We study an inverse problem which seeks to image the internal conductivity map of a body by one measurement of boundary and interior data. In our study the interior data is the magnitude of the current density induced by electrodes. Access to interior measurements has been made possible since the work of M. Joy et al. in early 1990s and couples two physical principles: electromagnetics and magnetic resonance. In 2007 Nachman et al. has shown that it is possible to recover the conductivity from the magnitude of one current density field inside. The method now known as Current Density Impedance Imaging is based on solving boundary value problems for the 1-Laplacian in an appropriate Riemann metric space. We consider two types of methods: the ones based on level sets and a variational approach, which aim to solve specific boundary value problem associated with the 1-Laplacian. We will address the Cauchy and Dirichlet problems with full and partial data, and also the Complete Electrode Model (CEM). The latter model is known to describe most accurately the voltage potential distribution in a conductive body, while taking into account the transition of current from the electrode to the body. For the CEM the problem is non-unique. We characterize the non-uniqueness, and explain which additional measurements fix the solution. Multiple numerical schemes for each of the methods are implemented to demonstrate the computational feasibility.

No description available.

No description available.

No description available.

Le but de cette thèse est triple : INÉGALITÉS DE SOBOLEV AVEC DES CONSTANTES EXPLICITES SUR DES VARIÉTÉS RIEMANNIENNES À DENSITÉ ET À BORD CONVEXE : On obtient des inégalités de Sobolev à densité, avec des constantes géométriques explicites pour des variétés à courbure de m-Bakry-Émery Ricci minorée par une constante positive et à bord convexe. Ceci permet de généraliser de nombreux résultats connus dans le cas riemannien aux variétés avec densité. Nous montrons aussi comment déduire des inégalités de Sobolev obtenues, un résultat d’isolement pour les applications f -harmoniques. Nous présenterons également une nouvelle et très simple méthode pour la preuve de l’inégalité de Moser-Trudinger-Onofri [Onofri, 1982] dans le cas du disque euclidien. / The purpose of this thesis is threefold: SOBOLEV INEQUALITIES WITH EXPLICIT CONSTANTS ON A WEIGHTED RIEMANNIAN MANIFOLD OF CONVEX BOUNDARY: We obtain weighted Sobolev inequalities with explicit geometric constants for weighted Riemannian manifolds of positive m-Bakry-Emery Ricci curvature and convex boundary. As a first application, we generalize several results of Riemannian manifolds to the weighted setting. Another application is a new isolation result for the f -harmonic maps. We also give a new and elemantry proof of the well-known Moser-Trudinger-Onofri [Onofri, 1982] inequality for the Euclidean disk.

The existence of pathwise stationary solutions of this stochastic partial differential equation (SPDE, for abbreviation) is demonstrated. In Part II, a connection between certain kind of state constrained controlled Forward-Backward Stochastic Differential Equations (FBSDEs) and Hamilton-Jacobi-Bellman equations (HJB equations) are demonstrated. The special case provides a probabilistic representation of some geometric flows, including the mean curvature flows. Part II includes also a probabilistic proof of the finite time existence of the mean curvature flows.

Nous étudions le déterminant du laplacien sur les fibrés vectoriels sur les graphes et l'utilisons, en lien avec des techniques d'analyse complexe discrète, pour comprendre des modèles de physique statistique. Nous calculons certaines constantes de réseaux, construisons des limites d'échelles d'excursions de la marche aléatoire à boucles effacées sur les surfaces, et étudions certains champs gaussiens et processus déterminantaux. / We study the determinant of the Laplacian on vector bundles on graphs and use it, combined with discrete complex analysis, to study models of statistical physics. We compute exact lattice constants, construct scaling limits for excursions of the loop-erased random walk on surfaces, and study some Gaussian fields and determinantal processes.

