Stability of single image self-calibration for tomographic PIV
Stefano Discetti and Tommaso AstaritaDipartimento di Ingegneria Industriale (Sezione Aerospazio), Università degli Studi di Napoli Federico II, Naples, Italy
stefano.discetti@unina.it
ABSTRACT
It is commonly assumed that a larger number of cameras is beneficial for the accuracy of multiple cameras system. This tautological assertion is most certainly true for Tomographic PIV and 3D Particle-Tracking-Velocimetry systems, where the search area on each image for the particle corresponding to a 3D trial position is small (typically less than the particle image diameter). On the other hand, when it comes to larger search areas (due to, for example, calibration uncertainties), quite surprisingly increasing the number of cameras might have a detrimental effect. Under some conditions this is the case of the volumetric self-calibration technique [1], in which the residual calibration error and cameras misalignments are corrected by statistically searching matching particles over a search area larger than the expected maximum calibration error. In this work the loss of signal in the self-calibration for systems with 4 or more cameras is discussed. Two readily implementable solutions are provided to reduce this source of error. The algorithms are tested on synthetic and real images.
INTRODUCTION
Multiple cameras systems are usually employed to extract information in a 3D environment (see, for example, the computation of the velocity component orthogonal to the laser sheet in planar Stereo-PIV, or the full 3D measurement of the velocity vector in Tomographic PIV). In all cases, the reliability of the optical calibration procedure, i.e. the construction of the correspondence between the 3D world reference system and the 2D image reference system for each camera, is of crucial importance (see for further details [2-5]).
In Stereo-PIV the misalignment between the laser sheet and the calibration target is the most significant source of error. In Tomographic PIV this aspect is not relevant, provided that the volume swept by the target in the calibration process covers reasonably well the illuminated volume. On the other hand, Tomo-PIV suffers of uncertainties due to inaccurate calibration plates, unstable calibration shift mechanisms, loose connections in the cameras system, vibrations, optical distortions that are not accounted by the mapping function (this aspect is particularly critical when using the pinhole camera model), thermal deformations, etc. Indeed, Tomo-PIV requires that the mapping functions have uncertainties possibly below 0.4 pixels [6], which might be difficult to achieve in case of large volumes and significant aberrations (for example when imaging through optical windows). When the error is large, lines of sight relative to the same particle could not intersect within the volume, thus providing cancellation (or intensity subtraction) of true particles. In the very first applications the problem was tackled by smoothing with a Gaussian filter the original images, thus increasing the particles size and the probability of successful reconstruction in case of calibration errors. The immediate drawback is the reduction of the reconstruction quality due to the increased particle image diameter.
Subsequently, the development of a Volume Self-Calibration (VSC) technique [1] determined a significant leap forward in this sense. The VSC is based on the correction of the mapping functions using the actual particles. The technique consists in locating the particles on the camera images and finding the 3D position of matching particles through triangulation, as in Particle Tracking Velocimetry (PTV) [7]. The residual disparity obtained by computing the distance between the projection of each particle and the correspondent positions on the camera images are used to correct the mapping functions. The application of this method works quite easily in case of very sparsely populated images (as in PTV), while the extension to Tomo-PIV images (with image density up to 0.1particles/pixel) is not straightforward at all, and often it is not possible to record a set of images with low density before the experiment. In [1] a clustering technique is proposed, consisting in an artificial reduction of the image density by taking only the brightest particles. A procedure to identify the matching particles is outlined in [1]. For each particle on the first camera image, the corresponding candidates for the matching on the second camera are those located within a strip centred on the epipolar line (i.e. the projection of the line of sight of the first camera on the second camera) and half-width equal to the search radius (that has to be larger than the expected maximum calibration error). A first guess position is computed for each candidate, and then projected on the other cameras to find the corresponding particle, and, eventually, the 3D particle position by triangulation. Finally, the projection of the 3D particle position is compared with the respective image particle positions; the disparity vectors are displayed in a histogram map to separate true matchings from ghost
particles. The disparity vectors are computed for a set of sub-volumes composing the measurement volume, and then used to correct the original mapping functions. In general the procedure is iterative, and the sub-volumes are gradually reduced in size; indeed, as the calibration error is reduced at each step of the process, the search radius can be progressively set smaller, thus reducing the number of spurious matchings and, consequently, the noise on the histogram maps.
