Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner.

  • [1]  Burak Aksoylu, Stephen Bond and , Michael Holst, An odyssey into local refinement and multilevel preconditioning. III. Implementation and numerical experiments, SIAM J. Sci. Comput., 25 (2003), 478–498 2058071 LinkISIGoogle Scholar

  • [2]  B. Aksoylu and M. Holst, An odyssey into local refinement and multilevel preconditioning II: Stabilizing hierarchical basis methods, SIAM J. Numer. Anal., submitted. Google Scholar

  • [3]  Randolph Bank, Todd Dupont and , Harry Yserentant, The hierarchical basis multigrid method, Numer. Math., 52 (1988), 427–458 89b:65247 CrossrefISIGoogle Scholar

  • [4]  Randolph Bank and , Jinchao Xu, An algorithm for coarsening unstructured meshes, Numer. Math., 73 (1996), 1–36 10.1007/s002110050181 97c:65055 CrossrefISIGoogle Scholar

  • [5]  D. Baraff and A. Witkin, Large steps in cloth simulation, in Proceedings of the SIGGRAPH, 1998, pp. 43–54. Google Scholar

  • [6]  James Bramble, Joseph Pasciak and , Jinchao Xu, Parallel multilevel preconditioners, Math. Comp., 55 (1990), 1–22 90k:65170 CrossrefISIGoogle Scholar

  • [7]  William Briggs, Van Henson and , Steve McCormick, A multigrid tutorial, Society for Industrial and Applied Mathematics (SIAM), 2000xii+193 2001h:65002 LinkGoogle Scholar

  • [8]  Tony Chan, Jinchao Xu and , Ludmil Zikatanov, An agglomeration multigrid method for unstructured grids, Contemp. Math., Vol. 218, Amer. Math. Soc., Providence, RI, 1998, 67–81 99h:65205 Google Scholar

  • [9]  U. Clarenz, M. Griebel, M. Rumpf, A. Schweitzer and , and A. Telea, A feature sensitive multiscale editing tool on surfaces, Visual Computer, 29 (2004), pp. 329–343. 8ua VICOE5 0178-2789 Visual Comput. CrossrefGoogle Scholar

  • [10]  B. Curless and M. Levoy, A volumetric method for building complex models from range images, in Proceedings of the SIGGRAPH, 1996, pp. 303–312. Google Scholar

  • [11]  E. de Sturler and J. Liesen, Block‐diagonal and constraint preconditioners for nonsymmetric indefinite linear systems, Part I: Theory, SIAM J. Sci. Comput., to appear. Google Scholar

  • [12]  Klaus Deckelnick and , Gerhard Dziuk, A fully discrete numerical scheme for weighted mean curvature flow, Numer. Math., 91 (2002), 423–452 2003k:65145 CrossrefISIGoogle Scholar

  • [13]  Klaus Deckelnick and , Gerhard Dziuk, Mean curvature flow and related topics, Universitext, Springer, Berlin, 2003, 63–108 2005e:65016 Google Scholar

  • [14]  James Demmel, Stanley Eisenstat, John Gilbert, Xiaoye Li and , Joseph Liu, A supernodal approach to sparse partial pivoting, SIAM J. Matrix Anal. Appl., 20 (1999), 720–755 10.1137/S0895479895291765 99m:65094 LinkISIGoogle Scholar

  • [15]  M. Desbrun, M. Meyer and , and P. Alliez, Intrinsic parameterizations of surface meshes, Computer Graphics Forum, 21 (2002), pp. 209–218. bjf ZZZZZZ 1067-7055 Comput. Graph. Forum CrossrefISIGoogle Scholar

  • [16]  M. Desbrun, M. Meyer, P. Schröder, and A. Barr, Implicit fairing of irregular meshes using diffusion and curvature flow, in Proceedings of the SIGGRAPH, 1999, pp. 317–324. Google Scholar

  • [17]  M. Desbrun, M. Meyer, P. Schröder, and A. Barr, Discrete differential‐geometry operators for triangulated 2‐manifolds, in VisMath’02, Berlin, Germany, 2002. Google Scholar

