Adaptive Contouring with Quadratic Tetrahedra

  • Benjamin F. Gregorski
  • David F. Wiley
  • Henry R. Childs
  • Bernd Hamann
  • Kenneth I. Joy
Part of the Mathematics and Visualization book series (MATHVISUAL)


We present an algorithm for adaptively extracting and rendering isosurfaces of scalar-valued volume datasets represented by quadratic tetrahedra. Hierarchical tetrahedral meshes created by longest-edge bisection are used to construct a multiresolution C0-continuous representation using quadratic basis functions. A new algorithm allows us to contour higher-order volume elements efficiently.


Quadratic Approximation Linear Element Tetrahedral Mesh Quadratic Element Volume Dataset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B.K. Bloomquist, Contouring Trivariate Surfaces, Masters Thesis, Arizona State University, Computer Science Department, 1990Google Scholar
  2. 2.
    P. Cignoni and P. Marino and C. Montani and E. Puppo and R. Scopigno Speeding Up Isosurface Extraction Using Interval Trees IEEE Transactions on Visualization and Computer Graphics 1991, 158–170Google Scholar
  3. 3.
    P. J. Davis Interpolation and Approximation Dover Publications, Inc., New York, NY. 2, 3Google Scholar
  4. 4.
    Klaus Engel and Rudiger Westermann and Thomas Ertl Isosurface Extraction Techniques For Web-Based Volume Visualization Proceedings of IEEE Visualization 1999, 139–146Google Scholar
  5. 5.
    Benjamin Gregorski, Mark Duchaineau, Peter Lindstrom, Valerio Pascucci, and Kenneth I. Joy Interactive View-Dependent Extraction of Large Isosurfaces Proceedings of IEEE Visualization 2002, 475–482Google Scholar
  6. 6.
    T. Gerstner Fast Multiresolution Extraction Of Multiple Transparent Isosurfaces, Data Visualization 2001 Proceedings of VisSim 2001Google Scholar
  7. 7.
    Thomas Gerstner and Renato Pajarola, Topology Preserving And Controlled Topology Simplifying Multiresolution Isosurface Extraction, Proceedings of IEEE Visualization 2000, 259–266Google Scholar
  8. 8.
    T. Gerstner and M. Rumpf, Multiresolution Parallel Isosurface Extraction Based On Tetrahedral Bisection, Volume Graphics 2000, 267–278Google Scholar
  9. 9.
    Leif P. Kobbelt, Mario Botsch, Ulrich Schwanecke, and Hans-Peter Seidel Feature-Sensitive Surface Extraction From Volume Data SIGGRAPH 2001 Conference Proceedings, 57–66Google Scholar
  10. 10.
    Y. Livnat and C. Hansen View Dependent Isosurface Extraction Proceedings of IEEE Visualization 1998, 172–180Google Scholar
  11. 11.
    Tao Ju, Frank Losasso, Scott Schaefer, and Joe Warren Dual contouring of hermite data SIGGRAPH 2002 Conference Proceedings, 339–346Google Scholar
  12. 12.
    Jian Huang, Yan Li, Roger Crawfis, Shao-Chiung Lu, and Shuh-Yuan Liou A Complete Distance Field Representation Proceedings of Visualization 2001, 247–254Google Scholar
  13. 13.
    Gerald Farin, Curves and Surfaces for CAGD, Fifth edition, Morgan Kaufmann Publishers Inc., San Francisco, CA, 2001Google Scholar
  14. 14.
    A.J. Worsey and G. Farin, Contouring a bivariate quadratic polynomial over a triangle, Computer Aided Geometric Design 7(1–4), 337–352, 1990MathSciNetGoogle Scholar
  15. 15.
    B. Hamann, I.J. Trotts, and G. Farin On Approximating Contours of the Piecewise Trilinear Interpolant using triangular rational-quadratic Bézier patches, IEEE Transactions on Visualization and Computer Graphics, 3(3), 315–337 1997CrossRefGoogle Scholar
  16. 16.
    Tom Roxborough and Gregory M. Nielson, Tetrahedron Based, Least Squares, Progressive Volume Models With Application To Freehand Ultrasound Data”, In Proceedings of IEEE Visualization 2000, 93–100Google Scholar
  17. 17.
    S. Marlow and M.J.D. Powell, A Fortran subroutine for plotting the part of a conic that is inside a given triangle, Report no. R 8336, Atomic Energy Research Establishment, Harwell, United Kingdom, 1976Google Scholar
  18. 18.
    R. Van Uitert, D. Weinstein, C.R. Johnson, and L. Zhukov Finite Element EEG and MEG Simulations for Realistic Head Models: Quadratic vs. Linear Approximations Special Issue of the Journal of Biomedizinische Technik, Vol. 46, 32–34, 2001.Google Scholar
  19. 19.
    David F. Wiley, H.R. Childs, Bernd Hamann, Kenneth I. Joy, and Nelson Max, Using Quadratic Simplicial Elements for Hierarchical Approximation and Visualization, Visualization and Data Analysis 2002, Proceedings, SPIE — The International Society for Optical Engineering, 32–43, 2002Google Scholar
  20. 20.
    David F. Wiley, H.R. Childs, Bernd Hamann, Kenneth I. Joy, and Nelson Max, Best Quadratic Spline Approximation for Hierarchical Visualization, Data Visualization 2002, Proceedings of VisSym 2002Google Scholar
  21. 21.
    D. F. Wiley, H. R. Childs, B. F. Gregorski, B. Hamann, and K. I. Joy Contouring Curved Quadratic Elements Data Visualization 2003, Proceedings of VisSym 2003Google Scholar
  22. 22.
    Jane Wilhelms and Allen Van Gelder Octrees for Faster Isosurface Generation ACM Transaction in Graphics, 201–227, July 1992Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Benjamin F. Gregorski
    • 1
  • David F. Wiley
    • 1
  • Henry R. Childs
    • 2
  • Bernd Hamann
    • 1
  • Kenneth I. Joy
    • 1
  1. 1.Institute For Data Analysis and VisualizationUniversity of CaliforniaDavis
  2. 2.B Division Lawrence Livermore National LaboratoryLivermore

Personalised recommendations