The motivation for this work comes from physics, when dealing with microstructures of polycrystalline materials. An adequate probabilistic model is a three-dimensional (3D) random tessellation. The original contribution of the author is dealing with the Gibbs-Voronoi and Gibbs- Laguerre tessellations in 3D, where the latter model is completely new. The energy function of the underlying Gibbs point process reflects interactions between geometrical characteristics of grains. The aim is the simulation, parameter estimation and degree-of-fit testing. Mathematical background for the methods is described and numerical results based on simulated data are presented in the form of tables and graphs. The interpretation of results confirms that the Gibbs-Laguerre model is promising for further investigation and applications.

Rendering in computer graphics is the process of converting a three dimensional scene description into a two dimensional image. In this work we focus on high quality rendering, which has numerous applications in entertainment and visualization. Many films today are created either entirely or in concert with computationally generated imagery and serve as a vivid example of the benefits of high quality rendering. This dissertation consists of two parts, each presenting novel work in the field of high quality, high performance rendering. The first part proposes the use of volumetric occluders, or a collection of axis-aligned boxes placed within a polygonal model, to accelerate the rendering of shadows cast by the model while producing images identical to the unaccelerated baseline. We show that our approach performs well on single object scenes and extend our approach for use with scenes from a professional open source movie. Although the technique has not yet proven itself on these multi-object scenes, we identify the scene characteristics which are hampering the approach and show that in some cases it is still possible to achieve an improvement in performance. The second part of the dissertation presents a new way to determine micropolygon tessellation rate within a Reyes style renderer. Our new scheme, called final approach tessellation, evaluates the tessellation rate close to a Reyes surface rather than upon entry into its bounding box. Our determination of the tessellation rate is more aggressive than previous approaches, producing a more compact tessellation which in turn is faster to compute and requires less memory. Our evaluation shows that although final approach tessellation is promising in theory, it ultimately fails to improve performance on actual test scenes. / text

In this thesis, we develop a new computational method using Voronoi vertex model and energy to describe the cell-cell interaction among the epithelial tissue. Several studies and simulations will be showed such as equilibrium states, wound closure process, and abnormal growth. We also perform analysis on circular epithelial wound closure process.

In this thesis, we develop a new computational method using Voronoi vertex model and energy to describe the cell-cell interaction among the epithelial tissue. Several studies and simulations will be showed such as equilibrium states, wound closure process, and abnormal growth. We also perform analysis on circular epithelial wound closure process.

[pt] Um dos maiores gargalos do pipeline gráfico hoje é a largura de banda disponível entre a GPU e CPU. Para minimizar esse gargalo funcionalidades programáveis foram inseridas nas placas de vídeo. Com o Geometry Shader é possível criar vértices em GPU, porém, este estágio da pipeline apresenta performance baixa. Com o lançamento das novas APIs gráficas (DirectX11 e OpenGL4) em 2009, foi adicionado o Tessellator, que permite a criação de vértices em massa na GPU. Esta dissertação estuda este novo estágio da pipeline, bem como apresenta algoritmos clássicos (PN-Triangles e Phong Tessellation) que originalmente foram feitos para CPU e propõe novos algoritmos (Renderização de Tubos e Terrenos em GPU) para tirar proveito deste novo paradigma. / [en] One of the main bottlenecks in the graphics pipeline nowadays is the memory bandwidth available between the CPU and the GPU. To avoid this bottleneck, programmable features were inserted into the video cards. With the Geometry Shader launch it is possible to create vertices in the GPU, however, this pipeline stage has a low performance. With the new graphic APIs (DirectX11 and OpenGL4) a Tessellator stage that allows massive vertex generation inside the GPU was created. This dissertation studies this new pipeline stage, as well as presents classic algorithms (PN-Triangles and Phong Tessellation) that were originally designed for CPU and proposes new algorithms (Tubes and Terrain rendering in the GPU) that takes advantage of this new paradigm.

By combining the high-order accuracy of spectral expansions with the locality and geometric flexibility of finite elements, spectral elements are an attractive option for the next generation of numerical climate models. Crucial to their construction is the configuration of nodes in an element — casual placement leads to polynomial fits exhibiting Runge phenomena manifested by wild spatial oscillations. I provide highorder triangular elements suitable for incorporation into existing spectral element codes. Constructed from a variety of measures of optimality, these nodes possess the best interpolation error norms discovered to date. Motivated by the need to accurately determine these error norms, I present an optimization method suitable for finding extrema in a triangle. It marries a branch and bound algorithm to a quadtree smoothing scheme. The resulting scheme is both robust and efficient, promising general applicability. In order to qualitatively evaluate these nodal distributions, I introduce the concept of a Lagrangian Voronoi tessellation. This partitioning of the triangle illustrates the regions over which each node dominates. I argue that distant and disconnected regions are undesirable as they exhibit a non-physical influence. Finally, I have discovered a link between point distributions in the simplex and on the hypersphere. Through a simple transformation, a distance metric is defined permitting the construction of Voronoi diagrams and the calculation of mesh norms.

