Ongoing developments in the field of medical imaging modalities have pushed the frontiers of modern medicine and biomedical engineering, prompting the need for new applications to improve diagnosis, treatment and prevention of diseases. Biomedical data visualization and modeling rely predominately on manual processing and utilization of voxel and facet based homogeneous models. Biological structures are naturally heterogeneous and in order to accurately design and biomimic biological structures, properties such as chemical composition, size and shape of biological constituents need to be incorporated in the computational biological models. Our proposed approach involves generating a density point cloud based on the intensity variations in a medical image slice, to capture tissue density variations through point cloud densities. The density point cloud is ordered and approximated with a set of cross-sectional least-squares B-Spline curves, based on which a skinned B-Spline surface is generated. The aim of this method is to capture and accurately represent density variations within the medical image data with a lofted surface function. The fitted B-Spline surface is sampled at uniformly distributed parameters, and our preliminary results indicate that the bio-CAD model preserves the density variations of the original image based point cloud. The resultant surface can thus be visualized by mapping the density in the parametric domain into color in pixel domain. The B-Spline function produced from each image slice can be used for medical visualization and heterogeneous tissue modeling. The process can be repeated for each slice in the medical dataset to produce heterogeneous B-Spline volumes. The emphasis of this research is placed on accuracy and shape fidelity needed for medical operations.

迴歸分析中，若變數間有非線性(nonlinear)的關係，此時我們可以用B-spline線性迴歸，一種無母數的方法，建立模型。Penalized spline是B-spline方法的一種改良，其想法是增加一懲罰項，避免估計函數時出現過度配適的問題。本文中，考慮三種方法:(a) Marginal Mixed Model approach, (b) Conditional Mixed Model approach, (c) 貝氏方法建立信賴帶，其中，我們對第一二種方法內的估計式作了一點調整，另外，懲罰項中的平滑參數也是我們考慮的問題。我們發現平滑參數確實會影響信賴帶，所以我們使用cross-validation來選取平滑參數。在調整的cross-validation下，Marginal Mixed Model的信賴帶估計不平滑的函數效果較好，Conditional Mixed Model的信賴帶估計平滑函數的效果較好，貝氏的信賴帶估計函數效果較差。 / In regression analysis, we can use B-spline to estimate regression function nonparametrically when the regression function is nonlinear. Penalized splines have been proposed to improve the performance of B-splines by including a penalty term to prevent over-fitting. In this article, we compare confidence bands constructed by three estimation methods: (a) Marginal Mixed Model approach, (b) Conditional Mixed Model approach, and (c) Bayesian approach. We modify the first two methods slightly. In addition, the selection of smoothing parameter of penalization is considered. We found that the smoothing parameter affects confidence bands a lot, so we use cross-validation to choose the smoothing parameter. Finally, based on the restricted cross-validation, Marginal Mixed Model performs better for less smooth regression functions, Conditional Mixed Model performs better for smooth regression functions and Bayesian approach performs badly.

Dans cette thèse, nous nous intéressons à développer des outils mathématiques, afin de traiter une gamme des signaux biologiques et économiques. En premier lieu, nous proposons la transformée Dynalet, considérée comme une alternative, pour des signaux de relaxation sans symétrie interne, à la transformée de Fourier et à la transformée ondelette. L'applicabilité de cette nouvelle approximation est illustrée sur des données réelles. Ensuite, nous corrigeons la ligne de base des signaux biologiques spectrométriques, à l'aide d'une régression expectile pénalisée, qui, sur les applications proposées, est plus performante qu'une régression quantile. Puis, afin d'éliminer le bruit blanc, nous adaptons aux signaux spectrométriques une nouvelle approche combinant ondelette, seuillage doux et composants PLS. Pour terminer, comme les signaux peuvent être considérés comme des données fonctionnelles, d'une part, nous développons une vraisemblance locale fonctionnelle dont le but est d'effectuer une classification supervisée des courbes, et, d'autre part, nous estimons l'opérateur de régression pour une réponse scalaire positive non nulle, par minimisation de l'erreur quadratique moyenne relative. De plus, les lois asymptotiques de notre estimateur sont établies et son efficacité est illustrée sur des données simulées et sur des données spectroscopiques et économiques. / In this thesis, we focus on developing mathematical tools to treat a range of biological and economic signals. First, we propose the Dynalet transform for non-symmetrical biological relaxation signals. This transform is considered as an alternative to the Fourier transform and the wavelet transform. The applicability of the new approximation approach is illustrated on real data. Then, for spectrometric biological signals, we correct the baseline using a penalized expectile regression. Thus, the proposed applications show that our proposed regression is more efficient than the quantile regression. Then to remove random noise, we adapt to spectrometric data a new denoising method that combine wavelets, soft thresholding rule and PLS components. Finally, note that the biological signals may be often regarded as functional data. On one hand, we develop a functional local likelihood aiming to perform a supervised classification of curves. On the other hand, we estimate the regression operator with positive responses, by minimizing the mean squared relative error. Moreover, The asymptotic distributions of our estimator are established and their efficiency is illustrated on a simulation study and on a spectroscopic and economic data set.

