19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

Volumetric Calibration Refinement using masked back projection andimage correlation superposition
Christoph Brücker1,*, David Hess2, Bo B. Watz2

1: Dept. of Aeronautical Engineering, City University London, Great Britain
2: Dantec Dynamics AS, Denmark* Correspondent author: Christoph.Bruecker@city.ac.uk
Keywords: PIV processing, Camera calibration, 3D-PIV, TOMO-PIV, Calibration, Calibration Refinement

ABSTRACTThis paper deals with a new, reconstruction based, approach of refining a volumetric calibration. The technique isbased on a 2D cross-correlation between particle images on the sensor plane with a planar back projection from atomographic reconstruction in the same sensor plane to determine potential disparities between the initial camera
calibration and the measurement. Additive superposition of the correlation maps from different sets or particle images
allows reducing the influence of noise and ghost particles such that the systematic errors in the calibration can becorrected. The different sections describe the theory, the principle processing steps and the convergence of theprocedure. Furthermore, the concept is proven by simulating the entire process of the measurement chain, with thehelp of a synthetic comparison. The results show that disparities of over 9 pixels could be corrected to an average of
below 0.1 pixels during the refinement steps. Finally, the technique demonstrates it´s potential to measured data,
where the numbers of outliers in the raw results are reduced after the volumetric calibration refinement.

1. Introduction
As a known fact, a proper camera calibration is key to a successful volumetric measurement.
Changes of the camera scene typically requires a new calibration, therefore an unintentional
movement of one or more cameras after initial calibration may lead to errors if the calibration
functions are not up-dated accordingly. On the other hand, the mapping functions between theworld and image coordinates (and the inverse of those) representing the mathematics of thecalibration process could be affected by errors in the target coordinates, modifications in the
optical properties along the viewing directions and errors in the image processing routines todetermine the mapping functions. Moreover, the approximation of the camera models such as
pinhole or lens distortion functions may not be ideal in all aspects for the actual recording
situation. The consequence of all of this a reduced accuracy of the measurement or even a complete19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

failure. This is especially important in voxel-based reconstructions of particle-filled volumes as thereconstruction quality largely depends on the calibration accuracy.
Consider a standard multi-camera arrangement for volumetric Particle Image Velocimetry suchas 3D Particle Tracking (3D-PTV) or Tomographic PIV (Tomo-PIV) see Fig. 1a. The minimum of
three cameras can view the scene in different planes or - for simplification - in the same plane but
from different angular displacements (Fig. 1b) or translational displacements. A particle in theworld coordinate system is then projected into the image plane according the mapping functions,describing the transfer from world to image coordinates. The mapping functions are typicallycalculated from calibration images of targets in the volume whose world coordinates are given apriori. For reconstruction of the particle field in the volume, one way is to calculate the inverse of
the mapping functions describing the lines of sights (LOS) for each pixel of each camera. It allows
then to calculate the tomographic MART- or SMART-type of voxel reconstructions, or in thesimplest way, to use the shape-of-silhouette method (SOS) in the voxel space. There, the particles
are reconstructed as clusters of voxels with the center of gravity crossed by all LOS originatingfrom the particle image centers in the image planes. Errors introduced in the calibration procedure
and/or changes in the camera positions between measurement situation and calibration may leadto unsuccessful reconstructions when using the original mapping function obtained in thecalibration. Consequently, disparities in the images planes need to be corrected in the calculationof the mapping functions if the LOS originating from image centers corresponding to the same
particle should again come to crossing of each other in the center of gravity in the reconstructedvolume.
In practice, this crossing is never accurate enough to hit exactly the center of gravity, even withperfect mapping. This, in addition to other issues depending on spatial resolution, number ofcameras, viewing angles and noise sources, leads to non-perfect reconstructions of the particles.
As a practical guide, Arroyo and Hinsch (2008) recommended a disparity below 0.4 pixels toensure that a typical tracer particle of 3px diameter in the image plane is still reconstructed as acompact cluster in the voxel space. In fact, these 0.4 pixel of uncertainty can already have asignificant influence on the 3D shape of reconstructed particles. Therefore, in Volumetric Particle
Image Velocimetry the calibration procedure must aim to get the disparity in all image planes
below a certain threshold. In 2008, Wieneke suggested to calculate disparity maps based oncomparison of triangulated particle coordinates and particle images in the image planes, which
can be used to correct the calibrations. The present work represents an alternative approach, which19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

uses iterative voxel-field reconstructions and back-projections to calculate the disparity maps.
Therefore, particle images are reconstructed using MinLos or SMART (see Atkinson and Soria2008) and small subsections of the volume are then back-projected to the image planes andcompared with the local image pattern in the original images. This is done stepwise selecting thesections along a 3D grid covering the box-size of the reconstructed volume. Disparities arecalculated from several realizations of particle recordings, which allow to remove the influence ofghost particles and noise from the final results.

