D. Tech. Electrical Engineering. / In the past few years, mesh representation of images has attracted a lot of research interest due to its wide area of applications in image processing. Mesh representation showed encouraging results for image segmentation, reconstruction and compression. The present work revisits the Laplacian mesh smoothing, a technique for fairing surfaces, almost exclusively applied to 3D meshes. The report is also based on the idea that while sampling points in an image are distributed uniformly, the information in an image is not following a uniform distribution. Instead of filtering the gray levels of the image, the proposed method, called grid smoothing, filter the coordinates of the sampling points of the image.

La capture du mouvement est un défi majeur dans le cadre de la modélisation d'objets articulés. Ce problème implique la recherche de correspondances entre objets vus dans des images différentes. On propose trois approches pour résoudre ce problème basé sur des techniques de vision par ordinateur et la théorie spectrale des graphes. La première consiste à modéliser une scène 3D à l'aide d'une collection de points. On propose deux extensions de l'algorithme de Lucas-Kanade pour tracker des caractéristiques de manière efficace et pour estimer le "scene-flow". La deuxième approche basée sur la théorie spectrale des graphes cherche à établir des correspondances entre des objets représentés par des graphes. Finalement on s'intéresse au problème de segmentation qui soit cohérente dans le temps et notre approche est basée sur une méthode de clustering spectral appliquée à une séquence temporelle.

Le spectre du Laplacien sans signe a fait l'objet de beaucoup d'attention dans la communauté scientifique ces dernières années. La principale raison est l'intuition, basée sur une étude des petits graphes et sur des propriétés valides pour des graphes de toutes tailles, que plus de graphes sont déterminés par le spectre de cette matrice que par celui de la matrice d'adjacence et du Laplacien. Les travaux présentés dans cette thèse ont apporté des éléments nouveaux sur les informations contenues dans le spectre cette matrice. D'une part, on y présente des relations entre les invariants de structure et une valeur propre du Laplacien sans signe. D'autre part, on présente des familles de graphes extrêmes pour deux de ses valeurs propres, avec et sans contraintes additionnelles sur la forme de graphe. Il se trouve que ceux-ci sont très similaires à ceux obtenus dans les mêmes conditions avec les valeurs propres de la matrice d'adjacence. Cela aboutit à la définition de familles de graphes pour lesquelles, le spectre du Laplacien sans signe ou une de ses valeurs propres, le nombre de sommets et un invariant de structure suffisent à déterminer le graphe. Ces résultats, par leur similitude avec ceux de la littérature viennent confirmer l'idée que le Laplacien sans signe détermine probablement aussi bien les graphes que la matrice d'adjacence.

In this research, we study two different geometric approaches, namely, the discrete exterior calculus and differential complexes, for developing numerical schemes for linear and nonlinear elasticity. Using some ideas from discrete exterior calculus (DEC), we present a geometric discretization scheme for incompressible linearized elasticity. After characterizing the configuration manifold of volume- preserving discrete deformations, we use Hamilton’s principle on this configuration manifold. The discrete Euler-Lagrange equations are obtained without using Lagrange multipliers. The main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field. We test the efficiency and robustness of this geometric scheme using some numerical examples. In particular, we do not see any volume locking and/or checkerboarding of pressure in our numerical examples. This suggests that our choice of discrete solution spaces is compatible. On the other hand, it has been observed that the linear elastostatics complex can be used to find very efficient numerical schemes. We use some geometric techniques to obtain differential complexes for nonlinear elastostatics. In particular, by introducing stress functions for the Cauchy and the second Piola-Kirchhoff stress tensors, we show that 2D and 3D nonlinear elastostatics admit separate kinematic and kinetic complexes. We show that stress functions corresponding to the first Piola-Kirchhoff stress tensor allow us to write a complex for 3D nonlinear elastostatics that similar to the complex of 3D linear elastostatics contains both the kinematics an kinetics of motion. We study linear and nonlinear compatibility equations for curved ambient spaces and motions of surfaces in R3. We also study the relationship between the linear elastostatics complex and the de Rham complex. The geometric approach presented in this research is crucial for understanding connections between linear and nonlinear elastostatics and the Hodge Laplacian, which can enable one to convert numerical schemes of the Hodge Laplacian to those for linear and possibly nonlinear elastostatics.

