## Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications (Google eBook)Topology-based methods are of increasing importance in the analysis and visualization of datasets from a wide variety of scientific domains such as biology, physics, engineering, and medicine. Current challenges of topology-based techniques include the management of time-dependent data, the representation of large and complex datasets, the characterization of noise and uncertainty, the effective integration of numerical methods with robust combinatorial algorithms, etc. .
The editors have brought together the most prominent and best recognized researchers in the field of topology-based data analysis and visualization for a joint discussion and scientific exchange of the latest results in the field.
This book contains the best 20 peer-reviewed papers resulting from the discussions and presentations at the third workshop on 'Topological Methods in Data Analysis and Visualization', held 2009 in Snowbird, Utah, US. The 2009 'TopoInVis' workshop follows the two successful workshops in 2005 (Slovakia) and 2007 (Germany). |

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### Contents

1 | |

Stripe Parameterization of Tubular Surfaces | 13 |

The Stability of the Apparent Contour of an Orientable 2Manifold | 27 |

Reconstructing Cell Complexes From Crosssections | 43 |

Substructure Topology Preserving Simplification of Tetrahedral Meshes | 55 |

Practical Considerations in MorseSmale Complex Computation | 67 |

Modeling and Simplifying Morse Complexes in Arbitrary Dimensions | 79 |

Simplification of Jacobi Sets | 91 |

TimeDependent Visualization of Lagrangian Coherent Structures by Grid Advection | 151 |

Topological Extraction and Tracking of Defects in Crystal Structures | 166 |

Extracting and Visualizing Structural Features in Environmental Point Cloud LiDaR Data Sets | 179 |

Topological Flow Structures in a Mathematical Model for RotationMediated Cell Aggregation | 193 |

A Categorical Approach to Contour Split and Join Trees with Application to Airway Segmentation | 205 |

Complementary Space for Enhanced Uncertainty and Dynamics Visualization | 217 |

Topological Feature Extraction for Comparison of Terascale Combustion Simulation Data | 229 |

Feature Tracking Using Reeb Graphs | 241 |

Combinatorial 2D Vector Field Topology Extraction and Simplification | 103 |

On the Extraction of Longliving Features in Unsteady Fluid Flows | 115 |

Stream Volume Segmentation of GridLess Flow Simulation | 127 |

Eigenvectorbased Interpolation and Segmentation of 2D Tensor Fields | 139 |

Author Index | 254 |

255 | |

### Common terms and phrases

2-manifold advection algorithm analysis approach arcs atoms boundary cancellation cells classiﬁcation combinatorial complementary space component Computer Graphics connected contour tree critical points curve data set deﬁned deﬁnition degenerate point discrete dislocations domain dynamics Edelsbrunner edge contraction edges efﬁcient eigenvector extraction Figure ﬁltering ﬁnite ﬁrst ﬁxed ﬂame ﬂip ﬂow ﬂuid FTLE ridge function Galilean invariant geometry gradient grid identiﬁed IEEE incidence graph input integration interpolation intersection isosurface Jacobi set label Lemma level set linear Lyapunov exponent manifold mapping method Morse complexes Morse function Morse theory Morse-Smale complex neighborhood nodes operators particles Pascucci path line point cloud Reeb graph represent sampling scalar ﬁeld Section segmentation separatrices simplicial simpliﬁcation simulation speciﬁc stacking faults step stream lines stream surfaces structure sublevel set substructures techniques tensor ﬁeld tensorlines tetrahedral threshold topological skeleton tracking triangle vector ﬁeld vector ﬁeld topology vertex vertices visualization