Konforme geometrische Algebra und deren Anwendungen auf stochastische Optimierungsprobleme im Bereich 3D-Vision

In the present work, the modeling capabilities of conformal geometric algebra (CGA) are harnessed to approach typical problems from the research field of 3D-vision. This increasingly popular methodology is then extended in a new fashion by the integration of a least squares technique into the framework of CGA. Specifically, choosing the linear Gauss-Helmert model as the basis, the most general variant of least squares adjustment can be brought into operation. The result is a new versatile parameter estimation, termed GH-method, that reconciles two different mathematical areas, that is algebra and stochastics, under the umbrella of geometry. The main concern of the thesis is to show up the advantages inhering with this combination. Monocular pose estimation, from the subject 3D-vision, is the applicational focus of this thesis; given a picture of a scene, position and orientation of the image capturing vision system with respect to an external coordinate system define the pose. The developed parameter estimation technique is applied to different variants of this problem. Parameters are encoded by the algebra elements, called multivectors. They can be geometric objects as a circle, geometric operators as a rotation or likewise the pose. In the conducted pose experiments, observations are image pixels with associated uncertainties. The high accuracy achieved throughout all experiments confirms the competitiveness of the proposed estimation technique. Central to this work is also the consideration of omnidirectional vision using a paracatadioptric imaging sensor. It is demonstrated that CGA provides the ideal framework to model the related image formation. Two variants of the perspective pose estimation problem are adapted to the omnidirectional case. A new formalization of the epipolar geometry of two images in terms of CGA is developed, from which new insights into the structures behind the essential and the fundamental matrix, respectively, are drawn. Renowned standard approaches are shown to implicitly make use of CGA. Finally, an invocation of the GH-method for estimating epipoles is presented. Experimental results substantiate the goodness of this approach. Next to the detailed elucidations on parameter estimation, this text also gives a comprehensive introduction to geometric algebra, its tensor representation, the conformal space and the respective conformal geometric algebra. A valuable contribution is especially the analytic investigation into the geometric capabilities of CGA.Die vorliegende Arbeit ist motiviert durch die im Forschungszweig Computer Vision (CV) der Informatik typisch auftretenden geometrischen Problemstellungen auf der Grundlage von digitalen Bildaufnahmen. Hierzu zählt die Berechnung einer optimal durch eine Menge von Bildpunkten verlaufende Kurve, die Bestimmung der Epipolargeometrie, das Schätzen der Pose eines Objektes oder die 3D-Rekonstruktion. Diese Klasse von Problemen