<p>In this thesis, we explore techniques in statistics and persistent homology, which detect features among data sets such as graphs, triangulations and point cloud. We accompany our theorems with algorithms and experiments, to demonstrate their effectiveness in practice.</p><p></p><p>We start with the derivation of graph scan statistics, a measure useful to assess the statistical significance of a subgraph in terms of edge density. We cluster graphs into densely-connected subgraphs based on this measure. We give algorithms for finding such clusterings and experiment on real-world data.</p><p></p><p>We next study statistics on persistence, for piecewise-linear functions defined on the triangulations of topological spaces. We derive persistence pairing probabilities among vertices in the triangulation. We also provide upper bounds for total persistence in expectation. </p><p></p><p>We continue by examining the elevation function defined on the triangulation of a surface. Its local maxima obtained by persistence pairing are useful in describing features of the triangulations of protein surfaces. We describe an algorithm to compute these local maxima, with a run-time ten-thousand times faster in practice than previous method. We connect such improvement with the total Gaussian curvature of the surfaces.</p><p></p><p>Finally, we study a stratification learning problem: given a point cloud sampled from a stratified space, which points belong to the same strata, at a given scale level? We assess the local structure of a point in relation to its neighbors using kernel and cokernel persistent homology. We prove the effectiveness of such assessment through several inference theorems, under the assumption of dense sample. The topological inference theorem relates the sample density with the homological feature size. The probabilistic inference theorem provides sample estimates to assess the local structure with confidence. We describe an algorithm that computes the kernel and cokernel persistence diagrams and prove its correctness. We further experiment on simple synthetic data.</p> / Dissertation

<p>This thesis explores a few geometric optimization problems that arise</p><p>in robotics and sensor networks. In particular we present efficient</p><p>algorithms for the hitting-set problem and the budgeted hitting-set problem.</p><p>Given a set of objects and a collection of subsets of the objects,</p><p>called ranges, the hitting-set problem asks for a minimum number of </p><p>objects that intersect all the subsets in the collection.</p><p>In geometric settings, objects are </p><p>typically a set of points and ranges are defined by a set of geometric</p><p>regions (e.g., disks or polygons), i.e., the subset of points lying in each </p><p>region forms a range.</p><p>The first result of this thesis is an efficient algorithm for an instance</p><p>of the hitting-set problem in which both the set of points and the set</p><p>of ranges are implicitly defined. Namely, we are given a convex</p><p>polygonal robot and a set of convex polygonal obstacles, and we wish</p><p>to find a small number of congruent copies of the robot that intersect</p><p>all the obstacles.</p><p>Next, motivated by the application of sensor placement in sensor networks,</p><p>we study the so-called ``art-gallery'' problem. Given a polygonal</p><p>environment, we wish to place the minimum number of guards so that</p><p>the every point in the environment is visible from at least one guard.</p><p>This problem can be formulated as a hitting-set problem. We present</p><p>a sampling based algorithm for this problem and study various extensions</p><p>of this problem.</p><p>Next, we study the geometric hitting-set problem in a dynamic setting,</p><p>where the objects and/or the ranges change with time and the goal is</p><p>to maintain a hitting set. We present algorithms </p><p>which maintain a small size hitting set with sub-linear update time.</p><p>Finally, we consider the budgeted hitting-set problem, in which we</p><p>are asked to choose a bounded number of objects that intersect as many</p><p>ranges as possible. Motivated by applications in network vulnerability</p><p>analysis we study this problem in a probabilistic setting.</p> / Dissertation

No description available.

