195 research outputs found

    Data Parallel Hypersweeps for in Situ Topological Analysis

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    Large bichromatic point sets admit empty monochromatic 4-gons

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    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

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    A Partially Randomized Approach to Trajectory Planning and Optimization for Mobile Robots with Flat Dynamics

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    Motion planning problems are characterized by huge search spaces and complex obstacle structures with no concise mathematical expression. The fixed-wing airplane application considered in this thesis adds differential constraints and point-wise bounds, i. e. an infinite number of equality and inequality constraints. An optimal trajectory planning approach is presented, based on the randomized Rapidly-exploring Random Trees framework (RRT*). The local planner relies on differential flatness of the equations of motion to obtain tree branch candidates that automatically satisfy the differential constraints. Flat output trajectories, in this case equivalent to the airplane's flight path, are designed using Bézier curves. Segment feasibility in terms of point-wise inequality constraints is tested by an indicator integral, which is evaluated alongside the segment cost functional. Although the RRT* guarantees optimality in the limit of infinite planning time, it is argued by intuition and experimentation that convergence is not approached at a practically useful rate. Therefore, the randomized planner is augmented by a deterministic variational optimization technique. To this end, the optimal planning task is formulated as a semi-infinite optimization problem, using the intermediate result of the RRT(*) as an initial guess. The proposed optimization algorithm follows the feasible flavor of the primal-dual interior point paradigm. Discretization of functional (infinite) constraints is deferred to the linear subproblems, where it is realized implicitly by numeric quadrature. An inherent numerical ill-conditioning of the method is circumvented by a reduction-like approach, which tracks active constraint locations by introducing new problem variables. Obstacle avoidance is achieved by extending the line search procedure and dynamically adding obstacle-awareness constraints to the problem formulation. Experimental evaluation confirms that the hybrid approach is practically feasible and does indeed outperform RRT*'s built-in optimization mechanism, but the computational burden is still significant.Bewegungsplanungsaufgaben sind typischerweise gekennzeichnet durch umfangreiche Suchräume, deren vollständige Exploration nicht praktikabel ist, sowie durch unstrukturierte Hindernisse, für die nur selten eine geschlossene mathematische Beschreibung existiert. Bei der in dieser Arbeit betrachteten Anwendung auf Flächenflugzeuge kommen differentielle Randbedingungen und beschränkte Systemgrößen erschwerend hinzu. Der vorgestellte Ansatz zur optimalen Trajektorienplanung basiert auf dem Rapidly-exploring Random Trees-Algorithmus (RRT*), welcher die Suchraumkomplexität durch Randomisierung beherrschbar macht. Der spezifische Beitrag ist eine Realisierung des lokalen Planers zur Generierung der Äste des Suchbaums. Dieser erfordert ein flaches Bewegungsmodell, sodass differentielle Randbedingungen automatisch erfüllt sind. Die Trajektorien des flachen Ausgangs, welche im betrachteten Beispiel der Flugbahn entsprechen, werden mittels Bézier-Kurven entworfen. Die Einhaltung der Ungleichungsnebenbedingungen wird durch ein Indikator-Integral überprüft, welches sich mit wenig Zusatzaufwand parallel zum Kostenfunktional berechnen lässt. Zwar konvergiert der RRT*-Algorithmus (im probabilistischen Sinne) zu einer optimalen Lösung, jedoch ist die Konvergenzrate aus praktischer Sicht unbrauchbar langsam. Es ist daher naheliegend, den Planer durch ein gradientenbasiertes lokales Optimierungsverfahren mit besseren Konvergenzeigenschaften zu unterstützen. Hierzu wird die aktuelle Zwischenlösung des Planers als Initialschätzung für ein kompatibles semi-infinites Optimierungsproblem verwendet. Der vorgeschlagene Optimierungsalgorithmus erweitert das verbreitete innere-Punkte-Konzept (primal dual interior point method) auf semi-infinite Probleme. Eine explizite Diskretisierung der funktionalen Ungleichungsnebenbedingungen ist nicht erforderlich, denn diese erfolgt implizit durch eine numerische Integralauswertung im Rahmen der linearen Teilprobleme. Da die Methode an Stellen aktiver Nebenbedingungen nicht wohldefiniert ist, kommt zusätzlich eine Variante des Reduktions-Ansatzes zum Einsatz, bei welcher der Vektor der Optimierungsvariablen um die (endliche) Menge der aktiven Indizes erweitert wird. Weiterhin wurde eine Kollisionsvermeidung integriert, die in den Teilschritt der Liniensuche eingreift und die Problemformulierung dynamisch um Randbedingungen zur lokalen Berücksichtigung von Hindernissen erweitert. Experimentelle Untersuchungen bestätigen, dass die Ergebnisse des hybriden Ansatzes aus RRT(*) und numerischem Optimierungsverfahren der klassischen RRT*-basierten Trajektorienoptimierung überlegen sind. Der erforderliche Rechenaufwand ist zwar beträchtlich, aber unter realistischen Bedingungen praktisch beherrschbar

