# A Comparative Study of Weighting Methods for Local Reference Frame
## 2 Overview of Five Weighting Methods  
The weighting methods are mainly related to the CA-based methods, so we employ the CA-based methods to perform the comparative study. Furthermore, the CA-based methods generally have higher computational efficiency in comparison with the GA-based methods and are easy to implement. The sign disambiguation technique [13] is applied to eliminate the sign ambiguity of the LRF for all the considered methods because our aim is to compare the weighting methods. In addition, we also use the keypoint to replace the barycenter when computing the covariance matrix, as proposed in reference [13]. The process diagram of building an LRF is shown in Figure 1.  
The covariance matrix is computed as  
\[C(\mathbf{p})=\frac{1}{\sum\limits_{\mathbf{q}_{i}\in\mathbf{N}(\mathbf{p})}w_{q_{i}}}\sum \limits_{\mathbf{q}_{i}\in\mathbf{N}(\mathbf{p})}w_{q_{i}}(\mathbf{q}_{i}-\mathbf{p})(\mathbf{q}_{i}- \mathbf{p})^{T}, \tag{1}\]  
where \(\mathbf{p}\) is the keypoint, \(N(\mathbf{p})\) is the radius neighborhood of \(\mathbf{p}\), \(\mathbf{q}_{i}\) is a neighboring point in \(N(\mathbf{p})\), \(w_{q_{i}}\) is the weight of \(\mathbf{q}_{i}\). After performing the eigenvalue decomposition on \(C(\mathbf{p})\), we can obtain three eigenvalues \((\lambda_{1},\lambda_{2},\lambda_{3})\) that satisfy \(\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\) and three corresponding eigenvectors \((\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3})\). The three eigenvectors provide a basis for LRF definition. Specifically, the sign disambiguation technique is applied to redetermine the signs of \(\mathbf{v}_{1}\) and \(\mathbf{v}_{3}\), producing two unambiguous vectors \(\overline{\mathbf{v}}_{1}\) and \(\overline{\mathbf{v}}_{3}\). \(\overline{\mathbf{v}}_{2}\) is defined as \(\overline{\mathbf{v}}_{1}\times\overline{\mathbf{v}}_{3}\), and \(\overline{\mathbf{v}}_{1}\), \(\overline{\mathbf{v}}_{2}\) and \(\overline{\mathbf{v}}_{3}\) define the directions of the x, y, and z-axes, respectively. Assigning different weights to the points in \(N(\mathbf{p})\) leads that each point makes different contribution to the covariance matrix. Several methods have been proposed to determine the weights, which are briefly described as follows. **EM**[26]: This method does not take any nuisances (e.g., occlusion and clutter) into consideration. The weights for all points are just set as \(1\). That is  
\[w_{q_{i}}=1. \tag{2}\]  
It means that all points in the local neighborhood make the same contribution to the covariance matrix. In spite of this, the method may obtain more repeatable and robust LRF for some data modalities, as will be shown in Section 4.1. **SHOT**[13]: This method treats the local points unequally to enhance the robustness under 3D object recognition scenarios. The weights are computed as  
\[w_{q_{i}}=R-\|\mathbf{q}_{i}-\mathbf{p}\|, \tag{3}\]  
where \(R\) is the support radius, \(\|\mathbf{q}_{i}-\mathbf{p}\|\) is the 3D Euclidean distance between \(\mathbf{q}_{i}\) and \(\mathbf{p}\). Thus, the distant points (i.e., the neighboring points have larger distances to the keypoint) are assigned smaller weights. In other words, the contributions of the distant points to the covariance matrix are weakened. The purpose is to increase repeatability in the presence of clutter. **BSC**[24]: [PERSON] et al. developed a different weighting strategy to determine the weights. It also assigns smaller weights to the distant points. The weights are computed as  
\[w_{q_{i}}=\frac{R-\|\mathbf{q}_{i}-\mathbf{p}\|}{R}. \tag{4}\]  
This weighting strategy is expected to improve the robustness of the LRF against occlusion and clutter because the distant points are easily influenced by occlusion and clutter. Unlike the SHOT method, this method limits the weight values to be between 0 and 1. **TOLDI**[18]: Although the method is designed for a GA-based LRF method, it also computes the weight values according to the distances. Therefore, we also include it in the comparison.