Our work applies techniques for blending and tightening solid shapes represented by sets. We require that the output contain one set and exclude a second set, and then we optimize the boundary separating the two sets. Working within that framework, we present mason, tightening, tight hulls, tight blends, and the medial cover, with details for implementation. Mason uses opening and closing techniques from mathematical morphology to smooth small features. By contrast, tightening uses mean curvature flow to minimize the measure of the boundary separating the opening of the interior of the closed input set from the opening of its complement, guaranteeing a mean curvature bound. The tight hull offers a significant generalization of the convex hull subject to volumetric constraints, introducing developable boundary patches connecting the constraints. Tight blends then use opening to replicate some of the behaviors from tightenings by applying tight hulls. The medial cover provides a means for adjusting the topology of a tight hull or tight blend, and it provides an implementation technique for two-dimensional polygonal inputs. Collectively, we offer applications for boundary estimation, three-dimensional solid design, blending, normal field simplification, and polygonal repair. We consequently establish the value of blending and tightening as tools for solid modeling.

Οι κινητικές δομές δεδομένων KDSs (kinetic data structures) είναι ένα νέο πλαίσιο εργασίας για το σχεδιασμό και την ανάλυση αλγορίθμων σχετικών με γεωμε- τρικά αντικείμενα (ευθύγραμμα τμήματα, πολύγωνα, δίσκοι κ.τ.λ.) σε κίνηση. Σκο- πός μας είναι να διατηρήσουμε ένα χαρακτηριστικό ενός συνόλου κινούμενων αντι- κειμένων, π.χ. την κυρτή θήκη ή το κοντινότερο ζευγάρι του. Η διατήρηση του χαρα κτηριστικού γίνεται μέσω ενός συνόλου συνθηκών που εγγυώνται την εγκυρότητα της δομής κάθε χρονική στιγμή και το οποίο μεταβάλλεται με το χρόνο λόγω της κίνησης. Οι συνθήκες αποθηκεύονται σε μια ουρά διατεταγμένες χρονολογικά. Κάθε φορά που αλλάζει το χαρακτηριστικό που μας ενδιαφέρει ενημερώνουμε τη δομή μας και την ουρά. Η πρώτη ενότητα της εργασίας είναι μια εισαγωγή στις KDSs. Αναφέρουμε βασικές έννοιες και ιδέες των KDSs όπως: συνάρτηση διαμόρφωσης, πιστοποιητικά, κρίσιμα γεγονότα. Επίσης, ασχολούμαστε και με τα μέτρα απόδοσής τους. Στη δεύτερη ενότητα ασχολούμαστε με τους δυαδικούς διαχωρισμούς χώρου BSPs, πρώτα σε στατικό και κατόπιν σε κινητικό περιβάλλον. Συγκεκριμένα παρουσιάζουμε τρεις αλγορίθμους για τη διατήρηση του BSP ενός συνόλου κινούμενων ευθυγράμμων τμημάτων στο επίπεδο. Σύμφωνα με τον πρώτο γνωστό αλγόριθμο που διατυπώθηκε για την αποτελεσματική διατήρηση του BSP για ένα σύνολο μη-τεμνόμενων ευθυγράμμων τμημάτων S στο επίπεδο χρησιμοποιώντας τη φιλοσοφία των KDSs, κατασκευάζουμε έναν BSP για το S θεωρώντας τα ευθύγραμμα τμήματα στάσιμα και στη συνέχεια τον διατηρούμε καθώς αυτά κινούνται. Ο δεύτερος αλγόριθμος είναι ουσιαστικά μια επέκταση του πρώτου καθώς ασχολείται με το ίδιο πρόβλημα, αλλά για τεμνόμενα ευθύγραμμα τμήματα. Αλλάζει το σύνολο των πιστοποιητικών και οι τρόποι με τους οποίους μπορεί να αλλάξει η δομή του BSP. Ο τρίτος αλγόριθμος χρησιμοποιεί ένα διαφορετικό τρόπο για την κατασκευή και διατήρηση του BSP για το σύνολο S βελτιώνοντας τον αρχικό. Στην τρίτη ενότητα ασχολούμαστε με τη διατήρηση του Voronoi διαγράμματος (VD) για ένα σύνολο κινούμενων, πιθανώς τεμνόμενων δίσκων στο επίπεδο και του συμπαγούς Voronoi διαγράμματος για ένα σύνολο μη-τεμνόμενων κυρτών πολυγώνων στο επίπεδο (το