# Volumetric Calibration Refinement of a Multi-Camera System Based on Tomographic Reconstruction of Particle Images

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}= −45°, α

_{2}= 0°, α

_{3}= +45°) (see also the data given in Table A1 in Appendix A). The projection of a particle from the world coordinate system into the image plane is described mathematically for each camera via its mapping functions—see the equations given in Appendix B. The coefficients of the mapping functions are typically computed from calibration images of known objects whose world coordinates are predetermined in the volume. Together with the inverse of the mapping functions, which is the calculation of the LOSs for each pixel in each camera, the computation of tomographic reconstructions using MART or SMART algorithms are carried out [14] (see Appendix B). The particles are reconstructed as contiguous clusters of voxels, ideally; the center of gravity is intersected by all the LOSs that originate from the particle image centers in the image planes. Since the particle images are finite in size, the reconstruction is always a spheroidal composite of cohesive voxels representing the particle. However, if larger deviations of the mapping functions from the actual optical conditions occur, the voxel cluster is deformed and the position of maximum intensity is dislocated from the error-free situation. The proposed method aims to correct the disparities in the image planes (and therefore the mapping functions) such that the LOSs intersect again and the spherical character of the particles is fully restored.

## 2. Methodology of Calibration Refinement

#### 2.1. The Shape of a Particle Reconstruction with Camera Mismatch

_{vx},Y

_{vx},Z

_{vx}) with a distance $r\le D/2$ to the center of the blob at (X

_{0},Y

_{0},Z

_{0}) is calculated according to the following equation:

#### 2.2. Cross-Correlation of Original Images with Back-Projected Images

#### 2.3. Ensemble Averaging of Correlation Maps from Snapshots

#### 2.4. Correction Steps

#### 2.5. Typical Iterative Correction Performance

#### 2.6. Treating Larger Mismatch with Image Pre-Processing Using Gaussian Blur

## 3. Numerical Assessment

^{3}with a simulated camera resolution of 800 × 500 pixels on a 2/3 inch sensor at a magnification of 1:5. The length of one voxel equals 0.1 mm in physical space. The synthetic volume is filled randomly with Gaussian blobs of a given diameter D (${e}^{-2}$ diameter) and number density to achieve a specified particle image diameter dP and ppp density. The initial mapping functions are linear functions in the X,Y, and Z-directions, simulating the situation of an ideal pinhole imaging system with no distortion (generated with the Soloff polynomials and the LOSs are calculated therefrom). The particle image diameter dP is directly proportional to D and is varied in this study by choosing different blob diameters. Initial images are generated from these voxel volumes by back-projection. Again, white noise is added to the images and thereafter image pre-processing is done (local salt and pepper filter and Gaussian smoothing with a 3 × 3 kernel). A tomographic MART procedure (10 iterations and a dampening factor of 1) re-computes the voxel volume from these images. These volumes are then the ground truth. All cameras are corrected simultaneously and the norm of all disparity vectors of all IV and all cameras is taken as an indicator of the overall performance of the system.

#### 3.1. Influence of Particle Image Diameter and Seeding Density

#### 3.2. Influence of Errors on All Cameras

#### 3.3. Performance Tests with Synthetic Velocity Data

_{0}of the vortex ring is adjusted such that the particle shift between successive timesteps corresponds to a displacement of 5 vx. The simulated flow is transferred into the observed-fixed reference system where the vortex is traveling from bottom to top with a velocity of U

_{0}and the outer velocity at infinity is zero. The synthetic voxel fields are filled with a higher seeding density, leading to typical particle image densities of about 0.045 ppp in the images. The initial and the refined calibrations described in Section 3.2 are used to reconstruct the voxel spaces back from the generated images. In a final step, pairs of voxel spaces are analyzed by 3D least squares matching (3D LSM, see Westfeld et al. [21], Maas et al. [22]) to generate the velocity vector maps of the Hill-type vortex (LSM conditions: cuboids of 30 × 30 × 30 voxel elements with a 50% overlap in all directions) (see Figure 8). The final result is a vector field of 51 × 31 × 19 vectors in the volume, transformed back into the right-handed Cartesian coordinate system (X,Y,Z) with the components (U,W,W). For VCR, the iso-surfaces of constant Q-criteria, a vortex detection method (see Hunt et al. [23]), shows a clear reconstruction of the vortex torus, which largely improved the results with the initial calibration (with the artificial camera mismatch). Figure 8b highlights the differences between the initial and refined calibration. The mismatch mainly affects the regions of higher velocity, which could not be reconstructed in the inner of the sphere.

