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J. Eur. Opt. Society-Rapid Publ. 22, 40( 2026)
Figure 7. a) Simulated localization errors for filter combinations of no temporal filtering, Gaussian filtering and matched filtering with an optional spatial LoG edge detection. b) Corresponding resulting target positions per temporal filter at 12,500 measurements per reconstruction and a laser pulse width six times the detector’ s temporal resolution and c) with additional LoG edge detection in the reconstruction volume. The red dots denote the nominal target position.
temporal domain data is examined. Filtering of the reconstruction is commonly used in ellipsoid based back-projection methods to increase the visibility of features in complex scenes in NLOS FBP imaging. There, the edge sensitive filters are applied to the reconstruction volume to compensate for uncorrelated ellipsoids from other parts of the scene that overlap with the target signal of interest. Although no other target is present in the localization scenario discussed here, edge filtering of the reconstruction result can still be used to compensate for noise and the pulse width induced extension of the target signal. By design, only the rising-edge of the pulse that interacts with the NLOS scene corresponds to the actual ToF of the threebounce scattering path. Therefore, filtering in the temporal domain of the measured transients is also investigated that preserves the rising-edge information.
To determine the influence of different filters on the localization result and error, further simulations were carried out. The simulations are based on a data set of 100 repetitions per measurement statistic and filter combination. The filter combination consists of filters used on the temporal domain data and spatial filtering of the reconstruction volume. The temporal filters are given by a Gaussian filter aiming for denoising and a matched filter for promoting the rising-edge of the received pulse from. The spatial filter is a Laplacian of Gaussian( LoG) edge sensitive filter to promote the edge of the ellipsoid intersection at the target position. The analysis of the resulting localization error after filtering is performed for different number of measurements per reconstruction( Fig. 7a), ranging from 1,250 to 125,000 measurements. Error bars denote the standard deviation of the determined target position and allow the distinction between systematic localization errors and the variation of extracted positions. Figures 7b and 7c visualize the extracted target positions for a statistic of 12,500 measurements per reconstruction, corresponding to 1 fps of data acquisition. For better visibility, the relay wall and thus the illumination positions are located at the bottom of each point cloud representation. Figure 7b corresponds to reconstruction with temporal filtering only, and Figure 7c shows results with additional LoG filtering of the reconstruction volume prior to the target position extraction.
Maximum detection on an unfiltered reconstruction yields the largest localization error for high statistics because the reconstruction forms an extended high signal volume behind the target position as discussed above( purple, solid outline, Fig. 7a). With Gaussian filtering in the temporal domain( green, solid outline, Fig. 7a), the standard deviation of the extracted position decreases at high statistics while the RMSE persists. This indicates the formation of a global maximum behind the actual target position at an increased distance from the relay wall( green dots, Fig. 7b). This maximum is formed because the Gaussian filter deforms the temporal top hat laser profile. Using a smaller width of the Gaussian filter can minimize this deformation and prevent a global maximum, but this resembles a transition to the unfiltered reconstruction and is therefore not pursued further. The matched filter( blue, solid outline, Fig. 7a), shows the best performance for results without spatial edge filtering. The filter is designed to generate a maximum at the target ToF in the detection point wise histograms representing the transients on the relay wall. Consequently, the overlapping ellipsoids will form a global maximum in the reconstruction as well( blue, solid outline, Fig. 7b). The position of this maximum corresponds to the actual target position, in contrast to the displaced maximum formed by Gaussian filtering. Remaining errors are due to temporal discretization and the finite voxel grid. It should be noted that the localization error still decreases more strongly along the axis defined by the illumination positions, where the geometric benefit of superposition comes into effect( y-axis in Figs. 7b and 7c).
Applying the edge filter to a reconstruction without temporal filtering yields the smallest systematic error at high statistics but performs worst at lower statistics( purple, dashed outline, Fig. 7a). For off-centered targets not shown explicitly, a position dependent systematic error is introduced due to shape sensitive edge detection of the high intensity area. With Gaussian filtering in the temporal domain beforehand, the rising edge of the pulse profile is