Topology-related issues are becoming increasingly important in Computer Graphics. This research examines the use of topological analysis for solving two important problems in 3D Graphics: surface parameterization, and vector field design on surfaces. Many applications, such as high-quality and interactive image synthesis, benefit from the solutions to these problems. Surface parameterization refers to segmenting a 3D surface into a number of patches and unfolding them onto a plane. A surface parameterization allows surface properties to be sampled and stored in a texture map for high-quality and interactive display. One of the most important quality measurements for surface parameterization is stretch, which causes an uneven sampling rate across the surface and needs to be avoided whenever possible. In this thesis, I present an automatic parameterization technique that segments the surface according to the handles and large protrusions in the surface. This results in a small number of large patches that can be unfolded with relatively little stretch. To locate the handles and large protrusions, I make use of topological analysis of a distance-based function on the surface. Vector field design refers to creating continuous vector fields on 3D surfaces with control over vector field topology, such as the number and location of the singularities. Many graphics applications make use of an input vector field. The singularities in the input vector field often cause visual artifacts for these applications, such as texture synthesis and non-photorealistic rendering. In this thesis, I describe a vector field design system for both planar domains and 3D mesh surfaces. The system provides topological editing operations that allow the user to control the number and location of the singularities in the vector field. For the system to work for 3D meshes surface, I present a novel piecewise interpolating scheme that produces a continuous vector field based on the vector values defined at the vertices of the mesh. I demonstrate the effectiveness of the system through several graphics applications: painterly rendering of still images, pencil-sketches of surfaces, and texture synthesis.

[pt] Campos vetoriais representam um papel principal em diversas aplicações científicas. Eles são comumente gerados via simulações computacionais. Essas simulações podem ser um processo custoso, dado que em muitas vezes elas requerem alto tempo computacional. Quando pesquisadores desejam quantificar a incerteza relacionada a esse tipo de aplicação, costuma-se gerar um conjunto de realizações de campos vetoriais, o que torna o processo ainda mais custoso. A Decomposição de Helmholtz-Hodge é uma ferramenta útil para a interpretação de campos vetoriais uma vez que ela distingue componentes conservativos (livre de rotação) de componentes que preservam massa (livre de divergente). No presente trabalho, vamos explorar a aplicabilidade de tal técnica na análise de incerteza de campos vetoriais 2D. Primeiramente, apresentaremos uma abordagem utilizando a Decomposição de Helmholtz-Hodge como uma ferramenta básica na análise de conjuntos de campos vetoriais. Dado um conjunto de campos vetoriais epsilon, obtemos os conjuntos formados pelos componentes livre de rotação, livre de divergente e harmônico, aplicando a Decomposição Natural de Helmholtz- Hodge em cada campo vetorial em epsilon. Com esses conjuntos em mãos, nossa proposta não somente quantifica, por meio de análise estatística, como cada componente é pontualmente correlacionado ao conjunto de campos vetoriais original, como também permite a investigação independente da incerteza relacionado aos campos livre de rotação, livre de divergente e harmônico. Em sequência, propomos duas técnicas que em conjunto com a Decomposição de Helmholtz-Hodge geram, de forma estocástica, campos vetoriais a partir de uma única realização. Por fim, propomos também um método para sintetizar campos vetoriais a partir de um conjunto, utilizando técnicas de Redução de Dimensionalidade e Projeção Inversa. Testamos os métodos propostos tanto em campos sintéticos quanto em campos numericamente simulados. / [en] Vector field plays an essential role in a large range of scientific applications. They are commonly generated through computer simulations. Such simulations may be a costly process because they usually require high computational time. When researchers want to quantify the uncertainty in such kind of applications, usually an ensemble of vector fields realizations are generated, making the process much more expensive. The Helmholtz-Hodge Decomposition is a very useful instrument for vector field interpretation because it traditionally distinguishes conservative (rotational-free) components from mass-preserving (divergence-free) components. In this work, we are going to explore the applicability of such technique on the uncertainty analysis of 2-dimensional vector fields. First, we will present an approach of the use of the Helmholtz-Hodge Decomposition as a basic tool for the analysis of a vector field ensemble. Given a vector field ensemble epsilon, we firstly obtain the corresponding rotational-free, divergence-free and harmonic component ensembles by applying the Natural Helmholtz-Hodge Decomposition to each1 vector field in epsilon. With these ensembles in hand, our proposal not only quantifies, via a statistical analysis, how much each component ensemble is point-wisely correlated to the original vector field ensemble, but it also allows to investigate the uncertainty of rotational-free, divergence-free and harmonic components separately. Then, we propose two techniques that jointly with the Helmholtz-Hodge Decomposition stochastically generate vector fields from a single realization. Finally, we propose a method to synthesize vector fields from an ensemble, using both the Dimension Reduction and Inverse Projection techniques. We test the proposed methods with synthetic vector fields as well as with simulated vector fields.

