## Abstract

This Letter presents a stereoscopic imaging concept for measuring the locations of particles in three-dimensional space. The method is derived from astigmatism particle tracking velocimetry (APTV), a powerful technique that is capable of determining 3D particle locations with a single camera. APTV locates particle $xy$ coordinates with high accuracy, while the particle $z$ coordinate has a larger location uncertainty. This is not a problem for 3D2C (i.e., three dimensions, two velocity components) measurements, but for highly three-dimensional flows, it is desirable to measure three velocity components with similar accuracy. The stereoscopic APTV approach discussed in this report has this capability. The technique employs APTV for giving an initial estimate of the particle locations. With this information, corresponding particle images on both sensors of the stereoscopic imaging system are matched. Particle locations are then determined by mapping the two particle image sensor locations to physical space. The measurement error of stereo APTV, determined by acquiring images of 1-μm DEHS particles in a $40\text{\hspace{0.17em}}\mathrm{mm}\times 40\text{\hspace{0.17em}}\mathrm{mm}\times 20\text{\hspace{0.17em}}\mathrm{mm}$ measurement volume in air at $\mathrm{\Delta}xyz\to 0$ between two frames, is less than 0.012 mm for $xy$ and 0.025 mm for $z$. This error analysis proves the excellent suitability of stereo APTV for the measurement of three-dimensional flows in macroscopic domains.

© 2014 Optical Society of America

Capturing three-dimensional flow fields is one of the key interests in experimental fluid mechanics. Various techniques have been developed, including multi camera techniques such as tomographic particle image velocimetry (tomographic PIV), and 3D particle tracking velocimetry (3D PTV) [1,2]. However, for measurement domains with limited optical access—like compressor, turbine, and combustion research—single camera approaches like defocusing PIV are better suited [3]. Beside their large range of application, they are also advantageous in terms of simplicity and robustness. Astigmatism PTV (APTV) is based on the astigmatic elongation of particle images according to their location within a three-dimensional measurement domain [4]. The particle $z$ location is coded in the vertical and horizontal axis lengths (${a}_{X}$, ${a}_{Y}$) of the elliptical particle image. APTV is well-established in microfluidics for accurate three-dimensional velocity measurements [5], but it is also feasible for macroscopic particle location estimation [6].

APTV locates particles very accurately in the $x$, and $y$ direction, but the particle $z$ coordinates, i.e., along the optical axis, have larger uncertainties. This is due to the fact that $xy$ particle locations are mainly derived from their particle image center locations $XY$, which are identified very accurately. In contrast, particle $z$ locations are derived from the axis lengths (${a}_{X}$, ${a}_{Y}$) of the particle images. The axis lengths determination is more affected by experimental conditions such as a low signal-to-noise ratio (SNR), and optical aberrations besides astigmatism. Hence, for highly three-dimensional flows, where a precise measurement of the velocity in the $z$ direction is important, it is desirable to extent the single-camera APTV measurement technique.

Stereo APTV provides a means to measure the three coordinates with similar accuracy, as illustrated qualitatively in Fig. 1. Particles are located by mapping the sensor coordinates of corresponding particle images to physical space by means of third-order polynomials. Unlike 3D PTV, the technique requires only two cameras for unambiguous particle image matching. At higher seeding concentrations, using 3D PTV, ambiguities in particle location arise, since the search area near the epipolar line on the other sensor likely contains multiple particle images [7]. These limitations can be overcome by increasing the number of views from at least three and taking time-series of images [8]. However, the necessity of time-resolved measurements strongly limits the applicability of the technique, especially for air flows. Stereo APTV, on the other hand, utilizes an initial estimation of particle locations by means of single-camera APTV processing to uniquely match corresponding particle images on the two camera sensors. In a second step, particles are located more accurately by coordinate mapping with stereoscopic view. Thus, the three-dimensional particle location is feasible with only two cameras. Furthermore, stereo APTV is a redundant measurement technique. Overlapping particle images on one sensor are unlikely to overlap in the second sensor. While coordinate mapping is not feasible with overlaps, the particle can still be located from a single view with lower $z$ accuracy. This is advantageous, especially for time-resolved measurements, since it is unlikely that particle tracks get lost.

