[pt] Campos vetoriais são resultados comuns em simuladores físicos. Simulações em modelos de reservatórios de petróleo podem nos fornecer, por exemplo, dados relativos ao fluxo de óleo, água e gás. Para um melhor entendimento de tais dados, entretanto, é interessante o uso de uma técnica de visualização que permita a identificação de características locais e tendências globais no campo. Este trabalho propõe uma técnica para visualização de campos vetoriais 3D baseada em GPU que utiliza o algoritmo de convolução de integral de linha (LIC) em 2D para a visualização da componente tangencial à superfície projetada no espaço da tela. Dados relativos à magnitude e componente normal são apresentados através de uma escala de cores bidimensional. Para fixar a imagem resultante do LIC no modelo é proposto um esquema simples baseado em coordenadas de texturas aleatórias, eliminando a necessidade de textura sólida 3D para armazenar o ruído branco. Filtros para animação da imagem de LIC foram adaptados para permitir velocidade variável de acordo com a magnitude do campo. Para melhoria da imagem final, o algoritmo de LIC é aplicado em duas passadas e o resultado é submetido a um filtro de passa-alta. O framework desenvolvido como parte do trabalho foi explorado no contexto da visualização de fluxos em modelos de reservatório de petróleo e de gradientes de altura em terrenos. No caso específico de reservatórios, é proposta uma variação da técnica que permite visualização simultânea de fluxos de óleo, gás e água. / [en] Vector fields are common results of physics simulators. Simulations over black-oil reservoirs, for instance, can generate oil, water and gas flow data. For a better understanding of such data, however, it s interesting to use a visualization technique that allows a better identification of local characteristics and global tendencies of the field. This work proposes a technique for visualization of 3D vector fields that is GPU-based and uses the 2D line integral convolution (LIC) algorithm to visualize the component tangential to the surface projected on screen space. Data related to magnitude and normal component are presented through a 2-dimensional color scale. A simple scheme based on randomly generated texture coordinates is proposed to fixate the resulting LIC image to the model, avoiding flickering during model manipulation and eliminating the need for a solid 3D texture noise. For animation, we adjust the use of filters to ensure that the animation speed varies in accordance to the field magnitude. To enhance the final image, the LIC algorithm is applied in two passes and the result is put through a high-pass filter. The framework developed as part of this work has been applied in the context of visualizing flow in black-oil reservoir models and height gradients in terrains. In the specific case of reservoirs, a variation from the main technique is proposed to allow simultaneous visualization of oil, gas and water flows.

Since the very beginning of the development of computers they have been used to accelerate the knowledge gain in science and research. Today they are a core part of most research facilities. Especially in natural and technical sciences they are used to simulate processes that would be hard to observe in real world experiments. Together with measurements from such experiments, simulations produce huge amounts of data that have to be analyzed by researchers to gain new insights and develop their field of science.

[pt] Tradicionalmente, para visualização de campos vetoriais em modelos discretos de reservatórios naturais de petróleo, traça-se linhas de fluxo resolvendo a sua equação diferencial ordinária célula-a-célula, seja através de soluções analíticas ou numéricas, considerando o campo de velocidade local de cada célula. Essa estratégia tem como desvantagem traçar a linha considerando um campo de velocidade discreto e portanto descontínuo. Além disso, para modelos massivos, resolver a equação célula-a-célula pode tornar o método ineficiente. Neste trabalho, exploramos uma estratégia diferente: ao traçar as linhas de fluxo considera-se um campo de velocidade contínuo, representado pelo modelo discreto do reservatório. Para tanto, propõe-se: (i) o uso de uma estrutura espacial para acelerar a localização de um ponto no modelo de reservatório; (ii) o uso de interpolação esférica para avaliação do campo de velocidade a partir do modelo discreto; (iii) o uso de um método numérico adaptativo para controlar o erro numérico da integração. Os resultados obtidos em modelos de reservatórios reais demonstram que o método proposto atende aos requisitos de precisão, mantendo um bom desempenho. / [en] Traditionally, streamlines in discrete models of natural oil reservoirs are traced by solving an ordinary differential equation in a cell-by-cell way, using analytical or numerical solutions, considering the local velocity of each cell. This strategy has a disadvantage: the streamline is traced considering a discrete, and so discontinuous, vector field. Furthermore, for massive models, to solve the equation in a cell-by-cell way may be inefficient. In this work, we explore a different strategy: the streamline tracing considers a continuous vector field represented by the discrete model. Therefore, we propose: (i) to use a spatial structure to speed up the point location process inside the reservoir model; (ii) to use spherical interpolation to obtain the velocity field from the discrete model; (iii) to use an adaptive numerical method to control the numerical error from the integration process. The results obtained for actual reservoir models demonstrate that the proposed method fulfills the precision requirements, keeping a good performance.

