# Vehicle Trajectory Reconstruction Using Lagrange-Interpolation-Based Framework
## 6 Conclusions  
This paper analyzes the existence of outliers and observation noise in the vehicle trajectory data and proposes a processing framework for vehicle trajectory reconstruction. The framework integrates the discrete continuous wavelet transform, Lagrange interpolation and the Kalman filter to identify outliers, re-estimate value after the outliers are eliminated and reduce observation noise in time series. Lagrange polynomial interpolation can be flexibly selected according to the number of trajectory points, which improves the migration of the algorithm. The Kalman filter method with optimized filtering parameters has better results compared with other filters. The proposed method is compared with a baseline method on the NGSIM dataset, and the results indicate that (1) the RMSE obtained by using the proposed method is 45.76% lower than that obtained by using the baseline method. (2) The SNR obtained by using the proposed method is 26.43%. (3) The advantage of the method in terms of the SNR is the largest when Q = 0.05 and R = 0.15. (4) The Pearson correlation coefficient obtained by using the proposed method is 25.58%. At least three interesting directions can be explored in further studies. Firstly, compared to the trajectory reconstruction based on a geometric curve function, a methodology based on deep learning (such as autoencoders, reinforcement learning, self-supervised learning) could be explored. Secondly, the adaptability of trajectory reconstruction methods should be further verified under complex road conditions. Finally, it is very interesting to combine trajectory reconstruction with urban-delivery route optimization for driverless delivery vehicles. Study conception and design: [PERSON] and J.T. ; Data collection: [PERSON]; Analysis and interpretation of results: [PERSON] and Z.W. ; Manuscript preparation: [PERSON] and J.W. All authors have read and agreed to the published version of the manuscript. This work is supported by the National Natural Science Foundation of China (Grant No. 52002281), the Major project of new generation of artificial intelligence (2023) (Grant No. 2022 ZD0115600) and the Autonomous Region Postgraduate Innovation project (Grant No. XJ2023G048). Not applicable. The Next Generation Simulation (NGSIM) Open Data datasets were used during the current study. They are available at [[https://datahub.transportation.gov/Automobiles/Next-Generation-Simulation-NGSIM-Vehicle-Trajectory/Sect-6](https://datahub.transportation.gov/Automobiles/Next-Generation-Simulation-NGSIM-Vehicle-Trajectory/Sect-6) jqj/data]([https://datahub.transportation.gov/Automobiles/Next-Generation-Simulation-NGSIM-Vehicle-Trajectory/Sect-6](https://datahub.transportation.gov/Automobiles/Next-Generation-Simulation-NGSIM-Vehicle-Trajectory/Sect-6) jqj/data), accessed on 26 July 2022. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Index** & \multicolumn{2}{c}{**Original Trajectory**} & **Baseline Method** & **Proposed Method** \\ \hline Jerk is greater than & mean value & 22.93 & 0.04 & 0.03 \\ \(\pm\)15 m/s\({}^{3}\)(\%) & standard deviation & 11.22 & 0.05 & 0.23 \\  & range & [4.70, 69.23] & [0.00, 1.75] & [0.00, 4.32] \\ \hline Maximum jerk & mean value & 832.16 & 33.14 & 16.83 \\  & standard deviation & 712.33 & 36.21 & 13.36 \\  & range & [36.33, 8326.91] & [3.96, 193.63] & [0.93, 169.65] \\ \hline Minimum jerk & mean value & \(-\)978.62 & \(-\)42.12 & \(-\)13.26 \\  & standard deviation & 1021.36 & 25.69 & 6.74 \\  & range & [\(-\)8795.41, \(-\)36.85] & [\(-\)121.16, \(-\)65.36] & [\(-\)86.52, \(-\)1.36] \\ \hline N & mean value & 56.32 & 13.65 & 7.49 \\  & standard deviation & 9.32 & 7.41 & 6.34 \\  & range & [42.75, 63.26] & [22.69, 43.77] & [12.34, 37.69] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Jerk analysis of trajectory reconstruction results. Conflicts of InterestWe confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed.