## 3 Results and Discussion
### Robustness Analysis  
The above experiments not only verify the correctness of the proposed method, but also prove the suitability of point cloud data. However, another point needs to be discussed by comparing experiment 1 and experiment 2 in this article: the reconstruction results of railway parameters such as the curve feature, main stake coordinates, and track mileage from the design data compared to actual measured data. In Section 3.1, the result of experiment 1 was very close to the design value, while the result of experiment 1 showed deviation from the resurvey values, despite reaching the construction demand. This indicates either that the railway may deviate from the original design position after operation for a period of time or that there is an inevitable error in point cloud data processing, especially in the extraction of the rail central line. In order to explore the robustness of the method proposed in this paper, the test data generated by the random function was used to simulate the error existing in the point cloud data based on the line design data. In this experiment, two indicators \(B_{1}\) and \(B_{2}\) were designed for robustness analysis.  
\[B_{1}=\frac{B_{P}+B_{I_{1}}+B_{I_{2}}}{3} \tag{14}\]  
\[B_{2}=\frac{\sqrt{x_{zh}{}^{2}+y_{zh}{}^{2}}+\sqrt{x_{hy}{}^{2}+y_{hy}{}^{2}}+ \sqrt{x_{hy}{}^{2}+y_{hy}{}^{2}}+\sqrt{x_{hz}{}^{2}+y_{hz}{}^{2}}}{4} \tag{15}\]  
where \(B_{1}\) is the mean of the curve parameter deviation and \(B_{2}\) is the mean of the main stake coordinate deviation \(B_{P},B_{I_{1}},B_{I_{2}}\), respectively, are the circular curve radius, first transition curve, and second transition curve deviation between the measured value and the design value.  
Figure 5a shows the relationship between random error and deviation (i.e., the \(B_{1}\)) of the curve unit. As the random error increases, the deviation of the four curve elements is also increased. In the range of 0-10 mm, the changes of \(B_{1}\) are moderate and do not exceed 100 m. When the random error is more than 15 mm, the value of \(B_{1}\) is divergent and exceeds the accuracy range. In Figure 5b, the trend of the four curve units is basically the same. When the random error is less than 10 mm, the value of \(B_{2}\) does not exceed 15 m. From the point of view of the point cloud data number, the difference between the calculated main stake point location and the design data point is less than two points. When the random error is bigger than 14 mm, the main stake point coordinates' deviation (or \(B_{2}\)) of values of the four curve units increase sharply and exceed 100 m.  
Figure 4: The data exhibition of calculation results and railway ledger data.  
Based on the above experiments, this paper can conclude that when the error of the railway centerline data is less than 14 mm, the curve parameters and the main stake coordinates obtained by the proposed method can be used to reach the requirements of railway reconstruction. The point cloud data obtained by the vehicle-borne 3D laser scanning technology can achieve a measurement accuracy of 10 mm after being processed and corrected, which is similar to the dynamic measurement accuracy of GNSS. Therefore, the proposed method is accurate for existing railway reconstruction from point cloud data.