Abelian Projective Planes of Square Order

We prove that, under certain conditions, multipliers of an abelian projective plane of square order have odd order modulo v*, where v* is the exponent of the underlying Singer group. As a consequence, we are able to establish the non-existence of an infinite number of abelian projective planes of square order

Constrained camera motion estimation and 3D reconstruction

The creation of virtual content from visual data is a tedious task which requires a high amount of skill and expertise. Although the majority of consumers is in possession of multiple imaging devices that would enable them to perform this task in principle, the processing techniques and tools are still intended for the use by trained experts. As more and more capable hardware becomes available, there is a growing need among consumers and professionals alike for new flexible and reliable tools that reduce the amount of time and effort required to create high-quality content. This thesis describes advances of the state of the art in three areas of computer vision: camera motion estimation, probabilistic 3D reconstruction, and template fitting. First, a new camera model geared towards stereoscopic input data is introduced, which is subsequently developed into a generalized framework for constrained camera motion estimation. A probabilistic reconstruction method for 3D line segments is then described, which takes global connectivity constraints into account. Finally, a new framework for symmetry-aware template fitting is presented, which allows the creation of high-quality models from low-quality input 3D scans. Evaluations with a broad range of challenging synthetic and real-world data sets demonstrate that the new constrained camera motion estimation methods provide improved accuracy and flexibility, and that the new constrained 3D reconstruction methods improve the current state of the art.Die Erzeugung virtueller Inhalte aus visuellem Datenmaterial ist langwierig und erfordert viel Geschick und Sachkenntnis. Obwohl der Großteil der Konsumenten mehrere Bildgebungsgeräte besitzt, die es ihm im Prinzip erlauben würden, dies durchzuführen, sind die Techniken und Werkzeuge noch immer für den Einsatz durch ausgebildete Fachleute gedacht. Da immer leistungsfähigere Hardware zur Verfügung steht, gibt es sowohl bei Konsumenten als auch bei Fachleuten eine wachsende Nachfrage nach neuen flexiblen und verlässlichen Werkzeugen, die die Erzeugung von qualitativ hochwertigen Inhalten vereinfachen. In der vorliegenden Arbeit werden Erweiterungen des Stands der Technik in den folgenden drei Bereichen der Bildverarbeitung beschrieben: Kamerabewegungsschätzung, wahrscheinlichkeitstheoretische 3D-Rekonstruktion und Template-Fitting. Zuerst wird ein neues Kameramodell vorgestellt, das für die Verarbeitung von stereoskopischen Eingabedaten ausgelegt ist. Dieses Modell wird in der Folge in eine generalisierte Methode zur Kamerabewegungsschätzung unter Nebenbedingungen erweitert. Anschließend wird ein wahrscheinlichkeitstheoretisches Verfahren zur Rekonstruktion von 3D-Liniensegmenten beschrieben, das globale Verbindungen als Nebenbedingungen berücksichtigt. Schließlich wird eine neue Methode zum Fitting eines Template-Modells präsentiert, bei der die Berücksichtigung der Symmetriestruktur des Templates die Erzeugung von Modellen hoher Qualität aus 3D-Eingabedaten niedriger Qualität erlaubt. Evaluierungen mit einem breiten Spektrum an anspruchsvollen synthetischen und realen Datensätzen zeigen, dass die neuen Methoden zur Kamerabewegungsschätzung unter Nebenbedingungen höhere Genauigkeit und mehr Flexibilität ermöglichen, und dass die neuen Methoden zur 3D-Rekonstruktion unter Nebenbedingungen den Stand der Technik erweitern

Affine multi-view modelling for close range object measurement

In photogrammetry, sensor modelling with 3D point estimation is a fundamental topic of research. Perspective frame cameras offer the mathematical basis for close range modelling approaches. The norm is to employ robust bundle adjustments for simultaneous parameter estimation and 3D object measurement. In 2D to 3D modelling strategies image resolution, scale, sampling and geometric distortion are prior factors. Non-conventional image geometries that implement uncalibrated cameras are established in computer vision approaches; these aim for fast solutions at the expense of precision. The projective camera is defined in homogeneous terms and linear algorithms are employed. An attractive sensor model disembodied from projective distortions is the affine. Affine modelling has been studied in the contexts of geometry recovery, feature detection and texturing in vision, however multi-view approaches for precise object measurement are not yet widely available. This project investigates affine multi-view modelling from a photogrammetric standpoint. A new affine bundle adjustment system has been developed for point-based data observed in close range image networks. The system allows calibration, orientation and 3D point estimation. It is processed as a least squares solution with high redundancy providing statistical analysis. Starting values are recovered from a combination of implicit perspective and explicit affine approaches. System development focuses on retrieval of orientation parameters, 3D point coordinates and internal calibration with definition of system datum, sensor scale and radial lens distortion. Algorithm development is supported with method description by simulation. Initialization and implementation are evaluated with the statistical indicators, algorithm convergence and correlation of parameters. Object space is assessed with evaluation of the 3D point correlation coefficients and error ellipsoids. Sensor scale is checked with comparison of camera systems utilizing quality and accuracy metrics. For independent method evaluation, testing is implemented over a perspective bundle adjustment tool with similar indicators. Test datasets are initialized from precise reference image networks. Real affine image networks are acquired with an optical system (~1M pixel CCD cameras with 0.16x telecentric lens). Analysis of tests ascertains that the affine method results in an RMS image misclosure at a sub-pixel level and precisions of a few tenths of microns in object space

Twistor actions for gauge theory and gravity

This is a review of recent developments in the study of perturbative gauge theory and gravity using action functionals on twistor space. It is intended to provide a user-friendly introduction to twistor actions, geared towards researchers or graduate students interested in learning something about the utility, prospects, and shortcomings of this approach. For those already familiar with the twistor approach, it should provide a condensed overview of the literature as well as several novel results of potential interest. This work is based primarily upon the author's D.Phil. thesis. We first consider four-dimensional, maximally supersymmetric Yang-Mills theory as a gauge theory in twistor space. We focus on the perturbation theory associated to this action, which in an axial gauge leads to the MHV formalism. This allows us to efficiently compute scattering amplitudes at tree-level (and beyond) in twistor space. Other gauge theory observables such as local operators and null polygonal Wilson loops can also be formulated twistorially, leading to proofs for several correspondences between correlation functions and Wilson loops, as well as a recursive formula for computing mixed Wilson loop / local operator correlators. We then apply the twistor action approach to general relativity, using the on-shell equivalence between conformal and Einstein gravity. This can be extended to N=4 supersymmetry. The perturbation theory of the twistor action leads to formulae for the MHV amplitude with and without cosmological constant, yields a candidate for the Einstein twistor action, and induces a MHV formalism on twistor space. Appendices include discussion of super-connections and Coulomb branch regularization on twistor space.Comment: 178 pages, 30 figures. Review based on the author's D.Phil. thesis. v2: references adde

On Algebraic Singularities, Finite Graphs and D-Brane Gauge Theories: A String Theoretic Perspective

In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes. We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NS-NS B-field, as well as the T-duals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay's Correspondence will also be considered. The present work is a transcription of excerpts from the first three volumes of the author's PhD thesis which was written under the direction of Prof. A. Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student.Comment: 513 pages, 71 figs, Edited Excerpts from the first 3 volumes of the author's PhD Thesi

On algebraic singularities, finite graphs and D-brane gauge theories: A String theoretic perspective

In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes. We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singularities, notably Gorenstein quotients and toric singularities. Attention will be paid to the general methodology of constructing gauge theories for these singular backgrounds, with and without the presence of the NS-NS B-field, as well as the T-duals to brane setups and branes wrapping cycles in the mirror geometry. Applications of such diverse and elegant mathematics as crepant resolution of algebraic singularities, representation of finite groups and finite graphs, modular invariants of affine Lie algebras, etc. will naturally arise. Various viewpoints and generalisations of McKay's Correspondence will also be considered. The present work is a transcription of excerpts from the first three volumes of the author's PhD thesis which was written under the direction of Prof. A. Hanany - to whom he is much indebted - at the Centre for Theoretical Physics of MIT, and which, at the suggestion of friends, he posts to the ArXiv pro hac vice; it is his sincerest wish that the ensuing pages might be of some small use to the beginning student

Binokulare Eigenbewegungsschätzung für Fahrerassistenzanwendungen

Driving can be dangerous. Humans become inattentive when performing a monotonous task like driving. Also the risk implied while multi-tasking, like using the cellular phone while driving, can break the concentration of the driver and increase the risk of accidents. Others factors like exhaustion, nervousness and excitement affect the performance of the driver and the response time. Consequently, car manufacturers have developed systems in the last decades which assist the driver under various circumstances. These systems are called driver assistance systems. Driver assistance systems are meant to support the task of driving, and the field of action varies from alerting the driver, with acoustical or optical warnings, to taking control of the car, such as keeping the vehicle in the traffic lane until the driver resumes control. For such a purpose, the vehicle is equipped with on-board sensors which allow the perception of the environment and/or the state of the vehicle. Cameras are sensors which extract useful information about the visual appearance of the environment. Additionally, a binocular system allows the extraction of 3D information. One of the main requirements for most camera-based driver assistance systems is the accurate knowledge of the motion of the vehicle. Some sources of information, like velocimeters and GPS, are of common use in vehicles today. Nevertheless, the resolution and accuracy usually achieved with these systems are not enough for many real-time applications. The computation of ego-motion from sequences of stereo images for the implementation of driving intelligent systems, like autonomous navigation or collision avoidance, constitutes the core of this thesis. This dissertation proposes a framework for the simultaneous computation of the 6 degrees of freedom of ego-motion (rotation and translation in 3D Euclidean space), the estimation of the scene structure and the detection and estimation of independently moving objects. The input is exclusively provided by a binocular system and the framework does not call for any data acquisition strategy, i.e. the stereo images are just processed as they are provided. Stereo allows one to establish correspondences between left and right images, estimating 3D points of the environment via triangulation. Likewise, feature tracking establishes correspondences between the images acquired at different time instances. When both are used together for a large number of points, the result is a set of clouds of 3D points with point-to-point correspondences between clouds. The apparent motion of the 3D points between consecutive frames is caused by a variety of reasons. The most dominant motion for most of the points in the clouds is caused by the ego-motion of the vehicle; as the vehicle moves and images are acquired, the relative position of the world points with respect to the vehicle changes. Motion is also caused by objects moving in the environment. They move independently of the vehicle motion, so the observed motion for these points is the sum of the ego-vehicle motion and the independent motion of the object. A third reason, and of paramount importance in vision applications, is caused by correspondence problems, i.e. the incorrect spatial or temporal assignment of the point-to-point correspondence. Furthermore, all the points in the clouds are actually noisy measurements of the real unknown 3D points of the environment. Solving ego-motion and scene structure from the clouds of points requires some previous analysis of the noise involved in the imaging process, and how it propagates as the data is processed. Therefore, this dissertation analyzes the noise properties of the 3D points obtained through stereo triangulation. This leads to the detection of a bias in the estimation of 3D position, which is corrected with a reformulation of the projection equation. Ego-motion is obtained by finding the rotation and translation between the two clouds of points. This problem is known as absolute orientation, and many solutions based on least squares have been proposed in the literature. This thesis