Real-valued functions have wide applications in various areas within computer graphics. In this work, we examine three representation of shapes using functions. In particular, we study the classical B-spline representation of piece-wise polynomials in the univariate domain. We provide a generalization of B-spline to the bivariate domain using intuition gained from the univariate construction. We also study the popular scheme of representing 3D density distribution using a uniform, rectilinear grid, where we provide a novel contouring scheme that culls occluded inner geometries. Lastly, we examine a ray-based representation for 3D indicator functions called ray-rep, for which we present a novel meshing scheme with multi-material extensions.

Geometric modeling of biomolecules plays an important role in the study of biochemical processes. Many simulation methods depend heavily on the geometric models of biomolecules. Among various studies, shape analysis is one of the most important topics, which reveals the functionalities of biomolecules.

abstract: Quad-dominant (QD) meshes, i.e., three-dimensional, 2-manifold polygonal meshes comprising mostly four-sided faces (i.e., quads), are a popular choice for many applications such as polygonal shape modeling, computer animation, base meshes for spline and subdivision surface, simulation, and architectural design. This thesis investigates the topic of connectivity control, i.e., exploring different choices of mesh connectivity to represent the same 3D shape or surface. One key concept of QD mesh connectivity is the distinction between regular and irregular elements: a vertex with valence 4 is regular; otherwise, it is irregular. In a similar sense, a face with four sides is regular; otherwise, it is irregular. For QD meshes, the placement of irregular elements is especially important since it largely determines the achievable geometric quality of the final mesh. Traditionally, the research on QD meshes focuses on the automatic generation of pure quadrilateral or QD meshes from a given surface. Explicit control of the placement of irregular elements can only be achieved indirectly. To fill this gap, in this thesis, we make the following contributions. First, we formulate the theoretical background about the fundamental combinatorial properties of irregular elements in QD meshes. Second, we develop algorithms for the explicit control of irregular elements and the exhaustive enumeration of QD mesh connectivities. Finally, we demonstrate the importance of connectivity control for QD meshes in a wide range of applications. / Dissertation/Thesis / Doctoral Dissertation Computer Science 2014

Organic Photovoltaic (OPV) solar cells provide a promising alternative for harnessing solar energy. However, the efficient design of OPV materials that achieve better performance requires support by better-tailored visualization tools than are currently available, which is the goal of this thesis. One promising approach in the OPV field is to control the effective material of the OPV device, which is known as the Bulk-Heterojunction (BHJ) morphology. The BHJ morphology has a complex composition. Current BHJ exploration techniques deal with the morphologies as black boxes with no perception of the photoelectric current in the BHJ morphology. Therefore, this method depends on a trial-and-error approach and does not efficiently characterize complex BHJ morphologies. On the other hand, current state-of-the-art methods for assessing the performance of BHJ morphologies are based on the global quantification of morphological features. Accordingly, scientists in OPV research are still lacking a sufficient understanding of the best material design. To remove these limitations, we propose a new approach for knowledge-assisted visual exploration and analysis in the OPV domain. We develop new techniques for enabling efficient OPV charge transport path analysis. We employ, adapt, and develop techniques from scientific visualization, geometric modeling, clustering, and visual interaction to obtain new designs of visualization tools that are specifically tailored for the needs of OPV scientists. At the molecular scale, the user can use semantic rules to define clusters of atoms with certain geometric properties. At the nanoscale, we propose a novel framework for visual characterization and exploration of local structure-performance correlations. We also propose a new approach for correlating structural features to performance bottlenecks. We employ a visual feedback strategy that allows scientists to make intuitive choices about fabrication parameters. We furthermore propose a visual analysis framework to help answer domain science questions through parameter space exploration for local morphology features. This framework is built on the shape-based clustering of local regions (patches), which for the first time enables local analysis of BHJ morphologies. Using our proposed system, domain experts can interactively create and visualize new BHJ structures of interest at both the molecular and nanoscale levels in a relatively short time.

