Alors que les surfaces géométriques sont essentiellement représentées à l'aide de maillages triangulaires dans le domaine de la géométrie numérique, les structures permettant d'interagir avec ces géométries sont variées et adaptées aux différents traitements visés par l'utilisateur. Cette thèse présente des travaux réalisés sur des structures de dimension et de représentation géométrique variées, allant de l'étude des structures internes comme les squelettes analytiques pour la modélisation géométrique, passant par les structures surfaciques pour la sélection automatiques de poignées de déformation, jusqu'aux structures externes de contrôle d'objet de type "cage" offrant une représentation haut niveau de séquences animées d'objets issues de systèmes de performance capture. Sont présentés également les résultats obtenus sur les coordonnées aux valeurs moyennes offrant une solution au problème de l'interpolation de conditions de Dirichlet, pour lesquelles les formules analytiques des gradients et Hessiens sont fournies, et les fonctions biharmoniques pour lesquelles une base d'éléments finis est formulée pour la résolution du problème de Laplace biharmonique avec conditions mixtes Dirichlet/Neumann, ainsi que leurs applications à la déformation de formes 3D. / While 3D surfaces are essentially represented using triangle meshes in the domain of digital geometry, the structures that allow to interact with those are various and adapted to the different geometry processing tasks that are targetted by the user.This thesis presents results on structures of various dimension and various geometrical representations, going from internal structures like analytical curve skeletons for shape modeling, to on-surface structures allowing automatic selection of feature handles for shape deformation, and external control structures known as “cages” offering a high-level representation of animated 3D data stemming from performance capture. Results on spatial functions are also presented, in particular for the Mean-Value Coordinates, for which the analytical formulae of the gradients and the Hessians are provided, and biharmonic functions, for which a finite elements basis is given for the resolution of the biharmonic Laplace problem with mixed Dirichlet/Neumann boundary conditions, as well as their applications to 3D shapes deformation.

No description available.

The importance of precise values for the parameters used in heat and mass transfer models has been demonstrated by many research studies. Thermal conductivity values used in previous models are usually empirical and fluctuate. Theoretical analysis and estimations of wood thermal conductivities in the radial and tangential directions were conducted with the geometric models built up from the macro- and micro-structure observations. Theoretically, thermal conductivity in the radial direction is about 1.5 to 2.5 times of the tangential direction for softwood species with moisture content (MC) below Fiber Saturation Point (FSP). When MC is over the FSP, tangential radial thermal conductivity both increase dramatically and are linear function of MC. The two thermal conductivity values are close with a ratio of near one estimated by the model for MC above the FPS. In hardwood species, radial thermal conductivity estimated by the model is 1.5 times of the tangential thermal conductivity. Validation tests for model estimations of thermal conductivities in the radial and tangential directions for three wood species showed the reliability of the geometric models developed in this project. Correlations between the wood thermal conductivity and structure parameters, such as latewood percentage and cell wall percentage, were examined. Linear relationships for the thermal conductivity and average temperature in wood were established in both radial and tangential directions of three wood species. A two-dimensional transient heat conduction model was developed utilizing thermal conductivity values derived from geometric models. The anisotropic material property affect on heat transport in radial and tangential directions was discussed using an assumed situation. The simulation run showed slightly faster heat flow in the radial direction than in the tangential direction due to higher thermal conductivity in the radial direction. Validation tests on practical wood blocks showed the 2D model with the use of theoretical thermal conductivity values can predict good temperature distribution in wood during the heating process. However, in the practical wood samples with curved rings on the cross section, no significant difference was found in the two transverse directions. Mathematica software was introduced in this study for the intense and complicated math calculations and model programming. Mathematica was found to be a powerful technique for solving sophisticated math problems. It had abundant and flexible plotting options for providing optimized presentations for the results. These advantages make Mathematica popular for engineering modeling research. / Ph. D.

