The primary emphasis of this thesis concerns the extraction and representation of intrinsic properties of three-dimensional (3D) unorganized point clouds. The points establishing a point cloud as it mainly emerges from LiDaR (Light Detection and Ranging) scan devices or by reconstruction from two-dimensional (2D) image series represent discrete samples of real world objects. Depending on the type of scenery the data is generated from the resulting point cloud may exhibit a variety of different structures. Especially, in the case of environmental LiDaR scans the complexity of the corresponding point clouds is relatively high. Hence, finding new techniques allowing the efficient extraction and representation of the underlying structural entities becomes an important research issue of recent interest. This thesis introduces new methods regarding the extraction and visualization of structural features like surfaces and curves (e.g. ridge-lines, creases) from 3D (environmental) point clouds. One main part concerns the extraction of curve-like features from environmental point data sets. It provides a new method supporting a stable feature extraction by incorporating a probability-based point classification scheme that characterizes individual points regarding their affiliation to surface-, curve- and volume-like structures. Another part is concerned with the surface reconstruction from (environmental) point clouds exhibiting objects that are more or less complex. A new method providing multi-resolutional surface representations from regular point clouds is discussed. Following the applied principles of this approach a volumetric surface reconstruction method based on the proposed classification scheme is introduced. It allows the reconstruction of surfaces from highly unstructured and noisy point data sets. Furthermore, contributions in the field of reconstructing 3D point clouds from 2D image series are provided. In addition, a discussion concerning the most important properties of (environmental) point clouds with respect to feature extraction is presented.
In this thesis we apply powerful mathematical tools such as interval arithmetic for applications in computational geometry, visualization and computer graphics, leading to robust, general and efficient algorithms. We present a completely novel approach for computing the arrangement of arbitrary implicit planar curves and perform ray casting of arbitrary implicit functions by jointly achieving, for the first time, robustness, efficiency and flexibility. Indeed we are able to render even the most difficult implicits in real-time with guaranteed topology and at high resolution. We use subdivision and interval arithmetic as key-ingredients to guarantee robustness. The presented framework is also well-suited for applications to large and unstructured data sets due to the inherent adaptivity of the techniques that are used. We also approach the topic of tensors by collaborating with mechanical engineers on comparative tensor visualization and provide them with helpful visualization paradigms to interpret the data.
The visualization of numerical fluid flow datasets is essential to the engineering processes that motivate their computational simulation. To address the need for visual representations that convey meaningful relations and enable a deep understanding of flow structures, the discipline of Flow Visualization has produced many methods and schemes that are tailored to a variety of visualization tasks. The ever increasing complexity of modern flow simulations, however, puts an enormous demand on these methods. The study of vortex breakdown, for example, which is a highly transient and inherently three-dimensional flow pattern with substantial impact wherever it appears, has driven current techniques to their limits. In this thesis, we propose several novel visualization methods that significantly advance the state of the art in the visualization of complex flow structures. First, we propose a novel scheme for the construction of stream surfaces from the trajectories of particles embedded in a flow. These surfaces are extremely useful since they naturally exploit coherence between neighboring trajectories and are highly illustrative in nature. We overcome the limitations of existing stream surface algorithms that yield poor results in complex flows, and show how the resulting surfaces can be used a building blocks for advanced flow visualization techniques. Moreover, we present a visualization method that is based on moving section planes that travel through a dataset and sample the flow. By considering the changes to the flow topology on the plane as it moves, we obtain a method of visualizing topological structures in three-dimensional flows that are not accessible by conventional topological methods. On the same algorithmic basis, we construct an algorithm for the tracking of critical points in such flows, thereby enabling the treatment of time-dependent datasets. Last, we address some problems with the recently introduced Lagrangian techniques. While conceptually elegant and generally applicable, they suffer from an enormous computational cost that we significantly use by developing an adaptive approximation algorithm. This allows the application of such methods on very large and complex numerical simulations. Throughout this thesis, we will be concerned with flow visualization aspect of general practical significance but we will particularly emphasize the remarkably challenging visualization of the vortex breakdown phenomenon.