<p>The first class of problem we study deals with geometric matchings. Given a set<br />of points in the plane, we study perfect matchings of those points by straight line<br />segments so that the segments do not cross. Bottleneck matching is such a matching that minimizes the length of the longest segment. We are interested in finding a bottleneck matching of points in convex position. In the monochromatic case, where any two points are allowed to be matched, we give an O(n ²)-time algorithm for finding a bottleneck matching, improving upon previously best known algorithm of O(n ³) time complexity. We also study a bichromatic version of this problem, where each point is colored either red or blue, and only points of different color can be matched. We develop a range of tools, for dealing with bichromatic non-crossing matchings of points in convex<br />position. Combining that set of tools with a geometric analysis enable us to solve the<br />problem of finding a bottleneck matching in O(n ²) time. We also design an O(n)-time<br />algorithm for the case where the given points lie on a circle. Previously best known results were O(n 3 ) for points in convex position, and O(n log n) for points on a circle.<br />The second class of problems we study deals with dilation of geometric networks.<br />Given a polygon representing a network, and a point p in the same plane, we aim to<br />extend the network by inserting a line segment, called a feed-link, which connects<br />p to the boundary of the polygon. Once a feed link is fixed, the geometric dilation<br />of some point q on the boundary is the ratio between the length of the shortest path<br />from p to q through the extended network, and their Euclidean distance. The utility of<br />a feed-link is inversely proportional to the maximal dilation over all boundary points.<br />We give a linear time algorithm for computing the feed-link with the minimum overall<br />dilation, thus improving upon the previously known algorithm of complexity that is<br />roughly O(n log n).</p> / <p>Prva klasa problema koju proučavamo tičee se geometrijskih mečinga. Za dat skup tačaaka u ravni, posmatramo savr&scaron;ene mečinge tih tačaka spajajućii ih&nbsp; dužima koje &nbsp; se ne smeju sećui. Bottleneck mečing je takav mečing koji minimizuje dužinu najduže duži. Na&scaron; cilj je da nađemo bottleneck mečiing tačaka u konveksnom položaju.Za monohromatski slučaj, u kom je dozvoljeno upariti svaki par tačaka, dajemo algoritam vremenske složenosti O(n ²) za nalaženje bottleneck mečinga. Ovo&nbsp; je bolje od prethodno najbolji poznatog algoritam, čiija je složenost O(n ³). Takođe proučavamo bihromatsku verziju ovog problema, u kojoj je svaka tačka&nbsp; obojena ili u crveno ili u plavo, i dozvoljeno je upariti samo tačke različite boje. Razvijamo niz alata za rad sa bihromatskim nepresecajućim mečinzima tačaka u konveksnom položaju. Kombinovanje ovih alata sa geometrijskom analizom omogućava nam da re&scaron;imo problem nalaženja bottleneck mečinga u O(n ² ) vremenu. Takođe, konstrui&scaron;emo algoritam vremenske složenosti O(n) za slučaj kada&nbsp; sve date tačkke leže na krugu. Prethodno najbolji poznati algoritmi su imali složenosti&nbsp; O(n ³ ) za tačkeke u konveksnom položaju i O(n log n) za tačke na krugu.<br />Druga klasa problema koju proučaavamo tiče se dilacije u geometrijskim mrežama. Za datu mrežu predstavljenu poligonom, i tačku p u istoj ravni, želimo pro&scaron;iriti mrežu&nbsp; dodavanjem duži zvane feed-link koja povezuje p sa obodom poligona. Kada je feed- link fiksiran, defini&scaron;emo geometrijsku dilaciju neke tačke q na obodu kao odnos izme&nbsp; đu&nbsp; dužine najkraćeg puta od p do q kroz pro&scaron;irenu mrežu i njihovog Euklidskog rastojanja. Korisnost feed-linka je obrnuto proporcionalna najvećoj dilaciji od svih ta čaka na obodu poligona. Konstrui&scaron;emo algoritam linearne vremenske složenosti koji nalazi feed-link sa najmanom sveukupnom dilacijom. Ovim postižemo bolji rezultat od prethodno najboljeg poznatog algoritma složenosti približno O(n log n).</p>

This thesis focuses on the analysis of the latest techniques for surface and solid binary voxelization of 3D models. It briefly describes current trends in this problematics and identifies a suitable method with an aim to parallelize the given solution on GPUs. It concretely explains the implementation process of the selected algorithm described in the paper Fast Parallel Surface and Solid Voxelization on GPUs , which produces a sparse voxel octree. The results are very close to those of the original authors. A new solution for extracting a smooth isosurface from this structure based on Marching Cubes is presented as well, providing up to 98 % reduction of the traversed cubes in higher resolutions. The resulting implementation is a framework usable for further voxel scene processing.