## 4. Performance Tests with Experimental Data

_{0}= 0.23 m/s, resulting in a jet Reynolds-number of approximately 2830. Under these conditions, the jet was in the transitional regime, where large-scale vortical structures still dominate the flow field in 3–4 jet diameters down of the orifice. The flow was measured by a set of four Speed-Sense M 310 (1280 × 800 px) cameras in an in-line configuration (see Table A3 in Appendix A). A DualPower 30–1000 Laser was used to illuminate a 10 mm thick light volume located along the middle of the tank in the vertical direction of the jet. This camera configuration yielded a resolution of approximately 15 px/mm. As tracer particles, hollow glass spheres (S-HGS-10 particles, Ø = 10 µm, Dantec Dynamics) were added to the water. In the following, we illustrate the improvement against the “standard” calibration often used in stereo PIV and Tomo-PIV is the so-called Soloff calibration method [1], where targets with known coordinates in 3D space are recorded. Herein the initial calibration was done with a dotted calibration target moving along the Z-axis and taking images at different positions. For the refinement step, it is possible to use either (a) extra particle recordings at lower particle density or (b) the original recordings for 3D PTV or Tomo-PIV with high particle density after filtering the images such that only the brightest particle images in all camera views remain. Herein, for refinement a set of five snapshots was taken with low particle density of 0.001 ppp. Then, the particle image density was increased to about 0.045 ppp and up to 4000 individual images were recorded with a repetition rate of 1 kHz for the Tomo-PIV processing. The raw images were processed by subtracting the background, performing a sliding minimum thresholding, and smoothing with a 3 × 3 Gaussian kernel.

^{3}was reconstructed with the initial calibration and after applying the VCR method described herein (five snapshots, IVs have a size of 50 × 50 ×50 vx with a mesh resolution of 3 × 6 × 3 positions). Velocity field processing was done using the 3D least square matching method with cuboids of 30 × 30 × 30 vx voxel elements with a 75% overlap in all directions. The result is a vector field of 30 × 100 × 30 vectors in the volume. Figure 11 shows an instantaneous snapshot of the velocity field. The vectors in the center slice show a nearly complete removal of outliers after VCR and a smooth field. In addition, the reconstructions of the surfaces of constant Q-value show less spotty appearance of smaller isosurfaces and stronger coherence of the structures, which agrees with the observations made for the simulated Hill-type vortex.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Cam #1 | Cam #2 | Cam #3 | Cam #4 | |
---|---|---|---|---|

Camera size full width × height (px) | 250 × 250 | 250 × 250 | 250 × 250 | |

Camera alpha α | α1 = −45° | α2 = 0° | α3 = +45° | |

Camera beta β | β1 = 0° | β2 = 0° | β3 = 0° | |

Camera pixel size (µm) | 10 | 10 | 10 | |

Magnification (mm/vx) | 1/10 | 1/10 | 1/10 | |

pixel to voxel ratio | 1 | 1 | 1 | |

Lens configuration | Ideal telecentric | Ideal telecentric | Ideal telecentric | |

Initial mapping function | Ideal rectilinear | Ideal rectilinear | Ideal rectilinear | |

Initial camera error translational | Δpx = +3 | - | - | |

Initial camera error rotational | Δγ1 = 0° | - | - | |

Refined mapping and LOS equation | 3rd order X,Y 2nd order Z | 3rd order X,Y 2nd order Z | 3rd order X,Y 2nd order Z |

Cam #1 | Cam #2 | Cam #3 | Cam #4 | |
---|---|---|---|---|

Camera size full width × height (px) | 800 × 500 | 800 × 500 | 800 × 500 | 800 × 500 |

Camera alpha α | α1 = 0° | α2 = +45° | α3 = 0° | α4 = −45° |

Camera beta β | β1 = −45° | β2 = 0° | β3 = +45° | Β4 = 0° |

Camera pixel size (µm) | 10 | 10 | 10 | |

Magnification (mm/vx) | 1/10 | 1/10 | 1/10 | 1/10 |

pixel to voxel ratio | 1 | 1 | 1 | 1 |

Lens configuration | Ideal pinhole | Ideal pinhole | Ideal pinhole | Ideal pinhole |

