Thesis (Ph. D.)--University of California, Riverside, 2009. / Includes abstract. Includes bibliographical references (leaves 74-76). Issued in print and online. Available via ProQuest Digital Dissertations.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Made available in DSpace on 2014-06-11T19:27:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-12-09Bitstream added on 2014-06-13T20:47:46Z : No. of bitstreams: 1 000734154.pdf: 2159172 bytes, checksum: b341e72df0a6a4c05a089066583aaf46 (MD5) / A história da Geometria Diferencial começa com o estudo de curvas. Noções de retas tangentes à curvas podem ser encontradas em Euclides, Arquimedes e Apolônio. Também, o Cálculo está baseado em ideias geométricas e, portanto, é natural encontrar investigações sobre curvas entre os tópicos tratados pelos pioneiros da Análise, Newton, Leibniz e Euler. Neste trabalho, serão apresentados os conceitos que fundamentam a teoria de curvas, bem como exemplos envolvendo algumas curvas clássicas, como a cicloide / The history of Differential Geometry begins with the study of curves. Notions of tangent lines to the curves can be found in Euclid, Archimedes and Apollonius. Also, the Calculus is based on geometrical ideas and therefore is natural to find researches on curves between topics treated by the pioneers of Analysis, Newton, Leibniz and Euler. In this work, the concepts that underlie the theory of curves and some examples involving classical curves are presented, as the cycloid

I investigate some algebras and calculi naturally associated with the symplectic and metric Clifford algebras. In particular, I reformulate the well known Lepage decomposition for the symplectic exterior algebra in geometrical form and present some new results relating to the simple subspaces of the decomposition. I then present an analogous decomposition for the symmetric exterior algebra with a metric. Finally, I extend this symmetric exterior algebra into a new calculus for the symmetric differential forms on a pseudo-Riemannian manifold. The importance of this calculus lies in its potential for the description of bosonic systems in Quantum Theory.

A "Geometrodynamical Analog to Electric Charge" (or "p-charge") is defined (as in the earlier paper by Unruh, [Gen. Rel. and Grav., 2, (1971), pp 27-33 ] to be the period on a p-cycle (p = 1, 2, or 3) of a p-form which is constructed out of only the Riemann tensor or its derivatives. A previously-unpublished proof by Unruh is briefly summarized which proves that no non-zero p-charges can exist on a completely unrestricted metric field. The metric field is then constrained to obey Einstein's equations for empty space, and sets of linearly-independent, purely-gravitational p-forms are analyzed to determine if p-charges can be defined under these conditions. A scheme is developed, based on the spin-tensor representation of the gravitational field, to generate complete sets of such p-forms, arid calculate their derivatives, with a symbolic-manipulation computer program. It is shown that no gravitational p-forms that are linear combinations of less than five Riemann tensors and less than nine derivatives will result in p-charges. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

published_or_final_version / Computer Science / Doctoral / Doctor of Philosophy

We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\mathfrak{p}$ in the infinitesimal isotropy representation $\mathbb{W}$ of a larger self-dual symmetric R-space $H\cdot\mathfrak{q}$. We also give a classification of projective parabolic geometries with $H\cdot\mathfrak{q}$ irreducible which, in addition to the aforementioned classical geometries, includes a geometry modelled on the Cayley plane $\mathbb{OP}^2$ and conformal geometries of various signatures. The larger R-space $H\cdot\mathfrak{q}$ severely restricts the Lie-algebraic structure of a projective parabolic geometry. In particular, by exploiting a Jordan algebra structure on $\mathbb{W}$, we obtain a $\mathbb{Z}^2$-grading on the Lie algebra of $H$ in which we have tight control over Lie brackets between various summands. This allows us to generalise known results from the classical theories. For example, which riemannian metrics are compatible with the underlying geometry is controlled by the first BGG operator associated to $\mathbb{W}$. In the final chapter, we describe projective parabolic geometries admitting a $2$-dimensional family of compatible metrics. This is the usual setting for the classical projective structures; we find that many results which hold in these settings carry over with little to no changes in the general case.

Abstract not available

The field of computer vision focuses on the goal of developing techniques to exploit and extract information from underlying data that may represent images or other multidimensional data. In particular, two well-studied problems in computer vision are the fundamental tasks of 2D image segmentation and 3D pose estimation from a 2D scene. In this thesis, we first introduce two novel methodologies that attempt to independently solve 2D image segmentation and 3D pose estimation separately. Then, by leveraging the advantages of certain techniques from each problem, we couple both tasks in a variational and non-rigid manner through a single energy functional. Thus, the three theoretical components and contributions of this thesis are as follows: Firstly, a new distribution metric for 2D image segmentation is introduced. This is employed within the geometric active contour (GAC) framework. Secondly, a novel particle filtering approach is proposed for the problem of estimating the pose of two point sets that differ by a rigid body transformation. Thirdly, the two techniques of image segmentation and pose estimation are coupled in a single energy functional for a class of 3D rigid objects. After laying the groundwork and presenting these contributions, we then turn to their applicability to real world problems such as visual tracking. In particular, we present an example where we develop a novel tracking scheme for 3-D Laser RADAR imagery. However, we should mention that the proposed contributions are solutions for general imaging problems and therefore can be applied to medical imaging problems such as extracting the prostate from MRI imagery

Our work applies techniques for blending and tightening solid shapes represented by sets. We require that the output contain one set and exclude a second set, and then we optimize the boundary separating the two sets. Working within that framework, we present mason, tightening, tight hulls, tight blends, and the medial cover, with details for implementation. Mason uses opening and closing techniques from mathematical morphology to smooth small features. By contrast, tightening uses mean curvature flow to minimize the measure of the boundary separating the opening of the interior of the closed input set from the opening of its complement, guaranteeing a mean curvature bound. The tight hull offers a significant generalization of the convex hull subject to volumetric constraints, introducing developable boundary patches connecting the constraints. Tight blends then use opening to replicate some of the behaviors from tightenings by applying tight hulls. The medial cover provides a means for adjusting the topology of a tight hull or tight blend, and it provides an implementation technique for two-dimensional polygonal inputs. Collectively, we offer applications for boundary estimation, three-dimensional solid design, blending, normal field simplification, and polygonal repair. We consequently establish the value of blending and tightening as tools for solid modeling.