συμπαγές VD είναι δυϊκό του VD, αλλά το μέγεθός του είναι συνάρτηση του αριθμού των πολυγώνων και όχι του αριθμού των κορυφών). Και στις δύο περιπτώσεις, η επίλυση του προβλήματος ανάγεται στη διατήρηση του δυϊκού του VD, της τριγωνοποίησης Delaunay DT . Η διατήρηση της DT βασίζεται στο γεγονός ότι ένα σύνολο τοπικών συνθηκών (έλεγχοι InCircle), πιστοποιούν την ολική ορθότητα της δομής και τοπικές επιδιορθώσεις είναι πάντα εφικτές. Έτσι, καθώς τα αντικείμενα κινούνται, έχουμε κάθε στιγμή μια έγκυρη DT και συνεπώς ένα έγκυρο VD. Τέλος, αναφέρουμε μια KDS για τον εντοπισμό συγκρούσεων μεταξύ δύο απλών πολυγώνων σε κίνηση. Ο αλγόριθμος διατηρεί μια υποδιαίρεση του ελεύθερου χώρου μεταξύ των πολυγώνων, που καλείται external relative geodesic triangulation, η οποία πιστοποιεί τη μη-σύγκρουσή των πολυγώνων. / Kinetic Data Structures (KDSs) are a new framework for designing and analyzing algorithms for geometrics objects (segments, polygons, disks etc.) in motion. Our goal is to maintain an attribute of a set of moving objects, for example the convex hull or the closest pair. The maintenance of the attribute is made through a set of conditions that guarantee the validity of the structure every moment. This set is changed with time due to the motion. The conditions are stored in a queue ordered chronologically. Every time the attribute is changed, we update the structure and the queue. The first chapter is an introduction to the KDSs. We mention basic notions and ideas of the KDSs, like: configuration function, certificates, critical events. Furthermore, we discuss their measure of performance. In the second chapter we deal with the Binary Space Partitions (BSPs), first in static and then in kinetic environment. Specifically, we present three algorithms for the maintenance of a BSP for a set of moving segments in the plane. According to the first known algorithm which was proposed for efficiently maintaining the BSP for a set of non-intersecting segments S in the plane using the philosophy of KDSs, we construct a BSP - considering that the segments are static - and then we maintain it as the segments move. The second algorithm is substantially an expansion of the first algorithm as it deals with the same problem, but for intersecting segments. The set of the certificates is changed as well as the set of critical events. The third algorithm uses a different technique for the construction and maintenance of the BSP for the set S. It is an improvement of the first algorithm. In the third chapter, we deal with the maintenance of the Voronoi diagram (VD) for a set of moving, probably intersecting disks in the plane and the maintenance of a compact Voronoi-like diagram for a set of non-intersecting, convex polygons in the plane (compact VD is dual to VD, except that its size is a function of the number of polygons and not of the number of vertices). In both cases, we solve the problem by maintaining the dual graph of VD, the Delaunay triangulation (DT ). The maintenance of the DT is based in the fact that a set of local conditions (InCircle tests) guarantee the total correctness of the structure and we are able to do only local changes. So, as the objects move, we have a valid DT every moment and consequently a valid VD. Finally, we mention a KDS for detecting collisions between two simple polygons in motion. In order to do so, we create a planar subdivision of the free space between the polygons, called External Relative Geodesic Triangulation, which certify their disjointness.

