# A New Monocular Pose Estimation Method for the Coplanar P4P Problem
## 3 Experimental Results
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**No.**} & \multirow{3}{*}{**Methods**} & \multicolumn{6}{c}{**Pose estimation Results**} & \multicolumn{6}{c}{**Difference between the Pose Results**} \\ \cline{3-14}  & & \multicolumn{3}{c}{**Translation Vector(mm)**} & \multicolumn{3}{c}{**Rotation Angles (\({}^{\circ}\))**} & \multicolumn{3}{c}{**Translation Vector**} & \multicolumn{3}{c}{**Rotation Angles (\({}^{\circ}\))**} & \multicolumn{3}{c}{**Mean Errors of the Vertices’**} \\  & & \multicolumn{3}{c}{**Translation Vector(mm)**} & \multicolumn{3}{c}{**Rotation Angles (\({}^{\circ}\))**} & \multicolumn{3}{c}{**(mm)**} & \multicolumn{3}{c}{**Rotation Angles (\({}^{\circ}\))**} & \multicolumn{3}{c}{**Coordinates (mm)**} \\ \cline{3-14}  & & \(\mathbf{T_{\text{BX}}}\) & \(\mathbf{T_{\text{BY}}}\) & \(\mathbf{T_{\text{BZ}}}\) & \(\mathbf{A_{\text{BX}}}\) & \(\mathbf{A_{\text{BY}}}\) & \(\mathbf{A_{\text{BZ}}}\) & \(\mathbf{T_{\text{BX}}}\) & \(\mathbf{T_{\text{BY}}}\) & \(\mathbf{T_{\text{BZ}}}\) & \(\mathbf{A_{\text{BX}}}\) & \(\mathbf{A_{\text{BY}}}\) & \(\mathbf{A_{\text{BZ}}}\) \\ \hline \multirow{3}{*}{**1**} & New P4P & 20.4 & \(-\)7.1 & 1981.9 & \(-\)0.8 & 10.0 & 0.9 & \multirow{3}{*}{\(-\)0.1} & \multirow{3}{*}{\(0.2\)} & \multirow{3}{*}{\(3.6\)} & \multirow{3}{*}{\(-\)0.7} & \multirow{3}{*}{\(2.3\)} & \multirow{3}{*}{\(0.2\)} & \multirow{3}{*}{\(0.4\)} \\ \cline{2-2} \cline{14-14}  & Old P4P & 20.5 & \(-\)7.3 & 1978.3 & \(-\)0.1 & 7.7 & & & & & & & & & & & \\ \hline \multirow{3}{*}{**2**} & New P4P & \(-\)5.1 & \(-\)7.4 & 1916.8 & 2.0 & \(-\)39.3 & \(-\)4.0 & \multirow{3}{*}{\(-\)0.6} & \multirow{3}{*}{\(-\)0.5} & \multirow{3}{*}{\(4.2\)} & \multirow{3}{*}{\(1.4\)} & \multirow{3}{*}{\(0.8\)} & \multirow{3}{*}{\(0.1\)} & \multirow{3}{*}{\(0.9\)} \\ \cline{2-2} \cline{14-14}  & Old P4P & \(-\)4.5 & \(-\)6.9 & 1912.6 & 0.6 & \(-\)40.1 & \(-\)4.1 & & & & & & & & & & & \\ \hline \multirow{3}{*}{**3**} & New P4P & 269.4 & 5.6 & 2048.2 & 3.0 & 10.4 & 0.1 & \multirow{3}{*}{\(0.1\)} & \multirow{3}{*}{\(0.1\)} & \multirow{3}{*}{\(-\)2.7} & \multirow{3}{*}{\(-\)0.8} & \multirow{3}{*}{\(-\)1.8} & \multirow{3}{*}{\(-\)0.1} & \multirow{3}{*}{\(0.6\)} \\ \cline{2-2} \cline{14-14}  & Old P4P & 269.3 & 5.5 & 2050.9 & 3.8 & 12.2 & 0.2 & & & & & & & & & & & & & \\ \hline \multirow{3}{*}{**4**} & New P4P & 323.9 & 19.0 & 2176.6 & 45.3 & 32.3 & 18.1 & \multirow{3}{*}{\(-\)0.4} & \multirow{3}{*}{\(-\)0.3} & \multirow{3}{*}{\(-\)3.2} & \multirow{3}{*}{\(0.6\)} & \multirow{3}{*}{\(-\)0.4} & \multirow{3}{*}{\(0.3\)} & \multirow{3}{*}{\(0.6\)} & \multirow{3}{*}{\(0.6\)} \\ \cline{2-2} \cline{14-14}  & Old P4P & 323.3 & 19.3 & 2179.8 & 44.7 & 32.7 & 17.8 & & & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Shape and pose estimation results of a parallelogram object in experiments. Figure 5: Experimental images for measuring a parallelogram object’ pose and shape parameters. In terms of estimating the parallelogram vertices' coordinates, under above conditions, the new method can do it but traditional methods cannot. In Table 1, the mean errors of the vertices' coordinates are calculated by  
\[E_{B}=\frac{\sum\limits_{i=1}^{4}\sqrt{\left(X_{Bi}-\widetilde{X}_{Bi}\right)^{2 }+\left(Y_{Bi}-\widetilde{Y}_{Bi}\right)^{2}}}{4} \tag{9}\]  
where, \(\left(X_{Bi},Y_{Bi}\right)\) and \(\left(\widetilde{X}_{Bi},\widetilde{Y}_{Bi}\right)\) are the estimation results and true values of the \(i\)th vertex' coordinates, respectively.