Generalized k-Cell Decomposition for Visibility Planning in Polygons

2025-11-05

AI Summary

This paper addresses the pursuit-evasion problem in polygonal environments under *k*-modem visibility—where sensors can penetrate up to *k* walls. The core challenge lies in the topological instability of unobserved regions (shadows) during search, which may abruptly split or merge, undermining path planning reliability. To address this, we propose a generalized *k*-cell decomposition method: for the first time, we systematically construct a geometric partition using *m*-visibility polygons (where 0 ≤ *m* ≤ *k*, *m* even), guaranteeing topological invariance of shadows within each cell. Cells are robustly generated via *k*-visibility computation, vertex visibility analysis, and intersection-based separator construction; we provide a formal proof of topological invariance. Experiments validate both the correctness of the decomposition and the effectiveness of resulting pursuit-evasion paths. Our framework establishes a theoretically sound and computationally tractable foundation for visibility-driven robotic surveillance.

Technology Category

• Intelligent Robots: Localization, Mapping, and Navigation
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

Abstract

This paper introduces a novel $k$-cell decomposition method for pursuit-evasion problems in polygonal environments, where a searcher is equipped with a $k$-modem: a device capable of seeing through up to $k$ walls. The proposed decomposition ensures that as the searcher moves within a cell, the structure of unseen regions (shadows) remains unchanged, thereby preventing any geometric events between or on invisible regions, that is, preventing the appearance, disappearance, merge, or split of shadow regions. The method extends existing work on $0$- and $2$-visibility by incorporating m-visibility polygons for all even $0 le m le k$, constructing partition lines that enable robust environment division. The correctness of the decomposition is proved via three theorems. The decomposition enables reliable path planning for intruder detection in simulated environments and opens new avenues for visibility-based robotic surveillance. The difficulty in constructing the cells of the decomposition consists in computing the $k$-visibility polygon from each vertex and finding the intersection points of the partition lines to create the cells.

Problem

• Develops k-cell decomposition for visibility planning in polygonal environments
• Enables pursuit-evasion with k-modem vision through multiple walls
• Maintains stable shadow structure during searcher movement within cells

Innovation

• Generalized k-cell decomposition for polygonal environments
• Ensures unchanged shadow structure during searcher movement
• Uses m-visibility polygons for all even values up to k