FoV-ESDF: Visibility Evaluation for Aerial Tracking

• Field of View ESDF (FoV-ESDF) is a precomputed, tracker-centric signed distance field limited to the camera’s view pyramid, enabling fast visibility evaluation.
• Its offline voxel grid and constant-time interpolation queries reduce runtime updates, supporting occlusion avoidance, observation-distance regulation, and target reacquisition.
• Integrated in a unified trajectory optimization framework, FoV-ESDF enhances planning speed and visibility robustness compared to conventional global ESDF methods.

Searching arXiv for the specified paper and closely related work on FoV-ESDF and visibility-aware aerial tracking. Field of View ESDF (FoV-ESDF) is a precomputed Euclidean Signed Distance Field tailored to a tracker’s field of view for visibility-aware aerial tracking. It was introduced in Eva-Tracker as a mechanism for rapid visibility evaluation that does not require runtime ESDF updates, and is used to support occlusion avoidance, observation-distance regulation, and target reacquisition within a unified planning and optimization framework (Lin et al., 13 Feb 2026). In contrast to a standard ESDF defined over all obstacle-occupied space, FoV-ESDF is restricted to the camera’s view pyramid and is computed once offline, with runtime usage reduced to coordinate transforms and constant-time interpolation queries.

1. Definition and conceptual scope

FoV-ESDF is defined relative to the tracker rather than the global workspace. The construction assumes a tracker-centric coordinate system $(x,y,z)$ whose optical axis is aligned with the $+x$ direction. Within that frame, the relevant spatial domain is not all of $\mathbb R^3$, but the camera’s quadrilateral pyramid. Eva-Tracker introduces this restriction explicitly in order to eliminate the need for frequent ESDF updates while preserving the information needed for visibility evaluation (Lin et al., 13 Feb 2026).

The contrast with the standard ESDF is central. Let $\mathcal O\subset\mathbb R^3$ denote the set of obstacle points in the world. The standard ESDF is the scalar field

$$d_{\mathrm{ESDF}}(x)= \begin{cases} \min_{o\in \mathcal O}\|x-o\|\,, & x\;\text{outside obstacles},\ -\min_{o\notin \mathcal O}\|x-o\|\,, & x\;\text{inside obstacles}. \end{cases}$$

As presented, this field must be updated whenever $\mathcal O$ changes, for example under dynamic obstacles, and such updates incur substantial run-time cost.

FoV-ESDF narrows the representational objective. It “cares only about visibility and ignores all space outside the camera’s FoV.” This makes it a task-specific distance field rather than a general-purpose geometric map. A common misconception would be to treat it as a drop-in replacement for a world ESDF; the formulation instead suggests a more limited but computationally targeted role: visibility-aware evaluation inside the field of view, not global signed-distance maintenance.

2. Geometric formulation

The FoV region is defined as the quadrilateral pyramid

$$\mathcal F \;=\;\Bigl\{(x,y,z)\Bigm|\;0 < x \le D,\;|y|/x\le\tan(\alpha/2),\;|z|/x\le\tan(\beta/2)\Bigr\},$$

where $\alpha,\beta$ are the horizontal and vertical FoV angles, and $D\!\approx\!d$ is the optimal observation distance chosen so that the point $(d,0,0)$ attains the maximum FoV-ESDF value (Lin et al., 13 Feb 2026).

On this domain, the FoV-ESDF is

$+x$0

An equivalent signed version is also described: $+x$1 if $+x$2 is inside the FoV interior and $+x$3 if it lies just outside. In practice, the non-negative magnitude is used to penalize obstacles within the FoV.

This formulation encodes distance to the boundary of the viewing pyramid rather than distance to obstacles in the world. The maximum value occurs around the preferred observation location near $+x$4, which is later exploited to define observation-distance objectives. This suggests that the field is simultaneously geometric and task-conditioned: its values reflect how well a point occupies the camera volume for tracking, rather than how far that point is from arbitrary scene surfaces.

3. Construction algorithm and data representation

FoV-ESDF is constructed offline on a regular voxel grid. A cuboid that bounds the pyramid $+x$5 is discretized into a grid of size $+x$6. Each voxel is marked as inside-FoV if its center lies in $+x$7, and a binary mask $+x$8 is formed with $+x$9 when the voxel lies on the boundary $\mathbb R^3$0, and $\mathbb R^3$1 otherwise. A 3D Euclidean Distance Transform à la Felzenszwalb–Huttenlocher is then run in $\mathbb R^3$2 time, where $\mathbb R^3$3, yielding

$\mathbb R^3$4

with $\mathbb R^3$5 the voxel size in meters (Lin et al., 13 Feb 2026).

The resulting scalar field is stored in a dense 3D array. At runtime, $\mathbb R^3$6 and $\mathbb R^3$7 at an arbitrary continuous point $\mathbb R^3$8 are evaluated via trilinear or cubic-linear interpolation among neighboring voxels. Each evaluation is $\mathbb R^3$9.

