Scalar fields are used to represent physical quantities measured over a domain of interest. Study of symmetric or repeating patterns in scalar fields is important in scientific data analysis because it gives deep insights into the properties of the underlying phenomenon. This thesis proposes three methods to detect symmetry in scalar fields. The first method models symmetry detection as a subtree matching problem in the contour tree, which is a topological graph abstraction of the scalar field. The contour tree induces a hierarchical segmentation of features at different scales and hence this method can detect symmetry at different scales. The second method identifies symmetry by comparing distances between extrema from each symmetric region. The distance is computed robustly using a topological abstraction called the extremum graph. Hence, this method can detect symmetry even in the presence of significant noise. The above methods compare pairs of regions to identify symmetry instead of grouping the entire set of symmetric regions as a cluster. This motivates the third method which uses a clustering analysis for symmetry detection. In this method, the contours of a scalar field are mapped to points in a high-dimensional descriptor space such that points corresponding to similar contours lie in close proximity to each other. Symmetry is identified by clustering the points in the descriptor space. We show through experiments on real world data sets that these methods are robust in the presence of noise and can detect symmetry under different types of transformations. Extraction of symmetry information helps users in visualization and data analysis. We design novel applications that use symmetry information to enhance visualization of scalar field data and to facilitate their exploration.

This thesis is about visualizing a kind of data that is trivial to process by computers but difficult to imagine by humans because nature does not allow for intuition with this type of information: high-dimensional data. Such data often result from representing observations of objects under various aspects or with different properties. In many applications, a typical, laborious task is to find related objects or to group those that are similar to each other. One classic solution for this task is to imagine the data as vectors in a Euclidean space with object variables as dimensions. Utilizing Euclidean distance as a measure of similarity, objects with similar properties and values accumulate to groups, so-called clusters, that are exposed by cluster analysis on the high-dimensional point cloud. Because similar vectors can be thought of as objects that are alike in terms of their attributes, the point cloud\''s structure and individual cluster properties, like their size or compactness, summarize data categories and their relative importance. The contribution of this thesis is a novel analysis approach for visual exploration of high-dimensional point clouds without suffering from structural occlusion. The work is based on implementing two key concepts: The first idea is to discard those geometric properties that cannot be preserved and, thus, lead to the typical artifacts. Topological concepts are used instead to shift away the focus from a point-centered view on the data to a more structure-centered perspective. The advantage is that topology-driven clustering information can be extracted in the data\''s original domain and be preserved without loss in low dimensions. The second idea is to split the analysis into a topology-based global overview and a subsequent geometric local refinement. The occlusion-free overview enables the analyst to identify features and to link them to other visualizations that permit analysis of those properties not captured by the topological abstraction, e.g. cluster shape or value distributions in particular dimensions or subspaces. The advantage of separating structure from data point analysis is that restricting local analysis only to data subsets significantly reduces artifacts and the visual complexity of standard techniques. That is, the additional topological layer enables the analyst to identify structure that was hidden before and to focus on particular features by suppressing irrelevant points during local feature analysis. This thesis addresses the topology-based visual analysis of high-dimensional point clouds for both the time-invariant and the time-varying case. Time-invariant means that the points do not change in their number or positions. That is, the analyst explores the clustering of a fixed and constant set of points. The extension to the time-varying case implies the analysis of a varying clustering, where clusters appear as new, merge or split, or vanish. Especially for high-dimensional data, both tracking---which means to relate features over time---but also visualizing changing structure are difficult problems to solve.

<p>The topic of the thesis is analysis of the topological structure of scalar fields and<br />shapes represented through Morse and cell complexes, respectively. This is<br />achieved by defining simplification and refinement operators on these<br />complexes. It is shown that the defined operators form a basis for the set of<br />operators that modify Morse and cell complexes. Based on the defined<br />operators, a multi-resolution model for Morse and cell complexes is constructed,<br />which contains a large number of representations at uniform and variable<br />resolution.</p> / <p>Тема дисертације је анализа тополошке структуре скаларних поља и<br />облика представљених у облику комплекса Морза и ћелијских комплекса,<br />редом. То се постиже дефинисањем оператора за симплификацију и<br />рафинацију тих комплекса. Показано је да дефинисани оператори чине<br />базу за скуп оператора на комплексима Морза и ћелијским комплексима.<br />На основу дефинисаних оператора конструисан је мулти-резолуциони<br />модел за комплексе Морза и ћелијске комплексе, који садржи велики број<br />репрезентација униформне и варијабилне резолуције.</p> / <p>Tema disertacije je analiza topološke strukture skalarnih polja i<br />oblika predstavljenih u obliku kompleksa Morza i ćelijskih kompleksa,<br />redom. To se postiže definisanjem operatora za simplifikaciju i<br />rafinaciju tih kompleksa. Pokazano je da definisani operatori čine<br />bazu za skup operatora na kompleksima Morza i ćelijskim kompleksima.<br />Na osnovu definisanih operatora konstruisan je multi-rezolucioni<br />model za komplekse Morza i ćelijske komplekse, koji sadrži veliki broj<br />reprezentacija uniformne i varijabilne rezolucije.</p>