## 1 Introduction  
By the end of 2017, the operation length of China's railways exceeded 127,000 km, including 25,000 km of high-speed railway. The maintenance and transformation of existing railway is necessary to ensure the efficiency and safety of railway operations [1]. Due to the effects of wheels and rails, ballast settlement, environmental change, and so forth, the railway is inevitably offset from the designed state after a period of operation. Railway reconstruction refers to recovering the railway to as close to the delivery status as possible based on the original design data and measured data. Meanwhile, using the results of railway reconstruction as a reference standard is an important aspect for the transformation of the existing railway and design of the new double-track railway [2]. The accuracy and quality of the reconstruction influence the time limit, cost, and safety of the engineering. Existing railway reconstruction is an important research topic in the fields of railway engineering [3], surveying and mapping engineering [4], reverse engineering [5], and computing science [6]. In recent years, scientists have conducted extensive investigations in the field of existingrailway reconstruction based on the coordinate method. [PERSON] et al. reconstructed an existing railway using the robust least squares method after each feature point was identified [7]. When the arc-diameter ratio is small, the circle parameters obtained by fitting are not accurate. [PERSON] et al. used the complex Simpson's Equation based on the curvature change of the centerline to develop the calculation method suitable for various line types [8]. Once the measurement error or measurement interval is large, curvature change would be too disordered to identify feature points. In 2009, [PERSON] and [PERSON] proposed a plane reconstruction algorithm that solved the minimal value of the function and corresponding three arguments using a direction acceleration method, and discussed how to determine the initial values and constraints from the design code [9]. However, the method of manually identifying the feature points cannot reach the actual requirements. At the same time as the development of the swarm intelligence algorithm, there was also the genetic algorithm (GA), ant colony algorithm (ACA), and particle swarm optimization (PSO) study on existing railway reconstruction [10]. [PERSON] et al. developed a GA that optimized the evaluation function by selection, crossover, and mutation operators based on the initial values calculated by a feasible region [11]. [PERSON] introduced the ACA into the existing railway reconstruction of the plane and longitudinal section [12]. Curve radius, transition curve length, and curve length were included as parameters in the model, and the optimization results that met the various constraints were calculated. [PERSON] presented a railway reconstruction method based on the PSO algorithm which calculated the reconstruction parameters in the shift distance calculation involving the limitation of the control point and the requirement for integer parameters. This method could calculate the curve parameters of multiple existing railway units [13]. The premise of railway reconstruction is to obtain the location and status of the track. Therefore, it is necessary to resurvey the railway. Various involute-based, coordinate-based, and even point cloud data-based methods have been widely conducted worldwide to actively promote its relevant applications. For example, the string lining method is applied to existing railway curve realignment [14]. The deflection method and coordinate method can achieve a higher precision compared to the former method [15]. Currently, the involute-based method has been gradually phased out in actual engineering because of the long track lining distance [16]. The coordinate-based method, which has the characteristics of less disturbance and improved safety, is popular in practical work. Obtaining the 3D coordinates of the railway is the main aim of the coordinate method [17].