2. Calibration Refinement Method
Assuming a situation of a three-camera set-up with angular displacement (cam#1 -45°, cam#2 0°,
cam#3 +45°) in one plane and a misalignment of the left-most camera in form of a shift in thecamera plane. The imaging system is such that all LOSs are straight lines and perpendicular to theimage plane (telecentric approximation). The original camera calibration induces then a shift of
the LOSs in direction of the red arrow (see Fig. 1b), equivalent to a virtual shift of the particle
image in this camera.
a)b)Fig. 1: Camera configuration for volumetric reconstruction; a) typical camera arrangement in different planes, b)camera arrangement in the horizontal plane with illustration of perfect calibration and with mismatch due to an
in-plane shift of the left camera. Blue filled rectangles represent the image planes. Blue solid lines show the LOS
for perfect calibration crossing in the true particle world coordinate in the center of gravity. Black squares show
the particle image projection in the image planes. An error introduced on the left cam#1 by an in-plane shift (redarrow) leads to a new center in the reconstruction (red circle) dislocated away from the original position.
Corrections for perfect crossing of the LOS affects all cameras seen by the disparity shift of the blue dashed lines.
Hence, the mismatch correction can be done either correcting only cam #1 or all cameras simultaneously.

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

The blue solid lines show the LOSs for perfect calibration crossing in the true particle world
coordinate system in the center of gravity. An error introduced on the left cam#1 by an in-planeshift (red arrow) leads to a new center in the reconstruction (red circle) dislocated away from theoriginal position. Corrections in form of disparity maps can either be applied only to cam#1 or to
all cameras together simultaneous (shown by the dashed blue lines). Note that both methods leadto perfect mapping, however the world coordinate system as a whole may translate relative to thecameras.
Particle representation in reconstructions with mismatch
The misalignments of the LOS lead to a change of the intensity distribution of the MART or
SMART reconstruction with a resulting shift of peak intensity location or - in SOS - a shift of thecenter of gravity, see Fig. 2. The fact used herein is that for Gaussian particle images and disparities
smaller than their diameter, the voxel reconstruction of the particle remains well represented by aGaussian blob of nearly spheroidal shape (ideally a sphere). This is demonstrated in a simulationof a synthetic Gaussian blob with 30vx diameter in a voxel volume, representing a situation of aten-fold super-resolution of a 3px diameter particle.
.
a)

b)

Fig. 2: Topview on the MART reconstruction of a simulated particle with artificially enlarged dP=30px diameter
and Gaussian intensity distribution, visualized via iso-lines of grey levels, using the camera configuration shownin Fig. 1b. a) zero mismatch, b) after disparity shift in the left camera cam#1. Note the shift of the intensity
maximum in the reconstruction away from the original center. The triangles indicate the LOSs starting at theparticle image centers for different pixel disparities of cam#1 (blue: 1dP, brown: 3/4dP, yellow: 1/2dP, red: 1/4dP).
The colored dots indicate the locations of maximum intensity for the different disparities.

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

The up-scaling is done herein to highlight the intensity distribution of the reconstruction. We
assume perfect linear mapping functions (straight LOS) and calculate the back-projections of theGaussian blob onto the image planes. Then, we shift the image in the left camera cam#1 stepwise
to introduce a calibration error and reconstruct the voxel volume again. The result given in Fig. 2
is shown in the horizontal center-plane of the reconstruction. In order to highlight the shape of theintensity distribution and the location of the peak, the images are shown as contours of constantintensity with constant incremental value.
In case of perfect calibration (zero mismatch), see Fig. 2a, the blob has a spherical structure (circularin the top-view) of Gaussian intensity distribution with the maximum very close to the geometriccrossing of the LOSs through the center of the particle images. Once the left camera cam#1 ismisaligned about a number of pixels to the right, this reconstruction deforms into an ellipticalshape with the intensity maximum shifted away from the original center, see Fig. 2b. Despite ashift in cam#1 of order of one particle diameter, the simulations show that the reconstruction still
provides a compact spheroidal contour with approximate Gaussian intensity distribution. This
fact allows us to use sub pixel analysis in the following correction step. The simulations fordifferent disparity shifts also show that the locations of maximum intensity in the reconstruction
remain always in the inner part of the triangle defined by the LOSs through the particle image
centers.
Back-projection and cross-correlationa)b)

