Searching Polyhedra by Rotating Half-Planes

The Searchlight Scheduling Problem was first studied in 2D polygons, where the goal is for point guards in fixed positions to rotate searchlights to catch an evasive intruder. Here the problem is extended to 3D polyhedra, with the guards now boundary segments who rotate half-planes of illumination. After carefully detailing the 3D model, several results are established. The first is a nearly direct extension of the planar one-way sweep strategy using what we call exhaustive guards, a generalization that succeeds despite there being no well-defined notion in 3D of planar "clockwise rotation". Next follow two results: every polyhedron with r>0 reflex edges can be searched by at most r^2 suitably placed guards, whereas just r guards suffice if the polyhedron is orthogonal. (Minimizing the number of guards to search a given polyhedron is easily seen to be NP-hard.) Finally we show that deciding whether a given set of guards has a successful search schedule is strongly NP-hard, and that deciding if a given target area is searchable at all is strongly PSPACE-hard, even for orthogonal polyhedra. A number of peripheral results are proved en route to these central theorems, and several open problems remain for future work.Comment: 45 pages, 26 figure

Polygon Exploration with Time-Discrete Vision

With the advent of autonomous robots with two- and three-dimensional scanning capabilities, classical visibility-based exploration methods from computational geometry have gained in practical importance. However, real-life laser scanning of useful accuracy does not allow the robot to scan continuously while in motion; instead, it has to stop each time it surveys its environment. This requirement was studied by Fekete, Klein and Nuechter for the subproblem of looking around a corner, but until now has not been considered in an online setting for whole polygonal regions. We give the first algorithmic results for this important algorithmic problem that combines stationary art gallery-type aspects with watchman-type issues in an online scenario: We demonstrate that even for orthoconvex polygons, a competitive strategy can be achieved only for limited aspect ratio A (the ratio of the maximum and minimum edge length of the polygon), i.e., for a given lower bound on the size of an edge; we give a matching upper bound by providing an O(log A)-competitive strategy for simple rectilinear polygons, using the assumption that each edge of the polygon has to be fully visible from some scan point.Comment: 28 pages, 17 figures, 2 photographs, 3 tables, Latex. Updated some details (title, figures and text) for final journal revision, including explicit assumption of full edge visibilit

Algorithms for fat objects : decompositions and applications

Computational geometry is the branch of theoretical computer science that deals with algorithms and data structures for geometric objects. The most basic geometric objects include points, lines, polygons, and polyhedra. Computational geometry has applications in many areas of computer science, including computer graphics, robotics, and geographic information systems. In many computational-geometry problems, the theoretical worst case is achieved by input that is in some way "unrealistic". This causes situations where the theoretical running time is not a good predictor of the running time in practice. In addition, algorithms must also be designed with the worst-case examples in mind, which causes them to be needlessly complicated. In recent years, realistic input models have been proposed in an attempt to deal with this problem. The usual form such solutions take is to limit some geometric property of the input to a constant. We examine a specific realistic input model in this thesis: the model where objects are restricted to be fat. Intuitively, objects that are more like a ball are more fat, and objects that are more like a long pole are less fat. We look at fat objects in the context of five different problems—two related to decompositions of input objects and three problems suggested by computer graphics. Decompositions of geometric objects are important because they are often used as a preliminary step in other algorithms, since many algorithms can only handle geometric objects that are convex and preferably of low complexity. The two main issues in developing decomposition algorithms are to keep the number of pieces produced by the decomposition small and to compute the decomposition quickly. The main question we address is the following: is it possible to obtain better decompositions for fat objects than for general objects, and/or is it possible to obtain decompositions quickly? These questions are also interesting because most research into fat objects has concerned objects that are convex. We begin by triangulating fat polygons. The problem of triangulating polygons—that is, partitioning them into triangles without adding any vertices—has been solved already, but the only linear-time algorithm is so complicated that it has never been implemented. We propose two algorithms for triangulating fat polygons in linear time that are much simpler. They make use of the observation that a small set of guards placed at points inside a (certain type of) fat polygon is sufficient to see the boundary of such a polygon. We then look at decompositions of fat polyhedra in three dimensions. We show that polyhedra can be decomposed into a linear number of convex pieces if certain fatness restrictions aremet. We also show that if these restrictions are notmet, a quadratic number of pieces may be needed. We also show that if we wish the output to be fat and convex, the restrictions must be much tighter. We then study three computational-geometry problems inspired by computer graphics. First, we study ray-shooting amidst fat objects from two perspectives. This is the problem of preprocessing data into a data structure that can answer which object is first hit by a query ray in a given direction from a given point. We present a new data structure for answering vertical ray-shooting queries—that is, queries where the ray’s direction is fixed—as well as a data structure for answering ray-shooting queries for rays with arbitrary direction. Both structures improve the best known results on these problems. Another problem that is studied in the field of computer graphics is the depth-order problem. We study it in the context of computational geometry. This is the problem of finding an ordering of the objects in the scene from "top" to "bottom", where one object is above the other if they share a point in the projection to the xy-plane and the first object has a higher z-value at that point. We give an algorithm for finding the depth order of a group of fat objects and an algorithm for verifying if a depth order of a group of fat objects is correct. The latter algorithm is useful because the former can return an incorrect order if the objects do not have a depth order (this can happen if the above/below relationship has a cycle in it). The first algorithm improves on the results previously known for fat objects; the second is the first algorithm for verifying depth orders of fat objects. The final problem that we study is the hidden-surface removal problem. In this problem, we wish to find and report the visible portions of a scene from a given viewpoint—this is called the visibility map. The main difficulty in this problem is to find an algorithm whose running time depends in part on the complexity of the output. For example, if all but one of the objects in the input scene are hidden behind one large object, then our algorithm should have a faster running time than if all of the objects are visible and have borders that overlap. We give such an algorithm that improves on the running time of previous algorithms for fat objects. Furthermore, our algorithm is able to handle curved objects and situations where the objects do not have a depth order—two features missing from most other algorithms that perform hidden surface removal

