In this thesis we present a wavelet-based geometry compression pipeline in the context of hierarchical surface and volume representations. Due to the increasing complexity of geometric models used in a vast number of different application fields, new methods have to be devised that enable one to store, transmit and manipulate large amounts of data. Based on a multi-resolution wavelet representation, we have developed a complete compression pipeline suitable for geometric data on uniform grids in two and three dimensions. Local and global oracles in wavelet space are employed to control the approximation error in lossy compression settings. Two novel geometry simplification schemes, which are able to build hierarchical mesh representations, are an essential part of the pipeline. The first method, a bottom-up vertex removal scheme, analyzes the detail information of the data at different levels after reconstruction from wavelet space. The resulting hierarchical quadtree data structure is triangulated subsequently using a look-up-table that stores the necessary connectivity information. The second method implements a topdown vertex insertion strategy that is capable of progressively reconstructing the model. Vertex connectivity is derived using Delaunay triangulation. This approach provides high flexibility for the construction of adaptive surface and volume approximations. Furthermore, it is possible to extract high quality iso-contours and to compress texture attributes along with the geometric surface model. In contrast to the two wavelet-based approximation schemes, we have devised the progressive tetrahedralization method, an extension of the popular progressive meshes into volumetric settings. These strategies can be used to transform unstructured input meshes into special representations which enable a model to be reconstructed progressively. In the volumetric setting, using tetrahedral mesh approximations, we have to account for potential mesh inconsistencies arising frequently during the course of the transformation. All methods developed in this dissertation have been implemented utilizing a software component-based approach. Several hundred components at different abstraction levels can be combined to build powerful prototype applications with high flexibility. We compare the three approximation schemes with each other using several two- and three-dimensional geometric models and provide an extensive error and performance analysis. These results emphasize the individual strengths of each of the introduced methods and concepts.

This paper describes some fundamental issues for robust implementations of progressively refined tetrahedralizations generated through sequences of edge collapses. We address the definition of appropriate cost functions and explain on various tests which are necessary to preserve the consistency of the mesh when collapsing edges. Although being considered a special case of progressive simplicial complexes [10], the results of our method are of high practical importance and can be used in many different applications, such as finite element meshing, scattered data interpolation, or rendering of unstructured volume data.

We describe and analyze a new technique for sonification of three-dimensional vector fields. This technique allows the user to use commodity hardware and widely available 3D sound interfaces to map vectors in a listener's local neighborhood into smooth wind-like sound (aerodynamic sound). The four types of information provided by this technique are flow direction, flow velocity, flow vorticity and local flow patterns. The result is a system that helps the user achieve an intuitive understanding of the data set by using their auditory sense.

We propose an acceleration scheme for real-time many-body dynamic collision detection. We kinetize the sweep and prune method for many-body collision pruning, extending its application to dynamic collision detection via kinetic data structures. In doing so, we modify the method from sample-rate driven to event-driven, with no more events than the original method processed, also removing the per-frame overhead, allowing our method to scale well in terms of frame-rates. Unlike many schemes for many-body collision pruning, ours performs well in both sparse and dense environments, with few or many collisions.

In this article, we present the blue-c application programming interface (API) and discuss some of its performance characteristics. The blue-c API is a software toolkit for media-rich, collaborative, immersive virtual reality applications. It provides easy to use interfaces to all blue-c technology, including immersive projection, live 3D video acquisition and streaming, audio, tracking, and gesture recognition. We emphasize on our performance-optimized 3D video handling and rendering pipeline, which is capable of rendering 3D video inlays consisting of up to 30,000 fragments updated at 10 Hz in real time, enabling remote users to meet inside our virtual environment.

We present a new method for adaptive surface meshing and triangulation which controls the local level-of-detail of the surface approximation by local spectral estimates. These estimates are determined by a wavelet representation of the surface data. The basic idea is to decompose the initial data set by means of an orthogonal or semi-orthogonal tensor product wavelet transform (WT) and to analyze the resulting coefficients. In surface regions, where the partial energy of the resulting coefficients is low, the polygonal approximation of the surface can be performed with larger triangles without loosing too much fine grain details. However, since the localization of the WT is bound by the Heisenberg principle the meshing method has to be controlled by the detail signals rather than directly by the coefficients. The dyadic scaling of the WT stimulated us to build an hierarchical meshing algorithm which transforms the initially regular data grid into a quadtree representation by rejection of unimportant mesh vertices. The optimum triangulation of the resulting quadtree cells is carried out by selection from a look-up table. The tree grows recursively as controlled by detail signals which are computed from a modified inverse WT. In order to control the local level-of-detail, we introduce a new class of wavelet space filters acting as magnifying glasses on the data.

Compression methods have become of fundamental importance in almost every subfield of scientific visualization. However, unlike image compression, advanced visualization applications impose manifold constraints on the design of appropriate algorithms, where progressiveness, multiresolution or topology preservation are some of the key issues. This paper demonstrates the importance of multiresolution compression methods for visualization using two examples: The first, compression domain volume rendering , enables one to visualize volume data progressively and instantaneously from its compressed data format and has been designed for WWW and networked applications. The second one is a multiresolution compression and reconstruction method that allows for progressive coding, transmission and geometric reconstruction of surfaces and volumes. Both of the presented methods are so-called transform coding schemes and use wavelets for data representation.

This paper presents a framework for multiresolution compression and geometric reconstruction of arbitrarily dimensioned data designed for distributed applications. Although being restricted to uniform sampled data, our versatile approach enables the handling of a large variety of real world elements. Examples include nonparametric, parametric and implicit lines, surfaces or volumes, all of which are common to large scale data sets. The framework is based on two fundamental steps: Compression is carried out by a remote server and generates a bitstream transmitted over the underlying network. Geometric reconstruction is performed by the local client and renders a piecewise linear approximation of the data. More precisely, our compression scheme consists of a newly developed pipeline starting from an initial B-spline wavelet precoding. The fundamental properties of wavelets allow progressive transmission and interactive control of the compression gain by means of global and local oracles. In particular we discuss the problem of oracles in semiorthogonal settings and propose sophisticated oracles to remove unimportant coefficients. In addition, geometric constraints such as boundary lines can be compressed in a lossless manner and are incorporated into the resulting bit-stream. The reconstruction pipeline performs a piecewise adaptive linear approximation of data using a fast and easy to use point removal strategy which works with any subsequent triangulation technique. As a result, the pipeline renders line segments, triangles or tetrahedra. Moreover, the underlying continuous approximation of the wavelet representation can be exploited to reconstruct implicit functions, such as isolines and isosurfaces more smoothly and precisely than commonplace methods. Although it scales straightforwardly to higher dimensions the performance of our framework is illustrated with results achieved on data very popular in practice: parametric curves and surfaces, digital terrain models, and volume data. CR Descriptors: E.4 [Coding and Information Theory]: Data compaction and compression; I.3.5 [Computational Geometry and Object Modeling]: Curve, surface, solid, and object representations; Splines; I.3.7 [Three-Dimensional Graphics and Realism]; I.4.5 [Reconstruction]: Transformation methods. Additional Keywords and Phrases: wavelets, isosurfaces, volumes, triangulation, tetrahedralization, meshing, oracles.