A novel approach for a robust computation of positional relations of two-dimensional geometric features is presented which guarantees reliable results, provided that the initial data is valid. The method is based on the use of integer coordinates and a method to generate a complete, gap-less and non-overlapping spatial decomposition. The spatial relationships of two geometric features are then represented using DE-9IM matrices. These allow the spatial relationships to be represented compactly. The DE-9IM matrices are based on the spatial decomposition using explicit neighborhood relations. No further geometric calculations are required for their computation. Based on comparative tests, it could be proven that this approach, up to a predictable limit, provides correct results and thus offers advantages over classical methods for the calculation of spatial relationships. This novel method can be used in all fields, especially where guaranteed reliable results are required.:Introduction Related Research Materials and Methods Results Discussion Outlook Author Contributions Funding Institutional Review Board Statement Informed Consent Statement Data Availability Statement Conflicts of Interest Abbreviations References

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No / This volume contains the papers presented at Computer Graphics International 2002, in July, at the University of Bradford, UK. These papers represent original research in computer graphics from around the world.

At the core of successful manipulation and computation over large geometric data is the notion of approximation, both structural and computational. The focus of this thesis will be on the combinatorial and algorithmic aspects of approximations of point-set data P in d-dimensional Euclidean space. It starts with a study of geometric data depth where the goal is to compute a point which is the 'combinatorial center' of P. Over the past 50 years several such measures of combinatorial centers have been proposed, and we will re-examine several of them: Tukey depth, Simplicial depth, Oja depth and Ray-Shooting depth. This can be generalized to approximations with a subset, leading to the notion of epsilon-nets. There we will study the problem of approximations with respect to convexity. Along the way, this requires re-visiting and generalizing some basic theorems of convex geometry, such as the Caratheodory's theorem. Finally we will turn to the algorithmic aspects of these problems. We present a polynomial-time approximation scheme for computing hitting-sets for disks in the plane. Of separate interest is the technique, an analysis of local-search via locality graphs. A further application of this technique is then presented in computing independent sets in intersection graphs of rectangles in the plane.

La biométrie du visage a suscité, ces derniers temps, l'intérêt grandissant de la communauté scientifique et des industriels de la biométrie vue son caractère naturel, sans contact et non-intrusif. Néanmoins, les performances des systèmes basés sur les images 2D sont affectées par différents types de variabilités comme la pose, les conditions d'éclairage, les occultations et les expressions faciales. Avec la disponibilité de caméras 3D capables d'acquérir la forme tridimensionnelle, moins sensibles aux changements d'illumination et de pose, plusieurs travaux de recherche se sont tournés vers l'étude de cette nouvelle modalité. En revanche, d'autres défis apparaissent comme les déformations de la forme faciales causées par les expressions et le temps de calcul que requièrent les approches développées. Cette thèse s'inscrit dans ce paradigme en proposant de coupler la géométrie Riemannienne avec les techniques d'apprentissage pour une biométrie faciale 3D efficace et robuste aux changements d'expressions. Après une étape de pré-traitement, nous proposons de représenter les surfaces faciales par des collections de courbes 3D qui captent localement leurs formes. Nous utilisons un cadre géométrique existant pour obtenir les déformations " optimales " entre les courbes ainsi que les distances les séparant sur une variété Riemannienne (espace des formes des courbes). Nous appliquons, par la suite, des techniques d'apprentissage afin de déterminer les courbes les plus pertinentes pour deux applications de la biométrie du visage : la reconnaissance d'identité et la classification du genre. Les résultats obtenus sur le benchmark de référence FRGC v2 et leurs comparaison avec les travaux de l'état de l'art confirment tout l'intérêt de coupler l'analyse locale de la forme par une approche géométrique (possibilité de calculer des moyennes, etc.) avec des techniques d'apprentissage (Basting, etc.) pour gagner en temps de calcul et en performances.

