# Iterative Camera Calibration Method Based on Concentric Circle Grids
## 3 Experimental Results and Analysis
### Experiments with Real Images  
To verify the accuracy of the circle center location method, a real template was designed, as shown in Figure 10. We show the center location results and the enlarged view. A '+' sign represents the center of the concentric circle feature point on the plane, with radii measuring 10 mm and 20 mm. The camera and the template were separated by a distance of 1500 mm, and the rotation angle was roughly 1 rad. Real images were taken using a 4M140 MCX camera with a resolution of \(2048\times 2048\) pixels, the pixel size was \(dx=dy=5.5\) um, and the focal length was \(f=12.5\) mm.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Ground Truth** & **Zhang** & **Conventional** & **Proposed** \\ \hline \(f_{x}\) & 1507 & 1506.989 & 1506.9983 & 1507 \\ \(f_{y}\) & 1507 & 1506.9889 & 1506.9952 & 1507 \\ \(u_{0}\) & 1045 & 1044.9253 & 1044.9268 & 1045.014 \\ \(v_{0}\) & 1010 & 1010.1394 & 1010.1385 & 1010.0848 \\ \(k_{1}\) & \(-0.1\) & \(-0.0998\) & \(-0.1001\) & \(-0.1\) \\ \(k_{2}\) & 0.08 & 0.0749 & 0.0823 & 0.08 \\ \(P_{1}\) & \(-0.0005\) & \(-0.0005\) & \(-0.0005\) & \(-0.0005\) \\ \(P_{2}\) & \(-0.0005\) & \(-0.0005\) & \(-0.0005\) & \(-0.0005\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Results of calibration using various methods in the presence of slight lens distortion(pixel).  
Figure 10: Circle center location on the real template.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Ground Truth** & **Zhang** & **Conventional** & **Proposed** \\ \hline \(f_{x}\) & 1507 & 1507.0557 & 1507.0552 & 1507 \\ \(f_{y}\) & 1507 & 1507.0402 & 1507.0396 & 1507 \\ \(u_{0}\) & 1045 & 1044.932 & 1044.9329 & 1044.9838 \\ \(v_{0}\) & 1010 & 1010.237  
In Figure 10, the conventional method is represented by the green '+' for the obtained imaged center, while the red '+' indicates the imaged center obtained using the proposed method. It can be seen that the projection determined using our approach aligned closely with the actual value.  
Then, the concentric circle grid was generated on an A4 sheet of paper and pasted onto the board. The board was moved to capture images at five different poses and the intrinsic parameters are shown in Table 3.  
Reprojection errors are usually the evaluation standard of camera calibration accuracy. Feature points are reprojected to the image space and the reprojection errors are shown in Figure 11. The concentration of the reprojection error distribution of the proposed method is evident.  
Figure 11: Reprojection errors distribution using different methods. (**a**) Reprojection errors distribution of Zhang calibration method; (**b**) Reprojection errors distribution of conventional calibration method; (**c**) Reprojection errors distribution of proposed calibration method.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Zhang** & **Conventional** & **Proposed** \\ \hline \(f_{x}\) & 2281.2359 & 2248.0639 & 2260.9497 \\ \(f_{y}\) & 2280.7966 & 2247.8106 & 2260.8361 \\ \(u_{0}\) & 1040.5597 & 1037.8423 & 1029.0437 \\ \(v_{0}\) & 1020.5549 & 1022.6404 & 1026.9882 \\ \(k_{1}\) & \(-\)0.1532 & \(-\)0.1539 & \(-\)0.1544 \\ \(k_{2}\) & 0.0835 & 0.1723 & 0.1791 \\ \(P_{1}\) & \(-\)0.0004 & \(-\)0.0004 & \(-\)0.0006 \\ \(P_{2}\) & \(-\)0.0007 & \(-\)0.0005 & \(-\)0.0006 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Evaluating intrinsic parameters using various methods in a real experiment(pixel).  
The mean reprojection error for the Zhang calibration method, the conventional calibration method, and the proposed method is 0.092 pixels, 0.0791 pixels, and 0.0523 pixels, respectively. The results indicate that the proposed method offers specific benefits in enhancing the precision of calibration under the same conditions.