Initial mapping function | Ideal rectilinear | Ideal rectilinear | Ideal rectilinear | Ideal rectilinear |

Initial camera error translational (px) | Δpy = +5 | Δpy = +5 | Δpx = −3 | Δpx = −3 |

Initial camera error rotational | Δγ1 = 0° | - | - | - |

Refined mapping and LOS equation | 3rd order X,Y 2nd order Z | 3rd order X,Y 2nd order Z | 3rd order X,Y 2nd order Z | 3rd order X,Y 2nd order Z |

Cam #1 | Cam #2 | Cam #3 | Cam #4 | |
---|---|---|---|---|

Camera size full width × height (px) | 1280 × 800 | 1280 × 800 | 1280 × 800 | 1280 × 800 |

Camera alpha α | α1 = −33.75° | α2 = −11.25° | α3 = 11.25° | α4 = 33.75° |

Camera beta β | β1 = 0° | β2 = 0° | β3 = 0° | β4 = 0° |

Camera pixel size (µm) | 20 | 20 | 20 | 20 |

Magnification (mm/vx) | 1/10 | 1/10 | 1/10 | 1/10 |

pixel to voxel ratio | 1:1.1 | 1:1.1 | 1:1.1 | 1:1.1 |

Lens configuration | Zeiss 100 mm | Zeiss 100 mm | Zeiss 100 mm | Zeiss 100mm |

Initial mapping function | Soloff | Soloff | Soloff | Soloff |

Initial camera error translational | unknown | unknown | unknown | unknown |

Initial camera error rotational | unknown | unknown | unknown | unknown |

Refined mapping and LOS equation | 3rd order X,Y 2nd order Z | 3rd order X,Y 2nd order Z | 3rd order X,Y 2nd order Z | 3rd order X,Y 2nd order Z |

## Appendix B

**Figure A1.**Illustration of mapping functions with non-linear contribution due to spatially varying disparity, illustrated by the inverse of mapping, which is the LOS progressing through the volume in a curved path.

## Appendix C

_{0}and the outer velocity at infinity is zero (subtracting −U

_{0}from the Equations (A8) and (A10)).

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**Figure 1.**Camera configuration for volumetric reconstruction of particles; (

**a**) typical camera arrangement in the horizontal X-Z-plane with different angular views, (

**b**) top view of the horizontal plane with illustration of perfect calibration and with mismatch due to an in-plane shift of the left camera. Gray solid lines show the LOSs for perfect initial calibration, which cross in the true particle world coordinate in the center of gravity. Black rectangles show the particle image projection in the image planes. An error introduced on the left cam #1 by an in-plane shift (red arrow) 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 dashed lines. Hence, the mismatch correction can either be done by correcting only the left camera or all cameras simultaneously.

**Figure 2.**(

**a**): Topview on the MART reconstruction of a simulated spherical Gaussian blob with a diameter of 30 vx, visualized via iso-lines of grey levels, using the camera configuration shown in Figure 1b at zero mismatch. (

**b**): Topview after applying a shift mismatch ∆px1 = ¾ dP to cam #1. The triangles indicate the LOSs starting at the particle image centers for different pixel shifts ∆px of cam#1 (blue: 1 dP, brown: ¾ dP, yellow: ½ dP, red: ¼ dP). The colored dots indicate the locations of maximum intensity for the different disparities.

**Figure 3.**Subdividing the voxel volume into a grid of smaller cuboids named interrogation volumes IV (

**a**). Particles behind or in front of the interrogation volumes as shown in (

**b**) influence the cross-correlation in the image plane after local IV back-projection.

**Figure 4.**Ensemble averaging of the correlation maps for the same IV and camera, obtained from different sets of particle calibration images. Adding the individual correlation maps to improve the peak elevation against the noise (

**a**). The resulting peak location in (

**b**) after each addition step i shows the convergence to the true correction value (total number of calibration experiments is 18, particle density 0.005 ppp, final correction −0.8 px in the x-direction, 0 px in the y-direction).