A \emph{simple} $n$-\emph{Venn diagram} is a collection of $n$ simple intersecting closed curves in the plane where exactly two curves meet at any intersection point; the curves divide the plane into $2^n$ distinct open regions, each defined by its intersection of the interior or exterior of each of the curves. A Venn diagram is \emph{reducible} if there is a curve that, when removed, leaves a Venn diagram with one less curve and \emph{irreducible} if no such curve exists. A Venn diagram is \emph{extendible} if another curve can be added, producing a Venn diagram with one more curve. Currently it is not known whether every simple Venn diagram is extendible by the addition of another curve. We show that all simple Venn diagrams with $5$ curves or less are extendible to another simple Venn diagram. We also show that for certain Venn diagrams, a new extending curve is relatively easy to produce. We define a new type of diagram of simple closed curves where each curve divides the plane into an equal number of regions; we call such a diagram a \emph{face-balanced} diagram. We generate and exhibit all face-balanced diagrams up to and including those with $32$ regions; these include all the Venn diagrams. Venn diagrams exist where the curves are the perimeters of polyominoes drawn on the integer lattice. When each of the $2^n$ intersection regions is a single unit square, we call these \emph{minimum area polyomino Venn diagrams}, or \emph{polyVenns}. We show that polyVenns can be constructed and confined in bounding rectangles of size $2^r \times 2^c$ whenever $r, c \ge 2$ and $n=r+c$. We show this using two constructive proofs that extend existing diagrams. Finally, for even $n$, we construct polyVenns with $n$ polyominoes in $(2^{n/2} - 1) \times (2^{n/2} + 1)$ bounding rectangles in which the empty set is not represented as a unit square. / Graduate / 0405 / 0984 / bbultena@uvic.ca

Μια γρήγορη επίλυση του Delaunay Τριγωνισμός (DT) πρόβληματος αποτελεί ένα από τα βασικά συστατικά σε πολλές θεωριτικές και πρακτικές εφαρμογές. Οι υπάρχουσες μονάδες επεξεργασίας γραφικών (GPU), με βάση τις εφαρμογές των αλγορίθμων DT πάσχουν από δύο σοβαρά μειονεκτήματα. Το πρώτο σχετίζεται με την εξάρτηση του αλγορίθμου καθοδήγηση της GPU από την CPU για τους υπολογισμούς. Το δεύτερο πιο σοβαρό μειονέκτημα είναι η εξάρτησή τους από τη διανομή του σημειοσύνολου εισόδου. Οι περισσότεροι αλγορίθμοι για GPU έχουν καλή απόδοση μόνο με ομοιόμορφες κατανομές σημειοσύνολον. Προτείνουμε ένα καινούριο αλγόριθμο που δεν πάσχουν από τα παραπάνω προβλήματα. / A Fast solver of Delaunay Triangulation (DT) problem constitutes one of the basic ingredients in many practical and sientific applications. Existing Graphics Processing Units (GPU) based implementations of DT algorithms suffer from two serious drawbacks. The first is related to the dependency of the CPU guidance algorithm on GPU calculations. Albeit the modern GPUs have high computational throughput, if the feedback from CPU is necessary for the algorithmic evolution, the overhead caused by CPU-GPU communication can seriously degrade the performance. The second most serious drawback is their dependency on the distribution of the given point-set. Most of the GPU-based implementations can optimally run only on uniformly distributed point-sets, however, in many practical applications this is not the case.

