# Fundamental Matrix Computing Based on 3D Metrical Distance
## 5 Experiment
### Numerical Result Comparison  
Table 1 compares metric criterion with traditional epipolar distance in 8-pt [6], 7-pt [6], 5-pt [13] and Gold [6] method. The second to sixth row of Table 1 are result based on the average distance from feature pixels to epipolar lines w.r.t the 5 methods to compute \(F\). The columns are three different scenes of \(Dino\), \(ZhangLan\) and \(Simu\). The total initial matching features number are 100, 123 and 100. The numbers under the column 'Error' are the average distance from feature pixels to epipolar lines while the numbers under the column 'Inliers #' are numbers of inliers. Here unit of the distance is pixel in image. Our method and the method Gold have the best re-projection error than the other four. Table 2 is the average re-projection error from feature pixels \(u,\ u^{\prime}\) to \(P_{E}X,\ P_{E}^{\prime}X\) and the 3D metric criterion \(\|u-P_{E}X\|\) and \(\|u^{\prime}-P_{E}^{\prime}X\|\) as described in Section 4.1, for the same datasets \(Dino\), \(ZhangLan\), and \(Simu\). The last column is our method, which outperforms the other four methods. In order to compare with each other, the same \(F\) in Table 1 were used to decompose out \(E\) with \(K\) to reconstruct the scene, i.e., their inliers are recorded to compute the average re-projection error. Our method has comparatively good results compared with the other four. Projective validation is a common way to judge the quality of \(F\). In this paper, we propose metric validation to perform the comparison. Metric validation is based on the assumption that the intrinsic matrix \(K\) is subject to the constraints in Section 4. Then, the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{2}{c}{\(Dino\)} & \multicolumn{2}{c}{\(ZhangLan\)} & \multicolumn{2}{c}{\(Simu\)} \\ \cline{2-7}  & **Error** & **Inliers \#** & **Error** & **Inliers \#** & **Error** & **Inliers \#** \\ \hline
8-pt & 1.50 & 86 & **0.50** & 100 & 0.45 & **100** \\
7-pt & 4.50 & **90** & 1.26 & 82 & 1.36 & **100** \\
5-pt & 5.24 & 88 & 4.35 & 101 & 5.65 & **100** \\ Gold & 1.38 & 89 & 1.69 & **123** & **0.41** & **100** \\ Ours & **1.05** & 88 & 1.22 & 98 & 1.05 & **100** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Error comparison based on the average distance from feature pixels to epipolar lines. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Scenes**} & **8-pt** & **7-pt** & **5-pt** & **Gold** & **Ours** \\ \hline \(Dino\) & 7.02 & 13.77 & 5.24 & 7.42 & **4.41** \\ \(ZhangLan\) & 6.15 & 15.29 & 4.35 & 6.91 & **4.07** \\ \(Simu\) & 1.54 & 5.36 & 5.65 & 2.31 & **1.05** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Error comparison based on the average re-projection error from feature pixels \(u,\ u^{\prime}\) to \(P_{E}X,\ P_{E}^{\prime}X\). It shows that our method outperform other methods. reconstruction is achieved by the deduced essential matrix \(E\). The criteria to judge the inlier becomes the average re-projection error of \(X\) on images to \([\hat{u},\ \hat{u}^{\prime}]\). If images are captured with the intrinsic matrix \(K\) under these constraints, the achieved result \(F\) will have the better final reconstruction in the metric coordinate. Actually, this is the case for most of the cameras in making metric validation work. The biggest benefit of our metric validation is that it can produce reasonable and visual metric reconstruction results rather than a sort of projective reconstruction with small numerical projection errors that could be used to compare. If a more precise result is required, the intrinsic matrix \(K\) should be measured in advance. Here, we focus on an automatic approach so that the pre-measured \(K\) is not adopted. As we can see in Table 1, our method has better results in metric space than the five-point method. In our experiments, thresholds of inliers have two types. (1) For metric validation, it is the metric re-projection errors of \(X\) on images to \(\hat{u}\) and \(\hat{u}^{\prime}\).