Gradient echo (GRE) imaging, a magnetic resonance imaging (MRI) technique that is sensitive to changes in the magnetic susceptibility property of tissues, has recently revealed significant signal heterogeneity in white matter (WM) at high magnetic field B0 ≥ 3T. Various aspects of the underlying white matter microstructure have been linked to the observed contrast between white matter regions. This thesis investigates the origins of the observed differences in GRE signal behaviour. We proposed an explicit multi-compartmental model of WM that incorporates realistic representation of the geometry and magnetic susceptibility of the underlying microstructure that can be used to study the effects of WM microstructural changes on GRE signal characteristics. In particular, we looked at the apparent transverse relaxation rate (R2*) and the resonance frequency, as well as their respective deviations from mono-exponential decay and linear phase evolution. Next, we investigated the effect of WM fiber orientation on GRE signal using healthy human volunteers at 3T by correlating the GRE signal from different WM regions with WM fiber orientation information. Using literature-based parameters, we demonstrated that the geometric model predicted similar trends. Lastly, we studied the effect of myelin on GRE signal using a cuprizone mouse model at 7T . An ex vivo study was used to correlate GRE signal in fixed mouse brain with normalized myelin stain intensity. Simulated GRE signal from hypothetical scenarios of demyelination were then compared with the experimental results. R2* and resonance frequency were then used in an in vivo longitudinal study to track myelin changes during demyelination and subsequent remyelination.

This dissertation delivers a new computational framework for the automatic generation of geometric feature patterns for industrial design and architectural facades on free-form surfaces. Such patterns include holes in a speaker grill, showerhead holes, protrusions on ceramics or bumpy textures on a panel. These patterns play a key role in making a designed object aesthetically pleasing as well as functional. Computer Aided Design (CAD) systems currently offer tools for generating simple patterns, such as uniformly spaced rectangular or radial patterns. However, they are not applicable to more general cases required in industrial design, including arbitrarily shaped target geometry and graded feature sizes. These tools are limited in several ways: (1) They cannot be applied to free-form geometries used in industrial design, (2) Patterning of these features happens within a single working plane and is not applicable to highly curved surfaces, and (3) Created features lack anisotropy and spatial variations, such as changes in the size and orientation of geometric features within a given region. This thesis proposes a new method of taking input for a target region along with sizing metrics. It will generate feature patterns automatically in three steps: (1) packing isotropic or anisotropic cells tightly in a target region, (2) scaling features according to the specified sizing metrics, and (3) adding features on the base geometry. This approach automatically generates complex patterns that conform to the boundary of any specified region. User input of a small number of geometric features (called “seed features”) of desired size and orientation in preferred locations also can be specified within the target domain. These geometric seed features are then transformed into tensors and used as boundary conditions to generate a Riemannian metric tensor field. A form of the Laplace heat equation is used to generate the field over the target domain, subject to specified boundary conditions. The field represents the anisotropic pattern of the geometric features. The system is implemented as a plugin module in a commercial CAD package to add geometric features to the target region of the model using two set operations, union and subtraction. This method facilitates the creation of a complex pattern of hundreds of geometric features in minutes. All the features are accessible from the CAD system and can be manipulated individually if required by the user. This allows the industrial designer or architect to explore more alternatives by avoiding the tedious and time-consuming manual generation of these geometric patterns.