**Figure 5.**Disparity shifts in all three cameras after inducing an initial +3 px shift in cam #1 in positive direction along the Y-axis of the camera (see Figure 1b) (particle density 0.001 ppp). (

**a**) correction done only for cam #1, (

**b**): correction done simultaneously for all cameras in each step.

**Figure 6.**Top view (X-Z plane) of MART reconstruction of particle field in the three-camera arrangement. (

**a**) reconstruction with initial error in calibration (+3 px error-shift in cam #1), (

**b**) reconstruction after final iteration step (correction only done for cam 1, Z-axis inverted, particle density in x-y-image plane 0.001 ppp).

**Figure 7.**Variations of different parameters to assess performance and ideal parameters to run the calibration refinement (

**a**) particle diameter (

**b**) particle density (

**c**) number of images (

**d**) interrogation volume size. If not varied, particle diameter dP = 2 px, particle density = 0.001 ppp, number of snapshots = 5, IV

_{Size}= 50 × 50 × 50 vx. The disparity shown here is the norm of all disparities for all IV and all cameras.

**Figure 8.**Comparison of the 3D LSM results for error-free calibration (rendered), initial calibration with artificial camera mismatch and after VCR of the initial calibration. (

**a**) isosurfaces of Q-Criteria showing the torus of the vortex ring, with color-coded velocity magnitude in the center-plane at Z = 0. (

**b**) velocity vector field in the X-Y plane crossing the center of the vortex ring (Z = 0).

**Figure 9.**Histogram of the differences in U, V and W components to the analysis of the rendered volume (error-free calibration). Results are shown as voxel shift from one to the next time-step (for the characteristic velocity U

_{0}of the vortex ring, the shift is 5 vx).

**Figure 10.**(

**a**) Front view picture of the 4-camera setup recording the vertical jet in the water tank, (

**b**) top view of the tank with polygonal cross-section to ensure that the surfaces are parallel to the image planes to avoid further optical distortion. The jet axis is at the center in vertical direction into the paper plane, which corresponds to the long axes of the cameras.

**Figure 11.**Comparison of the 3D LSM results for initial “standard” calibration and after VCR. (

**a**) velocity vector field in the X-Y plane crossing the center of the jet (Z = 0). (

**b**) isosurfaces of constant Q-value (Q = 1000) showing a row of two successive vortex rings in the jet.

**Table 1.**Parameter study of performance sensitivity to particle density, angular displacement and noise level for a three-camera arrangement in the same plane as defined in Table A1 and Appendix A.

Particle Density ppp | Angular Displacement of Cam #1 and Cam #3 | Noise Level Variance | Initial Correction Step of Cam 1 (from +3 px Error-Shift) | Minimum Number of Ensemble Additions to Achieve Peak Position within 0.1 px Radius | Iterations to Correct Cam #1 Down to 0.1 px Offset |
---|---|---|---|---|---|

0.001 | ±45° | 0 | −1.1 | 4 | 5 |

0.003 | ±45° | 0 | −1.0 | 6 | 7 |

0.005 | ±45° | 0 | −0.8 | 10 | 9 |

0.008 | ±45° | 0 | −0.4 | 20 | 13 |

0.010 | ±45° | 0 | −0.3 | 22 | - |

0.001 | ±22° | 0 | −0.6 | 4 | 10 |

0.001 | ±45° | 0.00 | −1.1 | 3 | 5 |

0.001 | ±45° | 0.01 | −0.6 | 6 | 7 |

0.001 | ±45° | 0.02 | −0.5 | 10 | 10 |

0.001 | ±45° | 0.03 | −0.6 | 20 | - |

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**MDPI and ACS Style**

Bruecker, C.; Hess, D.; Watz, B.
Volumetric Calibration Refinement of a Multi-Camera System Based on Tomographic Reconstruction of Particle Images. *Optics* **2020**, *1*, 114-135.
https://doi.org/10.3390/opt1010009

**AMA Style**

Bruecker C, Hess D, Watz B.
Volumetric Calibration Refinement of a Multi-Camera System Based on Tomographic Reconstruction of Particle Images. *Optics*. 2020; 1(1):114-135.
https://doi.org/10.3390/opt1010009

**Chicago/Turabian Style**

Bruecker, Christoph, David Hess, and Bo Watz.
2020. "Volumetric Calibration Refinement of a Multi-Camera System Based on Tomographic Reconstruction of Particle Images" *Optics* 1, no. 1: 114-135.
https://doi.org/10.3390/opt1010009