TEIXEIRA, Daniel Nascimento. Uma técnica de decomposição a priori para geração paralela de malhas bidimensionais. 2014. 95 f. Dissertação (Mestrado em ciência da computação)- Universidade Federal do Ceará, Fortaleza-CE, 2014. / Submitted by Elineudson Ribeiro (elineudsonr@gmail.com) on 2016-07-11T12:51:06Z No. of bitstreams: 1 2014_dis_dnteixeira.pdf: 17919971 bytes, checksum: 092ad12b33cf64a31552e6a839a5a5bc (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2016-07-15T12:49:53Z (GMT) No. of bitstreams: 1 2014_dis_dnteixeira.pdf: 17919971 bytes, checksum: 092ad12b33cf64a31552e6a839a5a5bc (MD5) / Made available in DSpace on 2016-07-15T12:49:53Z (GMT). No. of bitstreams: 1 2014_dis_dnteixeira.pdf: 17919971 bytes, checksum: 092ad12b33cf64a31552e6a839a5a5bc (MD5) Previous issue date: 2014 / This work describes a technique of two-dimensional domain decomposition for parallel mesh generation. This technique works for both distributed and shared memory and has the freedom to use any data structure that manages rectangular regions parallel to the axes to decompose the domain given as input, such as a quaternary tree (quadtree) or a binary space decomposition (bsp), for example. Any process of mesh generation that respects the prerequisites established can be used in the subdomains created, for instance, Delaunay or Advancing Front, among others. This technique is called a priori because the mesh on the interface of the subdomains is generated prior to the their internal meshes. The load estimation for each sub-domain in this work is performed with the aid of a refined quadtree, whose level of refinement guides the creation of edges that are defined from the bounderies of only inner cells. This way of estimate load produces results that accurately represent the number of elements to be generated in each subdomain. That contributes to a good partitioning of the domain, making the mesh generation in parallel be significantly faster than the serial generation. Furthermore, the quality of the generated mesh in parallel is qualitatively equivalent to that generated serially within acceptable limits. / Este trabalho descreve uma técnica de decomposição de domínios bidimensionais para geração em paralelo de malhas. Esta técnica funciona tanto para memória distribuída quanto compartilhada, além de permitir que se utilize qualquer estrutura de dados que gere regiões quadrangulares paralelas aos eixos para decompor o domínio dado como entrada. Pode se utilizar por exemplo, uma árvore quaternária (quadtree) ou uma partição binária do espaço (bsp). Além disso, qualquer processo de geração de malha que respeite os pré-requisitos estabelecidos pode ser empregado nos subdomínios criados, como as técnicas de Delaunay ou Avanço de Fronteira, dentre outras. A técnica proposta é dita a priori porque a malha de interface entre os subdomínios é gerada antes das suas malhas internas. A estimativa de carga de processamento associada a cada subdomínio é feita nesse trabalho com a ajuda de uma quadtree refinada, cujo nível de refinamento orienta a criação das arestas que são definidas a partir da discretização das fronteiras das células internas. Essa maneira de estimar carga produz resultados que representam, com boa precisão, o número de elementos a serem gerados em cada subdomínio. Isso contribui para um bom particionamento do domínio, fazendo com que a geração de malha em paralelo seja significativamente mais rápida do que a geração serial. Além disso, a qualidade da malha gerada em paralelo é qualitativamente equivalente àquela gerada serialmente, dentro de limites aceitáveis.

[pt] Este trabalho situa-se numa das linhas de pesquisa da PUC-Rio de projeto de unidades flutuantes tais como, navios e plataformas semi-submersíveis. Nesta linha de pesquisa estão sendo desenvolvidos os programas gráficos MG (Mesh Generator) e o Sstab. O primeiro programa é um modelador geométrico por meio de seções transversais e gerador de malhas para modelos de estruturas flutuantes. O segundo programa é utilizado para a análise de estabilidade estática dos modelos gerados pelo MG. Este trabalho propõe um ambiente integrado de modelagem e de análise estática e dinâmica de estruturas flutuantes. O principal diferencial deste ambiente está no fato de possibilitar a geração automática de variantes de um determinado modelo padrão, a fim de atingir uma configuração desejada, seja no aspecto geométrico ou com relação a sua estabilidade estática. Este ambiente faz uso da linguagem Lua e é possível definir variáveis globais para serem utilizadas como parâmetros de modelagem que extraem, ou modificam, dados como comprimento, largura, altura etc. É possível parametrizar um modelo qualquer, em função de variáveis escolhidas pelo usuário, o que possibilita uma modelagem automática, com a variação de alguns destes parâmetros. Foram ainda desenvolvidas algumas ferramentas auxiliares que facilitam a modelagem de uma estrutura