lässt sich durch den Einsatz der geometrischen Algebra (GA) – so werden unter geometrischen Aspekten besonders interessante Clifford Algebren bezeichnet – in überaus prägnanter und geschlossener Form modellieren. Dieser mit wachsender Akzeptanz verfolgte Ansatz, der beständig durch den Lehrstuhl „Kognitive Systeme“ der Universität Kiel weiterentwickelt wird, ist zentraler Bestandteile der Dissertation. Speziell wird die „konforme geometrische Algebra“ (CGA), die auf einer nicht-linearen Einbettung des euklidischen 3D-Raumes in einen fünfdimensionalen projektiven konformen Raum beruht, eingesetzt. Die Elemente dieser Algebra erlauben die Repräsentation geometrischer Basisentitäten, im wesentlichen Punkte, Linien, Kreise, Kugeln und Ebenen. Eine Vielzahl von Operationen ist möglich; besonders interessant sind die Transformationen der enthaltenen konformen Gruppe sowie die Möglichkeit algebraisch mit Unterräumen zu rechnen, d.h. diese zu vergrößern, zu schneiden oder Inzidenzen abzufragen. Den zweiten wichtigen Bestandteil der Arbeit stellt ein für die oben genannten Problemstellungen typisches stochastischen Verfahren dar – die Ausgleichsrechnung nach der Methode der kleinsten Quadrate. Deren allgemeinste Form erwächst aus der Verwendung des aus der Geodäsie bekannten linearen Gauß-Helmert (GH) Modells. Der resultierende GH-Schätzer zeigt alle Optimalitätseigenschaften wie minimale Varianz und Erwartungstreue. Eine der geometrischen Algebra inhärente Tensordarstellung stellt eine geeignete numerische Schnittstelle zwischen CGA und der GH-Schätzmethode zur Verfügung. Aufgrund der Bilinearität des Algebraprodukts lässt sich so ebenfalls das Konzept der Fehlerfortpflanzung, ein wichtiges Instrument der Ausgleichsrechnung, mit hoher Genauigkeit auf die Operationen der Algebra ausdehnen. Im Ergebnis entsteht ein neues universelles Parameterschätzverfahren zur Bestimmung der des jeweiligen Problems zugrundeliegenden Variablen. Ziel der vorliegenden Arbeit ist es auch, die aus der Verbindung von Algebra und Stochastik entstehenden Vorteile anhand von typischen CV-Anwendungen herauszustellen. Den Schwerpunkt hierfür bildet die Schätzung der Pose (Position und Orientierung eines Objekts bezüglich eines objektfremden Koordinatensystems), z.B. die eines Roboters anhand eines vom Roboter aufgenommenen Kamerabildes. Es wird ebenfalls gezeigt, dass CGA den optimalen Rahmen zur Modellierung omnidirektionaler Bildgebungsverfahren bietet, falls diese auf einem katadioptrischen System mit parabolischem Spiegel beruhen. Als omnidirektionale Anwendungen werden Posenschätzung sowie die Bestimmung der Epipolargeometrie präsentiert. Die erreichte Güte der GH-Parameterschätzung in den einzelnen Anwendungen wird jeweils durch experimentell gewonnene Resultate untermauert. Neben den umfangreichen Ausführungen zur Parameterschätzung liefert diese Arbeit auch eine detaillierte Einführung und Herleitung der geometrischen Algebra. Besonderes Augenmerk ist auch auf die analytische Darlegung der konformen geometrischen Algebra zu richten