reviews the available closed form solutions to the problem. The proposed framework is divided in three main blocks: 1) stereo and feature tracking computation, 2) ego-motion estimation and 3) estimation of 3D point position and 3D velocity. The first block solves the correspondence problem providing the clouds of points as output. No special implementation of this block is required in this thesis. The ego-motion block computes the motion of the cameras by finding the absolute orientation between the clouds of static points in the environment. Since the cloud of points might contain independently moving objects and outliers generated by false correspondences, the direct computation of the least squares might lead to an erroneous solution. The first contribution of this thesis is an effective rejection rule that detects outliers based on the distance between predicted and measured quantities, and reduces the effects of noisy measurement by assigning appropriate weights to the data. This method is called Smoothness Motion Constraint (SMC). The ego-motion of the camera between two frames is obtained finding the absolute orientation between consecutive clouds of weighted 3D points. The complete ego-motion since initialization is achieved concatenating the individual motion estimates. This leads to a super-linear propagation of the error, since noise is integrated. A second contribution of this dissertation is a predictor/corrector iterative method, which integrates the clouds of 3D points of multiple time instances for the computation of ego-motion. The presented method considerably reduces the accumulation of errors in the estimated ego-position of the camera. Another contribution of this dissertation is a method which recursively estimates the 3D world position of a point and its velocity; by fusing stereo, feature tracking and the estimated ego-motion in a Kalman Filter system. An improved estimation of point position is obtained this way, which is used in the subsequent system cycle resulting in an improved computation of ego-motion. The general contribution of this dissertation is a single framework for the real time computation of scene structure, independently moving objects and ego-motion for automotive applications.Autofahren kann gefährlich sein. Die Fahrleistung wird durch die physischen und psychischen Grenzen des Fahrers und durch externe Faktoren wie das Wetter beeinflusst. Fahrerassistenzsysteme erhöhen den Fahrkomfort und unterstützen den Fahrer, um die Anzahl an Unfällen zu verringern. Fahrerassistenzsysteme unterstützen den Fahrer durch Warnungen mit optischen oder akustischen Signalen bis hin zur Übernahme der Kontrolle über das Auto durch das System. Eine der Hauptvoraussetzungen für die meisten Fahrerassistenzsysteme ist die akkurate Kenntnis der Bewegung des eigenen Fahrzeugs. Heutzutage verfügt man über verschiedene Sensoren, um die Bewegung des Fahrzeugs zu messen, wie zum Beispiel GPS und Tachometer. Doch Auflösung und Genauigkeit dieser Systeme sind nicht ausreichend für viele Echtzeitanwendungen. Die Berechnung der Eigenbewegung aus Stereobildsequenzen für Fahrerassistenzsysteme, z.B. zur autonomen Navigation oder Kollisionsvermeidung, bildet den Kern dieser Arbeit. Diese Dissertation präsentiert ein System zur Echtzeitbewertung einer Szene, inklusive Detektion und Bewertung von unabhängig bewegten Objekten sowie der akkuraten Schätzung der sechs Freiheitsgrade der Eigenbewegung. Diese grundlegenden Bestandteile sind erforderlich, um viele intelligente Automobilanwendungen zu entwickeln, die den Fahrer in unterschiedlichen Verkehrssituationen unterstützen. Das System arbeitet ausschließlich mit einer Stereokameraplattform als Sensor. Um die Eigenbewegung und die Szenenstruktur zu berechnen wird eine Analyse des Rauschens und der Fehlerfortpflanzung im Bildaufbereitungsprozess benötigt. Deshalb werden in dieser Dissertation die Rauscheigenschaften der durch Stereotriangulation erhaltenen 3D-Punkte analysiert. Dies führt zu der Entdeckung eines systematischen Fehlers in der Schätzung der 3D-Position, der sich mit einer Neuformulierung der Projektionsgleichung korrigieren lässt. Die Simulationsergebnisse