It is quite evident that the problem of spurious matchings, i.e. the ambiguity in identifying the correct particle image on the cameras for each true particle within the volume, is much more severe in self-calibration than in Tomographic PIV. As a matter of fact, the problem of the ghost particles [7] in Tomo-PIV and 3D PTV can be assimilated to that of the spurious matchings of the VSC in case of small search regions (typically of the same order of magnitude of the average particle image area). In case of larger search region more than one particle can occur in each search area, thus causing an increasing growth of the random matchings with the number of cameras. This aspect is particularly interesting and counter-intuitive, as it leads to the anomalous detrimental effect of increasing the number of cameras on the accuracy and reliability of the self-calibration technique.
In this work the performances of the VSC for systems with 4 or more cameras are critically reviewed; furthermore, two possible approaches are proposed to reduce the uncertainty in case of high number of cameras. The algorithms are tested with synthetic images and for an application with experimental images.
SELF CALIBRATION WITH MORE THAN 3 CAMERAS
Consider a generic Ncam camera system, observing a volume with a depth of Lz voxels. The image density is expressed
in particles per pixel (Nppp) and the search radius is equal to εr (set to be greater than the expected calibration error).
Following the previously described procedure, for each particle in each sub-volume (suppose, without any loss of generality, that the depth of the sub-volume is the same of the measurement volume) one has to find the corresponding match candidates on the second camera M2 in a rectangular strip with length equal to the projection of the line of sight
on the second image and width equal to twice the search radius. These particles can be bundled in three different types of spurious matchings:
“Far” matchings, i.e. particles located within the strip but not in close proximity of the true particle;
“Close” matchings, i.e. particle images within a circle centred on the true particle. It is important to note that some of these particles are actually located in close proximity of the true one within the volume; the remaining particles are located elsewhere along the depth direction but their lines of sight pass very close to the 3D location of the true particle. In the following these two categories are referred as “spatially close” and “orientation-limited close”.
The far matchings can be easily estimated by multiplying the image density for the area of the rectangle minus the area of the circle enclosing the true particle. The close matchings are in principle slightly more complicate to be modelled, as one should take into account the occurrence of overlapping particles. For each particle on the first camera, the number of matching particles on the second camera can be written as follows:
M2 =1+Nppp AF+NSC+NOLC (1)
The symbols AF, NSC and NOLC indicate respectively the effective image area for the far matchings, the spatially close
matchings and the orientation-limited close matchings. Evidently, at least one matching has to be included, i.e. the one relative to the true particle.
For each of the matchings of Eq. (1) one can identify a 3D trial position. These positions are projected on the other cameras of the system to find possible candidates for the matching with the true particles. The behaviour of the three categories of ghost particles is quite different:
The far matchings will determine a 3D trial position that is very likely to be located far from the true 3D position. The corresponding search area on the other cameras will almost certainly include only incorrect random matchings, whose number can be estimated by multiplying the product of the image density Nppp and
the search area π εr2 for the number of far matchings of the previous camera. In practice, for a Ncam system:
MN_FAR =Nppp AF (Nppp π εr2)Ncam-2 (2)
It follows immediately from Eq. (2) that the number of far matchings increases with the number of cameras if
Nppp π εr2>1. In this case, adding cameras certainly determines a divergence of the number of spurious
matchings with the number of cameras. However, the significant presence of overlapping particles complicates the scenario, thus Eq. (2) can be considered valid in the case of small source density [8]. Incidentally, the phenomenon of ghost particles generation can be included in this argumentation by supposing that a ghost particle is formed whenever within a search area as large as the particle image one can find a particle on all the cameras of the set, and at least one of them is the image of another particle. Evidently, in this case the far matchings can be considered ghost particles. The formula (2) is quite accurate for the ghost particles number estimation, provided that the source density is not too large.
