## 4 Experiments
### Camera Calibration
#### 4.1.2 Calibration Results and Reprojection Error Analysis  
This section presents the calibrated intrinsic and extrinsic matrices, summarizes the distortion parameters with their uncertainties, and analyzes the reprojection error. For completeness, the optimized intrinsic matrix \(K\) is given in Equation (17). This \(3\times 3\) matrix encodes the camera's focal lengths and principal point. As shown, the focal length in the  
Figure 12: The checkerboard images captured from multiple angles were used for camera calibration.  
direction is approximately 748.29 pixels and in the \(y\) direction is 747.88 pixels, while the principal point is at \((640.24,356.21)\) in pixel coordinates (close to the image center):  
\[K=\begin{bmatrix}748.29024&0&640.23685\\ 0&747.87601&356.21422\\ 0&0&1\end{bmatrix} \tag{17}\]  
Likewise, the extrinsic calibration result is represented by the \(3\times 4\) matrix in Equation (18), which combines the rotation matrix \(R\) and translation vector \(T\) (in the form \([R|T]\)).  
\[[R|T]=\begin{bmatrix}0.9997&-0.0217&-0.0055&-372.1355\\ -0.0217&-0.9997&-0.0073&104.9251\\ -0.0054&0.0074&-1.0000&698.5552\end{bmatrix} \tag{18}\]  
Here, the first three columns form the rotation matrix \(R\), and the final column is the translation vector \(T\). These extrinsic parameters transform points from the world coordinate system to the camera's coordinate system.  
In addition to the intrinsics and extrinsics, the lens distortion coefficients were estimated as part of the calibration. Table 1 summarizes the five distortion parameters of the standard radial-tangential model--three radial coefficients (\(k_{1},k_{2},k_{3}\)) and two tangential coefficients (\(p_{1}\), \(p_{2}\))--along with their standard deviations (uncertainties). The results show that the primary radial distortion coefficients \(k_{1}\) and \(k_{2}\) are on the order of \(10^{-2}\) and have opposite signs (with \(k_{1}\) positive and \(k_{2}\) negative), indicating a mild barrel distortion that is slightly compensated by the second-term correction. The third radial coefficient \(k_{3}\) is effectively zero within the uncertainty, suggesting that higher-order radial distortion is negligible. Similarly, the tangential distortion coefficients \(p_{1}\) and \(p_{2}\) are very close to zero, implying negligible tangential distortion.  
The reprojection accuracy of the calibration is evaluated in Table 2 and visualized in Figure 13. The mean reprojection error is 0.07293 pixels in the x-direction and 0.07531 pixels in the y-direction, with the vast majority of corner points exhibiting errors below 0.1 pixels. As shown in Figure 13, the error distribution is tightly clustered around the origin, with no significant directional bias or outliers, indicating that the calibration model provides an excellent fit across all calibration images.  
\begin{table}
\begin{tabular}{l c} \hline
**Parameter** & **Value \(\pm\) Std. Dev.** \\ \hline \(k_{1}\) & \(0.06987\pm 0.00094\) \\ \(k_{2}\) & \(-0.07504\pm 0.00169\) \\ \(p_{1}\) & \(0.00025\pm 0.00023\) \\ \(p_{2}\) & \(-0.00030\pm 0.00025\) \\ \(k_{3}\) & \(0.00000\pm 0.00000\) \\ \hline \end{tabular}
\end{table}
Table 1: Estimated lens distortion coefficients (radial \(k_{1}\), \(k_{2}\), \(k_{3}\) and tangential \(p_{1}\), \(p_{2}\)) with their standard deviations.  
\begin{table}
\begin{tabular}{l c} \hline
**Image Axis** & **Mean Reprojection Error/Pixels** \\ \hline \(x\) & 0.07293 \\ \(y\) & 0.07531 \\ \hline \end{tabular}
\end{table}
Table 2: Mean reprojection error per image axis after calibration. Both values are well below one pixel, indicating high precision in the camera parameter estimation.