Ce mémoire a pour but d'étudier les propriétés des solutions à l'équation aux valeurs propres de l'opérateur de Laplace sur le disque lorsque les valeurs propres tendent vers l'in ni. En particulier, on s'intéresse au taux de croissance des normes ponctuelle et L1. Soit D le disque unitaire et @D sa frontière (le cercle unitaire). On s'inté- resse aux solutions de l'équation aux valeurs propres f = f avec soit des conditions frontières de Dirichlet (fj@D = 0), soit des conditions frontières de Neumann ( @f @nj@D = 0 ; notons que sur le disque, la dérivée normale est simplement la dérivée par rapport à la variable radiale : @ @n = @ @r ). Les fonctions propres correspondantes sont données par : f (r; ) = fn;m(r; ) = Jn(kn;mr)(Acos(n ) + B sin(n )) (Dirichlet) fN (r; ) = fN n;m(r; ) = Jn(k0 n;mr)(Acos(n ) + B sin(n )) (Neumann) où Jn est la fonction de Bessel de premier type d'ordre n, kn;m est son m- ième zéro et k0 n;m est le m-ième zéro de sa dérivée (ici on dénote les fonctions propres pour le problème de Dirichlet par f et celles pour le problème de Neumann par fN). Dans ce cas, on obtient que le spectre SpD( ) du laplacien sur D, c'est-à-dire l'ensemble de ses valeurs propres, est donné par : SpD( ) = f : f = fg = fk2 n;m : n = 0; 1; 2; : : :m = 1; 2; : : :g (Dirichlet) SpN D( ) = f : fN = fNg = fk0 n;m 2 : n = 0; 1; 2; : : :m = 1; 2; : : :g (Neumann) En n, on impose que nos fonctions propres soient normalisées par rapport à la norme L2 sur D, c'est-à-dire : R D F2 da = 1 (à partir de maintenant on utilise F pour noter les fonctions propres normalisées et f pour les fonctions propres quelconques). Sous ces conditions, on s'intéresse à déterminer le taux de croissance de la norme L1 des fonctions propres normalisées, notée jjF jj1, selon . Il est vi important de mentionner que la norme L1 d'une fonction sur un domaine correspond au maximum de sa valeur absolue sur le domaine. Notons que dépend de deux paramètres, m et n et que la dépendance entre et la norme L1 dépendra du rapport entre leurs taux de croissance. L'étude du comportement de la norme L1 est étroitement liée à l'étude de l'ensemble E(D) qui est l'ensemble des points d'accumulation de log(jjF jj1)= log : Notre principal résultat sera de montrer que [7=36; 1=4] E(B2) [1=18; 1=4]: Le mémoire est organisé comme suit. L'introdution et les résultats principaux sont présentés au chapitre 1. Au chapitre 2, on rappelle quelques faits biens connus concernant les fonctions propres du laplacien sur le disque et sur les fonctions de Bessel. Au chapitre 3, on prouve des résultats concernant la croissance de la norme ponctuelle des fonctions propres. On montre notamment que, si m=n ! 0, alors pour tout point donné (r; ) du disque, la valeur de F (r; ) décroit exponentiellement lorsque ! 