在迴歸分析中，若變數間具有非線性 (nonlinear) 的關係時，B-Spline線性迴歸是以無母數的方式建立模型。B-Spline函數為具有節點(knots)的分段多項式，選取合適節點的位置對B-Spline函數的估計有重要的影響，在希望得到B-Spline較好的估計量的同時，我們也想要只用少數的節點就達成想要的成效，於是Huang (2013) 提出了一種選擇節點的方式APLS (Adaptive penalized least squares)，在本文中，我們以此方法進行一些更一般化的設定，並在不同的設定之下，判斷是否有較好的估計效果，且已修正後的方法與基於BIC (Bayesian information criterion)的節點估計方式進行比較，在本文中我們將一般化設定的APLS法稱為GAPLS，並且經由模擬結果我們發現此兩種以B-Spline進行迴歸函數近似的方法其近似效果都很不錯，只是節點的個數略有不同，所以若是對節點選取的個數有嚴格要求要取較少的節點的話，我們建議使用基於BIC的節點估計方式，除此之外GAPLS法也是不錯的選擇。 / In regression analysis, if the relationship between the response variable and the explanatory variables is nonlinear, B-splines can be used to model the nonlinear relationship. Knot selection is crucial in B-spline regression. Huang (2013) propose a method for adaptive estimation, where knots are selected based on penalized least squares. This method is abbreviated as APLS (adaptive penalized least squares) in this thesis. In this thesis, a more general version of APLS is proposed, which is abbreviated as GAPLS (generalized APLS). Simulation studies are carried out to compare the estimation performance between GAPLS and a knot selection method based on BIC (Bayesian information criterion). The simulation results show that both methods perform well and fewer knots are selected using the BIC approach than using GAPLS.

Dans cette thèse nous avons traité le problème de l'estimation non paramétrique des lois de probabilités. Dans un premier temps, nous avons supposé que la densité inconnue f a été approchée par un mélange de base B-spline quadratique. Puis, nous avons proposé un nouvel estimateur de la densité inconnue f basé sur les fonctions B-splines quadratiques, avec deux méthodes d'estimation. La première est base sur la méthode du maximum de vraisemblance et la deuxième est basée sur la méthode d'estimation Bayésienne MAP. Ensuite, nous avons généralisé notre étude d'estimation dans le cadre du mélange et nous avons proposé un nouvel estimateur du mélange de lois inconnues basé sur les deux méthodes d'estimation adaptées. Dans un deuxième temps, nous avons traité le problème de la segmentation statistique semi supervisée des images en se basant sur le modèle de Markov caché et les fonctions B-splines. Nous avons montré l'apport de l'hybridation du modèle de Markov caché et les fonctions B-splines en segmentation statistique bayésienne semi supervisée des images. Dans un troisième temps, nous avons présenté une approche de fusion basée sur la méthode de maximum de vraisemblance, à travers l'estimation non paramétrique des probabilités, pour chaque pixel de l'image. Nous avons ensuite appliqué cette approche sur des images multi-spectrales et multi-temporelles segmentées par notre algorithme non paramétrique et non supervisé. / In this thesis we are treated the problem of nonparametric estimation probability distributions. At first, we assumed that the unknown density f was approximated by a basic mixture quadratic B-spline. Then, we proposed a new estimate of the unknown density function f based on quadratic B-splines, with two methods estimation. The first is based on the maximum likelihood method and the second is based on the Bayesian MAP estimation method. Then we have generalized our estimation study as part of the mixture and we have proposed a new estimator mixture of unknown distributions based on the adapted estimation of two methods. In a second time, we treated the problem of semi supervised statistical segmentation of images based on the hidden Markov model and the B-sline functions. We have shown the contribution of hybridization of the hidden Markov model and B-spline functions in unsupervised Bayesian statistical image segmentation. Thirdly, we presented a fusion approach based on the maximum likelihood method, through the nonparametric estimation of probabilities, for each pixel of the image. We then applied this approach to multi-spectral and multi-temporal images segmented by our nonparametric and unsupervised algorithm.

