (2010b).
The determination of a _{ x } and a _{ y } was done using an auto correlationbased algorithm. Although very sparse seeding was used, prior to this step each identified region of a possible particle image was checked for overlapping particles. The used criteria were a maximum allowed perimeter of the region where a particle is assumed and, a test, if the center of an identified region belongs to a particle image (i.e. has a higher intensity as the background). For ideal conditions, two of the four values of Eq. 4 are equal and give the depth position of the particles. However, due to small variations in the particle size distribution and the determination of their width and height, the data points scatter around the ideal solution. The determination of z was therefore made by finding the value that minimizes the Euclidean distance between the two measured points a _{ x } and a _{ y } to the calibration curve. The standard deviation between the estimated particle position \(z_{\rm est}\) and the position given by Eq. 4 gives an impression regarding the uncertainty of a single measurement and was calculated to be \(\sigma (zz_{\rm est}) = 3.14\,\upmu\hbox{m}\). Using this calibration, the maximum measurement depth was \(104\,\upmu\hbox{m}\). Therefore, the position uncertainty in the depth direction of a single measurement, without traversing, is 6% of the measurement depth. However, for a single measurement, this would result in an uncertainty of about \(15.7\,\hbox{mm/s}\) for the present conditions with a maximum volume depth of \(104\,\upmu\hbox{m}\). A reduction of the volume thickness would decrease this uncertainty significantly. To compare the results, one has to consider that for the crosscorrelation; approximately 6–10 particle images should be present in an interrogation area. The data were later interpolated on a Cartesian gird and showed a good convergence of the mean value in one volume element. The difference between the mean values of a certain number i = I of data points that belong to one grid volume and all the data points i = N in the same volume \((\Updelta x = \Updelta y = 10 \,\upmu\hbox{m}, \Updelta z = 10 \,\upmu\hbox{m}), \epsilon_w = \left\Upsigma\epsilon_{wi=I}/I  \Upsigma\epsilon_{wi=N}/N\right\) is a measure of convergence. Taking 10 particle images the difference is \(\epsilon_w = 1.4\,\hbox{mm/s}\). The average number of data points that contribute to a grid volume element was 50, which gives a difference of \(\epsilon_w \approx 0.38\,\hbox{mm/s}\). It should be mentioned at this point, that this approach leads to any desired accuracy, as the technique is free of systematic evaluation errors in contrast to PIV. The measurement volume’s depth depends on the microscope’s magnification and the focal length of the cylindrical lens, as well as on the detection level of the camera, the power of the laser, and the quality of the fluorescent dye.
For the study presented here, approximately 50% of the data points are within a span of \(34.5\,\upmu\hbox{m}\), centered at the midpoint between the two focal planes, and 90% fall between a span of \(59.6\,\upmu\hbox{m}\). To cover the whole channel, overlapping data were acquired at eight different zpositions.
For each zlevel, around 50,000 valid particle pairs were identified with a simple nearest neighbor algorithm. This gives a valid vector for 65% of the total particle images per frame, which was about 50–80. The 36% loss of pairs is due to the motion of particles out of the measurement volume in all directions, the excluding of overlapping particles in one of the two frames and due to the larger uncertainty for the determination of the position in zdirection. In the current study, a simple nearest neighbor algorithm was used for the tracking step. If a more sophisticated algorithm will be applied, it is supposed that the loss of pairs will significantly decrease. Less particle images per frame occur for the measurements closer to the wall, since a part of the measurement volume was already outside of the channel. The data of all individual particles were filtered by a global histogram filter in order to remove obvious outliers. A local universal outlier detection algorithm for PTV data proposed by Duncan et al. (2010) was additionally used. The authors proposed a weighting of the neighboring values by their distances. The normalized residuum or fluctuation at the position r _{0} ^{*} was set to be lower than 2 for valid data, taking the 10 closest neighboring points into consideration. Rejecting data with a residuum higher than 2 for one of the three velocity components result in an outlier removal of <4%. In total, 390,000 vectors were considered to be valid and were used for the following analysis. The data were then interpolated onto a Cartesian grid with a grid size of \(\Updelta x = \Updelta y = 10\,\upmu\hbox{m}, \Updelta z = 10\,\upmu\hbox{m}\) and 25% overlap. Using this interpolation, approximately 50 single PTV measurements points contribute to the mean at each point in the Cartesian grid. As a measure for the uncertainty of the measurements, the standard deviation of the single measurements \(\hbox{std}\left(u_iu_{\rm mean}\right)\) can be calculated. However, this quantity is strongly affected by the grid size in regions of high gradients. Therefore, it was evaluated upstream of the step (\(x <50\,\upmu\hbox{m}\)), where a laminar channel flow profile is present, and v and w have a zero mean, and the scatter of the single PTV data points is purely caused by the measurement technique. The mean standard deviation in that region was \(0.95\,\hbox{mm/s}\) for v and \(3.7\,\hbox{mm/s}\) for w. The uncertainty for the outofplane component is 4.9% of \(u_{\infty}\), which is four times higher than for the inplane component with 1.3% of \(u_{\infty}\).
3 Results
Standard deviation of the difference to the numerical simulation
Conventional μPIV  StereoscopicμPIV  AstigmaticμPTV  

