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2017 | Book

Introduction Read chapter Read first chapter

Topological Data Analysis for Scientific Visualization

Author: Prof. Dr. Julien Tierny

Publisher: Springer International Publishing

Book Series : Mathematics and Visualization

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Combining theoretical and practical aspects of topology, this book provides a comprehensive and self-contained introduction to topological methods for the analysis and visualization of scientific data.

Theoretical concepts are presented in a painstaking but intuitive manner, with numerous high-quality color illustrations. Key algorithms for the computation and simplification of topological data representations are described in detail, and their application is carefully demonstrated in a chapter dedicated to concrete use cases.

With its fine balance between theory and practice, "Topological Data Analysis for Scientific Visualization" constitutes an appealing introduction to the increasingly important topic of topological data analysis for lecturers, students and researchers.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
In early 2013, a group of researchers led by French scientists published in Nature a paper entitled “A vast, thin plane of corotating dwarf galaxies orbiting the Andromeda galaxy” [69]. This paper reported new intriguing observations that showed that a majority of the dwarf galaxies which orbit the larger Andromeda galaxy was actually rotating in a very thin, common plane structure. These observations then contradicted the state-of-art models which assumed that dwarf galaxies’ locations followed an isotropic random distribution. This discovery raised many fundamental open questions that can potentially reshape the entire understanding of the universe formation process, as it implies that a still-to-be-found phenomenon seems to control the geometry of cosmos gas flow.
Julien Tierny
Chapter 2. Background
Abstract
This chapter introduces all the theoretical preliminaries required for the reading of the rest of the book. First, the input data representation is formalized. Second, some of the core concepts of Topological Data Analysis are presented, including critical points, notions of Persistent Homology, Reeb graphs and Morse-Smale Complexes. Finally, a brief review of the state-of-the-art algorithms is presented. For the reader’s convenience, the most important definitions and properties are highlighted with boxes. For further readings, the reader is referred to the excellent introduction to Computational Topology by Edelsbrunner and Harer (Computational Topology: An Introduction. American Mathematical Society, Providence, 2009).
Julien Tierny
Chapter 3. Abstraction
Abstract
This chapter describes examples of reference algorithms for the efficient and robust computation of topological abstractions, for the purpose of data abstraction in scientific visualization. First, we present a combinatorial technique for the topological simplification of scalar data, given some user-defined or application-driven constraints. The algorithm slightly perturbs the input data such that only a constrained sub-set of critical points remains. Thus, this technique can serve in practice as a pre-processing step that significantly speeds up the subsequent computation of topological abstractions. Second, we present an efficient algorithm for the computation of Reeb graphs of PL scalar fields defined on PL 3-manifolds in \(\mathbb {R}^3\). This approach described the first practical algorithm for volumetric meshes, with virtually linear scalability in practice and up to three orders of magnitude speedups with regard to previous work. Such an algorithm enabled for the first time the generalization of contour-tree based interactive techniques to non simply-connected domains.
Julien Tierny
Chapter 4. Interaction
Abstract
This chapter describes examples of efficient algorithms for the interactive manipulation of topological abstractions. We start by describing an approach for the interactive simplification of isosurfaces on non-simply connected domains, for visual exploration purposes. This approach is enabled by our fast Reeb graph computation algorithm, described in the previous chapter. Next, we present two algorithms for the editing of topological abstractions in the context of data segmentation. First, we describe how to integrate user constraints in the construction of a discrete gradient to incorporate user knowledge in Morse-Smale complex based segmentations. Second, we describe an approach for the interactive editing of the geometry and topology of a Reeb-graph based segmentation for surface quadrangulation purposes.
Julien Tierny
Chapter 5. Analysis
Abstract
This chapter describes concrete application examples for the quantitative analysis of scientific data based on topological methods. Starting from precise application problems, we describe how these algorithms can be adapted to conduct interactive data exploration and quantitative analysis. First, we describe how the split tree can be used to extract, enumerate and track flames through time in turbulent combustion simulations. While this approach is accompanied with an exploration user interface capable of tracking individual flames, we describe how this approach can also be used to derive quantitative measurements helping in the interpretation of the simulation. Second, we describe how the segmentation capabilities of the join tree and the Morse-Smale complex can be combined to analyze covalent and non-covalent interactions in molecular systems. Such an approach enables not only to robustly extract these features, but also the atoms involved in each localized interaction. For simple systems, our analysis corroborates the observations made by the chemists while it provides new visual insights for larger molecular systems.
Julien Tierny
Chapter 6. Perspectives
Abstract
This chapter discusses important scientific challenges that remain to be addressed in Topological Data Analysis for Scientific Visualization. In particular, we describe the ongoing evolution of the constraints and usages in scientific computing, along with their consequences on data analysis. Based on this discussion, we present a list of key problems to be addressed in the upcoming years. We also present tentative research directions, whose relevance will be supported by preliminary recent results which show great promise for the resolution of the upcoming challenges of Topological Data Analysis.
Julien Tierny
Chapter 7. Conclusion
Abstract
This book presented an overview of Topological Data Analysis for Scientific Visualization. After a concise tutorial and survey on the core concepts of Topological Data Analysis (Chap. 2), it presented examples of reference algorithms in the field, in particular in each of the following topics:
Julien Tierny
Backmatter
Metadata
Title
Topological Data Analysis for Scientific Visualization
Author
Prof. Dr. Julien Tierny
Copyright Year
2017
Electronic ISBN
978-3-319-71507-0
Print ISBN
978-3-319-71506-3
DOI
https://doi.org/10.1007/978-3-319-71507-0

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