[Paper Review] Art Gallery Localization

Prosenjit Bose, Jean-Lou De Carufel+2 more|arXiv (Cornell University)|Jun 21, 2017

Computational Geometry and Mesh Generation7 references3 citations

TL;DR

This paper presents a novel tower-placement algorithm for self-localization in simple polygons using trilateration, where agents determine their coordinates via distance and position data from visible towers. It proves that at most ⌊2n/3⌋ towers are sufficient and necessary for full localization, improving the prior upper bound of ⌊8n/9⌋.

ABSTRACT

We study the problem of placing a set $T$ of broadcast towers in a simple polygon $P$ in order for any point to locate itself in the interior of $P$. Let $V(p)$ denote the visibility polygon of a point $p$, as the set of all points $q \in P$ that are visible to $p$. For any point $p \in P$: for each tower $t \in T \cap V(p)$ the point $p$ receives the coordinates of $t$ and the Euclidean distance between $t$ and $p$. From this information $p$ can determine its coordinates. We show a tower-positioning algorithm that computes such a set $T$ of size at most $\lfloor 2n/3 floor$, where $n$ is the size of $P$. This improves the previous upper bound of $\lfloor 8n/9 floor$ towers. We also show that $\lfloor 2n/3 floor$ towers are sometimes necessary.

Motivation & Objective

• To design a minimal set of broadcast towers in a simple polygon such that any interior point (agent) can determine its absolute coordinates via trilateration.
• To improve the upper bound on the number of required towers for localization, reducing it from ⌊8n/9⌋ to ⌊2n/3⌋.
• To adapt Tóth’s polygon partitioning method to handle non-general position subpolygons arising during decomposition, ensuring applicability to real-world polygons.
• To develop a localization algorithm that uses only tower coordinates and distances to resolve the agent’s position uniquely, even when only two towers are visible.
• To prove that ⌊2n/3⌋ towers are sometimes necessary, establishing the bound as tight.

Proposed method

• Adapts Tóth’s polygon partitioning method to decompose a simple polygon into at most ⌊n/3⌋ star-shaped subpolygons, each requiring a pair of towers for full visibility.
• Modifies the partitioning process to handle non-general position vertices introduced during decomposition, ensuring that each subpolygon’s kernel contains at least two distinct towers.
• Uses a 'parity trick' to resolve ambiguous trilateration solutions: when only two towers are visible, the algorithm selects the correct location based on the parity of the numerator of the distance between the two towers.
• Employs classical trilateration via intersection of circles centered at visible towers with radii equal to measured distances to the agent.
• Introduces a localization algorithm that operates solely on input from visible towers—coordinates and distances—without prior knowledge of the polygon’s structure.
• Validates the method by proving that the modified partitioning ensures each subpolygon is guarded by two towers whose visibility regions cover the entire subpolygon.

Experimental results

Research questions

• RQ1 What is the minimum number of towers required to enable full self-localization via trilateration in any simple polygon?
• RQ2 Can Tóth’s polygon partitioning method be adapted to ensure that each subpolygon contains a kernel with two distinct towers, even when vertices are not in general position?
• RQ3 How can ambiguous trilateration solutions (two possible locations from two circles) be resolved uniquely using geometric or arithmetic properties?
• RQ4 Is the upper bound of ⌊2n/3⌋ towers for localization tight, or can it be further reduced?
• RQ5 Is the problem of finding the optimal number of towers for polygon trilateration NP-hard?

Key findings

• The paper establishes an upper bound of ⌊2n/3⌋ towers for full localization in any simple polygon, improving the prior bound of ⌊8n/9⌋.
• It proves that ⌊2n/3⌋ towers are sometimes necessary, demonstrating that the bound is tight and cannot be improved asymptotically.
• The authors identify and correct errors in Tóth’s original claims (Claims 2 and 7), strengthening the theoretical foundation of the partitioning method.
• The proposed localization algorithm uniquely determines the agent’s position using only the coordinates and distances from visible towers, with ambiguity resolved via a parity-based rule.
• The method is robust to non-general position subpolygons introduced during partitioning, extending the applicability of Tóth’s approach.
• The paper suggests that the problem of computing the optimal number of towers for trilateration is likely NP-hard, proposing this as a direction for future work.