## 6 Conclusions  
This paper proposes a trajectory spatiotemporal reconstruction model for sparse target trajectories. This model can be divided into two steps. First, a spatial trajectory construction module is proposed to determine the spatial trajectory of the target. Then, a time prediction model based on historical trajectories and an attention mechanism is further proposed to predict the time information of targets traveling on roads based on known spatial trajectories. In the time prediction model, not only the spatiotemporal features within the trajectory are considered, but the spatiotemporal features of targets traveling on the path learned from historical trajectories are also considered.  
In this study, first, various comparative experiments were designed to verify the accuracy of our trajectory spatiotemporal reconstruction, including spatial reconstruction methods, such as RICK, MPR, and HMM, as well as time series prediction methods, including traditional methods such as history and linear methods, and deep learning methods such as the RNN and LSTM. The effectiveness of the spatiotemporal reconstruction model proposed in this work was verified by combining spatial and temporal reconstruction methods on a real dataset from Chongqing. This showed that our proposed method could more accurately perform spatiotemporal reconstruction on sparse vehicle trajectories compared to other comparative methods. Then, various ablation experiments were designed to verify the importance of historical trajectories in the proposed spatiotemporal reconstruction model. It was demonstrated that the existence of historical trajectories in the model can make the spatiotemporal reconstruction of trajectories more accurate. Finally, a module verification experiment was designed to verify the accuracy of the time prediction model, which focused on the time prediction part in the spatiotemporal reconstruction. Compared with other methods, it was demonstrated that our proposed time series prediction model could more effectively capture the spatiotemporal relationship between the trajectories and accurately predict the trajectory's time series. In addition to its accuracy, the method's ability to function without reliance on topological data significantly broadens its applicability, allowing it to handle diverse scenarios, including free-space environments and urban areas with limited or outdated mapping data. Furthermore, the integration of an attention mechanism to incorporate historical trajectory patterns makes the approach robust to data sparsity and adaptable to periodic and contextual variations in target behavior. Overall, the proposed method not only addresses the limitations of existing approaches but also lays the foundation for future research to build more sophisticated models capable of handling sparse and noisy trajectory data. These contributions are expected to advance applications in airspace security, urban traffic monitoring, and beyond.  
Figure 7: Case 2. The process of AVI trajectory reconstruction in a complex urban road scenario (a–c).  
Conceptualization, X.C. and Y.C.; Methodology, Y.C.; Software, Y.C.; Validation, Y.C., B.B. and X.C.; Formal analysis, X.C.; Investigation, Y.C.; Data curation, B.B.; Writing--original draft, Y.C.; Writing--review & editing, Y.C., B.B. and L.Z.; Visualization, Y.C.; Supervision, X.C. and L.Z. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
Restrictions apply to the availability of these data. Data was obtained from Chongqing University and are available from the corresponding authors with the permission of Chongqing University.  
The authors declare no conflicts of interest.