The spatially close matchings are far more dangerous than the orientation-limited close, as they systematically occur on all the cameras (i.e. two particles with a distance smaller than the search radius will appear in close proximity on all the other images). Consider, for example, two particles with relative small spatial separation, whose particle images occur on all the cameras of the system with distance smaller than the search radius. It is evident that the number of possible spatially-close matchings scales with 2Ncam-1 (for example, for a 8 cameras system, a pair of spatially-close particles will determine 128 matchings, with only 1 correct matching). It has to be noted, however, that the scaling is quite difficult, as close particles are very likely to be overlapped on one or more cameras of the set. Fortunately enough, this situation is quite rare. The occurrence of these matchings can be estimated by evaluating the particle concentration within the volume (in case of resolution ratio within pixels and voxels, one can use C=Nppp/Lz), and multiplying it by the volume of a sphere with radius equal to the
search radius of the self-cal procedure. For example, in case of a search radius of 6 pixels, Nppp=0.01 and
Lz=500 voxels, the mean number of particles within the sphere is about 0.02, i.e. 2% of the particles would
potentially lead to spatially-close matchings.
The orientation-limited close matchings can be placed halfway between the previous two extreme cases. While the spatially close matchings are fully systematic, and the far matchings are completely random (i.e. in order to have a spurious matching one needs to find at least one random particle within the search area of each camera, similarly to the logic AND operator), in this case one spurious matching is obtained if one finds an additional particle within at least one of the search areas on the camera images (similarly to a logic OR operator).
In this work two approaches are proposed to reduce the number of spurious matchings and increase the signal/noise ratio of the disparity maps:
1. Since a large number of cameras leads to a propagation of matchings, one can limit the self-calibration procedure to a maximum of 4 cameras. For example, 3 cameras can be considered as a reference, and the self-calibration can be performed independently on the systems composed by the cameras 4, 5, 1-2-3-6, and so on. With this method the number of spurious matchings is limited to that of the case of 4 cameras, independently on the total number of cameras of the imaging system. This method will be referred in the following as 4cam limited self-calibration.
2. Considering that the search radius is a leading parameter in determining the number of spurious matchings, one can perform the self-calibration only on three cameras of the system and correct the bulk of the error; in a second step, the first three cameras are considered as a reference, and the self-calibration is performed adding in sequence the other cameras, but using different search radii (i.e. very small on the first three cameras, which are used to determine the 3D position of the particles, while for the other cameras one has to use the original search radius). This approach has the advantage of reducing the number of starting trial particles of the first 3 cameras before adding the other cameras. In principle, one can extend this procedure by progressively adding one camera at a time. In this work a simplified version, in which the small search radius is used only on the first three cameras is implemented. This procedure will be referred in the following as piloted self-calibration. VALIDATION ON SYNTHETIC IMAGES
The proposed algorithm is validated with a synthetic experiment. Gaussian spherical particles are distributed in a volume of 50 x 50 x 10 mm3. The imaging system is composed by 6 cameras, covering a solid angle of 60° along the horizontal and vertical direction. The magnification and pixel pitch are set in order to have a resolution of 20pixels/mm. The total number of particles distributed into the volume is 50000, corresponding to Nppp=0.05, which is a commonly
used value for Tomo-PIV experiments. The particle intensities have a Gaussian distribution, with mean value and standard deviation of 800 and 200 counts, respectively. With this approach one can simulate the ideal conditions of the single image self-calibration (or the case of experiments in which it is not possible to collect a set of images with low image density), in which the particles density fulfils the requirement of the desired spatial resolution of the experiments, while the effective density for the self-calibration can be reduced by picking only the brightest particles. In this simulation only the 5000 brightest particles are considered, thus virtually reducing the particle image density to
Nppp=0.01. The self-calibration procedure is applied with a search radius of 3, 6 and 9 pixels. In the case of the piloted
self-calibration the search radius on the first three cameras after the correction is of 1 pixel. In all cases the entire
volume is considered, similarly to the first step of the single-image self-calibration implementation [9].
The disparity maps for the case of the 6 cameras system and the three different methods are reported in Figure 1-2 (search radius of 6 and 9 pixels). The maps are obtained by summing on the map for each matching a parabola with maximum value of 1 and radius of 0.1pixels. The maps are discretized with 20pix/pix, and presented in the following in non-dimensional form with respect to the maximum value attained in the centre.
In order to assess the peak detectability, a signal/noise ratio is defined as the ratio between the two tallest peaks, as it is commonly done in PIV for the cross-correlation maps. This parameter is much more solid than the median test, as it can be used at all the stages of the self-calibration process, since it does not require a matrix of vectors for spatial coherence comparisons. For simplicity an overall signal/noise ratio is considered for each imaging system by taking the average
value of the ratio for each camera. The obtained data are plotted as a function of the number of cameras for the three values of the search radius in figure 3.