  • [18]  Tamal Dey, Herbert Edelsbrunner, Sumanta Guha and , Dmitry Nekhayev, Topology preserving edge contraction, Publ. Inst. Math. (Beograd) (N.S.), 66(80) (1999), 23–45, Geometric combinatorics (Kotor, 1998) 2001f:57026 Google Scholar

  • [19]  David Dobkin and , David Kirkpatrick, A linear algorithm for determining the separation of convex polyhedra, J. Algorithms, 6 (1985), 381–392 10.1016/0196-6774(85)90007-0 87a:52005 CrossrefISIGoogle Scholar

  • [20]  T. Duchamp, A. Certain, T. DeRose, and W. Stuetzle, Hierarchical Computation of PL Harmonic Embeddings, manuscript. Google Scholar

  • [21]  G. Dziuk and J. E. Hutchinson, Finite Element Approximations to Surfaces of Prescribed Variable Mean Curvature, Preprint 03‐01, Fakultät für Mathematik und Physik, Universität Freiburg, Freiburg, Germany, 2003. Google Scholar

  • [22]  M. Eck, T. D. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle, Multiresolution analysis of arbitrary meshes, in Proceedings of the SIGGRAPH, 1995, pp. 173–182. Google Scholar

  • [23]  Joachim Escher, Uwe Mayer and , Gieri Simonett, The surface diffusion flow for immersed hypersurfaces, SIAM J. Math. Anal., 29 (1998), 1419–1433 10.1137/S0036141097320675 99f:58042 LinkISIGoogle Scholar

  • [24]  Michael Floater, Parametrization and smooth approximation of surface triangulations, Comput. Aided Geom. Design, 14 (1997), 231–250 98a:65018 CrossrefISIGoogle Scholar

  • [25]  Michael Floater, One‐to‐one piecewise linear mappings over triangulations, Math. Comp., 72 (2003), 685–696 2004k:65243 CrossrefISIGoogle Scholar

  • [26]  Michael Floater, Mean value coordinates, Comput. Aided Geom. Design, 20 (2003), 19–27 1968304 CrossrefISIGoogle Scholar

  • [27]  M. Garland and P. S. Heckbert, Surface simplification using quadric error metrics, in Proceedings of the SIGGRAPH, 1997, pp. 209–216. Google Scholar

  • [28]  T. Gieng, Unstructured Mesh Coarsening for Multilevel Methods, Master’s thesis, Multi‐Res Modeling Group, California Institute of Technology, Pasadena, CA, 2000. Google Scholar

  • [29]  T. S. Gieng, B. Hamann, K. L. Joy, G. L. Schussman and , and I. J. Trotts, Constructing hierarchies for triangle meshes, IEEE Trans. Visualization Comput. Graphics, 4 (1998), pp. 145–161. 9h9 ITVGEA 1077-2626 IEEE Trans. Vis. Comput. Graph. CrossrefISIGoogle Scholar

  • [30]  C. Gotsman, X. Gu and , and A. Sheffer, Fundamentals of spherical parameterization for 3D meshes, ACM Trans. Graphics, 22 (2003), pp. 358–363. atg ATGRDF 0730-0301 ACM Trans. Graphics CrossrefISIGoogle Scholar

  • [31]  E. Grinspun, P. Krysl and , and P. Schröder, CHARMS: A simple framework for adaptive simulation, ACM Trans. Graphics, 21 (2002), pp. 281–290. atg ATGRDF 0730-0301 ACM Trans. Graphics CrossrefISIGoogle Scholar

  • [32]  X. Gu, S. J. Gortler and , and H. Hoppe, Geometry images, ACM Trans. Graphics, 21 (2002), pp. 355–361. atg ATGRDF 0730-0301 ACM Trans. Graphics CrossrefISIGoogle Scholar

  • [33]  X. Gu and S.‐T. Yau, Global conformal surface parameterization, in Proceedings of the Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, 2003, pp. 127–137. Google Scholar