Les problèmes d'interpolation ont été largement étudiés dans la Conception Géométrique Assistée par Ordinateur. Ces problèmes consistent en la construction de courbes et de surfaces qui passent exactement par un ensemble de données. Dans ce cadre, l'objectif principal de cette thèse est de présenter des méthodes d'interpolation de données 2D et 3D au moyen de courbes Algébriques Trigonométriques à Hodographe Pythagorien (ATPH). Celles-ci sont utilisables pour la conception de modèles géométriques dans de nombreuses applications. En particulier, nous nous intéressons à la modélisation géométrique d'objets odontologiques. À cette fin, nous utilisons les courbes spatiales ATPH pour la construction de surfaces développables dans des volumes odontologiques. Initialement, nous considérons la construction de courbes planes ATPH avec continuité C² qui interpolent une séquence ordonnée de points. Nous employons deux méthodes pour résoudre ce problème et trouver la « bonne » solution. Nous étendons les courbes ATPH planes à l'espace tridimensionnel. Cette caractérisation 3D est utilisée pour résoudre le problème d'interpolation Hermite de premier ordre. Nous utilisons ces splines ATPH spatiales C¹ continues pour guider des facettes développables, qui sont déployées à l'intérieur de volumes tomodensitométriques odontologiques, afin de visualiser des informations d'intérêt pour le professionnel de santé. Cette information peut être utile dans l'évaluation clinique, diagnostic et/ou plan de traitement. / Interpolation problems have been widely studied in Computer Aided Geometric Design (CAGD). They consist in the construction of curves and surfaces that pass exactly through a given data set, such as point clouds, tangents, curvatures, lines/planes, etc. In general, these curves and surfaces are represented in a parametrized form. This representation is independent of the coordinate system, it adapts itself well to geometric transformations and the differential geometric properties of curves and surfaces are invariant under reparametrization. In this context, the main goal of this thesis is to present 2D and 3D data interpolation schemes by means of Algebraic-Trigonometric Pythagorean-Hodograph (ATPH) curves. The latter are parametric curves defined in a mixed algebraic-trigonometric space, whose hodograph satisfies a Pythagorean condition. This representation allows to analytically calculate the curve's arc-length as well as the rational-trigonometric parametrization of the offsets curves. These properties are usable for the design of geometric models in many applications including manufacturing, architectural design, shipbuilding, computer graphics, and many more. In particular, we are interested in the geometric modeling of odontological objects. To this end, we use the spatial ATPH curves for the construction of developable patches within 3D odontological volumes. This may be a useful tool for extracting information of interest along dental structures. We give an overview of how some similar interpolating problems have been addressed by the scientific community. Then in chapter 2, we consider the construction of planar C2 ATPH spline curves that interpolate an ordered sequence of points. This problem has many solutions, its number depends on the number of interpolating points. Therefore, we employ two methods to find them. Firstly, we calculate all solutions by a homotopy method. However, it is empirically observed that only one solution does not have any self-intersections. Hence, the Newton-Raphson iteration method is used to directly compute this \good" solution. Note that C2 ATPH spline curves depend on several free parameters, which allow to obtain a diversity of interpolants. Thanks to these shape parameters, the ATPH curves prove to be more exible and versatile than their polynomial counterpart, the well known Pythagorean-Hodograph (PH) quintic curves and polynomial curves in general. These parameters are optimally chosen through a minimization process of fairness measures. We design ATPH curves that closely agree with well-known trigonometric curves by adjusting the shape parameters. We extend the planar ATPH curves to the case of spatial ATPH curves in chapter 3. This characterization is given in terms of quaternions, because this allows to properly analyze their properties and simplify the calculations. We employ the spatial ATPH curves to solve the first-order Hermite interpolation problem. The obtained ATPH interpolants depend on three free angular values. As in the planar case, we optimally choose these parameters by the minimization of integral shape measures. This process is also used to calculate the C1 interpolating ATPH curves that closely approximate well-known 3D parametric curves. To illustrate this performance, we present the process for some kind of helices. In chapter 4 we then use these C1 ATPH splines for guiding developable surface patches, which are deployed within odontological computed tomography (CT) volumes, in order to visualize information of interest for the medical professional. Particularly, we construct piecewise conical surfaces along smooth ATPH curves to display information related to the anatomical structure of human jawbones. This information may be useful in clinical assessment, diagnosis and/or treatment plan. Finally, the obtained results are analyzed and conclusions are drawn in chapter 5.

