Advertisement

Advances in Computational Mathematics

, Volume 34, Issue 3, pp 253–277 | Cite as

Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms

  • S. Amat
  • K. Dadourian
  • J. LiandratEmail author
Article

Abstract

This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200–212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2):198–206, 2005; Serna and Marquina, J Comput Phys 194:632–658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165–187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).

Keywords

Nonlinear subdivision schemes Nonlinear multiresolution Approximation Stability Convergence 

Mathematics Subject Classifications (2010)

41A05 41A10 65D05 65D17 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amat, S., Donat, R., Liandrat, J., Trillo, J.C.: Analysis of a new nonlinear subdivision scheme. Applications in image processing. Found. Comput. Math 6(2), 193–225 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Amat, S., Liandrat, J.: On the stability of PPH nonlinear multiresolution. Appl. Comput. Harmon. Anal. 18(2), 198–206 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Arandiga, F., Donat, R.: Nonlinear multi-scale decompositions: the approach of A. Harten. Numer. Algorithms 23, 175–216 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Aspert, N., Ebrahimi, T., Vandergheynst, P.: Non-linear subdivision using local coordinates. Comput. Aided Geom. Des. 20, 165–187 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Beylkin, G.: Wavelets, Multiresolution Analysis and Fast Numerical Algorithms. INRIA Lectures, Manuscript (1991)Google Scholar
  6. 6.
    Belda, A.M.: Weighted ENO y Aplicaciones. Technical report, Universitat de Valencia (2004)Google Scholar
  7. 7.
    Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary subdivision. Mem. Am. Math. Soc. 93(453) (1991)Google Scholar
  8. 8.
    Cohen, A., Dyn, N., Matei, B.: Quasi linear subdivision schemes with applications to ENO interpolation. Appl. Comput. Harmon. Anal. 15, 89–116 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dadourian, K.: Schémas de subdivision et analyse multirésolution non-linéaire. Applications. Ph.D. thesis, Univ. de Provence. http://www.latp.univ-mrs.fr/~dadouria/these.html (2008)
  10. 10.
    Dadourian, K., Liandrat, J.: Analysis of some bivariate non-linear interpolatory subdivision schemes. Numer. Algorithms 48, 261–278 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Daubechies, I., Runborg, O., Sweldens, W.: Normal multiresolution approximation of curves. Constr. Approx. 20, 399–463 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. Constr. Approx. 5, 49–68 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Donoho, D., Yu, T.P.: Nonlinear pyramid transforms based on median interpolation. SIAM J. Math Anal. 31(5), 1030–1061 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dyn, N.: Subdivision Schemes in Computer Aided Geometric Design, vol. 20, no. 4, pp. 36–104. Oxford University Press, London (1992)Google Scholar
  15. 15.
    Floater, M.S., Michelli, C.A.: Nonlinear stationary subdivision. In: Govil, N.K, Mohapatra, N., Nashed, Z., Sharma, A., Szabados, J. (eds.) Approximation Theory: in Memory of A.K. Varna, pp. 209–224 (1998)Google Scholar
  16. 16.
    Grohs, P.: Stability of manifold-valued subdivision schemes and multi- scale transformations. Constr. Approx. doi:  10.1007/s00365-010-9085-8
  17. 17.
    Harizanov, S., Oswald, P.: Stability of nonlinear subdivision and multiscale transforms. Constr. Approx. 31(3), 359–393 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Harten, A.: Discrete Multiresolution analysis and generalized wavelets. Appl. J. Numer. Math. 12, 153–192 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Harten, A.: Multiresolution representation of data II: general framework. SIAM J. Numer. Anal. 33(3), 1205–1256 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Jiang, G.S., Shu, C.W.: Efficient Implementation of Weighted ENO Schemes. J. Comput. Phys. 126, 202–228 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kuijt, F., van Damme, R.: Convexity preserving interpolatory subdivision schemes. Constr. Approx. 14, 609–630 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Kuijt, F.: Convexity preserving interpolation. Stationary nonlinear subdivision and splines. Ph.D. thesis, University of Twente, The Netherlands (1998)Google Scholar
  23. 23.
    Levin, D.: Using Laurent polynomial representation for the analysis of non-uniform binary subdivision scheme. Adv. Comput. Math. 11, 41–54 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Marinov, M., Dyn, N., Levin, D.: Geometrically controlled 4-point interpolatory schemes. In: Le, A., Mehaute, Laurent, P.J., Schumaker, L.L. (eds.) Advances in Multiresolution for Geometric Modeling, pp. 301–315 (2005)Google Scholar
  26. 26.
    Matei, B.: Méthodes multirésolutions non-linéaires. Applications au traitement d’images. Ph.D. thesis, Université Paris VI (2002)Google Scholar
  27. 27.
    Oswald, P.: Smoothness of nonlinear median-interpolation subdivision. Adv. Comput. Math. 20(4), 401–423 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Serna, S., Marquina, A.: Power ENO methods: a fifth order accurate weighted power ENO method. J. Comput. Phys. 194, 632–658 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada y EstadísticaUniversidad Politécnica de Cartagenade CartagenaSpain
  2. 2.Ecole Centrale Marseille, laboratoire d’analyse topologie et probabilites (LATP)Marseille cedex 20France

Personalised recommendations