Multiresolution Analysis

  • Georges-Pierre Bonneau
  • Gershon Elber
  • Stefanie Hahmann
  • Basile Sauvage
Part of the Mathematics and Visualization book series (MATHVISUAL)

Multiresolution analysis has received considerable attention in recent years by researchers in the fields of computer graphics, geometric modeling and visualization. They are now considered a powerful tool for efficiently representing functions at multiple levels-ofdetail with many inherent advantages, including compression, Level-Of-Details (LOD) display, progressive transmission and LOD editing.

This survey chapter attempts to provide an overview of the recent results on the topic of multiresolution, with special focus on the work of researchers who are participating in the AIM@SHAPE European Networks of Excellence.


Computer Graphic Subdivision Scheme Multiresolution Analysis Freeform Surface Progressive Transmission 
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.
    M. Alexa, D. Cohen-Or, and D. Levin. As-rigid-as-possible shape interpolation. In SIGGRAPH ’00: Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pages 157-164, New York, NY, USA, 2000. ACM Press/AddisonWesley Publishing Co.CrossRefGoogle Scholar
  2. 2.
    R. Bartels and J. Beatty. A technique for the direct manipulation of spline curves. Graphics Interface ’89, pages 33-39, 1989.Google Scholar
  3. 3.
    G.-P. Bonneau. Multiresolution analysis with non-nested spaces. Computing Supplementum, 13:51-66, 1996.MathSciNetGoogle Scholar
  4. 4.
    G.-P. Bonneau. Multiresolution analysis on irregular surface meshes. IEEE Transactions on Graphics and Visualization, 4(4):365-378, 1998.CrossRefMathSciNetGoogle Scholar
  5. 5.
    G.-P. Bonneau and A. Gerussi. Hierarchical decomposition of datasets on irregular surface meshes. In Proceedings of CGI’98,, pages 59-63, Hannover, Germany, June 1998.Google Scholar
  6. 6.
    G.-P. Bonneau and A. Gerussi. Level of detail visualization of scalar data sets on irregular surface meshes. In Proceedings Visualization’98,, pages 73-77. IEEE, 1998.Google Scholar
  7. 7.
    G. P. Bonneau and H. Hagen. Variational design of rational b ézier curves and surfaces. In L. Laurent and L. Schumaker, editors, Curves and Surfaces, volume II, pages 51-58. AK Peters, 1994.Google Scholar
  8. 8.
    G.-P. Bonneau, S. Hahmann, and G.M. Nielson. Blac-wavelets: a multiresolution analysis with non-nested spaces. In Proceedings Visualization’96,, pages 43-48. IEEE, 1996.Google Scholar
  9. 9.
    P. Borel and A. Rappoport. Simple constrained deformations for geometric modeling and interactive design. ACM Transactions on Graphics, 13(2):137-155, 1994.CrossRefGoogle Scholar
  10. 10.
    M. P. Do Carmo. Differential Geometry of Curves and Surfaces. Cambridge University Press, 1976.Google Scholar
  11. 11.
    G. Celniker and D. Gossard. Deformable curve and surface finite-elements for free-form shape design. In ACM SIGGRAPH Conference Proceedings, pages 257-266. ACM, 1991.Google Scholar
  12. 12.
    G. Celniker and W. Welch. Linear constraints for deformable b-spline surfaces. In Symposium on Interactive 3D Graphics, pages 165-170, 1992.Google Scholar
  13. 13.
    A. Certain, J. Popovic, T. DeRose, T. Duchamp, D. Salesin, and W. Stuetzle. Interactive multiresolution surface viewing. Computer Graphics, 30(Annual Conference Series):91-98,1996.Google Scholar
  14. 14.
    C. K. Chui and E. G. Quak. Wavelets on a bounded interval. In D. Braess and L. Schumaker, editors, Numerical Methods of Approximation Theory, Volume 9, pages 53-75. Birkh äuser Verlag, Basel, 1992.Google Scholar
  15. 15.
    P. Cignoni, C. Montani, E. Puppo, and R. Scopigno. Multiresolution representation and visualization of volume data. IEEE Trans. on Visualization and Comp. Graph, 3(4):352-369,1997.CrossRefGoogle Scholar
  16. 16.
