# A Fast Point Clouds Registration Algorithm for Laser Scanners
## 3 Experimental Results
### Evaluation  
RMSE can roughly represent the registration result, but it is not accurate enough. The point clouds to be registered are often not complete models, some points have no corresponding points, so the calculated RMSE may not be suitable. Additionally, when debugging the parameters, only from the RMSE results, it is not known how many points in the point cloud are involved in the registration [22]. We improved RMSE to better describe the registration results of the incomplete models, and named it the effective root mean square error (ERMSE). We set parameters \(\delta(\delta=nC,n\in Z^{*})\) and \(\beta\). \(\delta\) is the distance threshold. If the calculated Euclidean distance of the nearest neighbor pair is less than \(\delta\), it is judged as a corresponding point, otherwise, it is a non-corresponding point. \(\beta\) is the ratio of the number of the corresponding points involved in the registration to that of the point cloud. The expressions of \(\beta\) and ERMSE are shown in Equation (13). \[\left\{ \begin{array}{l}\beta=(N-k)/N\\ ERMSE=\sqrt{\frac{1}{N-k}\sum_{i=1}^{N}\|p_{i}-q_{j}\|^{2}},(1\leq j\leq M) \end{array} \right. \tag{13}\]  
where \(k\) represents the number of non-corresponding points, \(N\) represents the number of points in the registered point cloud, and \(N-k\) represents the number of corresponding points, \(p_{i}\) and \(q_{j}\) represent the nearest neighbor pair in the two point clouds. As shown in Figure 10, we used ICP, SAC-IA+ICP, 3 DHoPD+ICP and our approach for the registration of Bed, Motorbike, Piano, Dragon1, Dragon2, and Bunny point clouds. For the Bed, Motorbike, Piano, and Dragon1 models with large initial pose deviations, the registration rate of the ICP algorithm is 0, and the ERMSEs are meaningless. However, its registration accuracies for Dragon2 and Bunny are quite high, which just reflects the advantages of ICP for small initial pose deviations. The SAC-IA+ICP algorithm, 3 DHoPD+ICP algorithm, and our proposed algorithm have almost the same registration rate for the six models with the increase of \(\delta\), and when \(\delta\)= 5\(C\), the registration rates are all over 90%. However, the registration accuracy of our approach is higher for models with large initial pose deviations. From the experimental results of Dragon2 and Bunny, we can see that for models with small initial pose deviations, the registration accuracies of ICP, SAC-IA+ICP, and our method algorithms are almost the same, while 3 DHoPD+ICP algorithm is higher than them. From Figure 10, we can also get that with the increase of \(\delta\), the values of \(\beta\) and ERMSE also increase gradually. This is because the point cloud is not completely matched, and the greater the distance threshold, the greater the error. Therefore, \(\delta\) needs to be limited. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Definition** \\ \hline Grid size & 0.03 m & The voxel grid of leaf size \\ C & 0.0043 m & Point cloud resolution \\ \(r_{iss\_frame}\) & 6C & Support radius for intrinsic frame \\ \(r_{density}\) & 4C & Weighted neighbors within the radius \\ \(\epsilon_{1}\) & 0.975 & Ratio of the second eigenvalue to the first \\ \(\epsilon_{2}\) & 0.975 & Ratio of the third eigenvalue to the second \\ \(R_{3 disc\_frame}\) & 0.3 m & Support radius for 3 DSC \\ \(r_{min}\) & 0.02 m & Minimum radius of radial division to avoid noise \\ \(\rho_{i}\) & 0.002 & Density of the corresponding local point \\ \(n\) & 3 & Sample points in surface \\ \(\varepsilon_{ploy}\) & 0.01 & Edge length similarity threshold \\ \(w\) & 0.01 & Inlier fraction \\ \(\varepsilon_{transac}\) & 0.3 m & Max-distance threshold of corresponding points for RANSAC \\ \(R_{iter}\) & 3000 & Max-iteration number for RANSAC \\ \(\varepsilon_{icp}\) & 0.3 m & Max-distance threshold of corresponding points for ICP \\ \(l_{iter}\) & 30 & Max-iteration number for ICP \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters set of this algorithm.