Fig. 3: a) subdividing the voxel volume into a grid of smaller cuboids named interrogation volumes IV and b)
masking a spherical box around the center of the IVs (red circle in top view) prior to back-projection of the IV into
the image planes. This ensures that only those voxels inside the spherical volume of the IVs are contributing to
the intensity in the projections.

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

The reconstructed voxel volume is divided into different Interrogation Volumes (IV) and each IV
is projected back into the image plane of all cameras. Prior to projection into the image planes, allvoxels outside of the sphere with diameter of the IV are set to zero (red circle in Fig. 3b) so that the
back-projection in the image plane is built only from intensities of voxels inside the IV. The image
contains then a circular region of gray-level pixels, centered to the location of the IW-center
projected into the image plane. This is the template window IWbp, which stands for "interrogation
window in back-projected image of IW". A corresponding interrogation window IWim is selected
at the same location in the original images (IWim stands for interrogation window in original
image). The calibration mismatch is assumed as a smooth function in space leading toapproximately a linear translational shift locally in the IWbp relative to the IWim. This shift vector
can be calculated from 2D cross-correlation between both windows. Because of preservation of
gaussian intensity distribution in the reconstructions and back-projections even under larger
disparity as shown above, this shift can be determined with sub-pixel accuracy in the image plane.

Fig. 4: Addition of the correlation maps for the same IV and camera, obtained from different sets of particle
images. The resulting peak represents the disparity shift in the image plane for the given center-location of the IV
in the reconstructed voxel volume.

Note that the IWim in the image plane can contain particle images, which result from locations infront or behind the IV along the viewing direction of the camera, see Fig. 4. The probability
increases with increasing particle number density. This effect together with other contributions
such as ghost particles or image noise lead to the fact that the particle images in the IWim and theparticle images of the IWbp from the masked back-projection cannot match perfect. On the other
hand, the calibration mismatch introduces a systematic shift, independent on the statistical19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

properties of ghosts or noise. Therefore, we use several realization of particle images and store thecross-correlation maps for each IV. The corresponding correlation maps at each IV center are thenadded, see Fig. 4, and finally the peak in the sum of the correlation maps is calculated with subpixel
accuracy. This value is stored as disparity vector for each camera and each IV center location in
the voxel volume. The core of the procedure to correct the mismatch is summarized in thefollowing chart Fig. 5.

Fig. 5: Flow chart of the major part of the correction methodology

Correction steps and convergence
The analyzed disparities are then used to correct the initial calibrations by updating the mapping
functions. Synthetic particle fields are generated to test the convergence of the iterative procedure.
To simplify the simulation, a thick voxel sheet (depth 160 vx, width 225 vx, height 5vx) is generated
and filled randomly with Gaussian blobs of diameter 3vx to generate particle images with adensity of 0.02ppp. All centers of the particles lie in the horizontal mid plane at a height of 3vx.
The initial mapping function assumes straight parallel LOS normal to the image planes (telecentric
conditions) at magnification of 1 without any optical distortion. First, the LOSs are determined19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