    D. Eberly and J. Lancaster. On gray scale image measurements: I. arc length and area. CVGIP: Graphical Models and Image Processing, 53(6):538-549, 1991.CrossRefGoogle Scholar
  17. 17.
    M. Eck, T. DeRose, T. Duchamp, H. Hoppe, T. Lounsbery, and W. Stuetzle. Multiresolution analysis of arbitrary meshes. In Computer Graphics Proceedings (SIGGRAPH 95), pages 173-182, 1995.Google Scholar
  18. 18.
    G. Elber. Symbolic and numeric computation in curve interrogation. Computer Graphics forum, 14(1):25-34, March 1995.CrossRefGoogle Scholar
  19. 19.
    G. Elber. Multiresolution curve editing with linear constraints. The Journal of Computing & Information Science in Engineering, 1(4):347-355, December 2001.CrossRefGoogle Scholar
  20. 20.
    G. Elber and C. Gotsman. Multiresolution control for nonuniform bspline curve editing. In The Third Pacific Graphics Conference on Computer Graphics and Applications, Seoul, Korea, pages 267-278, [August 1995.Google Scholar
  21. 21.
    G. Farin, G. Rein, N. Sapidis, and A. J. Worsey. Fairing cubic b-spline curves. Computer Aided Geometric Design, 4:91-103, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    A. Finkelstein and D. H. Salesin. Multiresolution curves. Computer Graphics Proceedings (SIGGRAPH 94), pages 261-268, 1994.Google Scholar
  23. 23.
    L. De Floriani. A pyramidal data structure for triangle-based surface description. IEEE Computer Graphics and Applications, 9(2):67-78, 1989.CrossRefGoogle Scholar
  24. 24.
    L. De Floriani, M. Alexa, Marie-Paule Cani, Paolo Cignoni, Emanuele Danovaro, Thomas Di Giacomo, HyungSeok Kim, Nadia Magnenat-Thalmann, and Enrico Puppo. Level-of-detail shape modeling. In chapter 5 of this book. Springer, 2005.Google Scholar
  25. 25.
    D. Forsey and R. Bartels. Hierarchical b-spline refinement. Proceedings of SIGGRAPH’88, ACM New York, pages 205-212, 1988.Google Scholar
  26. 26.
    B. Fowler. Geometric manipulation of tensor product surfaces. In 1992 Symposium on Interactive 3D Graphics, pages 101-108, 1992.Google Scholar
  27. 27.
    B. Fowler. Geometric manipulation of tensor product surfaces. In Proceedings of the 1992 symposium on Interactive 3D graphics, pages 101-108. ACM Press, 1992.Google Scholar
  28. 28.
    B. Fowler and R. Bartels. Constraint-based curve manipulation. IEEE Computer Graphics and Applications, 13(5):43-49, 1993.CrossRefGoogle Scholar
  29. 29.
    M. Gleicher. Integrating constraints and direct manipulation. In Proceedings of the 1992 symposium on Interactive 3D graphics, pages 171-174. ACM Press, 1992.Google Scholar
  30. 30.
    E. Goldstein and Craig Gotsman. Polygon morphing using a multiresolution representation. In Graphics Interface ’95, pages 247-254. Canadian Inf. Process. Soc., 1995.Google Scholar
  31. 31.
    C. Gonzalez-Ochoa, S. Mccammon, and J. Peters. Computing moments of objects enclosed by piecewise polynomial surfaces. ACM Transaction on Graphics, 17(3):143-157, July 1998.CrossRefGoogle Scholar
  32. 32.
    C. Gonzalez-Ochoa and J. Peters. Localized-hierarchy surface splines (less). In Pro- ceedings of the 1999 symposium on Interactive 3D graphics, pages 7-15. ACM Press, 1999.Google Scholar
  33. 33.
    S. Gortler and M. Cohen. Hierarchical and variational geometric modeling with wavelets. In 1995 Symposium on 3D Interactive Graphics, pages 35-41, 1995.Google Scholar
  34. 34.
