Geometric trimming of surfaces results in a new mathematical description of the matching surface. This matching surface is required to closely resemble the remaining portion of the original surface. Typically, the approximation error in such cases is measured with a view to minimize it. The data associated with the error between two matching surfaces is large and needs to be filtered into meaningful information.This research looks at suitable norms for achieving this data reduction or abstraction with a view to provide quantitative feedback about the approximation error. Also, the differences between geometric shapes are easily discerned by the human eye but are difficult to characterize or describe. Error visualization tools have been developed to provide effective visual inputs that the designer can interpret into meaningful information. / Master of Science

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.

Das Hauptgebiet der Arbeit stellt die Approximation der Lösungen partieller Differentialgleichungen mit Dirichlet-Randbedingungen durch Splinefunktionen dar. Partielle Differentialgleichungen finden ihre Anwendung beispielsweise in Bereichen der Elektrostatik, der Elastizitätstheorie, der Strömungslehre sowie bei der Untersuchung der Ausbreitung von Wärme und Schall. Manche Approximationsaufgaben besitzen keine eindeutige Lösung. Durch Anwendung der Penalized Least Squares Methode wurde gezeigt, dass die Eindeutigkeit der gesuchten Lösung von gewissen Minimierungsaufgaben sichergestellt werden kann. Unter Umständen lässt sich sogar eine höhere Stabilität des numerischen Verfahrens gewinnen. Für die numerischen Betrachtungen wurde ein umfangreiches, effizientes C-Programm erstellt, welches die Grundlage zur Bestätigung der theoretischen Voraussagen mit den praktischen Anwendungen bildete. / This work focuses on approximating solutions of partial differential equations with Dirichlet boundary conditions by means of spline functions. The application of partial differential equations concerns the fields of electrostatics, elasticity, fluid flow as well as the analysis of the propagation of heat and sound. Some approximation problems do not have a unique solution. By applying the penalized least squares method it has been shown that uniqueness of the solution of a certain class of minimizing problems can be guaranteed. In some cases it is even possible to reach higher stability of the numerical method. For the numerical analysis we have developed an extensive and efficient C code. It serves as the basis to confirm theoretical predictions with practical applications.

A doença de Chagas possui uma importante relevância clínica, sendo uma das principais causas de mortalidade e morbidade na América Latina. As relações entre a lesão tecidual miocárdica e os defeitos na inervação autonômica na doença de Chagas são pouco conhecidas. Este trabalho descreve o desenvolvimento e aplicação de métodos de segmentação, corregistro e análise de imagens capazes de prover uma análise integrada das lesões teciduais através do imageamento de ressonância magnética (MRI), perfusão miocárdica e inervação autonômica, disponíveis através da tomografia de emissão de fótons (SPECT). O método proposto é baseado na segmentação e corregistro entre as imagens MRI e imagens SPECT usando 99mTc-MIBI e 123I-MIBG. Para realizar a segmentação do miocárdio, foi utilizada a técnica de Contorno Ativo Geodésico. A segmentação de fibrose em imagens MRI foi realizada com base na maximização da entropia de Tsallis. O corregistro não-rígido foi realizado através do método B-Spline. Os resultados de quantificação indicam correlações entre a presença de fibrose, desnervação e isquemia, além de mostrar a presença de regiões de miocárdio vivo, isquêmico e desnervado. Assim, a ferramenta desenvolvida fornece uma análise integrada de informação, permitindo uma melhor compreensão da relação entre o dano ao tecido do miocárdio e defeitos de inervação autonômica causadas pela doença de Chagas. / Chagas disease is of major clinical relevance, and a major cause of morbidity and mortality in Latin America. The relations between the myocardial tissue damage, myocardial perfusion and defects in autonomic innervations are poorly understood. This study proposes the development and application of image analysis methods capable of providing an integrated visualization and analysis of tissue injuries through enhanced magnetic resonance imaging (MRI), autonomic innervations and myocardial perfusion, available through photon emission tomography (SPECT). The proposed method is based on segmentation and registration between MRI images and SPECT images using 99mTc-MIBI and 123I-MIBG. To perform the segmentation of myocardium, we used Geodesic Active Contour. Fibrosis segmentation in MRI images was performed based on the algorithm of maximum Tsallis entropy. Nonrigid registrations was performed based on B-Spline method. The quantification results showed correlations between the presence of fibrosis, denervation and ischemia, as well as showing the regarded presence of regions of healthy myocardium, ischemic and denervated. Thus, the developed tool provides an integrated analysis of information contributing to a better understanding of the relationship between myocardial tissue damage and autonomic innervations injuries caused by Chagas disease.

