Dissertations / Theses on the topic 'Methods of global Riemannian geometry'

Pas de résumé<br>In this thesis, we study the problem of registering 2D image sets and 3D point clouds under threedifferent acquisition set-ups. The first set-up assumes that the image sets are captured using 2Dcameras that are fully calibrated and coupled, or rigidly attached, with a 3D sensor. In this context,the point cloud from the 3D sensor is registered directly to the asynchronously acquired 2D images.In the second set-up, the 2D cameras are internally calibrated but uncoupled from the 3D sensor,allowing them to move independently with respect to each other. The registration for this set-up isperformed using a Structure-from-Motion reconstruction emanating from images and planar patchesrepresenting the point cloud. The proposed registration method is globally optimal and robust tooutliers. It is based on the theory Sum-of-Squares polynomials and a Branch-and-Bound algorithm.The third set-up consists of uncoupled and uncalibrated 2D cameras. The image sets from thesecameras are registered to the point cloud in a globally optimal manner using a Branch-and-Prunealgorithm. Our method is based on a Linear Matrix Inequality framework that establishes directrelationships between 2D image measurements and 3D scene voxels

Dans ce manuscrit, nous présentons un modèle à effets mixtes, présenté dans un cadre Bayésien, permettant d'estimer la progression temporelle d'un phénomène biologique à partir d'observations répétées, à valeurs dans une variété Riemannienne, et obtenues pour un individu ou groupe d'individus. La progression est modélisée par des trajectoires continues dans l'espace des observations, que l'on suppose être une variété Riemannienne. La trajectoire moyenne est définie par les effets mixtes du modèle. Pour définir les trajectoires de progression individuelles, nous avons introduit la notion de variation parallèle d'une courbe sur une variété Riemannienne. Pour chaque individu, une trajectoire individuelle est construite en considérant une variation parallèle de la trajectoire moyenne et en reparamétrisant en temps cette parallèle. Les transformations spatio-temporelles sujet-spécifiques, que sont la variation parallèle et la reparamétrisation temporelle sont définnies par les effets aléatoires du modèle et permettent de quantifier les changements de direction et vitesse à laquelle les trajectoires sont parcourues. Le cadre de la géométrie Riemannienne permet d'utiliser ce modèle générique avec n'importe quel type de données définies par des contraintes lisses. Une version stochastique de l'algorithme EM, le Monte Carlo Markov Chains Stochastic Approximation EM (MCMC-SAEM), est utilisé pour estimer les paramètres du modèle au sens du maximum a posteriori. L'utilisation du MCMC-SAEM avec un schéma numérique permettant de calculer le transport parallèle est discutée dans ce manuscrit. De plus, le modèle et le MCMC-SAEM sont validés sur des données synthétiques, ainsi qu'en grande dimension. Enfin, nous des résultats obtenus sur différents jeux de données liés à la santé<br>We propose a generic Bayesian mixed-effects model to estimate the temporal progression of a biological phenomenon from manifold-valued observations obtained at multiple time points for an individual or group of individuals. The progression is modeled by continuous trajectories in the space of measurements, which is assumed to be a Riemannian manifold. The group-average trajectory is defined by the fixed effects of the model. To define the individual trajectories, we introduced the notion of « parallel variations » of a curve on a Riemannian manifold. For each individual, the individual trajectory is constructed by considering a parallel variation of the average trajectory and reparametrizing this parallel in time. The subject specific spatiotemporal transformations, namely parallel variation and time reparametrization, are defined by the individual random effects and allow to quantify the changes in direction and pace at which the trajectories are followed. The framework of Riemannian geometry allows the model to be used with any kind of measurements with smooth constraints. A stochastic version of the Expectation-Maximization algorithm, the Monte Carlo Markov Chains Stochastic Approximation EM algorithm (MCMC-SAEM), is used to produce produce maximum a posteriori estimates of the parameters. The use of the MCMC-SAEM together with a numerical scheme for the approximation of parallel transport is discussed. In addition to this, the method is validated on synthetic data and in high-dimensional settings. We also provide experimental results obtained on health data

Neural network training algorithms have always suffered from the problem of local minima. The advent of natural gradient algorithms promised to overcome this shortcoming by finding better local minima. However, they require additional training parameters and computational overhead. By using a new formulation for the natural gradient, an algorithm is described that uses less memory and processing time than previous algorithms with comparable performance.

