**4. Conclusions**

This paper presented a method for spatio-temporal synchronization and microscopic traffic data reconstruction from cross-section-based detection devices. The required input data consisted of the timestamp and speed of individual vehicles. The three main problems that were been overcome were: unknown offset of the recording devices, unknown vehicle matches, and unknown vehicle trajectories. The proposed methods were based on the assumption of limited vehicle dynamics between the recording devices. It has been shown how under these assumptions, the iterative EM-algorithm can be used to register the individual detections of the devices and estimate the previously unknown offset simultaneously. In a further step, the registered datasets were used to reconstruct trajectory data between the devices by means of quintic Beziér curves.

After presenting the basic methodology, a comprehensive validation was presented to demonstrate the capabilities of the method. Using synthetically-generated datasets, different error possibilities, like location variance, speed measurement error, and outliers in the input data, were discussed, and their effect on the outcome of the method was thoroughly investigated. It was shown that errors in the offset estimation can be reduced by increasing the number of vehicles in the datasets, which led to a higher computational load. Thus, one must find a trade-off between these two parameters. Additionally, on the synthetic datasets, a validation based on empirical data was also conducted, where was shown that the algorithm not only precisely estimated the offsets of the used devices, but also accurately reconstructed vehicle trajectories.

Based on this work, further validation of the matching performance could be conducted by precisely recording individual vehicle matches. This could be done by using detectors based on number plate recognition. Furthermore, the applicability of the proposed methods should also be demonstrated by using other types of cross-section detectors like inductive loops and computer vision systems. In order to provide more information of the vehicle passes, the method could be applied in a multidimensional space to also reconstruct the lateral movements of vehicles rather than only longitudinal trajectories. Additionally, an iterative version of the method could be developed to be able to optimize continuously the parameters of the cross-sections with a continuously-growing data volume. Finally, a very interesting and promising extension of this work would be a thorough analysis on the possibility of overcoming the limitations regarding dynamic behavior. If the propagation of shock waves in congested traffic is of interest, many dynamic maneuvers between cross-sections need to be taken into account. This could be done by incorporating driver modeling and microscopic traffic simulation in combination with data on the vehicle lengths in order to derive complex vehicle interactions.

**Author Contributions:** Conceptualization, methodology, software, validation, formal analysis, investigation, resources, data curation, writing, original draft preparation, and writing, review and editing and visualization, A.F.; supervision, project administration, and funding acquisition, M.O.

**Funding:** This research is a part of the project "Highly automated tunnel surveillance for catastrophe managemen<sup>t</sup> and regular operation" (Grant Number 13N13874) funded by the German Federal Ministry of Education and Research (Bundesministerium für Bildung und Forschung (BMBF)). The APC was funded by the Post-Grant-Fund of the BMBF. This method was adapted and used within the project "Basic Evaluation for Simulation-Based Crash-Risk-Models: Multi-Scale Modeling Using Dynamic Traffic Flow States" (project number 280497386) funded by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)).

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of the data; in the writing of the manuscript; nor in the decision to publish the results.