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2017-12-27T14:12:36Z No. of bitstreams: 2 Tese - Ubirajara José Gama de Castro - 2017.pdf: 14188106 bytes, checksum: 942882692cd259cae5e8d267f6ac1188 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-12-28T09:43:26Z (GMT) No. of bitstreams: 2 Tese - Ubirajara José Gama de Castro - 2017.pdf: 14188106 bytes, checksum: 942882692cd259cae5e8d267f6ac1188 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-12-28T09:43:26Z (GMT). No. of bitstreams: 2 Tese - Ubirajara José Gama de Castro - 2017.pdf: 14188106 bytes, checksum: 942882692cd259cae5e8d267f6ac1188 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-11-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study reversible vector fields in two zones and equivariant vector fields in two zones. Our main result is the classification of the symmetric singularities of codimensions 0,1 and 2 of such vector fields. More precisely, in the reversible case in R3, where the dimension of the fixed points variety of the involution associated to the vector field is 2, we present all bifurcation diagram of the codimensions 1 and 2 singularities, describing the changes in the behavior of the symmetric singularities and tangents of the vector field with the transition manifold, S, according to the variation of the bifucartion parameter. We also show the existence of invariant cylinders and, in this case, doing small perturbations we determine invariant manifolds that persisted and we determine the number of limit cycles that were born. When the vector field defined on two zones is equivariant, the dynamic is enriched with the emergence of the sliding vector field and we also do a local study and the classification of singularities (and pseudo-singularities) of codimensions 0,1 and 2. We show the existence of homoclinic sliding orbit and that it is a codimension one phenomenon. Moreover, provided the symmetry we get a double Shilnikov sliding orbit. / Neste trabalho, estudamos campos vetoriais em duas zonas reversíveis e campos vetoriais em duas zonas equivariantes. Nosso resultado principal é a classificação das singularidades simétricas de codimensões 0, 1 e 2 de tais campos vetoriais. Mais precisamente, no caso reversível em R3, onde a dimensão da variedade de pontos fixos da involução associada ao campo vetorial é 2, apresentamos todos os diagramas de bifurcação das singularidades de codimensão 1 e 2, descrevendo as mudanças no comportamento das singularidades simétricas e das tangências do campo vetorial com a variedade de transição S, de acordo com a variação do parâmetro de bifurcação. Mostramos também a existência de cilindros invariantes e, nesse caso, fazendo pequenas perturbações determinamos variedades invariantes que persistiram e determinamos o número de ciclos limites que surgiram. Quando o campo vetorial definido em duas zonas é equivariante, a dinâmica é enriquecida com o surgimento do campo vetorial deslizante e também fazemos um estudo local e a classificação das singularidades (e pseudossingularidades) de codimensões 0, 1 e 2. Mostramos a existência de órbitas homoclínicas deslizantes e que esse é um fenômeno de codimensão 1 e devido à simetria do campo vetorial equivariante, teremos um duplo Shilnikov deslizante.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2017-03-22T12:19:52Z No. of bitstreams: 2 Tese - Oscar Alexander Ramírez Cespedes - 2017.pdf: 14139665 bytes, checksum: 57b8ba9047422f62ed013be3b7bf660e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-03-22T12:20:24Z (GMT) No. of bitstreams: 2 Tese - Oscar Alexander Ramírez Cespedes - 2017.pdf: 14139665 bytes, checksum: 57b8ba9047422f62ed013be3b7bf660e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-03-22T12:20:24Z (GMT). No. of bitstreams: 2 Tese - Oscar Alexander Ramírez Cespedes - 2017.pdf: 14139665 bytes, checksum: 