# Vehicle Trajectory Reconstruction Using Lagrange-Interpolation-Based Framework
## 2 Problem Statement  
Vehicle trajectory data are very important to analyze the microscopic phenomena in transportation systems. GPS devices have usually been used to collect trajectory data in the last decades, but vehicle location sensors can generate a large amount of outlier and observation noise due to a relatively low sampling rate and location error [22; 24; 31; 36]. With the development of multimedia equipment, many transportation researchers have used high-resolution images to extract vehicle trajectory data, and the high-resolution images are from videos and cameras, which are installed on an unmanned aerial vehicle, a moving car or transportation infrastructures. But the immature computer vision extraction techniques in complex traffic environments (e.g., vehicles are obscured by buildings or bridges in some images) cause some unexpected data in extracted vehicle trajectory data [1; 2; 32].  
The Next Generation Simulation (NGSIM) program uses videos and cameras to collect high-quality traffic datasets including vehicle trajectory data. It is a classic and publicly available dataset to support research on microscopic traffic state estimation and driving behavior. The paper uses the NGSIM dataset to test the proposed framework of vehicle trajectory reconstruction. According to previous studies [9; 13; 26; 37], outliers and observation noise are common in trajectory data in the NGSIM dataset. The original data in NGSIM have recorded 25 types of information, such as vehicle ID, global time, velocity, acceleration and so on. Figure 1 presents the outliers and observation noises of the trajectory data of vehicle 1882 in the I-80 data subset. The speed is the longitudinal instantaneous speed obtained from the first derivative of the longitudinal position data.  
Figure 1 shows that the original trajectory data generally have outliers and noises. The data points indicated by arrows are not much different from neighbor data, but the change in speed has exceeded the limit of vehicle dynamic performance and range of the body tolerance limit (\(-8\) m/s\({}^{2}\), \(5\) m/s\({}^{2}\)). These rapidly changing data are considered as detection errors. The data points indicated by the rectangular are significantly different from neighboring data. The values do not conform to the rule of vehicle operation. The data from frames 518 to 522 are extracted to illustrate this problem in Table 1.  
Table 1 shows that the data of frame 520 are much larger than the values of previous frames. This kind of outlier occurs for two reasons: the vehicle in previous frames is not recognized, and the value is accumulated from previous frames and the current frame. On the other hand, the data drift when the collection equipment is interfered with by factors such as signal occlusion. The outliers mainly exist in speed and acceleration and do not conform to vehicle dynamic performance. The outliers and observation noises in the raw NGSIM trajectory data will seriously hinder further research of the microscopic behavioral algorithm for modeling and simulation. The paper proposes a Lagrange-interpolation-based framework to deal with these outliers and observation noises.