# A Multivariate Local Descriptor Registration Method for Surface Topography Evaluation
## 3 Part-in-Whole Point Cloud Registration
### Coarse Registration Based on Multivariate Descriptors (MD)  
Based on spheres obtained from the above-mentioned method, each sphere's feature needs to be described. Feature descriptors are generally used for the initial alignment of the point cloud in the coarse registration stage. Common point cloud descriptors can be divided into global and local descriptors. Using a global descriptor requires that the size of the local measurement be much smaller than that of the global point cloud. However, some measurement data obtained for small workpieces are one-sided projections. When the  
Figure 8: (**a**) multi-scale RSCS of measurement data; (**b**) multi-scale RSCS of CAD model. whole point cloud is divided according to the spherical scale of the local point cloud, the whole point cloud will be divided into a sphere, leading to registration failure. Therefore, point cloud local descriptors will be adopted in this study. This study proposed a multivariate descriptor to improve registration accuracy. The combination of two classical descriptors: Fast Point Feature Histogram (FPFH) [33], and Shot descriptor, which is based on a local reference frame, is used in a subsequent experiment. The local descriptors used are not limited to the above two manual descriptors; it can also use some descriptors based on deep learning, such as Spinnet [34]. This study presented an optimised scheme to improve the description ability of descriptors. The descriptors do not belong to the research subject of this paper. Therefore, based on the hypothesis of the scheme, assuming that \(n\) local feature descriptors are eventually used, each sphere will get n descriptor vectors. Coarse registration adopts sample consensus initial alignment (SAC-IA) [33], and the main contents of SAC-IA are as follows:  
1. The descriptor sets of multi-scale RSCS of LMPC and global point cloud are calculated, \(\hat{p}_{n}=\left\{d_{1}^{i},d_{2}^{i},\cdots,d_{k}^{i}\right\}\), \(\hat{q}_{n}=\left\{d_{1}^{i},d_{2}^{i},\cdots,d_{K}^{i}\right\}\), where \(n\) represents different descriptors, \(i\) is RSCS of different scales, \(k\) and \(K\) are the number of LMPC and whole CAD point cloud at a certain scale. 2. \(n\) descriptors are randomly selected from \(\hat{p}_{n}\) and \(\hat{q}_{n}\) to determine the \(n\) corresponding relationship. The centre of RSCS is defined as the corresponding point. Supposing n descriptors in \(\hat{p}_{n}\) determine the most similar corresponding \(n\) points in \(\hat{q}_{n}\), the weighted average points between the corresponding points of \(n\) different descriptors are calculated as the final corresponding points. Then the final corresponding point position in \(\hat{q}_{n}\) is defined by Equation (11), where \(w_{i}\) is the normalised weight of the matching score in corresponding points from \(n\) descriptor and \(c_{i}\) is the position of the corresponding point from a different descriptor. Figure 9 is a sketch of a weighted average corresponding point with three descriptors, in which the blue is the local point cloud, and the grey is the whole DMPC. In the whole DMPC, different shapes correspond to the corresponding points calculated by different descriptors, and the circular points are the final weighted corresponding points. 3. After determining the corresponding points, the corresponding points of the s group are randomly selected to obtain the corresponding point set \(\hat{p}_{c}=\left\{c_{1},c_{2},\cdots,c_{s}\right\}\) and \(\hat{q}_{c}=\left\{l_{1},l_{2},\cdots,l_{s}\right\}\), according to Equations (15) and (16), the transformation matrices \(R\) and \(T\) are calculated, where U and V are singular matrices; it is obtained by performing a singular value decomposition on Equation (14). 4.