Geodesic Active Fields:A Geometric Framework for Image Registration

Image registration is the concept of mapping homologous points in a pair of images. In other words, one is looking for an underlying deformation field that matches one image to a target image. The spectrum of applications of image registration is extremely large: It ranges from bio-medical imaging and computer vision, to remote sensing or geographic information systems, and even involves consumer electronics. Mathematically, image registration is an inverse problem that is ill-posed, which means that the exact solution might not exist or not be unique. In order to render the problem tractable, it is usual to write the problem as an energy minimization, and to introduce additional regularity constraints on the unknown data. In the case of image registration, one often minimizes an image mismatch energy, and adds an additive penalty on the deformation field regularity as smoothness prior. Here, we focus on the registration of the human cerebral cortex. Precise cortical registration is required, for example, in statistical group studies in functional MR imaging, or in the analysis of brain connectivity. In particular, we work with spherical inflations of the extracted hemispherical surface and associated features, such as cortical mean curvature. Spatial mapping between cortical surfaces can then be achieved by registering the respective spherical feature maps. Despite the simplified spherical geometry, inter-subject registration remains a challenging task, mainly due to the complexity and inter-subject variability of the involved brain structures. In this thesis, we therefore present a registration scheme, which takes the peculiarities of the spherical feature maps into particular consideration. First, we realize that we need an appropriate hierarchical representation, so as to coarsely align based on the important structures with greater inter-subject stability, before taking smaller and more variable details into account. Based on arguments from brain morphogenesis, we propose an anisotropic scale-space of mean-curvature maps, built around the Beltrami framework. Second, inspired by concepts from vision-related elements of psycho-physical Gestalt theory, we hypothesize that anisotropic Beltrami regularization better suits the requirements of image registration regularization, compared to traditional Gaussian filtering. Different objects in an image should be allowed to move separately, and regularization should be limited to within the individual Gestalts. We render the regularization feature-preserving by limiting diffusion across edges in the deformation field, which is in clear contrast to the indifferent linear smoothing. We do so by embedding the deformation field as a manifold in higher-dimensional space, and minimize the associated Beltrami energy which represents the hyperarea of this embedded manifold as measure of deformation field regularity. Further, instead of simply adding this regularity penalty to the image mismatch in lieu of the standard penalty, we propose to incorporate the local image mismatch as weighting function into the Beltrami energy. The image registration problem is thus reformulated as a weighted minimal surface problem. This approach has several appealing aspects, including (1) invariance to re-parametrization and ability to work with images defined on non-flat, Riemannian domains (e.g., curved surfaces, scalespaces), and (2) intrinsic modulation of the local regularization strength as a function of the local image mismatch and/or noise level. On a side note, we show that the proposed scheme can easily keep up with recent trends in image registration towards using diffeomorphic and inverse consistent deformation models. The proposed registration scheme, called Geodesic Active Fields (GAF), is non-linear and non-convex. Therefore we propose an efficient optimization scheme, based on splitting. Data-mismatch and deformation field regularity are optimized over two different deformation fields, which are constrained to be equal. The constraint is addressed using an augmented Lagrangian scheme, and the resulting optimization problem is solved efficiently using alternate minimization of simpler sub-problems. In particular, we show that the proposed method can easily compete with state-of-the-art registration methods, such as Demons. Finally, we provide an implementation of the fast GAF method on the sphere, so as to register the triangulated cortical feature maps. We build an automatic parcellation algorithm for the human cerebral cortex, which combines the delineations available on a set of atlas brains in a Bayesian approach, so as to automatically delineate the corresponding regions on a subject brain given its feature map. In a leave-one-out cross-validation study on 39 brain surfaces with 35 manually delineated gyral regions, we show that the pairwise subject-atlas registration with the proposed spherical registration scheme significantly improves the individual alignment of cortical labels between subject and atlas brains, and, consequently, that the estimated automatic parcellations after label fusion are of better quality

NASA thesaurus. Volume 1: Hierarchical Listing

There are over 17,000 postable terms and nearly 4,000 nonpostable terms approved for use in the NASA scientific and technical information system in the Hierarchical Listing of the NASA Thesaurus. The generic structure is presented for many terms. The broader term and narrower term relationships are shown in an indented fashion that illustrates the generic structure better than the more widely used BT and NT listings. Related terms are generously applied, thus enhancing the usefulness of the Hierarchical Listing. Greater access to the Hierarchical Listing may be achieved with the collateral use of Volume 2 - Access Vocabulary and Volume 3 - Definitions

NASA Thesaurus. Volume 1: Hierarchical listing

There are 16,713 postable terms and 3,716 nonpostable terms approved for use in the NASA scientific and technical information system in the Hierarchical Listing of the NASA Thesaurus. The generic structure is presented for many terms. The broader term and narrower term relationships are shown in an indented fashion that illustrates the generic structure better than the more widely used BT and NT listings. Related terms are generously applied, thus enhancing the usefulness of the Hierarchical Listing. Greater access to the Hierarchical Listing may be achieved with the collateral use of Volume 2 - Access Vocabulary

NASA thesaurus. Volume 2: Access vocabulary

The Access Vocabulary, which is essentially a permuted index, provides access to any word or number in authorized postable and nonpostable terms. Additional entries include postable and nonpostable terms, other word entries, and pseudo-multiword terms that are permutations of words that contain words within words. The Access Vocabulary contains 40,738 entries that give increased access to the hierarchies in Volume 1 - Hierarchical Listing

