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J. Eur. Opt. Society-Rapid Publ. 22, 40( 2026)
Figure 5. a) Simulated illumination dependent localization errors per number of measurements on the symmetry plane, b) near laser position 1. c) Localization errors for lateral target displacement at 1,250 measurements, and d) at 50,000 measurements.
the superposed reconstruction for practical measurement counts.
For other target positions the benefit of superimposed reconstructions varies and can even degrade localization performance. For example, a target at the same relay wall distance but outside the uniformly illuminated center is best localized using only the nearest laser position when the number of measurements is small, see Figure 5b. This lack of improvement from superposition is caused by the imbalance in signal to noise ratio( SNR) between the two reconstructions, as visualized in Figure 4e. For large numbers of measurements and resulting higher SNR the confined target region is again reduced geometrically by superposition, compare Figure 2c. The lower localization error for the nearest illumination compared with central illumination is due to the higher receivable signal caused by minimized free-space propagation losses.
Figure 5c shows how the localization error evolves with lateral target displacement from the centered position and reveals the performance difference between central illumination and off-center illumination. A centered target benefits from central illumination, while a target displacement reduces the receivable signal in two ways. First, a longer free space path and the angular dependence of Lambertian illumination by the laser spot reduces the signal reaching the target. Second, the view factor from the target to the detection area decreases at non-central positions. For a centered target, illumination by a non-central laser position is comparable to central illumination except for the slight reduction in signal reaching the target. With lateral target displacement, this loss decreases for the approached laser position and increases for the other position. The increased signal at the target for the closer laser position counteracts the reduced view factor towards the detection area, which results in a more stable localization error across the scene. This is indicated by the nearly constant RMSE for illumination position 1( blue curve, Fig. 5c). Note that although the distant laser position 2 does not provide reliable localization, superposition is nevertheless advantageous for limited displacements( yellow curve, Fig. 5c). The general geometric confinement benefit of superposition is valid for high measurement counts as illustrated in Figure 5d since it resembles the model of Figure 2c. However, the reduction of the localization error for low statistics is more important regarding the overall systems performance.
Summarizing, the multi-spot illumination scheme improves the localization system not only through spatially more balanced scene illumination but also through geometrical confinement of the feasible target region, even when individual illumination positions show limited standalone performance. While the confinement gain was analyzed with two illumination positions for an exemplary axis, the use of four positions at the middle of each edge of the rectangular field of view naturally leads to confinement along two orthogonal axes. Further confinement could be achieved with more illumination points, although a finite number of illuminations is desirable due to limited detector frame rates and the requirement that only one illumination position can be used in every measurement cycle.
3.2 Measurement validation
The localization benefit of superposed reconstructions is validated with measurement data shown in Figure 6. The designed measurement system is compact and eye-safe. It uses a SPAD-array detector with integrated timing and two laser diodes with nanosecond pulses as sketched in the Measurement Setup section. The system is placed on the floor of a room with a height of 3 m imaging the flat ceiling at an angle of 15 ° in order to localize a target that is not directly visible to the system. An occluding wall that prevents direct imaging of the target( Fig. 1a) would potentially contribute an additional NLOS signal, so it is omitted. A 20 20 cm 2 flat target positioned in parallel to the ceiling is used, in contrast to the point-like scatterer assumed in the simulations. Consequently, not only the pulse width but also its spatial extend influences the localization error by introducing multiple ToF per target and detection point. The target is placed near the center of the scene at a distance of 1.5 m from the relay wall( ceiling).
As predicted by the simulations, the localization error decreases as the number of measurements increases( Fig. 6a). When the number of measurements rises from 1,250 to 5,000, reducing the possible repetition rate of a measurement from 10 frames per second( fps) to 4 fps, the localization error decreases from 1 m to 0.74 m and 0.68 m for laser positions 1 and 2, respectively. With more measurements, the localization error for laser position 2 decreases stronger than for position 1; at 12,500 measurements( 1 fps), the error is 0.32 m for position 1 and