## Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data ExplorationTorsten Möller, Bernd Hamann, Robert D. Russell The goal of visualization is the accurate, interactive, and intuitive presentation of data. Complex numerical simulations, high-resolution imaging devices and incre- ingly common environment-embedded sensors are the primary generators of m- sive data sets. Being able to derive scienti?c insight from data increasingly depends on having mathematical and perceptual models to provide the necessary foundation for effective data analysis and comprehension. The peer-reviewed state-of-the-art research papers included in this book focus on continuous data models, such as is common in medical imaging or computational modeling. From the viewpoint of a visualization scientist, we typically collaborate with an application scientist or engineer who needs to visually explore or study an object which is given by a set of sample points, which originally may or may not have been connected by a mesh. At some point, one generally employs low-order piecewise polynomial approximationsof an object, using one or several dependent functions. In order to have an understanding of a higher-dimensional geometrical “object” or function, ef?cient algorithms supporting real-time analysis and manipulation (- tation, zooming) are needed. Often, the data represents 3D or even time-varying 3D phenomena (such as medical data), and the access to different layers (slices) and structures (the underlying topology) comprising such data is needed. |

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### Contents

1 | |

Computation and Presentation of Multiresolution Topology | 19 |

Isocontour Based Visualization of TimeVarying Scalar Fields | 41 |

DeBruijn Counting for Visualization Algorithms | 69 |

Topological Methods for Visualizing Vortical Flows | 89 |

A StateoftheArt Report | 109 |

An Ants Perspective | 126 |

TensorFields Visualization Using a Fabriclike Texture Applied to Arbitrary Twodimensional Surfaces | 139 |

Constructing 3D Elliptical Gaussians for Irregular Data | 212 |

From Sphere Packing to the Theory of Optimal Lattice Sampling | 227 |

Reducing Interpolation Artifacts by Globally Fairing Contours | 256 |

Time and SpaceEfficient Error Calculation for Multiresolution Direct Volume Rendering | 271 |

A Survey | 284 |

Compression and Occlusion Culling for Fast Isosurface Extraction from Massive Datasets | 303 |

Volume Visualization of Multiple Alignment of Large Genomic DNA | 325 |

Computing Perceptually Optimal Visualizations | 343 |

Flow Visualization via Partial Differential Equations | 156 |

Iterative Twofold Line Integral Convolution for TextureBased Vector Field Visualization | 191 |

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algorithm alignment alpha blending applications approximation boundary box spline branch cancellation Cartesian clustering color components compression Computer Graphics conﬁguration construct Contour Tree corresponding critical points curve dataset deﬁned deﬁnition density diamonds diffusion discrete display domain efﬁcient eigenvalues encoding equation error extraction ﬁlter ﬁnd ﬁnite ﬁrst ﬁxed ﬂow ﬂow ﬁeld ﬂow visualization Fourier geodesic geometry gradient Hausdorff distance hierarchy Hilbert curve histogram IEEE Visualization integral interactive interpolation isocontour isosurface isovalue kernel lattice level set linear Mathematical matrix medial axis mesh method Morse function Morse–Smale complex multiresolution node occlusion culling occlusion queries octree optimal parameter permutation Proc Proceedings reconstruction Reeb graph reﬁned reﬁnement regions representation saddle points sampling scalar ﬁeld sequence shape shows simpliﬁcation space speciﬁc splatting split streamlines structure surface techniques tensor ﬁeld tetrahedra time-varying tion topology transform triangles twofold convolution vertex vertices volume rendering Voronoi cell