## 3 Methodology
### 1-N Mapping Relationship between TLS Point Cloud and Digital Images  
As the point in the TLS point cloud may be projected to more than one image, this paper defines a 1-\(N\) mapping relationship between the TLS point cloud and digital images. The coordinate transformation mathematical model is the key to get the 1-\(N\) mapping relationship. Suppose one point \(P_{i}=\left(X_{\text{TLS}},Y_{\text{TLS}},Z_{\text{TLS}}\right)^{\text{T}}\) in the TLS point cloud, and the corresponding point \(P_{i}^{\prime}=\left(X_{\text{TLS}}^{\prime},Y_{\text{TLS}}^{\prime},Z_{ \text{TLS}}^{\prime}\right)^{\text{T}}\) in the SfM point cloud generated by digital images could be obtained according to the rigid transformation model of registration, which is shown as Equation (5). \[P_{i}^{\prime}=\left[ \begin{array}{c}X_{\text{TLS}}^{\prime}\\ Y_{\text{TLS}}^{\prime}\\ Z_{\text{TLS}}^{\prime}\end{array} \right]=\mu\mathbf{R}\left[ \begin{array}{c}X_ {\text{TLS}}\\ Y_{\text{TLS}}\\ Z_{\text{TLS}}\end{array} \right]+\mathbf{T}=\mu\mathbf{R}P_{i}+\mathbf{T}, \tag{5}\]  
where \(\mathbf{R}\) and \(\mathbf{T}\) respectively denotes the rotation and translation matrix; \(\mu\) means the scale factor. Let the rotation matrix \(\mathbf{R}_{j}\) and translation matrix \(\mathbf{T}_{j}\) of image \(I_{j}\), then in the camera coordinate system of image \(I_{j}\), coordinates of the point \({P_{i}^{(j)}}^{\prime}\) which corresponds to the point \(P_{i}\) could be calculated by Equation (6):  
\[P_{i}^{(j)\prime}=\left[ \begin{array}{c}X_{\text{TLS}}^{(j)}\\ Y_{\text{TLS}}^{(j)}\\ Z_{\text{TLS}}^{(j)}\end{array} \right]=\mathbf{R}_{j}\left[ \begin{array}{c}X_{ \text{TLS}}^{\prime}\\ Y_{\text{TLS}}^{\prime}\\ Z_{\text{TLS}}^{\prime}\end{array} \right]+\mathbf{T}_{j}=\mathbf{R}_{j}P_{i}^{\prime}+ \mathbf{T}_{j}, \tag{6}\]  
All elements of \({P_{i}^{(j)}}^{\prime}\) are divided by \(\left(-Z_{\text{TLS}}^{(j)}\right)^{\prime}\), and denotes as Equation (7):  
\[p=-\left[ \begin{array}{c}X_{\text{TLS}}^{(j)}\,^{\prime}/Z_{\text{TLS}}^{(j) }\,^{\prime}\\ Y_{\text{TLS}}^{(j)}/Z_{\text{TLS}}^{(j)}\end{array} \right]=\left[ \begin{array} []{c}x^{\prime}\\ y^{\prime}\end{array} \right], \tag{7}\]  
Let \(f\) denote the camera focal length, and \(k_{1}\) as well as \(k_{2}\) denote radial distortion parameters, then coordinates of the point \(p^{\prime}\) in image coordinate system corresponding to the point \(P_{i}\) could be obtained by Equation (8). \[p^{\prime}=\left[ \begin{array}{c}x\\ y\end{array} \right]=f\!\cdot\!r(p)\!\cdot\!\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\end{array} \right]=f\!\cdot\!r(p)\!\cdot\!p, \tag{8}\]  
where,  
\[r(p)=1+k_{1}\!\cdot\!\|p\|^{2}+k_{2}\!\cdot\!\|p\|^{4}, \tag{9}\]  
The image coordinate system \(o\)-\(xy\) takes the image principle point as the origin, the \(x\) axis is to the right, and the \(y\) axis is vertical upward. However, instead of the above image coordinate system, the pixel coordinate system \(o^{\prime}\)-\(uv\) is commonly used in the process of calculation, the origin of which is located in the upper-left corner of the image. The relation between image coordinate system and pixel coordinate system is shown as Figure 3.  
In Figure 3, it can be concluded that for the point \(P_{i}\) in the TLS point cloud, its corresponding point \(p^{\prime\prime}\) in pixel coordinate system is expressed as Equation (10). \[p^{\prime\prime}=\left[ \begin{array}{c}u\\ v\end{array} \right]=\left[ \begin{array}{c}(W/2)+x\\ (H/2)-y\end{array} \right], \tag{10}\]  
where \(W\) and \(H\) respectively means the width and height of an image. Both the intrinsic and extrinsic parameters for each image as well as transformation parameters between the TLS point cloud and SfM dense point cloud could be obtained with the G-Super4 PCS registration algorithm introduced in reference [9]. According to all of the above process, the 1-\(N\) mapping relationship between the Point cloud and images could be concluded.