Generalized k-Cell Decomposition for Visibility Planning in Polygons

TL;DR

The paper addresses pursuit-evasion in polygonal environments with a $k$-modem capable of seeing through up to $k$ walls. It builds a $k$-cell decomposition by aggregating the lines of all even-valued $0,2,\dots,k$ visibility polygons at every vertex and proves shadow invariance within each cell via three theorems, enabling robust intra-cell planning. The main contributions are the formal invariance proofs, the $O(k^2 n^4)$ complexity bound for the decomposition, and a pathway to k-modem–driven path planning using the decomposition. This approach advances visibility-based surveillance by enabling reliable intruder detection in general polygons under $k$-visibility.

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.

Figures (10)

• Figure 1: Visibility between two points $p$ and $q$
• Figure 2: The four geometric events that may occur to the pursuer's shadow as the pursuer moves within the polygon
• Figure 3: The cell decomposition with all decomposition lines drawn from every vertex, for $k=2$
• Figure 4: The shadow of type 1 - a shadow that includes a vertex
• Figure 5: The shadow of type 2 (edge shadow) - a shadow that occurs between two edges

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 2
• Theorem 3