Subdivision Directional Fields

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure-preserving: it reproduces curl-free fields precisely, and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover's distance computation, for efficient and robust computation

Vector field processing on triangle meshes

While scalar fields on surfaces have been staples of geometry processing, the use of tangent vector fields has steadily grown in geometry processing over the last two decades: they are crucial to encoding directions and sizing on surfaces as commonly required in tasks such as texture synthesis, non-photorealistic rendering, digital grooming, and meshing. There are, however, a variety of discrete representations of tangent vector fields on triangle meshes, and each approach offers different tradeoffs among simplicity, efficiency, and accuracy depending on the targeted application. This course reviews the three main families of discretizations used to design computational tools for vector field processing on triangle meshes: face-based, edge-based, and vertex-based representations. In the process of reviewing the computational tools offered by these representations, we go over a large body of recent developments in vector field processing in the area of discrete differential geometry. We also discuss the theoretical and practical limitations of each type of discretization, and cover increasingly-common extensions such as n-direction and n-vector fields. While the course will focus on explaining the key approaches to practical encoding (including data structures) and manipulation (including discrete operators) of finite-dimensional vector fields, important differential geometric notions will also be covered: as often in Discrete Differential Geometry, the discrete picture will be used to illustrate deep continuous concepts such as covariant derivatives, metric connections, or Bochner Laplacians

Doctor of Philosophy

dissertationShape analysis is a well-established tool for processing surfaces. It is often a first step in performing tasks such as segmentation, symmetry detection, and finding correspondences between shapes. Shape analysis is traditionally employed on well-sampled surfaces where the geometry and topology is precisely known. When the form of the surface is that of a point cloud containing nonuniform sampling, noise, and incomplete measurements, traditional shape analysis methods perform poorly. Although one may first perform reconstruction on such a point cloud prior to performing shape analysis, if the geometry and topology is far from the true surface, then this can have an adverse impact on the subsequent analysis. Furthermore, for triangulated surfaces containing noise, thin sheets, and poorly shaped triangles, existing shape analysis methods can be highly unstable. This thesis explores methods of shape analysis applied directly to such defect-laden shapes. We first study the problem of surface reconstruction, in order to obtain a better understanding of the types of point clouds for which reconstruction methods contain difficulties. To this end, we have devised a benchmark for surface reconstruction, establishing a standard for measuring error in reconstruction. We then develop a new method for consistently orienting normals of such challenging point clouds by using a collection of harmonic functions, intrinsically defined on the point cloud. Next, we develop a new shape analysis tool which is tolerant to imperfections, by constructing distances directly on the point cloud defined as the likelihood of two points belonging to a mutually common medial ball, and apply this for segmentation and reconstruction. We extend this distance measure to define a diffusion process on the point cloud, tolerant to missing data, which is used for the purposes of matching incomplete shapes undergoing a nonrigid deformation. Lastly, we have developed an intrinsic method for multiresolution remeshing of a poor-quality triangulated surface via spectral bisection

An Empirical Evaluation of Force-Directed Graph Layout

Force-directed graph layout is a widely used algorithm for the automatic layout of graphs. Little experimental work has been done exploring the behaviour of the algorithm under a variety of conditions. This thesis carries out three large-scale metric-based experiments. The first explores how the core algorithm behaves under changes to initial conditions. The second looks at extending the force-directed layout algorithm with additional forces to reduce overlaps. The third develops a novel symmetry metric for graphs and uses that to explore the symmetries of graphs. This thesis also carries out a user study to show that the differences reported by metrics in the graphs are reflected in a difference in user performance when using graphs for a free-form selection task

Rekonstruktion, Analyse und Editierung dynamisch deformierter 3D-Oberflächen

Dynamically deforming 3D surfaces play a major role in computer graphics. However, producing time-varying dynamic geometry at ever increasing detail is a time-consuming and costly process, and so a recent trend is to capture geometry data directly from the real world. In the first part of this thesis, I propose novel approaches for this research area. These approaches capture dense dynamic 3D surfaces from multi-camera systems in a particularly robust and accurate way. This provides highly realistic dynamic surface models for phenomena like moving garments and bulging muscles. However, re-using, editing, or otherwise analyzing dynamic 3D surface data is not yet conveniently possible. To close this gap, the second part of this dissertation develops novel data-driven modeling and animation approaches. I first show a supervised data-driven approach for modeling human muscle deformations that scales to huge datasets and provides fine-scale, anatomically realistic deformations at high quality not attainable by previous methods. I then extend data-driven modeling to the unsupervised case, providing editing tools for a wider set of input data ranging from facial performance capture and full-body motion to muscle and cloth deformation. To this end, I introduce the concepts of sparsity and locality within a mathematical optimization framework. I also explore these concepts for constructing shape-aware functions that are useful for static geometry processing, registration, and localized editing.Dynamisch deformierbare 3D-Oberflächen spielen in der Computergrafik eine zentrale Rolle. Die Erstellung der für Computergrafik-Anwendungen benötigten, hochaufgelösten und zeitlich veränderlichen Oberflächengeometrien ist allerdings äußerst arbeitsintensiv. Aus dieser Problematik heraus hat sich der Trend entwickelt, Oberflächendaten direkt aus Aufnahmen der echten Welt zu erfassen. Dazu nötige 3D-Rekonstruktionsverfahren werden im ersten Teil der Arbeit entwickelt. Die vorgestellten, neuartigen Verfahren erlauben die Erfassung dynamischer 3D-Oberflächen aus Mehrkamera-Aufnahmen bei hoher Verlässlichkeit und Präzision. Auf diese Weise können detaillierte Oberflächenmodelle von Phänomenen wie in Bewegung befindliche Kleidung oder sich anspannende Muskeln erfasst werden. Aber auch die Wiederverwendung, Bearbeitung und Analyse derlei gewonnener 3D-Oberflächendaten ist aktuell noch nicht auf eine einfache Art und Weise möglich. Um diese Lücke zu schließen beschäftigt sich der zweite Teil der Arbeit mit der datengetriebenen Modellierung und Animation. Zunächst wird ein Ansatz für das überwachte Lernen menschlicher Muskel-Deformationen vorgestellt. Dieses neuartige Verfahren ermöglicht eine datengetriebene Modellierung mit besonders umfangreichen Datensätzen und liefert anatomisch-realistische Deformationseffekte. Es übertrifft damit die Genauigkeit früherer Methoden. Im nächsten Teil beschäftigt sich die Dissertation mit dem unüberwachten Lernen aus 3D-Oberflächendaten. Es werden neuartige Werkzeuge vorgestellt, die eine weitreichende Menge an Eingabedaten verarbeiten können, von aufgenommenen Gesichtsanimationen über Ganzkörperbewegungen bis hin zu Muskel- und Kleidungsdeformationen. Um diese Anwendungsbreite zu erreichen stützt sich die Arbeit auf die allgemeinen Konzepte der Spärlichkeit und Lokalität und bettet diese in einen mathematischen Optimierungsansatz ein. Abschließend zeigt die vorliegende Arbeit, wie diese Konzepte auch für die Konstruktion von oberflächen-adaptiven Basisfunktionen übertragen werden können. Dadurch können Anwendungen für die Verarbeitung, Registrierung und Bearbeitung statischer Oberflächenmodelle erschlossen werden