## 4 Results
### Single-Frame Symmetry Plane Estimation  
To evaluate the effectiveness of our proposed method for estimating the orientation of symmetric objects based on single-frame images, we took each valid observation of the symmetric object orientation before multi-frame optimization and computed their average orientation errors with respect to the ground-truth value, i.e, the minimum rotation angle required to align the estimated object's three rotation axes with any axis of the ground-truth object to a straight line.  
Two methods were chosen as our benchmark. One was quadric-SLAM [11], which only considers the position and size of the object detection box, without considering the specific texture in the detection box. Therefore, although the SVD (singular value decomposition) method can be used to obtain an initialization estimate of the ellipsoid, the orientation of the ellipsoid principal axis is not meaningful in practice. The other method is the object initialization method used in cube-SLAM [10], which considers the line features of the object image and, in turn, infers the orientation of the object.  
We need to point out that these methods are not strictly similar--quadric-SLAM requires at least three frames to initialize the object, and cube-SLAM obtains not only the orientation, but also the position and size of the object by single-frame inference. So we actually give quadric-SLAM more than one frame of data and record the results of quadric-SLAM initialization and the results of multi-frame optimization. Moreover, since this subsection discusses the effectiveness of our single-frame orientation observation method, we only compare the object orientation accuracy--the overall accuracy of the object landmarks will be discussed in the next subsection.  
The measurement results of our method and [10] using the same images are shown in Figure 7. Since the orientation measurement method of [10] relies mainly on the vanish points of straight-edged lines, its results will be inaccurate when there is too much texture inside the object (TUM-keyboard) or when the edges of the object are curved (ICL-chair). On the contrary, our method can still obtain the exact orientation in the above cases, because the symmetry property still exists. However, we also found some failure cases, such as the third column of Figure 7, where the chair was mistaken for a diamond shape by our method and, in turn, received an incorrect estimated orientation, due to the ambiguous information of the single frame observation.  
Figure 7: Single image 3D orientation detection examples. The first row is the result using [10]. The second row is our result.  
The quantitative results are presented in Table 1. As quadric-SLAM does not explicitly constrain the orientation of the object, the average orientation error is relatively large, reaching 38.7 degrees and 33.0 degrees, respectively. Cube-SLAM achieves the most accurate orientation in the case of ideal cube-like objects, such as cabinets and books, and we outperform other methods in most of the remaining cases. The orientation error of our method is reduced by 60.6% compared to quadric-SLAM and 46.5% compared to cube-SLAM, reaching an average error of 13.07 degrees.  
In summary, the experiments demonstrate that the properties of symmetry can help achieve finer orientation estimation of the object landmark. Considering that the vanishing point-based estimation method is more effective on small and square objects, combining the two methods may lead to more accurate results