flutuante. Estas ferramentas verificam a consistência topológica de uma malha, gerar uma subdivisão gradativa das curvas cortadas e simplificar as novas malhas geradas. É possível também detectar se o modelo possui simetria num determinado plano e realizar, de forma automática, cortes do modelo final para diferentes calados. / [en] This work is related to the PUC-Rio research area of vessel´s designs such as ships and semi-sub platforms. In this research area two softwares have been developed: MG and Sstab. The first is a geometric modeler based on cross sections and also on a mesh generator; the second is a software for the analysis of static and dynamic stability of MG models. This work proposes an integrated environment for modeling, and static and dynamic analysis of vessels. The main advantage of the proposed environment is that it is possible to obtain automatically variants of a specific model in order to achieve a desired configuration, not only in relation to geometry but also concerning the static stability aspect. This environment uses the Lua programming language and it is possible to define global variables to be used as parameters which retrieve or modify modeling values such as length, width, height, and so on. Any model can be parameterized, as a function of user chosen variables, which allows an automatic modeling with the variation of those parameters. There has been also developed some auxiliary tools which help the modeling of floating structures. Those tools verify the topological consistency of a mesh, generate a gradual subdivision of intersected curves and simplify the new generated meshes. They are also able to detect if the model has symmetry in relation to a certain plane, and sections can be automatically obtained according to different draughts.

Malhas simpliciais são utilizadas em várias áreas da Computação Gráfica e Engenharia, como por exemplo, em vizualização, simulação, prototipação, além de outras aplicações. Este tipo de malha é, geralmente, utilizada como aproximações discretas de espaços contínuos, onde eles oferecem representações flexíveis e eficientes. Muito esforço é gasto visando gerar malhas de boa qualidade, porém, em alguns casos as malhas acabam sendo modificadas. Entretanto, este tipo de operação é geralmente custosa e inflexível, o que pode resultar na geraão de malhas bem diferentes das originais. A habilidade de manipular cenas dinâmicas revela-se um dos problemas mais desafiadores da computação gráfica. Este trabalho propõe um método alternativo para atualizar malhas simpliciais que vai além de mudanças geométricas e topológicas. Tal método explora uma das propriedade das Tringulações de Delaunay com Pesos, que permite a usá-las para definir implicitamente as relações de conectividade de uma malha. Ao contrário de manter as informações de conectividade explicitamente, a atual abordagem simplesmente armazena uma coleção de pesos associados a cada vértice. Além disso, criamos um algoritmo para calcular uma Tringulação de Delaunay com Pesos a partir de uma dada triangulação. O algoritmo consiste em uma busca em largura que atribui pesos aos vértices, e uma estratégia de de subdivisão para assegurar que a triangulação reconstruída será correspondente à original. Este método apresenta diversas aplicações e, em particular, permite a criação de um sistema simples de realizar combinação entre triangulações, que será ilustrada com exemplos em 2D e 3D. / Simplicial meshes are used in many fields of Computer Graphics and Engineering, for instance, in visualization, simulation, prototyping, among other applications. This kind of mesh is often used as discrete approximations of continuous spaces, where they offer flexible and efficient representations. Considerable effort is spent in generating good quality meshes, but in some applications the meshes can be modified over time. However, this kind of operation is often very expensive and inflexible, sometimes leading to results very different from the original meshes. The ability to handle dynamic scenes reveals itself as one of the most challenging problems in computer graphics. This work proposes an alternative technique for updating simplicial meshes that undergo geometric and topological changes. It explores the property that a Weighted Delaunay Triangulation (WDT) can be used to implicitly define the connectivity of a mesh. Instead of explicitly maintaining connectivity information, this approach simply keeps a collection of weights associated to each vertex. It consists of an algorithm to compute a WDT from any given triangulation, which relies on a breadth-first traversal to assign weights to vertices, and a subdivision strategy to ensure that the reconstructed triangulation conforms with the original one. This technique has many applications and, in particular, it allows for a very simple method of merging triangulations, which is illustrated with both 2D and 3d examples.