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J. Eur. Opt. Society-Rapid Publ. 22, 40( 2026)
Figure 2. a) NLOS measurement process with two Lambertian scatterings indicated with red arrows. Dashed arrows indicate pulse propagation. b) Resulting ellipses of possible target reflections that could have produced the measured data of detection positions 4 and N( blue circles) from illumination position 1( red circle). The intersection indicates the actual target position. c) Overlapping ellipses for finite pulse width of a centrally positioned target. Ellipses I and II originate from laser position 1( red circle) and detection position 1 and N respectively. Ellipse III originates from laser position 2 and detection position N, confining the light grey area to the dark grey area.
Gaussian( LoG) filter which is typical for FBP [ 21, 24 ] can optionally be applied to the reconstruction, with the Gaussian width matched to the voxel size to detect the rising edge of the broad overlapping ellipsoids. The temporal data can optionally be filtered before reconstruction with a Gaussian filter with a sigma matched to the pulse width. A matched filter with an appropriate temporal shift can also be applied to concentrate the received pulse form on the rising edge of the original signal to mimic a discrete ToF besides the finite pulse width.
3 Results and discussion
3.1 Illumination schemes
A central illumination position within the detection area on the relay wall may seem obvious in terms of scene illumination but strongly reduces the sensitivity for multiply scattered NLOS signals due to detector saturation by direct scattering in the employed first photon detection scheme. Hence, illumination positions outside the detection area are examined. Possible illumination positions are placed on an exemplary axis across the center of the detection area as shown in Figure 1b along which a target will be placed with an offset to the relay wall. A plane of interest is defined that is perpendicular to the relay wall and intersects with the aforementioned illumination axis. Within this plane, the distribution of receivable signal is computed using view factors that describe the radiation from a surface into a specific solid angle. The receivable signal quantifies the photon flux that reaches the detection area on the relay wall given by the photons that impinge on the target. The latter depends on both the illumination and the target position. Since the receivable signal from a target is directly linked to the signal-to-noise ratio( SNR) of a NLOS measurement it is a figure of merit for achievable reconstruction performance.
For a non-central illumination located at the left edge of the detection area the distribution of receivable signal is highly inhomogeneous across the scene as shown in
Figure 3a. An additional illumination position shown in Figure 3b is used to compensate for the disadvantaged right side of the scene. At the cost of an additional laser source and alternating measurements of each illumination position with doubled detection rate, the photon flux into the NLOS scene can be doubled while complying with eye safety restrictions. All illumination schemes( Figs. 3a – 3c) show that the receivable signal is maximized for target positions in front of the illumination positions. For larger target distances from the relay wall, the receivable signal decreases due to free-space propagation losses. Non-central targets benefit from non-central illumination, as indicated by higher values at lateral target positions for the dual illumination scheme compared to the central scheme where the receivable light decreases visibly towards the side edges( cf. Figs. 3b, 3c). Conversely, the advantage of central illumination( Fig. 3c) for central target positions is confined to the near field of the relay wall, mostly covered by the direct line-of-sight and thus of little relevance for the NLOS discussion. Fig. 3d) shows the regions where central illumination outperforms dual illumination, presenting the ratio of receivable light. At all positions colored white, the receivable light under dual illumination equals or surpasses that of the central scheme with single illumination. Consequently, non-central illumination can be used to illuminate the NLOS scene with minimal loss of receivable light, preventing detector saturation due to directly scattered laser light.
While the receivable signal is only an indicator for the relative localization performance across the NLOS scene, simulated transients are used to further assess the non-confocal reconstruction method. This provides a qualitative insight into the reconstruction performance under the dual-illumination scheme. A NLOS scene with a single target at a distance of 1.5 m from the relay wall is considered. The target is either centered within the detection area, and thus symmetric with respect to both illumination positions, or laterally displaced toward illumination position 2, as depicted in Figure 4a. The aforementioned plane of interest of the three-dimensional back-projection result is shown in Figures 4b and 4c for a centered target and each