A New Class of Non-stationary Interpolatory Subdivision Schemes Based on Exponential Polynomials

  • Yoo-Joo Choi
  • Yeon-Ju Lee
  • Jungho Yoon
  • Byung-Gook Lee
  • Young J. Kim
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4077)


We present a new class of non-stationary, interpolatory subdivision schemes that can exactly reconstruct parametric surfaces including exponential polynomials. The subdivision rules in our scheme are interpolatory and are obtained using the property of reproducing exponential polynomials which constitute a shift-invariant space. It enables our scheme to exactly reproduce rotational features in surfaces which have trigonometric polynomials in their parametric equations. And the mask of our scheme converges to that of the polynomial-based scheme, so that the analytical smoothness of our scheme can be inferred from the smoothness of the polynomial based scheme.


Parametric Surface Trigonometric Polynomial Subdivision Scheme Klein Bottle Lift Scheme 
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  1. 1.
    Zorin, D., Schröder, P.: Subdivision for modeling and animation. SIGGRAPH Course Notes (2000)Google Scholar
  2. 2.
    Sweldens, W.: The lifting scheme: a construction of second generation wavelets. SIAM J. Math. Anal., 511–546 (1998)Google Scholar
  3. 3.
    Guskov, I., Sweldens, W., Schroder, P.: Multiresolution signal processing for meshes. In: Proc. of ACM SIGGRAPH, pp. 325–334 (1999)Google Scholar
  4. 4.
    Dyn, N., Levin, D., Luzzatto, A.: Refining Oscillatory Signals by Non-Stationary Subdivision Schemes. In: Modern Developments in Multivariate Approximation, Internat. Ser. Numer. Math., vol. 145, Birkhäuser (2002)Google Scholar
  5. 5.
    Dyn, N., Gregory, J.A., Levin, D.: A butterfly subdivision scheme for surface interpolation with tension control. ACM Trans. Graph. 9, 160–169 (1990)MATHCrossRefGoogle Scholar
  6. 6.
    Hoppe, H., DeRose, T., Duchamp, T., Halstead, M., Jin, H., McDonald, J., Schweitzer, J., Stuetzle, W.: Piecewise smooth surface reconstruction. In: Proceedings of ACM SIGGRAPH, pp. 295–302 (1994)Google Scholar
  7. 7.
    Loop, C.: Smooth subdivision surfaces based on triangles. Master’s thesis, Department of Mathematics, University of Utah (1987)Google Scholar
  8. 8.
    Morin, G., Warren, J., Weimer, H.: A subdivision scheme for surfaces of revolution. Comp. Aided Geom. Design 18, 483–502 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Jena, M.J., Shunmugaraj, P., Das, P.J.: A sudivision algorithm for trigonometric spline curves. Comp. Aided Geom. Desig. 19, 71–88 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Warren, J., Weimer, H.: Subdivision methods for geometric design. Academic Press, London (2002)Google Scholar
  11. 11.
    Jena, M.J., Shunmugaraj, P., Das, P.J.: A non-stationary subdivision scheme for generalizing trigonometric spline surfaces to arbitrary meshes. Comp. Aided Geom. Desig. 20, 61–77 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    McClellan, J.M., Schafer, R.W., Yoder, M.A.: DSP First: A Multimedia Approach. Prentice-Hall, Englewood Cliffs (1998)Google Scholar
  13. 13.
    Chenny, E., Light, W., Light, W.: A Course in Approximation Theory. Brooks Cole (1999)Google Scholar
  14. 14.
    Yoon, J.: Analysis of non-stationary interpolatory subdivision schems based on exponential polynomials. Ewha womans university tech. document (2005),
  15. 15.
    Dyn, N.: Subdivision Schemes in Computer-Aided Geometric Design. In: Advances in Numerical Analysis. Wavelets, Subdivision Algorithms and Radial Basis Functions, vol. II, Oxford University Press, Oxford (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Yoo-Joo Choi
    • 1
  • Yeon-Ju Lee
    • 2
  • Jungho Yoon
    • 2
  • Byung-Gook Lee
    • 3
  • Young J. Kim
    • 4
  1. 1.Dept. of CS.Seoul Univ. of Venture and Info.SeoulKorea
  2. 2.Dept. of Math.Ewha Womans Univ.SeoulKorea
  3. 3.Div. of Internet EngineeringDongseo Univ.BusanKorea
  4. 4.Dept. of CS.Ewha Womans Univ.SeoulKorea

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