Relative Pose Estimation Using Non-overlapping Multicamera Clusters

This thesis considers the Simultaneous Localization and Mapping (SLAM) problem using a set of perspective cameras arranged such that there is no overlap in their fields-of-view. With the known and fixed extrinsic calibration of each camera within the cluster, a novel real-time pose estimation system is presented that is able to accurately track the motion of a camera cluster relative to an unknown target object or environment and concurrently generate a model of the structure, using only image-space measurements. A new parameterization for point feature position using a spherical coordinate update is presented which isolates system parameters dependent on global scale, allowing the shape parameters of the system to converge despite the scale parameters remaining uncertain. Furthermore, a flexible initialization scheme is proposed which allows the optimization to converge accurately using only the measurements from the cameras at the first time step. An analysis is presented identifying the configurations of the cluster motions and target structure geometry for which the optimization solution becomes degenerate and the global scale is ambiguous. Results are presented that not only confirm the previously known critical motions for a two-camera cluster, but also provide a complete description of the degeneracies related to the point feature constellations. The proposed algorithms are implemented and verified in experiments with a camera cluster constructed using multiple perspective cameras mounted on a quadrotor vehicle and augmented with tracking markers to collect high-precision ground-truth motion measurements from an optical indoor positioning system. The accuracy and performance of the proposed pose estimation system are confirmed for various motion profiles in both indoor and challenging outdoor environments

NASA thesaurus. Volume 2: Access vocabulary

The access vocabulary, which is essentially a permuted index, provides access to any word or number in authorized postable and nonpostable terms. Additional entries include postable and nonpostable terms, other word entries and pseudo-multiword terms that are permutations of words that contain words within words. The access vocabulary contains almost 42,000 entries that give increased access to the hierarchies in Volume 1 - Hierarchical Listing

The Probabilistic Active Shape Model: From Model Construction to Flexible Medical Image Segmentation

Automatic processing of three-dimensional image data acquired with computed tomography or magnetic resonance imaging plays an increasingly important role in medicine. For example, the automatic segmentation of anatomical structures in tomographic images allows to generate three-dimensional visualizations of a patient’s anatomy and thereby supports surgeons during planning of various kinds of surgeries. Because organs in medical images often exhibit a low contrast to adjacent structures, and because the image quality may be hampered by noise or other image acquisition artifacts, the development of segmentation algorithms that are both robust and accurate is very challenging. In order to increase the robustness, the use of model-based algorithms is mandatory, as for example algorithms that incorporate prior knowledge about an organ’s shape into the segmentation process. Recent research has proven that Statistical Shape Models are especially appropriate for robust medical image segmentation. In these models, the typical shape of an organ is learned from a set of training examples. However, Statistical Shape Models have two major disadvantages: The construction of the models is relatively difficult, and the models are often used too restrictively, such that the resulting segmentation does not delineate the organ exactly. This thesis addresses both problems: The first part of the thesis introduces new methods for establishing correspondence between training shapes, which is a necessary prerequisite for shape model learning. The developed methods include consistent parameterization algorithms for organs with spherical and genus 1 topology, as well as a nonrigid mesh registration algorithm for shapes with arbitrary topology. The second part of the thesis presents a new shape model-based segmentation algorithm that allows for an accurate delineation of organs. In contrast to existing approaches, it is possible to integrate not only linear shape models into the algorithm, but also nonlinear shape models, which allow for a more specific description of an organ’s shape variation. The proposed segmentation algorithm is evaluated in three applications to medical image data: Liver and vertebra segmentation in contrast-enhanced computed tomography scans, and prostate segmentation in magnetic resonance images

Proceedings. 9th 3DGeoInfo Conference 2014, [11-13 November 2014, Dubai]

It is known that, scientific disciplines such as geology, geophysics, and reservoir exploration intrinsically use 3D geo-information in their models and simulations. However, 3D geo-information is also urgently needed in many traditional 2D planning areas such as civil engineering, city and infrastructure modeling, architecture, environmental planning etc. Altogether, 3DGeoInfo is an emerging technology that will greatly influence the market within the next few decades. The 9th International 3DGeoInfo Conference aims at bringing together international state-of-the-art researchers and practitioners facilitating the dialogue on emerging topics in the field of 3D geo-information. The conference in Dubai offers an interdisciplinary forum of sub- and above-surface 3D geo-information researchers and practitioners dealing with data acquisition, modeling, management, maintenance, visualization, and analysis of 3D geo-information