In this dissertation we present a generalization of Principal Component Analysis (PCA) to Riemannian manifolds called Barycentric Subspace Analysis and show some applications. The notion of barycentric subspaces has been first introduced first by X. Pennec. Since they lead to hierarchy of properly embedded linear subspaces of increasing dimension, they define a generalization of PCA on manifolds called Barycentric Subspace Analysis (BSA). We present a detailed study of the method on the sphere since it can be considered as the finite dimensional projection of a set of probability densities that have many practical applications. We also show an application of the barycentric subspace method for the study of cardiac motion in the problem of image registration, following the work of M.M. Rohé.

Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2021.<br>Considerar una G2-estructura definida en una variedad diferenciable de dimensión siete. Hay múltiples maneras de hacerla evolucionar con el objeto de anular su torsión, facilitando la búsqueda de variedades riemannianas con holonomía igual al grupo de Lie excepcional G2. Entre estas evoluciones, el así llamado flujo isométrico tiene la característica distintiva de preservar la métrica subyacente inducida por esta G2-estructura. Dicho flujo está construido a partir de la divergencia del tensor de torsión total de la G2-estructura en evolución de manera tal que sus puntos críticos son precisamente las G2-estructuras con tensor de torsión total con divergencia nula. En este trabajo se estudian dos grandes familias de G2-estructuras no cerradas no equivalentes definidas sobre grupos de Lie solubles simplemente conexos previamente analizados en la literatura y se calcula la divergencia del tensor de torsión total de las mismas, hallándose en ambos casos que ésta es idénticamente nula.<br>Take a G2-structure defined on a seven-dimensional manifold. There are many possible ways of making it evolve with the aim of making it torsion-free, easing in turn the search for Riemannian manifolds with holonomy equal to the exceptional Lie group G2. Among those evolutions, the so-called isometric flow has the distinctive feature of preserving the underlying metric induced by that G2-structure. This flow is built upon the divergence of the full torsion tensor of the flowing G2-structures in such a way that its critical points are precisely G2-structures with divergence-free full torsion tensor. In this work we study two large families of non-equivalent non-closed G2-structures defined on simply connected solvable Lie groups previously scrutinized in the literature and compute the divergence of their full torsion tensor, finding that it is identically zero in both cases.<br>Fil: Garrone, Agustín Nicolás. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.

Motivated by various higher dimensional theories in high-energy-physics and cosmology, we consider the local and global isometric embeddings of pseudo-Riemannian manifolds into manifolds of higher dimensions. We provide the necessary background in general relativity, topology and differential geometry, and present the technique for local isometric embeddings. Since an understanding of the local results is key to the development of global embeddings, we review some local existence theorems for general pseudo-Riemannian embedding spaces. In order to gain insight we recapitulate the formalism required to embed static spherically symmetric space-times into fivedimensional Einstein spaces, and explicitly treat some special cases, obtaining local and isometric embeddings for the Reissner-Nordstr¨om space-time, as well as the null geometry of the global monopole metric. We also comment on existence theorems for Euclidean embedding spaces. In a recent result, it is claimed (Katzourakis 2005a) that any analytic n-dimensional space M may be globally embedded into an Einstein space M × F (F an analytic real-valued one-dimensional field). As a corollary, it is claimed that all product spaces are Einsteinian. We demonstrate that this construction for the embedding space is in fact limited to particular types of embedded spaces. We analyze this particular construction for global embeddings into Einstein spaces, uncovering a crucial misunderstanding with regard to the form of the local embedding. We elucidate the impact of this misapprehension on the subsequent proof, and amend the given construction so that it applies to all embedded spaces as well as to embedding spaces of arbitrary curvature. This study is presented as new theorems.<br>Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2007.

Tsang, Man Ho.<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 2013.<br>Includes bibliographical references (leaves 57-60).<br>Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.<br>Abstracts also in Chinese.

Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2021.<br>Una ecuación especialmente sofisticada para evolucionar variedades casi-Kähler es el flujo de curvatura simpléctico, introducido por Streets-Tian. Los puntos fijos de este flujo, que reciben el nombre de estructuras estáticas, son objetos de gran interés y han presentado dificultades en su estudio. En dimensión 4, Streets-Tian y Kelleher probaron que estas estructuras presentan ciertas condiciones de rigidez. En este trabajo se muestra que a partir de dimensión 6 esas propiedades de rigidez ya no son válidas, y se dan los primeros ejemplos de estructuras estáticas que no son ni Kähler ni Einstein<br>A specially sophisticated equation that evolves almost-Kähler manifolds is the symplectic curvature flow, introduced by Streets-Tian. The fixed points of this flow, which are called static structures, are objects of interest whose study has presented difficulties. In dimension 4, Streets-Tian and Kelleher have proved certain conditions of rigidity that hold for these structures. We show that in dimension 6 and above, these rigidity properties are no longer valid, and we give the first examples of static structures that are not Kähler nor Einstein.<br>Fil: Molina, Camilla. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía, Física y Computación; Argentina.

