# Newly Designed Identification Scheme for Monitoring Ice Thickness on Power Transmission Lines
## 4 Ice Thickness Calculation Using 3D Measurement
 \tag{25}\]  
Among them, \(s_{l}\) and \(s_{r}\) are scale factors that satisfy \(\frac{s_{l}}{z}=1\) and \(\frac{s_{r}}{z}=1\), respectively. \((x_{l},y_{l})\) and \((x_{r},y_{r})\), respectively, represent the image coordinate system of the projection points \(p_{1}\) and \(p_{2}\) of point P on the left and right images. \((x,y,z)\) and \((x_{r},y_{r},z_{r})\) represent the left and right camera coordinate systems, respectively. The left and right camera coordinate systems \(O-xyz\) and \(O_{r}-x_{r}y_{r}\)\(z_{r}\) can establish the positional relationship by expressing the conversion relationship of the binocular structure. The space conversion matrix can express \(M_{lr}\) as follows:  
\[ \begin{bmatrix}x_{r}\\ y_{r}\\ z_{r}\end{bmatrix} =M_{lr} \begin{bmatrix}x\\ y\\ z\\ 1\end{bmatrix}\begin{bmatrix}R_{11}&R_{12}&R_{13}&T_{x}\\ R_{21}&R_{22}&R_{23}&T_{y}\\ R_{31}&R_{32}&R_{33}&T_{z}\end{bmatrix}\begin{bmatrix}x\\ y\\ z\\ 1\end{bmatrix} . \tag{26}\]  
Among them is the space transformation matrix \(M_{lr} \begin{bmatrix}R&T\end{bmatrix} \), where R and T are the rotation and translation matrices between the \(O-xyz\) coordinate system and the \(O_{r}-x_{r}y_{r}\)\(z_{r}\) coordinate system, respectively. According to Formulas (25) and (26), for the spatial point in the \(O-xyz\) coordinate system, the corresponding relationship between the image plane points of the left and right cameras can be transformed into:  
\[ \begin{bmatrix}x_{r}\\ y_{r}\\ 1\end{bmatrix} =\frac{1}{s_{r}} \begin{bmatrix}f_{r}R_{11}&f_{r}R_{12}&f_{r}R_{13 }&f_{r}T_{x}\\ f_{r}R_{21}&f_{r}R_{22}&f_{r}R_{23}&f_{r}T_{y}\\ R_{31}&R_{32}&R_{33}&T_{z}\end{bmatrix}\begin{bmatrix}zX_{l}/f_{l}\\ zY_{l}/f_{l}\\ z\\ 1\end{bmatrix} . \tag{27}\]  
Therefore, according to Formulas (24) and (27), the 3D coordinates of the spatial point can be obtained:  
Figure 16: The 3D measurement model for binocular stereo vision in space. \[\left\{ \begin{array}{l}x=\frac{zX_{l}}{f_{l}}\\ y=\frac{zY_{l}}{f_{l}}\\ z=\frac{f_{l}(f_{l}T_{x}-x_{r}T_{z})}{x_{r}(R_{31}X_{l}+R_{32}Y_{l}+f_{l}R_{33})- f_{r}(R_{11}X_{l}+R_{12}Y_{l}+f_{l}R_{13})}\\ =\frac{f_{l}(f_{r}T_{y}-Y_{r}T_{z})}{Y_{r}(R_{31}X_{l}+R_{32}Y_{l}+f_{l}R_{33}) -f_{r}(R_{21}X_{l}+R_{2}Y_{l}+f_{l}R_{23})}\end{array} \right.. \tag{28}\]  
In summary, the internal parameters' focal length \(f_{l}\) and \(f_{r}\) of the two cameras, the external parameters rotation matrix \(R\) and translation matrix \(T\), and the image coordinates \((x_{l},y_{l})\) and \((x_{r},y_{r})\) of the feature point \(P\) in the left and right cameras are known. In the case of 3D space, the coordinates of \(P\) can be obtained. As we all know, the internal and external parameters of the camera can be obtained through camera calibration. Therefore, only the coordinates of the image coordinate system of the point \(P\) in the left and right cameras, that is, the corresponding relationship between the points in the left and right images, can be used to obtain the 3D space coordinates of \(P\). Subsequently, iced transmission line identification is used to find the top and bottom boundaries of ice formed on power transmission lines. Next, the iced transmission line's 3D coordinates for the key points are computed using the key point matching method. Finally, the ice thickness is calculated. The ice thickness is estimated after the 3D coordinate points \((x,y,x)\) are obtained. The 3D coordinate graphed in the \(xyz-space\) is more difficult than in the \(xy-plane\) because depth perception is required. One can use projections onto the coordinate planes to simplify plotting points. The projection of a point \((x,y,x)\) onto \(xy-plane\) is obtained by connecting the point to the \(xy-plane\) by a line segment perpendicular to the plane and calculating the intersection of the line segment with the plane. In this investigation, ice thickness was calculated using a three-dimensional Euclidean distance. Figure 17 shows a schematic diagram of 3D Euclidean distances.