## 2 Related Works  
The method of trajectory reconstruction is highly correlated with the characteristics of the trajectory data, which in turn are closely related to the collection and sensing technology of the data. Methods for the collection of trajectory data include GPS, traffic video, radio frequency identification (RFID), and 4G-based mobile sensors, among others. Therefore, trajectory data can be divided into two categories based on their collection methods: trajectory data obtained from mobile sensors and trajectory data collected by fixed-point devices. These two types of data exhibit significant differences. Trajectory data obtained from mobile sensors, such as taxi GPS data, have a high sampling rate but a low penetration rate, meaning that the trajectory sampling points are dense but only include the trajectory data of the taxi population and not those of private cars, ride-hailing cars, trucks, and other vehicles. The GPS device installed on a taxi records information every few seconds or longer, and, although there may be some measurement errors, the accuracy and reliability of taxi trajectory data are still very high. In contrast, data collected by fixed collection points, such as automatic vehicle recognition data based on fixed collection points, typically have a high penetration rate but a very low sampling rate due to environmental and equipment limitations. This type of data can include the trajectory data of all vehicles in a city, including taxis, private cars, ride-hailing cars, trucks, and others, but the trajectory points are sparse, with only one trajectory point every few hundred or even thousand meters. Due to the different characteristics exhibited by these two types of trajectory data, their reconstruction methods differ significantly. For trajectory data collected by mobile sensors, such as taxi GPS data, the data themselves would form a complete trajectory if there were no errors. However, due to environmental limitations and device errors, there may be missing and offset points in the taxi trajectory. Therefore, reconstructing the taxi trajectory usually involves restoring these missing and offset points. A common solution to this problem is to treat the trajectory as a two-dimensional spatiotemporal sequence and use interpolation to fill in missing values [6; 7]. Based on this, a smoothing filter [6; 7] model was proposed to fill in missing values in the trajectory, and its performance is acceptable when there are few missing points. However, when the missing rate is high, the model's performance will be significantly reduced. Another direction is to observe and model the transition patterns of vehicles between different locations and then interpolate the missing points in the trajectory based on the transition rules between trajectory points [8; 9]. However, since observation records are usually unevenly distributed, the transition patterns cannot be used to infer the locations in the trajectory that have never been observed, so this method is still not sufficient. Therefore, [PERSON] et al. [10] proposed a model based on Markov chains to synthesize low-sampling trajectories into dense trajectories. [PERSON] et al. [11] used a logistic regression model and inferred the vehicle's driving route based on a Markov model. This method still has accuracy issues because it does not consider the spatiotemporal constraints of the trajectory, and the path selection may be different at different times. Therefore, [PERSON] et al. [12] proposed a network mobility model to infer vehicle trajectories by capturing the spatial patterns and temporal characteristics of movement from historical trajectories. In recent years, the development of deep learning methods has provided powerful tools for us to model complex data relationships. Recurrent neural networks (RNNs) and their variants, such as long short-term memory (LSTM), have been widely used to model trajectory data.