# 3D Reconstruction of a Single Bubble in Transparent Media Using Three Orthographic Digital Images
## 2 Experimental Apparatus and Visualization
### Measurement Accuracy  
All steps of visualization produced uncertainties. First, high precision monox calibration chessboard was used to improve the measurement accuracy, the accuracy reached \(\pm\)0.01 mm. Then, in order to eliminate the error produced by limited calibration pictures, this paper considers 30 different calibration postures as plentiful enough. We can easily know that one pixel created an measurement error of 0.0048 mm. Thus, the errors of hole filling as well as edge extraction could be measured and each of them exerted a negligible 0.1% on the measurement bubble length, as background division and binarization. Further, both front and side cameras induced uncertainty of local length through camera calibration process, which was [37.48217, 40.70242] and [56.57221, 64.13330] in pixels. Hence, the calibration errors in \(x\) and \(y\) pixel axes cold be calculated, respectively. In this way, the uncertainty on the measurement of bubble diameter was accumulated. According to the propagation law of errors:  
\[m_{d}=\sqrt{\text{mean}\left[\sum_{i,j=1,2}\left(\left(\frac{1}{f_{ij}\cdot px }\right)^{2}\cdot m_{D}^{2}+\left(-\frac{D_{j}}{f_{ij}^{2}\cdot px^{2}} \right)^{2}\cdot m_{fij}^{2}+\left(\frac{D_{j}}{f_{ij}\cdot px}\right)^{2} \cdot m_{c}^{2}\right)\right]} \tag{1}\]  
where, \(m_{d}\) is the objective mean square error of bubble diameter, \(D_{j}\) is the distance between the cameras and the center of the box, and was measured by standard ruler. Therefore, the mean square error  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Calibration Parameters** & **Calibration Results** \\ \hline \([f_{11}\),\(f_{21}]\) & [2618.40929 2663.82735] \\ \(\gamma\) & [0.00000] \\ \(k_{c}\) & [0.33266 \(-\)0.52175 0.07873 0.08099 0.00000] \\ \hline \hline \end{tabular}
\end{table}
Table 1: The calibration parameters of front camera (pixel).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Calibration Parameters** & **Calibration Results** \\ \hline \([f_{12}\),\(f_{22}]\) & [2702.94005 2677.29985] \\ \(\gamma^{\prime}\) & [0.00000] \\ \(k_{c}\) & [0.13499 \(-\)0.35469 0.00509 \(-\)0.07338 0.0000] \\ \hline \hline \end{tabular}
\end{table}
Table 2: The calibration parameters of side camera (pixel).  
\(m_{D}\) was \(\pm 1\) mm on the basis of ruler precision, \(m_{f}\) is the calibration mean square errors, \(px\) is the length of a pixel, \(m_{c}\) is the precision of calibration chessboard. Thus, after all the above operations, the measurement uncertainty on bubble diameter during camera calibration and image processing was estimated to be \(\pm 0.46\) mm. The mean square error of spherical bubble volume can be deduced roughly as following:  
\[m_{V}=\frac{\pi}{2}d^{2}\big{|}m_{d}\big{|} \tag{2}\]  
where, \(m_{V}\) is the objective mean square error of bubble volume, of which the measurement is the uncertainty evolution of bubble geometrical characteristics. Then the relative mean square error (\(K\)) could be written as:  
\[K=\big{|}m_{V}\big{|}/V_{b} \tag{3}\]  
where, \(V_{b}\) is the volume of spherical bubbles. It can be seen in Figure 3 that \(K\) was the decreasing function of bubble diameter. With the bubble growth, the K met a relative stable level and the measurement accuracy increased. Pleased results (given the expected threshold \(\leqq K3\%\)) could be met in the interval [4, +\(\infty\)], which was quite acceptable in our experiments, where most of the bubble diameters were in this interval.