and the corresponding images of the camera setup (Fig. 1b) are generated by back-projection. The
ground truth is then the MART reconstruction from these images (the ground truth therefore canalready contain ghost particles, if particle density is high). In a second step, a shift is introducedon cam#1 as indicated in Fig. 1b and a new voxel volume is reconstructed based on this mismatch.
The shift can be understood either as a misalignment of the camera after calibration or equivalently
as an error in the original mapping function of cam#1. To correct this, the volumetric calibration
refinement procedure is started by back-projection of the masked IVs and cross-correlation is donewith the IWbp and IWim in the original (zero-shifted) images. The same procedure can be
repeated with different random arrangements of particles in the voxel sheet such that 10
correlation maps exist for each IV, which then can be summed up. However, as the LOSs are
straight parallel lines and the shift is constant over the entire image of cam#1, the number of 10
IVs in one reconstruction allows to get the added correlation maps just from one single simulation.
The resulting peak-locations in the added correlation maps correspond to the disparities in the
different cameras. As seen in Fig. 1b, the corrections for perfect crossing of the LOS can affect allcameras illustrated by the disparity shift of the blue dashed lines in Fig. 1b. Hence, the mismatchcorrection can either be done correcting only cam#1 or all cameras together simultaneous. Fig. 6
shows the iterative correction procedure if only cam#1 with the largest disparity is corrected after
each step. The initial shift of cam#1 is 3px in positive direction of the long axis of the image, seeFig 1b. After the first step of correction, the disparities from the added correlation maps result tovalues of -1.1 for cam#1, +0.5px for cam#2 and 0 for cam#3. Hence, the largest correction is for
cam#1 to achieve a better match of the back-projections with the original images. Interestingly,
cam#3, which is looking perpendicular to the view of cam#1 where the error shift is introduced,
has a zero disparity. This is expected as the error shift vector is parallel to the viewing direction of
cam#3. Now, the initial error shift of 3px on cam#1 is correct with a disparity of -1.1px, while allother original image are not changed. Thus, we generate a new image of cam#1 with an error shiftof +1.9px (+3px-1.1px) in direction of the long axis compared to the original image. This isequivalent to updating the mapping function of cam#1 after the first iteration step.
Further correction steps are done in the same way until the final residual compared to the original
image is less than 0.1px. The resulting iteration steps with the correction values of all cameras are
given in Fig. 6. Note that the largest disparity remains in all steps for cam#1, which is the one
where the calibration mismatch was introduces in form of an error shift. The correction steps leadto a mismatch of cam#1 from the original shift of +3px in step 1 to +1.9px (+3px -1.1px), in step 2

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

to +1.1px (+3px -1.1px -0.8px), in step 3 to +0.5px (+3px -1.1px -0.8px -0.6px), in step 4 to +0.1px(+3px -1.1px -0.8px -0.6px -0.4px) and finally in step 5 to 0 px (+3px -1.1px -0.8px -0.6px - 0.4px 0.1px). That means the image of cam#1 with the imposed 3px error shift has been shifted back to
the original image after 5 iteration steps and the mapping functions now correspond to a perfectcalibration.

Fig. 6: Iterative reduction of residual disparity shifts in all three cameras after inducing an initial 3px mismatch incam#1 in positive direction along the long axis of the camera, see Fig 1b (blue: cam#1, orange: cam#2, gray: cam#3).

Fig. 7 shows the reconstructions of particle fields starting from a +3px mismatch along the longaxis of cam#1 before and after the final correction step (top-view of the MART volume) of the error
shift indicated in Fig 1b. Due to the mismatch of cam#1 representing a systematic error in the
mapping function of cam#1, the reconstruction shows a larger number of ghosts near original
particle images. Note that the original images were blurred with a 3x3 Gaussian kernel prior toreconstruction to highlight this effect. After the final correction step, all original particles arereconstructed in their true shape and location and no ghosts are seen.
The method has been further tested for synthetic images including noise and has been provenrobust and practicable. For most cases, a total number of 5 iteration steps is sufficient to reduce
the residual below 0.1px. Note that the number of iterations depends on the number density ofparticles in the calibration images and the number of IV in the total volume. The latter needs alsoa sufficient grid resolution so that the determination of improved transfer functions can cope withcamera rotation, translation and optical distortions. A practical number is about 5x5x3 IV in a
typical volume. Though, it is recommended to keep the number of IV constant with the aspectratio of the volume.

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018a)b)

Fig. 7: Top view of MART reconstruction of particle field in the three-cam arrangement shown in Fig. 1. a)reconstruction with initial error in calibration with 3px error shift in cam#1, showing larger number of ghosts. b)reconstruction after final iteration step in Fig. 6.

