## 4 Evaluation
### Real Data
In each sub-figure, green circle is rendered with a radius equal to the mean re-projection error. In order to alternatively examine the proposed technique, different measurements on displacement and attitude are carried out using the CMM and camera mentioned above. Firstly, for displacement measurement, a target (6 \(\times\) 6 grid of points, 20 \(\times\) 20 mm cell size) fixed on the end of the probe is moved with the guide and imaged by the camera placed in front of the CMM, for measuring a distance \(d\) between two different positions as an evaluation factor (Figure 12). Secondly, for attitude measurement, two targets, T\({}_{1}\) and T\({}_{2}\), mounted on the base with the same grid and cell as that of the above measurement, are imaged by the camera (Figure 13). Among the three axis vectors, only the one in the z direction can be perfectly measured using the probe (Renishaw, SP600), by scanning the pattern plane of each target, due to a restriction that makes it hard to capture 3-D coordinates of a corner accurately, by means of contact measurements. Thus, the included angle \(\theta\) between two normal vectors is adopted as another evaluation factor more suitably. Fifteen independent trials are performed to localize sub-pixel corners, employing both the proposed and referenced methods, and estimate the camera poses from their respective intrinsic parameters listed in Table 2. The metrics \(d\) and \(\theta\) are then computed for investigating discrepant deviations with respect to the CMM data. Figure 14 presents the results from the above measurements. Under the premise that the CMM provides baselines with a higher accuracy, the RMS errors of \(d\) and \(\theta\) are 0.032 mm and 0.010\({}^{\circ}\) for [16], 0.021 mm and 0.009\({}^{\circ}\) for [20], 0.037 mm and 0.013\({}^{\circ}\) for [24], and 0.014 mm and 0.006\({}^{\circ}\) for the proposed approach. Although there are many estima  
Figure 12: Experiment for measuring displacement. (**a**) Determining \(d\) via CMM and camera. (**b**) A merged image of two positions with located corners. Figure 13: Experiment for measuring attitude. (**a**) Determining \(\theta\) via CMM and camera. (**b**) One shot in 15 trials with located corners. the results are mainly dependent on the accuracies of intrinsic parameters and corner coordinates, and essentially subject to the performance of each sub-pixel localization method because the camera is also calibrated from the respective corner set. From a synthetical point of view, exact values of \(d\) and \(\theta\) are derived from reliable estimations of camera poses predetermined by accurate corner coordinates. As an apparent outcome of the comparison, the proposed technique presents a higher performance than others. As shown in Figure 15, in order to test the proposed approach in terms of its robustness to real-world data gathering, four images of a stationary chessboard are captured by the mentioned camera under underexposed, overexposed, indoor light interfered, and outdoor light interfered scenarios. For each corner in a 6 \(\times\) 6 array, its maximin deviation between different scenarios is computed and gathered for an overall evaluation. Table 3 lists the overall evaluation result for four different approaches. The RMS deviations are 0.419 pixels for [16], 0.287 pixels for [20], 0.396 pixels for [24], and 0.241 pixels for the proposed approach. Considering the fact that the relative pose between the target and camera is stationary, the variability of each detected corner is mainly subject to the robustness of corner localization in the presence of the ambient light changes. The smallest RMS deviation proves that the proposed approach has higher interference immunity, resulting from a more robust corner model. Figure 14: Measurement results of (**a**) displacement \(d\) and (**b**) attitude angle \(\theta\).