With the increasing use of geometry scanners to create 3D models, there is a rising need for fast and robust mesh smoothing to remove inevitable noise in the measurements. While most previous work has favored diffusion-based iterative techniques for feature-preserving smoothing, we propose a radically different approach, based on robust statistics and local first-order predictors of the surface. The robustness of our local estimates allows us to derive a non-iterative feature-preserving filtering technique applicable to arbitrary "triangle soups". We demonstrate its simplicity of implementation and its efficiency, which make it an excellent solution for smoothing large, noisy, and non-manifold meshes. / Singapore-MIT Alliance (SMA)

Constructing spline models for isogeometric analysis is important in integrating design and analysis. Converting designed CAD (Computer Aided Design) models with B-reps to analysis-suitable volumetric T-spline is fundamental for the integration. In this thesis, we work on two directions to achieve this: (a) using Boolean operations and skeletons to build polycubes for feature-preserving high-genus volumetric T-spline construction; and (b) developing weighted T-splines with arbitrary degree for T-spline surface and volume modeling which can be used for analysis. In this thesis, we first develop novel algorithms to build feature-preserving polycubes for volumetric T-spline construction. Then a new type of T-spline named the weighted T-spline with arbitrary degree is defined. It is further used in converting CAD models to analysis-suitable volumetric T-splines. An algorithm is first developed to use Boolean operations in CSG (Constructive Solid Geometry) to generate polycubes robustly, then the polycubes are used to generate volumetric rational solid T-splines. By solving a harmonic field with proper boundary conditions, the input surface is automatically decomposed into regions that are classified into topologically either a cube or a torus. Two Boolean operations, union and difference, are performed with the primitives and polycubes are generated by parametric mapping. With polycubes, octree subdivision is carried out to obtain a volumetric T-mesh. The obtained T-spline surface is C2-continuous everywhere except the local region surrounding irregular nodes, where the surface continuity is elevated from C0 to G1. B´ezier elements are extracted from the constructed solid T-spline models, which are further used in isogeometric analysis. The Boolean operations preserve the topology of the models inherited from design and can generate volumetric T-spline models with better quality. Furthermore, another algorithm is developed which uses skeleton as a guidance to the polycube construction. From the skeleton of the input model, initial cubes in the interior are first constructed. By projecting corners of interior cubes onto the surface and generating a new layer of boundary cubes, the entire interior domain is split into different cubic regions. With the splitting result, octree subdivision is performed to obtain T-spline control mesh or T-mesh. Surface features are classified into three groups: open curves, closed curves and singularity features. For features without introducing new singularities like open or closed curves, we preserve them by aligning to the parametric lines during subdivision, performing volumetric parameterization from frame field, or modifying the skeleton. For features introducing new singularities, we design templates to handle them. With a valid T-mesh, we calculate rational trivariate T-splines and extract B´ezier elements for isogeometric analysis. Weighted T-spline basis functions are designed to satisfy partition of unity and linear independence. The weighted T-spline is proved to be analysis-suitable. Compared to standard T-splines, weighted T-splines have less geometrical constraint and can decrease the number of control points significantly. Trimmed NURBS surfaces of CAD models are reparameterized with weighted T-splines by a new edge interval extension algorithm, with bounded surface error introduced. With knot interval duplication, weighted T-splines are used to deal with extraordinary nodes. With B´ezier coefficient optimization, the surface continuity is elevated from C0 to G1 for the one-ring neighborhood elements. Parametric mapping and sweeping methods are developed to construct volumetric weighted T-splines for isogeometric analysis. Finally, we develop an algorithm to construct arbitrary degree T-splines. The difference between odd degree and even degree T-splines are studied in detail. The methods to extract knot intervals, calculate new weights to handle extraordinary nodes, and extract B´ezier elements for analysis are investigated with arbitrary degrees. Hybrid degree weighted Tspline is generated at designated region with basis functions of different degrees, for the purpose of performing local p-refinement. We also study the convergence rate for T-spline models of different degrees, showing that hybrid degree weighted T-splines have better performance after p-refinement. In summary, we develop novel methods to construct volumetric T-splines based on polycube and sweeping methods. Arbitrary degree weighted T-spline is proposed, with proved analysis-suitable properties. Weighted T-spline basis functions are used to reparameterize trimmed NURBS surfaces, handling extraordinary nodes, based on which surface and volumetric weighted T-spline models are constructed for isogeometric analysis.