\(\sigma(u_{\rm exp}u_{\rm sim})/\rm mms^{1}\)  7.75  4.60  5.00 
\(\sigma(v_{\rm exp}v_{\rm sim})/{\rm mms}^{1}\)  1.44  0.73  0.73 
\(\sigma(w_{\rm exp}w_{\rm sim})/{\rm mms}^{1}\)  –  4.20  2.38 
On the lower part of Fig. 6, the streamwise velocity is presented. Conventional μPIV is included as well but shows poor performance in the region above the step. Beyond the step, the velocity is significantly underestimated although the profile was taken at \(z = 60\,\upmu\hbox{m}\). The boundary layer of the bottom wall, prior to the step, cannot be well resolved, and the measured velocity is much too low. SμPIV performs slightly better but overpredicts the velocity upstream of the step. Nevertheless, beyond the step, the profile matches the simulation quite well. The best match between experiments and simulation upstream of the step is achieved by the AμPTV although the velocity is slightly underestimated downstream of it.
4 Conclusion and outlook

μPIV gives reliable 2D2C results, but in regions of strong gradients (inplane and outofplane direction) and strong outofplane motion, the technique fails at providing reliable results.

SμPIV gives reliable 2D3C results, but due to the unavoidable mismatch of the focal planes, systematic errors appear that cannot be compensated digitally. To minimize the error, or at least to know the extent to which the results will be biased, the focus function has to be evaluated for both cameras.

By scanning the measurement plane, average 3D3C velocity fields can be estimated with SμPIV but no instantaneous 3D3C velocity fields can be obtained.

AμPTV provides instantaneous 3D3C velocity information and allows for the study of unsteady volumetric flow phenomena.

For AμPTV, the accuracy of the velocity components in x and ydirection is not effected by the measurement volume depth and is comparable to correlationbased methods.

The uncertainty of the instantaneous outofplane velocity estimation by AμPTV increases with increasing measurement volume depth but, for mean values, it decreases below the corresponding value of the μPIV due to the absence of systematic evaluation errors.

An advantage of the AμPTV technique, compared to the correlationbased methods, is that the results do not suffer from the influence of the depth of correlation and a higher resolution in depth direction can be achieved.

Since with longer measurement time the mean distance between the vectors is decreasing for the AμPTV technique, it is possible to increase the spatial resolution for the average flow fields in all three dimensions. In case of SμPIV, the spatial resolution cannot be increased by acquiring more images. Thus high gradients are always underestimated.

Since the particle positions are known in the whole volume, the particle distribution can be used to reconstruct interfaces between fluids to characterize the mixing process at the microscale (Mastrangelo et al. 2010; Rossi et al. 2011).

Using timeresolved AμPTV imaging, the Lagrangian trajectories of the particles can be determined and the complete motion (velocity and acceleration) or the interaction of particles in time and space can be fully reconstructed (Kumar et al. 2011).

AμPTV can be used to determine the full 3D velocity information as well as a scalar distributions such as temperature, phvalue, or pressure fields by combining the underlying imaging technique with particles whose fluorescent emission is a function of these physical properties.
Notes
Acknowledgments
Financial support from Deutsche Forschungsgemeinschaft (DFG) in frame of the priority program SPP 1147 and the research group FOR 856 (Mikropart) is gratefully acknowledged. The authors also would like to thank Stefanie Demming and the Institute for Microtechnology at the Technische Universität Braunschweig for their kind support which is gratefully appreciated.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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