    29th International Symposium on Algorithms and Computation: ISAAC 2018, December 16-19, 2018, Jiaoxi, Yilan, Taiwan

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    Discrete Geometry

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    Image Processing and Pattern Recognition Applied to Soil Structure

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    This thesis represents a collaborative research between the Department of Electronics & Electrical Engineering and the Department of Civil Engineering, University of Glasgow. The project was initially aimed at development of some theories and techniques of image processing and pattern recognition for the study of soil microstructures. More specifically, the aim was to study the shapes, orientations, and arrangements of soil particles and voids (i.e. pores): these three are very important properties, which are used both for description, recognition and classification of soils, and also for studying the relationships between the soil structures and physical, chemical, geological, geographical, and environmental changes. The work presented here was based principally on a need for analysing the structure of soil as recorded in two-dimensional images which might be conventional photographs, optical micrographs, or electron-micrographs. In this thesis, first a brief review of image processing and pattern recognition and their previous application in the study of soil microstructures is given. Then a convex hull based shape description and classification for soil particles is presented. A new algorithm, SPCH, is proposed for finding the convex hull of either a binary object or a cluster of points in a plane. This algorithm is efficient and reliable. Features of pattern vectors for shape description and classification are obtained from the convex hull and the object. These features are invariant with respect to coordinate rotation, translation, and scaling. The objects can then be classified by any standard feature-space method: here minimum-distance classification was used. Next the orientation analysis of soil particles is described. A new method, Directed Vein, is proposed for the analysis. Another three methods: Convex Hull, Principal Components, and Moments, are also presented. Comparison of the four methods shows that the Directed Vein method appears the fastest; but it also has the special property of estimating an 'internal preferred orientation' whereas the other methods estimate an 'elongation direction'. Fourth, the roundness/sharpness analysis of soil particles is presented. Three new algorithms, referred to as the Centre, Gradient Centre, and Radius methods, all based on the Circular Hough Transform, are proposed. Two traditional Circular Hough Transform algorithms are presented as well. The three new methods were successfully applied to the measurement of the roundness (sharpness of comers) of two-dimensional particles. The five methods were compared from the points of view of memory requirement, speed, and accuracy; and the Radius method appears to be the best for the special topic of sharpness/roundness analysis. Finally the analysis and classification of aggregates of objects is introduced. A new method. Extended Linear Hough Transform, is proposed. In this method, the orientations and locations of the objects are mapped into extended Hough space. The arrangements of the objects within an aggregate are then determined by analysing the data distributions in this space. The aggregates can then be classified using a tree classifier. Taken together, the methods developed or tested here provide a useful toolkit for analysing the shapes, orientation, and aggregation of particles such as those seen in two-dimensional images of soil structure at various scales

    Big Data Computing for Geospatial Applications

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    The convergence of big data and geospatial computing has brought forth challenges and opportunities to Geographic Information Science with regard to geospatial data management, processing, analysis, modeling, and visualization. This book highlights recent advancements in integrating new computing approaches, spatial methods, and data management strategies to tackle geospatial big data challenges and meanwhile demonstrates opportunities for using big data for geospatial applications. Crucial to the advancements highlighted in this book is the integration of computational thinking and spatial thinking and the transformation of abstract ideas and models to concrete data structures and algorithms
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