zeigen, dass eine bedeutende Verringerung des Fehlers in der geschätzten 3D-Punktposition möglich ist. Die Eigenbewegungsschätzung wird gewonnen, indem die Rotation und Translation zwischen Punktwolken geschätzt wird. Dieses Problem ist als „absolute Orientierung” bekannt und viele Lösungen auf Basis der Methode der kleinsten Quadrate sind in der Literatur vorgeschlagen worden. Diese Arbeit rezensiert die verfügbaren geschlossenen Lösungen zu dem Problem. Das vorgestellte System gliedert sich in drei wesentliche Bausteine: 1. Registrierung von Bildmerkmalen, 2. Eigenbewegungsschätzung und 3. iterative Schätzung von 3D-Position und 3D-Geschwindigkeit von Weltpunkten. Der erster Block erhält eine Folge rektifizierter Bilder als Eingabe und liefert daraus eine Liste von verfolgten Bildmerkmalen mit ihrer entsprechenden 3D-Position. Der Block „Eigenbewegungsschätzung” besteht aus vier Hauptschritten in einer Schleife: 1. Bewegungsvorhersage, 2. Anwendung der Glattheitsbedingung für die Bewegung (GBB), 3. absolute Orientierungsberechnung und 4. Bewegungsintegration. Die in dieser Dissertation vorgeschlagene GBB ist eine mächtige Bedingung für die Ablehnung von Ausreißern und für die Zuordnung von Gewichten zu den gemessenen 3D-Punkten. Simulationen werden mit gaußschem und slashschem Rauschen ausgeführt. Die Ergebnisse zeigen die Überlegenheit der GBB-Version über die Standardgewichtungsmethoden. Die Stabilität der Ergebnisse hinsichtlich Ausreißern wurde analysiert mit dem Resultat, dass der „break down point” größer als 50% ist. Wenn die vier Schritte iterativ ausgeführt, werden wird ein Prädiktor-Korrektor-Verfahren gewonnen.Wir nennen diese Schätzung Multi-frameschätzung im Gegensatz zur Zweiframeschätzung, die nur die aktuellen und vorherigen Bildpaare für die Berechnung der Eigenbewegung betrachtet. Die erste Iteration wird zwischen der aktuellen und vorherigen Wolke von Punkten durchgeführt. Jede weitere Iteration integriert eine zusätzliche Punktwolke eines vorherigen Zeitpunkts. Diese Methode reduziert die Fehlerakkumulation bei der Integration von mehreren Schätzungen in einer einzigen globalen Schätzung. Simulationsergebnisse zeigen, dass obwohl der Fehler noch superlinear im Laufe der Zeit zunimmt, die Größe des Fehlers um mehrere Größenordnungen reduziert wird. Der dritte Block besteht aus der iterativen Schätzung von 3D-Position und 3D-Geschwindigkeit von Weltpunkten. Hier wird eine Methode basierend auf einem Kalman Filter verwendet, das Stereo, Featuretracking und Eigenbewegungsdaten fusioniert. Messungen der Position eines Weltpunkts werden durch das Stereokamerasystem gewonnen. Die Differenzierung der Position des geschätzten Punkts erlaubt die zusätzliche Schätzung seiner Geschwindigkeit. Die Messungen werden durch das Messmodell gewonnen, das Stereo- und Bewegungsdaten fusioniert. Simulationsergebnisse validieren das Modell. Die Verringerung der Positionsunsicherheit im Laufe der Zeit wird mit einer Monte-Carlo Simulation erzielt. Experimentelle Ergebnisse werden mit langen Sequenzen von Bildern erzielt. Zusätzliche Tests, einschließlich einer 3D-Rekonstruktion einer Waldszene und der Berechnung der freien Kamerabewegung in einem Indoor-Szenario, wurden durchgeführt. Die Methode zeigt gute Ergebnisse in allen Fällen. Der Algorithmus liefert zudem akzeptable Ergebnisse bei der Schätzung der Lage kleiner Objekte, wie Köpfe und Beine von realen Crash-Test-Dummies

Mathematical surfaces models between art and reality

In this paper, I want to document the history of the mathematical surfaces models used for the didactics of pure and applied “High Mathematics” and as art pieces. These models were built between the second half of nineteenth century and the 1930s. I want here also to underline several important links that put in correspondence conception and construction of models with scholars, cultural institutes, specific views of research and didactical studies in mathematical sciences and with the world of the figurative arts furthermore. At the same time the singular beauty of form and colour which the models possessed, aroused the admiration of those entirely ignorant of their mathematical attraction