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Context. The geometric complexity of objects in games and other real-time applications is a crucial aspect concerning the performance of the application. Such applications usually redraw the screen between 30-60 times per second, sometimes even more often, which can be a hard task in an environment with a high number of geometrically complex objects. The concept called Level of Detail, often abbreviated LoD, aims to alleviate the load on the hardware by introducing methods and techniques to minimize the amount of geometry while still maintaining the same, or very similar result. Objectives. This study will compare four of the often used techniques, namely Static LoD, Unpopping LoD, Curved PN Triangles, and Phong Tessellation. Phong Tessellation is silhouette-based, and since the silhouette is considered one of the most important properties, the main aim is to determine how it performs compared to the other three techniques. Methods. The four techniques are implemented in a real-time application using the modern rendering API Direct3D 11. Data will be gathered from this application to use in several experiments in the context of both performance and image space metrics. Conclusions. This study has shown that all of the techniques used works in real-time, but with varying results. From the experiments it can be concluded that the best technique to use is Unpopping LoD. It has good performance and provides a good visual result with the least average amount of popping of the compared techniques. The dynamic techniques are not suitable as a substitute to Unpopping LoD, but further research could be conducted to examine how they can be used together, and how the objects themselves can be designed with the dynamic techniques in mind.

Coordinated missions with platoons of autonomous agents are rapidly becoming popular because of technological advances in computing, networking, miniaturization and combination of electromechanical systems. These multi-agents networks coordinate their actions to perform challenging spatially-distributed tasks such as search, survey, exploration, and mapping. Environmental monitoring and locational optimization are among the main applications of the emerging technology of wireless sensor networks where the optimality refers to the assignment of sub-regions to each agent, in such a way that a suitable coverage metric is maximized. Usually the coverage metric encodes a distribution of risk defined on the area, and a measure of the performance of individual robots with respect to points inside the region of interest. The risk density can be used to quantify spatial distributions of risk in the domain. The solution of the optimal control problem in which the risk measure is not time varying is well known in the literature, with the optimal con figuration of the robots given by the centroids of the Voronoi regions forming a centroidal Voronoi tessellation of the area. In other words, when the set of mobile robots converge to the corresponding centroids of the Voronoi tessellation dictated by the coverage metric, the coverage itself is maximized. In this work, it is considered a time-varying risk density evolving according to a diffusion equation with varying boundary conditions that quantify a time-varying risk on the border of the workspace. Boundary conditions model a time varying flux of external threats coming into the area, averaged over the boundary length, so that rather than considering individual kinematics of incoming threats it is considered an averaged, distributed effect. This approach is similar to the one commonly adopted in continuum physics, in which kinematic descriptors are averaged over spatial domain and suitable continuum fields are introduced to describe their evolution. By adopting a first gradient constitutive relation between the flux and the density, a simple diffusion equation is obtained. Asymptotic convergence and optimality of the non-autonomous system are studied by means of Barbalat's lemma and connections with varying boundary conditions are established. Some criteria on time-varying boundary conditions and evolution are established to guarantee the stabilities of agents' trajectories. A set of numerical simulations illustrate theoretical results.

Generating and rendering ribbons from a stream of particles is common in games and VFX to visualize flow. To create detailed smooth particle ribbons this work combines tessellation and B-splines, which gives smooth surfaces with contiguous normals. The traditional pipeline for doing hardware tessellation with hull- and domain shaders is limited to a maximum tessellation level, which limits how much detail a ribbon can get. In Nvidia’s Turing architecture, mesh- and task shaders were introduced, which provide support for tessellating geometry while not having any constraints on the amount of geometry that can be generated. This work evaluates three implementations for generating and drawing smooth particle ribbons by measuring performance in terms of execution time. The implementations are based on mesh shaders, mesh + task shaders and the traditional pipeline. In turn, three optimizations are proposed. The optimizations evaluated are adaptive level of detail, culling and Gouraud shading. The results show that the traditional pipeline is faster than using mesh shader based tessellation up to its maximum tessellation factor. For larger tessellation factors, when combining adaptive LOD, culling and Gouraud shading, mesh and task shaders combined can generate ribbons at higher quality than the traditional tessellation pipeline, with comparable performance. / Att generera och rendera band från partikelströmmar är vanligt i spel och VFX för att visualisera flöden. För att skapa detaljrika mjuka partikelband kombinerar det här arbetet tessellering och B-splines, vilket ger mjuka ytor med kontinuerliga normaler. Den traditionella pipelinen för att göra hårdvarutessellering med hull- och domain shaders är begränsade till en maximal tesselleringsnivå, vilket begränsar hur detaljrik ett band kan bli. Med Nvidias turingarkitektur introducerades mesh- och task shaders, vilket också stödjer tessellering av geometri med fördelen att inte vara begränsade till en maximal tesselleringsnivå. Det här arbetet evaluerar tre implementationer av mjuka partikelband när de genereras och ritas genom att mäta prestanda genom exekveringstid. Implementationerna är baserade på mesh shaders, mesh + task shaders och den traditionella pipelinen. Utöver detta föreslås tre optimeringar. Dessa optimeringar är adaptiv LOD, culling och Gouraud shading. Resultaten visar att den traditionella pipelinen är snabbare än att använda mesh shaderbaserad tessellering upp till och med dess maximala tesselleringsfaktor. För större tesselleringsfaktorer, när adaptiv LOD, culling och Gouraud shading kombineras, kan mesh och task shaders tillsammans generera band med högre kvalitet än den traditionella tesselleringspipelinen, med jämförbar prestanda.