In the last few years, spatial particle location using astigmatic aberrations was applied increasingly to measure three-dimensional flow fields in microfluidics [1]. The astigmatism paradigm was first introduced in 1994 to track single fluorescent particles in artificial solutions and living cells [2]. Astigmatism leads to the appearance of two separated focal planes. This distance depends mainly on the amount of astigmatism induced by a cylindrical lens placed in front of the camera sensor. Because of this aberration, a tracer particle forms an elliptical particle image with a distinct geometry corresponding to its distance from the camera (Fig. 1). Hence, the $z$ position of a particle is coded in the horizontal and vertical axis length ($a_x$ and $a_y$) of its elliptical particle image [4].

 

Fig. 1. Optical setup for APTV. A cylindrical lens acting in the $yz$ plane leads to the appearance of a second, separated focal plane. Particle image axis lengths, $a_x$ and $a_y$, vary with distance to the camera (figure adopted from [3]).

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There is considerable interest to employ astigmatism particle tracking velocimetry (APTV) and its unique capabilities for the measurement of macroscopic flows. Unlike multicamera techniques such as tomographic PIV [5], APTV employs only a single camera. Thus, APTV is best suited for measurement domains with limited optical access (e.g. compressor, turbine, combustion chamber, and engine research). In a study presented by Towers and Towers [6] astigmatic aberrations are used in a macroscopic particle position determination approach. Geometrical optics approaches are used to deduce depth information from particle image dimensions. Aberrations besides astigmatism are not accounted for in this geometrical model, leading to errors in depth position determination. Sensitivity to illumination disparities introduces further inaccuracies. The measurement depth range is limited to regions well within the two focal planes as the geometrical model does not apply in near-focal regions. Therefore, the distance of the focal planes has to be increased to measure a certain depth range, resulting in larger particle images with lower SNRs.

This Letter presents a theoretical and experimental image formation study in the presence of astigmatic aberrations. Particles are simulated by pinholes as they have a similar light emission behavior. In the second part, a three-dimensional, macroscopic location scheme of micrometer-sized particles is introduced for the APTV technique. The technique’s feasibility to measure macroscopic flows is proven by an experimental accuracy analysis of the particle $z$ position determination, using state-of-the-art PIV equipment.

The image formation study provides a qualitative description of particle image shapes in the presence of astigmatism. In macroscopic APTV setups astigmatic aberrations, induced by a cylindrical lens placed in front of the camera sensor, can be considered as large aberrations, i.e., the deviation of the actual wavefront from the ideal, spherical wavefront is large. Thus, geometrical optics describe image formation relatively accurately. Nonetheless, for a comprehensive analysis of the image formation, the influence of diffraction on image formation has to be accounted for. According to the Huygens–Fresnel principle, the diffraction integral for a disturbance at a point $P$ in image space, in the presence of aberrations, is given by [7]

In the presence of astigmatism, the location of the plane of least distortion, i.e., midway between the focal planes, is at $p=0$. Particle images show asteroid shapes with a circular ring of higher intensity (Fig. 2, and [5,7,8]). The image of the 5 μm pinhole in Fig. 2(b) matches very well with the simulated image in Fig. 2(a). The asteroid shape can hardly be observed with a lower SNR [image of the 1 μm pinhole, Fig. 2(c)]. The image has a circular shape of almost equi-distributed intensity with a less distinct outer ring. The geometrical theory of aberrations predicts a circular image of constant intensity in the central plane, quite similar to the image analyzed in Fig. 2(c). The distance of the two focal planes is determined by the amount of astigmatism. Their location is denoted by $p=\pm {\beta }_{\mathrm{ast}}$ along the optical axis. Images at the focal planes form thin lines of high intensity [Figs. 3(a) and 3(b)], whereas geometrical optics predicts one axis length to be zero (provided ideal lenses are present) [10]:

 

Fig. 2. Intensity coded pinhole/particle images in the central plane (intensity inverted; note that the intensity scaling differs between the images). Particle/pinhole images show an asteroid shaped diffraction pattern (high SNR provided).