Locating particles using coordinate mapping requires knowledge of the corresponding particle images of the two views of the stereoscopic imaging system. Thus, the input of the third-order polynomial mapping functions are the particle image center locations on the two sensors ${\mathbf{X}}_{1}={X}_{1}{Y}_{1}$ and ${\mathbf{X}}_{2}={X}_{2}{Y}_{2}$, whereas the output is the particle location in physical space: $\mathbf{x}=xyz$ (see Fig. 2; capital letters denote image/sensor coordinates, lowercase letters denote physical coordinates). Altogether, the polynomials have 35 coefficients for every spatial coordinate, yielding the following equation for the $x$ coordinate:

Before the determination of the particle locations by means of coordinate mapping, the coefficients of the polynomials have to be calculated. Thus, a calibration target has to be moved through the measurement volume in steps of $\mathrm{\Delta}z$. The target consists of a backlight illuminated pinhole matrix (see Fig. 3), since the light emission behavior of pinholes is equal to that of particles of the same diameter (Babinet’s principle). From imaging these pinholes, the information for calibrating the stereo APTV system is derived. First, the physical coordinates of the pinholes are given by their location on the target ($x$, and $y$ location) and the $z$ position of the target ($z$ location). Their pinhole image coordinates, ${X}_{1}{Y}_{1}$ and ${X}_{2}{Y}_{2}$, are denoted by the pinhole image center locations on the two sensors. All this information is required to calibrate the stereoscopic system. A $n\times 35$ matrix, $S$, is established, where $n$ denotes the number of processed pinhole images. Typically, at every $z$ position of the calibration target (50–100 positions along the $z$ axis), 400 pinholes are analyzed, so that $n$ lies in the range of 20,000–40,000. The 35 entries represent the right side of equation 1, where ${X}_{1}$, ${Y}_{1}$, ${X}_{2}$, and ${Y}_{2}$ are the sensor coordinates of the pinhole images of the ${n}^{\mathrm{t}\mathrm{h}}$ pinhole. The 35 coefficients $\mathbf{c}$ are estimated by solving the following set of equations in the least squares sense (in this particular case, the coefficients ${\mathbf{c}}_{x}$ for the $x$ coordinate):

where $\mathbf{x}$ is an $n\times 1$ vector containing the $x$ locations of $n$ pinholes (the coefficients ${\mathbf{c}}_{y}$ and ${\mathbf{c}}_{z}$ are determined accordingly). Furthermore, the pinhole matrix provides the information to calibrate the APTV imaging system. For APTV, a relation between the geometry of the particle images, i.e., the axis lengths ${a}_{X}$ and ${a}_{Y}$, their center locations $XY$ on the sensor, and the spatial location $xyz$ of the particles has to be established. The pinhole images incorporate the information of the axis lengths and the center locations, while the locations of the pinholes on the target, as well as the calibration target $z$ position, denote the $xyz$ coordinate (more information on APTV calibration in macroscopic domains is given in [6]; for microfluidics the reader is referred to [11,12]). This yields calibration functions of the type, to estimate particle locations, $\mathbf{x}=xyz$. Hence, employing a pinhole matrix enables a simultaneous calibration of both imaging systems, saving measurement and processing time. Additionally, an error-prone alignment of different calibration targets is avoided.It was outlined before that APTV is used for an initial estimate of the particle locations on both cameras. This is necessary in order to match corresponding particle images, since the stereoscopic imaging system does not have this capability. The matching procedure establishes small search radii around the spatial particle locations to find intersecting, and thus matching, particles imaged by the two cameras independently. With the known particle intersections, the corresponding particle images are determined. After the matching procedure, to locate the particles, again a $n\times 35$ matrix, $S$, is set up. Now $S$ contains the sensor locations ${X}_{1}{Y}_{1}$, and ${X}_{2}{Y}_{2}$ of $n$ imaged particles in the measurement volume. With the knowledge of the coefficients, $\mathbf{c}$, the particles locations are determined by,

where $\mathbf{x}$ is a $n\times 1$ vector containing the $x$ location of the particles. The particles’ $y$ and $z$ locations are calculated accordingly. Figure 4 gives an overview of the outlined stereo APTV particle location scheme.To prove the feasibility of stereo APTV for accurate three-dimensional particle location, a comprehensive accuracy analysis was carried out. In a first analysis, the pinhole locations were reconstructed using the stereo APTV processing algorithms, as introduced before. These estimated locations, ${\mathbf{x}}_{\mathrm{est}}$, were then compared to the known locations of the pinholes leading to the deviations $\mathrm{\Delta}\mathbf{x}={\mathbf{x}}_{\mathrm{est}}-{\mathbf{x}}_{\mathrm{real}}$ in a $40\text{\hspace{0.17em}}\mathrm{mm}\times 40\text{\hspace{0.17em}}\mathrm{mm}\times 20\text{\hspace{0.17em}}\mathrm{mm}$ measurement volume. From these deviations we calculated the deviation error,