The complexity claims are explicit. Precomputation is a one-time offline cost of $\mathcal O\subset\mathbb R^3$0, with memory $\mathcal O\subset\mathbb R^3$1. At runtime, no ESDF updates are required. Dynamic obstacles therefore incur only $\mathcal O\subset\mathbb R^3$2 per sample, where $\mathcal O\subset\mathbb R^3$3 is the number of obstacle points inside $\mathcal O\subset\mathbb R^3$4 at that sample. In the pseudocode summary, outside-region voxels are treated as zero-ESDF during construction, reinforcing that the field is intentionally bounded to the view pyramid rather than extended as a conventional signed-distance representation.

4. Role in visibility-aware trajectory optimization

Eva-Tracker embeds FoV-ESDF in a trajectory optimization pipeline whose state is parameterized by a MINCO spline,

$\mathcal O\subset\mathbb R^3$5

The optimization problem is

$\mathcal O\subset\mathbb R^3$6

solved via L-BFGS, where $\mathcal O\subset\mathbb R^3$7 is the visibility cost, $\mathcal O\subset\mathbb R^3$8 is energy, and $\mathcal O\subset\mathbb R^3$9 contains safety and kinematic penalties (Lin et al., 13 Feb 2026).

The visibility cost is decomposed as

$$d_{\mathrm{ESDF}}(x)= \begin{cases} \min_{o\in \mathcal O}\|x-o\|\,, & x\;\text{outside obstacles},\ -\min_{o\notin \mathcal O}\|x-o\|\,, & x\;\text{inside obstacles}. \end{cases}$$0

with each term integrated over spline segments.

The occlusion penalty $$d_{\mathrm{ESDF}}(x)= \begin{cases} \min_{o\in \mathcal O}\|x-o\|\,, & x\;\text{outside obstacles},\ -\min_{o\notin \mathcal O}\|x-o\|\,, & x\;\text{inside obstacles}. \end{cases}$$1 uses obstacle points transformed into the tracker frame,

$$d_{\mathrm{ESDF}}(x)= \begin{cases} \min_{o\in \mathcal O}\|x-o\|\,, & x\;\text{outside obstacles},\ -\min_{o\notin \mathcal O}\|x-o\|\,, & x\;\text{inside obstacles}. \end{cases}$$2

For each transformed point, $$d_{\mathrm{ESDF}}(x)= \begin{cases} \min_{o\in \mathcal O}\|x-o\|\,, & x\;\text{outside obstacles},\ -\min_{o\notin \mathcal O}\|x-o\|\,, & x\;\text{inside obstacles}. \end{cases}$$3 is queried; if $$d_{\mathrm{ESDF}}(x)= \begin{cases} \min_{o\in \mathcal O}\|x-o\|\,, & x\;\text{outside obstacles},\ -\min_{o\notin \mathcal O}\|x-o\|\,, & x\;\text{inside obstacles}. \end{cases}$$4, the point lies inside the FoV pyramid. The instantaneous cost is

$$d_{\mathrm{ESDF}}(x)= \begin{cases} \min_{o\in \mathcal O}\|x-o\|\,, & x\;\text{outside obstacles},\ -\min_{o\notin \mathcal O}\|x-o\|\,, & x\;\text{inside obstacles}. \end{cases}$$5

Its gradients with respect to spline coefficients, yaw parameters, and segment times use $$d_{\mathrm{ESDF}}(x)= \begin{cases} \min_{o\in \mathcal O}\|x-o\|\,, & x\;\text{outside obstacles},\ -\min_{o\notin \mathcal O}\|x-o\|\,, & x\;\text{inside obstacles}. \end{cases}$$6 through the chain rule.

The observation-distance penalty $$d_{\mathrm{ESDF}}(x)= \begin{cases} \min_{o\in \mathcal O}\|x-o\|\,, & x\;\text{outside obstacles},\ -\min_{o\notin \mathcal O}\|x-o\|\,, & x\;\text{inside obstacles}. \end{cases}$$7 drives the target toward the field maximum at the preferred slot $$d_{\mathrm{ESDF}}(x)= \begin{cases} \min_{o\in \mathcal O}\|x-o\|\,, & x\;\text{outside obstacles},\ -\min_{o\notin \mathcal O}\|x-o\|\,, & x\;\text{inside obstacles}. \end{cases}$$8: $$d_{\mathrm{ESDF}}(x)= \begin{cases} \min_{o\in \mathcal O}\|x-o\|\,, & x\;\text{outside obstacles},\ -\min_{o\notin \mathcal O}\|x-o\|\,, & x\;\text{inside obstacles}. \end{cases}$$9 The yaw-angle centering penalty $\mathcal O$0 aligns the tracker yaw with the target bearing,

$\mathcal O$1

All three terms are sampled and trapezoid-integrated along the spline. The stated reason for the speed of the method is threefold: no ESDF updates, a unified framework in which occlusion, distance, and angle are handled by the same $\mathcal O$2 field plus a simple angular cost, and the use of MINCO for fast gradient propagation. A plausible implication is that FoV-ESDF is valuable not merely as a visibility heuristic, but as a differentiable objective substrate for continuous-time tracking optimization.