  • [34]  I. Guskov, A. Khodakovsky, P. Schröder, and W. Sweldens, Hybrid meshes: Multiresolution using regular and irregular refinement, in Proceedings of the Eighteenth Annual Symposium on Computational Geometry, 2002, pp. 264–272. Google Scholar

  • [35]  I. Guskov, W. Sweldens, and P. Schröder, Multiresolution signal processing for meshes, in Proceedings of the SIGGRAPH, 1999, pp. 325–334. Google Scholar

  • [36]  I. Guskov, K. Vidimče, W. Sweldens, and P. Schröder, Normal meshes, in Proceedings of the SIGGRAPH, 2000, pp. 95–102. Google Scholar

  • [37]  Wolfgang Hackbusch, Multigrid methods and applications, Springer Series in Computational Mathematics, Vol. 4, Springer‐Verlag, 1985xiv+377 87e:65082 CrossrefGoogle Scholar

  • [38]  S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro and , and M. Halle, Conformal surfaces parameterization for texture mapping, IEEE Trans. Vizualization Comput. Graphics, 6 (2000), pp. 181–189. 9h9 ITVGEA 1077-2626 IEEE Trans. Vis. Comput. Graph. CrossrefISIGoogle Scholar

  • [39]  M. Holst, Adaptive numerical treatment of elliptic systems on manifolds, Adv. Comput. Math., 15 (2001), 139–191, A posteriori error estimation and adaptive computational methods 10.1023/A:1014246117321 2003a:65108 CrossrefISIGoogle Scholar

  • [40]  H. Hoppe, Progressive meshes, in Proceedings of the SIGGRAPH, 1996, pp. 99–108. Google Scholar

  • [41]  H. Hoppe, Smooth view‐dependent level‐of‐detail control and application to terrain rendering, in IEEE Visualization, 1998, pp. 35–42. Google Scholar

  • [42]  K. Hormann and G. Greiner, MIPS: An efficient global parametrization method, in Curve and Surface Design: Saint‐Malo 1999, Vanderbilt University Press, Nashville, TN, 2000, pp. 153–162. Google Scholar

  • [43]  Jim Jones and , Panayot Vassilevski, AMGe based on element agglomeration, SIAM J. Sci. Comput., 23 (2001), 109–133 10.1137/S1064827599361047 2002g:65035 LinkISIGoogle Scholar

  • [44]  A. Khodakovsky, N. Litke and , and P. Schröder, Globally smooth parameterizations with low distortion, ACM Trans. Graphics, 22 (2003), p. 350–357. atg ATGRDF 0730-0301 ACM Trans. Graphics CrossrefISIGoogle Scholar

  • [45]  David Kirkpatrick, Optimal search in planar subdivisions, SIAM J. Comput., 12 (1983), 28–35 10.1137/0212002 84b:68042 LinkISIGoogle Scholar

  • [46]  L. Kobbelt, S. Campagna, and H.‐P. Seidel, A General framework for mesh decimation, in Proceedings the of Graphics Interface Conference, 1998, pp. 43–50. Google Scholar

  • [47]  L. Kobbelt, S. Campagna, J. Vorsatz, and H.‐P. Seidel, Interactive multi‐resolution modeling on arbitrary meshes, in Proceedings of the SIGGRAPH, 1998, pp. 105–114. Google Scholar

  • [48]  L. P. Kobbelt, J. Vorsatz, U. Labsik and , and H.‐P. Seidel, A shrink wrapping approach to remeshing polygonal surfaces, Computer Graphics Forum, 18 (1999), pp. 119–130. bjf ZZZZZZ 1067-7055 Comput. Graph. Forum CrossrefISIGoogle Scholar

  • [49]  V. Krishnamurthy and M. Levoy, Fitting smooth surfaces to dense polygon meshes, in Proceedings of the SIGGRAPH, 1996, pp. 313–324. Google Scholar

  • [50]  A. W. F. Lee, W. Sweldens, P. Schröder, L. Cowsar, and D. Dobkin, MAPS: Multiresolution adaptive parameterization of surfaces, in Proceedings of the SIGGRAPH, 1998, pp. 95–104. Google Scholar

