In projective geometry, the common self-polar triangle has often been used to discuss the location relationship of two planar conics. However, there are few researches on the properties of the common self-polar triangle, especially when the two planar conics are special conics. In this thesis, the properties of the common self-polar triangle of special conics are studied and their applications to computer vision are presented. Specifically, the applications focus on the two topics of the computer vision: camera calibration and homography estimation. This thesis first studies the common self-polar triangle of two sphere images and also that the common self-polar triangle of two concentric circles, and exploits its properties to solve the problem of camera calibration. For the sphere images, by recovering the constraints on the imaged absolute conic from the vertices of the common self-polar triangles, a novel method for estimating the intrinsic parameters of a camera from an image of three spheres has been developed. For the other case of concentric circles, it is shown in this thesis that the imaged circle center and the vanishing line of the support plane can be recovered simultaneously. Furthermore, many orthogonal vanishing points can be obtained from the common self-polar triangles. Consequently, two novel calibration methods have been developed. Based on our method, one of the state-of-the-art calibration methods has been well interpreted. This thesis then studies the common self-polar triangle of two separate ellipses, and applies it to planar homography estimation. For two images of the separate ellipses, by inducing four corresponding lines from the common self-polar triangle, a homography estimation method has been developed without ambiguity. Based on these results, a special case of planar rectification with two identical circles is also studied. It is shown that given one image of the two identical circles, the vanishing line of the support plane can be recovered from the common self-polar triangle and the imaged circle points can be obtained by intersecting the vanishing line with the image of the circle. Accordingly, a novel method for estimating the rectification homography has been developed and experimental results show the feasibility of our method.

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Abstract Not Provided / Thesis / Master of Science (MSc)

<p> In this thesis we examine a generalized notion of ordinary two dimensional affine and projective geometries The first six chapters deal very generally with coordinatization methods for these geometries and a direct construction of the analytic model for the affine case. The last two chapters are concerned with a discussion of these structures viewed as topological geometries. </p> / Thesis / Doctor of Philosophy (PhD)

An interesting thing happens when one begins with the axioms of a field, but does not require the associative and commutative properties. The resulting nonassociative division ring is referred to as a ``semifield" in this paper. Semifields have intimate ties to finite projective planes. In short, a finite projective plane with certain restrictions gives rise to a semifield, and, in turn, a finite semifield can be used via a coordinate construction, to build a special finite projective plane. It is also shown that two finite semifields provide a coordinate system for isomorphic projective planes if and only if the semifields are isotopic, where isotopy is a relationship similar to but weaker than isomorphism. Before we prove those results, we explore the nature of isotopy to get a little better feel for the concept. For example, we look at isotopy for associative algebras. We will also examine a particular family of semifields and gather concrete information about solutions to linear equations and isomorphisms. / Master of Science

One of the marvelous phenomena of coordinate geometry is the equivalence of Desargues' Theorem to the presence of an underlying division ring in a projective plane. Supplementing this correspondence is the general theory of intersection theorems, which, restricted to desarguian projective planes P, corresponds precisely to the theory of integral rational identities, restricted to division rings D. The first chapter of this paper introduces projective planes, develops the concept of an intersection theorem, and expounds upon the Theorem of Desargues; the discussion culminates with a proof of the desarguian phenomenon in the second chapter. The third chapter characterizes the automorphisms of P and introduces the theory of polynomial identities; the fourth chapter expands this discussion to rational identities and cements the ``dictionary''. The last section describes a measure of complexity for these intersection theorems, and the paper concludes with a curious spawn of the correspondence. / Master of Science

Objectifs : Cette recherche vise une meilleure compréhension de la construction identitaire des frères et soeurs de personnes autistes. Elle porte sur l'évolution de la représentation de soi de ces membres de fratries à l'adolescence, et l'analyse est centrée sur l'image du corps, l'investissement narcissico-objectal et le processus de subjectivation. Une revue théorique et clinique des travaux francophones et anglo-saxons met à jour l'état des connaissances dans ce domaine clinique encore peu étudié.Méthode : Nous procédons par approche comparative en constituant un groupe témoin de fratries sans condition de handicap en parallèle au groupe d'étude clinique de fratries avec autisme, chacun constitué de 24 sujets âgés de 13 à 18 ans. Nous adoptons une méthode clinique et projective en utilisant lentretien clinique, deux tests projectifs (Rorschach et TAT), une activité d'écriture originale créée pour la recherche, deux échelles standardisées (R-CMAS : anxiété ; SEI : estime de soi). Des analyses statistiques sont proposées en complément de l'analyse des données cliniques.Résultats : Nous dégageons une spécificité des caractéristiques de la représentation de soi chez les adolescents de fratrie avec autisme qui se traduit à un triple niveau : celui de l'intégration psychique du corps en transformation ; celui de l'engagement en direction de l'autre sexué ; celui de l'élaboration de projets d'avenir personnels. Une constellation de traits particuliers apparaît plus ou moins fortement selon les sujets et nous amène à proposer des perspectives préventives et thérapeutiques tenant compte de la variabilité des fonctionnements psychiques individuels. / Objective: The aim of this research is to understand psychic functioning of siblings of children with autism. It focuses on the evolution of self image of these members of siblings in adolescence. Literature review shows that this clinical area has hardly been studied. Methods: We compare two samples: clinical group of adolescent of siblings with autism versus control group of typical adolescent, each consisting of 24 teenagers aged from 13 to 18. We adopt a clinical and projective method by using clinical interviews, projective tests (Rorschach and TAT), an original writing activity created for the research, and standardized scales (R-CMAS: anxiety; SEI: self-esteem). Statistic methods and clinical analyses are used to discuss clinical data.Results: We observe a specificity of the characteristics of self image in the clinical group. A real psychological suffering exists for adolescents of siblings with autism that can be describe by three processes: integration of changing body, evolution of object-relationship, development of personal plans for the future. A cluster of special features appears depending on the teenagers and leads us to propose preventive and therapeutic perspectives taking into account the variability of individual psychic functioning.

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.

This thesis is concerned with the theory of invariant bilinear differential pairings on parabolic geometries. It introduces the concept formally with the help of the jet bundle formalism and provides a detailed analysis. More precisely, after introducing the most important notations and definitions, we first of all give an algebraic description for pairings on homogeneous spaces and obtain a first existence theorem. Next, a classification of first order invariant bilinear differential pairings is given under exclusion of certain degenerate cases that are related to the existence of invariant linear differential operators. Furthermore, a concrete formula for a large class of invariant bilinear differential pairings of arbitrary order is given and many examples are computed. The general theory of higher order invariant bilinear differential pairings turns out to be much more intricate and a general construction is only possible under exclusion of finitely many degenerate cases whose significance in general remains elusive (although a result for projective geometry is included). The construction relies on so-called splitting operators examples of which are described for projective geometry, conformal geometry and CR geometry in the last chapter. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1339548 / Thesis (Ph.D.) - University of Adelaide, School of Mathematical Sciences, 2008

Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (leaf [106]).