# A Fast and Simple Method for Absolute Orientation Estimation Using a Single Vanishing Point
## 3 Experiments and Results
### Real Images  
The ground truth of camera orientation is unknown in the scenarios and thus we cannot test the performance of our proposed method directly. In this section, an indirect experiment on real images is performed. The accuracy of the orientation will affect the accuracy of the 3D position measured using stereo vision with two cameras, whose orientations are given by our proposed method, P3P solver, and RPnP solver. Hence, after estimating the orientation, two cameras are used to measure the 3D position of a control point whose ground truth can be given through the total station or RTK. The error between the measuring value and the ground truth is used to reflect the performance of our proposed method in practice. In Section 2.3, we give the limitation of parallel lines whose direction vector could not be \( \begin{pmatrix}0&0&1\end{pmatrix} \). Hence, we select two categories of typical parallel lines to test our proposed method thoroughly, and their direction vectors are \( \begin{pmatrix}1&0&0\end{pmatrix} \) and \( \begin{pmatrix}0&1&0\end{pmatrix} \), respectively, i.e., which are parallel to the \(x\)-axis (the first case) and \(y\)-axis (the second case) of the world frame, respectively. In addition, some control points are placed in the same FOV of two cameras to test the other two methods, namely the P3P solver and RPnP solver. The real images captured by the cameras (MV-CS016, the CMOS is IMX296 of Sony and focal length is 12 mm) in our lab are shown in Figure 6.  
From Figure 6, we select the sides of the ceiling tile as the two categories of parallel lines, which are perpendicular to each other and strictly parallel to the \(x\)-axis and \(y\)-axis of the world frame, respectively. In this paper, the Hough transform is used to extract the lines in the images and then the equations of the lines are given. Last, we use the equations for computation of the vanishing point. We use the intersection points of the two categories of parallel lines as the 3D control points, whose positions are measured by RTK (NTS-362R, measuring precision better than 0.5 cm) [54], which are used as ground truths. First, the parallel lines of the \(x\)-axis are selected for our proposed method to estimate the camera orientation, and three 3D control points are selected for the P3P solver and five 3D control points are selected for the RPnP solver. When the camera orientation is estimated using three methods individually, stereo vision [55,56] is used to measure the positions of the other 3D control points. In stereo vision (also called resection), two cameras are selected to measure the 3D spatial position of the feature point, and the accuracy is high if the extraction accuracy has a feature point. For improving the extraction accuracy, we use the intersection points of the two categories of parallel lines as the feature points, which are \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Our Proposed Method** & **P3P** & **RPnP** \\ \hline
**Computational time/ms** & 0.2625 & 0.7500 & 0.8063 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational time. extremely easy to extract and many methods [57; 58] can be selected to complete this job. In general, the accuracy of extraction is better than 0.1 pixels, which means it affects the accuracy of the position by stereo vision slightly. Hence, the accuracy of positions given by the stereo vision is determined by the camera orientation estimated via the three methods. The ground truths \(P_{i}\) given by the total station and measuring values \(P^{\prime}_{i}\) given by the stereo vision are compared and the position mean error can be obtained using  
\[E_{position}=\frac{\sum\limits_{i=1}^{n}\left|P^{\prime}_{i}-P_{i}\right|}{n} \tag{23}\]  
Here, \(n\) is the number of the control points. In addition, the image reprojections \(p_{i}\) of the other 3D control points can be obtained when the camera orientation is estimated, and we can also obtain the projection \(p^{\prime}_{i}\) of the control point from the image.