# A Comparative Study of Weighting Methods for Local Reference Frame
## 1 Introduction  
Local shape description has been proved to be very successful in the field of photogrammetric engineering, computer vision and graphics. Numerous applications are performed by the local shape description, such as 3D object recognition [1; 2; 3], simultaneous localization and mapping [4], 3D modeling and scene reconstruction [5; 6], shape retrieval [7], and 3D object categorization [8]. In comparison with the global shape descriptor, the local shape descriptor (LSD), which is an important component in the local shape description, holds a number of advantages, including being robust against clutter, occlusion, and missing regions [9]. This leads to that many scholars start to do research on the LSD. A large number of LSDs have been proposed, which encode the geometric information of the local neighborhood around a specific keypoint by calculating geometric features such as distance [10], local depth [11], and normal deviation [12]. Essentially, the LSD should be invariant to rigid transformation, because point-to-point correspondences are established by matching the LSDs. However, in real applications, the LSD is easily influenced by noise, point density variation, clutter, occlusion, and missing regions. Therefore, an ideal LSD should have the ability to resist these nuisances. Many efforts have been made to deal with these difficulties, such as signature ofhistograms of orientations (SHOT) [13], rotational projection statistics (RoPS) [1], Tri-Spin-Image (TriSI) [14], and rotational contour signatures (RCS) [15]. These descriptors can be broadly classified into two categories: with or without a local reference frame (LRF). The descriptors without LRF (e.g., spin image [16] and fast point features histograms (FPFH) [17] mainly take the geometric information into consideration but discard the spatial information, thus they usually suffer from low descriptiveness. In contrast, the LRF-based descriptors (e.g., RoPS and TriSI) first build an LRF for the local surface and then characterize the geometric and spatial information with respect to the LRF [18]. A comprehensively comparative study of several LSDs had been performed, and the results showed that the LRF-based descriptors generally outperform those descriptors without LRF in most public datasets [19]. An LRF built on the local surface is a 3D coordinate system independent of the world coordinate system. The aim of building the LRF for the LSD is at least twofold. The first is to make the LSD invariant to rigid transformation, because the LRF is independent of the world coordinate system. The second is to provide fully spatial information for local shape description. This makes the descriptors with LRF more descriptive. As an important component of the LSD, the repeatability and robustness of the LRF directly affect the descriptiveness and robustness of the LSD. Large LRF errors could result in very poor descriptor matching performance [20]. In addition, the LRF also has many other uses. For example, in reference [21], the 1-point random sample consensus (RANSAC) algorithm was proposed to produce robust transformation parameters. A pair of points rather than three pairs of points are used to calculate the transformation parameters because the LRFs of the pair of points can also provide convenience for the calculation of the transformation parameters. This largely reduces the computational complexity. In reference [22], the 3D Hough voting method based on the LRF was proposed to detect 3D objects by using a set of transformations, and in reference [23], the authors used the LRF to remove outliers for shape correspondence problems. The performance of all these methods largely depends on a repeatable and robust LRF. In order to improve the descriptor matching performance, a wide variety of LRF methods have been proposed.