3.2.5. Outliers

We have already discussed many possible measurement errors given a vehicle has correctly been detected. Nevertheless, depending on the circumstances of the recording and on the traffic density, the detectors will also record to some extent false negative and positive detections. For the presented algorithm, the false negatives of one sensor led to outliers in the dataset of the other sensor. Thus, in this validation step, we first examined how these outliers reduced the resulting accuracies. For the

analysis, we varied a percentage of false negative detections for both sensors. We simply removed detections from both sensors at random accordingly to the false negative rate chosen. For each value, we again applied many runs with our algorithm, in each run generating new vehicle trajectories and deleting some of them as described. In Figure 12, we can see that the offset estimation error was zero up to a False-Negative-Rate (FNR) value of 0.25, after which the results became very unreliable. We must note here that in order to be able to examine the effect of outliers specifically, we neglected all other sources of error, which led to the estimation error value of zero. The number of required iterations also remained at a fairly low level up to the FNR value of 0.25.

**Figure 12.** Influence of false negatives in the offset estimation (**a**) and on the number of required iterations (**b**). Continuous lines show the mean values, while dashed lines show the standard deviation of the result.

A very interesting effect can be seen in Figure 13, where the left plot shows that the matching sensitivity decreased quite fast. This means that out of all available matches still available in the dataset after deleting random vehicles, the algorithm can only find a portion of vehicle matches due to the very narrow Gaussian distribution after convergence. On the other hand, the right plot clearly shows that a very large portion of matches resulting from the algorithm were true matches, which explains the ability of precisely estimating the sensor offsets. This of course is a trade-off that can further be adapted. If we limit the convergence thresholds in the EM algorithm, one could achieve a vehicle registration sensitivity at the cost of more pairs of vehicles being falsely matched and consequently a higher offset estimation error.

Similar to the false negative detections, we also simulated false positives. From the recorded radar data, it can be observed that false positive detections occurred mostly in the presence of vehicles. It did for example occasionally happen that a large vehicle like a bus or a truck was wrongfully detected as two small vehicles. In the absence of traffic, false positive detections were extremely rare. To simulate this fact, we specifically generated false positive detections at both sensors around the already existing detections. Thus, we extracted a number of vehicles according to the chosen false positive rate and copied the data of those vehicles also adding a random time to the original timestamp with a minimum and maximum absolute difference of 2 s and 3 s, respectively. In order to examine the combination between both false negatives and positives, we additionally set the false negative rate to 0, 0.1 and 0.2. The sensor distance was fixed to 100 m. Figure 14 shows that the matching relevance was significantly higher than the sensitivity, although the false positive rate seemed to have a slightly more negative effect on the resulting relevance than the false negative rate.

**Figure 13.** Influence of false negative detections on the matching sensitivity (**a**) and on the relevance (**b**). Continuous lines show the mean values, while dashed lines show the standard deviation of the result.

**Figure 14.** Influence of false positive detections on the matching sensitivity (**a**) and on the relevance (**b**). Continuous lines show the mean values, while dashed lines show the standard deviation of the result.

#### *3.3. Real Data Experiment*

In addition to validation with synthetic data, in this subsection, we want to demonstrate the capabilities of the proposed methods using real measurements at two cross-sections. In order to be able to validate device offset reconstruction, we used high-precision RTK-GNSS sensors for two purposes: to record the exact location of the radar devices with a precision of a few centimeters and to record the vehicles' paths along the curve. We were able to record the device cross-sections with a high precision with static GNSS data recording at the exact location of the radar devices. The GNSS data recorded when the sensor was mounted on the vehicle showed a high amount of jitter due to driving beneath the bridge seen in Figure 4, where satellite coverage was lost and could only be regained after 5–10 s, which limited our path reconstruction possibilities. Thus, instead of using the GNSS, our reference data of vehicle positions came from an optical distance measurement device mounted at the back of the vehicle (Figure 15). The sensor was based on laser Doppler velocimetry to measure the velocity and length of moving surfaces [28]. With the laser pointed downwards, the measured surface in this case was the road. The technique used had a very high accuracy, being capable of recording the distance and speed with an error less than ±0.1%. Additionally, the laser was coupled with a control box, which enabled recording the data with a connected notebook. On the other hand, the control box also

provided the possibility of triggering a measurement reset either by a starter button or by an infrared light barrier. In the latter case, an additional laser device was directed sideways with reflecting markers being installed along the road. When the laser beam was reflected to the device, the measurement of the velocimeter was reset. We installed reflective markers at the location of the radar devices so that passing the relevant cross-sections triggered new distance measurements, and thus, we had the same coordinate system along the road section. At each new section between two radars, the record of the velocimeter started with zero.

**Figure 15.** Laser Doppler velocimeter used to generate reference data for the validation of the trajectory reconstruction.

For the validation of the reconstruction of microscopic traffic data, we drove with the equipped vehicle 33 times within three hours. For this period of time, we recorded both radar data and vehicle paths from the distance measurement device. As we saw in the previous subsection, our proposed algorithm was unable to converge to a good result when there were more than 25% outliers in the recorded data. When looking at Figure 4, we can see that the first radar device was located at a cross-section with three lanes, while the other devices only monitored the exit lane. This can also be verified in the radar data, as the first devices recorded 2928 vehicles in the same amount of time as the others recorded 444, 286, and 316 vehicles in the order of the devices. It is practically impossible to find feasible reconstruction between Radar 1 on the other devices. Thus, we validate the algorithm based on the data from Radar 2 and Radar 4. The distance between these two devices was 70 m along the curve, while all the devices were synchronized via Bluetooth using a handheld tablet, which connected to each radar device separately. The synchronization with the tablet was possible up to a 1-s deviation from the clock of the tablet. Additionally the radar devices were only capable of detecting vehicles with a resolution of 1 s.

We used the EM algorithm with both spatial and temporal optimization, as we could not guarantee a very precise time synchronization between the radar devices. The calculated offset after convergence was −0.63 s and 69.7 m between Radar 2 and 4, which matched very well with our measured reference. This indeed was in accordance with the results from the synthetic data validation. As the vehicles appeared randomly within the time windows of 1 s, the spatio-temporal offset error balanced out over many matched vehicle pairs. In other words, for a number of vehicle pairs that would correct the temporal offset towards the lower second value, their was a similar number of pairs that corrected the offset towards the higher second value. The results also delivered 274 matches between the two cross-sections. Moreover, from the 33 passes recorded with our equipped vehicle, we could successfully find 25 passes in the resulting matches. Figure 16 shows an outline of the reconstructed

trajectories, where we can see that even with the limited accuracy and resolution specifications of the recording devices used, very good trajectory reconstruction was feasible. To be more precise, the resulting mean RMS trajectory error was 3.48 m with a standard deviation of 2.08 m over these 25 trajectories. This result also fit our expectations as the vehicle passes showed speeds of 10–14 m/s. With a quantization of 1 s, the detection time error for a vehicle could be at most 0.5 s (either rounding up or down), but the mean RMS time error was about 0.3 s, which led to the found RMS error for the mentioned speed values.

**Figure 16.** Longitudinal microscopic trajectory of the reference vehicle and the reconstructed data. The subplots (**<sup>a</sup>**–**<sup>c</sup>**) show three different vehicle passes with blue lines, while the red, dashed lines show the corresponging reconstructed trajectories (data recorded on the date of 30-November-2018).