Nonlinear singular partial differential equations arise naturally when studying models from such areas as Riemannian geometry, applied probability, mathematical physics and biology. The purpose of this thesis is to develop analytical methods to investigate a large class of nonlinear elliptic PDEs underlying models from physical and biological sciences. These methods advance the knowledge of qualitative properties of the solutions to equations of the form &Delta u= &fnof(x,u) where &Omega is a smooth domain in R^N (bounded or possibly unbounded) with compact (possibly empty) boundary &part&Omega. A non-negative solution of the above equation subject to the singular boundary condition u(x)&rarr &infin as dist(x,&part&Omega)&rarr 0 (if &Omega&ne R^N), or u(x)&rarr &infin as | x | &rarr &infin (if &Omega=R^N) is called a blow-up or large solution; in the latter case the solution is called an entire large solution. Issues such as existence, uniqueness and asymptotic behavior of blow-up solutions are the main questions addressed and resolved in this dissertation. The study of similar equations with homogeneous Dirichlet boundary conditions, along with that of ODEs, supplies basic tools for the theory of blow-up. The treatment is based on devices used in Nonlinear Analysis such as the maximum principle and the method of sub and super-solutions, which is one of the main tools for finding solutions to boundary value problems. The existence of blow-up solutions is examined not only for semilinear elliptic equations, but also for systems of elliptic equations in R^N and for singular mixed boundary value problems. Such a study is motivated by applications in various fields and stimulated by very recent trends in research at the international level. The influence of the nonlinear term &fnof(x,u) on the uniqueness and asymptotics of the blow-up solution is very delicate and still eludes researchers, despite a very extensive literature on the subject. This challenge is met in a general setting capable of modelling competition near the boundary (that is, 0&sdot &infin near &part &Omega), which is very suitable to applications in population dynamics. As a special feature, we develop innovative methods linking, for the first time, the topic of blow-up in PDEs with regular variation theory (or Karamata's theory) arising in applied probability. This interplay between PDEs and probability theory plays a crucial role in proving the uniqueness of the blow-up solution in a setting that removes previous restrictions imposed in the literature. Moreover, we unveil the intricate pattern of the blow-up solution near the boundary by establishing the two-term asymptotic expansion of the solution and its variation speed (in terms of Karamata's theory). The study of singular phenomena is significant because computer modelling is usually inefficient in the presence of singularities or fast oscillation of functions. Using the asymptotic methods developed by this thesis one can find the appropriate functions modelling the singular phenomenon. The research outcomes prove to be of significance through their potential applications in population dynamics, Riemannian geometry and mathematical physics.

Geometric sensor data alignment - the problem of finding the rigid transformation that correctly aligns two sets of sensor data without prior knowledge of how the data correspond - is a fundamental task in computer vision and robotics. It is inconvenient then that outliers and non-convexity are inherent to the problem and present significant challenges for alignment algorithms. Outliers are highly prevalent in sets of sensor data, particularly when the sets overlap incompletely. Despite this, many alignment objective functions are not robust to outliers, leading to erroneous alignments. In addition, alignment problems are highly non-convex, a property arising from the objective function and the transformation. While finding a local optimum may not be difficult, finding the global optimum is a hard optimisation problem. These key challenges have not been fully and jointly resolved in the existing literature, and so there is a need for robust and optimal solutions to alignment problems. Hence the objective of this thesis is to develop tractable algorithms for geometric sensor data alignment that are robust to outliers and not susceptible to spurious local optima. This thesis makes several significant contributions to the geometric alignment literature, founded on new insights into robust alignment and the geometry of transformations. Firstly, a novel discriminative sensor data representation is proposed that has better viewpoint invariance than generative models and is time and memory efficient without sacrificing model fidelity. Secondly, a novel local optimisation algorithm is developed for nD-nD geometric alignment under a robust distance measure. It manifests a wider region of convergence and a greater robustness to outliers and sampling artefacts than other local optimisation algorithms. Thirdly, the first optimal solution for 3D-3D geometric alignment with an inherently robust objective function is proposed. It outperforms other geometric alignment algorithms on challenging datasets due to its guaranteed optimality and outlier robustness, and has an efficient parallel implementation. Fourthly, the first optimal solution for 2D-3D geometric alignment with an inherently robust objective function is proposed. It outperforms existing approaches on challenging datasets, reliably finding the global optimum, and has an efficient parallel implementation. Finally, another optimal solution is developed for 2D-3D geometric alignment, using a robust surface alignment measure. Ultimately, robust and optimal methods, such as those in this thesis, are necessary to reliably find accurate solutions to geometric sensor data alignment problems.