## 4 Results
### Multi-Frame Object-Mapping  
Indoors, the visual odometry (or SLAM) system has been able to give fairly accurate trajectory estimates. Since we are mainly concerned with the accuracy of object landmark establishment, we fix all the camera nodes in the back-end optimization and optimize only the 9-DOF object landmark. The mapping results are shown in Figure 8.  
We use the indicators translation, rotation and shape error to fully evaluate the mapping effects. The translation error (m) measures the center distance between the estimated \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Datasets** & **Label** & **Quadric-SVD** & **Quadric-SLAM** & **Cube-SLAM** & **Ours** \\ \hline \multirow{3}{*}{ICL\_home} & Bench & 38.14 & 31.91 & 8.20 & **5.82** \\  & Chair1 & 38.08 & 41.65 & 17.17 & **3.73** \\  & Chair2 & 34.79 & 43.22 & 35.78 & **5.45** \\ \hline \multirow{3}{*}{Fr1\_desk} & TV & 49.30 & 49.30 & 33.96 & **17.51** \\  & Book & 38.46 & 40.09 & **14.36** & 19.75 \\  & Keyboard & 34.70 & 36.50 & 19.74 & **6.97** \\ \hline \multirow{3}{*}{Fr2\_desk} & Mouse & 54.11 & 59.13 & **22.03** & 29.08 \\  & Keyboard & 40.82 & 24.89 & 15.31 & **10.72** \\  & TV & 41.65 & 20.88 & 42.76 & **19.22** \\  & Book & 53.37 & 44.96 & **7.28** & 12.85 \\ \hline \multirow{3}{*}{Fr3\_cabi} & Cabinet & 19.98 & 18.31 & **4.17** & 11.44 \\ \hline \multirow{3}{*}{Real\_robot} & Chair & 10.87 & 11.31 & 24.73 & **10.07** \\  & Bench1 & 41.53 & **11.01** & 36.64 & 11.58 \\ \cline{1-1}  & Bed & 52.38 & 30.67 & 43.18 & **24.62** \\ \cline{1-1}  & TV & 31.92 & 30.58 & 40.09 & **7.28** \\ \hline \multicolumn{2}{c}{Average} & 38.67 & 32.96 & 24.36 & **13.07** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Error of single frame orientation detection (°). Figure 8: Landmark estimation results from different datasets, the ground-truth objects and point clouds are shown as references. For symmetric objects, the circumscribed cubes of ellipsoidal landmarks are shown to better demonstrate orientation. A part of the detected symmetry planes is marked in purple. object and the ground-truth object. The shape error evaluates the 3D intersection over union (IoU) between the two circumscribed cubes after aligning the center and rotation of the estimated object and the ground-truth object. For objects with symmetry, rotation error (deg) is the orientation error defined in the subsection above. We conducted experiments using the original SVD initialization and joint optimization method in quadric-SLAM, as well as our method with the symmetry factor. The shape error of cube-SLAM is cited from [10]. As shown in Table 2, the translation and shape of quadric-SLAM after joint optimization is improved compared to SVD initialization. With the symmetry factor added in the back-end factor graph, our method achieved better average translation, rotation and shape error values of 0.15 m/12.93 deg/0.47, improved by 63.4%/61.7%/42.4%, respectively, compared with quadric-SLAM. In the Fr3_cabinet, the result of cube-SLAM obtained an IoU of 0.64, 48% higher than ours, as the cabinet has a tight cubic shape without other texture disturbance. For the results with the other tested datasets, we outperformed the two SOTA object-level SLAM systems. We implemented the algorithm in C++ and used the gtsam library for the graph optimization. The object detection runs on GTX 1660s with 33 Hz and can work in parallel with other visual odometry threads as well as the back-end optimization. On a PC with an Intel Core i5-9400 2.9 GHz CPU, 16 GB RAM, our orientation detection method has an average time cost of 9 ms per object in each frame, because only a 4 \(\times\) 4 matrix multiplication and inverse calculation are required. This only brings an additional 29% time cost and the front-end tracking can still run in real time.