Vision-Based Localization Algorithm Based on Landmark Matching, Triangulation, Reconstruction, and Comparison

Many generic position-estimation algorithms are vulnerable to ambiguity introduced by nonunique landmarks. Also, the available high-dimensional image data is not fully used when these techniques are extended to vision-based localization. This paper presents the landmark matching, triangulation, reconstruction, and comparison (LTRC) global localization algorithm, which is reasonably immune to ambiguous landmark matches. It extracts natural landmarks for the (rough) matching stage before generating the list of possible position estimates through triangulation. Reconstruction and comparison then rank the possible estimates. The LTRC algorithm has been implemented using an interpreted language, onto a robot equipped with a panoramic vision system. Empirical data shows remarkable improvement in accuracy when compared with the established random sample consensus method. LTRC is also robust against inaccurate map data

On realistic target coverage by autonomous drones

Low-cost mini-drones with advanced sensing and maneuverability enable a new class of intelligent sensing systems. To achieve the full potential of such drones, it is necessary to develop new enhanced formulations of both common and emerging sensing scenarios. Namely, several fundamental challenges in visual sensing are yet to be solved including (1) fitting sizable targets in camera frames; (2) positioning cameras at effective viewpoints matching target poses; and (3) accounting for occlusion by elements in the environment, including other targets. In this article, we introduce Argus, an autonomous system that utilizes drones to collect target information incrementally through a two-tier architecture. To tackle the stated challenges, Argus employs a novel geometric model that captures both target shapes and coverage constraints. Recognizing drones as the scarcest resource, Argus aims to minimize the number of drones required to cover a set of targets. We prove this problem is NP-hard, and even hard to approximate, before deriving a best-possible approximation algorithm along with a competitive sampling heuristic which runs up to 100× faster according to large-scale simulations. To test Argus in action, we demonstrate and analyze its performance on a prototype implementation. Finally, we present a number of extensions to accommodate more application requirements and highlight some open problems

Large-scale Geometric Data Decomposition, Processing and Structured Mesh Generation

Mesh generation is a fundamental and critical problem in geometric data modeling and processing. In most scientific and engineering tasks that involve numerical computations and simulations on 2D/3D regions or on curved geometric objects, discretizing or approximating the geometric data using a polygonal or polyhedral meshes is always the first step of the procedure. The quality of this tessellation often dictates the subsequent computation accuracy, efficiency, and numerical stability. When compared with unstructured meshes, the structured meshes are favored in many scientific/engineering tasks due to their good properties. However, generating high-quality structured mesh remains challenging, especially for complex or large-scale geometric data. In industrial Computer-aided Design/Engineering (CAD/CAE) pipelines, the geometry processing to create a desirable structural mesh of the complex model is the most costly step. This step is semi-manual, and often takes up to several weeks to finish. Several technical challenges remains unsolved in existing structured mesh generation techniques. This dissertation studies the effective generation of structural mesh on large and complex geometric data. We study a general geometric computation paradigm to solve this problem via model partitioning and divide-and-conquer. To apply effective divide-and-conquer, we study two key technical components: the shape decomposition in the divide stage, and the structured meshing in the conquer stage. We test our algorithm on vairous data set, the results demonstrate the efficiency and effectiveness of our framework. The comparisons also show our algorithm outperforms existing partitioning methods in final meshing quality. We also show our pipeline scales up efficiently on HPC environment

Resolução do problema da galeria de arte : um método prático e robusto para o posicionamento ótimo de guardas-ponto

