Geometry-Driven Adequate Zone Detection

• Geometry-driven adequate zone detection defines regions using rigorous geometric consistency and reachability models to ensure safety and effective perception.
• The methodology integrates Hamilton–Jacobi reachability, temporal convolution for maneuver refinement, and sensor-based polygonal modeling to optimize zone accuracy.
• Empirical results indicate up to 76% reduction in conservatism while preserving safety-critical completeness across autonomous and perception systems.

Geometry-driven adequate zone detection encompasses a set of methodologies for partitioning sensor, state, or image space into regions deemed geometrically “adequate” for safety, perception, collision avoidance, or task-specific utility. At its core, this paradigm leverages explicit geometric consistency, contrast, and reachability properties to produce zones that are provably sound for downstream planning or evaluation metrics, while minimizing the inclusion of unnecessary or irrelevant regions. This article surveys theoretical foundations, algorithmic constructions, application domains, and quantitative outcomes of geometry-driven adequate zone detection across robotics, autonomous driving, image analysis, and 3D affordance reasoning.

1. Hamilton–Jacobi Reachability and Safety Zone Construction

The formalism of Hamilton–Jacobi (HJ) reachability yields a rigorous mechanism for defining safety-critical “zones” in ego–contender interaction systems. Given the joint state $x=(E,C)$, with $E \in \mathbb{R}^{n_E}$ (ego vehicle) and $C \in \mathbb{R}^{n_C}$ (contender/obstacle), the system evolves under joint controls: $\dot{x} = f(x, u_E, u_C)$. The collision set $\mathcal{T} = \{x \mid h_\text{col}(x) \leq 0\}$ codifies unsafe separations or overlaps. The zero-cost two-player game solution is computed via the backward-in-time HJ PDE:

$$\frac{\partial V}{\partial \tau} + \min_{u_E \in \mathcal{U}_E} \min_{u_C \in \mathcal{U}_C} \nabla_x V(x, \tau)^T f(x, u_E, u_C) = 0$$

with terminal condition $V(x,0) = h_\text{col}(x)$.

The baseline safety zone is extracted as the zero-sublevel set at initial time,

$$Z_\text{baseline} = \{E \mid \exists\, C: V((E,C),0) \leq 0\}$$

guaranteeing completeness for all possible interactive evolutions within horizon $T$ (Topan et al., 2023, Topan et al., 2022).

2. Maneuver-Driven Zone Refinement via Temporal Convolution

To reduce unnecessary conservatism, geometry-driven adequate zone detection incorporates knowledge of the planned ego maneuver $m \in \mathcal{M}$ (e.g., lane change, fixed-radius turn). The maneuver constraint set $\mathcal{C}_m \subset \mathbb{R}^{n_E}$ specifies the terminal conditions (lateral lane position, final heading, etc.).

A two-stage reachability decomposition is formulated:

• Ego-only value $E(E,t)$ solves

$$\frac{\partial E}{\partial t} + \min_{u_E \in \mathcal{U}_E} \nabla E(E,t)^T f_E(E, u_E) = 0,\quad E(E,0) = h_\text{maneuver}(E)$$

where $E(E,t)<0$ denotes states from which the maneuver is feasible within $t$.

• Collision at time $s \in [0,T]$ with leftover time for maneuver completion leads to terminal

$$G_s(x) = \max\{h_\text{col}(x),\, E^{\uparrow}(x,s)\}$$

and the two-agent HJ-PDE is solved over $\tau \in [0,s]$.

Composing all such possibilities produces the temporal convolution metric:

$V_m(x) = \min_{s \in [0,T]} V^s(x,s)$

with the maneuver-aware safety zone

$Z_m = \{x \mid V_m(x) \leq 0\}$

Empirical results demonstrate that maneuver-aware zones are up to $76\%$ smaller than the baseline, while full completeness (no missed unsafe regions) is preserved (Topan et al., 2023).

3. Algorithmic and Numerical Aspects

HJ-based adequate zone construction is solved numerically on multidimensional grids (e.g., 7D for lane change, 6D for fixed-path turn). Offline computation proceeds via level-set or PDE integration toolkits; online evaluation involves state interpolation on the discretized zone value function. This approach supports real-time operation for perception modules and evaluation metrics in safety-critical systems (Topan et al., 2022).

Comparison of zone volumes and boundaries is performed by varying the maneuver constraints and interaction models. Adversarial MPC simulation validates completeness: all actual collision trajectories under constrained maneuvers fall within the predicted adequate zone. Sparse false positives are attributed to numerical grid limitations rather than metric deficiencies (Topan et al., 2023).

