## 4 Case Study
### The Experiment Platform  
#### 4.2.1 First Experiment  
As shown in Figure 5, the Cartesian space coordinate system is established. The side length of the black and white square is 100 mm. The side length of the black and white checkerboard is 20 mm. The height of the cylinder is 18 mm, its diameter is 300 mm, and the angles of the 12 sectors are equal. Points 1, 2, 3, 4, 5 and 6 are control points; points \(A\), \(B\), \(C\) and \(D\) are to be measured. Point \(G\) is the center of the circle. There is a certain error between the fitting coordinates and the true coordinates, which is used to measure the accuracy of the equal weight midperpendicular algorithm and the weighted midperpendicular algorithm. Figure 4: Flowchart of experiment. The camera is a Canon 5 DS, and its parameters are shown in Table 1; \(f\) represents the focal length. \(\Delta x\) and \(\Delta y\) represent deviations of principal point of the photograph in the direction of \(X\) and \(Y\), respectively. \(k_{1}\) and \(k_{2}\) are radial distortion parameters, respectively. There are two photography sites, and one image is taken at each site. As shown in Figure 6, according to the basic principle of photogrammetry [31], two images are used to complete the calculation of points. \begin{table}
\begin{tabular}{c c c c c c} \hline \(f\) (mm) & Resolution & Pixel Size (\(\mu\)m) & \(\Delta x\) (mm) & \(\Delta y\) (mm) & \(k_{1}\) & \(k_{2}\) \\ \hline
48.19 & \(8688\times 5792\) & 4.14365 & 0.01892 & \(-0.15786\) & \(-0.08748\) & \(-0.73372\) \\ \hline \end{tabular}
\end{table} 
Table 1: Camera parameters of first experiment. \begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{1}{c}{**Points**} & \(X\) (mm) & \(Y\) (mm) & \(Z\) (mm) \\ \hline
1 & \(-20\) & 340 & 0 \\
2 & 0 & 0 & 0 \\
3 & 280 & 0 & 0 \\
4 & 0 & 0 & 200 \\
5 & 200 & 180 & 0 \\
6 & 0 & 0 & 100 \\ \(A\) & 125 & 50.0962 & 18 \\ \(B\) & 179.9038 & 105 & 18 \\ \(C\) & 200 & 180 & 18 \\ \(D\) & 125 & 309.9038 & 18 \\ \(G\) & 50 & 180 & 18 \\ \hline \end{tabular}
\end{table} 
Table 2: The object space coordinates of first experiment. Figure 5: The experimental model. The image plane coordinates of the control points and the points to be measured on the two images are shown in Table 3.  
The six elements of exterior orientation of the two images are shown in Table 4. \(X_{S}\), \(Y_{S}\), and \(Z_{S}\) are the spatial coordinates of the center of the image projection, respectively; \(\varphi\), \(\omega\), and \(\kappa\) are the angle of the camera relative to the spatial coordinate system, and these six parameters together determine the spatial position and attitude of the image. The calculated coordinates of space points are shown in Table 5. Points \(A\), \(B\), \(C\) and \(D\) are all on the spatial circle. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **First Image** & **Second Image** \\ \hline \(X_{S}\) (mm) & 1191.7739 & 1181.1086 \\ \(Y_{S}\) (mm) & 209.1936 & 6.4546 \\ \(Z_{S}\) (mm) & 742.3528 & \(-\)7.3536 \\ \(\varphi\) (rad) & 14.5999 & \(-\)7.3536 \\ \(\omega\) (rad) & 0.0382 & 3.2725 \\ \(\kappa\) (rad) & \(-\)7.8937 & 20.5072 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The elements of exterior orientation of the two images in first experiment. Figure 6: Images of the model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**First Image**} & \multicolumn{2}{c}{**Second Image**} \\ \hline
**Points** & \(x\) **(mm)** & \(y\) **(mm)** & \(x\) **(mm)** & \(y\) **(mm)** \\ \hline
1 & 6.1867 & 4.4868 & 5.1725 & 1.6796 \\
2 & \(-\)5.4985 & 4.3206 & \(-\)6.3170 & 3.0732 \\
3 & \(-\)7.2376 & 10.4184 & \(-\)5.7582 & 9.3107 \\
4 & \(-\)5.8047 & \(-\)1.9510 & \(-\)6.9455 & \(-\)3.2973 \\
5 & 0.2790 & 8.8237 & 1.0681 & 6.5986 \\
6 & \(-\)5.6505 & 1.2744 & \(-\)6.6369 & \(-\)0.0267 \\ \(A\) & \(-\)4.3736 & 6.3158 & \(-\)4.2450 & 4.8427 \\ \(B\) & \(-\)2.5396 & 7.6163 & \(-\)1.9431 & 5.8007 \\ \(C\) & 0.3132 & 8.2337 & 1.0973 & 5.9870 \\ \(D\) & 5.3836 & 6.7833 & 5.3952 & 3.9046 \\ \hline \hline \end{tabular}
\end{table}