As b-splines são utilizadas em sistemas CAD/CAM/CAE para representar e definir curvas e superfícies complexas, sendo adotada pelos principais padrões da computação gráfica devido a características como representação matemática de forma compacta, flexibilidade e transformações afins. Em sistemas de aquisição de dados 3D e sistemas CAM-CNC integrados, a utilização da b-spline na transferência de informações geométricas e na reconstrução da superfície de objetos resulta em um significativo incremento na eficiência do processo, geralmente implementado em sistemas embarcados. Nestes sistemas embarcados, integrados no auxílio a máquinas de manufatura, a utilização de FPGAs é incipiente, sem circuitos para b-splines disponibilizados em lógica reconfigurável de circuito aberto (open core), razão pela qual este projeto propõe o desenvolvimento de um circuito de geração b-spline aberto, em um sistema embarcado FPGA, utilizando algoritmos adaptados para os circuitos, elaborados em linguagem Verilog HDL, padronizada para a síntese de circuitos em lógica reconfigurável. Os circuitos foram desenvolvidos, utilizando-se um barramento de dados padronizado em circuito aberto, nas seguintes implementações para processamento paralelo das b-splines: o BFEA, o método baseado em funções base fixas, ambos projetados para circuitos integrados, e o fast Cox-de Boor, desenvolvido para FPGAs. Foram comparados o tempo de execução e o consumo de recursos disponíveis no FPGA utilizado, entre cada implementação. Os resultados evidenciaram que os circuitos de funções base fixas apresentaram o processamento mais rápido para a geração de b-splines em um FPGA, com um tempo de execução em média 20% menor em relação às outras implementações. Os circuitos BFEA apresentaram a menor utilização de elementos lógicos, em média 50% menor em relação aos outros circuitos implementados. O circuito fast Cox-de Boor apresentou a melhor escalabilidade, devido à modularidade da implementação, com tempos de execução similares aos circuitos de funções base fixas. / The b-splines are used in CAD/CAM/CAE systems to represent and define complex curves and surfaces, being adopted by the main computer graphics standards due to features like compact mathematic representation, flexibility and affine transformations. In 3D acquisition systems and integrated CAM-CNC systems, the use of the b-spline in the geometric information data transfer and in the object surface reconstruction results in a increase in the process efficiency, generally implemented in embedded systems. In these embedded systems, integrated in the aid to manufacturing machines, the use of FPGAs is incipient, without available b-splines open core circuits in reconfigurable logic, the reason why this project propose the development of a b-spline generation open core circuit, in a FPGA embedded system, using adaptated algorithms for the circuits, made in Verilog HDL language, standardized for the circuit synthesis in reconfigurable logic. The circuits were developed, using an open core standardized data bus, in the following implementations of b-spline parallel processing: the BFEA, fixed basis functions based method, both designed for integrated circuits, and the fast Cox-de Boor, developed for FPGAs. The execution time and available resource consumption in the FPGA were compared, between each implementation. The results show that the fixed basis functions circuits presented the fastest processing for the b-splines generation in a FPGA, with a 20% mean execution time reduction in relation to the other implementations. The BFEA circuits presented the lowest logic elements use, in mean 50% fewer in relation to the other implemented circuits. The fast Cox-De Boor circuit presented the best scalability, due to the implementation modularity, with execution times similar to the fixed basis functions circuits.