De nos jours, les objets virtuels sont devenus omniprésents. On les trouve dans de nombreux domaines comme le divertissement (cinéma, jeux vidéo, etc.), la conception assistée par ordinateur ou encore la réalité virtuelle. Nous nous intéressons en particulier à la modélisation d'objets 3D dans le domaine de la création artistique. Ici, la création d'images riches nécessite de faire appel à des modèles très détaillés et donc extrêmement complexes. Les surfaces de subdivision, traditionnellement utilisées dans ces domaines, voient leur complexité croître rapidement lorsqu'on ajoute des détails, et la gestion de la connectivité du maillage de contrôle devient trop contraignante. Une approche standard pour gérer la complexité de tels modèles est d'utiliser des représentations différentes pour la forme générale de la surface et les détails. Cependant, ces détails sont représentés par des cartes matricielles qui ne possèdent pas la plupart des avantages des représentations vectorielles, et cela complexifie certaines tâches, comme par exemple l'animation. Dans cette thèse, nous proposons deux nouvelles représentations vectorielles, la première pour les surfaces de base, la deuxième pour les détails. Nous utilisons pour cette dernière une représentation vectorielle appelée images de diffusion permettant de créer des variations lisses à l'aide d'un ensemble réduit de contraintes. Cela nous permet de représenter aussi bien la géométrie que la couleur ou d'autres paramètres nécessaires au rendu de façon purement vectoriel, en conservant des contrôles de haut niveau.Notre première contribution est une représentation de surfaces, baptisée LS3, issue de la combinaison entre surfaces de subdivision et -point set surfaces. Cette approche réduit notablement les artefacts des surfaces de subdivision aux alentours de sommets dits extraordinaires, qui sont connus pour poser problème. Nous présentons une analyse numérique des propriétés de ces surfaces, qui tend à montrer que du point de vue de la continuité elles se comportent au moins aussi bien que les schémas de subdivision linéaires traditionnels. Notre deuxième contribution est un solveur pour les images de diffusion dont le principal avantage est de produire en sortie une autre représentation vectorielle légère et très rapide à évaluer. Nous illustrons la force de note solveur sur de nombreux exemples difficiles ou impossibles à réaliser avec les méthodes précédentes. Pour conclure, nous montrons comment combiner nos deux contributions pour obtenir une représentation de surface entièrement vectorielle capable de représenter des détails sans avoir à manipuler la connectivité d'un maillage. / Nowadays, virtual objects have become omnipresent. We can find them in various domains such as entertainment (movies, video games, etc.), computer-aided design or virtual reality. Our main focus in this document is the modeling of 3D objects in the domain of artistic creation, where rich images creation requires highly detailed and complex models.Subdivision surfaces, the most used surface representation in this domain, quickly become very dense as the user add details, and manual handling of the connectivity becomes too cumbersome. A standard approach to handle the complexity of such models is to separate the overall shape of the surface and the details. Although, these detail maps are often stored in bitmap images that does not provide the advantages of vectorial representation, which complicate some tasks, like animation.In this document, we present two new vectorial representations: the first one for the base surface, the second one for the detail maps. For the later, we use a vectorial representation called diffusion images that allow to create smooth or sharp variations from a small set of constraints. This enables us to represent geometry as well as color or any other parameter required for rendering, while keeping high-level controls.Our first contribution is a surface representation, called LS3, based on the combination of subdivision surfaces and point set surfaces. This approach reduces notably artifacts that subdivision surfaces produce around so called extraordinary vertices. We also present a numerical analysis of the mathematical properties of these surfaces, that show that they behave at least as well as classical subdivision schemes.Our second contribution is a solver for diffusion images that has the particularity to produce as output a denser vectorial representation which is light and fast to evaluate. We show the advantages of this approach on several examples that would be hard or impossible to produce with former methods.To conclude, we show how these two contributions can be used together to obtain a fully vectorial surface representation able to produce detailed surfaces without needing to deal with complex connectivity.

O presente trabalho buscou investigar as contribuições da utilização de tecnologias touchscreen para o desenvolvimento do pensamento matemático. Mais especificamente, propôs uma investigação de singularidades no desenvolvimento dos hábitos do pensamento em atividades de Modelagem Geométrica implementadas a partir do uso da tecnologia touchscreen. No que se refere ao desenvolvimento de hábitos do pensamento, propostos por Paul Goldenberg, é entendido como uma ação que contribui diretamente para o desenvolvimento do pensamento matemático. A escolha de trabalhar com Modelagem Geométrica é explorada neste trabalho com a finalidade de possibilitar aos sujeitos envolvidos o estudo de fenômenos reais a partir do uso de ferramentas matemáticas. Quanto à interação touchscreen, nos baseamos na teoria da Cognição Corporificada que parte do princípio de que corpo e mente estão diretamente relacionados, ou seja, no gesto com a mão os estudantes tornam evidentes as suas intenções e pensamentos. Para realização da pesquisa, a metodologia escolhida foi o estudo de casos múltiplos (dois), mas com apenas uma unidade de análise. Configura um estudo exploratório, em virtude da escassez de estudos na área por tratar-se da integração de uma nova tecnologia ao ensino Como suporte tecnológico, a escolha pelo Sketchometry, software de geometria dinâmica disponível para smartphones, se justifica, visto que este possui um importante destaque no desenvolvimento da proposta de pesquisa, uma vez que, através do seu uso, é possível que os sujeitos elaborem e construam modelos geométricos de forma corporificada (interação touchscreen), o que é uma ação necessária para a compreensão do fenômeno investigado. Ao final da tese, mostramos que, no uso da tecnologia touchscreen, é possível identificar singularidades no desenvolvimento do pensamento do sujeito. Notou-se que a singularidade no desenvolvimento do pensamento está no início da implementação de um modelo geométrico. O aluno segue um caminho guiado por movimentos espontâneos e, desta forma, estabelece a construção e funcionamento de seu modelo. / The present work has aimed to investigate the contributions of the use of touchscreen technologies for the development of mathematical thinking. More specifically, it has proposed an investigation of singularities in the development of habits of thought in Geometric Modeling activities implemented through the use of touchscreen technology. With regard to the development of habits of thought, proposed by Paul Goldenberg, it is understood as an action that contributes directly to the development of mathematical thinking. The choice of working with Geometric Modeling has been explored in this work with the purpose of enabling the subjects involved to study real phenomena that can be investigated, assimilated and better understood from the use of mathematical tools. As for the touchscreen interaction, the work was based on the theory of Embodied Cognition which assumes that body and mind are directly related, which means that, students make evident their intentions and thoughts in gesture with the hand. The methodology chosen to carry out the research was the study of multiple cases (two), but with only one unit of analysis. The study has benn configured as an exploratory one, due to the scarcity of studies in the area, since it is the integration of a new technology in teaching As technological support, the choice of Sketchometry, dynamic geometry software available for smartphones, has been justified, because it has an important emphasis in the development of the research proposal, since, through its use, it is possible for the individuals to elaborate and construct geometric models of an embodied form (touchscreen interaction), which is a necessary action for the understanding of the investigated phenomenon. At the end of the thesis, it has been shown that it is possible to identify singularities in the development of the subject's thinking in the use of touchscreen technology. It has been noted that the uniqueness in the development of thought is at the beginning of the implementation of a geometric model. The student follows a path guided by spontaneous movements and establishes the construction and functioning of his/her model.

