# MPCR-Net: Multiple Partial Point Clouds Registration Network Using a Global Template
## 4 Experiments
### Experiments Based on Untrained Models
#### 4.2.1 Evaluation Criteria for Experiments  
1. Estimation accuracy  
The estimation accuracy of correspondence points refers to the proportion of correspondence points correctly estimated by the algorithm during the experiment to the actual number of correspondence points. Taking MPCR-Net as an example, the estimation process of TPCC-Net is as follows:  
Suppose the point clouds input into the network are partial point cloud A and global template point cloud B, and the point number of B is \(N_{B}\). Encode data points in B and extract the index addresses \(M_{Tue}\in\mathbb{R}^{N_{x}\times 2}\) of all correspondence points of A and B. Input A and B to TPCC-Net to estimate the index address \(M_{ES}\in\mathbb{R}^{N_{x}\times 2}\). Count the number of elements that are the same in \(M_{Tue}\) and \(M_{ES}\), and denote it as \(N_{ES}\). The actual number of correspondence points is equal to \(N_{B}\); thus, the estimation accuracy of correspondence points of TPCC-Net is:  
\[P_{1}=\frac{N_{ES}}{N_{B}} \tag{14}\]  
2. Registration error  
Rigid body transformation is composed of translation and rotation transformations. The registration error is subdivided into rotation and translation transformation errors. The registration error is calculated as follows:  
Suppose the actual rotation angle of the partial point cloud relative to the template point cloud in the three Cartesian coordinate axis directions is \(R=\{a,\beta,\gamma\}\), and the actual translation distance is \(t=\{d_{0},d_{1},d_{2}\}\). Input the global template point cloud and the corresponding point set of the partial point cloud in the global template point cloud into TMPE-Net and estimate the parameter vector \(Z=\left[ \begin{array}{cccccc}a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&a_{5}&a_{6}\end{array} \right]\). Elements \(a_{0},\ldots,a_{3}\) in \(Z\) represent the four parameters for the quaternion rotation matrix. The rotation Euler angles in the three directions corresponding to the parameter vector can be obtained according to the relationship between the quaternion rotation matrix and the Euler angle rotation matrix:  
\[R^{\prime}=\left\{ \begin{array}{c}\alpha^{\prime}\\ \beta^{\prime}\\ \gamma^{\prime}\end{array} \right\}=\left\{ \begin{array}{c}\arctan\frac{2(a_ {0}a_{1}+a_{2}a_{3})}{1-2(a_{1}^{2}-a_{2}^{2})}\\ 2\arcsin\left(a_{0}a_{2}-a_{3}a_{1}\right)\\ \arctan\frac{2(a_{0}a_{3}+a_{1}a_{2})}{1-2(a_{2}^{2}-a_{3}^{2})}\end{array}  \right\} \tag{15}\]  
Calculate the mean absolute error \(\Delta R\) between \(R^{\prime}\) and \(R\):  
\[\Delta R=\frac{|\alpha^{\prime}-\alpha|+|\beta^{\prime}-\beta|+|\gamma-\gamma^ {\prime}|}{3} \tag{16}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Environment** & \multicolumn{2}{c}{**Configuration**} \\ \hline \multirow{4}{*}{Software} & Operating system & Windows 10 \\  & Deep learning framework & \multirow{2}{*}{Pytorch 1.8.1 + CUDA 11.0 + cuDNN} \\  & Programming language & & \\  & Point cloud processing library & Open3D \\  & CPU & Intel(R) Core(TM) i5-9400F \\ \multirow{2}{*}{Hardware} & Memory & \multirow{2}{*}{16 GB} \\  & Graphics card & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental environment. \(\Delta R\) is the rotation transformation error of TMPE-Net. Elements \(a_{4},a_{5},a_{6}\) in \(Z\) represents the translation distance \(t^{\prime}=\{a_{4},a_{5},a_{6}\}\) of the partial point cloud relative to the template point cloud in the three directions estimated by TMPE-Net, and the average absolute difference \(\Delta t\) between \(t^{\prime}\) and \(t\) is calculated as follows:  
\[\Delta t=\frac{|a_{4}-d_{0}|+|a_{5}-d_{1}|+|a_{6}-d_{3}|}{3} \tag{17}\]  
\(\Delta t\) is the translation transformation error of TMPE-Net. 3.