Nous avons assisté au développement rapide de la technologie de l'information au cours de la dernière décennie. D'une part, la capacité du traitement et du stockage des appareils numériques est en constante augmentation grâce aux progrès des méthodes de construction. D'autre part, l'interaction entre ces dispositifs puissants a été rendue possible grâce à la technologie de réseautage. Une conséquence naturelle de ces progrès, est que le volume des données générées dans différentes applications a grandi à un rythme sans précédent. Désormais, nous sommes confrontés à de nouveaux défis pour traiter et représenter efficacement la masse énorme de données à notre disposition. Cette thèse est centrée autour des deux axes de recommandation du contenu pertinent et de sa visualisation correcte. Le rôle des systèmes de recommandation est d'aider les utilisateurs dans le processus de prise de décision pour trouver des articles avec un contenu pertinent et une qualité satisfaisante au sein du vaste ensemble des possibilités existant dans le Web. D'autre part, la représentation correcte des données traitées est un élément central à la fois pour accroître l'utilité des données pour l'utilisateur final et pour la conception des outils d'analyse efficaces. Dans cet exposé, les principales approches des systèmes de recommandation ainsi que les techniques les plus importantes de la visualisation des données sous forme de graphes sont discutées. En outre, il est montré comment quelques-unes des mêmes techniques appliquées aux systèmes de recommandation peuvent être modifiées pour tenir compte des exigences de visualisation.

選舉是實行民主政治最有效的方法之一，而選區劃分的方式將直接或間接的影響投票結果與民主政治理念的施行。 然而在選舉法規或行政區域發生變動時，舊有的選區劃分方式需要隨之調整。而傳統人工的方式具有許多缺點，如：耗費人力資源、人口分配不均、難以兼顧形狀及行政區完整等等。若每次行政區域發生變動，都需要重新劃分，將花費許多不必要的人力、物力及時間，因此利用電腦以完成自動劃分的技術逐漸受到重視。 本論文中我們打破現有的政治與人文鴻溝，嘗試以系統化的方法對選區劃分作全面性的查驗。我們利用計算幾何學的特性與人工智慧搜尋的技巧，儘量找出可能的劃分方式再進行評估。我們依據中選會的建議採用村裡為劃分之最小行政區域，從數以十萬計之合理解中，根據形狀等客觀條件篩選出較佳之劃分方式，進而將歷史投票行為加入考量，以對篩選出的劃分方式作進一步評估與分析。 實作中我們以台南市為對象，在不同的人口限制及形狀條件下，分別比較所能找到的合理解數目。同時選出一部分的劃分方式，和中選會的劃分方式比較，結果顯示我們的方法可以全面性的分析選區劃分，不同的劃分方式可能產生不同的選舉結果。 / Election is one of the most effective way of conducting democratic politics, and mean of electoral redistricting shall post effect, either directly or indirectly, on electoral outcome as well as delivering ideas of democratic politics. As election regulations or administrational districts experience alterations, the present electoral districting is forcefully accompanied with adjustments. Electoral redistricting using traditional human labor works reveal several flaws such as: human resource wastage, uneven population distributions, hard to maintain shape contiguity and compactness, as well as the completeness of administration districts. Every single alteration experience in administration district requires redistribution, thus expensing on unnecessary human labor, resources and time. As such, it had brought great attention on techniques of automatic redistribution by means of modern computer technologies. In this thesis, we shall breakthrough a giant gap between politics and humanity; conduct a thorough examination on systematic approach on electoral redistricting. We are going to utilize characteristics of computational geometry and artificial intelligence searching techniques to find out every conceivable means of redistricting then evaluation the performance of them. By recommendation of Central Election Commission (hence CEM), we will adopt the classification of township as basic unit of administrational district, from counts of thousand adequate explanations, by objective factors of shape accordance and others, select the better means of redistricting methods, and afterward put into concern of historical voting behavior, conduct a further evaluation and analysis upon chosen redistricting method. In actual practices we had selected Tainan City as the experiment target, under different population limitations and factors of form, compare the searchable numbers of decent explanation respectively. We choose some redistricting outcomes, and put into comparison with redistricting method of the CEM. The results indicated our approach is able to conduct a thorough redistricting analysis, as well as more diversified comparing to CEM's outcome. The result of this experiment also reveals different election outcome with adoption of different redistricting methods.

