arXiv subjects

Yeganeh Bahoo

Publications and source records attributed to Yeganeh Bahoo.

12 recordsLinked to original sources

arXiv2026

Efficient K-Visibility Query in Polygons

This paper investigates $k$-visibility, where a line of sight can penetrate up to $k$ obstacles. While computing the $k$-visibility polygon from a single query point is well-studied, existing spatial preprocessing approaches rely on full $O(n^2)$ line arrangements through all vertex pairs without characterizing the minimal set of topological boundaries. We present a refined cell decomposition framework that isolates the exact geometric events governing $k$-visibility: primary vertex horizon lines and secondary mutually critical hinge lines. We prove that this minimal set of partition lines yields a spatial decomposition of $Θ(n^4)$ cells within which the combinatorial structure of the $k$-visibility polygon remains strictly invariant. By leveraging a combinatorial $δ$-compression scheme across cell boundaries, we achieve an overall storage complexity of $\mathcal{O}(n^4)$ while supporting optimal $\mathcal{O}(\log n + m)$ query time to reconstruct explicit $k$-visibility polygons of size $m$. Our framework naturally extends to polygons containing holes.

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arXiv2026

Generalized k-Cell Decomposition for Visibility Planning in Polygons

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.

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arXiv2026

An Exact Generalized k-Cell Decomposition

This paper introduces an exact $k$-cell decomposition for visibility planning in polygonal environments for agents equipped with $k$-modems, devices that can see through up to $k$ walls. Unlike prior decompositions that may include redundant partition lines, our proposed method ensures that visibility events (appear, disappear, merge, and split) are guaranteed to occur on every line of the decomposition. By eliminating these redundancies, we achieve an $O(n^4)$ complexity , representing a potentially quadratic improvement over the previous best $O(k^2n^4)$ result. This decomposition explicitly identifies the locations of all critical visibility events and extends to polygons with holes. It has practical applications in tasks such as optimal pursuit-evasion under $k$-visibility and agent counting in invisible regions.

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arXiv2026

Guaranteed Escape for a Bouncing Robot in Pipe Chains

We study the symmetric bouncing of a point robot within orthogonally-joined rectangles with equal width, which we refer to as pipes. We provide an exhaustive case analysis of every trajectory pattern inside a single rectangular pipe segment, identifying the conditions under which the robot exits. We then extend the analysis to L-shaped pipes and, more generally, to linear chains of $k$ orthogonally connected pipe segments. We prove exit guarantees for the special angle $α= π/4$. Furthermore, these results extend to pipes with curved joints.

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arXiv2025

Inverse k-visibility for RSSI-based Indoor Geometric Mapping

In recent years, the increased availability of WiFi in indoor environments has gained interest in the robotics community to utilize WiFi signals for indoor simultaneous localization and mapping algorithms. This paper discusses the challenges of achieving high-accuracy geometric map building using WiFi signals. The paper introduces the concept of inverse k-visibility, developed from the k-visibility algorithm, to identify free space in an unknown environment, used for planning, navigation, and obstacle avoidance. Comprehensive experiments, including those utilizing single and multiple RSSI signals, were conducted in both simulated and real-world environments to demonstrate the robustness of the proposed algorithm. Additionally, a detailed analysis comparing the resulting maps with ground-truth LiDAR-based maps is provided to highlight the algorithm's accuracy and reliability.

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arXiv2025

M-Guarding in K-Visibility

We explore the problem of $M$-guarding polygons with holes using $k$-visibility guards, where a set of guards is said to $M$-guard a polygon if every point in the polygon is visible to at least $M$ guards, with the constraint that there may only be 1 guard on each edge. A $k$-visibility guard can see through up to $k$ walls, with $k \geq 2$. We present a theorem establishing that any polygon with holes can be 2-guarded under $k$-visibility where $k \geq 2$, which expands existing results in 0-visibility. We provide an algorithm that $M$-guards a polygon using a convex decomposition of the polygon. We show that every point in the polygon is visible to at least four $2$-visibility guards and then extend the result to show that for any even $k \geq 2$ there exists a placement of guards such that every point in the polygon is visible to $k + 2$ guards.

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arXiv2024

DPPE: Dense Pose Estimation in a Plenoxels Environment using Gradient Approximation

We present DPPE, a dense pose estimation algorithm that functions over a Plenoxels environment. Recent advances in neural radiance field techniques have shown that it is a powerful tool for environment representation. More recent neural rendering algorithms have significantly improved both training duration and rendering speed. Plenoxels introduced a fully-differentiable radiance field technique that uses Plenoptic volume elements contained in voxels for rendering, offering reduced training times and better rendering accuracy, while also eliminating the neural net component. In this work, we introduce a 6-DoF monocular RGB-only pose estimation procedure for Plenoxels, which seeks to recover the ground truth camera pose after a perturbation. We employ a variation on classical template matching techniques, using stochastic gradient descent to optimize the pose by minimizing errors in re-rendering. In particular, we examine an approach that takes advantage of the rapid rendering speed of Plenoxels to numerically approximate part of the pose gradient, using a central differencing technique. We show that such methods are effective in pose estimation. Finally, we perform ablations over key components of the problem space, with a particular focus on image subsampling and Plenoxel grid resolution. Project website: https://sites.google.com/view/dppe