Imagens digitais são utilizadas para diversas finalidades, variando de uma simples foto com os amigos até a identificação de doenças em exames médicos. Por mais que as tecnologias de captura de imagens tenham evoluído, toda imagem adquirida digitalmente possui um ruído intrínseco a ela que normalmente é adquirido durante os processo de captura ou transmissão da imagem. O grande desafio neste tipo de problema consiste em recuperar a imagem perdendo o mínimo possível de características importantes da imagem, como cantos, bordas e texturas. Este trabalho propõe uma abordagem baseada em um Algoritmo Genético Híbrido (AGH) para lidar com este tipo de problema. O AGH combina um algoritmo genético com alguns dos melhores métodos de supressão de ruídos em imagens encontrados na literatura, utilizando-os como operadores de busca local. O AGH foi testado em imagens normalmente utilizadas como benchmark corrompidas com um ruído branco aditivo Gaussiano (N; 0), com diversos níveis de desvio padrão para o ruído. Seus resultados, medidos pelas métricas PSNR e SSIM, são comparados com os resultados obtidos por diferentes métodos. O AGH também foi testado para recuperar imagens SAR (Synthetic Aperture Radar), corrompidas com um ruído Speckle multiplicativo, e também teve seus resultados comparados com métodos especializados em recuperar imagens SAR. Através dessa abordagem híbrida, o AGH foi capaz de obter resultados competitivos em ambos os tipos de testes, chegando inclusive a obter melhores resultados em diversos casos em relação aos métodos da literatura. / Digital images are used for many purposes, ranging from a simple picture with friends to the identification of diseases in medical exams. Even though the technology for acquiring pictures has been evolving, every image digitally acquired has a noise intrinsic to it that is normally gotten during the processes of transmission or capture of the image. A big challenge in this kind of problem consists in recovering the image while losing the minimum amount of important features of the image, such as corners, borders and textures. This work proposes an approach based on a Hybrid Genetic Algorithm (HGA) to deal with this kind of problem. The HGA combines a genetic algorithm with some of the best image denoising methods found in literature, using them as local search operators. The HGA was tested on benchmark images corrupted with an additive white Gaussian noise (N;0) with many levels of standard deviation for the noise. The HGAs results, which were measured by the PSNR and SSIM metrics, were compared to the results obtained by different methods. The HGA was also tested to recover SAR (Synthetic Aperture Radar) images that were corrupted by a multiplicative Speckle noise and had its results compared against the results by other methods specialized in recovering with SAR images. Through this hybrid approach, the HGA was able to obtain results competitive in both types of tests, even being able to obtain the best results in many cases, when compared to the other methods found in the literature.

Imagens digitais são utilizadas para diversas finalidades, variando de uma simples foto com os amigos até a identificação de doenças em exames médicos. Por mais que as tecnologias de captura de imagens tenham evoluído, toda imagem adquirida digitalmente possui um ruído intrínseco a ela que normalmente é adquirido durante os processo de captura ou transmissão da imagem. O grande desafio neste tipo de problema consiste em recuperar a imagem perdendo o mínimo possível de características importantes da imagem, como cantos, bordas e texturas. Este trabalho propõe uma abordagem baseada em um Algoritmo Genético Híbrido (AGH) para lidar com este tipo de problema. O AGH combina um algoritmo genético com alguns dos melhores métodos de supressão de ruídos em imagens encontrados na literatura, utilizando-os como operadores de busca local. O AGH foi testado em imagens normalmente utilizadas como benchmark corrompidas com um ruído branco aditivo Gaussiano (N; 0), com diversos níveis de desvio padrão para o ruído. Seus resultados, medidos pelas métricas PSNR e SSIM, são comparados com os resultados obtidos por diferentes métodos. O AGH também foi testado para recuperar imagens SAR (Synthetic Aperture Radar), corrompidas com um ruído Speckle multiplicativo, e também teve seus resultados comparados com métodos especializados em recuperar imagens SAR. Através dessa abordagem híbrida, o AGH foi capaz de obter resultados competitivos em ambos os tipos de testes, chegando inclusive a obter melhores resultados em diversos casos em relação aos métodos da literatura. / Digital images are used for many purposes, ranging from a simple picture with friends to the identification of diseases in medical exams. Even though the technology for acquiring pictures has been evolving, every image digitally acquired has a noise intrinsic to it that is normally gotten during the processes of transmission or capture of the image. A big challenge in this kind of problem consists in recovering the image while losing the minimum amount of important features of the image, such as corners, borders and textures. This work proposes an approach based on a Hybrid Genetic Algorithm (HGA) to deal with this kind of problem. The HGA combines a genetic algorithm with some of the best image denoising methods found in literature, using them as local search operators. The HGA was tested on benchmark images corrupted with an additive white Gaussian noise (N;0) with many levels of standard deviation for the noise. The HGAs results, which were measured by the PSNR and SSIM metrics, were compared to the results obtained by different methods. The HGA was also tested to recover SAR (Synthetic Aperture Radar) images that were corrupted by a multiplicative Speckle noise and had its results compared against the results by other methods specialized in recovering with SAR images. Through this hybrid approach, the HGA was able to obtain results competitive in both types of tests, even being able to obtain the best results in many cases, when compared to the other methods found in the literature.