Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs

AbstractIn this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs.We transform the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a properly defined triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in its 2-dominating set. We show that: (1) ⌊n+13⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊2n+15⌋ edge guards are always sufficient and sometimes necessary, in order to 2-dominate a triangulation graph. Furthermore, we show how to compute: (1) a diagonal 2-dominating set of size ⌊n+13⌋ in linear time and space, (2) an edge 2-dominating set of size ⌊2n+15⌋ in O(n2) time and O(n) space, and (3) an edge 2-dominating set of size ⌊3n7⌋ in O(n) time and space.Based on the above-mentioned results, we prove that, for piecewise-convex polygons, we can compute: (1) a mobile guard set of size ⌊n+13⌋ in O(nlogn) time, (2) an edge guard set of size ⌊2n+15⌋ in O(n2) time, and (3) an edge guard set of size ⌊3n7⌋ in O(nlogn) time. All space requirements are linear. Finally, we show that ⌊n3⌋ mobile or ⌈n3⌉ edge guards are sometimes necessary.When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: ⌈n+14⌉ edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most ⌈n+14⌉, can be computed in O(n) time and space

Systems and Algorithms for Automated Collaborative Observation using Networked Robotic Cameras

The development of telerobotic systems has evolved from Single Operator Single Robot (SOSR) systems to Multiple Operator Multiple Robot (MOMR) systems. The relationship between human operators and robots follows the master-slave control architecture and the requests for controlling robot actuation are completely generated by human operators. Recently, the fast evolving advances in network and computer technologies and decreasing size and cost of sensors and robots enable us to further extend the MOMR system architecture to incorporate heterogeneous components such as humans, robots, sensors, and automated agents. The requests for controlling robot actuation are generated by all the participants. We term it as the MOMR++ system. However, to reach the best potential and performance of the system, there are many technical challenges needing to be addressed. In this dissertation, we address two major challenges in the MOMR++ system development. We first address the robot coordination and planning issue in the application of an autonomous crowd surveillance system. The system consists of multiple robotic pan-tilt-zoom (PTZ) cameras assisted with a fixed wide-angle camera. The wide-angle camera provides an overview of the scene and detects moving objects, which are required for close-up views using the PTZ cameras. When applied to the pedestrian surveillance application and compared to a previous work, the system achieves increasing number of observed objects by over 210% in heavy traffic scenarios. The key issue here is given the limited number (e.g., p (p > 0)) of PTZ cameras and many more (e.g., n (n >> p)) observation requests, how to coordinate the cameras to best satisfy all the requests. We formulate this problem as a new camera resource allocation problem. Given p cameras, n observation requests, and [epsilon] being approximation bound, we develop an approximation algorithm running in O(n/[epsilon]³ + p²/[epsilon]⁶) time, and an exact algorithm, when p = 2, running in O(n³) time. We then address the automatic object content analysis and recognition issue in the application of an autonomous rare bird species detection system. We set up the system in the forest near Brinkley, Arkansas. The camera monitors the sky, detects motions, and preserves video data for only those targeted bird species. During the one-year search, the system reduces the raw video data of 29.41TB to only 146.7MB (reduction rate 99.9995%). The key issue here is to automatically recognize the flying bird species. We verify the bird body axis dynamic information by an extended Kalman filter (EKF) and compare the bird dynamic state with the prior knowledge of the targeted bird species. We quantify the uncertainty in recognition due to the measurement uncertainty and develop a novel Probable Observation Data Set (PODS)-based EKF method. In experiments with real video data, the algorithm achieves 95% area under the receiver operating characteristic (ROC) curve. Through the exploration of the two MOMR++ systems, we conclude that the new MOMR++ system architecture enables much wider range of participants, enhances the collaboration and interaction between participants so that information can be exchanged in between, suppresses the chance of any individual bias or mistakes in the observation process, and further frees humans from the control/observation process by providing automatic control/observation. The new MOMR++ system architecture is a promising direction for future telerobtics advances

