This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Simplification of Tetrahedral Meshes with Error Bounds
July-September 1999 (vol. 5 no. 3)
pp. 224-237

Abstract—We present a method for the construction of multiple levels of tetrahedral meshes approximating a trivariate scalar-valued function at different levels of detail. Starting with an initial, high-resolution triangulation of a three-dimensional region, we construct coarser representation levels by collapsing edges of the mesh. Each triangulation defines a linear spline function, where the function values associated with the vertices are the spline coefficients. Error bounds are stored for individual tetrahedra and are updated as the mesh is simplified. Two algorithms are presented that simplify the mesh within prescribed error bounds. Each algorithm treats simplification on the mesh boundary. The result is a hierarchical data description suited for efficient visualization of large data sets at varying levels of detail.

[1] W.J. Schroeder, J.A. Zarge, and W.E. Lorensen, “Decimation of Triangle Meshes,” Proc. SIGGRAPH '92, pp. 65-70, 1992.
[2] K.J. Renze and J.H. Oliver, Generalized Unstructured Decimation IEEE Computer Graphics and Applications, vol. 16, no. 6, pp. 24-32, Nov. 1996.
[3] H. Hoppe, “Progressive Meshes,” Proc. SIGGRAPH '96, pp. 99-108, 1996.
[4] H. Hoppe, “View-Dependent Refinement of Progressive Meshes,” Proc. SIGGRAPH '97, pp. 189-198, 1997.
[5] J. Popovic and H. Hoppe, “Progressive Simplicial Complexes,” Proc. SIGGRAPH '97, pp. 217-224, 1997.
[6] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Mesh Optimization,” Proc. SIGGRAPH '93, pp. 19-26, 1993.
[7] J. Xia and A. Varshney, "Dynamic View-Dependent Simplification for Polygonal Models," Proc. IEEE Visualization 96, ACM Press, New York, 1996, pp. 327-334.
[8] M. Garland and P.S. Heckbert, "Surface Simplification Using Quadric Error Metrics," Proc. Siggraph 97, ACM Press, New York, 1997, pp. 209-216.
[9] O.G. Staadt and M.H. Gross, “Progressive Tetrahedralizations,” Proc. IEEE Visualization '98, pp. 397-402, 1998.
[10] P. Cignoni, L. De Floriani, C. Montoni, E. Puppo, and R. Scopigno, "Multiresolution Modeling and Visualization of Volume Data Based on Simplicial Complexes," Proc. 1994 Symp. Volume Visualization, pp. 19-26, 1994.
[11] P. Cignoni, C. Montani, E. Puppo, and R. Scopigno, Multiresolution Modeling and Visualization of Volume Data IEEE Trans. Visualization and Computer Graphics, vol. 3, no. 4, pp. 352-369, Oct.-Dec. 1997.
[12] B. Hamann, "A Data Reduction Scheme for Triangulated Surfaces," Computer Aided Geometric Design, vol. 11, no. 2, pp. 197-214 1994.
[13] B. Hamann and J.-L. Chen, “Data Point Selection for Piecewise Linear Curve Approximation,” Computer-Aided Geometric Design, vol. 11, no. 3, pp. 289-301, June 1994.
[14] T.S. Gieng, B. Hamann, K.I. Joy, G.L. Schussman, and I.J. Trotts, Smooth Hierarchical Surface Triangulations Proc. Visualization '97, R. Yagel and H. Hagen, eds., pp. 379-386, 1997.
[15] T.S. Gieng, B. Hamann, K.I. Joy, G.L. Schussman, and I.J. Trotts, Constructing Hierarchies for Triangle Meshes IEEE Trans. Visualization and Computer Graphics, vol. 4, no. 2, pp. 145-161, Apr.-June 1998.
[16] J. Cohen, M. Olano, and D. Manocha, Simplifying Polygonal Models Using Successive Mappings Proc. Visualization '97, pp. 395-402, Oct. 1997.
[17] C.L. Bajaj and D.R. Schikore, “Topology Preserving Data Simplification with Error Bounds,” Computers and Graphics, vol. 22, no. 1, pp. 3-12, 1998.
[18] H. Hoppe, “Smooth View-Dependent Level-of-Detail Control and Its Application to Terrain Rendering,” Proc. IEEE Visualization '98, pp. 35-42, Oct. 1998.
[19] R. Klein, G. Liebich, and W. Straßer, "Mesh Reduction With Error Control," Proc. Visualization, 1996.
[20] I.J. Trotts, B. Hamann, K.I. Joy, and D.F. Wiley, “Simplification of Tetrahedral Meshes,” Proc. Visualization '98, D. Ebert, H. Hagen, and H.E. Rushmeier, eds., pp. 287-295, 1998.
[21] Scientific Visualization: Overviews, Methodologies, and Techniques, G. Nielson, H. Müller, and H. Hagen, eds. Academic Press, 1997.
[22] B. Wyvill, C. McPheeters, and G. Wyvill, “Animating of Soft Objects,” The Visual Computer, vol. 2, no. 4, pp. 235-242, 1986.
[23] S.-M. Lei, "Forward Error Correction Codes for MPEG-2 over ATM," IEEE Trans. Circuits and Systems for Video Tech., Vol. 4, No. 2, Apr. 1994, pp. 200-203.

Index Terms:
Mesh simplification, hierarchical representation, multiresolution method, scattered data, spline, tetrahedral mesh, visualization.
Citation:
Issac J. Trotts, Bernd Hamann, Kenneth I. Joy, "Simplification of Tetrahedral Meshes with Error Bounds," IEEE Transactions on Visualization and Computer Graphics, vol. 5, no. 3, pp. 224-237, July-Sept. 1999, doi:10.1109/2945.795214
Usage of this product signifies your acceptance of the Terms of Use.