In the case of the standard self-calibration, the peaks are quite fragmented for a search radius of 6 pixels, but still quite distinguishable (figure 1, first row), with a signal/noise ratio of about 1.7. The 4cam-limited self-calibration (figure 1, second row) provides a significant improvement of the quality of the maps, with a signal/noise of 4.1. It has to be noted that in this case the first three disparity maps are characterized by a slightly better quality, as they benefit from adding all the other cameras of the set one at the time (thus reducing the divergence of the spurious matchings); on the other hand, the remaining cameras are corrected singularly. However, the quality of the disparity maps of the cameras 4-5-6 is significantly better than the case of the standard self-calibration. The signal/noise ratio of the piloted self-calibration is in average comparable to that of the 4cam limited procedure.
Increasing the search radius up to 9 pixels leads to a significant deterioration of the performances of the standard self-calibration procedure, with an abrupt fragmentation of the disparity peaks (the average signal/noise ratio is less than 1.1). The 4cam-limited algorithm obtains more distinguishable peaks on the first three cameras, while the disparity maps for the cameras 4-5-6 presents a much wider dispersion of the matchings contributions; however, the overall signal/noise ratio is 2.1. The piloted self-calibration benefits of the significant advantage of the small radius on the first three cameras, which significantly reduces the number of starting matchings before adding the other cameras to the system. The performance improvement is remarkable, as the obtained signal/noise ratio is 2.6.
Figure 1 Disparity maps for the case of search radius of 6 pixels: standard method; 4-cam limited algorithm; piloted self-calibration procedure.
Figure 2 Disparity maps for the case of search radius of 9 pixels: standard method; 4-cam limited algorithm; piloted self-calibration procedure.
Figure 3 Signal/noise ratio as a function of the number of cameras for a search radius of 3 (a), 6 (b) and 9 (c) pixels. standard self-cal 4cam-limited self-cal piloted self-cal.
In Figure 3 the signal/noise ratio is plotted as a function of the number of cameras for the 3 values of the search radius of 3, 6 and 9 pixels. For the smallest search radius, increasing the number of cameras is always beneficial since the ambiguity in determining the matching particles is quite small (the probability of occurrence of random particles within the search area is much lower, as discussed in the previous section). The piloted self-calibration has the slight advantage of a small search radius in the first three cameras, enabling a further reduction of the spurious contributions to the disparity maps. For the case of a search radius of 6 pixels, the performances of the standard self-calibration degrade with the number of cameras, as expected. The 4cam-limited method is characterized by a weakly increasing signal/noise ratio with the number of cameras, mainly due to the improvement of the quality of the disparity maps of the first three cameras by adding the other cameras of the set. On the other hand, the piloted self-calibration obtains a significant increase of the signal/noise ratio for the case of 4 cameras, while the advantage decreases by adding further cameras. This is due to the non-optimized algorithm, in which the maximum potential improvement can be obtained by adding one camera at the time and progressively reduce the search radius on the added cameras. Finally, for the case of a search radius equal to 9 pixels, the signal/noise ratio of the standard self-calibration is 1.1 for the six cameras system, while the proposed alternative algorithms can guarantee a signal/noise ratio larger than 2.
Figure 4 Schematic view of the illumination and imaging setup of the experiments
EXPERIMENTAL APPLICATION
The experiment is carried out in a water facility at the University of Naples Federico II. A double radial swirl injector (with exit diameter D equal to 40mm) designed by Avio S.p.A. is installed at the centre of the bottom wall of a nonagonal Plexiglas tank (allowing full optical access for both illumination and camera imaging) with circumscribed diameter of 16 D and height of 18 D. A schematic view of the experimental setup is shown in Fig. 4; more details are provided in [10].
The flow is seeded with neutrally buoyant polyamide particles with average diameter of 56µm. The particles are illuminated by a double-cavity Gemini PIV Nd:YAG system (light wavelength equal to 532nm, 200mJ/pulse@15Hz, 5ns pulse duration). The exit beam of about 5mm diameter is shaped into a parallelepiped volume using a three lenses system, i.e. a diverging and a converging spherical lens (with focal length equal to -75mm and 100mm, respectively), and a diverging cylindrical lens (with focal length equal to -50mm). A mask is placed along the laser path in order to set obtain an intensity profile similar to a top-hat, and to set the volume thickness to 46mm.