The basic planar cam mechanisms and link mechanisms are widely used in industrial automatic machines. In determining the design method and design procedure for the cam-link mechanism, the basic kinematic synthesis and motion curve generation method require effective design procedure and optimization method to determine the kinematic structure of the mechanism and its kinematic performance clearly. In order to determine the result of the multi-objective optimization problem for the cam-link mechanism, the genetic algorithm is defined as the problem solver and begins this dissertation. By considering the influences of the parameters in the evolving procedure and by defining the conditions of the parameters and variables properly, the best solutions of the multi-objective optimization problem can then be solved successfully. By comparing the curves for the motion synthesis of the cam-link mechanism, the existing motion functions with their kinematic characteristics used in cam mechanisms are introduced and the rational B-spline motion function is proposed. By using the genetic algorithm to approximate the motion curves that is combined with trigonometric functions, the flexibility of the rational B-spline is demonstrated. Furthermore, to minimize different kinematic characteristics of the single-objective minimization problems, these problems are also searched by using rational B-splines with genetic algorithm for having better results. For synthesizing different structures of cam-link mechanisms, first of all is to derive the kinematics of the two planar link mechanisms and four planar cam mechanisms and then the genetic algorithm is used here to find out the minimal cam dimension with the limits of the motion curves, the pressure angles, and the radius of curvatures. Then, the kinematic synthesis problem of the function generation slider-crank mechanisms as the slider starts at the toggle position is discussed. Through the analysis finds out that when using the traditional motion functions with acceleration continuity to synthesize the slider motion, the angular acceleration of the crank cannot be continuous. To achieve the acceleration continuity of the crank motion, the curve that contains the fourth derivatives of the displacement with respect to time are set to be zeros can fulfill the continuity requirement. Then using the structural synthesis design procedure, by following the input and output relations of the link mechanisms and cam mechanisms with design constraints to select the proper structures of the mechanisms. To apply the cam-link mechanism in real application, a machine containing a slider-crank mechanism as toggle mechanism is introduced. Through the design constraints of space and motion limits to find out the possible mechanism structure and define the dimensions and then analyze the kinematics and kinetostatics of the machine. By using the genetic algorithm to solve the multi-objective optimization problem, the parameters of the rational B-spline are adjusted to have optimal kinematics and minimal kinetostatics to reduce the contact stress and to improve the fatigue life of the cam follower. Due to the existing problem of the slider-crank mechanism that contains discontinuous acceleration at the toggle position, to prove the correctness of the theoretical results, the experimental tests are measured and verified with the theoretical results in high similarity. The results show that when the slider motion curves begin at the toggle position with boundary motion constraints up to fourth or more than fourth derivatives of the displacement with respect to time that are set to be zeros, the angular accelerations of the cranks are continuous. In summary, this dissertation provides suggestions of the kinematic characteristics for the designer to design cam-link mechanisms that contain a slider-crank mechanism as the toggle mechanism and design methods for the synthesis, analysis and experimental test of the simple function generation cam-link mechanism.

In this thesis, the author studies recursIve subdivision algorithms for curves and surfaces. Several subdivision algorithms are constructed and investigated. Some graphic examples are also presented. Inspired by the Chaikin's algorithm and the Catmull-Clark's algorithm, some non-uniform schemes, the non-uniform corner cutting scheme and the recursive subdivision algorithm for non-uniform B-spline curves, are constructed and analysed. The adapted parametrization is introduced to analyse these non-uniform algorithms. In order to solve the surface interpolation problem, the Dyn-Gregory-Levin's 4-point interpolatory scheme is generalized to surfaces and the 10-point interpolatory subdivision scheme for surfaces is formulated. The so-called Butterfly Scheme, which was firstly introduced by Dyn, Gregory Levin in 1988, is just a special case of the scheme. By studying the Cross-Differences of Directional Divided Differences, a matrix approach for analysing uniform subdivision algorithms for surfaces is established and the convergence of the 10-point scheme over both uniform and non-uniform triangular networks is studied. Another algorithm, the subdivision algorithm for uniform bi-quartic B-spline surfaces over arbitrary topology is introduced and investigated. This algorithm is a generalization of Doo-Sabin's and Catmull-Clark's algorithms. It produces uniform Bi-quartic B-spline patches over uniform data. By studying the local subdivision matrix, which is a circulant, the tangent plane and curvature properties of the limit surfaces at the so-called Extraordinary Points are studied in detail.

