1st Edition

Isosurfaces Geometry, Topology, and Algorithms

By Rephael Wenger Copyright 2013
    488 Pages 228 Color & 228 B/W Illustrations
    by A K Peters/CRC Press

    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.

    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.


    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