Esta dissertação aborda a natureza e o desenvolvimento da lógica geométrica do espaço envolvida na execução de representações gráficas, tanto manuais quanto digitais, de objetos tridimensionais. Para este propósito, analisou-se as condutas, os desenhos das Vistas Ortográficas e a construção de um Modelo Geométrico tridimensional feito no Autocad, apresentados durante as entrevistas realizadas com os alunos do curso de Design de Móveis do CEFET/RS. Para fundamentar a análise dos dados colhidos, utilizou-se os estudos psicogenéticos de Piaget, referentes à Imagem Mental e à Representação do Espaço. Estes dados deram condições de se verificar a presença das relações espaciais na construção da imagem mental dos objetos apresentados ao longo das entrevistas e, a partir desta constatação, buscou-se analisar como estas relações projetivas e euclidianas se coordenavam de modo a garantir suas representações gráficas. Verificou-se que os desenhos apresentados dependem da evolução da lógica geométrica operatória, respeitando o desenvolvimento do sujeito epistêmico. Acredita-se que o estudo sobre os processos cognitivos envolvidos no ato de representar objetos graficamente, contribui para a educação da Expressão Gráfica, no sentido de auxiliar numa melhor adequação dos materiais e recursos didáticos, além dos programas curriculares de ensino do desenho e consequentemente da computação gráfica, na direção de uma aprendizagem imbuída no espírito construtivista. / This research approaches the nature and development of space geometric logic involved in both manual and digital graphic representation of tree dimension objects. The research data was collected by analyzing the attitudes, the drawings of the Orthographic Views and the construction of a tree dimension Geometric Modeling on AutoCAD, presented during interviews with “Design de Móveis” students from CEFET/RS. Psychogenetic studies of Piaget on Mental Image and Space Representation were used as basic tools on the analysis of the collected data. It was possible to verify the presence of space relations on the mental image construction of the objects presented during the interviews. Through this evidence, it was possible to investigate how the projective and Euclidian relations coordinate with each other in a way to guarantee their graphic representations. It was verified that the drawings presented depend on the operating geometric logic evolution, respecting the development of the epistemic subject. It is believed that the study on the cognitive processes, engendered in the act of representing objects graphically, contributes to the Graphic Expression education by helping to provide more adequate materials and didactic resources, also in drawing curricular programs and graphic computation, towards a constructivist perspective of learning.