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Fig. 3. Intensity coded pinhole/particle images in the focal plane and well beyond (intensity inverted; note that the intensity scaling differs between the images). Near the focal planes, the particle images form thin lines. Beyond the focal planes, the axis lengths increase in both directions.

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Spherical aberrations have only limited influence on image formation in macroscopic domains, as the amount of spherical aberrations is small compared to the induced astigmatic aberration (${\beta }_{\mathrm{ast}}\gg {\beta }_{\mathrm{sph}}$). Therefore, spherical aberrations can be neglected for the image formation analysis in macroscopic domains. In microscopic domains, the influence of spherical aberrations on the image formation is significant, as the focal plane distances are small (${\beta }_{\mathrm{ast}}\approx {\beta }_{\mathrm{sph}}$). In macroscopic domains, the appearance of particle images can be considered to be symmetric with respect to $p=0$ but rotated by 90 degrees. In microscopic domains, particle images beyond one focal plane ($p>{\beta }_{\mathrm{ast}}$) have a bright outer ring and beyond the other focal plane ($p<-{\beta }_{\mathrm{ast}}$) they have a bright central spot with decreasing intensity toward the particle image edge. With a sufficiently small amount of astigmatism, the asteroid-shaped diffraction pattern cannot be observed anymore. Instead particle images have cross-shaped central spots in the center. In this case, the location of the focal planes cannot be determined as easily, as particle images do not form thin lines anymore.

To account for particle image shapes as observed in macroscopic APTV setups, a suitable image processing algorithm is established. Local intensity distributions at the four vertices of the elliptical particle images are fitted by means of a thin-plate spline. An intensity threshold denotes the subpixel locations of ellipse vertices, from which center locations and axis lengths, $a_x$ and $a_y$, are determined. Local intensity distributions are normalized by particle images’ mean intensities. Hence, differing illumination intensities and distributions have a small impact on the position determination, making the processing much more robust. The image formation study has proven that in macroscopic APTV setups image formation is dominated by astigmatism, leading to similar-shaped particle images. As a consequence, the macroscopic APTV processing and calibration procedures are applicable independently of the measurement setup.

In contrast to the qualitative image formation study presented in the previous section, the object of the accuracy analysis is to give a quantitative measure of the particle position determination error in a macroscopic domain ($20\mathrm{mm}\times 20\mathrm{mm}\times 40\mathrm{mm}$, $x\times y\times z$; coordinates in physical space are denoted by lowercase letters, coordinates in image space, i.e., on the sensor, by capital letters). To prove the applicability of APTV to measure macroscopic velocity fields in air, the analysis is conducted with state-of-the-art PIV recording and illumination equipment. We focus on the particle $z$ position determination, as this coordinate along the optical axis has the lowest accuracy. For the systematic investigation, particles are simulated by pinholes, as both have similar light emission behavior. A pinhole matrix ($d_p=1.1\pm 0.1\mathrm{\mu m}$, vertical and horizontal pinhole displacement of 4 mm), backlight-illuminated by an Innolas SpitLight 400 PIV laser, is moved through the measurement volume depth of 40 mm in steps of $\mathrm{\Delta }z=0.5\pm 0.006\mathrm{mm}$. At every position, a LaVision sCMOS camera recorded 20 images. The cylindrical lens is placed in front of the camera sensor to induce astigmatism. It has a focal length of 1000 mm, leading to a focal plane distance of $\mathrm{\Delta }{z}_{\mathrm{FP}}\approx 25\mathrm{mm}$. The experiments were conducted at a working distance of 290 mm, measured from the camera sensor to the central plane of the measurement volume.

