GEOMETRIC ANALYSIS TOOLS FOR MESH SEGMENTATION

Surface segmentation, a process which divides a surface into parts, is the basis for many surface manipulation applications which include model metamorphosis, model simplifica- tion, model retrieval, model alignment and texture mapping. This dissertation discusses novel methods for geometric surface analysis and segmentation and applications for these methods. Novel work within this dissertation includes a new 3D mesh segmentation algo- rithm which is referred to as the ridge-walking algorithm. The main benefit of this algo- rithm is that it can dynamically change the criteria it uses to identify surface parts which allows the algorithm to be adjusted to suit different types of surfaces and different segmen- tation goals. The dynamic segmentation behavior allows users to extract three different types of surface regions: (1) regions delineated by convex ridges, (2) regions delineated by concave valleys, and (3) regions delineated by both concave and convex curves. The ridge walking algorithm is quantitatively evaluated by comparing it with competing algo- rithms and human-generated segmentations. The evaluation is accompanied with a detailed geometrical analysis of a select subset of segmentation results to facilitate a better under- standing of the strengths and weaknesses of this algorithm. The ridge walking algorithm is applied to three domain-specific segmentation prob- lems. The first application uses this algorithm to partition bone fragment surfaces into three semantic parts: (1) the fracture surface, (2) the periosteal surface and (3) the articular surface. Segmentation of bone fragments is an important computational step necessary in developing quantitative methods for bone fracture analysis and for creating computational tools for virtual fracture reconstruction. The second application modifies the 3D ridge walking algorithm so that it can be applied to 2D images. In this case, the 2D image is modeled as a Monge patch and principal curvatures of the intensity surface are computed iv for each image pixel. These principal curvatures are then used by ridge walking algorithm to segment the image into meaningful parts. The third application uses the ridge walking algorithm to facilitate analysis of virtual 3D terrain models. Specifically, the algorithm is integrated as a part of a larger software system designed to enable users to browse, visualize and analyze 3D geometric data generated by NASA’s Mars Exploratory Rovers Spirit and Opportunity. In this context, the ridge walking algorithm is used to identify surface features such as rocks in the terrain models

Local Regularity of the Complex Monge-Ampere Equation

In this thesis, we present a self-contained account of the current development in the local regularity theory of the complex Monge-Ampere equation through the modern fully-nonlinear PDE point of view. We have apply the modern elliptic techniques to establish new local regularity results. These includes: regularity of small perturbed solutions, Holder regularity of the Hessian of the W^{2,p} solutions and a Liouville-type theorem

Gauduchon metrics with prescribed volume form

We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.Comment: 29 pages. Final version to appear in Acta Mat

Integral Geometry and its Applications

In recent years there has been a series of striking developments in modern integral geometry which has, in particular, lead to the discovery of new relations to several branches of pure and applied mathematics. A number of examples were presented at this meeting, e.g. the work of Bernig, Solanes, and Fu on kinematic formulas on complex projective and complex hyperbolic spaces, that of Schneider and Vedel Jensen on tensor valuations and a series of results on convex body valued valuations by Abardia, Ludwig, Parapatits, and Wannerer

Logarithmically-concave moment measures I

We discuss a certain Riemannian metric, related to the toric Kahler-Einstein equation, that is associated in a linearly-invariant manner with a given log-concave measure in R^n. We use this metric in order to bound the second derivatives of the solution to the toric Kahler-Einstein equation, and in order to obtain spectral-gap estimates similar to those of Payne and Weinberger.Comment: 27 page

Affine deformations of quasi-divisible convex cones

For any subgroup of $\mathrm{SL}(3,\mathbb{R})\ltimes\mathbb{R}^3$ obtained by adding a translation part to a subgroup of $\mathrm{SL}(3,\mathbb{R})$ which is the fundamental group of a finite-volume convex projective surface, we first show that under a natural condition on the translation parts of parabolic elements, the affine action of the group on $\mathbb{R}^3$ has convex domains of discontinuity that are regular in a certain sense, generalizing a result of Mess for globally hyperbolic flat spacetimes. We then classify all these domains and show that the quotient of each of them is an affine manifold foliated by convex surfaces with constant affine Gaussian curvature. The proof is based on a correspondence between the geometry of an affine space endowed with a convex cone and the geometry of a convex tube domain. As an independent result, we show that the moduli space of such groups is a vector bundle over the moduli space of finite-volume convex projective structures, with rank equaling the dimension of the Teichm\"uller space.Comment: 37 pages, 6 figures. Comments welcom

Stability of complex hyperbolic space under curvature-normalized Ricci flow

Using the maximal regularity theory for quasilinear parabolic systems, we prove two stability results of complex hyperbolic space under the curvature-normalized Ricci flow in complex dimensions two and higher. The first result is on a closed manifold. The second result is on a complete noncompact manifold. To prove both results, we fully analyze the structure of the Lichnerowicz Laplacian on complex hyperbolic space. To prove the second result, we also define suitably weighted little H\"{o}lder spaces on a complete noncompact manifold and establish their interpolation properties.Comment: Some typos in version 2 are correcte

Around stability for functional inequalities

Les inégalités fonctionnelles sont des inégalités qui encodent beaucoup d'information, tant de nature probabiliste (concentration de la mesure), qu'analytique (théorie spectrale des opérateurs) ou encore géométrique (profil isopérimétrique). L'inégalité de Poincaré en est un exemple fondamental. Dans cette thèse, nous obtenons des résultats de stabilité dans le cadre d'hypothèses de normalisation de moments, ainsi que dans le cadre de conditions de courbure-dimension. Un résultat de stabilité est une façon de quantifier la différence entre deux situations dans lesquelles les mêmes inégalités fonctionnelles sont presque vérifiées. Les résultats de stabilité obtenus dans cette thèse sont en particulier basés sur la méthode de Stein, qui est une méthode en plein développement ces dernières années, provenant du domaine des statistiques et permettant d'établir des estimations quantitatives sur des résultats de convergence. Par ailleurs, une partie de cette thèse est consacrée à l'étude des constantes optimales des inégalités de Bobkov, qui sont des inégalités fonctionnelles à caractère isopérimétrique.Functional inequalities are inequalities that encode a lot of information, both of a probabilistic (the concentration of measure phenomenon), analytical (the spectral theory of operators) and geometric (isoperimetric profile) nature. The Poincaré inequality is a fundamental example. In this thesis, we obtain stability results under moment normalisation assumptions, as well as under curvature-dimension conditions. A stability result is a way to quantify the difference between two situations where almost the same functional inequalities are verified. The stability results obtained in this thesis are in particular based on the Stein method, which is a method in full development in recent years, coming from the field of statistics and allowing to establish quantitative estimates on convergence results. In addition, a part of this thesis is devoted to the study of the optimal constants of Bobkov inequalities, which are functional inequalities of isoperimetric character