Este documento apresenta uma dissertação sobre o desenvolvimento de um sistema de integração de dados para geração de estruturas tridimensionais a partir de vídeo 3D. O trabalho envolve a extensão de um sistema de vídeo 3D em tempo real proposto recentemente. Esse sistema, constituído por projetor e câmera, obtém imagens de profundidade de objetos por meio da projeção de slides com um padrão de faixas coloridas. Tal procedimento permite a obtenção, em tempo real, tanto do modelo 2,5 D dos objetos quanto da textura dos mesmos, segundo uma técnica denominada luz estruturada. Os dados são capturados a uma taxa de 30 quadros por segundo e possuem alta qualidade: resoluções de 640 x 480 pixeis para a textura e de 90 x 240 pontos (em média) para a geometria. A extensão que essa dissertação propõe visa obter o modelo tridimensional dos objetos presentes em uma cena por meio do registro dos dados (textura e geometria) dos diversos quadros amostrados. Assim, o presente trabalho é um passo intermediário de um projeto maior, no qual pretende-se fazer a reconstrução dos modelos por completo, bastando para isso apenas algumas imagens obtidas a partir de diferentes pontos de observação. Tal reconstrução deverá diminuir a incidência de pontos de oclusão (bastante comuns nos resultados originais) de modo a permitir a adaptação de todo o sistema para objetos móveis e deformáveis, uma vez que, no estado atual, o sistema é robusto apenas para objetos estáticos e rígidos. Até onde pudemos averiguar, nenhuma técnica já foi aplicada com este propósito. Este texto descreve o trabalho já desenvolvido, o qual consiste em um método para detecção, rastreamento e casamento espacial de componentes conexas presentes em um vídeo 3D. A informação de imagem do vídeo (textura) é combinada com posições tridimensionais (geometria) a fim de alinhar partes de superfícies que são vistas em quadros subseqüentes. Esta é uma questão chave no vídeo 3D, a qual pode ser explorada em diversas aplicações tais como compressão, integração geométrica e reconstrução de cenas, dentre outras. A abordagem que adotamos consiste na detecção de características salientes no espaço do mundo, provendo um alinhamento de geometria mais completo. O processo de registro é feito segundo a aplicação do algoritmo ICP---Iterative Closest Point---introduzido por Besl e McKay em 1992. Resultados experimentais bem sucedidos corroborando nosso método são apresentados. / This document presents a MSc thesis focused on the development of a data integration system to generate tridimensional structures from 3D video. The work involves the extension of a recently proposed real time 3D video system. This system, composed by a video camera and a projector, obtains range images of recorded objects using slide projection of a coloured stripe pattern. This procedure allows capturing, in real time, objects´ texture and 2,5 D model, at the same time, by a technique called structured light. The data are acquired at 30 frames per second, being of high quality: the resolutions are 640 x 480 pixels and 90 x 240 points (in average), respectively. The extension that this thesis proposes aims at obtaining the tridimensional model of the objects present in a scene through data matching (texture and geometry) of various sampled frames. Thus, the current work is an intermediary step of a larger project with the intent of achieving a complete reconstruction from only a few images obtained from different viewpoints. Such reconstruction will reduce the incidence of occlusion points (very common on the original results) such that it should be possible to adapt the whole system to moving and deformable objects (In the current state, the system is robust only to static and rigid objects.). To the best of our knowledge, there is no method that has fully solved this problem. This text describes the developed work, which consists of a method to perform detection, tracking and spatial matching of connected components present in a 3D video. The video image information (texture) is combined with tridimensional sites (geometry) in order to align surface portions seen on subsequent frames. This is a key step in the 3D video that may be explored in several applications such as compression, geometric integration and scene reconstruction, to name but a few. Our approach consists of detecting salient features in both image and world spaces, for further alignment of texture and geometry. The matching process is accomplished by the application of the ICP---Iterative Closest Point---algorithm, introduced by Besl and McKay in 1992. Succesful experimental results corroborating our method are shown.