Both man-made or natural objects contain repeated geometric elements that can be interpreted as primitive shapes. Plants, trees, living organisms or even crystals, showcase primitives that repeat themselves. Primitives are also commonly found in man-made environments because architects tend to reuse the same patterns over a building and typically employ simple shapes, such as rectangular windows and doors. During my PhD I studied geometric primitives from three points of view: their composition, simulation and autonomous discovery. In the first part I present a method to reverse-engineer the function by which some primitives are combined. Our system is based on a composition function template that is represented by a parametric surface. The parametric surface is deformed via a non-rigid alignment of a surface that, once converged, represents the desired operator. This enables the interactive modeling of operators via a simple sketch, solving a major usability gap of composition modeling. In the second part I introduce the use of a novel primitive for real-time physics simulations. This primitive is suitable to efficiently model volume-preserving deformations of rods but also of more complex structures such as muscles. One of the core advantages of our approach is that our primitive can serve as a unified representation to do collision detection, simulation, and surface skinning. In the third part I present an unsupervised deep learning framework to learn and detect primitives. In a signal containing a repetition of elements, the method is able to automatically identify the structure of these elements (i.e. primitives) with minimal supervision. In order to train the network that contains a non-differentiable operation, a novel multi-step training process is presented. / Graduate

The goal of this thesis is to introduce new methods to create intricate perforated shapes in a computing environment. Modeling shapes with a large number of holes and handles, while requiring minimal human interaction, is an unsolved research problem in computer graphics. In this thesis, we have developed two methods for interactively modeling such shapes. Both methods developed create perforated shapes by building a framework of tube like elements, such that each edge of a given mesh is replaced by a pipe. The first method called Wire modeling replaces each edge with a pipe that has a square cross-section. The result looks like a shape that is created by a framework of matchsticks. The second method, called Column modeling allows more rounded cross-sections for the pipes. The cross-sections can be any uniform polygon, and the users are able to control the number of the segments in the cross-section. These methods are implemented as an extension to an existing modeling system guaranteeing that the pipes are connected and the resulting shape can be physically constructed. Our methods require an initial input mesh that can either be imported from a commercially available software package, or created directly in this modeling system. The system also allows the users to export the models in obj file format, so that the models can be animated and rendered in other software packages.

Scales are a visually striking feature that grows on many animals. These small, rigid plates embedded in the skin form an integral part of our description of fish and reptiles, some plants, and many extinct animals. Scales exist in many shapes and sizes, and serve as protection, camouflage, and plumage for animals. The variety of scales and the animals they grow from pose an interesting problem in the field of Computer Graphics. This dissertation presents a method for generating scales and scale-like structures on a polygonal mesh through surface replacement. A triangular mesh was covered with scales and one or more proxy-models were used as the scales shape. A user began scale generation by drawing a lateral line on the model to control the distribution and orientation of scales on the surface. Next, a vector field was created over the surface to control an anisotropic Voronoi tessellation, which represents the region occupied by each scale. Then these regions were replaced by cutting the proxy model to match the boundary of the Voronoi region and deform the cut model onto the surface. The final result is a fully connected 2-manifold that is suitable for subsequent post-processing applications, like surface subdivision.

The goal of this thesis is to introduce new methods to create intricate perforated shapes in a computing environment. Modeling shapes with a large number of holes and handles, while requiring minimal human interaction, is an unsolved research problem in computer graphics. In this thesis, we have developed two methods for interactively modeling such shapes. Both methods developed create perforated shapes by building a framework of tube like elements, such that each edge of a given mesh is replaced by a pipe. The first method called Wire modeling replaces each edge with a pipe that has a square cross-section. The result looks like a shape that is created by a framework of matchsticks. The second method, called Column modeling allows more rounded cross-sections for the pipes. The cross-sections can be any uniform polygon, and the users are able to control the number of the segments in the cross-section. These methods are implemented as an extension to an existing modeling system guaranteeing that the pipes are connected and the resulting shape can be physically constructed. Our methods require an initial input mesh that can either be imported from a commercially available software package, or created directly in this modeling system. The system also allows the users to export the models in obj file format, so that the models can be animated and rendered in other software packages.

