Algumas variantes do problema de iluminação de qualidade

Mestrado em Matemática e Aplicações - Ciências da ComputaçãoA nocao de visibilidade surge no nosso dia-a-dia como censo comum, por exemplo quando nos desviamos de obstaculos enquanto caminhamos. Os algoritmos de visibilidade surgem na decada de 70 como uma subarea da Geometria Computacional. Hoje em dia o conceito de visibilidade surge em areas como a Computacao Grafica, Robotica e os Sistemas de Informacao Geografica e em muitas aplicacoes as nocoes de visibilidade e iluminacao aparecem naturalmente relacionadas. Nesta dissertacao, abordamos algumas variantes de iluminacao de qualidade, dando uma especial atencao a t-boa iluminacao [Canales, 2004]. Segundo este conceito, um elemento geometrico so esta bem iluminado se todos os focos de luz que o iluminam estao gbem distribuidos h em seu redor. Assim, vamos focar-nos em quatro tipos de problemas: t-boa iluminacao sem obstaculos, t-boa iluminacao em poligonos convexos, 2-boa iluminacao em poligonos nao convexos e 1-boa iluminacao com obstaculos convexos. Para alem dos problemas referidos, abordamos tambem os seguintes: 3-boa iluminacao em poligono nao convexo, 2-boa iluminacao com obstaculo convexo (ainda em aberto) e iluminacao por triangulos. Foi desenvolvida uma aplicacao, com uma interface grafica, que permite determinar regioes t-bem iluminadas para os primeiros quatro problemas de t-boa iluminacao referidos acima. Para tal consideramos que a amplitude de um foco de luz e 2ƒÎ e que o seu alcance e ilimitado. No entanto, na aplicacao e possivel simular o alcance limitado das luzes. Sobre a aplicacao, descrevemos ainda a implementacao usada e um conjunto de testes que avaliam os algoritmos.The notion of visibility is used in our daily life by common sense, for instance, when we avoid an obstacle on our way. Visibility algorithms first appeared in the 70's as a branch of Computational Geometry. Nowadays, the concept of visibility can be found in areas such as Graphic Computation/Computing, Robotics and Geographic Information Systems. Visibility and Illumination rise as natural applications in several other applications. In this thesis, we study some variants of quality illumination, giving special attention to t-good illumination [Canales, 2004]. According to this concept, a geometrical element only is well-illuminated if all the lights in the plane are “well distributed” around it. According to that, we focus our study in four types of problems: t-good illumination without obstacles, t-good illumination in convex polygons, 2-good illumination in non convex polygons and 1-good illumination with convex obstacles. Furthermore, we also make an approach to the following: 3-good illumination in non convex polygons, 2-good illumination with convex obstacles (still open) and illumination by triangles. We developed an application with graphical interface that determines t-good illuminated regions for the first four problems that were mentioned above. Having that in mind, we consider that each light has amplitude 2π and its range is unlimited. Nevertheless, in this application it is possible to simulate limited illumination range. About the application, we also describe its implementation and a battery of tests to evaluate the algorithms

AUTOMATED 3D CAMERA PLACEMENT

This project aims to find the minimum number of cameras needed to observe given three-dimensional (3D) environment and the cameras placement. This report traces the structure of the algorithm used to find the optimal number and placement of the cameras, the deployment of the algorithm in camera placement system, and presents the results of testing done on the system. This study related to the well known Art Gallery Problem (AGP) that addressed the problem of finding minimum number of guards necessary to guard the art gallery. Since this problem was posed, much research has been done on solving the problem from two dimensional (2D) perspectives. Not much research is done from 3D perspective and only recently more researchers are interested to study this problem in 3D environment.

Approximation Algorithms for Geometric Clustering and Touring Problems

Clustering and touring are two fundamental topics in optimization that have been studied extensively and have ``launched a thousand ships''. In this thesis, we study variants of these problems for Euclidean instances, in which clusters often correspond to sensors that are required to cover, measure or localize targets and tours need to visit locations for the purpose of item delivery or data collection. In the first part of the thesis, we focus on the task of sensor placement for environments in which localization is a necessity and in which its quality depends on the relative angle between the target and the pair of sensors observing it. We formulate a new coverage constraint that bounds this angle and consider the problem of placing a small number of sensors that satisfy it in addition to classical ones such as proximity and line-of-sight visibility. We present a general framework that chooses a small number of sensors and approximates the coverage constraint to arbitrary precision. In the second part of the thesis, we consider the task of collecting data from a set of sensors by getting close to them. This corresponds to a well-known generalization of the Traveling Salesman Problem (TSP) called TSP with Neighborhoods, in which we want to compute a shortest tour that visits at least one point from each unit disk centered at a sensor. One approach is based on an observation that relates the optimal solution with the optimal TSP on the sensors. We show that the associated bound can be improved unless we are in certain exceptional circumstances for which we can get better algorithms. Finally, we discuss Maximum Scatter TSP, which asks for a tour that maximizes the length of the shortest edge. While the Euclidean version admits an efficient approximation scheme and the problem is known to be NP-hard in three dimensions or higher, the question of getting a polynomial time algorithm for two dimensions remains open. To this end, we develop a general technique for the case of points concentrated around the boundary of a circle that we believe can be extended to more general cases

