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No doubt: Mathematics has become a technology in its own right, maybe even a key technology. Technology may be defined as the application of science to the problems of commerce and industry. And science? Science maybe defined as developing, testing and improving models for the prediction of system behavior; the language used to describe these models is mathematics and mathematics provides methods to evaluate these models. Here we are! Why has mathematics become a technology only recently? Since it got a tool, a tool to evaluate complex, "near to reality" models: Computer! The model may be quite old - Navier-Stokes equations describe flow behavior rather well, but to solve these equations for realistic geometry and higher Reynolds numbers with sufficient precision is even for powerful parallel computing a real challenge. Make the models as simple as possible, as complex as necessary - and then evaluate them with the help of efficient and reliable algorithms: These are genuine mathematical tasks.

Sound surrounds us all the time and in every place in our daily life, may it be pleasant music in a concert hall or disturbing noise emanating from a busy street in front of our home. The basic properties are the same for both kinds of sound, namely sound waves propagating from a source, but we perceive it in different ways depending on our current mood or if the sound is wanted or not. In this thesis both pleasant sound as well as disturbing noise is examined by means of simulating the sound and visualizing the results thereof. However, although the basic properties of music and traffic noise are the same, one is interested in different features. For example, in a concert hall, the reverberation time is an important quality measure, but if noise is considered only the resulting sound level, for example on ones balcony, is of interest. Such differences are reflected in different methods of simulation and required visualizations, therefore this thesis is divided into two parts. The first part about room acoustics deals with the simulation and novel visualizations for indoor sound and acoustic quality measures, such as definition (original "Deutlichkeit") and clarity index (original "Klarheitsmaß"). For the simulation two different methods, a geometric (phonon tracing) and a wave based (FEM) approach, are applied and compared. The visualization techniques give insight into the sound behaviour and the acoustic quality of a room from a global as well as a listener based viewpoint. Furthermore, an acoustic rendering equation is presented, which is used to render interference effects for different frequencies. Last but not least a novel visualization approach for low frequency sound is presented, which enables the topological analysis of pressure fields based on room eigenfrequencies. The second part about environmental noise is concerned with the simulation and visualization of outdoor sound with a focus on traffic noise. The simulation instruction prescribed by national regulations is discussed in detail, and an approach for the computation of noise volumes, as well as an extension to the simulation, allowing interactive noise calculation, are presented. Novel visualization and interaction techniques for the calculated noise data, incorporated in an interactive three dimensional environment, enabling the easy comprehension of noise problems, are presented. Furthermore additional information can be integrated into the framework to enhance the visualization of noise and the usability of the framework for different usages.

Feature Based Visualization
(2007)

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.

Computer-based simulation and visualization of acoustics of a virtual scene can aid during the design process of concert halls, lecture rooms, theaters, or living rooms. Because, not only the visual aspect of the room is important, but also its acoustics. In factory floors noise reduction is important since noise is hazardous to health. Despite the obvious dissimilarity between our aural and visual senses, many techniques required for the visualization of photo-realistic images and for the auralization of acoustic environments are quite similar. Both applications can be served by geometric methods such as particle- and ray tracing if we neglect a number of less important effects. By means of the simulation of room acoustics we want to predict the acoustic properties of a virtual model. For auralization, a pulse response filter needs to be assembled for each pair of source and listener positions. The convolution of this filter with an anechoic source signal provides the signal received at the listener position. Hence, the pulse response filter must contain all reverberations (echos) of a unit pulse, including their frequency decompositions due to absorption at different surface materials. For the room acoustic simulation a method named phonon tracing, since it is based on particles, is developed. The approach computes the energy or pressure decomposition for each particle (phonon) sent out from a sound source and uses this in a second pass (phonon collection) to construct the response filters for different listeners. This step can be performed in different precision levels. During the tracing step particle paths and additional information are stored in a so called phonon map. Using this map several sound visualization approaches were developed. From the visualization, the effect of different materials on the spectral energy / pressure distribution can be observed. The first few reflections already show whether certain frequency bands are rapidly absorbed. The absorbing materials can be identified and replaced in the virtual model, improving the overall acoustic quality of the simulated room. Furthermore an insight into the pressure / energy received at the listener position is possible. The phonon tracing algorithm as well as several sound visualization approaches are integrated into a common system utilizing Virtual Reality technologies in order to facilitate the immersion into the virtual scene. The system is a prototype developed within a project at the University of Kaiserslautern and is still a subject of further improvements. It consists of a stereoscopic back-projection system for visual rendering as well as professional audio equipment for auralization purposes.