4. Adequate Zone Detection in Perception and Occupancy Modeling

Geometry-driven approaches in perception interpret “adequate zones” as free-space or collision-safe polygons constructed directly from sensor data. In radar-based polygonal occupancy modeling, each time step produces a deformable polygon $\mathcal{P}_t$ whose vertices are derived from radar pointclouds, SNR-based evidence measures, and Doppler velocities:

• Sector-wise vertex selection uses proximity and SNR cue, verified by weighted detection probability and local evidence maps.
• Polygon assembly chains sector-select points (and possible virtual insertions) in azimuth order.
• Deformation prediction extrapolates future shapes using per-vertex Doppler velocity information (Xiangyu et al., 2022).

In LIDAR-based occupancy grid mapping, the pipeline utilizes:

• DBSCAN filtering to remove clutter and interpolate virtual endpoints.
• Bayesian log-odds updates per cell, distinguished between occupied, free, and virtual states.
• Polygon-based ray-casting for efficient grid filling.
• Circle-based grid alignment for spatial coverage and computational reduction.
• In-sight edge detection via radial Bresenham rays and subsequent Douglas–Peucker simplification to generate a free-space polygon of fixed vertex count (Eraqi et al., 2018).

These geometric zone constructions provide compact, interpretable, and robust representations directly consumable by safety arbitration, motion planning, or collision avoidance modules.

5. Geometry-Driven Adequate Zone Detection in Image Stitching

In multi-view computer vision, geometry-driven adequate zone detection addresses parallax-minimization for image stitching. SENA introduces zone selection based on disparity consistency among RANSAC-filtered inlier matches in the overlap region:

• Matches $M = \{(p_S^i, p_T^i)\}$ are partitioned into $m$ horizontal “abscissa classes.”
• For each class, compute mean horizontal disparity $d_j$ and perform thresholded clustering: classes are grouped into clusters where consecutive means differ by less than $v$.
• Clusters are scored via $w_k = C_k/(σ_k + λΔμ_k + ε)$, balancing point count, intra-cluster variance, and deviation from global mean.
• The highest scoring cluster (adequate zone $Z_\text{pf}$) is chosen for anchor-based seamline cutting and one-to-one geometrical correspondence (Tchana et al., 3 Jan 2026).

The approach is robust, model-free, and empirically superior to semantic-segmentation methods, providing visually optimal seam selection and artifact minimization, especially in scenes with depth variation and parallax.

6. Adequate Zone Detection in 3D Affordance Reasoning

Geometry-driven “adequate zones” in 3D environments are instantiated by the one-shot interaction tensor methodology. For a query object $Q$ and scene object $S$:

• The Interaction Bisector Surface (IBS), $IBS(Q,S) = \{x \mid D_Q(x) = D_S(x)\}$, is sampled and parameterized via provenance vectors.
• The Interaction Tensor $iT(x)$ encodes both $v_Q(x)$ and $v_S(x)$ at sampled IBS locations.
• For novel scenes, candidate placements are evaluated by matching provenance vectors at seed points and candidate orientations; score normalization is based on the fraction of IBS points satisfactorily matched within a spatial radius.
• Adequate zones for affordance are defined as those placements (pose, orientation) where the interaction tensor matches exceed a threshold (Ruiz et al., 2019).

Performance is characterized by high throughput, low position and orientation error, and generalizability across affordance types and environments. The limitations pertain to missing geometric data, non-rigid or articulated objects, and occlusions.

7. Quantitative Benchmarks and Limitations

• Maneuver-based decomposition achieves state-space volume reductions of up to $76\%$ while maintaining completeness for safety-critical interactions (Topan et al., 2023).
• Deformable radar polygons significantly surpass baseline occupancy models in IoU-gt and IoU-smooth visual metrics (Xiangyu et al., 2022).
• Circle-based grid alignment and polygonal extraction in LIDAR systems operate at $20$–$25$ Hz with total memory footprint under $1$ MB (Eraqi et al., 2018).
• Disparity-consistency-driven zones provide empirically higher alignment scores and lower artifacts compared to homography- or semantic-seam methods for image stitching (Tchana et al., 3 Jan 2026).
• Interaction tensor affordance detection yields $0.80$ precision and $0.83$ recall for “place bowl” and similar tasks, with typical pose error $4.5 \pm 1.2$ cm on real and synthetic scenes (Ruiz et al., 2019).

Limitations in all methods arise from sensor density, numerical grid coarseness, rigid-body assumptions, and outlier sensitivity; extensions may address dynamic, articulated, or non-geometric cues.

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Geometry-driven adequate zone detection, across all surveyed domains, achieves mathematically explicit, data-efficient, and high-fidelity network-free segmentation of spaces critical to safety, perception, and task accomplishment. The underlying principle—extraction of essential regions via geometric consistency, reachability, or structural constraints—continues to inform advances in robotics, autonomous systems, computer vision, and physical human-object interaction.