57b8ba9047422f62ed013be3b7bf660e (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-03-07 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / In this work, we analize the version of Hilbert’s 16th problem for a piecewise linear differential system, PWLS, in R2. More precisely,we determinete the maximum number of certain types of limit cycles when the system is define in two zones separated by a straight line. Some results on the maximum number of cycles of a PWLS defined in two sectors were established. In addition, we classify typical singularities of a piecewise smooth systemin R3, taking into account the behavior of the associated sliding field. / Neste trabalho, estudamos uma versão do 16◦ Problema de Hilbert para sistemas de equações diferenciais lineares por partes, PWLS, em R2. Mais precisamente, determinamos o número máximo de certos tipos de ciclos limite quando o sistema está definido em duas zonas separadas por uma linha reta. Alguns resultados sobreo número de máximo de ciclos de um PWLS definido em dois setores são estabelecidos. Além disso, estudamos e classificamos singularidades típicas de um sistema suave por partes em R3 levando em consideração o comportamento do campo deslizante associado.

The goal of this thesis is the development of a fast and robust algorithm that is able to detect patterns in flow fields independent from their orientation and adequately visualize the results for a human user. This thesis is an interdisciplinary work in the field of vector field visualization and the field of pattern recognition. A vector field can be best imagined as an area or a volume containing a lot of arrows. The direction of the arrow describes the direction of a flow or force at the point where it starts and the length its velocity or strength. This builds a bridge to vector field visualization, because drawing these arrows is one of the fundamental techniques to illustrate a vector field. The main challenge of vector field visualization is to decide which of them should be drawn. If you do not draw enough arrows, you may miss the feature you are interested in. If you draw too many arrows, your image will be black all over. We assume that the user is interested in a certain feature of the vector field: a certain pattern. To prevent clutter and occlusion of the interesting parts, we first look for this pattern and then apply a visualization that emphasizes its occurrences. In general, the user wants to find all instances of the interesting pattern, no matter if they are smaller or bigger, weaker or stronger or oriented in some other direction than his reference input pattern. But looking for all these transformed versions would take far too long. That is why, we look for an algorithm that detects the occurrences of the pattern independent from these transformations. In the second part of this thesis, we work with moment invariants. Moments are the projections of a function to a function space basis. In order to compare the functions, it is sufficient to compare their moments. Normalization is the act of transforming a function into a predefined standard position. Moment invariants are characteristic numbers like fingerprints that are constructed from moments and do not change under certain transformations. They can be produced by normalization, because if all the functions are in one standard position, their prior position has no influence on their normalized moments. With this technique, we were able to solve the pattern detection task for 2D and 3D flow fields by mathematically proving the invariance of the moments with respect to translation, rotation, and scaling. In practical applications, this invariance is disturbed by the discretization. We applied our method to several analytic and real world data sets and showed that it works on discrete fields in a robust way.