  • [51]  M. Levoy, K. Pulli, B. Curless, S. Rusinkiewicz, D. Koller, L. Pereira, M. Ginzton, S. Anderson, J. Davis, J. Ginsberg, J. Shade, and D. Fulk, The digital Michelangelo project: 3D scanning of large statues, in Proceedings of the SIGGRAPH, 2000, pp. 131–144. Google Scholar

  • [52]  B. Lévy, S. Petitjean, N. Ray and , and J. Maillot, Least squares conformal maps for automatic texture atlas generation, ACM Trans. Graphics, 21 (2002), pp. 362–371. atg ATGRDF 0730-0301 ACM Trans. Graphics CrossrefISIGoogle Scholar

  • [53]  J. Liesen, E. de Sturler, A. Sheffer, Y. Aydin, and C. Siefert, Preconditioners for indefinite linear systems arising in surface parameterization, in Proceedings of the 10th International Meshing Round Table, 2001, pp. 71–82. Google Scholar

  • [54]  W. E. Lorensen and  and H. E. Cline, Marching cubes: A high resolution 3D surface construction algorithm, Computer Graphics, 21 (1987), pp. 163–169. bjf ZZZZZZ 1067-7055 Comput. Graph. Forum CrossrefGoogle Scholar

  • [55]  J. Maillot, H. Yahia, and A. Verroust, Interactive texture mapping, in Proceedings of the SIGGRAPH, 1993, pp. 27–34. Google Scholar

  • [56]  Uwe Mayer, A numerical scheme for moving boundary problems that are gradient flows for the area functional, European J. Appl. Math., 11 (2000), 61–80 2000m:76086 CrossrefISIGoogle Scholar

  • [57]  Uwe Mayer, Numerical solutions for the surface diffusion flow in three space dimensions, Comput. Appl. Math., 20 (2001), 361–379 2004f:65153 ISIGoogle Scholar

  • [58]  Uwe Mayer and , Gieri Simonett, A numerical scheme for axisymmetric solutions of curvature‐driven free boundary problems, with applications to the Willmore flow, Interfaces Free Bound., 4 (2002), 89–109 2003g:53119 CrossrefISIGoogle Scholar

  • [59]  G. L. Miller, D. Talmor and , and S.‐H. Teng, Optimal coarsening of unstructured meshes, J. Algorithms, 31 (1999), pp. 29–65. aug JOALDV 0196-6774 J. Algorithms CrossrefISIGoogle Scholar

  • [60]  E. Morano, D. Mavriplis and , V. Venkatakrishnan, Coarsening strategies for unstructured multigrid techniques with application to anisotropic problems, SIAM J. Sci. Comput., 20 (1998), 393–415 99e:76077 LinkISIGoogle Scholar

  • [61]  Johannes Nitsche, Boundary value problems for variational integrals involving surface curvatures, Quart. Appl. Math., 51 (1993), 363–387 94c:58036 CrossrefISIGoogle Scholar

  • [62]  Stanley Osher and , Ronald Fedkiw, Level set methods and dynamic implicit surfaces, Applied Mathematical Sciences, Vol. 153, Springer‐Verlag, 2003xiv+273 2003j:65002 CrossrefGoogle Scholar

  • [63]  Ulrich Pinkall and , Konrad Polthier, Computing discrete minimal surfaces and their conjugates, Experiment. Math., 2 (1993), 15–36 94j:53009 CrossrefGoogle Scholar

  • [64]  E. Praun and  and H. Hoppe, Spherical parameterization and remeshing, ACM Trans. Graphics, 22 (2003). atg ATGRDF 0730-0301 ACM Trans. Graphics CrossrefISIGoogle Scholar

  • [65]  M.‐Cecilia Rivara, Algorithms for refining triangular grids suitable for adaptive and multigrid techniques, Internat. J. Numer. Methods Engrg., 20 (1984), 745–756 85h:65258 CrossrefISIGoogle Scholar

  • [66]  María‐Cecilia Rivara, Local modification of meshes for adaptive and/or multigrid finite‐element methods, J. Comput. Appl. Math., 36 (1991), 79–89 10.1016/0377-0427(91)90227-B 92e:65164 CrossrefISIGoogle Scholar

