J. Eur. Opt. Society-Rapid Publ. 22, 40( 2026) 393
subtracted but requires increased measurement statistics to detect targets near this interfering wall, as the signal strength of such large surfaces exceeds the target signal.
2.3 Reconstruction and filters
For the reconstruction of the NLOS scene based on the transient data, that corresponds to individual ToF per target position and detection position, several methods have been reported. Individual requirements on illumination and detection positions, being uniformly distributed [ 17 ] or in confocal arrangements [ 8, 18 ], prevents their applicability to the non-confocal system under study without error inducing transformation of the measurement data [ 19 ]. In contrast, the filtered back-projection( FBP) [ 20 – 23 ] used here is an ellipsoid-based reconstruction method that can handle arbitrary measurement schemes and gives insights into pulse width induced errors directly. The formation of the transients used for the reconstruction is described by the three-bounce path of the laser pulse. The time-of-flight( ToF) of the laser pulse from its scatter on the relay wall to its return to the wall after target reflection is measured( Fig. 2). This ToF is proportional to the sum of the two paths a and b indicated in Figure 2a. Possible target positions that could have produced the measured ToF are all located on an ellipsoid, or on an ellipse in two dimensions, that satisfies a + b = a 0 + b 0. The foci of these ellipsoids are the illumination position and the corresponding detection point on the relay wall. By introducing multiple detection points or illumination points, the intersection of the superimposed ellipsoids indicates the target position( Fig. 2b).
For a resolution estimation of the non-confocal setup under study, the signal contribution of finite pulse widths within back projection reconstruction is examined. In Figure 2c, three relevant ellipses are shown with an effective width of approximately half the pulse width times the speed of light. The target is centered at p T. ThecentralellipseII describes the signal contribution of a laser position and the detection point at the opposite border of the detection area, for example illumination position 1 and detection point N. Because this arrangement is symmetric, both laser position yield the same ellipsoid. The ellipse I results from laser position 1 and its nearest detection point 1 and is approximated by a circle around the laser position. The possible ellipses of all other pairs of laser position 1 and the remaining detection points are not shown but they all contribute to the regime hatched in grey, which is defined by the overlap of these two specific ellipsoids. The circle III belonging to illumination position 2 defines the dark grey region where all ellipsoids of the laser position 1 and 2 contribute. The lateral width l single( 5) is halved( 6) by restricting the feasible target area from illumination position 1 through superposition with the reconstruction from illumination position 2. This is observed by the lateral distance of the intersections at points p 1( 2) and p 2( 3) for a target at position p T compared with the boundary p 3( 4) of the more confined area( dark grey, Fig. 2c).
p 1; lateral ¼ t p t ToF 4 w c2 þ t 2
p
4 w c2; ð2Þ p 2; lateral ¼� t p t ToF 4 w c2; ð3Þ p 3; lateral ¼ t p t ToF 8 w c2 þ t 2
p
16 w c2; ð4Þ
l single ¼ p 1; lateral � p 2; lateral ¼ t p t ToF 2 w c2 þ t 2
p
4 w c2; ð5Þ l double ¼ 2 p 3; lateral ¼ t p t ToF 4 w c2 þ t 2
p
8 w c2 ¼ 1 2 l single:
Here, c denotes the speed of light. These expressions follow from a geometrical model with ellipse foci at ± w, which correspond to the illumination offset to the scene center. The semi-major axis a of the ellipse corresponding to II in Figure 2c and the radius r of the intersecting circle( 7) centered at the focus point are defined by the target ToF for the rising edges( 9) with additional dependence on the pulse width for the falling edge( 10). The lateral intersection position of two circles at p 3( 4) with a spacing of 2w are given in( 8) used for calculation of p 3.
p ellipse; circle ¼ a ða � rÞ; ð7Þ w ð6Þ p circle; circle ¼ r 1 2 2
� r 2 4 w; ð8Þ d rising ¼ t ToF 2 c; ð9Þ
d falling ¼ t ToF þ t P c: ð10Þ
2
According to the geometric model, the lateral spread of feasible target positions is expected to be halved along the axis defined by the illumination points. The resulting reduction in three-dimensional localization error due to this superposition is examined in the Results section.
The filtering applied to the reconstruction result and the underlying temporal domain data discussed is not intended to enable general imaging capabilities but to compensate the effects caused by the finite temporal laser pulse width, which is several times larger than the detector’ stemporal resolution. The extracted target position from the reconstruction result corresponds to the maximum-valued voxel in the reconstruction volume. To use this position extraction, only a single target may be simulated or measured at a time to avoid multiple local maxima. Moreover, partial occlusion by additional targets and phantom targets resulting from signal overlap of individual targets would complicate the extraction of target positions. Partial occlusion of the relay wall by the target is also omitted, so that all points of detection are accessible to the system and the reconstruction scenario discussed above remains valid. A Laplacian-of-