3. Synthetic data analysis
In order to judge about the quality of this refinement method under more realistic conditions, acomplete experiment with optical transfer functions including noise and lens distortion issimulated. In the first step, the initial calibrations of the original camera positions are defined andused for the calculations of the LOSs and back-projection of randomly distributed particles in thevoxel volume. In the second step we simulate a sparse set of particle images, where the camera
positions are modified and the images are distorted. These images are used for calibration
refinement. Afterwards, the initial and the refined calibrations are used to reconstruct voxel spaces
for dense particle fields at different time-steps of a given stationary flow field. In a final step, thevelocities of the different reconstructions can be analyzed by 3D Least Squares Matching (3D LSM,
see Westfeld et al 2010). Thus the resulting vector-maps of the refined and non-refined calibrations
can be compared and improvements can be quantified.
Four synthetic cameras (800x500pixels) are arranged in a cross-type configuration, observing asegment of a Hill-type ring vortex in a volume of 65x45x15 mm. This set-up also represents theinitial calibration. The volume is reconstructed in isotropic voxels of 0.1mm length, totaling in
650x450x150 vx. To introduce an error, the cameras are moved away from their theoretical initial
position in the following way: the top and right cameras are moved up by 0.5 mm; similarly, the19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

bottom and the left cameras are shifted 0.5 mm to the right. With a magnification factor of 1 pix =1 vox = 0.1 mm, each camera is moved with a shift of 5 pixel. Additionally, distortions are added
to the images by the radial parameters of the pinhole camera model. This distortion introduces a
maximum shift of additional 1.5 pixel, summing the maximum potential calibration error to morethan 9 pixel.
For the calibration refinement process, a set of particle images with moderate particle density issimulated. To better represent an experimental environment, noise is added to the images. The
maximum particle gray values are in the range of 35 counts to represent a low light situation, theadded noise level is 6 counts with a standard deviation of 2 Sigma. During image pre-processing
steps, the noise is reduced by a local average subtraction, a threshold and final Gaussiansmoothing with a 3x3 kernel. For processing the refined calibration, a set of 10 images is used. The
number of cuboidal IVs for the correlations is set to 10x6x3 with each IV of the size of a cuboid of
80vx length. This results in one disparity vector every 63rd voxel in x- and y axis and every 35th
voxel in z-direction, respectively. Note that for the z-direction the IVs are overlapping each other.

2.5

7.0

2

6.0
Disparity [pix]Disparity [pix]For the refinement, all cameras were corrected at the same time in each iteration.

1.5
1
0.5
0

5.0
4.0
3.0
2.0
1.0

0

5

Cam 1

# of iteration
Cam 2
Cam 3

10
Cam 4

Fig. 8: Average disparity error over the number ofrefinement iterations

0.0

0

Cam 1

5
# of iteration
Cam 2
Cam 3

10
Cam 4

Fig. 9: Maximum dispartiy error over the number ofrefinement iterations

The results of the refinement shows that within 5 iterations the average projection error drops
from 1.88 pixel to 0.1 pixel and the maximum disparity is reduced from 6.4 to 0.5 pixel (see also
Fig. 8 and Fig. 9). After 10 iterations, the maximum disparity is about 0.36 pixel and the average
disparities are 0.057 pixel. This is by far less than the required 0.4 pixel of uncertainty claimed by
Arroyo and Hinsch in 2008 and in the same range as described by Wieneke in 2008.

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

In the final step, the refined calibrations are used for volume reconstruction and are comparedwith the initial calibration as well as with a simulated volume of the known particle positions.

Fig. 10: Z-axis intensities of the reconstructed voxel spaces indicates the large improvement in ghost level intensities
between the reconstructions with the outdated initial- (red line) and the reconstruction using the refined calibration(yellow line).

Fig. 10 shows the intensity profiles along the z-axis of the voxel space. The normalized signal levelin the illuminated area -7.5 < z < +7.5 mm is about 0.87, and the average normalized noise level isabout 0.65. This results to an average signal to noise ratio (SNR) of 1.3. For the enhanced
calibration, the values are about 0.85 for the signal level and about 0.085 for the noise level,
resulting in an SNR of about 10. Hence, the reconstruction using the volumetric calibration offers
more than 7 times higher signal quality. The benefit of lower ghost level intensities afterrefinement has been discussed, for instance, by Novara et al (2010) and de Silva et al. (2012). A
side effect of the refinement is a shift of the z-location of the measurement domain as discussed
above and also discussed by Cornic et al. (2015), which makes a voxel-wise comparison difficult.
The direct impact of the enhanced calibration on the 3D velocity measurements is shown byprocessing the displacements in three successive particle volumes in time. The method of 3D Least
Squares Matching (3D LSM) was applied on the synthetic voxel space as well as the tworeconstructed ones. Fig. 12 shows the results. Here, the yellow-colored vectors are from the voxel
spaces that used the refined calibration and show a clear improvement over the vector quality
against the red-colored vectors from the analysis with the initial calibration. Furthermore, the19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

yellow-colored vectors are in good agreement with the velocities obtained from the simulated
voxel space (green-colored vectors).

