J. Eur. Opt. Society-Rapid Publ. 22, 40( 2026) 397
Figure 6. a) Measured localization errors for different measurement statistics. b – d) Extracted lateral target positions of 100 repetitions with 1,250( b), 5,000( c) and 50,000( d) measurements per reconstruction. Histograms show the distribution of extracted target positions for a single illumination( top and right, blue histograms) and superposed reconstructions( bottom and left, yellow histograms) of each lateral axis. The black cross indicates the actual target position.
0.26 m for position 2. This difference is attributed to imperfect Lambertian reflection at the primary reflection of the laser pulse on the ceiling. While the specular component for laser position 1 can geometrically contribute to the target illumination, the direct reflection from position 2 is directed away from the NLOS scene, to the right in Figure 1a. At 50,000 measurements, which corresponds to four seconds of measurement time, the localization error and the standard deviation of the extracted target locations fall even below the values predicted by the simulations. Independent of the individual laser positions, the combination of both illumination points shows the smallest error for all measurement statistics. Therefore, the simulations and the experimental measurements agree concerning the performance of the illumination concept proposed in this work: to achieve a small error in NLOS localization measurements, it is necessary to use more than one illumination points. This is especially important for low measurement statistics or fast measurements, but even at high statistics, using two illumination points can further reduce the localization error.
Nevertheless, there are discrepancies between the simulation and the measurement concerning the generation of the feasible target region that is assessed by the localization error. In contrast to a single-valued plateau approached with increasing measurement statistic for the simulation, the physical setup causes a global maximum to be formed within the laser pulse width induced feasible target region. On the one hand, the spatial extent of the physical target generates confined phantom objects through the overlap of signals from different reflection points on the target. On the other hand, system dependent factors lead to a global maximum of intersections in the reconstruction, which supports convergent localization. The imperfect top hat laser profile affects the signal distribution along the width of the ellipsoid and prevents the formation of an equal valued volume behind the actual target position as discussed above. Furthermore, pixel specific dark count rates and sensitivities influence the signal contribution of each ellipsoid orientation depending on the detection position relative to the illumination position. This artificially confines the overlapping signal volume by favoring reconstruction voxels along individual ellipsoids.
Since the distance from the relay wall of the target position is measured with little error even for low measurement statistics, a two-dimensional map of extracted lateral target positions is used to illustrate the three-dimensional localization in Figures 6b – 6d. For each measurement statistic, the target position is measured 100 times and its extracted position is plotted. This naturally results in multiple occupancy of the discrete reconstruction voxel space. To better illustrate convergent localization, the distribution of determined target positions is also indicated by histograms along both lateral axes. Opposite histograms represent the results of superposition( yellow dots) and laser position 1( blue dots) respectively and are equally scaled. In agreement with the simulations, the reduced localization error is mainly observed along the illumination axis defined by the line connecting the separated illumination points. This can be observed by the narrower distribution in the horizontal histograms compared with the vertical histograms of the super positioned results( yellow).
The compact measurement setup locates a NLOS target successfully under eye-safe conditions and validates the simulated benefits of superposed reconstructions of individual illumination points. The deviation from the homogenous feasible target region predicted by the simulation, resembles the formation of a global maximum in the actual measurement result. Although this effectively decreases the standard deviation of extracted positions, it results in a systematic error depending on the pulse form and target position. This indicates the need for filtering to achieve localization that is independent of pulse width.
3.3 Spatial and temporal filtering
Our demonstrated eye-safe NLOS localization exhibits an extended feasible target region that leads to localization errors larger than the temporal detector resolution and the finite reconstruction grid presets, caused mainly by the temporal laser pulse width. The achievable localization performance, limited by the laser pulse width and measurement statistics, has been addressed through scene illumination schemes and their superposition. To directly address the extended pulse width, associated with the compact laser diodes, filtering of the reconstruction result and the