## 2 Methodology
### Registration
#### 2.3.2 Fine Registration  
After coarse registration, the centroid coordinate of each point sets group will be calculated as the center of matching point to find the closest matched one in TLS point clouds. Then fine registration will be carried out. ICP is adopted as fine registration method, which is an algorithm for accurate registration of two point sets. ICP looks for points with a corresponding relationship between the two point sets and puts forward the points with a corresponding relationship to form a point set and regards the point set as a rigid body. After calculating the centroids of two rigid bodies, the centroid of two rigid bodies are rotated and shifted for making them close to each other. Then through iteration, the distance between the two point sets is gradually reduced and the direction tends to be the same. From a mathematical point of view. By repeatedly determining the corresponding point relation and calculating the rigid body transformation, two points are successfully matched when the distance between two point sets is less than a certain convergence threshold. ICP algorithm processing flow is shown in Figure 11.  
Let \(\mathbf{P}\) and \(\mathbf{Q}\) be the corresponding point sets in the UAV photogrammetric point clouds and TLS point clouds, respectively. The main process of ICP algorithm includes the following steps:  
1. Constructing the objective function of the least square method and setting the threshold as \(\Delta\): \[E(\mathbf{R},\mathbf{T})=\frac{1}{n}\sum_{i=1}^{n}\left\|q_{i}-(\mathbf{R}p_{i}+\mathbf{T}) \right\|^{2}\] (10) where, the objective function \(E\) is the mean of the sum of each distances between corresponding point pair, \(p_{i}\) is in the point set \(\mathbf{P}\) and \(q_{i}\) is in the point set \(\mathbf{Q}\), \(\mathbf{R}\) and \(\mathbf{T}\) are the rotation matrix and translation matrix respectively. 2. Find the centroid \(\mu_{p}\) and \(\mu_{q}\) of the point set \(\mathbf{P}\) and \(\mathbf{Q}\), and convert the position relation of each point into the relative centroid position relation: \[\mu_{p}=\frac{1}{n}\sum_{i=1}^{n}p_{i}\] (11) \[\mu_{q}=\frac{1}{n}\sum_{i=1}^{n}q_{i}\] (12) \[p_{i}^{\prime}=p_{i}-\mu_{p},\ q_{i}^{\prime}=q_{i}-\mu_{q}\] (13)
3. The matrix \(\mathbf{M}\) is constructed according to \(\mathbf{P}^{\prime}\) and \(\mathbf{Q}^{\prime}\) obtained after transformation, so that: \[\mathbf{M}=\sum_{i=1}^{n}p_{i}^{\prime}q_{i}^{\prime T}\] (14)
4. Singular value decomposition of matrix \(\mathbf{M}\) can be obtained: \[\mathbf{M}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{T}=\mathbf{U}\left[ \begin{array}{ccc}\sigma_{1 }&0&0\\ 0&\sigma_{2}&0\\ 0&0&\sigma_{3}\end{array} \right]\mathbf{V}^{T}\] (15)  
Figure 11: ICP algorithm processing flow. \[\mathbf{D}=\left\{ \begin{array}{c}\mathbf{I}_{3}=\left[\begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right]&\det(\mathbf{U})\text{det}(\mathbf{V})\geq 0\\ \\ \mathbf{diag}(1,1,-1)=\left[\begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&-1\end{array}\right]&\det(\mathbf{U})\text{det}(\mathbf{V})<0\end{array} \right. \tag{5}\]  
When \(rank(\mathbf{D})\geq 2\), \(\mathbf{R}=\mathbf{U}\mathbf{D}\mathbf{V}^{T}\):  
\[\mathbf{R}=\mathbf{U}\mathbf{D}\mathbf{V}^{T},\ \mathbf{T}=\mu_{q}-\mathbf{R}\mu_{p} \tag{17}\]  
When \(rank(\mathbf{M})=3\), there is a unique optimal rotation matrix \(\mathbf{R}_{opt}\) and translation matrix \(\mathbf{T}_{opt}\), making the objective function \(E\left(\mathbf{R},\mathbf{T}\right)\) minimum. \[\mathbf{R}_{opt}=\mathbf{U}\mathbf{V}^{T},\ \mathbf{T}_{opt}=\mu_{q}-\mathbf{R}_{opt}\mu_{p} \tag{18}\]  
(6)  
Finally, the value of the objective function was calculated under the current rotation matrix and translation matrix, and be compared with the threshold. If the value is less than the threshold, the iteration ends; otherwise, the cycle is repeated step (2). Point clouds after matching and registration are shown in Figure 11. Compared with the point clouds generated by UAV, the density of registration point clouds is well (Figure 12).