The geometry of a pinhole image does not only depend on the distance to the camera ($z$ position) but also on its $x$ and $y$ position ($X$ and $Y$ position, respectively), as the focal planes are curved. In addition, distortions in the optical path have to be accounted for. Hence, it is necessary to establish a calibration function depending on the geometry of a pinhole image and the spatial coordinates of the corresponding pinhole. The first step of the calibration procedure is to calculate an intensity averaged image of the 20 recordings at each $z$ position. In a second step, spline fits are applied to the axis ratio values of the pinhole images detected on the averaged recordings. From these spline fits, unique calibration functions for every $X$ and $Y$ position, as seen in Fig. 4, can be determined. According to this calibration procedure, $z$ positions of particle/pinhole images are estimated as follows: center locations and axis ratios are determined first. A calibration function ($X$ and $Y$ position denoted by center location) for each single particle/pinhole image is derived from the spline fits of the averaged images. With this unique calibration function, $z$ positions of particle/pinhole images are approximated by the corresponding axis ratios. When measuring beyond focal planes, the calibration functions become ambiguous. These ambiguities can be overcome by analyzing the axis lengths of particle images. Whereas axis ratios have a minimum or maximum at focal planes, the larger axis of a particle image changes linearly near the corresponding focal plane. Thus, the length of the larger particle image axis determines whether the particle position is located within or beyond the focal planes.

 

Fig. 4. Calibration function for a specific $XY$ position. Both the minimum and maximum values of the axis ratio, $a_x/a_y$, denote focal planes ($\mathrm{\Delta }{z}_{\mathrm{FP}}\approx 25\mathrm{mm}$).

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For the $z$ position accuracy analysis, the absolute deviations, $\mathrm{\Delta }{z}_p$, of the estimated positions compared to the actual positions are determined. At every $z$ location the standard deviation of all $\mathrm{\Delta }{z}_p$ values to the exact position (zero deviation) is calculated, denoted by the $z$ position determination error $E_z$. For each $z$ location, a data set of about 300 particle images is analyzed. Figure 5(a) shows the distribution of $E_z$ along $z$, at a measurement depth of 40 mm. At both focal planes (approximate $z$ positions: 8 and 32 mm), $E_z$ has peaks even though SNRs are large. This can be explained by the calibration function that has a maximum or minimum at the respective focal planes (Fig. 4). Generally, $E_z$ increases with decreasing slopes of the calibration functions. Lower SNRs yield higher $E_z$ values as well. The average error, ${\overline{E}}_z$, is 0.133 mm (0.33%). An adjustment of the processing parameters improves the $E_z$ values of the gray backgrounded area, i.e., only the area between the focal planes is considered. Large particle images with low SNRs located beyond the focal planes cannot be processed with these parameters. Distribution of $E_z$ is then relatively uniform and follows the slope of the calibration function [Fig. 5(b)]. For a resulting measurement depth of 23.5 mm ${\overline{E}}_z$ is 0.075 mm (0.32%).

 

Fig. 5. Position error, $E_z$, depending on $z$ location. (a) Average error, ${\overline{E}}_z$, is 0.133 mm (0.33%) at a measurement depth of 40 mm. (b) Using adjusted processing parameters, ${\overline{E}}_z$ decreases to 0.075 mm (0.32%), when only the region between focal planes is considered (gray backgrounded area).

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The relative $z$ position determination accuracy of the proposed macroscopic APTV technique lies in the same range of the accuracy of the aforementioned anamorphic approach [5]. For the latter approach no pulsed laser was used, allowing for longer exposures (higher SNRs) of the sensor. These long exposures, however, are not feasible for the measurement of macroscopic flows.

A theoretical and experimental image formation study was presented. Particle image processing algorithms were optimized on the basis of image formation observations. A three-dimensional, macroscopic location scheme of micrometer-sized particles, introduced in the second part of the Letter, showed small errors in particle $z$ position determination. Average errors were as low as 0.33%, at a measurement depth of 40 mm. These accuracies prove the ability of APTV to measure volumetric velocity fields in macroscopic domains with only a single camera for applications with limited optical access.