    S. Gortler, P. Schr öder, M. Cohen, and P. Hanrahan. Wavelet radiosity. Computer Graphics Proceedings (SIGGRAPH 93), pages 221-230, 1993.Google Scholar
  35. 35.
    S. J. Gortler. Private communications.Google Scholar
  36. 36.
    S. J. Gortler. Wavelet methods in computer graphics. PhD thesis, Department of Computer Science, Princeton, 1994.Google Scholar
  37. 37.
    G. Greiner. Variational design and fairing of spline surfaces. In Proc. Eurographics 1994, pages 143-154, 1994.Google Scholar
  38. 38.
    G. Greiner and J. Loos. Data dependent thin plate energy and its use in interactive surface modeling. Eurographics ’96 (1996), 15:176-185, 1996.Google Scholar
  39. 39.
    M. Gross, L. Lippert, R. Dietrich, and S. H äring. Two methods or wavelet-based volume rendering. Computers & Graphics, 21(2):237-252, 1997.CrossRefGoogle Scholar
  40. 40.
    I. Guskov, A. Khodakovsky, P. Schr öder, and W. Sweldens. Hybrid meshes: Multiresolution using regular and irregular refinement. In Proceedings of SoCG 2002, 2000.Google Scholar
  41. 41.
    I. Guskov, K. Vidimce, W. Sweldens, and P. Schr öder. Normal meshes. In Kurt Akeley, editor, Siggraph 2000, Computer Graphics Proceedings, pages 95-102. ACM Press / ACM SIGGRAPH / Addison Wesley Longman, 2000.Google Scholar
  42. 42.
    H. Hagen and P. Santarelli. Variational design of smooth b-spline surfaces. In H. Hagen, editor, Topics in Geometric Modeling, pages 85-94. SIAM Philadelphia, 1992.Google Scholar
  43. 43.
    H. Hagen and G. Schulze. Automatics smoothing with geometric surface patches. Computer Aided Geometric Design, pages 231-236, 1987.Google Scholar
  44. 44.
    S. Hahmann. Shape improvement of surfaces. Computing Suppl., 13:135-152, 1998.MathSciNetGoogle Scholar
  45. 45.
    S. Hahmann and G.-P. Bonneau. Polynomial surfaces interpolating arbitrary triangulations. IEEE Transactions on Visualization and Computer Graphics, 9(1):99-109, 2003.CrossRefGoogle Scholar
  46. 46.
    S. Hahmann, G.-P. Bonneau, B. Caramiaux, and M. Cornillac. Multiresolution morphing of planar curves. Computing, 2007. to appear.Google Scholar
  47. 47.
    S. Hahmann, B. Sauvage, and G.-P. Bonneau. Area preserving deformation of multireso-lution curves. Computer Aided Geometric Design, 22(4):349-367, 2005.zbMATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    H. Hoppe. Progressive meshes. Computer Graphics Proceedings (SIGGRAPH 96), pages 99-108, 1996.Google Scholar
  49. 49.
    H. Hoppe. View-dependent refinement of progressive meshes. Computer Graphics Proceedings (SIGGRAPH 97), pages 189-198, 1997.Google Scholar
  50. 50.
    W. M. Hsu, J. F. Hughes, and H. Kaufman. Direct manipulation of free-form deformations. In Computer Graphics (SIGGRAPH 92 Proceedings), pages 177-184, July 1992.Google Scholar
  51. 51.
    P. D. Kaklis and N. S. Sapidis. Convexity-preserving interpolatory parametric splines of nonuniform polynomial degree. Comput. Aided Geom. Des., 12(1):1-26, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    R. Kazinnik and G. Elber. Orthogonal decomposition of non-uniform bspline spaces using wavelets. Computer Graphics forum, 16(3):27-38, September 1997.CrossRefGoogle Scholar
  53. 53.
    A. Khodakovsky and I. Guskov. Compression of normal meshes, 2003.Google Scholar
  54. 54.
    A. Khodakovsky, P. Schr öder, and W. Sweldens. Progressive geometry compression. In Kurt Akeley, editor, Siggraph 2000, Computer Graphics Proceedings, pages 271-278. ACM Press / ACM SIGGRAPH / Addison Wesley Longman, 2000.Google Scholar
  55. 55.
