Isosurfaces : Geometry, Topology, and Algorithms book cover
1st Edition

Geometry, Topology, and Algorithms

ISBN 9781466570979
Published June 24, 2013 by A K Peters/CRC Press
488 Pages 228 Color & 228 B/W Illustrations

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Book Description

Ever since Lorensen and Cline published their paper on the Marching Cubes algorithm, isosurfaces have been a standard technique for the visualization of 3D volumetric data. Yet there is no book exclusively devoted to isosurfaces. Isosurfaces: Geometry, Topology, and Algorithms represents the first book to focus on basic algorithms for isosurface construction. It also gives a rigorous mathematical perspective on some of the algorithms and results.

In color throughout, the book covers the Marching Cubes algorithm and variants, dual contouring algorithms, multilinear interpolation, multiresolution isosurface extraction, isosurfaces in four dimensions, interval volumes, and contour trees. It also describes data structures for faster isosurface extraction as well as methods for selecting significant isovalues.

For designers of visualization software, the book presents an organized overview of the various algorithms associated with isosurfaces. For graduate students, it provides a solid introduction to research in this area. For visualization researchers, the book serves as a reference to the vast literature on isosurfaces.

Table of Contents

What Are Isosurfaces?
Applications of Isosurfaces
Isosurface Properties
Isosurface Construction
Limitations of Isosurfaces
Multi-Valued Functions and Vector Fields
Definitions and Basic Techniques

Marching Cubes and Variants
Marching Squares
Marching Cubes
Marching Tetrahedra

Dual Contouring
Surface Nets
Dual Marching Cubes
Comparison with Marching Cubes

Multilinear Interpolation
Bilinear Interpolation: 2D
The Asymptotic Decider: 3D
Trilinear Interpolation

Isosurface Patch Construction
Definitions and Notation
Isosurface Patch Construction
Isosurface Table Construction
Marching Polyhedra Algorithm

Isosurface Generation in 4D
Definitions and Notation
Isosurface Table Generation in 4D
Marching Hypercubes
Marching Simplices
Marching Polytopes
4D Isohull
4D Surface Nets

Interval Volumes
Definitions and Notation
Automatic Table Generation
MCVol Interval Volume Properties
Tetrahedral Meshes
Convex Polyhedral Meshes

Data Structures
Uniform Grid Partitions
Span Space Priority Trees
Seed Sets

Multiresolution Tetrahedral Meshes
Bisection of Tetrahedra
Multiresolution Isosurfaces

Multiresolution Polyhedral Meshes
Multiresolution Convex Polyhedral Mesh
Multiresolution Surface Nets
Multiresolution in 4D

Counting Grid Vertices
Counting Grid Edges and Grid Cubes
Measuring Gradients

Contour Trees
Examples of Contour Trees
Definition of Contour Tree
Join, Split and Merge Trees
Constructing Join, Split and Merge Trees
Constructing Contour Trees
Theory and Proofs
Simplification of Contour Trees

Appendix A: Geometry
Appendix B: Topology
Appendix C: Graph Theory
Appendix D: Notation



Notes and Comments appear at the end of each chapter.

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Rephael Wenger is an associate professor in the Department of Computer Science and Engineering at the Ohio State University. He earned a Ph.D. from McGill University. He has published over fifty papers in computational geometry, computational topology, combinatorics, geometric modeling, and visualization.


"Visualization has long needed a solid, standard and detailed text on the algorithmic aspects of isosurface construction and use. This text will become the standard entry point into this vast literature for at least the next decade, even for researchers already accustomed to working with isosurfaces. It belongs on every professional’s shelf."
—Hamish Carr, University of Leeds

"Isosurfaces are one of the most prevalent ways to visualize three-dimensional data. This wonderful book is the first that nicely summarizes the foundations as well as the state of the art on isosurfaces. Everyone, from the novice to the expert, will find something new and interesting in this book. This book's treatment of isosurfaces goes way beyond the surface, deep into the heart and soul of this rich topic situated in between the fields of graphics, visualization, and computational geometry."
—Torsten Möller, University of Vienna (Universität Wien)

"…well written, well illustrated, and extensively referenced."
—Lyuba S. Alboul, Mathematical Reviews Clippings, January 2015