In this work we study the curvature visualization problem on surfaces implicitly defined by functions f: [0,1]³ → [0,1], using the ray casting technique. As we usually know only sampled values of f, we study the tricubic interpolation method to compute second order derivatives accurately. This work's implementation was designed as modules to the framework for volume rendering and image processing named Voreen, that uses the processing capability of graphics cards to improve the rendering tasks. / Fundação de Amparo a Pesquisa do Estado de Alagoas / Neste trabalho, estudaremos a visualização da curvatura de superfícies definidas implicitamente por funções do tipo f:[0,1]³ [0,1], usando a técnica de lançamento de raios (ray casting). Como em geral conhecemos apenas valores amostrados de f, estudaremos um método de interpolação tricúbica, a fim de calcular as derivadas de segunda ordem precisamente. A implementação computacional deste trabalho foi desenvolvida na forma de módulos do framework de visualização e processamento de imagens Voreen, o qual se beneficia do poder de processamento das placas gráficas atuais para acelerar o processo de visualização.

Nous étudions la régression bayésienne sous contraintes de régularité et de forme. Pour cela,on considère une base de B-spline pour obtenir une courbe lisse et nous démontrons que la forme d'une spline engendrée par une base de B-spline est contrôlée par un ensemble de points de contrôle qui ne sont pas situés sur la courbe de la spline. On propose différents types de contraintes de forme (monotonie, unimodalité, convexité, etc). Ces contraintes sont prises en compte grâce à la loi a priori. L'inférence bayésienne a permis de dériver la distribution posteriori sous forme explicite à une constante près. En utilisant un algorithme hybride de type Metropolis-Hastings avec une étape de Gibbs, on propose des simulations suivant la distribution a posteriori tronquée. Nous estimons la fonction de régression par le mode a posteriori. Un algorithme de type recuit simulé a permis de calculer le mode a posteriori. La convergence des algorithmes de simulations et du calcul de l'estimateur est prouvée. En particulier, quand les noeuds des B-splines sont variables, l'analyse bayésienne de la régression sous contrainte devient complexe. On propose des schémas de simulations originaux permettant de générer suivant la loi a posteriori lorsque la densité tronquée des coefficients de régression prend des dimensions variables. / We investigate the Bayesian regression under shape and smoothness constraints. We first elicita Bayesian method for regression under shape restrictions and smoothness conditions. Theregression function is built from B-spline basis that controls its regularity. Then we show thatits shape can be controlled simply from its coefficients in the B-spline basis. This is achievedthrough the control polygon whose definition and some properties are given in this article.The regression function is estimated by the posterior mode. This mode is calculated by asimulated annealing algorithm which allows to take into account the constraints of form inthe proposal distribution. A credible interval is obtained from simulations using Metropolis-Hastings algorithm with the same proposal distribution as the simulated annealing algorithm.The convergence of algorithms for simulations and calculation of the estimator is proved. Inparticular, in the case of Bayesian regression under constraints and with free knots, Bayesiananalysis becomes complex. we propose original simulation schemes which allows to simulatefrom the truncated posterior distribution with free dimension.