  • [67]  J. Ruge and , K. Stüben, Algebraic multigrid, Frontiers Appl. Math., Vol. 3, SIAM, Philadelphia, PA, 1987, 73–130 972756 Google Scholar

  • [68]  P. Sander, S. Gortler, J. Snyder, and H. Hoppe, Signal‐specialized parameterization, in Eurographics Workshop on Rendering, 2002, pp. 87–100. Google Scholar

  • [69]  P. V. Sander, J. Snyder, S. J. Gortler, and H. Hoppe, Texture mapping progressive meshes, in Proceedings of the SIGGRAPH, 2001, pp. 409–416. Google Scholar

  • [70]  W. J. Schroeder, J. A. Zarge and , and W. E. Lorensen, Decimation of triangle meshes, Computer Graphics, 26 (1992), pp. 65–70. cog CGRADI 0097-8930 Comput. Graph. CrossrefGoogle Scholar

  • [71]  A. Sheffer and E. de Sturler, Surface parameterization for meshing by triangulation flattening, in Proceedings of the 9th International Meshing Roundtable, 2000, pp. 161–172. Google Scholar

  • [72]  J. R. Shewchuk, What is a good linear element? Interpolation, conditioning, and quality measures, in Proceedings of the 11th International Meshing Roundtable, 2002, pp. 115–126. Google Scholar

  • [73]  O. Sorkine, D. Cohen‐Or, R. Goldenthal, and D. Lischinski, Bounded‐distortion piecewise mesh parameterization, in IEEE Visualization Conference, 2002, pp. 355–362. Google Scholar

  • [74]  Wim Sweldens, The lifting scheme: a construction of second generation wavelets, SIAM J. Math. Anal., 29 (1998), 511–546 10.1137/S0036141095289051 99e:42052 LinkISIGoogle Scholar

  • [75]  W. Sweldens and P. Schröder, Building your own wavelets at home, in Wavelets in Computer Graphics, Course Notes, ACM Proceedings of the SIGGRAPH, 1996, pp. 15–87. Google Scholar

  • [76]  W. Tutte, How to draw a graph, Proc. London Math. Soc. (3), 13 (1963), 743–767 28:1610 CrossrefGoogle Scholar

  • [77]  P. Vaněk, J. Mandel and , M. Brezina, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing, 56 (1996), 179–196, International GAMM‐Workshop on Multi‐level Methods (Meisdorf, 1994) 97c:65207 CrossrefISIGoogle Scholar

  • [78]  W. L. Wan, T. F. Chan, and B. Smith, An Energy‐Minimizing Interpolation for Robust Multigrid, Tech. report, Department of Mathematics, UCLA, Los Angeles,1998. Google Scholar

  • [79]  Z. Wood, M. Desbrun, P. Schröder, and D. Breen, Semi‐regular mesh extraction from volumes, in IEEE Visualization Conference, 2000, pp. 275–282. Google Scholar

  • [80]  J. C. Xia and A. Varshney, Dynamic view‐dependent simplification for polygonal models, in IEEE Visualization Conference, 1996, pp. 327–334. Google Scholar

  • [81]  G. Xu, Convergent discrete Laplace–Beltrami Operators over Triangular Surfaces, Tech. report, ICMSEC, China, 2003. Google Scholar

  • [82]  G. Xu, Q. Pan, and C. Bajaj, Discrete Surface Modeling Using Geometric Flows, Tech. report, Dept. of Computer Sciences, University of Texas, Austin, 2003. Google Scholar

  • [83]  Jinchao Xu and , Jinshui Qin, Some remarks on a multigrid preconditioner, SIAM J. Sci. Comput., 15 (1994), 172–184 10.1137/0915012 95a:65210 LinkISIGoogle Scholar

  • [84]  H. Yserentant, On the multilevel splitting of finite element spaces, Numer. Math., 49 (1986), pp. 379–412. num NUMMA7 0029-599X Numer. Math. CrossrefISIGoogle Scholar