Fig. 11: comparison of z-axis intensity profiles in the 3
different voxel-spaces. Red: Voxel space from initial
calibration. Green: Voxel space from simulated voxel
space. Yellow: Voxel space from refined calibration.

Fig. 12: Simulated comparison vector results
demonstrate the improvement Volumetric
Calibration Refinement. Red: Velocities fromoutdated initial calibration. Green: Velocities fromsimulated voxel space. Yellow: Velocities fromrefined calibration.

Though, the green and the yellow vector fields look similar in both direction and magnitude, as
seen with other refinement methods (Cornic et al. 2015), the coordinate system of the refinedcalibration gets slightly shifted away from the original during the refinement step. This makes adirect comparison of the vectors from the refined calibration and the simulated ones difficult.

4. Experimental data analysis
For the experimental case, a transitional water jet-flow is used. The jet emits from a nozzle withdiameter (D) of 12.36 mm at a flow rate of 100 l/h, resulting in an average flow velocity at theorifice of approximately 0.232 m/s. With the given velocity and viscosity, the Reynolds number isapproximately 3000. The flow is measured by a set of 4 Speed-Sense M 310 (1280x800 pix) cameras
in an inline configuration up to 4000 individual images that were illuminated by a DualPower 301000 Laser and recorded with a repetition rate between 800 and 1200 Hz.
A volume of 27 x 65 x 27 mm is reconstructed with the initial calibration and with a refinedcalibration. The result of the z-axis intensities can be found in Fig. 13 . An improvement of thesignal to noise ratio can be observed again, though it is not as pronounced as in the synthetic
simulated data fields. Comparing the vectors in Fig. 14 also shows a reduced number of outliers19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

between the red-colored ones where the reconstruction was based on the initial calibration,compared to the reconstruction with the refined calibration (yellow-colored vectors).

Fig. 13: Comparison of average z-plane intensity of thejet flow experiment. Red - with initial calibration,
yellow - with VCR.

Fig. 14: Comparison of vectors with the initial
calibration (red) and with VCR (yellow).

5. Summary
This study presents a new approach to enhance a 3D camera calibration based on particle images,
which is based on a statistical approach of reconstructed particles in sub volumes (Interrogation
Volumes IV) and their masked back-projected images in the image planes. The initial total volume,
represented as a voxel-based grid, is generated from the parameters of an initial classical multicamera calibration. This volume is subdivided into smaller cuboids for each of which a backprojection into the camera planes is calculated. Voxels outside of the IV are blanked out and theresulting back-projections are compared with the original images around the location of the centerof the IV projected back into the image plane. Cross-correlations between both image regions(back-projected IWbp and original IWim) and successive adding of the correlation maps ofdifferent realizations of calibration experiments then leads to accumulated peaks in the maps
which represent the average disparity, assigned to the centers of the IVs. The calibration
parameters are then corrected in an iterative procedure improving the transfer functions between
image- and world coordinates. The process converges after 5-10 iteration steps down to residual
disparities less than 0.1 px for all cameras, based on simulations with synthetic images. The new
method is robust and practical in its use as it does not require any user input and can be
automatized with a given set of design rules.

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

The study of the entire process from calibration over the volumetric reconstruction to the final
vector analysis using 3D LSM routines shows a great potential of the new refinement method. In
the synthetic case, a disparity of more than 9 pixels in length was corrected to an average disparity
error in the range of 0.05 pixel, despite the simulated added noise of the simulated particle images.
When comparing the reconstructed voxel volumes, a clear advantage can be seen. The z-intensity
profiles of the different volumes show that the SNR improved over 7 times when the refinedcalibrations are used instead of the initial calibrations. Hence, the reconstruction quality is
drastically improved. Comparing the 3D-LSM results of the volumes from the initial calibrationand the refined calibrations, a clear reduction in erroneous vectors is seen. Furthermore,comparing the improved results with the 3D-LSM results obtained for ideal simulated voxel
spaces, a good agreement of the vectors is obvious. Though, similar as with other refinement
methods, the coordinate systems gets slightly shifted making it difficult to directly compare thevector results.
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