1. Au chapitre 4, on montre plusieurs résultats sur la croissance de la norme L1. Le probl ème avec conditions frontières de Neumann est discuté au chapitre 5 et on présente quelques résultats numériques au chapitre 6. Une brève discussion et un sommaire de notre travail se trouve au chapitre 7. / The goal of this master's thesis is to explore the properties of the solutions of the eigenvalue problem for the Laplace operator on a disk as the eigenvalues go to in nity. More speci cally, we study the growth rate of the pointwise and the L1 norms of the eigenfunctions. Let D be the unit disk and @D be its boundary (the unit circle). We study the solutions of the eigenvalue problem f = f with either Dirichlet boundary condition (fj@D = 0) or Neumann boundary condition ( @f @nj@D = 0; note that for the disk the normal derivative is simply the derivative with respect to the radial variable: @ @n = @ @r ). The corresponding eigenfunctions are given by: f (r; ) = fn;m(r; ) = Jn(kn;mr)(Acos(n ) + B sin(n )) (Dirichlet) fN (r; ) = fN n;m(r; ) = Jn(k0 n;mr)(Acos(n ) + B sin(n )) (Neumann) where Jn is the nth order Bessel function of the rst type, kn;m is its mth zero and k0 n;m is the mth zero of its derivative (here we denote the eigenfunctions for the Dirichlet problem by f and those for the Neumann problem by fN). The spectrum of the Laplacian on D, SpD( ), that is the set of its eigenvalues, is given by: SpD( ) = f : f = fg = fk2 n;m : n = 0; 1; 2; : : :m = 1; 2; : : :g (Dirichlet) SpN D( ) = f : fN = fNg = fk0 n;m 2 : n = 0; 1; 2; : : :m = 1; 2; : : :g (Neumann) Finally, we normalize the L2 norm of the eigenfunctions on D, namely: R D F2 da = 1 (here and further on we use the notation F for the normalized eigenfunctions and f for arbitrary eigenfunctions). Under these conditions, we study the growth rate of the L1 norm of the normalized eigenfunctions, jjF jj1, in relation to . It is important to mention that the L1 norm of a function on a given domain corresponds to the iv maximum of its absolute value on the domain. Note that depends on two parameters, m and n, and the relation between and the L1 norm depends on the regime at which m and n change as goes to in nity. Studying the behavior of the L1 norm is linked to the study of the set E(D) which is the set of accumulation points of log(jjF jj1)= log : One of our main results is that [7=36; 1=4] E(B2) [1=18; 1=4]: The thesis is organized as follows. Introduction and main results are presented in chapter 1. In chapter 2 we review some well-known facts regarding the eigenfunctions of the Laplacian on the disk and the properties of the Bessel functions. In chapter 3 we prove results on pointwise growth of eigenfunctions. In particular, we show that, if m=n ! 0, then, for any xed point (r; ) on D, the value of F (r; ) decreases exponentially as ! 1. In chapter 4 we study the growth of the L1 norm. Eigenfunctions of the Neumann problem are discussed in chapter 5. Some numerical results are presented in chapter 6. A discussion and a summary of our work could be found in chapter 7.

