# A Multivariate Local Descriptor Registration Method for Surface Topography Evaluation
## 5 Experimental Verification and Comparisons
### The Analysis of Registration  
In point cloud registration experiments, two point clouds are sampled with a voxel of size 1, and the registration error is calculated as  
\[RMS=\frac{1}{n}\sqrt{\sum(p_{i}-f(q_{i}))^{2}} \tag{18}\]  
where \(p_{i}\) represents the target point cloud, \(q_{i}\) represents the measurement point cloud, and \(f\) represents the point cloud after registration. First, workpiece b is used to compare the registration accuracy of single-scale RSCS and multi-scale RSCS, and the results are shown in Table 1. Figure 14a-c shows the coarse registration results of spheres with scales of \(20,50\) and \(80\), respectively. It can be seen that coarse registration results are relatively superior when the scale is \(80\), and using multiple scales can increase the accuracy of registration compared to a single scale. In general, the scale of the sphere in RSCS can be made according to the volume of the measured data. There is an empirical formula for setting a sphere's scale, expressed in Equation (19), where \(V\) represents the volume of the minimal circumscribed sphere, and m is the number of spheres in RSCS.  
Figure 13: (**a**) Digital model point cloud and a measurement point cloud of part 1; (**b**) Digital model point cloud and a measurement point cloud of part 2.  
The registration results of parts 1 and 2 are shown in Figures 15 and 16 and Table 2. The labels a and b represent the coarse and fine registration results. Table 2, _\"part number_registration scheme\"_, shows the results of different registration schemes with different parts. Although only two descriptors are used in the multivariate descriptors in the experiment, the result is better than using a single descriptor. In the proposed coarse algorithm, the matching relation of the point cloud patch in the final RSCS is shown in Figure 16, where the black point is the weighted average corresponding point, the different colour of the point cloud patch is the sphere of the final RSCS, the line represents the corresponding relation. Although the proposed scheme can reduce the registration error and increase the probability of registration success, it can be seen from the broken line figure in Figure 17 that with the increase of sphere candidate scale and descriptor, especially with the rise of sphere candidate scale, the time consumption presents an exponential growth, so it needs a tradeoff in terms of actual use.  
\[Radius=\sqrt[3]{(\frac{3}{4\pi}*\frac{0.7}{2m}*V)} \tag{19}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**The Scale of RSCS** & **Registration Error** \\ \hline
20-scale RSCS+MD+ICP & 2.095 \\
50-scale RSCS+MD+ICP & 4.233 \\
80-scale RSCS+MD+ICP & 4.856 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Influence of RSCS at different scales on parts registration error (mm).  
Figure 14: Registration results of RSCS at different scales. (**a**) Scale of 20; (**b**) Scale of 50; (**c**) Scale of 80.  
Figure 15: Results of different registration schemes for part 1: (**a**) coarse registration; (**b**) fine registration.