## 4 Evaluation
### Synthetic Data  
In order to acquire synthetic chessboard image, a pin-hole camera is simulated with the properties: \([f_{x},f_{y}]\) = [7000, 7000], \([u_{0},\,v_{0}]\) = [1296, 972]. The image resolution is set to 2592 \(\times\) 1944. A single chessboard pattern with 20 mm cell size in both directions is projected to the image plane. Since optical paths are reversible, an ideal projection from the pattern center can be found and defined as ground truth. Gaussian blur with the window parameter \(\sigma_{\text{f}}\) and Gaussian noise with 0 mean and standard deviation \(\sigma_{\text{n}}\) are added to make the image similar in appearance to a real one (Figure 5). For each given \(\sigma_{\text{f}}\) and \(\sigma_{\text{n}}\), 100 independent trials are performed, with other simulation parameters varied and limited in their ranges (Table 1), under the premise of ensuring faultless projections. Figure 6 depicts the RMS error of sub-pixel localization as a function of \(\sigma_{\text{f}}\) and \(\sigma_{\text{n}}\). The proposed technique performs significantly better than the referenced ones. Although it results in a higher error due to the increase of \(\sigma_{\text{f}}\) and \(\sigma_{\text{n}}\), the performance drop is not as pronounced as for the others. Concretely, for the poorest image quality (\(\sigma_{\text{f}}\) = 3, \(\sigma_{\text{n}}\) = 0.2), the result shows that the errors are about 0.154, 0.041, 0.077, and 0.024 pixels for [16; 20; 24], and the proposed technique, respectively. Remarkably, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(g_{\text{max}}\) & \(g_{\text{min}}\) & \(\mathbf{yaw,pitch, roll}\) & \(\mathbf{t_{x}}\) (mm) & \(\mathbf{t_{y}}\) (mm) & \(\mathbf{t_{z}}\) (mm) \\ \hline
[191; 255] & [0; 63] & \([-\pi/4,\pi/4]\) & \([-40,\,40]\) & \([-30,\,30]\) & [950, 1050] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Range set of simulation parameters. _yaw, pitch_, and _roll_ are the Euler angles related to \(\mathbf{r}\). \(t_{x}\), \(t_{y}\), and \(t_{x}\) are the dimensional elements of \(\mathbf{t}\). Figure 5: Chessboard images captured by (**a**) simulation and (**b**) real device. [PERSON] et al. [24] yields a stable, but significant, error in the presence of the change of \(\sigma_{\rm f}\) and \(\sigma_{\rm n}\) for taking filtered images as inputs. That is to say, it not only eliminates noise distinctly, but also leads to an extra uncertainty of sub-pixel localization. In addition, sub-pixel localization errors from all trials (the total number is 40,000) are gathered for an overall evaluation represented by boxplots. As shown in Figure 7, for the proposed and referenced methods, interquartile ranges (IQRs) are highly symmetrical about medians pretty close to zero. In detail, the IQRs are about 0.18, 0.13, 0.32, and 0.04 pixels in both directions for [16; 20; 24], and the proposed method, respectively. The smaller IQR reflects the better performance of sub-pixel localization. Again, using filtered images as inputs lead to a particular outcome, that there are no outliers to be distinguished with the largest IQR for [24]. Figure 6: Localization error with respect to blur strength \(\sigma_{\rm f}\) and noise level \(\sigma_{\rm n}\) for (**a**) [16], (**b**) [20], (**c**) [24], and (**d**) the proposed technique. The above simulation relies on the assumption that edges defining a corner are completely straight in the observation area, or region of interest, where the corner is going to be found. However, it is well known that lenses inevitably have distortions. To obtain maximum allowable distortions for the method, another simulation is conducted, with the fixed blur strength and noise level (\(\sigma_{\text{f}}=1.5\), \(\sigma_{\text{n}}=0.1\)), and the first order radial distortion with the degree \(k_{1}\) is added to the image (Figure 8).