Topological Methods in Data Analysis and Visualization II: Theory, Algorithms, and Applications

✍ Scribed by Jan Reininghaus, Ingrid Hotz (auth.), Ronald Peikert, Helwig Hauser, Hamish Carr, Raphael Fuchs (eds.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2012
• Tongue: English
• Leaves: 312
• Series: Mathematics and Visualization
• Edition: 1
• Category: Library

✦ Synopsis

When scientists analyze datasets in a search for underlying phenomena, patterns or causal factors, their first step is often an automatic or semi-automatic search for structures in the data. Of these feature-extraction methods, topological ones stand out due to their solid mathematical foundation. Topologically defined structures—as found in scalar, vector and tensor fields—have proven their merit in a wide range of scientific domains, and scientists have found them to be revealing in subjects such as physics, engineering, and medicine.

Full of state-of-the-art research and contemporary hot topics in the subject, this volume is a selection of peer-reviewed papers originally presented at the fourth Workshop on Topology-Based Methods in Data Analysis and Visualization, TopoInVis 2011, held in Zurich, Switzerland. The workshop brought together many of the leading lights in the field for a mixture of formal presentations and discussion. One topic currently generating a great deal of interest, and explored in several chapters here, is the search for topological structures in time-dependent flows, and their relationship with Lagrangian coherent structures. Contributors also focus on discrete topologies of scalar and vector fields, and on persistence-based simplification, among other issues of note. The new research results included in this volume relate to all three key areas in data analysis—theory, algorithms and applications.

✦ Table of Contents

Front Matter....Pages i-xi
Front Matter....Pages 1-1
Computational Discrete Morse Theory for Divergence-Free 2D Vector Fields....Pages 3-14
Efficient Computation of a Hierarchy of Discrete 3D Gradient Vector Fields....Pages 15-29
Computing Simply-Connected Cells in Three-Dimensional Morse-Smale Complexes....Pages 31-45
Combinatorial Vector Field Topology in Three Dimensions....Pages 47-59
Front Matter....Pages 61-61
Topological Cacti: Visualizing Contour-Based Statistics....Pages 63-76
Enhanced Topology-Sensitive Clustering by Reeb Graph Shattering....Pages 77-90
Efficient Computation of Persistent Homology for Cubical Data....Pages 91-106
Front Matter....Pages 107-107
Visualizing Invariant Manifolds in Area-Preserving Maps....Pages 109-124
Understanding Quasi-Periodic Fieldlines and Their Topology in Toroidal Magnetic Fields....Pages 125-140
Consistent Approximation of Local Flow Behavior for 2D Vector Fields Using Edge Maps....Pages 141-159
Cusps of Characteristic Curves and Intersection-Aware Visualization of Path and Streak Lines....Pages 161-175
Glyphs for Non-Linear Vector Field Singularities....Pages 177-190
2D Asymmetric Tensor Field Topology....Pages 191-204
Front Matter....Pages 205-205
On the Elusive Concept of Lagrangian Coherent Structures....Pages 207-220
Ridge Concepts for the Visualization of Lagrangian Coherent Structures....Pages 221-235
Filtering of FTLE for Visualizing Spatial Separation in Unsteady 3D Flow....Pages 237-253
A Variance Based FTLE-Like Method for Unsteady Uncertain Vector Fields....Pages 255-268
On the Finite-Time Scope for Computing Lagrangian Coherent Structures from Lyapunov Exponents....Pages 269-281
Scale-Space Approaches to FTLE Ridges....Pages 283-296
Back Matter....Pages 297-299

✦ Subjects

Visualization; Algorithms; Computing Methodologies; Computer Graphics