# Vehicle Trajectory Reconstruction Using Lagrange-Interpolation-Based Framework
## 1 Introduction
[PERSON] et al. used trajectory historical information and network topology geometry information to construct a weighting-based map matching method to detect notable location errors and proposed an interpolation method with path determination and trajectory interpolation to solve the low sampling rate of vehicle trajectory data by GPS devices [22]. [PERSON] et al. proposed a stacked autoencoder for detecting outliers of monitoring data. Based on the difference between the input and the output of the trained stacked autoencoder, the Grubbs criterion and the PauTa criterion are used to evaluate whether data points are outliers [23]. [PERSON] et al. proposed unsupervised deep neural network models based on stacked autoencoders to detect the outliers among position, speed and angular position [24]. [PERSON] et al. proposed a three-step vehicle trajectory reconstruction method with the wavelet transform, Gaussian kernel and Savitzky-Golay filter to reconstruct the trajectory data from 15 intersections. The framework used Gaussian-kernel-based locally weighted linear regression to interpolate the data after outliers were removed [25]. Observation noise removal has also received some research attention. [PERSON] et al. analyzed the reason for the observation noise and used a low pass filter to reduce the observation noise [26]. [PERSON] et al. proposed a smoothing algorithm to denoise the data of the beginning segment and the ending segment of a vehicle trajectory [11]. [PERSON] et al. used the Kalman filter to reduce the observation noise of vehicle trajectory data [27]. [PERSON] et al. used the wavelet filter method to filter the observation noise of vehicle trajectory data [28]. [PERSON] et al. used a symmetric index moving average filter method to denoise in the NGSIM (US-101) dataset [29]. [PERSON] et al. combined empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD) and the wavelet transform to achieve noise suppression [30]. [PERSON] et al. used the wavelet transform to denoise trajectory data [17]. It is known that the denoise and smoothing methods of trajectory data mainly include the moving average filter, Kalman filter and symmetric exponential average filter. Each denoising method has its advantages, but its parameters remain unchanged. [PERSON] et al. proposed a multiple-model filtering adaptive method to infer the observation noise and predict vehicle trajectories. The method integrated maximum-likelihood multiple-model filtering and the unscented Kalman filter (UKF). The adaptive UKF had been pretrained in the constant velocity model, constant turn rate model, constant acceleration model and constant turn rate and acceleration model [31]. [PERSON] et al. used smoothing filters and extreme gradient boosting with adaptive regularization to reduce the observation noise between speed and acceleration. The method had better robustness in processing raw data from an unmanned aerial system compared with manual thresholds [32]. [PERSON] et al. proposed a multimodel-based extended Kalman filter (EKF) to reduce the vehicle location noise using the vehicle trajectory. The EKF used the velocity, position and distance of the vehicle to construct a state vector matrix and used probability calculation to obtain the predicted results [33]. [PERSON] et al. used a Kalman filter to reduce the noise of prediction uncertainty while predicting real-time cooperative vehicle trajectory with the kinematic and data-driven models [34]. [PERSON] et al. proposed a multi-objective optimization control method based on expanded state observers in order to keep the accuracy and stability, which are disturbed by a variety of factors in curved roads [35]. Built on the state of the art, the innovations of this study are summarized as follows: (1) A Lagrange-interpolation-based framework for vehicle trajectory reconstruction is proposed, which integrates the discrete continuous wavelet transform, Lagrange interpolation and the Kalman filter, which can obtain a smooth trajectory.