The paper describes basic principles of the finite element method. Further, basic properties of triangulation are discussed. The thesis is primarily focused on the Delaunay and Greedy triangulation. Greedy triangulation is simple to implement, but may not produce the optimal shape of triangles. Delaunay method is used due to its robustness in a wide range of fields especially in computer graphics. This method is relatively easy to implement and provides triangles with high quality.

Les artistes apprécient les images vectorielles car elles sont compactes et facilement manipulables. Cependant, beaucoup d’artistes expriment leur créativité en dessinant, en peignant ou encore en prenant des photographies. Digitaliser ces contenus produit des images rasterisées. L’objectif de cette thèse est de convertir des images rasterisées en images vectorielles qui sont facilement manipulables. Nous avons formulé le problème de vectorisation comme un problème de minimisation d’énergie que nous avons défini par deux termes. Le premier terme, plutôt classique, mesure la fidélité de l’image vectorielle générée avec l’image rasterisée d’origine. La nouveauté principale est le second terme qui mesure la simplicité de l’image vectorielle générée. Le terme de simplicité est global et contient des variables discrètes, ce qui rend sa minimisation difficile. Nous avons proposé deux algorithmes de vectorisation : un pour la vectorisation de croquis et un autre pour la vectorisation multicouches d’images couleurs. Ces deux algorithmes commencent par extraire des primitives géométriques (un squelette pour les croquis et une segmentation pour les images couleurs) qu’ils assemblent ensuite pour former l’image vectorielle. Dans la dernière partie de la thèse, nous proposons un nouvel algorithme qui est capable de vectoriser des croquis sans étapes préliminaires : on extrait et assemble les primitives simultanément. Nous montrons le potentiel de ce nouvel algorithme pour une variété de problèmes de vision par ordinateur comme l’extraction de réseaux linéiques, l’extraction d’objets et la compression d’images. / Artists appreciate vector graphics for their compactness and editability. However many artists express their creativity by sketching, painting or taking photographs. Digitizing these images produces raster graphics. The goal of this thesis is to convert raster graphics into vector graphics that are easy to edit. We cast image vectorization as an energy minimization problem. Our energy is a combination of two terms. The first term measures the fidelity of the vector graphics to the input raster graphics. This term is a standard term for image reconstruction problems. The main novelty is the second term which measures the simplicity of the vector graphics. The simplicity term is global and involves discrete unknowns which makes its minimization challenging. We propose two stochastic optimizations for this formulation: one for the line drawing vectorization problem and another one for the color image vectorization problem. These optimizations start by extracting geometric primitives (skeleton for sketches and segmentation for color images) and then assembling these primitives together to form the vector graphics. In the last chapter we propose a generic optimization method for the problem of geometric shape extraction. This new algorithm does not require any preprocessing step. We show its efficiency in a variety of vision problems including line network extraction, object contouring and image compression.

Mobile robotics has been very discussed and wide spread topic recently.   This due to the development in the computer technology that allows us to create   better and more sophisticated robots. The goal of this effort is to create robots   that will be able to autonomously move in the chosen environment. To achieve this goal,   it is necessary for the robot to create the map of its environment, where   the motion planning will occur. Nowadays, the probabilistic algorithms based   on the SLAM algorithm are considered standard in the mapping in these times.   This Phd. thesis deals with the proposal of the motion planning of the robot with   stereocamera placed on the pan-and-tilt unit. The motion planning is designed with   regard to the use of algorithms, which will look for the significant features   in the pair of the images. With the use of the triangulation the map, or a model will be created.     The benefits of this work can be divided into three parts. In the first one the way   of marking the free area, where the robot will plan its motion, is described. The second part   describes the motion planning of the robot in this free area. It takes into account   the properties of the SLAM algorithm and it tries to plan the exploration in order to create   the most precise map. The motion of the pan-and-tilt unit is described in the third part.   It takes advantage of the fact that the robot can observe places that are in the different   directions than the robot moves. This allows us to observe much bigger space without   losing the information about the precision of the movements.

This thesis aims to seek interesting patterns from massive amounts of Flickr data in Sweden with pro- posed new clustering strategies. The aim can be further divided into three objectives. The first one is to acquire large amount of timestamped geolocation data from Flickr servers. The second objective is to develop effective and efficient methods to process the data. More specifically, the methods to be developed are bifold, namely, the preprocessing method to solve the “Big Data” issue encountered in the study and the new clustering method to extract spatio-temporal patterns from data. The third one is to analyze the extracted patterns with scaling analysis techniques in order to interpret human social activities underlying the Flickr Data within the urban envrionment of Sweden. During the study, the three objectives were achieved sequentially. The data employed for this study was vector points downloaded through Flickr Application Programming Interface (API). After data ac- quisition, preprocessing was performed on the raw data. The whole dataset was firstly separated by year based on the temporal information. Then data of each year was accumulated with its former year(s) so that the evovling process can be explored. After that, large datasets were splitted into small pieces and each piece was clipped, georeferenced, and rectified respectively. Then the pieces were merged together for clustering. With respect to clustering, the strategy was developed based on the Delaunay Triangula- tion (DT) and head/tail break rule. After that, the generated clusters were analyzed with scaling analysis techniques and spatio-temporal patterns were interpreted from the analysis results. It has been found that the spatial pattern of the human social activities in the urban environment of Sweden generally follows the power-law distribution and the cities defined by human social activities are evolving as time goes by. To conclude, the contributions of this research are threefold and fulfill the objectives of this study, respectively. Firstly, large amount of Flickr data is acquired and collated as a contribution to other aca- demic researches related to Flickr. Secondly, the clustering strategy based on the DT and head/tail break rule is proposed for spatio-temporal pattern seeking. Thirdly, the evolving of the cities in terms of human activities in Sweden is detected from the perspective of scaling. Future work is expected in major two aspects, namely, data and data processing. For the data aspect, the downloaded Flickr data is expected to be employed by other studies, especially those closely related to human social activities within urban environment. For the processing aspect, new algorithms are expected to either accelerate the processing process or better fit machines with super computing capacities.

