Natallia Kotava

• 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 cost. 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.