5. Target reacquisition and visibility recovery

The FoV-ESDF is not used in isolation. Within Eva-Tracker, it appears alongside a target trajectory prediction method and a visibility-aware initial path generation algorithm that maintain an appropriate observation distance, avoid occlusions, and enable rapid replanning to reacquire the target when it is lost (Lin et al., 13 Feb 2026).

This integration matters because aerial tracking can fail not only through collision or kinematic infeasibility, but also through temporary target disappearance. The paper explicitly frames the overall system as incorporating a recovery-capable path generation method for target reacquisition. In that setting, FoV-ESDF supplies a precomputed visibility prior over the tracker-centric field of view, and the same field is then reused both for normal visibility maintenance and for recovery behavior after target loss.

The reported design therefore treats visibility maintenance and reacquisition as variants of one planning problem rather than as disjoint modules. This suggests a shift away from update-heavy map-centric visibility reasoning toward tracker-centric, field-conditioned replanning. The significance is methodological: the same scalar field $\mathcal O$3 underlies regular trajectory optimization and rapid recovery-oriented path generation.

6. Empirical performance and practical significance

The empirical claims reported for Eva-Tracker are directly tied to the FoV-ESDF-based formulation (Lin et al., 13 Feb 2026). In simulation benchmarks with a random 100-obstacle scene and a 200 s trajectory, the path-generation time is 0.02 ms for Eva-Tracker, compared with approximately 0.8 ms for Vis-Planner and SF-Tracker. The trajectory optimization time is 10.7 ms for Eva-Tracker, approximately 36 ms for Vis-Planner and SF-Tracker, and 6.6 ms for Elastic-Tracker.

Visibility robustness is reported over the same 200 s horizon. The occlusion rate (OR) is 4.36% for Eva-Tracker, 8.71% for Elastic-Tracker, 17.3% for Vis-Planner, and 27.5% for SF-Tracker. The failure rate (FR, target lost) is 6.75% for Eva-Tracker, 12.27% for Elastic, 28.5% for Vis-Planner, and 40.6% for SF-Tracker.

Metric	Eva-Tracker	Comparison values
Path-generation time	0.02 ms	Vis-Planner, SF-Tracker: $\mathcal O$4 ms
Trajectory optimization time	10.7 ms	Vis-Planner / SF-Tracker: $\mathcal O$5 ms; Elastic-Tracker: 6.6 ms
Occlusion rate (OR)	4.36%	Elastic-Tracker: 8.71%; Vis-Planner: 17.3%; SF-Tracker: 27.5%
Failure rate (FR)	6.75%	Elastic: 12.27%; Vis-Planner: 28.5%; SF-Tracker: 40.6%

The summary given in the source states that Eva-Tracker is approximately $\mathcal O$6 faster than other ESDF-based planners, with a $\mathcal O$7 reduction in occlusion and failure. Real-world validation is reported on an onboard platform using NVIDIA Jetson Orin NX, LiDAR, and camera, with tests in indoor and outdoor settings. The described experiments include continuous visibility maintenance in a door-closing indoor scenario and a tree-dense outdoor scenario, as well as rapid recovery from temporary target loss using the same FoV-ESDF-guided path generation.

7. Interpretation, boundaries, and relation to standard ESDF usage

FoV-ESDF is best understood as a specialized visibility field. It is precomputed, bounded to the camera pyramid, and never updated. Standard ESDF, by contrast, covers all 3D space and must be updated on each LiDAR or depth-map update. This difference is not merely an implementation detail; it defines the computational and representational boundary of the method (Lin et al., 13 Feb 2026).

Because the field is tracker-centric and zero outside the FoV region, it does not encode free-space geometry in the same way as a conventional world ESDF. Its practical value arises from the fact that dynamic obstacles need only be transformed into the tracker frame and queried against $\mathcal O$8, rather than fused into a globally maintained distance map. This is why the method can support rapid visibility evaluation without requiring updates.

The principal limitation, as implied by the formulation, is the same specialization that gives it efficiency. FoV-ESDF is tailored to the tracker’s field of view and to visibility-aware planning objectives. A plausible implication is that it is most effective in planners where visibility is the primary geometric concern and where the camera model, FoV angles, and preferred observation distance are known in advance. Within that scope, it functions as a compact differentiable proxy for visibility structure rather than as a universal signed-distance representation of the environment.