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To estimate the delivered dose to the patient during intra-fraction or throughout the whole treatment, it is important to determine the contribution of dose accumulated at different patient geometries to the overall dose. Dose mapping utilizes deformable image registration to map doses deposited on patient geometries at different times. Inputs to the dose mapping process are the irradiated and reference images, the displacement vector field, and a dose mapping algorithm. Thus accuracy of the mapped dose depends on the DVF and dose mapping algorithm. Dose mapping had been the subject of many research studies however, up to now there is no gold standard DIR or dose mapping algorithm. This thesis compares current dose mapping algorithms under different conditions such as choosing the planning target and dose grid size, and introduces new tool to estimate the required spatial accuracy of a DVF. 11 lung patients were used for this thesis work. IMRT plans were generated on the end of inhale breathing phases with 66 Gy as the prescription dose. Demons DVF’s were generated using the Pinnacle treatment planning system DIR interface. Dtransform, Tri-linear with sub-voxel division, and Pinnacle dose mapping algorithms were compared to energy transfer with mass sub-voxel mapping. For breathing phase 50% on 11 patients, tissue density gradients were highest around the edge of the tumor compared to the CTV and the PTV edge voxels. Thus treatment plans generated with margin equal to zero on the tumor might yield the highest dose mapping error (DME). For plans generated on the tumor, there was no clinical effect of DME on the MLD, lung V20, and Esophagus volume indices. Statistically, MLD and lung V20 DME were significant. Two patients had D98 Pinnacle-DME of 4.4 and 1.2 Gy. In high dose gradient regions DVF spatial accuracy of ~ 1 mm is needed while 8 to 10 mm DVF accuracy can be tolerated before introducing any considerable dose mapping errors inside the CTV. By using ETM with mass sub-voxel mapping and adapting the reported DVF accuracy, the findings of this thesis have the potential to increase the accuracy of 4D lung planning.

Modern big data often emerge as tensors. Standard statistical methods are inadequate to deal with datasets of large volume, high dimensionality, and complex structure. Therefore, it is important to develop algorithms such as low-rank tensor decomposition for data compression, dimensionality reduction, and approximation. With the advancement in technology, high-dimensional images are becoming ubiquitous in the medical field. In lung radiation therapy, the respiratory motion of the lung introduces variabilities during treatment as the tumor inside the lung is moving, which brings challenges to the precise delivery of radiation to the tumor. Several approaches to quantifying this uncertainty propose using a model to formulate the motion through a mathematical function over time. [Li et al., 2011] uses principal component analysis (PCA) to propose one such model using each image as a long vector. However, the images come in a multidimensional arrays, and vectorization breaks the spatial structure. Driven by the needs to develop low-rank tensor decomposition and provided the 4DCT and Displacement Vector Field (DVF), we introduce two tensor decompositions, Population Value Decomposition (PVD) and Population Tucker Decomposition (PTD), to estimate the respiratory lung motion with high levels of accuracy and data compression. The first algorithm is a generalization of PVD [Crainiceanu et al., 2011] to higher order tensor. The second algorithm generalizes the concept of PVD using Tucker decomposition. Both algorithms are tested on clinical and phantom DVFs. New metrics for measuring the model performance are developed in our research. Results of the two new algorithms are compared to the result of the PCA algorithm.

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.