# A Comparative Study of Weighting Methods for Local Reference Frame
## 1 Introduction
For instance, an LRF method was proposed in reference [13], in which a weighting method was developed to increase repeatability of the LRF in the presence of clutter, and the sign ambiguity of all the axes of the LRF was eliminated for the first time. In reference [24], the authors proposed an LRF method for point cloud data and developed a different weighting method to improve the robustness of the LRF to occlusion and clutter. In reference [1], an LRF method was formulated for dealing with triangular mesh data. In reference [25], an evaluation study of six LRF methods was presented in registration and object recognition context, and a novel LRF method was proposed. Only a small subset of the neighboring points around keypoint were used to determine the \(z\)-axis, and only a subset of the points lying at the periphery of the local surface were used to determine the \(x\)-axis. Subsequently, in reference [18], an LRF method was also proposed, in which the neighboring points with distances to the keypoint being smaller than a threshold were used to determine the \(z\)-axis, and all the points in the local neighborhood were projected on the tangent plane of keypoint to obtain the projection vectors, which were then integrated to form the \(x\)-axis. The \(y\)-axis was determined as the cross-product of the other two axes. In reference [20], these methods were classified into two categories: covariance analysis (CA)-based and geometric attribute (GA)-based methods, and a comprehensive performance assessment of eight LRF methods was also presented. The CA-based methods use all the neighboring points to construct the covariance matrix. The \(x\), \(y\), and \(z\) axes of the LRF are obtained by performing the eigenvalue decomposition on the covariance matrix. In contrast, the GA-based methods exploit a subset of the neighboring points to get the \(z\)-axis. Then, the \(x\)-axis is achieved according to geometric attributes (e.g., signed distance [25]). Finally, the \(y\)-axis is obtained by the cross-product of the x and z axes. The experimental results showed that no method is always suitable for different data modalities and application scenarios, and the GA-based methods are generally more time consuming compared to the CA-based methods. Although the comparative studies of both the LSDs and the LRF methods have been presented in the literature, the comparative study of weighting methods for the LRF has not been conducted. The performance of the LRF is affected by multiple factors, so comparing the LRF methods directly is very hard to determine which factors make an improvement to the LRF. A weighting method plays an important role in an LRF method. Assigning proper weight values to the points in the neighborhood can improve the repeatability and robustness of the LRF. Therefore, in this paper, we focus on the comparison of weighting methods. Majority of these methods calculate the weight values for the points in the local neighborhood according to the distances from the keypoint to these points, but which methods are better is unknown. In this paper, five weighting methods are compared, and a quantitative assessment of them is presented. Six datasets with different data modalities and application contexts are selected to perform the comparative study. The repeatability of the LRF against various nuisances is tested. In addition, the Gaussian function is introduced to build the LRF. Unlike other weighting methods, as long as the weighting function is determined, the weight of each point is fixed. In the Gaussian function, the weight of each point can be changed to adjust its contribution to the LRF. Through the comparative study, the main conclusions are summarized as follows:  
1. When the wights of the distant points in the local neighborhood are too large, the LRF is sensitive to occlusion, clutter, partial overlap, noise, outliers, and keypoint localization error. By contrast, when the wights of the distant points are too small, the LRF is susceptible to noise, varying point density, and keypoint localization error. 2.