Limited range coverage problems

Doutoramento em MatemáticaTal como o título indica, esta tese estuda problemas de cobertura com alcance limitado. Dado um conjunto de antenas (ou qualquer outro dispositivo sem fios capaz de receber ou transmitir sinais), o objectivo deste trabalho é calcular o alcance mínimo das antenas de modo a que estas cubram completamente um caminho entre dois pontos numa região. Um caminho que apresente estas características é um itinerário seguro. A definição de cobertura é variável e depende da aplicação a que se destina. No caso de situações críticas como o controlo de fogos ou cenários militares, a definição de cobertura recorre à utilização de mais do que uma antena para aumentar a eficácia deste tipo de vigilância. No entanto, o alcance das antenas deverá ser minimizado de modo a manter a vigilância activa o maior tempo possível. Consequentemente, esta tese está centrada na resolução deste problema de optimização e na obtenção de uma solução particular para cada caso. Embora este problema de optimização tenha sido investigado como um problema de cobertura, é possível estabelecer um paralelismo entre problemas de cobertura e problemas de iluminação e vigilância, que são habitualmente designados como problemas da Galeria de Arte. Para converter um problema de cobertura num de iluminação basta considerar um conjunto de luzes em vez de um conjunto de antenas e submetê-lo a restrições idênticas. O principal tema do conjunto de problemas da Galeria de Arte abordado nesta tese é a 1-boa iluminação. Diz-se que um objecto está 1-bem iluminado por um conjunto de luzes se o invólucro convexo destas contém o objecto, tornando assim este conceito num tipo de iluminação de qualidade. O objectivo desta parte do trabalho é então minimizar o alcance das luzes de modo a manter uma iluminação de qualidade. São também apresentadas duas variantes da 1-boa iluminação: a iluminação ortogonal e a boa !-iluminação. Esta última tem aplicações em problemas de profundidade e visualização de dados, temas que são frequentemente abordados em estatística. A resolução destes problemas usando o diagrama de Voronoi Envolvente (uma variante do diagrama de Voronoi adaptada a problemas de boa iluminação) é também proposta nesta tese.As the title implies, this thesis studies limited range coverage problems. Given a set of antennas (or any wireless device able to send or receive some sort of signal), the objective of the discussion that follows is to calculate the antennas’ minimum range so that a path between two points within a region is covered by the antennas, a path known as a safe route. The definition of coverage is variable and depends on the applications. In some instances, for example, when monitoring is critical as in the case of fires or military, the definition of coverage necessarily involves the use of multiple antennas to increase the effectiveness of monitoring. However, it is also desirable to extend a network’s lifespan, normally achieved by minimising the antennas’ range. Therefore the focus of this thesis will be the resolution of this dual problem and an affective solution is offered for each case. Although this question has been researched as an issue of coverage, it is also possible to establish a relation between coverage and illumination and visibility, known as Art Gallery problems. To conceptualise coverage problems as Art Gallery problems, all that is needed is to consider a set of lights instead of a set of antennas, which are subject to a similar set of restrictions. The main focus of the Art Gallery problems addressed in this thesis is 1-good illumination. An object is 1-well illuminated if it is fully contained by the convex hull of a set of lights, making this a type of quality illumination. The objective of the discussion that follows is therefore to minimise the lights’ range whilst maintaining a quality illumination. Moreover, two variants of 1-good illumination are also presented: orthogonal good illumination and good ! -illumination. The latter being related to data depth problems and data visualisation that are frequently used in statistics. The resolution of these problems using the Embracing Voronoi diagram (a variant of Voronoi diagrams adapted to good illumination) is also discussed in this thesis

Placement and motion planning algorithms for robotic sensing systems

University of Minnesota Ph.D. dissertation. October 2014. Major: Computer Science. Advisor: Prof. Ibrahim Volkan Isler. I computer file (PDF); xxiii, 226 pages.Recent technological advances are making it possible to build teams of sensors and robots that can sense data from hard-to-reach places at unprecedented spatio-temporal scales. Robotic sensing systems hold the potential to revolutionize a diverse collection of applications such as agriculture, environmental monitoring, climate studies, security and surveillance in the near future. In order to make full use of this technology, it is crucial to complement it with efficient algorithms that plan for the sensing in these systems. In this dissertation, we develop new sensor planning algorithms and present prototype robotic sensing systems.In the first part of this dissertation, we study two problems on placing stationary sensors to cover an environment. Our objective is to place the fewest number of sensors required to ensure that every point in the environment is covered. In the first problem, we say a point is covered if it is seen by sensors from all orientations. The environment is represented as a polygon and the sensors are modeled as omnidirectional cameras. Our formulation, which builds on the well-known art gallery problem, is motivated by practical applications such as visual inspection and video-conferencing where seeing objects from all sides is crucial. In the second problem, we study how to deploy bearing sensors in order to localize a target in the environment. The sensors measure noisy bearings towards the target which can be combined to localize the target. The uncertainty in localization is a function of the placement of the sensors relative to the target. For both problems we present (i) lower bounds on the number of sensors required for an optimal algorithm, and (ii) algorithms to place at most a constant times the optimal number of sensors. In the second part of this dissertation, we study motion planning problems for mobile sensors. We start by investigating how to plan the motion of a team of aerial robots tasked with tracking targets that are moving on the ground. We then study various coverage problems that arise in two environmental monitoring applications: using robotic boats to monitor radio-tagged invasive fish in lakes, and using ground and aerial robots for data collection in precision agriculture. We formulate the coverage problems based on constraints observed in practice. We also present the design of prototype robotic systems for these applications. In the final problem, we investigate how to optimize the low-level motion of the robots to minimize their energy consumption and extend the system lifetime.This dissertation makes progress towards building robotic sensing systems along two directions. We present algorithms with strong theoretical performance guarantees, often by proving that our algorithms are optimal or that their costs are at most a constant factor away from the optimal values. We also demonstrate the feasibility and applicability of our results through system implementation and with results from simulations and extensive field experiments