[pt] Esta dissertação tem por objetivo estudar um método para aproximação de dados esparços multivariados denominado o método Mínimos Quadrados Móveis por Aproximações Iteradas (Iterated Approximate Moving Least-Square Approximation – IAMLA). Este método é baseado no método de interpolação por funções de base radial (RBF) e no método de aproximação AMLS. Mas diferentemente do método RBF, ele não requer a solução de um sistema de equações lineares. O método IAMLS no limite converge para o interpolante RBF sob certas condições. / [en] The objective of this work is to study an approximation method for multivariate sparse data named Iterated Approximate Moving Least Square Approximation – IAMLS. This method is based on the Radial Basis Functions (RBF) interpolation method and on the AMLS approximation method. Differently from the RBF interpolation method, the IAMLS does not requires to solve a system of linear equations. The IAMLS method converges to the RBF interpolant under some conditions.

[pt] O presente trabalho de pesquisa, apresenta a aplicação de um novo conceito de visualização e interpretação da informação geológica. Os mapas geológicos que, até hoje em dia, são apresentados em papel, agora podem ser vistos, em computador, de forma tridimensional. Esta representação tridimensional é realizada a partir do software GOCAD (Geological Object Computer Aided Draw) (grupo de Ciência da Computação da Escola Nacional de Geologia, Nancy, França, 1989). Este software foi inicialmente desenvolvido para aplicação e representação de bacias sedimentares na indústria do petróleo. O presente trabalho de pesquisa apresenta uma extensão do GOCAD à representação tridimensional de maciços rochosos, como os encontrados na região litorânea do Estado do Rio de Janeiro. Uma área na zona sul da cidade do Rio de Janeiro foi escolhida como área piloto de estudo para a modelagem tridimensional. Esta área, o maciço Dois Irmãos, foi inicialmente caracterizada do ponto de vista da geologia estrutural e as informações coletadas foram utilizadas na modelagem geométrica usando GOCAD. Os resultados obtidos são comentados e são também mostrados o potencial e limitações do software GOCAD para este tipo particular de modelagem geométrica de maciços rochosos. / [en] The present research presents the application of a new concept in visualization and interpretation of geological information. The geological maps that until now are still represented on paper, now can be seen tridimensionally, in computer screen. This 3D representation is accomplished through the software GOCAD (Geological Object Computer Aided Draw), developped by a team of Computer Science of the National School of Geology, Nancy, France. The software was developed originally for application and representation of sedimentary basins basically for use in the petroleum industry. The present research, presents an extension of the GOCAD software to the 3D representation of rock masses found on the coastal region of the state of Rio de Janeiro. An area in the southern area of the city of Rio de Janeiro was chosen to be a pilot study area for the 3D geometrical modelling. This area, the Morro Dois Irmãos massive, was initially characterized from the structural geology point of view and the collected information was used in the 3D geometrical modelling with GOCAD . The obtained results are commented and the potential and limitations of the GOCAD software for this particular type of geometrical modelling of rock masses are discussed. / [es] Este trabajo de investigación presenta la aplicación de un nuevo concepto de visualización e interpretación de la información geológica. Los mapas geológicos que, hasta hoy en día, son presentados en papel, pueden ser vistos ahora en computador, de forma tridimensional. Esta representación tridimensional se realiza a partir del software GOCAD (Geological Object Computer Aided Draw) (grupo de Ciencias de la Computación de la Escuela Nacional de Geología, Nancy, Francia, 1989). Este software fue inicialmente desarrollado para la aplicación y representación de bacias sedimentares en la industria del petróleo. EL presente trabajo de investigación presenta una extensión del GOCAD a la representación tridimensional de macizos rocosos, como los encontrados en la región litoránea del Estado del Rio de Janeiro. Cierta área situada en la zona sul de la ciudad de Rio de Janeiro fue escogida como área piloto de estudio para la modelaje tridimensional. Esta área, el macizo Dois Irmãos, fue inicialmente caracterizada desde el punto de vista de la geología extructural y las informaciones recolectadas fueron utilizadas en el modelo geométrico usando GOCAD. Se comentan los resultados obtenidos y se muestra el potencial y limitaciones del software GOCAD para este tipo particular de modelo geométrico de macizos rocosos.