# Off-Line Camera-Based Calibration for Optical See-Through Head-Mounted Displays
## 5 Experiments and Results
### Results and Discussion  
Table 1 shows the results of the eight calibrations in terms of overlay error (\(o_{\text{2D}}^{err}\)), angular error (\(\alpha^{err}\)), and absolute error (\(a_{\text{3D}}^{err}\)). The overall mean, standard deviation, and max values of \(o_{\text{2D}}^{err}\) were 2.23, 0.96, and 5.03 px; the same values for \(\alpha^{err}\) were 5.98, 2.58, and 13.47 arcmin and for \(a_{\text{3D}}^{err}\) 1.13, 0.49, and 2.55 mm. Figure 8: Image grabbed from the viewpoint camera after the calibration. The accuracy of the calibration is measured as the overlay error between the ground truth real corners of the validation checkerboard and the centers of the virtual yellow circles. The zoomed details of the image shows the on-image displacements between the real and the virtual landmarks. Red arrows indicate the direction and magnitude of overlay error in pixels. The average overlay error for the reference position was comparable to that obtained by [PERSON] et al. [14] and by [PERSON] et al. [16], whereas for the remaining positions, the average overlay error was comparable to that obtained in [9] and in [8]. It should be also noted that the results of the angular error are reported considering the rather low angular resolution of the viewpoint camera (\(\approx\)2.67 arcmin/pixel). This explained the values of the angular errors obtained. If we considered the absolute error at the distance where the validation checkerboard was placed (\(\approx\)650 mm), our results were comparable to those obtained with the camera based triangulation approach proposed by [PERSON] et al. [20]. The experiment results suggested that the calibration refinement step accounting for the user's eye position was paramount to achieve accurate results also for eye positions different from the reference one for all those OST HMDs with finite focal distances (i.e., not parallax-free OST HMDs). This was in accordance with what was suggested by [PERSON] et al. [14] and contrary to what was hypothesized by [PERSON] et al. and by [34] et al. It should be also stressed that this assumption was clearly different from what was suggested in [20] (page 58), where the classification of the OST HMDs was made based on the level of complexity of the OST optical combiner: \"these results show \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Overlay Error**} & \multicolumn{3}{c}{**Angular Error**} & \multicolumn{3}{c}{**Absolute Error**} \\ \hline
**Camera** & \multicolumn{3}{c}{**(pixel)**} & \multicolumn{3}{c}{**(arcmin)**} & \multicolumn{3}{c}{**(mm)**} \\ \hline
**Position** & **Mean** & **Std Dev** & **Max** & **Mean** & **Std Dev** & **Max** & **Mean** & **Std Dev** & **Max** \\ \hline
1-L & 1.08 & 0.52 & 1.98 & 2.88 & 1.39 & 5.3 & 0.55 & 0.26 & 1 \\
2-UL & 1.43 & 0.78 & 3.23 & 3.82 & 2.08 & 8.66 & 0.72 & 0.39 & 1.64 \\
3-UR & 2.03 & 0.87 & 3.48 & 5.43 & 2.32 & 9.33 & 1.03 & 0.44 & 1.76 \\
4-T & 2.06 & 0.58 & 3.08 & 5.52 & 1.56 & 8.25 & 1.04 & 0.3 & 1.56 \\
5-LL & 2.74 & 0.82 & 4.34 & 7.35 & 2.2 & 11.61 & 1.39 & 0.42 & 2.19 \\
6-LR & 3.29 & 0.6 & 5.03 & 8.82 & 1.6 & 13.47 & 1.67 & 0.3 & 2.55 \\
7-D & 2.61 & 0.66 & 4 & 6.99 & 1.76 & 10.72 & 1.32 & 0.33 & 2.03 \\
8-R & 2.61 & 0.67 & 4.05 & 6.99 & 1.8 & 10.85 & 1.32 & 0.34 & 2.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of the accuracy analysis in pixels (overlay error), arcmin (angular error), and mm (absolute error). Figure 9: Viewpoint camera frames. (**a**) Camera frame of the augmented scene. (**b**) Camera frame of the real-world scene. (**c**) Camera frame of the virtual scene. that the triangulation approach only works for OSTG with simple optics and that it is not a universal approach\". Overall, there were at least two sources of calibration inaccuracies in our method, both of which could be easily addressed in the future.