Geometric optimization on visibility problems: metaheuristic and exact solutions

Doutoramento em MatemáticaOs problemas de visibilidade têm diversas aplicações a situações reais. Entre os mais conhecidos, e exaustivamente estudados, estão os que envolvem os conceitos de vigilância e ocultação em estruturas geométricas (problemas de vigilância e ocultação). Neste trabalho são estudados problemas de visibilidade em estruturas geométricas conhecidas como polígonos, uma vez que estes podem representar, de forma apropriada, muitos dos objectos reais e são de fácil manipulação computacional. O objectivo dos problemas de vigilância é a determinação do número mínimo de posições para a colocação de dispositivos num dado polígono, de modo a que estes dispositivos consigam “ver” a totalidade do polígono. Por outro lado, o objectivo dos problemas de ocultação é a determinação do número máximo de posições num dado polígono, de modo a que quaisquer duas posições não se consigam “ver”. Infelizmente, a maior parte dos problemas de visibilidade em polígonos são NP-difíceis, o que dá origem a duas linhas de investigação: o desenvolvimento de algoritmos que estabelecem soluções aproximadas e a determinação de soluções exactas para classes especiais de polígonos. Atendendo a estas duas linhas de investigação, o trabalho é dividido em duas partes. Na primeira parte são propostos algoritmos aproximados, baseados essencialmente em metaheurísticas e metaheurísticas híbridas, para resolver alguns problemas de visibilidade, tanto em polígonos arbitrários como ortogonais. Os problemas estudados são os seguintes: “Maximum Hidden Vertex Set problem”, “Minimum Vertex Guard Set problem”, “Minimum Vertex Floodlight Set problem” e “Minimum Vertex k-Modem Set problem”. São também desenvolvidos métodos que permitem determinar a razão de aproximação dos algoritmos propostos. Para cada problema são implementados os algoritmos apresentados e é realizado um estudo estatístico para estabelecer qual o algoritmo que obtém as melhores soluções num tempo razoável. Este estudo permite concluir que as metaheurísticas híbridas são, em geral, as melhores estratégias para resolver os problemas de visibilidade estudados. Na segunda parte desta dissertação são abordados os problemas “Minimum Vertex Guard Set”, “Maximum Hidden Set” e “Maximum Hidden Vertex Set”, onde são identificadas e estudadas algumas classes de polígonos para as quais são determinadas soluções exactas e/ou limites combinatórios.Visibility problems have several applications to real-life problems. Among the most distinguished and exhaustively studied visibility problems are the ones involving concepts of guarding and hiding on geometrical structures (guarding and hiding problems). This work deals with visibility problems on geometrical structures known as polygons, since polygons are appropriate representations of many real-world objects and are easily handled by computers. The objective of the guarding problems studied in this thesis is to find a minimum number of device positions on a given polygon such that these devices collectively ''see'' the whole polygon. On the other hand, the goal of the hiding problems is to find a maximum number of positions on a given polygon such that no two of these positions can “see" each other. Unfortunately, most of the visibility problems on polygons are NP-hard, which opens two lines of investigation: the development of algorithms that establish approximate solutions and the determination of exact solutions on special classes of polygons. Accordingly, this work is divided in two parts where these two lines of investigation are considered. The first part of this thesis proposes approximation algorithms, mainly based on metaheuristics and hybrid metaheuristics, to tackle some visibility problems on arbitrary and orthogonal polygons. The addressed problems are the Maximum Hidden Vertex Set problem, the Minimum Vertex Guard Set problem, the Minimum Vertex Floodlight Set problem and the Minimum Vertex k-Modem Set problem. Methods that allow the determination of the performance ratio of the developed algorithms are also proposed. For each problem, the proposed algorithms are implemented and a statistical study is performed to determine which of the developed methods obtains the best solution in a reasonable amount of time. This study allows to conclude that, in general, the hybrid metaheuristics are the best approach to solve the studied visibility problems. The second part of this dissertation addresses the Minimum Vertex Guard Set problem, the Maximum Hidden Set problem and the Maximum Hidden Vertex Set problem, where some classes of polygons are identified and studied and for which are determined exact solutions and/or combinatorial bounds

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version