This thesis discusses several applications of computational topology to the visualization
of scalar fields. Scalar field data come from different measurements and simulations. The
intrinsic properties of this kind of data, which make the visualization of it to a complicated
task, are the large size and presence of noise. Computational topology is a powerful tool
for automatic feature extraction, which allows the user to interpret the information contained
in the dataset in a more efficient way. Utilizing it one can make the main purpose of
scientific visualization, namely extracting knowledge from data, a more convenient task.
Volume rendering is a class of methods designed for realistic visual representation of 3D
scalar fields. It is used in a wide range of applications with different data size, noise
rate and requirements on interactivity and flexibility. At the moment there is no known
technique which can meet the needs of every application domain, therefore development
of methods solving specific problems is required. One of such algorithms, designed for
rendering of noisy data with high frequencies is presented in the first part of this thesis.
The method works with multidimensional transfer functions and is especially suited for
functions exhibiting sharp features. Compared with known methods the presented algorithm
achieves better visual quality with a faster performance in presence of mentioned
features. An improvement on the method utilizing a topological theory, Morse theory, and
a topological construct, Morse-Smale complex, is also presented in this part of the thesis.
The improvement allows for performance speedup at a little precomputation and memory
The usage of topological methods for feature extraction on a real world dataset often
results in a very large feature space which easily leads to information overflow. Topology
simplification is designed to reduce the number of features and allow a domain expert
to concentrate on the most important ones. In the terms of Morse theory features are
represented by critical points. An importance measure which is usually used for removing
critical points is called homological persistence. Critical points are cancelled pairwise
according to their homological persistence value. In the presence of outlier-like noise
homological persistence has a clear drawback: the outliers get a high importance value
assigned and therefore are not being removed. In the second part of this thesis a new
importance measure is presented which is especially suited for data with outliers. This
importance measure is called scale space persistence. The algorithm for the computation
of this measure is based on the scale space theory known from the area of computer
vision. The development of a critical point in scale space gives information about its
spacial extent, therefore outliers can be distinguished from other critical points. The usage
of the presented importance measure is demonstrated on a real world application, crater
identification on a surface of Mars.
The third part of this work presents a system for general interactive topology analysis
and exploration. The development of such a system is motivated by the fact that topological
methods are often considered to be complicated and hard to understand, because
application of topology for visualization requires deep understanding of the mathematical
background behind it. A domain expert exploring the data using topology for feature
extraction needs an intuitive way to manipulate the exploration process. The presented
system is based on an intuitive notion of a scene graph, where the user can choose and
place the component blocks to achieve an individual result. This way the domain expert
can extract more knowledge from given data independent on the application domain. The
tool gives the possibility for calculation and simplification of the underlying topological
structure, Morse-Smale complex, and also the visualization of parts of it. The system also
includes a simple generic query language to acquire different structures of the topological
structure at different levels of hierarchy.
The fourth part of this dissertation is concentrated on an application of computational
geometry for quality assessment of a triangulated surface. Quality assessment of a triangulation
is called surface interrogation and is aimed for revealing intrinsic irregularities
of a surface. Curvature and continuity are the properties required to design a visually
pleasing geometric object. For example, a surface of a manufactured body usually should
be convex without bumps of wiggles. Conventional rendering methods hide the regions
of interest because of smoothing or interpolation. Two new methods which are presented
here: curvature estimation using local fitting with B´ezier patches and computation of reflection
lines for visual representation of continuity, are specially designed for assessment
problems. The examples and comparisons presented in this part of the thesis prove the
benefits of the introduced algorithms. The methods are also well suited for concurrent visualization
of the results from simulation and surface interrogation to reveal the possible
intrinsic relationship between them.
In urban planning, both measuring and communicating sustainability are among the most recent concerns. Therefore, the primary emphasis of this thesis concerns establishing metrics and visualization techniques in order to deal with indicators of sustainability.
First, this thesis provides a novel approach for measuring and monitoring two indicators of sustainability - urban sprawl and carbon footprints – at the urban neighborhood scale. By designating different sectors of relevant carbon emissions as well as different household categories, this thesis provides detailed information about carbon emissions in order to estimate impacts of daily consumption decisions and travel behavior by household type. Regarding urban sprawl, a novel gridcell-based indicator model is established, based on different dimensions of urban sprawl.
Second, this thesis presents a three-step-based visualization method, addressing predefined requirements for geovisualizations and visualizing those indicator results, introduced above. This surface-visualization combines advantages from both common GIS representation and three-dimensional representation techniques within the field of urban planning, and is assisted by a web-based graphical user interface which allows for accessing the results by the public.
