Reciprocal Visibility

TL;DR

The paper tackles occlusion removal in aerial imaging of forests by leveraging reciprocal visibility (RV), a Helmholtz-reciprocity-inspired duality that links ground-point visibility to airborne sampling using pre-scanned data. It formalizes visibility with a matrix $\boldsymbol{V}$ and shows that the integral visibility from airborne positions can be expressed as $\boldsymbol{i}= \frac{1}{Z}\boldsymbol{V}\boldsymbol{p}$, while the reciprocal (bottom-up) view is $\boldsymbol{i}_{\uparrow}= \boldsymbol{V}^{\top}\boldsymbol{p}_{\uparrow}$. Practically, it uses sparse LiDAR point clouds to approximate the bottom-up visibility map $\boldsymbol{U}_{\uparrow} \approx \boldsymbol{V}^{\top}$, and introduces a coding approach with binary weights $2^k$ to decode per-ground-point visibility from the integrated map $\boldsymbol{i}_{\uparrow}$. A greedy sampling algorithm then selects flight positions to maximize visibility while preserving uniform reconstruction, outperforming unguided particle swarm optimization in simulated forest scenarios. The work demonstrates scalable, pre-computed RV guidance for real-time occlusion removal, with potential to coordinate drone swarms and improve detection in static or slowly changing woodland environments.

Abstract

We propose a guidance strategy to optimize real-time synthetic aperture sampling for occlusion removal with drones by pre-scanned point-cloud data. Depth information can be used to compute visibility of points on the ground for individual drone positions in the air. Inspired by Helmholtz reciprocity, we introduce reciprocal visibility to determine the dual situation - the visibility of potential sampling position in the air from given points of interest on the ground. The resulting visibility map encodes which point on the ground is visible by which magnitude from any position in the air. Based on such a map, we demonstrate a first greedy sampling optimization.

Figures (7)

• Figure 1: Synthetic aperture imaging and integration principle (a). Examples of conventional aerial images and AOS integral images of forest at different synthetic focal distances and captured at various spectral ranges (visible red/green/blue and far-infrared, FIR)(b). This examples illustrates a search and rescue use-cases where in particular thermal measurements are relevant. People on the ground become visible in the FIR channel after occlusion removal (i.e., by focusing computationally on the ground). This data has been recorded with a single drone in a 30m x 30m waypoint grid and at an altitude of 35m above ground level (AGL) Schedl20b.
• Figure 2: 2D illustration of the 4D visibility field in $\boldsymbol{V}$. While same columns of $\boldsymbol{V}$ represent same positions on the synthetic aperture plane (a), same rows of $\boldsymbol{V}$ represent same directions (b).
• Figure 3: 2D visualizations of components in \ref{['Eq:ReciprocalVisibility']} using procedural forest simulation. Single visibility masks $\boldsymbol{V_n}$ (a) indicate visibility (binary 1) and occlusion (binary 0) as seen top-down from the synthetic aperture plane to the ground. For selected points of interest on the ground $\boldsymbol{p}_{\uparrow}$ (b,e,g), the integrated visibility maps $\boldsymbol{i}_{\uparrow}$ (c,f,h) indicate the visibility of the selected ground points from all positions at the synthetic aperture plane (higher values correspond to higher visibility). The inlay in (h) is a contrast enhanced version for better visibility. An overlay of (a,b,c) illustrates that the resulting integrated visibility spatially aligns correctly with the occluder situation of the forest (here, darker blue indicates lower visibility of the selected ground points) -- keeping tree heights (20m on average in this example) and altitude of the synthetic aperture plane (35m above ground level in this example) in mind. Simulated density: 100$trees/ha$ (birch) on a 32x32$m^2$ area.
• Figure 4: A single bottom-up visibility mask (one column of $\boldsymbol{U}_{\uparrow}$) computed from a ground point of interest -- center ground position in this example -- (a) is the high-resolution counter-part of the integrated visibility map $i_{\uparrow}$ for the sample point in \ref{['Eq:ReciprocalVisibility']} - see Fig. \ref{['FIG:RV']}f. Corresponding high-resolution integrated visibility maps (b for Fig. \ref{['FIG:RV']}c, d for Fig. \ref{['FIG:RV']}h) and the overlay visualization (d) that corresponds to Fig. \ref{['FIG:RV']}d. Here, the integrated visibility map in (b,c) has been computed for only 21 ground points of interest that approximates the shape of our rectangular pattern shown in (c). Simulated density: 100$trees/ha$ on 32x32$m^2$.
• Figure 5: Coded integrated visibility map $\boldsymbol{i}_{\uparrow}$ for the example shown in Fig. \ref{['FIG:bottom-upRV']}b (a) and the overlay (b) that corresponds to Fig. \ref{['FIG:bottom-upRV']}d. In this case, unique binary codes for $K=21$ ground points of interest (individual codes are now visualized in our target area) are integrated. The possible $2^{21}=2.097.152$ different integrated combinations in $\boldsymbol{i}_{\uparrow}$ are color coded. Simulated density: 100$trees/ha$ on a 32x32$m^2$ area.