# A Comparative Study of Weighting Methods for Local Reference Frame
## 4 Experimental Results and Analysis
### Performance Summary and Suggestions  
We have presented the comprehensive performance of the LRF obtained by the five weighting methods. Several observations can be given. (1) The weighting methods lack universality to different data modalities. For example, GF(\(\sigma=0.4\)) and TOLDI achieve good performance on the Laser Scanner and LiDAR Registration datasets but poor performance on the Kinect and Kinect Registration datasets. GF(\(\sigma=0.8\)) behaves well on the Retrieval dataset but performs poorly on the Laser Scanner dataset. (2) The application scenario has little influence on the performance of the weighting method because, for different application scenarios with the same data modality, the most suitable method is basically the same. For instance, the Laser Scanner dataset is created for object recognition, and the LiDAR Registration dataset is created for 3D registration. GF(\(\sigma=0.4\)) and TOLDI obtain the best performance on both of them. The Kinect dataset is acquired for object recognition, and the Kinect Registration dataset is acquired for 3D registration. GF(\(\sigma=0.8\)) and EM are the superior methods on both of them. (3) The weighting method is always designed to reduce the contributions of the distant points for improving the robustness to shape incompleteness. However, this fails on some datasets such as the Kinect and Kinect Registration datasets. Even the occlusion, clutter, and missing regions are present, EM and GF(\(\sigma=1\)) are the best methods. Because the point density variation is obvious in the two datasets, the contributions of the distant points should be enhanced, as shown in Figure 4b. (4) The weights of the distant points can be adjusted by varying the value of \(\sigma\) in the GF method. We can always find a proper value of \(\sigma\) for different data modalities to get a good performance. From this point of view, the GF is a generalized method, and therefore we suggest using this method to replace all the  
Figure 5: The weights calculated by the five weighting methods, including (**a**) weights, (**b**) normalized weights.  
other methods. The value of \(\sigma\) can be determined according to the descriptor matching performance to get the robustness to noise and keypoint localization error, as used in many papers. Then, if shape incompleteness is present, the value of \(\sigma\) needs to be reduced properly to get a balanced robustness to noise, keypoint localization error, and shape incompleteness. It is worth noting that the value of \(\sigma\) can also be set as 0.3, 0.5, 0.7, 0.9, 1.1, and bigger values. Better performance may be obtained than that presented in the above experiments. In addition, finding a method to adaptively determine the value of \(\sigma\) for each keypoint is also desired. (5) No method is always suitable for different support radii. As shown in Figure 3d, when the support radius is \(15 pr\), the repeatability of the LRF is the highest. Therefore, we suggest that the support radius is set as \(15 pr\). (6) As shown in Figure 3a-c, with the increase of the degree of shape incompleteness, the ranking of the methods is unchanged on the whole. Hence, the weighting method should be mainly designed to get robustness to other nuisances, and then it should be properly adjusted to get the robustness to shape incompleteness. (7) Another weighting method to improve the robustness against point density variation is necessary because the ranking of the method under the nuisance is different from that under noise and keypoint localization error, as shown in Figure 4. (8) Similarly, a mechanism should be applied to detect and eliminate outliers.