In addition, by focusing on local neighborhoods, this thesis provides an alternative approach in measuring and visualizing both indicators by utilizing a Neighborhood Relation Diagram (NRD), based on weighted Voronoi diagrams. Thus, the user is able to a) utilize original census data, b) compare direct impacts of indicator results on the neighboring cells, and c) compare both indicators of sustainability visually.
In its rather short history robotic research has come a long way in the half century since it started to exist as a noticeable scientic eld. Due to its roots in engineering, computer science, mathematics, and several other 'classical' scientic branches,a grand diversity of methodologies and approaches existed from the very beginning. Hence, the researchers in this eld are in particular used to adopting ideas that originate in other elds. As a fairly logical consequence of this, scientists tended to biology during the 1970s in order to nd approaches that are ideally adapted to the conditions of our natural environment. Doing so allows for introducing principles to robotics that have already shown their great potential by prevailing in a tough evolutionary selection process for millions of years. The variety of these approaches spans from efficient locomotion, to sensor processing methodologies and all the way to control architectures. Thus, the full spectrum of challenges for autonomous interaction with the surroundings while pursuing a task can be covered by such means. A feature that has proven to be amongst the most challenging to recreate is the human ability of biped locomotion. This is mainly caused by the fact that walking,running and so on are highly complex processes involving the need for energy efficient actuation, sophisticated control architectures and algorithms, and an elaborate mechanical design while at the same time posting restrictions concerning stability and weight. However, it is of special interest since our environment is favoring this specic kind of locomotion and thus promises to open up an enormous potential if mastered. More than the mere scientic interest, it is the fascination of understanding and recreating parts of oneself that drives the ongoing eorts in this area of research. The fact that this is not at all an easy task to tackle is not only caused by the highly dynamical processes but also has its roots in the challenging design process. That is because it cannot be limited to just one aspect like e.g. the control architecture, actuation, sensors, or mechanical design alone. Each aspect has to be incorporated into a sound general concept in order to allow for a successful outcome in the end. Since control is in this context inseparably coupled with the mechanics of the system, both has to be dealt with here.
Modern digital imaging technologies, such as digital microscopy or micro-computed tomography, deliver such large amounts of 2D and 3D-image data that manual processing becomes infeasible. This leads to a need for robust, flexible and automatic image analysis tools in areas such as histology or materials science, where microstructures are being investigated (e.g. cells, fiber systems). General-purpose image processing methods can be used to analyze such microstructures. These methods usually rely on segmentation, i.e., a separation of areas of interest in digital images. As image segmentation algorithms rarely adapt well to changes in the imaging system or to different analysis problems, there is a demand for solutions that can easily be modified to analyze different microstructures, and that are more accurate than existing ones. To address these challenges, this thesis contributes a novel statistical model for objects in images and novel algorithms for the image-based analysis of microstructures. The first contribution is a novel statistical model for the locations of objects (e.g. tumor cells) in images. This model is fully trainable and can therefore be easily adapted to many different image analysis tasks, which is demonstrated by examples from histology and materials science. Using algorithms for fitting this statistical model to images results in a method for locating multiple objects in images that is more accurate and more robust to noise and background clutter than standard methods. On simulated data at high noise levels (peak signal-to-noise ratio below 10 dB), this method achieves detection rates up to 10% above those of a watershed-based alternative algorithm. While objects like tumor cells can be described well by their coordinates in the plane, the analysis of fiber systems in composite materials, for instance, requires a fully three dimensional treatment. Therefore, the second contribution of this thesis is a novel algorithm to determine the local fiber orientation in micro-tomographic reconstructions of fiber-reinforced polymers and other fibrous materials. Using simulated data, it will be demonstrated that the local orientations obtained from this novel method are more robust to noise and fiber overlap than those computed using an established alternative gradient-based algorithm, both in 2D and 3D. The property of robustness to noise of the proposed algorithm can be explained by the fact that a low-pass filter is used to detect local orientations. But even in the absence of noise, depending on fiber curvature and density, the average local 3D-orientation estimate can be about 9° more accurate compared to that alternative gradient-based method. Implementations of that novel orientation estimation method require repeated image filtering using anisotropic Gaussian convolution filters. These filter operations, which other authors have used for adaptive image smoothing, are computationally expensive when using standard implementations. Therefore, the third contribution of this thesis is a novel optimal non-orthogonal separation of the anisotropic Gaussian convolution kernel. This result generalizes a previous one reported elsewhere, and allows for efficient implementations of the corresponding convolution operation in any dimension. In 2D and 3D, these implementations achieve an average performance gain by factors of 3.8 and 3.5, respectively, compared to a fast Fourier transform-based implementation. The contributions made by this thesis represent improvements over state-of-the-art methods, especially in the 2D-analysis of cells in histological resections, and in the 2D and 3D-analysis of fibrous materials.
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.