Preface; Contents; Part I Topology-Based Analysis of Multi-Variate Data Sets; Theory of Singular Fibers and Reeb Spaces for Visualization; 1 Introduction; 2 Morse Functions; 2.1 Functions on Manifolds Without Boundary; 2.2 Functions on Manifolds with Boundary; 2.3 Deformations of Morse Functions; 3 Stable Maps; 3.1 Notion of Stable Maps; 3.2 Characterization of Stable Maps for Specific Cases; 4 Singular Fibers; 4.1 Concept; 4.2 Ehresmann Fibration Theorem; 4.3 Classification Results; 5 Reeb Space; 5.1 Notion of Reeb Space; 5.2 Deformation of Reeb Spaces; 6 Future Problems

6.1 Impact on Mathematics6.2 Impact on Computational Topology and Visualization; References; Topology-Based Analysis for Multimodal Atmospheric Data of Volcano Eruptions; 1 Introduction; 2 Related Work; 3 Description of the Application and Data Sources; 3.1 Input Data Sources; 4 Analysis of the Common Reference Domain; 4.1 Construction of the Common Reference Domain; 4.2 Topological Analysis in the Common Reference Domain; 4.3 Topology-Based Analysis and Visualization; 5 Results; 5.1 Limitations; 6 Conclusion; References; A Comparison of Joint Contour Nets and Pareto Sets; 1 Introduction

2 Previous Work3 Joint Contour Nets; 4 Pareto Sets; 5 Runtime Complexity Comparison; 6 Theoretical Comparison; 7 Visual Comparison; 8 Conclusion; References; Part II Topological Techniques for High-Dimensional Data; Visualizing Topological Properties of the Search Landscape of Combinatorial Optimization Problems; 1 Introduction; 2 Background; 3 Related Work; 4 Topological Analysis of the Search Landscape; 4.1 Alternative Approaches; 5 Topological Visualization of the Search Landscape; 5.1 Design Alternatives; 6 Confirmation of the Method: Topological Analysis of the Search Landscape of TSP

7 Conclusions and Future WorkReferences; Computing and Visualizing Time-Varying Merge Treesfor High-Dimensional Data; 1 Introduction; 2 Related Work; 3 Background; 4 Overview; 5 Merge Tree Transformation; 6 Feature Tracking; 7 Simplification; 8 Prototype Visualization and Examples; 8.1 2-D Example; 8.2 High-Dimensional Example; 9 Conclusion; References; Agreement Analysis of Quality Measures for DimensionalityReduction; 1 Introduction; 2 Related Work; 3 Quality Measures; 4 Methods; 4.1 Scalar Field Decomposition Using Persistent Homology; 4.1.1 The 1-Dimensional Case

4.1.2 The High-Dimensional Case4.1.3 Threshold Selection; 4.2 Similarity Measure and Quality Visualization; 5 Results; 5.1 Swiss Roll; 5.2 Handwritten Digits; 5.3 Concrete Compressive Strength; 6 Conclusion; References; Part III Scalar Field Topology; Fast Similarity Search in Scalar Fields using Merging Histograms; 1 Introduction and Related Work; 2 Background and Notation; 3 Merging Histograms; 3.1 Histograms as Feature Descriptors; 3.2 Computation; 4 Similarity Search Using Merging Histograms; 4.1 Distance Measure; 4.2 Querying the Distance Matrix; 5 Results; 6 Discussion