In the area of steering behaviours of autonomous agents and crowd simulations, there is a plethora of methods for executing the simulations. A very hard-to-achieve goal of crowd simulations is to make them seem natural and accurately reflect real-life crowds. A very important criterion for this goal is to have the agents avoid collisions, both with each other and with the environment. A less important, but important nonetheless, criterion is to not let the time taken or distance covered to reach the goal in the simulation be too high, compared with when not implementing collision avoidance. This paper proposes and explores a novel method of enhancing vector field-based steering with rule-based deviations to implement collision avoidance. This method is called ’DevVec’ (’Deviation + Vector Field steering’). The rules which are used for the deviations are extracted from a user survey, and they describe what the agent should do in different collision avoidance scenarios. The viability of DevVec is tested by comparing it with another already established method, called ’Gradient-based Steering’, in terms of fulfilling the criteria mentioned above. Both methods are used to simulate pedestrians moving throughout different scenes. The results suggest that DevVec has potential, but would require additional time and resources, and perhaps a few changes in future works to be presented in its best possible version. / Inom ämnesområdet för styrbeteenden hos autonoma agenter och simuleringar av folkmassor finns det många metoder för att framställa dessa simuleringar. Ett väldigt svåruppnåeligt mål för denna typ av simuleringar är få dem att verka naturliga och verklighetstrogna. Ett viktigt kriterie för detta mål är att få agenterna att undvika kollisioner, både med varandra och med den kringliggande omgivningen. Ett mindre viktigt, men viktigt oavsett, kriterie är att inte låta en agent ta för lång tid eller gå för långt för att nå sitt mål i simuleringen, i jämförelse med när de inte försöka undvika hinder. Denna studie presenterar och utforskar en ny metod som utökar en vektorfältsbaserat styralgoritm med regelbaserade avvikelser för att ta hänsyn till att undvika kollisioner. Denna nya metod kallas för ’DevVec’ (’Deviation + Vector Field steering’). Reglerna som används för avvikelserna är framtagna från en enkät, och de beskriver vad en agent borde göra vid olika kollision-scenarion. Användbarheten av DevVec prövas genom att jämföra den med en redan etablerad metod som kallas för ’Gradientbaserad styrning’, med avseende på de ovan nämnda kriterierna. Båda metoderna används för att simulera fotgängare i olika omgivningar. Resultaten antyder att DevVec har potential, men att det krävs ytterligare tid och resurser, och troligtvis några ändringar i framtiden för att framställa den bästa möjliga versionen.

The first aim of this PhD Thesis is devoted to the study of geodesic distances defined on a subdomain of a Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot–Carathéodory distance. Then one shows that the uniform convergence, on compact sets, of these distances can be equivalently characterized in terms of Gamma-convergence of several kinds of variational problems. Moreover, it investigates the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle. The second purpose is to obtain the integral representation of some classes of local functionals, depending on a family of vector fields, that satisfy a weak structure assumption. These functionals are defined on degenerate Sobolev spaces and they are assumed to be not translations-invariant. Then one proves some Gamma-compactness results with respect to both the strong topology of L^p and the strong topology of degenerate Sobolev spaces.

[pt] Arnold no seu trabalho The asymptotic Hopf Invariant and its applications de 1986, considerou sobre um domínio (ômega maiúsculo) compacto de R3 com bordo suave e homología trivial campos X e Y de divergência nula e tangentes ao bordo de (ômega maiúsculo) e definiu o índice de enlaçamento assintótico lk(X; Y ) e o invariante de Hopf associados a X e Y pela integral I(X; Y ) igual a (integral em ômega maiúsculo de alfa produto d-beta), onde (d-alfa) igual a iX-vol e (d-beta) igual a iy-vol, e mostrou que I(X; Y ) igual a lk(X; Y ). Agora, no presente trabalho estenderemos estas definições de índices de enlaçamento assintótico lk(fi maiúsculo,xi maiúsculo) e de invariante de Hopf I(fi maiúsculo,xi maiúsculo), onde (fi maiúsculo) e (xi maiúsculo) são ações de Rk e de Rs, k mais s igual a n-1, respectivamente de difeomorfismos que preservam volume em (ômega maiúsculo n) a bola unitária fechada em Rn e mostraremos que lk (fi maiúsculo, xi maiúsculo) igual a I(fi maiúsculo,xi maiúsculo). / [en] V.I. Arnold, in his paper The algebraic Hopf invariant and its applications published in 1986, considered a compact domain (ômega maiúsculo) in R3 with a smooth boundary and trivial homology and two divergence free vector fields X and Y in (ômega maiúsculo) tangent to the boundary. He defined an asymptotic linking invariant lk(X; Y ) and a Hopf invariant associated to X and Y by the integral I(X; Y ) equal (integral em ômega maiúsculo de alfa produto d-beta) where (d-alfa) equal iX-vol e (d-beta) equal iy- vol. He showed that que I(X; Y ) equal lk(X; Y ). In the present work we extend these definitions of the asymptotic linking invariant lk(fi capital letter,xi capital letter) and the Hopf invariant I(fi maiúsculo,xi capital letter) where (fi capital letter) and (xi capital letter) are actions Rk and Rs, k plus s equal n-1 by volume preserving diffeomorphisms, on the closed unit ball (ômega capital letter n) in and we show lk (fi capital letter, xi capital lette r equal I(ficapital letter ,xi capital letter).

