## 4 Experiments
### Experiments on the Performance of Descriptors
#### 4.2.3 Comparison between Different Feature Vectors for Covariance Descriptor  
Random variables in the feature vector play a significant part in the generation of our proposed covariance matrix descriptor, which impacts not only the encapsulation ability of information in the covariance matrix but also the length of the feature vector. The following feature vectors are concatenated by random variables in this paper:  
\[F_{3}=\left(\cos(\alpha(p_{i}p)),\cos(\beta(p_{i}p)),\cos(\gamma(p_{i}p))\right)\]  
\[F_{4}=\left(\sigma(p_{i}p),\lambda_{1}(p_  
vector information of neighboring points. These normal vectors lack rotation translation invariance as well. We find keypoints from a source point cloud using our proposed keypoint detection method with boundary removal in these experiments. Simultaneously, to ensure that the entire number of positives is known, we must construct keypoints of the target point cloud that match to keypoints of the source point cloud. The following description depicts the method of producing corresponding keypoints of target point clouds:  
* Transform the source point cloud's keypoints using the ground truth matrix;
* For every keypoint in transformed keypoints, find the closest point to it in the target point cloud and designate the point as a keypoint of the target point cloud. In the process of generating the covariance matrix descriptor, the support radius is set to 6 mr, 7 mr, 8 mr, and 9 mr, correspondingly. The average recall and 1-precision are determined under three Gaussian noise situations. Figure 6 depicts the experimental results, where Figure 7a-c represent RP curves at 0.1 mr, 0.3 mr, and 0.5 mr, respectively. Figure 6 shows that under the three noises, the performance of the covariance matrix descriptor formed by the feature vectors \(F_{3}\), \(F_{4}\), and \(F_{7}\) is better than the performance of the covariance matrix formed by the feature vectors \(F_{10}\) and \(F_{13}\), and the performance of the covariance matrix formed by the feature vectors \(F_{10}\) and \(F_{13}\) will decline rapidly as the noise deviation increases. This is due to two factors. First, the random variables in \(F_{3}\), \(F_{4}\), and \(F_{7}\) have rotation and translation invariance, but the random variables in \(F_{10}\) and \(F_{13}\) do not have rotation and translation invariance, such as absolute spatial coordinates x, y, z, and so on. Second, the random variables \(F_{3}\), \(F_{4}\), and \(F_{7}\) include the geometric structure, surface variation gradient, and eigenvalue variation gradient, which are all robust to noise, whereas the random variables \(F_{10}\) and \(F_{13}\) include normal vector and spatial coordinates that are not robust to noise. In the meantime, when the performance of the covariance matrix descriptor constructed from the feature vectors \(F_{3}\) and \(F_{4}\) is compared, it can be seen that the performance of the covariance matrix descriptor formed by \(F_{4}\) is better than the performance of the covariance matrix descriptor formed by \(F_{3}\), which is due to the fact that surface variation gradient and eigenvalue variation gradient are more stable and more robust to noise than geometric structure. Additionally, by comparing the performance of the covariance matrix descriptor constructed by the feature vector \(F_{3}\), \(F_{4}\), and \(F_{7}\), it can also be shown that the performance of the covariance matrix descriptor will be enhanced by adding more random variables with strong robustness to noise and invariance to rotation and translation to the feature  
Figure 7: Performance of covariance matrix descriptor formed by different feature vectors under different noise conditions. vector. [PERSON] et al. [37] obtain a similar result and recommend that the number of random variables in the feature vector be limited to roughly 15. In this study, the length of the random variables is 7. Despite the limited number of random variables, the reduced dimensionality of the covariance matrix descriptor can increase time efficiency and minimize space overhead.