## 4 Evaluation
### Real Data  
In contrast to simulations, real data experiments cannot directly evaluate the accuracy of sub-pixel localization via the observed corner coordinates, due to their undetermined ground truth data. An alternative and indirect way is examining it based on camera calibration technique. Figure 10 shows that a camera (JPLY, G1 GD05C) with 16 mm lens and 2592 \(\times\) 1944 image resolution is employed for conducting a camera calibration experiment based on a coordinate measuring machine (CMM) (Brown & Sharpe, Global Image 7107) with a single chessboard pattern (20 \(\times\) 20 mm cell size) mounted on the end of its probe. 3-D control points are achieved by programmatically driving the probe to a set of specially designed positions, and provided with a dimensional error of less than 0.003 mm in both directions. For each position, the chessboard pattern is recorded by the camera for capturing a corresponding corner. Since all corners are located at the sub-pixel level, the camera can be calibrated based on bundle adjustment [4; 6]. Table 2 lists the result of intrinsic parameters calibrated from the corners based on four different approaches. According to the definition of radial distortion coefficients detailed in [2], for [16; 20; 24] and the proposed method, the maximum distortions evaluated using the image point furthest from the principal point are 22.78, 25.92, 29.33, and 24.54 pixels in the radial direction, respectively. Among them,  
Figure 10: Camera calibration using (**a**) single chessboard and CMM for achieving (**b**) 3-D control points with specially designed positions. Figure 9: RMS error between localized and standard values as a function of \(k_{1}\) for four different approaches. the contributions of \(k_{2}\) are 2.03 pixels for [16], 6.06 pixels for [20], 10.06 pixels for [24], and 4.62 pixels for the proposed method. Therefore, \(k_{2}\) has a much smaller influence on the pixel offsets than \(k_{1}\). Or rather, the estimator of \(k_{2}\) is more sensitive to noise in the corner coordinates. In spite of the fact that the result cannot intuitively demonstrate the performance of each approach, it is pivotal for the following investigations. Figure 11 represents four scatter plots of re-projection errors. For general examinations, the maximum and mean re-projection errors for the proposed method are 0.22 pixels and 0.11 pixels, evidently less than 0.32 pixels and 0.15 pixels for [16], 0.29 pixels and 0.15 pixels for [20], 0.39 pixels and 0.18 pixels for [24]. From the standpoint of addressing perspective-n-point problem, the re-projection errors, assessing the validity of calibration, are subjected to some optical indications, e.g., image and lens resolutions, and integrated with certain methodologies, including calibration model, target geometry, and sub-pixel localization. The mentioned experiment employs a robust model with stereo points establishing correspondences between world and image frames accurately and, therefore, the lower re-projection errors not only reflect the better solution of perspective-n-point, but also testify the higher accuracy of sub-pixel localization. Therefore, the corners obtained using the proposed technique are better suited for camera calibration. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \([\mathbf{u_{0}},\mathbf{v_{0}}]\) & \([\mathbf{f_{\mathbf{\kappa}}},\mathbf{f_{y}}]\) & \([\mathbf{k_{1}},\mathbf{k_{2}}]\) \\ \hline
[16] & [1344.42, 937.37] & [7296.06, 7298.85] & [0.2332083, 0.4313581] \\
[20] & [1342.78, 936.71] & [7298.67, 7302.12] & [0.2237942, 1.2905459] \\
[24] & [1340.02, 937.04] & [7297.59, 7300.76] & [0.2118124, 2.2750773] \\ Proposed & [1343.70, 936.62] & [7299.13, 7302.50] & [0.2253345, 0.9926340] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calibration result for four different approaches. \(k_{1}\) and \(k_{2}\) denote the 1 st and 2 nd order radial distortion coefficients. Figure 11: Scatter plots of re-projection errors (red dots) for (**a**) [16], (**b**) [20], (**c**) [24], and (**d**) proposed method.