## 2 Methodology
### Generating UAV Photogrammetric Point Clouds
#### 2.1.2 Spatial Conversion  
Before registration of two types of point clouds, the coordinate needs to be unified as world coordinate system due to UAV photographic measurement point clouds is generated in its own coordinate system. In a high-resolution image, every pixel, such as \(P(u,v)\), is marked in image coordinate system. However, not all pixels have position associated mapping with the actual distance on the position, so the position coordinate system is introduced to bind pixel-position relationship and reflect the actual distance between any two pixels. For instance, there is a point in image which can be represented as \((x,y)\) in the pixel position coordinate system, as Figure 3 shows. The interactive transforming relation can be presented as Equation (1). In the Equation (1), \(\Delta x\) and \(\Delta y\) are the unit distance on \(x\)-axis and \(y\)-axis between two adjacent pixels.  
\[\left\{\begin{array}{l}u=x/\Delta x+u_{0}\\ v=y/\Delta y+v_{0}\end{array}\right.\text{or}\left\{\begin{array}{l}x=(u-u_{ 0})\Delta x\\ y=(v-v_{0})\Delta y\end{array}\right. \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No** & **1** & **2** & **3** & **4** & **5** \\ \hline Width \(\times\) height/pixel & 5472 \(\times\) 3684 & & 4608 \(\times\) 3405 & \\ SIFT time-consuming/s & 4.313 & 4.582 & 2.750 & 2.752 & 2.576 \\ The number of feature points extracted & 426,067 & 448,317 & 13,796 & 29,538 & 9216 \\ SURF time-consuming/s & 3.915 & 4.833 & 1.734 & 1.910 & 1.517 \\ The number of feature points extracted & 139,640 & 127,072 & 4850 & 9156 & 3089 \\ by SURF & & & & & \\ \hline \end{tabular}
\end{table}
Table 1: The comparison of SIFT and SURF algorithms.  
Figure 3: Transformation to pixel position coordinate.  
Figure 2: The identification of interest points by using SIFT and SURF algorithms. **(a)** SIFT; **(b)** SURF.  
Figure 4 then any pixel coordinate in camera coordinate system can be calculated as Equation (2). In the Equation (2), \(z\) is the vertical distance from the camera to the origin of pixel position coordinate system.  
\[\left\{\begin{array}{c}x_{c}=x=(u-u_{0})\Delta x\\ y_{c}=y=(v-v_{0})\Delta y\\ z_{c}=z_{O_{c}}+z=z_{O_{c}}+\left|O_{c}O_{p}\right|\end{array}\right. \tag{2}\]  
The transformation from camera coordinate system to world coordinate system can be achieved by rotation matrix and translation matrix, presented as \(\mathbf{R}\) and \(\mathbf{T}\), respectively.  
The camera on UAV automatically records the coordinate of it in world coordinate system, so the transformation can be executed itself. The final transformation relationship from image coordinate to pixel position coordinate to camera coordinate and finally to world coordinate system can be expressed as a series of equation matrix (Equations (3)-(5)).  
\[\left[\begin{array}{c}u\\ v\\ 1\end{array}\right]=\left[\begin{array}{cc}\
icefrac{{1}}{{\Delta x}}&0&u_{ 0}\\ 0&\
icefrac{{1}}{{\Delta y}}&v_{0}\\ 0&0&1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right] \tag{3}\]  
\[z_{c}\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]=\left[\begin{array}{cc}\left|O_{c}O_{p}\right|&0&0&0\\ 0&\left|O_{c}O_{p}\right|&0&0\\ 0&0&1&0\end{array}\right]\left[\begin{array}{c}x_{c}\\ y_{c}\\ z_{c}\\ 1\end{array}\right] \tag{4}\]  
\[\left[\begin{array}{c}x_{c}\\ y_{c}\\ z_{c}\\ 1\end{array}\right]=\left[\mathbf{R}|\mathbf{T}\right]\left[\begin{array}{c}x_{w}\\ y_{w}\\ z_{w}\\ 1\end{array}\right] \tag{5}\]  
With the real coordinate of UAV photography, the actual distance and position relationship can be factually reflected, then the successive research work can go on.