In this thesis, an approach is presented for the generation of grids suitable for the simulation of high Reynolds number viscous flows in complex three-dimensional geometries. The automatic and reliable generation of such grids is today on the biggest bottlenecks in the industrial CFD simulation environment. In the proposed approach, unstructured tetrahedral grids are employed for the regions far from the viscous boundaries of the domain, while semi-structured layers of high aspect ratio prismatic and hexahedral elements are used to provide the necessary grid resolution inside the boundary layers and normal to the viscous walls. The definition of the domain model is based on the STEP ISO standard and the topological information contained in the model is used for applying the hierarchical grid generation parameters defined by the user. An efficient, high-quality and robust algorithm is presented for the generation of the unstructured simplicial (triangular of tetrahedral) part of the grid. The algorithm is based on the Delaunay triangulation and the internal grid points are created following a centroid or frontal approach. For the surface grid generation, a hybrid approach is also proposed similar to the volume. Semi-structured grids are generated on the surface grid (both on the edges and faces of the domain) to improve the grid resolution around convex and concave ridges and corners, by aligning the grid elements in the directions of high solution gradients along the surface. A method is also developed for automatically setting the grid generation parameters related to the surface grid generation based on the curvature of the surface in order to obtain an accurate and smooth surface grid. Finally, a semi-structured prismatic/hexahedral grid generation algorithm is presented for the generation of the part of grid close to the viscous walls of the domain. The algorithm is further extended with improvements meant to increase the grid quality around concave and convex ridges of the domain, where the semi-structured grids are known to be inadequate. The combined methodology is demonstrated on a variety of complex examples mainly from the automotive and aeronautical industry.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

This work deals with the description and analysis of image data, which related to the initial stages of the thin film growth. The introductory retrieval section includes a description of thin films and methods of their deposition. The following part is an overview of the growth models of thin layers. The heart of my thesis is the analysis and modification of morphological methods and interpretation of their results. The emphasis is placed on the statistical aspect of methods and their optimal implementation due to the accuracy of the results. The work shows how to modify the radial distribution function and methods based on so-called Voronoi and Delaunay triangulation tessellation so that they can better affect the character of test data. New methods are tested both on the experimental and model data. Then we examine their robustness, sensitivity and their mutual independence. At the conclusion it is introduced and analyzed a new model of thin film growth.

Mesh generation is a fundamental precursor to finite element implementations for solution of partial differential equations in engineering and science. This dissertation advances the field in three distinct but coupled areas. A robust and fast three dimensional mesh generator for arbitrarily shaped geometries was developed. It deploys nodes throughout the domain based upon user-specified mesh density requirements. The system is integer and pixel based which eliminates round off errors, substantial memory requirements and cpu intensive calculations. Linked, but fully detachable, to the mesh generation system is a physical boundary recovery routine. Frequently, the original boundary topology is required for specific boundary condition applications or multiple material constraints. Historically, this boundary preservation was not available. An algorithm was developed, refined and optimized that recovers the original boundaries, internal and external, with fidelity. Finally, a node repositioning algorithm was developed that maximizes the minimum solid angle of tetrahedral meshes. The highly coveted 2D Delaunay property that maximizes the minimum interior angle of a triangle mesh does not extend to its 3D counterpart, to maximize the minimum solid angle of a tetrahedron mesh. As a consequence, 3D Delaunay created meshes have unacceptable sliver tetrahedral elements albeit composed of 4 high quality triangle sides. These compromised elements are virtually unavoidable and can foil an otherwise intact mesh. The numerical optimization routine developed takes any preexisting tetrahedral mesh and repositions the nodes without changing the mesh topology so that the minimum solid angle of the tetrahedrons is maximized. The overall quality enhancement of the volume mesh might be small, depending upon the initial mesh. However, highly distorted elements that create ill-conditioned global matrices and foil a finite element solver are enhanced significantly.

Orientador: Márcio Fabiano da Silva / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática Aplicada, 2013

In this thesis, we study the Euclidean Steiner tree problem (ESTP) which arises in the field of combinatorial optimization. The ESTP asks for a network of minimal total edge length spanning a set of given terminal points in Rd with the ability to add auxiliary connecting points (Steiner points) to decrease the overall length of the network. The graph theory literature contains extensive studies of exact, approximation, and heuristic algorithms for ESTP in the plane, but less is known in higher dimensions. The contributions of this thesis include a heuristic algorithm and enhancements to an exact algorithm for solving the ESTP. We present a local search heuristic for the ESTP in Rd for d ≥ 2. The algorithm utilizes the Delaunay triangulation to generate candidate Steiner points for insertion, the minimum spanning tree to create a network on the inserted points, and second order cone programming to optimize the locations of Steiner points. Unlike other ESTP heuristics relying on the Delaunay triangulation, the algorithm inserts Steiner points probabilistically into Delaunay triangles to achieve different subtrees on subsets of terminal points. The algorithm extends effectively into higher dimensions, and we present computational results on benchmark test problems in Rd for 2 ≤ d ≤ 5. We develop new geometric conditions derived from properties of Steiner trees which bound below the number of Steiner points on paths between terminals in the Steiner minimal tree. We describe conditions for trees with a Steiner topology and show how these conditions relate to the Voronoi diagram. We describe more restrictive conditions for trees with a full Steiner topology and their implementation into the algorithm of Smith (1992). We present computational results on benchmark test problems in Rd for 2 ≤ d ≤ 5.