FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico / CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Este trabalho tem como principal objetivo estudar as mÃtricas do funcional volume, volume mÃnimo e curvatura mÃnima em variedades compactas de dimensÃo quatro. Na primeira parte o objetivo à investigar as mÃtricas crÃticas do funcional volume sob a condiÃÃo de tais mÃtricas serem Bach-flats em uma variedade compacta com bordo ∂M. Provamos que uma mÃtrica crÃtica do funcional volume Bach-flat em uma variedade simplesmente conexa de dimensÃo quatro com bordo isomÃtrico a uma esfera padrÃo à necessariamente isomÃtrico a uma bola geodÃsica em um espaÃo forma simplesmente conexo R4, H4 ou S4. AlÃm disso, mostramos que em dimensÃo trÃs o resultado continua valido substituindo a condiÃÃo Bach-flat pela condiÃÃo mais fraca de M ter o tensor de Bach harmÃnico. Na segunda parte estudamos os invariantes geomÃtricos: volume mÃnimo e curvatura mÃnima. Em 1982, Gromov introduziu o conceito de volume mÃnimo para uma variedade suave como sendo o Ãnfimo de todos os volumes sob as mÃtricas de curvatura seccional limitada, em valor absoluto, por 1. Enquanto a curvatura mÃnima, que foi introduzido por Yun, à o menor pinching da curvatura seccional dentre as mÃtricas de volume 1. Em ambos os casos damos estimativas inferiores envolvendo alguns invariantes diferenciÃveis e topolÃgicos. Dentre elas mostraremos exemplos em que as estimativas sÃo Ãtimas. AlÃm disso, obtemos uma caracterizaÃÃo para o caso da igualdade em algumas estimativas. / This aim of this is to study the critical metrics of the volume functional, minimal volume and minimal curvature on four-dimensional compact manifolds. In the first part, we investigate Bach-flat critical metrics of the volume functional on a compact manifold M with boundary ∂M. Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form R4, H4 or S4. Moreover, we show that in dimension three the result even is true replacing the Bach-flat condition by the weaker assumption that M has divergence-free Bach tensor. In the second part we investigate the geometric invariants: minimal volume and minimal curvature. In 1982, Gromov introduced the concept of minimal volume for a smooth manifold as the greatest lower bound of the total volumes of Mn with respect to complete Riemannian metrics whose sectional curvature is bounded above in absolute value by 1. While the minimal curvature, introduced by G. Yun in 1966, is the smallest pinching of the sectional curvature among metrics of volume 1. In both cases we give below estimates to minimal volume and minimal curvature on 4-dimensional compact manifolds involving some differential and topological invariants. Among these ones, we get some sharp estimates. Moreover, we deduce characterizations for the equality case in some estimates.

Cette thèse aborde le démélange en-ligne d’images hyperspectrales acquises par un imageur pushbroom, pour la caractérisation en temps réel du matériau bois. La première partie de cette thèse propose un modèle de mélange en-ligne fondé sur la factorisation en matrices non-négatives. À partir de ce modèle, trois algorithmes pour le démélange séquentiel en-ligne, fondés respectivement sur les règles de mise à jour multiplicatives, le gradient optimal de Nesterov et l’optimisation ADMM (Alternating Direction Method of Multipliers) sont développés. Ces algorithmes sont spécialement conçus pour réaliser le démélange en temps réel, au rythme d'acquisition de l'imageur pushbroom. Afin de régulariser le problème d’estimation (généralement mal posé), deux sortes de contraintes sur les endmembers sont utilisées : une contrainte de dispersion minimale ainsi qu’une contrainte de volume minimal. Une méthode pour l’estimation automatique du paramètre de régularisation est également proposée, en reformulant le problème de démélange hyperspectral en-ligne comme un problème d’optimisation bi-objectif. Dans la seconde partie de cette thèse, nous proposons une approche permettant de gérer la variation du nombre de sources, i.e. le rang de la décomposition, au cours du traitement. Les algorithmes en-ligne préalablement développés sont ainsi modifiés, en introduisant une étape d’apprentissage d’une bibliothèque hyperspectrale, ainsi que des pénalités de parcimonie permettant de sélectionner uniquement les sources actives. Enfin, la troisième partie de ces travaux consiste en l’application de nos approches à la détection et à la classification des singularités du matériau bois. / This PhD dissertation addresses the problem of on-line unmixing of hyperspectral images acquired by a pushbroom imaging system, for real-time characterization of wood. The first part of this work proposes an on-line mixing model based on non-negative matrix factorization. Based on this model, three algorithms for on-line sequential unmixing, using multiplicative update rules, the Nesterov optimal gradient and the ADMM optimization (Alternating Direction Method of Multipliers), respectively, are developed. These algorithms are specially designed to perform the unmixing in real time, at the pushbroom imager acquisition rate. In order to regularize the estimation problem (generally ill-posed), two types of constraints on the endmembers are used: a minimum dispersion constraint and a minimum volume constraint. A method for the unsupervised estimation of the regularization parameter is also proposed, by reformulating the on-line hyperspectral unmixing problem as a bi-objective optimization. In the second part of this manuscript, we propose an approach for handling the variation in the number of sources, i.e. the rank of the decomposition, during the processing. Thus, the previously developed on-line algorithms are modified, by introducing a hyperspectral library learning stage as well as sparse constraints allowing to select only the active sources. Finally, the third part of this work consists in the application of these approaches to the detection and the classification of the singularities of wood.