The imaging system is composed of four LaVision Imager sCMOS 5.5 megapixels cameras (2560 x 2160 pixels resolution, pixel pitch 6.5µm, 16bit intensity resolution). The cameras are equipped with 100mm EX objectives, set at
f#=16. Lens-tilt adapters are installed between the image plane and the lens plane to allow properly focused particle
images throughout the volume by achieving the Scheimpflug condition.
For the self-calibration experiment the particle concentration is set in order to have an average particle image density of 0.03 particles/pixels. In the experiment described by [10] the swirling injector is tested in condition and confined outflow. The latter configuration is obtained by confining the swirling jet with a Plexiglass cylinder with inner diameter equal to D. The calibration is performed without the presence of the Plexiglas hollow cylinder; this additional difficulty in the case of the confined jet is due to physical access restrictions. A two-levels target with white dots on dark background is imaged at 5 Z-locations covering the entire measurement domain (±23mm). The 3D mapping functions are modelled by fitting the correspondence between spatial and image coordinates with a 3rd order polynomial function in X and Y, and 2nd order in Z. The root mean square (rms) of the residual calibration error is equal to about 0.3 pixels. Despite of the very similar values of the refraction index of water and Plexiglas, the optical distortions can cause significant errors (up to about 3 pixels near the tube walls) since the viewing angle on the cylinder surface varies across the image. In this kind of application, a solid self-calibration procedure is of crucial importance.
The clustering method proposed in [1] is applied, taking only the brightest 10000 particles (resulting in approximately 0.0025particles/pixel). The first step of the self-calibration procedure (i.e. considering the entire volume) is performed with a search radius of 6 pixels. The disparity maps obtained by averaging over 20 images are reported in Fig. 5 for the three methods. The standard self-calibration provides maps with unsatisfactorily elongated and fragmented disparity peaks. The 4cam limited procedure, on the other hand, is capable of providing a quite reliable estimation of the disparity vector using only 20 images. Even better results are obtained by the piloted self-calibration algorithm, with strong disparity peaks on the first three cameras and only a slightly larger measurement noise on the fourth camera.
Figure 5 Disparity maps of the 4 cameras for: standard method (first row); 4-cam limited algorithm (second row); piloted self-calibration procedure (third row).
Figure 6 Signal/noise ratio as a function of the number of cameras: standard self-cal 4cam-limited self-cal piloted self-cal.
The signal/noise ratio obtained by varying the number of images is reported in Fig. 6. For the standard self-calibration procedure the quality of the disparity maps is quite poor even after 50 images; the 4cam limited algorithm, on the other hand provides a significant improvement, even though with a small rate of increase of the signal/noise ratio with the number of images. Much better results are obtained with the piloted self-calibration, which largely benefits by adding further images. Indeed in this case the main source of noise is due to the far matchings, that are completely random, and for this reason it is rapidly weakened by adding images.
CONCLUSIONS
The performances of the volumetric self-calibration procedure in case of imaging systems with 4 or more cameras have been reviewed. In case of large search radius, adding cameras can significantly affect the quality of the disparity maps due to the contribution generated by random and systematic spurious matchings. Two methods have been proposed to increase the signal/noise ratio of the measured disparity maps. The method refereed as 4cam limited is featured by a much better quality of the disparity maps of the first three cameras with respect to the other cameras, which are added one at a time in the procedure; however, the overall peak detectability is much better than the standard self-calibration. An even larger potential improvement can be obtained using the piloted self-calibration, in which the first three cameras are corrected in an initial step and then are used as a reference before introducing the other cameras in the procedure. The application of these methods is recommended in case of large number of cameras, especially in cases in which a set of images with purposely low image density cannot be captured, or when vibrations makes it necessary to implement a single-image self-calibration.
ACKNOWWLEDGEMENTS
The authors kindly acknowledge LaVision GmbH for providing the tomographic hardware the cameras used in the experimental test case. The research leading to these results is supported by the AFDAR project (Advanced Flow Diagnostics for Aeronautical Research) funded by the European Community’s Seventh Framework programme (FP7/2007-2013) under grant agreement no 265695.
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