# Vehicle Trajectory Reconstruction Using Lagrange-Interpolation-Based Framework
## 1 Introduction  
Outliers and observation noises of vehicle trajectory data deteriorate the value of the data in traffic state prediction and traffic management. [PERSON] et al. pointed out that vehicle trajectory data could be used to construct the driving behavior model, such as car following, lane changing, cooperative driving, distance control and safe driving, and proposed a data transcription and extraction method from video data to vehicle trajectories [1]. [PERSON] et al. proposed a non-parameter clustering Dirichlet process Gaussian mixture model (DPGMM) to extract vehicle trajectories from 70-h traffic video data at two intersections in Brooklyn and used hidden Markov models (HMMs) to recognize the rear-end conflict risk for adjacent vehicles [2]. [PERSON] et al. researched driver heterogeneity behavior and situation-dependent behavior in car-following scenes using large vehicle trajectory datasets and proposed a dynamic time-warping algorithm with a time parameter to calibrate the microscopic simulation model [3]. [PERSON] et al. used heterogeneous traffic data to estimate the state of spatial-temporal traffic and capture traffic congestion [4]. [PERSON] et al. studied the cooperative perception framework of the vehicle microtraffic state based on freeway and arterial vehicle trajectory datasets [5]. [PERSON] et al. used a microemissionmodel to solve the problem of insufficient trajectory data [6]. [PERSON] et al. proposed a deep convolutional neural network for the accurate estimation of space-time traffic speeds [7]. However, the vehicle trajectory data collected suffered from outliers and observation noises by the communication signal occlusion, severe weather environment and so on. Outliers are defined as data points that are significantly larger or smaller than the neighbor points. [PERSON] et al. identified obvious trajectory outliers and observation noises in the Next Generation Simulation (NGSIM) trajectory dataset [8]. [PERSON] et al. used jerk analysis, consistency analysis and spectral analysis to assess the data accuracy of vehicle trajectory data, and the results of error statistics indicated that trajectory outliers and observation noises are common in many datasets, such as NGSIM [9]. [PERSON] et al. took the I-80 data to illustrate that there were many errors such as large acceleration and bumper position marking errors [10]. Identifying and correcting outliers of vehicle trajectory data is one of the topics. [PERSON] et al. took the vehicle dynamic performance (such as maximum speed, acceleration time) as the judgment standard of abnormal speed and acceleration and used the moving symmetric index average method to repair outliers [11]. [PERSON] et al. detected outliers based on outlying scores. An outlying score of a vehicle trajectory is calculated as the similarity between the direction vector of a trajectory and those of other trajectories [12]. [PERSON] et al. added a jerk and frequency spectrum to the threshold discrimination method based on the vehicle dynamic performance threshold [13]. [PERSON] et al. proposed an expectation maximization (EM) algorithm to re-estimate the outliers [14]. [PERSON] et al. identified and deleted outliers by resampling missing data [15]. [PERSON] et al. used a time-varying completion method to estimate trajectory data [16]. [PERSON] et al. used data quality evaluation to remove outliers in the Waymo Open Dataset (WOD) [17]. Many machine learning methods, such as the convolutional neural network (CNN), support vector machine (SVM) and so on, have been applied to predict trajectory data [18]. [PERSON] et al. calculated the similarity between original trajectories and trajectories of a dataset by attention-based learning [19]. [PERSON] et al. developed a clustering algorithm to cluster the potential outliers and a KNN algorithm to identify outliers from the cluster [20]. The methods deal with various types of outliers but require many calibrated parameters. [PERSON] et al. illustrated the simple structure of the Lagrange interpolation algorithm [21].