Analytic solutions to the static and stationary boundary value field problems relative to an arbitrary configuration of parallel cylinders are obtained by using translational addition theorems for scalar Laplacian polar functions, to express the field due to one cylinder in terms of the polar coordinates of the other cylinders such that the boundary conditions can be imposed at all the cylinder surfaces. The constants of integration in the field expressions of all the cylinders are obtained from a truncated infinite matrix equation. Translational addition theorems are available for scalar cylindrical and spherical wave functions but such theorems are not directly available for the general solution of the Laplace equation in polar coordinates. The purpose of deriving these addition theorems and applying them to field problems involving systems of cylinders is to obtain exact analytic solutions with controllable accuracies, thereby, yielding benchmark solutions to validate other approximate numerical methods.

<p>We introduce Hypoelliptic Diffusion Maps (HDM), a novel semi-supervised machine learning framework for the analysis of collections of anatomical surfaces. Triangular meshes obtained from discretizing these surfaces are high-dimensional, noisy, and unorganized, which makes it difficult to consistently extract robust geometric features for the whole collection. Traditionally, biologists put equal numbers of ``landmarks'' on each mesh, and study the ``shape space'' with this fixed number of landmarks to understand patterns of shape variation in the collection of surfaces; we propose here a correspondence-based, landmark-free approach that automates this process while maintaining morphological interpretability. Our methodology avoids explicit feature extraction and is thus related to the kernel methods, but the equivalent notion of ``kernel function'' takes value in pairwise correspondences between triangular meshes in the collection. Under the assumption that the data set is sampled from a fibre bundle, we show that the new graph Laplacian defined in the HDM framework is the discrete counterpart of a class of hypoelliptic partial differential operators.</p><p>This thesis is organized as follows: Chapter 1 is the introduction; Chapter 2 describes the correspondences between anatomical surfaces used in this research; Chapter 3 and 4 discuss the HDM framework in detail; Chapter 5 illustrates some interesting applications of this framework in geometric morphometrics.</p> / Dissertation

We establish the upper bounds for the difference between the first two eigenvalues of the relative and absolute eigenvalue problems. Relative and absolute boundary conditions are generalization of Dirichlet and Neumann boundary conditions on functions to differential forms respectively. The domains are taken to be a family of symmetric regions in Rn consisting of two cavities joined by a straight thin tube. Our operators are Hodge Laplacian operators acting on k-forms given by the formula Δ(k) = dδ+δd, where d and δ are the exterior derivatives and the codifferentials respectively. A result has been established on Dirichlet case (0-forms) by Brown, Hislop, and Martinez [2]. We use the same techniques to generalize the results on exponential decay of eigenforms, singular perturbation on domains [1], and matrix representation of the Hodge Laplacian restricted to a suitable subspace [2]. From matrix representation, we are able to find exponentially small upper bounds for the difference between the first two eigenvalues.

In 1976, Uhlenbeck used transversality theory to show that for certain families of elliptic operators, the property of having only simple eigenvalues is generic. As one application, she proved that on a closed Riemannian manifold, the eigenvalues of the Laplace-Beltrami operator Δg are all simple for a residual set of Cr metrics. In 2012, Enciso and Peralta-Salas established an analogue of Uhlenbeck's theorem for differential forms, showing that on a closed 3-manifold, there exists a residual set of Cr metrics such that the nonzero eigenvalues of the Hodge Laplacian Δg(k) on k-forms are all simple for 0 ≤ k ≤ 3. In this dissertation, we continue to address the question of whether Uhlenbeck's theorem can be extended to differential forms. In particular, we prove that for a residual set of Cr metrics, the nonzero eigenvalues of the Hodge Laplacian Δg(2) acting on coexact 2-forms on a closed 5-manifold have multiplicity 2. To prove our main result, we structure our argument around a study of the Beltrami operator *gd, which is related to the Hodge Laplacian by Δg(2) = -(*gd)2 when the operators are restricted to coexact 2-forms on a 5-manifold. We use techniques from perturbation theory to show that the Beltrami operator has only simple eigenvalues for a residual set of metrics. We further establish even eigenvalue multiplicities for the Hodge Laplacian acting on coexact k-forms in the more general setting n = 4 ℓ + 1 and k = 2 ℓ for ℓ ϵ N.

In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami operator when the Finsler metric is Riemannian. We compute explicit spectral data for some Katok-Ziller metrics. When the Finsler metric is negatively curved, we show, thanks to a result of Ancona that the Martin boundary is Hölder-homeomorphic to the visual boundary. This allow us to deduce the existence of harmonic measures and some ergodic preoperties. In the second part of this dissertation, we study Anosov flows in 3-manifolds, with leaf-spaces homeomorphic to .... When the manifold is hyperbolic, Thurston showed that the (un)stable foliations induces an "orthogonal" flow. We use this second flow to study isotopy class of periodic orbits of the Anosov flow and existence of embedded cylinders.