Stuttgart, Univ., Diss., 2007.

Submitted by Daniel Santos (danielbarbedo.teto@gmail.com) on 2018-02-05T17:19:02Z No. of bitstreams: 1 Dissertacao Daniel Barbedo versao final corrigida.pdf: 1964343 bytes, checksum: 65c0f4da238b8e1285dc35e992eb7432 (MD5) / Approved for entry into archive by Vanessa Reis (vanessa.jamile@ufba.br) on 2018-02-06T10:18:48Z (GMT) No. of bitstreams: 1 Dissertacao Daniel Barbedo versao final corrigida.pdf: 1964343 bytes, checksum: 65c0f4da238b8e1285dc35e992eb7432 (MD5) / Made available in DSpace on 2018-02-06T10:18:48Z (GMT). No. of bitstreams: 1 Dissertacao Daniel Barbedo versao final corrigida.pdf: 1964343 bytes, checksum: 65c0f4da238b8e1285dc35e992eb7432 (MD5) / CAPES / Esta dissertação tem como objetivo apresentar um embasamento teórico sobre a formulação de problemas de contato mecânico com atrito para sólidos tridimensionais utilizando o método da superfície B-Spline para a suavização de superfície. Este é baseado na mecânica do contínuo. Inicialmente é apresentada uma breve descrição dos tensores da mecânica do contínuo necessários à formulação, bem como as suas relações entre si. Parte-se então para a definição da formulação da mecânica dos sólidos utilizando a forma fraca, conhecida como o Princípio dos Trabalhos Virtuais, onde a equação de equilíbrio é obtida pela equação do balanço dos momentos, utilizando-se do modelo de equação de energia de um material Neo-Hookiano, com propriedades hiperelásticas. Descreve-se o processo de linearização das funções, necessário para a obtenção da matriz de rigidez em elementos finitos, e prossegue-se para a discretização em elementos finitos utilizando os elementos hexaédricos de oito nós. Para a solução numérica do problema de contato (com e sem atrito) é apresentado o método do Lagrangiano Aumentado. Em seguida é apresentada a formulação para contato nó-superfície. É importante mencionar que nas formulações padrões do contato mecânico, as superfícies de contato são discretizadas por elementos planos, o que resulta na descontinuidade do vetor normal entre as superfícies adjacentes. Neste tipo de abordagem, para que seja possível o deslizamento do nó escravo entre uma superfície mestre e outra adjacente, existe a necessidade de utilizar três tipos de formulações para o contato mecânico, i.e., contato nó-superfície, nó-seguimento e nó-nó. O objetivo desta dissertação é utilizar uma superfície curva e suave, criada a partir da superfície mestre original, que represente as superfícies de contato. Com isto, a direção da normal passa a ser contínua, sendo necessária apenas a formulação do contato com atrito nó-superfície. As superfícies de contato são suavizadas pela superfície B-Spline. Por este motivo, é apresentada a descrição do método da construção da superfície B-Spline para tornar a superfície de contato uma superfície única e suave, eliminando assim, a necessidade da formulação do contato nó-aresta e nó-nó. Ao final do trabalho, são apresentados exemplos numéricos para evidenciar a eficiência e o desempenho da formulação proposta. / This dissertation has as objective to show theoretical background about contact mechanical problems with friction for tridimensional solids, using the B-Spline surface algorithm for surface smoothing. This is based on continuum mechanics. At first, a short descrition of continuum mechanics is presented, as it is the basis for the posterior solid element and contact B-Spline formulation. In a second moment, the virtual work principle in its weak form is shown, where the equilibrium equation is obtained through the balance of momentum, utilizing the energy density equation of a Neo-Hookean hyperelastic material. Afterwards, the linearization process is presented, which is necessary for obtaining the finite element stiffness matrix and, finally, the finite element discretization for the 8 node hexaedrical element is shown. At first a short description of continuum mechanics tensors necessary to the formulation is shown, as well as their relations between themselves. Then the definition for the weak form of solid mechanics, the virtual work principle is shown, where the equilibrium equation is obtained through the balance of momentum, utilizing the energy equation of a Neo-hookean material, with hiperelastic properties. The linearization process is presented, which is necessary for obtaining the finite element stiffness matrix, and then the finite element discretization for the 8 node hexaedrical element is shown. Regarding the numerical solution for the contact problem (with and without friction), the Augmented Lagrangean method is presented. Afterwards, the formulation for node-surface contact is shown. In the standard mechanical contact formulations, the contact surfaces are discretized by adjacent planes, resulting in discontinuity of the normal vector between adjacent surfaces. In the classical approach, to allow the sliding of a slave node over multiple master surfaces, three different contact mechanical formulations are needed, i.e., node-surfarce, node-segment and node-node contact. This work’s objective is to compose a curve and smooth surface to represent the master surface, utilizing the B-Spline surface algorithm. Achieving that, the normal vector direction is continuous, with only the node-surface contact formulation being necessary. The contact surfaces are discretized with a B-Spline surface, and for that reason, the B-Spline curve construction method is shown. At the end of the work, numerical examples are shown to evidence the efficiency and performance of the proposed solution.