Neste trabalho estudamos a equação de ondas do tipo p-Laplaciano \'u IND. tt\' - \'DELTA\' IND.p u + \'(- \'DELTA\' POT. alpha\' u IND. t\' = \' [u] POT.q - 2 u, definida num domínio limitado limitado do \'R POT. n\', com 2 \' > ou = \' p < q e 0 < \' alpha\' < 1. Utilizando o método de Faedo-Galerkin provamos a existência de soluções fracas globais para dados iniciais pequenos. Para essas soluções estudamos também o decaimento polinomial da energia associada. A questão da não existência de soluções globais é considerada para o caso em que a energia inicial do sistema é negativa / In this work we study the p-Laplacian wave equation \'u IND. tt\' - \' DELTA\' IND p u + \'(- \'DELTA\' POT. \'alpha\' \' u IND. t\' = \'[u] POT. q - 2 u, defined in a bounded domain of \'R POT n\', with 2 \'> or =\' p < q and 0 < \' alpha\' < 1. By using the Faedo-Galerkin method we prove the existence of weak global solutions for small initial data. We also study the polynomial decay of the associate energy. The blow-up of solutions in finite time is considered for negative initial energy

Der Laplace-Operator auf kompakten Riemannschen Mannigfaltigkeiten besitzt eine natürliche Verallgemeinerung auf kompakte Riemannsche Orbifolds und das Spektrum des so gewonnenen Operators besteht ausschließlich aus Eigenwerten endlicher Vielfachheit. Die Feststellung, dass das Spektrum Informationen über die Geometrie einer Mannigfaltigkeit (oder, allgemeiner, einer Orbifold) enthält, begründete ein ganzes Teilgebiet der Mathematik. Es ist eine offene Frage der sogenannten Spektralgeometrie, ob eine Mannigfaltigkeit und eine singuläre Orbifold isospektral sein (d.h., dasselbe Spektrum mitsamt den Vielfachheiten der Eigenwerte besitzen) können. Angesichts diverser Obstruktionen zur Existenz eines solchen Beispiels für die bekannten Beispiele isospektraler guter Orbifolds, soll diese Arbeit die Spektralgeometrie schlechter Orbifolds erhellen. Zu diesem Zweck geben wir die ersten Beispiele für isospektrale Metriken auf schlechten Orbifolds an. Diese basieren auf bestimmten gewichteten projektiven Räumen, auf denen wir mittels einer Verallgemeinerung von Schüths Version der Torus-Methode nicht-trivial isospektrale Metriken konstruieren. / The Laplace Operator on compact Riemannian manifolds naturally generalizes to compact Riemannian orbifolds and the spectrum of the resulting operator consists only of eigenvalues with finite multiplicities. The observation that the spectrum contains information about the geometry of a manifold (and, more generally, an orbifold) gave rise to a whole field of mathematics. It is an open question of so-called spectral geometry, whether a manifold and a singular orbifold can be isospectral (i.e., have the same spectrum with the same multiplicities of the eigenvalues). Given the various obstructions to the existence of such an example for the known examples of isospectral good orbifolds, this work is an attempt to shed light on the spectral geometry of bad orbifolds by giving the first examples of isospectral Riemannian metrics on bad orbifolds. In our case these are particular fixed weighted projective spaces equipped with non-trivially isospectral metrics obtained by a generalization of Schüth''s version of the torus method.

Image segmentation is a process to identify regions of interest from digital images. Image segmentation plays an important role in medical image processing which enables a variety of clinical applications. It is also a tool to facilitate the detection of abnormalities such as cancerous lesions in the brain. Although numerous efforts in recent years have advanced this technique, no single approach solves the problem of segmentation for the large variety of image modalities existing today. Consequently, brain MRI segmentation remains a challenging task. The purpose of this thesis is to demonstrate brain MRI segmentation for delineation of tumors, ventricles and other anatomical structures using Insight Segmentation and Registration Toolkit (ITK) routines as the foundation. ITK is an open-source software system to support the Visible Human Project. Visible Human Project is the creation of complete, anatomically detailed, three-dimensional representations of the normal male and female human bodies. Currently under active development, ITK employs leading-edge segmentation and registration algorithms in two, three, and more dimensions. A goal of this thesis is to implement those algorithms to facilitate brain segmentation for a brain cancer research scientist.