Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration

Visualization is one of the most active and exciting areas of Mathematics and Computing Science, and indeed one which is only beginning to mature. Current visualization algorithms break down for very large data sets. While present approaches use multi-resolution ideas, future data sizes will not be handled that way. New algorithms based on sophisticated mathematical modeling techniques must be devised which will permit the extraction of high-level topological structures that can be visualized. For these reasons a workshop was organized at the Banff International Research Station, focused specifically on mathematical issues. A primary objective of the workshop was to gather together a diverse set of researchers in the mathematical areas relevant to the recent advances in order to discuss the research challenges facing this field in the next several years. The workshop was organized into five different thrusts: - Topology and Discrete Methods - Signal and Geometry Processing - Partial Differential Equations - Data Approximation Techniques - Massive Data Applications This book presents a summary of the research ideas presented at this workshop.

Here is a summary of cutting-edge research ideas on visualization presented at a key workshop in Banff, Canada, where new algorithms were discussed based on sophisticated modeling techniques that permit the extraction of high-level topological structures.

1. Maximizing Adaptivity in Hierarchical Topological Models Using Cancellation Trees Peer-Timo Bremer, Valerio Pascucci, and Bernd Hamann.- 2. The TOPORRERY: computation and presentation of multi-resolution topology Valerio Pascucci, Kree Cole-McLaughlin, and Giorgio Scorzelli.- 3. Isocontour based Visualization of Time-varying Scalar Fields Ajith Mascarenhas, and Jack Snoeyink.- 4. DeBruijn Counting for Visualization Algorithms David C. Banks and Paul K. Stockmeyer.- 5. Topological Methods for Visualizing Vortical Flows Xavier Tricoche and Christoph Garth.- 6. Stability and Computation of Medial Axes - a State-of-the-Art Report Dominique Attali, Jean-Daniel Boissonnat, and Herbert Edelsbrunner.- 7. Local Geodesic Parametrization: An Ant's Perspective Lior Shapira and Ariel Shamir.- 8. Tensor-fields Visualization using a Fabric like Texture to Arbitrary two-dimensional Surfaces Ingrid Hotz, Louis Feng, Bernd Hamann, and Kenneth Joy.- 9. Flow Visualization via Partial Differential Equations T. Preusser, M. Rumpf, and A. Telea.- 10. Iterative Twofold Line Integral Convolution for Texture-Based Vector Field Visualization Daniel Weiskopf.- 11. Constructing 3D Elliptical Gaussians for Irregularly Gridded Data Wei Hong, Neophytos Neophytou, Klaus Mueller and Arie Kaufman.- 12. From Sphere Packing to the Theory of Optimal Lattice Sampling Alireza Entezari, Ramsay Dyer, and Torsten Möller.- 13. Reducing Interpolation Artifacts by Globally Fairing Contours Martin Bertram and Hans Hagen.- 14. Time- and Space-efficient Error Calculation for Multiresolution Direct Volume Rendering Attila Gyulassy, Lars Linsen, and Bernd Hamann.- 15. Massive Data Visualization: A Survey Kenneth I. Joy.- 16. Compression and Occlusion Culling for Fast Isosurface Extraction from Massive Datasets Benjamin Gregorski, Joshua Senecal, Mark Duchaineau, and Kenneth I. Joy.- 17. Volume Visualization of Multiple Alignment of Large Genomic DNA Nameeta Shah, Scott E. Dillard, Gunther H. Weber, Bernd Hamann.- 18. Model-based Visualization - Computing Perceptually Optimal Visualizations J.J. van Wijk.