    D. Kirkpatrick. Optimal search in planar subdivisions. SIAM Journal on Computing, 12(1):28-35, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    L. Kobbelt. A variational approach to subdivision. Computer Aided Geometric Design, 13:743-761, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    L. Kobbelt, S. Campagna, J. Vorsatz, and HP. Seidel HP. Interactive multiresolution modeling on arbitrary meshes. In Computer Graphics Proceedings (SIGGRAPH 98), pages 105-114, 1998.Google Scholar
  58. 58.
    L. Kobbelt and P. Schr öder. A multiresolution framework for variational subdivision. ACM Trans. on Graph., 17(4):209-237, 1998.CrossRefGoogle Scholar
  59. 59.
    A. W. F. Lee, D. Dobkin, W. Sweldens, and P. Schr öder. Multiresolution mesh morphing. Computer Graphics Proceedings (SIGGRAPH 99), pages 343-350, 1999.Google Scholar
  60. 60.
    Aaron W. F. Lee, Wim Sweldens, Peter Schr öder, Lawrence Cowsar, and David Dobkin. MAPS: Multiresolution adaptive parameterization of surfaces. Computer Graphics, 32 (Annual Conference Series):95-104, 1998.Google Scholar
  61. 61.
    M. Lounsbery, T. De Rose, and J. Warren. Multiresolution analysis for surfaces of arbitrary topological type. ACM Transaction on Graphics, 16(1):34-73, 1997.CrossRefGoogle Scholar
  62. 62.
    T. Lyche and K. Morken. Spline wavelets of minimal support. In D. Braess and L. Schumaker, editors, Numerical Methods of Approximation Theory, pages 177-194. Birkh äuser Verlag, Basel, 1992.Google Scholar
  63. 63.
    Tom Lyche and Knut Morken. Knot removal for parametric b-spline curves and surfaces. Comput. Aided Geom. Des., 4(3):217-230, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    S. Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11:674-693, 1989.zbMATHCrossRefGoogle Scholar
  65. 65.
    E. Mehlum. Non-linear spline. In R. E. Barnhill and R. F.R iesenfeld, editors, Computer Aided Geometric Design, pages 173-208. Academic Press, 1974.Google Scholar
  66. 66.
    M. Gross, O. Staadt, and R. Gatti. Efficient triangular surface approximations using wavelets and quadtree data structures. IEEE Transactions on Visualization and Computer Graphics, 2(2):130-143, 1996.CrossRefGoogle Scholar
  67. 67.
    H. P. Moreton and C. H. S équin. Functional optimisation for fair surface design. Computer Graphics, 26(2):167-176, 1992.CrossRefGoogle Scholar
  68. 68.
    G. Nielson, IH. Jung, and J. Sung. Haar-wavelets over triangular domains with applications to multiresolution models for flow over a sphere. In IEEE Visualization’97, pages 143-150, november 1997.Google Scholar
  69. 69.
    F. Payan and M. Antonini. An efficient bit allocation for compressing normal meshes with an error-driven quantization. Computer Aided Geometric Design, 22:466-486, July 2005.zbMATHCrossRefMathSciNetGoogle Scholar
  70. 70.
    F. Payan and M. Antonini. Mean square error approximation for wavelet-based semiregular mesh compression. IEEE Transactions on on Visualization and Computer Graphics (TVCG), 2006. to appear.Google Scholar
  71. 71.
    J. P. Pernot, S. Guillet, J. C. Leon, F. Giannini, B. Falcidieno B., and E. Catalano. A shape deformation tool to model character lines in the early design phases. In Proceedings Shape Modeling International 2002, Banff, Canada, 2002.Google Scholar
  72. 72.
    J. Peters. C1-surface splines. SIAM J. Numer. Anal., 32(2):645-666, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  73. 73.
    M. Plavnik and G. Elber. urface design using global constraints on total curvature. In The VIII IMA Conference on Mathematics of Surfaces, September 1998.Google Scholar
  74. 74.
