# Point-Graph Neural Network Based Novel Visual Positioning System for Indoor Navigation
## 4 Experimental Results
### VPS Results
#### 4.2.3 VPS Error Rate  
The pose estimation error of the improved point-GNN is proportional to the object's center coordinates (x, y, z) and rotation values, namely the pitch, yaw, and roll. Furthermore, it is proportional to the distance of the camera's origin coordinates (0, 0, 0). However, as VPS is performed considering the voxel address, the coordinates of the camera in the database space are identical to those of the position in the cube of the voxel labeling. Figure 9 depicts the VPS error rate when the position of the camera coordinates and the actual camera coordinates are not the same voxels in the database space estimated by the VPS. On average, the error rate is as much as the distance between the centers of the two voxels.  
The actual camera position of the selected object in the indoor space can be represented as (x\({}_{2}\), y\({}_{2}\), z\({}_{2}\)) \(\in\) Voxel No. x\({}_{2}\). The camera position predicted using VPS based on the pose estimation of the improved point-GNN can be represented as (x\({}_{1}\), y\({}_{1}\), z\({}_{1}\)) \(\in\) Voxel No.n When x\({}_{1}\) is the range of Equation (7) and is included in the same voxel, the proposed method provides the same position by correcting the error of the voxel space of size: length, width, height = \(\alpha\) cm. The final actual error distance in the same voxel space does not exceed Equation (7). However, when the voxel numbers do not match, the error distance can be obtained using Equation (8).  
\[(\text{VPS Error Distance}=\text{Max}\sqrt{3\alpha^{2}}) \in \{\text{Voxel}_{\text{No.x2}}=\text{Voxel}_{\text{No.x1}}\} \tag{7}\] \[(\text{VPS Error Distance}=\sqrt{\left(x_{2}-x_{1}\right)^{2}+ \left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}) \in \{\text{Voxel}_{\text{No.x2}}\
eq\text{Voxel}_{\text{No.x1}}\} \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Log**} & \multicolumn{2}{c}{**Details**} \\ \cline{2-4}  & data min \_ point & \([-9.5453,\,-1.4278,\,-12.1236]\) & Min coordinates in the point cloud \\ \cline{2-4} Point cloud & data max \_ point & \([13.6986,\,2.6894,\,21.7361]\) & Max coordinates in the point cloud \\ \cline{2-4} (Indoor) & database min \_ point & \([-10,\,-2,\,-13]\) & Size of the space expands to the smallest integer \\ \cline{2-4}  & database max \_ point & \([14,\,3,\,22]\) & Maximum size of the space expands to an integer \\ \hline \multirow{2}{*}{Voxel} & voxel\_size & 0.2 m & Voxel size setting (possible up to 0.1 m) \\ \cline{2-4}  & max\_dim & \([120,\,25,\,175]\) & voxel\_x y z\_size \\ \hline \multirow{5}{*}{VPS} & deep Learning\_object\_xyz & \([-0.0102,\,-0.0112,\,-0.4209]\) & Center coordinates of an object estimated \\ \cline{2-4}  & converted\_ & camera\_origin & \([-0.0190,\,-0.0132,\,-0.4205]\) & Camera position coordinates in the voxel \\ \cline{2-4}  & converted\_VPS & \([8.2809,\,1.2867,\,2.8794]\) & VPS coordinates of the voxel database \\ \cline{2-4}  & voxel\_points\_xyz & \([17.8262,\,2.7146,\,15.0030]\) & VPS coordinates from the voxel database origin \\ \cline{2-4}  & voxels\_numbers & 553,696 & Number of voxels in the voxel database \\ \cline{2-4}  & VPS\_voxels\_address & 47,450 & User location VPS\_voxel address \\ \hline \hline \end{tabular}
\end{table}
Table 4: Visual positioning system (VPS) log.  
Figure 9: Visual positioning system (VPS) error and measurement.  
Table 5 summarizes the VPS error distance calculated using Equation (7) in the experiment described in Section 4.1. The proposed method can localize locations within a sub-meter-level accuracy in our dataset and to test on indoor scenes we use the publicly available 7-Scenes dataset with scenes shown in Figure 10. This dataset contains significant variation in camera height and was designed for RGB-D relocalization. We compared our method with the state-of-the-art method by creating a point cloud from RGB-D images. As indicated in Tables 5 and 6, more than 80% of the test data were successfully located within 0.25 m of the ground truth location.