Made available in DSpace on 2014-06-12T15:51:52Z (GMT). No. of bitstreams: 2 arquivo2941_1.pdf: 3552776 bytes, checksum: 6651915523db744871d183f17c632edc (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2009 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Os algoritmos evolucionários são métodos de otimização e busca global baseados em populações. Como nas populações biológicas, um algoritmo evolucionário perde diversidade, ao longo de gerações, restringindo a busca em uma região restrita do espaço de soluções e prejudicando a busca global. Em ambientes complexos, multimodais e dinâmicos, a perda de diversidade torna-se um problema ainda mais crítico, pois a busca deve ser abrangente e o algoritmo se adaptar o mais rápido possível. Um algoritmo evolucionário possui parâmetros cujos valores influenciam tanto o resultado do processo quanto a diversidade da população. Esta tese apresenta dois novos métodos de controle de parâmetros de algoritmos evolucionários, o controle adaptativo e o controle da função de distribuição de probabilidade. O objetivo desses métodos é controlar a diversidade da população de acordo com funções pré-determinadas. O processo evolucionário é, portanto, tratado como um problema de controle, cujos parâmetros do algoritmo evolucionário são as entradas de controle e a diversidade da população é a saída do processo. No método de controle adaptativo, a estratégia de controle é baseada no sistema adaptativo por modelo de referência, onde uma diversidade de referência é utilizada como modelo de comportamento para a diversidade do processo evolucionário. O segundo método tem como objetivo manter a função de distribuição de probabilidade da diversidade da população próxima de uma distribuição determinada. Assim, a distribuição da população no espaço de soluções é também indiretamente controlada. Para esse método manter um controle de baixo custo computacional, utiliza-se uma rede neural B-spline para modelar o processo evolucionário. Em problemas de controle, é necessário conhecer o modelo do processo para se elaborar uma estratégia de controle. Assim, foi proposto um novo modelo de dinâmica de populações que descreve o comportamento da frequência gênica e da diversidade de populações. Baseado nesse modelo, o processo evolucionário é formalizado matematicamente. Portanto, o método de controle adaptativo proposto utiliza esse modelo de dinâmica de populações na estratégia de controle. Os dois métodos de controle de diversidade propostos foram validados em estudos de casos. Todos os problemas utilizados tiveram características multimodais e dinâmicas, com comportamentos que variaram de uniforme, pequenas e grandes variações, a caótica. Os desempenhos dos métodos propostos foram comparadas com um algoritmo genético padrão e outros seis algoritmos evolucionários adaptativos

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This thesis deals with a nonrigid image registration for movement correction in thoracic dynamic contrast CT data. The deformation field is initialized by the analysis of disparities based on nonlinear matched filter, which defines local movement deformation. The values of control points are optimized by the Nelder-Mead method. The transformation model is based on a 4D (3D + time) free-form B-spline deformation for feature of movement distortion. The first part of the thesis briefly discusses the theory of image registration. Knowledge of this theory is necessary for understanding the remaining chapters, which describe the proposed method and its realization. The large part of this thesis is devoted to the geometrical image transformations, that is very important for the image registration. The thesis also describes a simplex method for function minimization. Three publicated methods of registration of medical 4D CT data are given. In the following chapter are individual parts of the purposed nonrigid registration including possible problems and their solution described.