    A. Rappoport, A. Sheffer, and M. Bercovier. Volume-preserving free-form solids. In Proceedings of Solid Modeling 95, pages 361-372, May 1995.Google Scholar
  75. 75.
    A. Raviv and G. Elber. Three dimensional freeform sculpting via zero sets of scalar trivariate functions. CAD, 32(8/9):513-526, July/August 2000.Google Scholar
  76. 76.
    C. H. Reinsch. Smoothing by spline functions ii. Num. Math., 16:451-454, 1967.CrossRefMathSciNetGoogle Scholar
  77. 77.
    B. Sauvage. D éformation de courbes et surfaces multir ésolution sous contraintes. Phd, Institut National Polytechnique de Grenoble (INPG), December 2005.Google Scholar
  78. 78.
    B. Sauvage, S. Hahmann, and G.-P. Bonneau. Length preserving multiresolution editing of curves. Computing, 72:161-170, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  79. 79.
    B. Sauvage, S. Hahmann, and G.-P. Bonneau. Length constrained multiresolution deformation for surface wrinkling. In International Conference on Shape Modeling and Applications, SMI’06, pages 113-121, Matsushima, June 2006. IEEE Computer Society Press.Google Scholar
  80. 80.
    P. Schr öder and W. Sweldens. Spherical wavelets: Efficiently representing functions on the sphere. Computer Graphics Proceedings (SIGGRAPH 95), pages 161-172, 1995.Google Scholar
  81. 81.
    T.W. Sederberg, P. Gao, G. Wang, and H. Mu. 2-d shape blending: An intrinsic solution to the vertex path problem. Computer Graphics,(SIGGRAPH 93 Proceedings), 27:15-18, 1993.CrossRefGoogle Scholar
  82. 82.
    M. Shapira and A. Rappoport. Shape blending using the star-skeleton representation. IEEE Comput. Graph. Appl., 15(2):44-50, 1995.CrossRefGoogle Scholar
  83. 83.
    E. Stollnitz, T. DeRose, and D. Salesin. Wavelets for Computer Graphics: Theory and Applications. Morgan-Kaufmann, 1996.Google Scholar
  84. 84.
    Eric J. Stollnitz, Tony D. DeRose, and David H. Salesin. Wavelets for computer graphics: A primer, part 2. IEEE Computer Graphics and Applications, 15(4):75-85, 1995.Google Scholar
  85. 85.
    W. Sweldens. The lifting scheme: A construction of second generation wavelets. SIAM J. Math. Anal., 29(2):511-546, 1997.MathSciNetGoogle Scholar
  86. 86.
    J. Warren and H. Weimer. Variational subdivision for natural cubic splines. Approximation Theory IX, 2:345-352, 1998.MathSciNetGoogle Scholar
  87. 87.
    H. Weimer and J. Warren. Subdivision schemes for thin plate splines. Computer Graphics Forum (Proceedings of Eurographics 98), pages 303-313, 1998.Google Scholar
  88. 88.
    H. Weimer and J. Warren. Subdivision schemes for fluid flow. Computer Graphics (SIGGRAPH 99 Conference Proceedings), pages 111-120, August 1999.Google Scholar
  89. 89.
    W. Welch and A. Witkin. Variational surface modeling. Computer Graphics (SIGGRAPH ’92 proceedings), 26:157-166, July 1992.CrossRefGoogle Scholar
  90. 90.
    A. Yvart, S. Hahmann, and G.-P. Bonneau. Hierarchical triangular splines. ACM Transactions on Graphics, 24(4):1374-1391, 2005.CrossRefGoogle Scholar
  91. 91.
    D. Zorin, P. Schr öder, and W. Sweldens. Interactive multiresolution mesh editing. Computer Graphics Proceedings (SIGGRAPH 97), pages 259-268, 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Georges-Pierre Bonneau
    • 1
  • Gershon Elber
    • 2
  • Stefanie Hahmann
    • 3
  • Basile Sauvage
    • 3
  1. 1.Université Joseph FourierGrenobleFrance
  2. 2.Technion - Israel Institute of TechnologyHaifaIsrael
  3. 3.Laboratoire Jean KuntzmannInstitut National Polytechnique de GrenobleFrance

Personalised recommendations