This Article 
 Bibliographic References 
 Add to: 
Multiresolution Analysis on Irregular Surface Meshes
October-December 1998 (vol. 4 no. 4)
pp. 365-378

Abstract—Wavelet-based methods have proven their efficiency for the visualization at different levels of detail, progressive transmission, and compression of large data sets. The required core of all wavelet-based methods is a hierarchy of meshes that satisfies subdivision-connectivity: This hierarchy has to be the result of a subdivision process starting from a base mesh. Examples include quadtree uniform 2D meshes, octree uniform 3D meshes, or 4-to-1 split triangular meshes. In particular, the necessity of subdivision-connectivity prevents the application of wavelet-based methods on irregular triangular meshes. In this paper, a "wavelet-like" decomposition is introduced that works on piecewise constant data sets over irregular triangular surface meshes. The decomposition/reconstruction algorithms are based on an extension of wavelet-theory allowing hierarchical meshes without subdivision-connectivity property. Among others, this approach has the following features: It allows exact reconstruction of the data set, even for nonregular triangulations, and it extends previous results on Haar-wavelets over 4-to-1 split triangulations.

[1] G.-P. Bonneau, S. Hahmann, and G.M. Nielson, “BLaC-Wavelets: A Multi-Resolution Analysis with Non-Nested Spaces,” Proc. Visualization '96, R. Yagel and G.M. Nielson, eds., pp. 43-48, 1996.
[2] M. de Berg and K.T.G. Dobrindt, "On Levels of Detail in Terrains," Technical Report UU-CS-1995-12, Utrecht Univ., 1995.
[3] L. De Floriani,“A pyramidal data structure for triangle-based surface description,” IEEE Computer Graphics Applications, pp. 67-78, Mar. 1989.
[4] S.J. Gortler, P. Schroder, M.F. Cohen, and P. Hanrahan, "Wavelet Radiosity," Computer Graphics Proc., Ann. Conf. Series: SIGGRAPH '93,Anaheim, Calif., pp. 221-230, Aug. 1993.
[5] M.H. Gross, L. Lippert, R. Dittrich, and S. Häring, "Two Methods for Wavelet-Based Volume Rendering," Computers&Graphics, vol. 21, no. 2, pp. 237-252, 1997.
[6] H. Hoppe, “Progressive Meshes,” Proc. SIGGRAPH '96, pp. 99-108, 1996.
[7] H. Hoppe, “View-Dependent Refinement of Progressive Meshes,” Proc. SIGGRAPH '97, pp. 189-198, 1997.
[8] B. Jawerth and W. Sweldens, "An Overview of Wavelet Based Multiresolution Analyses," SIAM Rev., vol. 36, no. 3, pp. 377-412, 1994.
[9] D. Kirkpatrick, "Optimal Search in Planar Subdivisions," SIAM J. Computing, vol. 12, no. 1, pp. 28-35, Feb. 1983.
[10] J.M. Lounsbery, T. DeRose, and J. Warren, “Multiresolution Analysis for Surfaces of Arbitrary Topological Type,” ACM Trans. Graphics, vol. 16, no. 1, pp. 34-73, Jan. 1997.
[11] G.M. Nielson, I.-H. Jung, and J. Sung, Haar Wavelets over Triangular Domains with Applications to Multiresolution Models for Flow over a Sphere Proc. Visualization '97, R. Yagel and H. Hagen, eds., pp. 143-149, 1997.
[12] P. Schröder and W. Sweldens, "Spherical Wavelets: Efficiently Representing Functions on the Sphere," Proc. SIGGRAPH '95 Conf., pp. 161-172, 1995.
[13] P. Schroeder and W. Sweldens, "Spherical Wavelets: Texture Processing," Proc. Eurographics Rendering Workshop 1995, June 1995.
[14] E.J. Stollnitz, T.D. DeRose, and D.H. Salesin, Wavelets for Computer Graphics: Theory and Applications. Morgan Kaufmann, 1996.
[15] G. Strang, Introduction to Applied Mathematics. Wellesley-Cambridge Press, 1986.

Index Terms:
Wavelets, nonregular triangulations, compression, visualization.
Georges-Pierre Bonneau, "Multiresolution Analysis on Irregular Surface Meshes," IEEE Transactions on Visualization and Computer Graphics, vol. 4, no. 4, pp. 365-378, Oct.-Dec. 1998, doi:10.1109/2945.765329
Usage of this product signifies your acceptance of the Terms of Use.