## 2 Methodology
### Registration
#### 2.3.1 Coarse Registration  
Coarse registration is carried out with PCA. PCA is used to reduce the dimensionality of data. It retains some characteristics of large variance in point clouds, and is used to ensure that the coarse registration process reaches better efficiency and accuracy. Assuming two point sets: UAV photogrammetric point clouds as \(\mathbf{P}=\{p_{1},p_{2},p_{3},\ldots,p_{n}\}\), TLS as \(\mathbf{Q}=\{q_{1},q_{2},q_{3},\ldots,q_{m}\}\), calculate the centroid \(\overline{p}\) (Equation (6)), \(\overline{q}\) (Equation (7)) and covariance matrix _cov_ (Equation (8)).  
\[\overline{p}=\
icefrac{{1}}{{n}}\sum_{1}^{n}p_{i},\text{as}\left[\begin{array} []{c}\overline{x}\\ \overline{y}\\ \overline{z}\end{array}\right]=\
icefrac{{1}}{{n}}\left[\begin{array}{c} \sum_{i}^{n}x_{i}\\ \sum_{1}^{n}y_{i}\\ \sum_{1}^{n}z_{i}\end{array}\right] \tag{6}\]  
\[\overline{q}=\
icefrac{{1}}{{n}}\sum_{1}^{n}q_{i},\text{ as }\left[\begin{array}{c} \overline{x}\\ \overline{y}\\ \overline{z}\end{array}\right]=\
icefrac{{1}}{{n}}\left[\begin{array}{c} \sum_{i}^{n}x_{i}\\ \sum_{1}^{n}y_{i}\\ \sum_{1}^{n}z_{i}\end{array}\right] \tag{7}\]  
Figure 9: Registered point clouds on the edge of the view-blocked area. (**a**,**c**) are not registered point clouds. (**b**,**d**) are registered point clouds based on (**a**,**c**), respectively.  
\[cov=\left[\begin{array}{ccc}cov(x,x)&cov(x,y)&cov(x,z)\\ cov(y,x)&cov(y,y)&cov(y,z)\\ cov(z,x)&cov(z,y)&cov(z,z)\end{array}\right] \tag{8}\]  
The eigenvalues and eigenvectors were calculated based on \(cov\), and the original coordinates were redecomposed long the direction of the eigenvectors to obtain the UAV photogrammetric point clouds in the eigenvector coordinate system. Similarly, the centroid \(\overline{q}\) of TLS point clouds is shifted to make it coincide with the centroid \(\overline{p}\) of UAV photography point clouds, and TLS point clouds in the feature vector coordinate system are transformed to the direction of the UAV photography point clouds through rotation, so that the two coordinate systems are unified. At this time, point clouds of the two parts are converted to the same coordinate system and a preliminary registration of the overall fusion point clouds is obtained. Finally, the position of the two point clouds is judged by the point clouds minimum boundary box [35]. Minimum bounding box is an algorithm to solve the optimal enclosing space of discrete point sets. It replaces complex geometric objects with slightly larger volume and simple characteristics, as shown in Figure 10. Since the point clouds in the minimum enveloping box all meet the conditions:  
\[\left\{\begin{array}{l}x_{min}\leq x_{c}\leq x_{max}\\ y_{min}\leq y_{c}\leq y_{max}\\ z_{min}\leq z_{c}\leq z_{max}\end{array}\right. \tag{9}\]  
Therefore, each vertex (\((x_{max},\ y_{max},\ z_{max})\), \((x_{min},\ y_{max},\ z_{max})\), \((x_{max},\ y_{min},\ z_{max})\), \((x_{max},\ y_{min},\ z_{max})\), \((x_{max},\ y_{max},\ z_{min})\), \((x_{min},\ y_{min},\ z_{max})\), \((x_{min},\ y_{max},\ z_{min})\), \((x_{max},\ y_{min},\ z_{min})\), \((x_{min},\ y_{min},\ z_{min})\)) of the minimum bounding box can be found by reading the maximum and minimum values of \(x\), \(y\) and \(z\) in the point clouds, then, the region of the bounding box is delineated. Finally, the overlap rate of between the two point clouds bounding box can be used to judge whether the coordinate direction of the two part point clouds is opposite. When the overlap rate (The ratio of the number of point clouds in the overlapping region of the enveloping box to the total number of point clouds in itself) of the two point clouds exceeds a threshold, the coordinates of the two point clouds are in the same direction, and the coarse registration of the two part point clouds is completed. In this study, the overlap rate threshold was set at 90%.  
Figure 10: The TLS point clouds minimum bounding box.