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arXiv2024

Structure from WiFi (SfW): RSSI-based Geometric Mapping of Indoor Environments

With the rising prominence of WiFi in common spaces, efforts have been made in the robotics community to take advantage of this fact by incorporating WiFi signal measurements in indoor SLAM (Simultaneous Localization and Mapping) systems. SLAM is essential in a wide range of applications, especially in the control of autonomous robots. This paper describes recent work in the development of WiFi-based localization and addresses the challenges currently faced in achieving WiFi-based geometric mapping. Inspired by the field of research into k-visibility, this paper presents the concept of inverse k-visibility and proposes a novel algorithm that allows robots to build a map of the free space of an unknown environment, essential for planning, navigation, and avoiding obstacles. Experiments performed in simulated and real-world environments demonstrate the effectiveness of the proposed algorithm.

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arXiv2020

Compatible Paths on Labelled Point Sets

Let $P$ and $Q$ be finite point sets of the same cardinality in $\mathbb{R}^2$, each labelled from $1$ to $n$. Two noncrossing geometric graphs $G_P$ and $G_Q$ spanning $P$ and $Q$, respectively, are called compatible if for every face $f$ in $G_P$, there exists a corresponding face in $G_Q$ with the same clockwise ordering of the vertices on its boundary as in $f$. In particular, $G_P$ and $G_Q$ must be straight-line embeddings of the same connected $n$-vertex graph. Deciding whether two labelled point sets admit compatible geometric paths is known to be NP-complete. We give polynomial-time algorithms to find compatible paths or report that none exist in three scenarios: $O(n)$ time for points in convex position; $O(n^2)$ time for two simple polygons, where the paths are restricted to remain inside the closed polygons; and $O(n^2 \log n)$ time for points in general position if the paths are restricted to be monotone.

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arXiv2019

A Time-Space Trade-off for Computing the k-Visibility Region of a Point in a Polygon

Let $P$ be a simple polygon with $n$ vertices, and let $q \in P$ be a point in $P$. Let $k \in \{0, \dots, n - 1\}$. A point $p \in P$ is $k$-visible from $q$ if and only if the line segment $pq$ crosses the boundary of $P$ at most $k$ times. The $k$-visibility region of $q$ in $P$ is the set of all points that are $k$-visible from $q$. We study the problem of computing the $k$-visibility region in the limited workspace model, where the input resides in a random-access read-only memory of $O(n)$ words, each with $Ω(\log{n})$ bits. The algorithm can read and write $O(s)$ additional words of workspace, where $s \in \mathbb{N}$ is a parameter of the model. The output is written to a write-only stream. Given a simple polygon $P$ with $n$ vertices and a point $q \in P$, we present an algorithm that reports the $k$-visibility region of $q$ in $P$ in $O(cn/s+c\log{s} + \min\{\lceil k/s \rceil n,n \log{\log_s{n}}\})$ expected time using $O(s)$ words of workspace. Here, $c \in \{1, \dots, n\}$ is the number of critical vertices of $P$ for $q$ where the $k$-visibility region of $q$ may change. We generalize this result for polygons with holes and for sets of non-crossing line segments.

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arXiv2018

Polygon Simplification by Minimizing Convex Corners

Let $P$ be a polygon with $r>0$ reflex vertices and possibly with holes and islands. A subsuming polygon of $P$ is a polygon $P'$ such that $P \subseteq P'$, each connected component $R$ of $P$ is a subset of a distinct connected component $R'$ of $P'$, and the reflex corners of $R$ coincide with those of $R'$. A subsuming chain of $P'$ is a minimal path on the boundary of $P'$ whose two end edges coincide with two edges of $P$. Aichholzer et al. proved that every polygon $P$ has a subsuming polygon with $O(r)$ vertices, and posed an open problem to determine the computational complexity of computing subsuming polygons with the minimum number of convex vertices. We prove that the problem of computing an optimal subsuming polygon is NP-complete, but the complexity remains open for simple polygons (i.e., polygons without holes). Our NP-hardness result holds even when the subsuming chains are restricted to have constant length and lie on the arrangement of lines determined by the edges of the input polygon. We show that this restriction makes the problem polynomial-time solvable for simple polygons.

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arXiv2017

Exploring Increasing-Chord Paths and Trees

A straight-line drawing $Γ$ of a graph $G=(V,E)$ is a drawing of $G$ in the Euclidean plane, where every vertex in $G$ is mapped to a distinct point, and every edge in $G$ is mapped to a straight line segment between their endpoints. A path $P$ in $Γ$ is called increasing-chord if for every four points (not necessarily vertices) $a,b,c,d$ on $P$ in this order, the Euclidean distance between $b,c$ is at most the Euclidean distance between $a,d$. A spanning tree $T$ rooted at some vertex $r$ in $Γ$ is called increasing-chord if $T$ contains an increasing-chord path from $r$ to every vertex in $T$. In this paper we prove that given a vertex $r$ in a straight-line drawing $Γ$, it is NP-complete to determine whether $Γ$ contains an increasing-chord spanning tree rooted at $r$. We conjecture that finding an increasing-chord path between a pair of vertices in $Γ$, which is an intriguing open problem posed by Alamdari et al., is also NP-complete, and show a (non-polynomial) reduction from the 3-SAT problem.

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