Simple Computation of the Eigencomponents of a Subdivision Matrix in the Fourier Domain

  • Loïc Barthe
  • Cédric Gérot
  • Malcolm Sabin
  • Leif Kobbelt
Part of the Mathematics and Visualization book series (MATHVISUAL)


After demonstrating the necessity and the advantage of decomposing the subdivision matrix in the frequency domain when analysing a subdivision scheme, we present a general framework based on a method, introduced by Ball and Storry, which computes the Discrete Fourier Transform of a subdivision matrix. The efficacy of the technique is illustrated by performing the analysis of Kobbelt's \(\sqrt 3 \) scheme in a very simple manner.


Discrete Fourier Transform Rotational Frequency Tangent Plane Subdivision Scheme Fourier Domain 
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.
    Ball, A.A., Storry, D.J.T.: Conditions for tangent plane continuity over recursiveley generated B-spline surfaces. ACM Trans. Graphics, 7(2):83–102 (1988)CrossRefGoogle Scholar
  2. 2.
    de Boor, C., Hollig, D., Riemenschneider, S.: Box Splines. Springer-Verlag, New York (1994).Google Scholar
  3. 3.
    Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Computer Aided Design, 10(6):350–355 (1978)CrossRefGoogle Scholar
  4. 4.
    Dodgson N.A., Ivrissimtzis, I.P., Sabin, M.A.: Characteristics of dual triangular \(\sqrt 3 \) subdivision. Curve and Surface Fitting: Saint-Malo 2002, Nashboro Press, Brentwood, 119–128 (2003)Google Scholar
  5. 5.
    Doo, D., Sabin, M.A.: Analysis of the behaviour of recursive subdivision surfaces near extraordinary points. Computer Aided Design, 10(6):356–360 (1978)CrossRefGoogle Scholar
  6. 6.
    Dyn, N.: Subdivision schemes in Computer-Aided Geometric Design. Advances in Numerical Analysis II: Wavelets, Subdivision Algorithms and Radial Basis Functions, W. Light (ed.), Clarendon Press, Oxford, 36–104 (1992)Google Scholar
  7. 7.
    Dyn, N.: Analysis of convergence and smoothness by the formalism of Laurent polynomials. Tutorials on Multiresolution in Geometric Modelling, A. Iske, E. Quak and M. Floater (eds.), Springer, 51–68 (2002)Google Scholar
  8. 8.
    Dyn, N., Levin, D., Gregory, J.: A butterfly subdivision scheme for surface interpolation with tension control. ACM Trans. Graphics, 9(2):160–169 (1990)CrossRefGoogle Scholar
  9. 9.
    Farin, G.: Curves and Surfaces for CAGD. 5th Edition, Academic Press (2002)Google Scholar
  10. 10.
    Kobbelt, L.: \(\sqrt 3 \)-Subdivision. Proc. ACM SIGGRAPH 2000, 103–112 (2000)Google Scholar
  11. 11.
    Loop, C.: Smooth subdivision surfaces based on triangles. Master's thesis, University of Utah (1987)Google Scholar
  12. 12.
    Peters, J., Reif, U.: Analysis of algorithms generalizing B-spline subdivision. SIAM Journal of Num. Anal., 35(2):728–748 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Prautzsch, H.: Analysis of Ck-Subdivision surfaces at extraordinary points. Technical Report 98/4, Fakultät für Informatik, University of Karlsruhe, Germany (1998)Google Scholar
  14. 14.
    Prautzsch, H., Umlauf, G.: A G2-subdivision algorithm. Geometric Modelling, Dagstuhl 1996, Computing supplement 13, Springer-Verlag, 217–224 (1998)Google Scholar
  15. 15.
    Reif, U.: A unified approach to subdivision algorithms near extraordinary vertices. Computer Aided Design, 12:153–174 (1995)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Sabin, M.A.: Eigenanalysis and artifacts of subdivision curves and surfaces. Tutorials on Multiresolution in Geometric Modelling, A. Iske, E. Quak and M. Floater (ed.), Springer, 69–97 (2002)Google Scholar
  17. 17.
    Stam, J.: Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. Proc. ACM SIGGRAPH '98, 395–404 (1998)Google Scholar
  18. 18.
    Warren, J., Weimer, H.: Subdivision methods for geometric design: A constructive approach. San Francisco: Morgan Kaufman, (2002)Google Scholar
  19. 19.
    Zorin, D.: Stationary subdivision and multiresolution surface representations. PhD thesis, Caltech, Pasadena, California (1997)Google Scholar
  20. 20.
    Zorin, D.: A method for analysis of C1-continuity of subdivision surfaces. SIAM Journal of Num. Anal., 35(5):1677–1708 (2000)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zorin, D., Schröder, P: Subdivision for modeling and animation. SIGGRAPH 2000 Course Notes. (2000)Google Scholar
  22. 22.
    Zorin, D., Schröder, P., Sweldens, W.: Interpolating subdivision for meshes with arbitrary topology. Proc. ACM SIGGRAPH '97, 189–192 (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Loïc Barthe
    • 1
  • Cédric Gérot
    • 2
  • Malcolm Sabin
    • 3
  • Leif Kobbelt
    • 4
  1. 1.Computer Graphics Group, IRIT/UPSToulouseFrance
  2. 2.Laboratoire des Images et des SignauxDomaine UniversitaireGrenobleFrance
  3. 3.Numerical Geometry Ltd.CambridgeUK
  4. 4.Computer Graphics GroupRWTH AachenGermany

Personalised recommendations