Orientadores: Cid Carvalho de Souza, Pedro Jussieu de RezendeDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Nesta dissertação, apresentamos nossa pesquisa sobre o Problema da Galeria de Arte (AGP), um dos problemas mais estudados em Geometria Computacional. O AGP, que é um problema NP-difícil, consiste em encontrar o número mínimo de guardas suficiente para garantir a cobertura visual de uma galeria de arte representada por um polígono. Na versão do problema tratada neste trabalho, usualmente chamada de Problema da Galeria de Arte com Guardas-Ponto, os guardas podem ser posicionados em qualquer lugar do polígono e o objetivo é cobrir toda a região, que pode ou não conter buracos. Nós estudamos como aplicar conceitos e algoritmos de Geometria Computacional, bem como Técnicas de Programação Inteira, com a finalidade de resolver o AGP de forma exata. Este trabalho culminou na criação de um novo algoritmo para o AGP, cuja ideia é gerar, de forma iterativa, limitantes superiores e inferiores para o problema através da resolução de versões discretizadas do AGP, que são reduzidas a instâncias do Problema de Cobertura de Conjuntos. O algoritmo foi implementado e testado em mais de 2800 instâncias, de diferentes tamanhos e classes. A técnica foi capaz de resolver, em minutos, mais de 90% de todas as instâncias consideradas, incluindo polígonos com milhares de vértices, e ampliou em muito o conjunto de casos para os quais são conhecidas soluções exatas. Até onde sabemos, apesar do extensivo estudo do AGP nas últimas quatro décadas, nenhum outro algoritmo demonstrou a capacidade de resolver o AGP de forma tão eficaz como a técnica aqui descritaAbstract: In this dissertation, we present our research on the Art Gallery Problem (AGP), one of the most investigated problems in Computational Geometry. The AGP, which is a known NP-hard problem, consists in finding the minimum number of guards sufficient to ensure the visibility coverage of an art gallery represented as a polygon. In the version of the problem treated in this work, usually called Art Gallery Problem with Point Guards, the guards can be placed anywhere in the polygon and the objective is to cover the whole region, which may or not have holes. We studied how to apply Computational Geometry concepts and algorithms as well as Integer Programming techniques in order to solve the AGP to optimality. This work culminated in the creation of a new algorithm for the AGP, whose idea is to iteratively generate upper and lower bounds for the problem through the resolution of discretized versions of the AGP, which are reduced to instances of the Set Cover Problem. The algorithm was implemented and tested on more than 2800 instances of different sizes and classes of polygons. The technique was able to solve in minutes more than 90% of all instances considered, including polygons with thousands of vertices, greatly increasing the set of instances for which exact solutions are known. To the best of our knowledge, in spite of the extensive study of the AGP in the last four decades, no other algorithm has shown the ability to solve the AGP as effectively as the one described hereMestradoCiência da ComputaçãoMestre em Ciência da Computaçã

How Do the Largest and Smallest Baboon Species Compete for Reproductive Success in a Natural Hybrid Zone?

This dissertation examines hybridization between two of the most divergent baboons, the kinda baboon (Papio kindae) and the grayfooted chacma baboon (P. ursinus griseipes), which differ markedly in body size and in some social behavior. Preliminary research revealed hybridization between males of the smaller species (kinda) and females of the larger species (grayfoots), but not the reverse. Using behavioral, phenotypic, and genetic data collected from a single hybrid group in Kafue National Park from May 2012 to July 2013, I evaluated whether a similar asymmetry was borne out in this group and whether phenotypic markers of species assignation matched genotypic groupings based upon seven microsatellite markers. I assessed what factors were influential for male mating success in this group and explored whether mating and reproductive success could be explained by the priority-of-access model, whereby the dominant male realizes the most reproductive success. I investigated whether a modified form of this priority-of-access model, female preference for unusually friendly kinda males, and/or genetic or obstetric incompatibility might explain this proposed asymmetry. I found that while asymmetry is present in the overall hybrid zone, it was not found in this group. Phenotypic markers of species assignation did not match genotypic groupings. As in most other baboon species, dominance rank and mate guarding were the most influential factors in male reproductive success in this group, supporting the priority-of-access model. However, since Kinda-like males in this group groom more and have higher reproductive success, the asymmetry may be in part due to this “friendly” behavior. While mating occurred across all genotypes and phenotypes, the lack of hybrid offspring resulting from parents having opposite genetic backgrounds suggests possible genetic or obstetric incompatibility. Results from this study reveal that being a genetically Kinda-like male (regardless of phenotype) confers some sort of reproductive advantage. This study has helped clarify the importance of friendly male behavior in this unexpected asymmetry, provided insight into issues related to mate choice in hybrids, and revealed possible reproductive barriers to hybridization, as well as contributed to the corpus of knowledge of baboon diversity in general