The development of a three dimensional B-Spline based method, which is suitable for the steady-state potential flow analysis of free surface piercing bodies in hydrodynamics, is presented. The method requires the B-Spline or Non Uniform Rational B-Spline (NURBS) representation of the body as an input. In order to solve for the unknown potential, the source surface, both for the body as well as the free surface, is represented by NURBS surfaces. The method does not require the body surface to be discritized into flat panels. Therefore, instead of a mere panel approximation, the exact body geometry is utilized for the computation. The technique does not use a free surface Green's function, which already satisfies the linear free surface boundary conditions, but uses a separate source patch for the free surface. By eliminating the use of a free surface Green's function, the method can be extended to considering non-linear free surface conditions, thus providing the possibility for wave resistance calculations. The method is first applied to the double body flow problem around a sphere and a Wigley hull. Some comparisons are made with exact solutions to validate the accuracy of the method. Results of linear free surface conditions are then presented.

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.

Clinico-genomic modeling refers to the statistical analysis that incorporates both clinical data such as medical test results, demographic information and genomic data such as gene expression profiles. It is an emerging research area in biomedical science and has been shown to be able to extend our understanding of complex diseases. We describe a general statistical modeling strategy for the integrated analysis of clinical and genomic data in which the clinical data are longitudinal observations. Our modeling strategy is aimed at the identification of disease-associated genes and it consists of two stages. In the first stage, we propose a hierarchical B spline model to estimate the disease severity trajectory based on the clinical variables. This disease severity trajectory is a functional summary of the disease progression. We can extract any characteristics of interest from the trajectory. In the second stage, combinations of the extracted characteristics are included in the gene-wise linear model to detect the genes that are responsible for variations in the disease progression. We illustrate our modeling approach in the context of two biomedical studies of complex diseases: tuberculosis (Tb) and colitis-associated carcinoma. The animal experimental subjects were measured longitudinally for clinical information and biological samples were extracted at the final points of the subjects to determine the gene expression profiles. Our results demonstrate that the incorporation of the longitudinal clinical data increases the value of information extracted from the expression profiles and contributes to the identification of predictive biomarkers. / acase@tulane.edu

這篇論文提出了比較兩組資料曲線在對齊後是否有相似外形的分析方法。在 functional data analysis 中，可能會有多條曲線具有相同外形但是時間轉換卻不一樣的情形。這篇論文檢定了兩組資料曲線在對齊後是否有相似外形，論文中並提出一個檢定統計量，再藉由模擬得到檢定的 p-value 和檢定力。 / This thesis proposed an analysis comparing whether the shape function for two groups of curves are similar after alignment. In functional data analysis, it is common to have curves of the same pattern but with variation in time. The common pattern can be characterized by a shape function. The problem considered in this thesis is to test whether the shape functions for two groups of curves are essentially the same. A test statistic is proposed and the p-value is obtained via simulation. Simulation results indicate that the test performs well.

Medical imaging technologies offer the benefits of faster and accurate diagnosis. When the medical imaging combined with the digitization, they offer the advantage of permanent storage and fast transmission to any geographical location. However, there is a need for efficient compression algorithms that alleviate the taxing burden of both large storage space and transmission bandwidth requirements. The Perceptually Lossless Medical Image Coder is a new image compression technique. It provides a solution to challenge of delivering clinically critical information in the shortest time possible. It embeds the visual pruning into the JPEG 2000 coding framework to achieve the optimal compression without losing the visual integrity of medical images. However, the performance of the PLMIC under certain medical image operation is still unknown. In this thesis, we investigate the performance of the PLMIC by applying linear, quadratic and cubic standard and centered B-spline interpolation filters. In order to evaluate the visual performance, a subjective assessment consisting of 30 medical images and 6 image processing experts was conducted. The perceptually lossless medical image coder was compared to the state-of-the-art JPEG-LS compliant LOCO and NLOCO image coders. The results have shown overall, there were no perceivable differences of statistical significance when the medical images were enlarged by a factor of 2. The findings of the thesis may help the researchers to further improve the coder. Additionally, it may also notify the radiologists the performance of the PLMIC coder to help them with correct diagnosis.