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2018-03-28T11:58:56Z No. of bitstreams: 2 Dissertação - Jeferson Arley Poveda Contreras - 2018.pdf: 763599 bytes, checksum: 6800571168e0aa9de85d151e4c912725 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-03-29T11:29:24Z (GMT) No. of bitstreams: 2 Dissertação - Jeferson Arley Poveda Contreras - 2018.pdf: 763599 bytes, checksum: 6800571168e0aa9de85d151e4c912725 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-03-29T11:29:24Z (GMT). No. of bitstreams: 2 Dissertação - Jeferson Arley Poveda Contreras - 2018.pdf: 763599 bytes, checksum: 6800571168e0aa9de85d151e4c912725 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-03-07 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / The goal of this work is study limit cycles of piecewise smooth vector fields. First, we present the basic theory, passing through the areas of analysis, qualitative theory of differential equations and algebra. We also present basic concepts of Filippov fields, which are indispensable for the study of piecewise smooth fields. In chapter one, was the main topic, a general method for finding limit cycles will be described; in the second chapter limit cycles are found in a piecewise smooth vector field with non-degenerate center being perturbed by a piecewise polynomial vector field. In the fourth chapter, we study limit cycles in piecewise smooth Hamiltonian fields. / O objetivo deste trabalho é estudar ciclos limite de campos de vetores suaves por parte. Primeiro apresentaremos a teoria básica, passando pelas áreas de análise, teoria qualitativa das equações diferenciais e álgebra. Apresentamos também conceitos básicos de campos de Filippov, os quais são imprescindíveis para o estudo dos campos suaves por partes. No capítulo dos, como tópico principal, será descrito um método geral para encontrar ciclos limite; no segundo três são encontrados ciclos limites em um campo de vetores suave por partes com um centro não degenerado sendo perturbado por um polinômio. No quarto capitulo estudaremos os ciclos limites de campos de vetores Hamiltonianos por parte.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-03-12Bitstream added on 2014-06-13T19:06:40Z : No. of bitstreams: 1 cardin_pt_me_sjrp.pdf: 780321 bytes, checksum: 2c76fcd2cf98ce623cf8bc779edb3379 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nesta dissertação estudamos critérios para determinar a existência, a não existência e a unicidade de ciclos limites de campos de vetores planares. Mais especificamente, estudamos equações de Lienard Äx + f(x; _ x) _ x + g(x) = 0; onde f e g satisfazem determinadas hip¶oteses. Em particular estudamos a equa»c~ao de van der Pol Äx + (x2 ¡ 1) _ x + x = 0; a qual é conhecida da teoria dos circuitos elétricos. Provamos a existência e a unicidade de ciclos limites para estas equações. Por fim estudamos a equação de van der Pol com o parâmetro 1 e o fenômeno canard que ocorre ao considerarmos um parâmetro adicional ®: As técnicas utilizadas s~ao as usuais de Análise Assintótica. / In this work we study the existence, the non existence and the uniqueness of limit cycles of planar vector felds. More specifically, we study Lienard equations Äx+f(x; _ x) _ x+g(x) = 0; where f and g satisfy some hypothesis. In particular we study the van der Pol equation Äx + (x2 ¡ 1) _ x + x = 0; which is knew of the circuit theory. We prove the existence and the uniqueness of limit cycles for these equations. In the last part we study the van der Pol equation with the parameter 1 and the canard phenomenon which appears when we consider an additional parameter ®: The techniques employed are the usual in the Asymptotic Analysis.