This Article 
 Bibliographic References 
 Add to: 
A Topological Hierarchy for Functions on Triangulated Surfaces
July/August 2004 (vol. 10 no. 4)
pp. 385-396

Abstract—We combine topological and geometric methods to construct a multiresolution representation for a function over a two-dimensional domain. In a preprocessing stage, we create the Morse-Smale complex of the function and progressively simplify its topology by cancelling pairs of critical points. Based on a simple notion of dependency among these cancellations, we construct a hierarchical data structure supporting traversal and reconstruction operations similarly to traditional geometry-based representations. We use this data structure to extract topologically valid approximations that satisfy error bounds provided at runtime.

[1] A. Cayley, On Contour and Slope Lines London, Edinburgh and Dublin Phil. Mag. J. Sci., vol. XVIII, pp. 264-268, 1859.
[2] J.C. Maxwell, On Hills and Dales London, Edinburgh and Dublin Phil. Mag. J. Sci., vol. XL, pp. 421-427, 1870.
[3] J. Pfaltz, Surface Networks Geographical Analysis, vol. 8, pp. 77-93, 1976.
[4] J. Pfaltz, A Graph Grammar that Describes the Set of Two-Dimensional Surface Networks, 1979.
[5] M. Morse, Relations between the Critical Points of a Real Function of$n$Independent Variables Trans. Am. Math. Soc., vol. 27, pp. 345-396, 1925.
[6] J. Milnor, Morse Theory. Princeton Univ. Press, 1963.
[7] H. Hoppe, Progressive Meshes Computer Graphics (Proc. SIGGRAPH), vol. 30, pp. 99-108, 1996.
[8] J. Popovic and H. Hoppe, Progressive Simplicial Complexes Computer Graphics (Proc. SIGGRAPH), vol. 31, pp. 209-216, 1997.
[9] M. Garland and P.S. Heckbert, Surface Simplification Using Quadric Error Metrics Computer Graphics (Proc. SIGGRAPH), vol. 31, pp. 209-216, 1997.
[10] P. Lindstrom and G. Turk, Fast and Memory Efficient Polygonal Simplification Proc. IEEE Visualization, pp. 279-286, 1998.
[11] T. He, L. Hong, A. Varshney, and S. Wang, "Controlled Topology Simplification," IEEE Trans. Visualization and Computer Graphics, vol. 2, no. 2, pp. 171-184, June 1996.
[12] J. El-Sana and A. Varshney, “Topology Simplification for Polygonal Virtual Environments,” IEEE Trans. Visualization and Computer Graphics, vol. 4, no. 2, pp. 133-144, Apr.-June 1997.
[13] J.L. Helman and L. Hesselink, "Visualization of Vector Field Topology in Fluid Flows," IEEE Computer Graphics and Applications, vol. 11, no. 3, pp. 36-46, 1991.
[14] W.C. de Leeuw and R. van Liere, Collapsing Flow Topology Using Area Metrics Proc. Visualization '99, pp. 349-354, Oct. 1999.
[15] X. Tricoche, G. Scheuermann, and H. Hagen, A Topology Simplification Method for 2D Vector Fields Proc. Visualization '00, pp. 359-366, Oct. 2000.
[16] X. Tricoche, G. Scheuermann, and H. Hagen, Continuous Topology Simplification of Planar Vector Fields Proc. Visualization '01, pp. 159-166, Oct. 2001.
[17] Topological Modeling for Visualization, A.T. Fomenko and T.L. Kunii, eds. Springer-Verlag, 1997.
[18] C.L. Bajaj and D.R. Schikore, Topology Preserving Data Simplification with Error Bounds Computers and Graphics, vol. 22, pp. 3-12, 1998.
[19] K. Hormann, Morphometrie der Erdoberfläche Schrift. Univ. Kiel, 1971.
[20] D.M. Mark, Topological Properties of Geographic Surfaces Proc. Advanced Study Symp. Topolological Data Structures, 1977.
[21] H. Edelsbrunner, D. Letscher, and A. Zomorodian, Topological Persistence and Simplification Discrete Computer Geometry, vol. 28, pp. 511-533, 2002.
[22] H. Edelsbrunner, J. Harer, and A. Zomorodian, Hierarchical Morse-Smale Complexes for Piecewise Linear 2-Mmanifolds Discrete Computer Geometry, vol. 30, pp. 87-107, 2003.
[23] H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci, Morse-Smale Complexes for Piecewise Linear 3-Manifolds Proc. 19th Ann. Symp. Computer Geometry, pp. 361-370, 2003.
[24] Y. Matsumoto, An Introduction to Morse Theory. Am. Math. Soc., 2002.
[25] J.R. Munkres, Elements of Algebraic Topology. Redwood City, Calif.: Addison-Wesley, 1984.
[26] P.S. Alexandrov, Combinatorial Topology. New York: Dover, 1998.
[27] T.F. Banchoff, Critical Points for Embedded Polyhedral Surfaces Am. Math. Monthly, vol. 77, pp. 457-485, 1970.
[28] H. Edelsbrunner and E.P. Mücke, Simulation of Simplicity: A Technique to Cope with Degenerate Cases in Geometric Algorithms ACM Trans. Graphics, vol. 9, pp. 66-104, 1990.
[29] J.C. Xia and A. Varshney, Dynamic View-Dependent Simplification for Polygonal Meshes Proc. Visualization '96, R. Yagel and G.M. Nielson, eds., pp. 327-334, 1996.
[30] H. Hoppe, View-Dependent Refinement of Progressive Meshes Computer Graphics (Proc. SIGGRAPH), vol. 31, pp. 189-198, 1997.
[31] R. Carlson and F.N. Fritsch, Monotone Piecewise Bicubic Interpolation J. Numerical Analysis, vol. 22, pp. 386-400, 1985.
[32] H. Greiner, A Survey on Univariate Data Interpolation and Approximation by Splines of Given Shape Math. Computer Modeling, vol. 15, pp. 97-106, 1991.
[33] G. Taubin, A Signal Processing Approach to Fair Surface Design Computer Graphics (Proc. SIGGRAPH), pp. 351-358, 1995.
[34] M.S. Floater, Mean Value Coordinates Computer Aided Geometric Design, vol. 20, pp. 19-27, 2003.
[35] M. Desbrun, M. Meyer, and P. Alliez, Intrinsic Parameterizations of Surface Meshes Computer Graphics Forum (Proc. Eurographics), vol. 21, 2002.
[36] A. Balázs, M. Guthe, and R. Klein, Fat Borders: Gap Filling for Efficient View-Dependent LOD Rendering Technical Report CG-2003-2, Univ. Bonn, Germany, 2003.
[37] E. Echekki and J.H. Chen, Direct Numerical Simulation of Autoignition in Non-Homogeneous Hydrogen-Air Mixtures Combustion Flame, 2003.

Index Terms:
Critical point theory, Morse-Smale complex, terrain data, simplification, multiresolution data structure.
"A Topological Hierarchy for Functions on Triangulated Surfaces," IEEE Transactions on Visualization and Computer Graphics, vol. 10, no. 4, pp. 385-396, July-Aug. 2004, doi:10.1109/TVCG.2004.3
Usage of this product signifies your acceptance of the Terms of Use.