at every $z$ position of the pinhole matrix for the three spatial coordinates. The error $E$ decreases slightly with distance to the camera for all coordinates (see Fig. 1 for the coordinate system; the $y$ coordinate is perpendicular to the sketch). The error, ${E}_{x}$, of the $x$ coordinate yields values between 0.0025 and 0.005 mm, while ${E}_{z}$ lies between 0.005 and 0.015 mm, such that the $z$ coordinate has the highest uncertainty (Fig. 5). This is due to the fact that the measurements were conducted at an angle of $\beta =30\xb0$ ($\beta <90\xb0$: ${E}_{x}<{E}_{z}$; $\beta =90\xb0$: ${E}_{x}={E}_{z}$; $\beta >90\xb0$: ${E}_{x}>{E}_{z}$;). Consequently, ${E}_{y}=0.0025\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ is the smallest, since the cameras are aligned at an angle of 0° with respect to the $y$ axis.In addition to the pinhole location uncertainty analysis, we determined the particle displacement error in a $40\text{\hspace{0.17em}}\mathrm{mm}\times 40\text{\hspace{0.17em}}\mathrm{mm}\times 20\text{\hspace{0.17em}}\mathrm{mm}$ measurement volume in air. This was performed by means of the measurement of an “motionless” flow, using DEHS seeding particles with a diameter of around 1 μm. These particles were imaged in a glass tank at two instances, which were separated in time by 0.5 μs. Since there was no driven flow in the glass tank, the displacement of the particles between the light pulses approaches zero ($\mathrm{\Delta}\mathbf{x}\to 0$). However, processing the particle images, a displacement $\mathrm{\Delta}\mathbf{x}\ne 0$ is likely estimated. This results from external influences, such as misalignments of the laser light volume illumination, image noise, and light reflections. The particle displacement error, $\sigma $, is calculated like $E$ in Eq. 5, only that here, $\mathrm{\Delta}\mathbf{x}$ is the displacement of a particle between the two light pulses. For the $x$ coordinate, ${\sigma}_{x}$ is 0.012 mm, while ${\sigma}_{z}$ is larger with 0.025 mm (the measurements were conducted at $\beta =30\xb0$). Again, the error of the $y$ coordinate is the lowest with 0.005 mm. However, assuming a maximum particle image displacement of 10 pixels between the two frames, these uncertainties yield velocity errors in the range of 1%–3% for $u$, $v$, and $w$, relative to the maximum flow velocity, for every single particle track. Note, that the latter analysis provides a true measure of the stereo APTV uncertainty, since the experiments were conducted under realistic measurement conditions, i.e., same cameras, lasers, seeding particles, etc., except that the light pulses had a small separation in time.

To reduce the uncertainty of the third velocity component of the APTV technique with a minimum of cost and effort, a stereoscopic APTV technique was developed and qualified. The results of a comprehensive accuracy analysis show the excellent suitability of stereo APTV for 3D3C flow measurements in air. Stereo APTV directly maps the particle image’s sensor coordinates to physical space by means of third-order polynomials. Thus, in contrast to 3D PTV, a cumbersome modeling of the light path, to account for aberrations, different refractive indices of the media, and distortions due to windows, are not necessary. The third-order polynomials compensate for these influences. Unlike tomographic PIV, stereo APTV is not affected by spatial averaging, enabling the technique to resolve small-scale flow structures more accurately. Furthermore, less equipment is required, which reduces the costs and simplifies the experimental setup, the calibration, and the image processing. Stereo APTV is also less sensitive to vibrations and noise. Consequently, the technique is more robust and applicable in situations where other 3D3C methods fail.

The investigations were conducted as part of the joint research programme AG Turbo 2020 in the frame of AG Turbo. The work was supported by the Bundesministerium für Wirtschaft und Technologie (BMWi) as per resolution of the German Federal Parliament under grant number 03ET2013M. The authors gratefully acknowledge AG Turbo and MTU Aero Engines AG for their support and permission to publish this Letter. The responsibility for the content lies solely with its authors.

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