A doença de Chagas possui uma importante relevância clínica, sendo uma das principais causas de mortalidade e morbidade na América Latina. As relações entre a lesão tecidual miocárdica e os defeitos na inervação autonômica na doença de Chagas são pouco conhecidas. Este trabalho descreve o desenvolvimento e aplicação de métodos de segmentação, corregistro e análise de imagens capazes de prover uma análise integrada das lesões teciduais através do imageamento de ressonância magnética (MRI), perfusão miocárdica e inervação autonômica, disponíveis através da tomografia de emissão de fótons (SPECT). O método proposto é baseado na segmentação e corregistro entre as imagens MRI e imagens SPECT usando 99mTc-MIBI e 123I-MIBG. Para realizar a segmentação do miocárdio, foi utilizada a técnica de Contorno Ativo Geodésico. A segmentação de fibrose em imagens MRI foi realizada com base na maximização da entropia de Tsallis. O corregistro não-rígido foi realizado através do método B-Spline. Os resultados de quantificação indicam correlações entre a presença de fibrose, desnervação e isquemia, além de mostrar a presença de regiões de miocárdio vivo, isquêmico e desnervado. Assim, a ferramenta desenvolvida fornece uma análise integrada de informação, permitindo uma melhor compreensão da relação entre o dano ao tecido do miocárdio e defeitos de inervação autonômica causadas pela doença de Chagas. / Chagas disease is of major clinical relevance, and a major cause of morbidity and mortality in Latin America. The relations between the myocardial tissue damage, myocardial perfusion and defects in autonomic innervations are poorly understood. This study proposes the development and application of image analysis methods capable of providing an integrated visualization and analysis of tissue injuries through enhanced magnetic resonance imaging (MRI), autonomic innervations and myocardial perfusion, available through photon emission tomography (SPECT). The proposed method is based on segmentation and registration between MRI images and SPECT images using 99mTc-MIBI and 123I-MIBG. To perform the segmentation of myocardium, we used Geodesic Active Contour. Fibrosis segmentation in MRI images was performed based on the algorithm of maximum Tsallis entropy. Nonrigid registrations was performed based on B-Spline method. The quantification results showed correlations between the presence of fibrosis, denervation and ischemia, as well as showing the regarded presence of regions of healthy myocardium, ischemic and denervated. Thus, the developed tool provides an integrated analysis of information contributing to a better understanding of the relationship between myocardial tissue damage and autonomic innervations injuries caused by Chagas disease.

On propose des conditions de bout pour les fonctions spines polynomiales d'interpolation de degré p (p≥2) associées aux abscisses équidistantes qui économisent le calcul et entraînent un ordre de convergence optimal. Cette fonction spline peut être interprétée comme une fonction spline cardinale tronquée avec une correction convenable. La technique utilisée pour les fonctions splines polynomiales est applicable dans le cas des fonctions splines sous tension. On donne aussi quelques résultats pour les fonctions splines cubiques de lissage

In spite of increasingly sophisticated workforce management tools, a significant gap remains between the goal of effective staffing and the present difficulty predicting the stochastic demand of inbound calls. We have investigated the hypothesized nonhomogeneous Poisson process model of modem pool callers of the University community. In our case, we tested if the arrivals could be approximated by a piecewise constant rate over short intervals. For each of 1 and 10-minute intervals, based on the close relationship between the Poisson process and the exponential distribution, the test results did not show any sign of homogeneous Poisson process. We have examined the hypothesis of a nonhomogeneous Poisson process by a transformed statistic. Quantitative and graphical goodness-of-fit tests have confirmed nonhomogeneous Poisson process. <p>Further analysis on the intensity function revealed that linear rate intensity was woefully inadequate in predicting time varying arrivals. For sinusoidal rate model, difficulty arose in setting the period parameter. Spline models, as an alternative to parametric modelling, had more control of balance between data fitting and smoothness, which was appealing to our analysis on call arrival process.

In spite of increasingly sophisticated workforce management tools, a significant gap remains between the goal of effective staffing and the present difficulty predicting the stochastic demand of inbound calls. We have investigated the hypothesized nonhomogeneous Poisson process model of modem pool callers of the University community. In our case, we tested if the arrivals could be approximated by a piecewise constant rate over short intervals. For each of 1 and 10-minute intervals, based on the close relationship between the Poisson process and the exponential distribution, the test results did not show any sign of homogeneous Poisson process. We have examined the hypothesis of a nonhomogeneous Poisson process by a transformed statistic. Quantitative and graphical goodness-of-fit tests have confirmed nonhomogeneous Poisson process. <p>Further analysis on the intensity function revealed that linear rate intensity was woefully inadequate in predicting time varying arrivals. For sinusoidal rate model, difficulty arose in setting the period parameter. Spline models, as an alternative to parametric modelling, had more control of balance between data fitting and smoothness, which was appealing to our analysis on call arrival process.

Zugl.: Kaiserslautern, Techn. Univ., Diss., 2007