[pt] Diversas aplicações, em matemática, computação gráfica, medicina, geofísica e outras áreas, têm explorado a representação de sólidos por superfícies de contorno, em particular malhas poligonais. As malhas podem aproximar com muita precisão as propriedades geométricas da superfície de contorno de um sólido e ainda guardar importantes propriedades topológicas das superfícies como gênero, bordo e conexidade. Devido à grande complexidade dessas malhas, elas são geralmente processadas em meios computacionais usando alguma estrutura de dados. Essas estruturas guardam, além da geometria da malha, informações de incidências e adjacências entre os elementos da malha e exigem uma capacidade de armazenamento e processamento em função da complexidade da malha. Apesar da evolução dos recursos computacionais disponíveis para a manipulação destas estruturas, malhas extremamente complexas com milhões de elementos inviabilizam o armazenamento, processamento e transmissão de sua estrutura de dados nos meios computacionais. Muitas pesquisas recentes estão voltadas para a obtenção de processos de simplificação de malhas que permitam representar a mesma superfície com menos elementos na estrutura de dados e processos de compressão que codifiquem os modelos em formatos menores para efeitos de transmissão e armazenamento em mídia. Neste trabalho, desenvolvemos operadores, em uma estrutura de dados compacta, para a simplificação de malhas através da decimação de células da superfície. Objetivamos, com esses operadores, obter uma malha menos complexa que preserve as propriedades topológicas da superfície original e ainda, controlar as propriedades geométricas como volume, área e aspecto visual da mesma. Apresentamos ainda algumas aplicações para os processos de simplificação desenvolvidos com esses operadores. / [en] Many applications, in mathematics, computer graphics, medical imaging, geophysics and others, have used the representation of solids by their boundary surface, usually polygonal meshes. Those meshes can represent, with high precision, the geometric properties of the boundary surface of solid and also store important topological surface properties as genus, boundary and connected components. Because of the high complexity of such meshes, they are usually processed by the computers using specific data structures. These structures store, beyond the mesh geometry, information about incidence and adjacency relations among the mesh elements. They require computational resources for storage and processing according to the mesh complexity. Even with the development of the computational resources available for handling such structures, very large meshes with millions of elements are hard to store, to process and to exchange through the web. Many recent researches are looking for mesh simplification process that allows to represent the same surface with fewer elements and compression process to encode it in compact ways for transmition and storage. In this work, we develop topological operators, in a concise data structure, for simplifying meshes by the decimation of its cells. One of our goals, with these operators, is to obtain a mesh with a low complexity that preserves the topological properties from the original surface without loosing the control of the geometric proprieties as volume, area and visual aspect.

Les modifications physiologiques dans la configuration spatiale des organes pelviens sont de plus en plus diagnostiquées et traitées pour améliorer la qualité de vie des patientes. Le projet MoDyPe (ANR-09-SYSC-008) a été créé pour concevoir un simulateur chirurgical patiente-spécifique, et quantifier le geste médical en mode pré-opératoire. La modélisation géométrique des organes se fait à partir de nuages de points épars bruités. Les formes sont considérées fermées, lisses, creuses et à membrane épaisse. Le processus est décomposé en deux étapes : la construction de la surface et l'ajout d'une épaisseur.Afin de respecter les contraintes physiologiques, de manipuler la géométrie et de localiser précisément un point sur la surface, une B-spline de genre 0 au moins C1-continue est ajustée aux données. La fonction à minimiser est basée sur une énergie bidirectionnelle, caractérisant la dissimilarité des données sur l'échantillonnage et inversement. Sa réduction repose sur un schéma alternant re-paramétrisation et descente de gradient à pas optimal.Une surface-offset est ensuite générée vers l'intérieur de l'organe, via un maillage discret, en traitant le problème d'auto-intersection. Elle exploite la forme allongée des organes, grâce à un axe curviligne décrivant leur diamètre généralisé.Finalement, un maillage hexaédrique est créé à partir de la surface ajustée et de l'offset, qui sert à la simulation mécanique du comportement des organes à l'étape suivante du projet. / Physiological changes in the spatial configuration of the organs in the pelvic area are increasingly taken into account and treated to enhance the comfort of patients. MoDyPe project (ANR-09-SYSC-008 french support) has been created to develop a patient-specific simulator and to quantify the surgical gesture for preoperative purposes. The geometric modeling of the organs starts with noisy scattered point clouds. The shapes have been considered closed, smooth, hollow with a thick membrane. The process can be divided into two main parts: the construction of the surface and the addition of a thickness.In order to meet the physiological constraints, to manipulate the geometry and to accurately localize a point on the surface, a 0-genus B-spline surface is fitted to the data. It minimizes a bidirectional energy, characterizing the dissimilarities between the surface sampling and the input dataset. Its reduction is based on an alternate scheme between re-parametrization and optimal steepest descent step.Once achieved, an offset-surface is generated inwards, helped by a mesh to overcome self-intersection problems. The method created takes into account the elongated shapes of the organs, based on a curvilinear axis describing their generalized diameter.Finally, a hexahedral mesh is created from the fitted surface and its offset. It is the start point for the next step of the project consisting in mechanically simulating the dynamic behavior of the organs.