J. Eur. Opt. Society-Rapid Publ. 22, 40( 2026) 395
Figure 3. Receivable signal on the detection area per target position a) for one illumination position to the left of the detection area, b) for two illumination positions around the detection area, c) for a central illumination. d) Target positions where the central illumination yields more signal than the dual illumination scheme.
Figure 4. a) Top-view schematic of the NLOS setup with indicated target and illumination positions. b – d) 2D slices of the simulated unfiltered ellipsoid based reconstruction volume for laser position 1 and 2 and for the superposition of both, e) for a target off the symmetry axis by 0.5 m to the right. White arrows indicate the target position.
illumination position. In Figure 4b, with the relay wall at the top and a single laser positions, the overlapping ellipsoids produce a spatially extended high-intensity region that indicates the target position. Because of the finite pulse width, this intersection of ellipsoids does not reduce to a single point. Moreover, due to limited statistics and added noise in the simulated transients, the overlap is even more extended than predicted by the geometric approximation introduced earlier. The alternative illumination position in Figure 4c yields the same result, except for a horizontal mirror due to the shift of the fixed focus point to the new illumination position. The sum of these reconstructions( Fig. 4d) produces a more confined, higher-contrast region at the target position. For a single illumination position, all ellipsoids share one focus and are therefore similar, whereas the tangents of the ellipsoids at the target position differ more strongly for individual illumination positions. For targets off the symmetry axis, this geometric confinement by superposed reconstructions persists, but the resulting high-intensity region is less pronounced( Fig. 4e). This reduced confinement is attributed to unequal ellipsoid intensities in each reconstruction caused by different distances to the illumination position. Under the same simulated-measurement parameters of the transient simulation, but for a lateral target displacement of 0.5 m toward illumination position 2, the ellipsoids of illumination point 1 are barely visible. The visualized reconstruction results indicate, that the dual-illumination provides geometric benefits beyond more uniform scene illumination.
A root-mean-squared-error( RMSE) analysis is performed to quantitively assess localization errors and the
confinement gain by superimposed reconstructions from individual illumination positions. The confined high-intensity region corresponds to feasible target positions, which are extracted from the unfiltered reconstruction result by maximum detection. Despite the two-dimensional depiction in Figure 4, the simulation covers a three-dimensional volume, so that the RMSE refers to the three-dimensional offset between the extracted target position and the ground truth.
For a central target at a distance of 1.5 m from the relay wall, the localization error decreases with an increasing number of measurements per laser position, as expected( Fig. 5a). A localization error of more than one meter for the lowest count of 125 simulated measurements per reconstruction is not considered a successful localization. Nevertheless, the residual RMSE for the largest statistic considered reflects the uncertainty in the feasible target area induced by the pulse width and is therefore consistent with a valid localization within the constraints given by the assumed system parameters. As indicated in Figure 4, superposition of the two reconstructions reduces the localization error compared with using only one illumination position. This enhancement cannot be explained simply by the increased number of receivable signal photons by the additional measurement, which would only follow the general trend of decreased errors with increased statistics. This is evident because the superposed result at 1,250 measurements outperforms single illumination results that use 10 more measurements, see Figure 5a. The central illumination scheme yields the maximum receivable signal among single illumination positions, yet it is still outperformed by