## 3 Results
### Quantitative Validation
#### 3.1.2 Results  
The experiment was designed to perform on the three training videos with the can and 14 markers. Several markers were randomly occluded in some images because of the camera movement, and only \(T_{M}\) of the valid markers were involved in the accuracy validation experiment. At the initial state, the LCS of the can was assumed to coincide with the center of each marker. For each experiment video, 14 offset matrices \(\mathcal{T}=\{\Gamma_{\Omega 1},T_{\Omega 2},\cdots,T_{\Omega 14}\}\) of each marker were calibrated once using the proposed method. The calibration was performed on 3-5 selected image frames to cover all the surrounding markers. The results were stored as a vector of matrices according to the marker ID. For all visible markers in the images of the experimental videos, the registration matrix was calculated by Equation (1), where \(T_{M}\) was given by the tracking pose results of one ARUCO marker, and \(T_{O}\) was given by \(\mathcal{T}\) and the detected marker ID. The composed camera pose was compared with the ground truth pose by calculating the absolute error of the translation vector and the rotation vector. The absolute error results are shown in Table 1, where the unit of the \(\left|\Delta x\right|\), \(\left|\Delta y\right|\), \(\left|\Delta z\right|\) is mm and the \(\left|\Delta\alpha\right|\), \(\left|\Delta\beta\right|\), \(\left|\Delta\gamma\right|\) is degree respectively. The average registration residue \(E(T_{R})\) of the 14 markers and the consuming time of the calibration process are also presented in Table 1. The error distribution results on the 6 degrees of freedom are shown in Figure 9.  
According to the mean absolute error value in Table 1, the mean error value on all 6 degree of freedoms of the three experiment videos are 1.28 mm, 1.2 mm, 3.8 mm, 2.3\({}^{\circ}\), 1.73\({}^{\circ}\), 0.61\({}^{\circ}\) respectively, with the standard deviation of 0.92 mm, 0.51 mm, 2.18 mm, 0.63\({}^{\circ}\), 0.58\({}^{\circ}\), 0.20\({}^{\circ}\). In Table 1 and Figure 9, \(\left|\Delta\alpha\right|\) and \(\left|\Delta\beta\right|\) are much higher than \(\left|\Delta\gamma\right|\), for the estimation of the normal direction is directly determined by the supporting surface assumption while the other two rotation freedoms are the result of non-linear optimization. The error in Video2 and Video3 is obviously much higher than that of the video1 because of the blur and occlusion of the marker in the calibration image. The \(E(T_{R})\) reveals that the calibration error is mainly caused by the appearance domain differences between the CAD model and the real image of the target object. To compare the influence of the calibration error, the relative scale error is defined in terms of the absolute size of the object as \(\left|\Delta t\right|/l_{diagonal}\). As the diagonal of the bounding box is 105.4 mm, the relative scale error on translation is 1.9%, which is relatively small and neglectable to the human eye's visual perception of the spatial position, meaning it fulfills the precision requirement for industrial AR applications. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Experiment**} & \multicolumn{5}{c}{**Mean Absolute Error**} & \multirow{2}{*}{\(E(T_{R})\)} & \multirow{2}{*}{**Time (s)**} \\ \cline{2-2} \cline{5-8}  & & & & & & & & \\ \cline{2-2} \cline{5-8}  & \(\left|\Delta x\right|\) & \(\left|\Delta y\right|\) & \(\left|\Delta z\right|\) & \(\left|\Delta\alpha\right|\) & \(\left|\Delta\beta\right|\) & \(\left|\Delta\gamma\right|\) & \(E(T_{R})\) \\ \hline Video 1 & 0.43 & 0.69 & 0.93 & 2.25 & 1.39 & 0.49 & 0.18 & 6.24 \\ Video 2 & 2.24 & 1.53 & 5.32 & 1. In the aspect of time consumption, Video2 and Video3 also cost higher than Video1, because the initial normal and translation accuracy estimated by the DOT template matching was greatly influenced by the cluttered scene background and the relatively large error of the initial parameter guess further lowers the convergence speed of the non-linear optimization.