Advertisement

Constructive Approximation

, Volume 31, Issue 3, pp 359–393 | Cite as

Stability of Nonlinear Subdivision and Multiscale Transforms

  • S. Harizanov
  • P. Oswald
Article

Abstract

Extending upon the work of Cohen, Dyn, and Matei (Appl. Comput. Harmon. Anal. 15:89–116, 2003) and of Amat and Liandrat (Appl. Comput. Harmon. Anal. 18:198–206, 2005), we present a new general sufficient condition for the Lipschitz stability of nonlinear subdivision schemes and multiscale transforms in the univariate case. It covers the special cases (weighted essentially nonoscillatory scheme, piecewise polynomial harmonic transform) considered so far but also implies the stability in some new cases (median interpolating transform, power-p schemes, etc.). Although the investigation concentrates on multiscale transforms
$$\bigl\{v^0,d^1,\ldots,d^J\bigr\}\longmapsto v^J,\quad J\ge1,$$
in (ℤ) given by a stationary recursion of the form
$$v^{j}=Sv^{j-1}+d^{j},\quad j\ge1,$$
involving a nonlinear subdivision operator S acting on (ℤ), the approach is extendable to other nonlinear multiscale transforms and norms, as well.

Keywords

Nonlinear subdivision and multiscale transforms Lipschitz stability Finite differences Derived subdivision schemes Spectral radius conditions 

Mathematics Subject Classification (2000)

65D15 65D17 65T50 26A16 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amat, S., Arandiga, F., Cohen, A., Donat, R., Garcia, G., von Oehsen, M.: Data compression with ENO schemes: A case study. Appl. Comput. Harmon. Anal. 11, 273–288 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Amat, S., Liandrat, J.: On the stability of the PPH nonlinear multiresolution. Appl. Comput. Harmon. Anal. 18, 198–206 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Baraniuk, R., Janssen, M., Lavu, S.: Multiscale approximation of piecewise smooth two-dimensional functions using normal triangulated meshes. Appl. Comput. Harmon. Anal. 19, 92–130 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision, vol. 93. American Mathematical Society, Providence (1991) Google Scholar
  6. 6.
    Cohen, A., Dyn, N., Matei, B.: Quasilinear subdivision schemes with applications to ENO interpolation. Appl. Comput. Harmon. Anal. 15, 89–116 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmon. Anal. 21, 5–30 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dadourian, K., Liandrat, J.: Analysis of some bivariate non-linear interpolatory subdivision schemes. Numer. Algorithms 48(1–3), 261–278 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Daubechies, I., Runborg, O., Sweldens, W.: Normal multiresolution approximation of curves. Constr. Approx. 20, 399–463 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Donoho, D.L., Yu, T.P.-Y.: Nonlinear pyramid transforms based on median-interpolation. SIAM J. Math. Anal. 31(5), 1030–1061 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dyn, N.: Subdivision schemes in CAGD. In: Light, W.A. (ed.) Advances in Numerical Analysis, vol. II, pp. 36–104. Oxford University Press, London (1992) Google Scholar
  12. 12.
    Dyn, N., Grohs, P., Wallner, J.: Approximation order of interpolatory nonlinear subdivision schemes. J. Comput. Appl. Math. 223, 1697–1703 (2010) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Dyn, N., Levin, D.: Subdivision schemes in geometric modelling. Acta Numer. 11, 73–144 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Floater, M.S., Micchelli, C.A.: Nonlinear stationary subdivision. In: Govil, N.K., Mohapatra, R.N., Nashed, Z., Sharma, A., Szabados, J. (eds.) Approximation Theory: in Memory of A.K. Varma, pp. 209–224. Dekker, New York (1998) Google Scholar
  15. 15.
    Grohs, P.: Smoothness analysis of subdivision schemes on regular grids by proximity. SIAM J. Numer. Anal. 46, 2169–2182 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Grohs, P.: Smoothness equivalence properties of univariate subdivision schemes and their projection analogues. Numer. Math. 113(2), 163–180 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Guskov, I., Vidimce, K., Sweldens, W., Schröder, P.: Normal meshes. In: Akeley, K. (ed.) Computer Graphics (SIGGRAPH ’00: Proceedings), pp. 95–102. ACM, New York (2000) Google Scholar
  18. 18.
    Harizanov, S.: Stability of nonlinear multiresolution analysis. PAMM 8(1), 10933–10934 (2008) CrossRefGoogle Scholar
  19. 19.
    Harizanov, S.: Stability of nonlinear subdivision schemes and multiresolutions. Master’s Thesis, Jacobs University Bremen (2008) Google Scholar
  20. 20.
    Harten, A., Engquist, B., Osher, S., Chakravarthy, S.: Uniformly high order essentially non-oscillatory schemes. J. Comput. Phys. 71, 231–303 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Heijmans, H.J.A.M., Goutsias, J.K.: Nonlinear multiresolution signal decomposition schemes. I. Morphological pyramids. IEEE Trans. Image Process. 9(11), 1862–1876 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Heijmans, H.J.A.M., Goutsias, J.K.: Nonlinear multiresolution signal decomposition schemes. ii. Morphological wavelets. IEEE Trans. Image Process. 9(11), 1897–1913 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Khodakovsky, A., Guskov, I.: Compression of normal meshes. In: Brunnett, G., Hamann, B., Müller, H., Linsen, L. (eds.) Geometric Modeling for Scientific Visualization, pp. 189–207. Springer, Berlin (2003) Google Scholar
  24. 24.
    Khodakovsky, A., Schröder, P., Sweldens, W.: Progressive geometry compression. In: Akeley, K. (ed.) Computer Graphics (SIGGRAPH ’00: Proceedings), pp. 271–278. ACM, New York (2000) Google Scholar
  25. 25.
    Kuijt, F., van Damme, R.: Convexity preserving interpolatory subdivision schemes. Constr. Approx. 14, 609–630 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Kuijt, F., van Damme, R.: Stability of subdivision schemes. TW Memorandum 1469, Faculty of Applied Mathematics, University of Twente, The Netherlands (1998) Google Scholar
  27. 27.
    Kuijt, F., van Damme, R.: Monotonicity preserving interpolatory subdivision schemes. J. Comput. Appl. Math. 101(1–2), 203–229 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Marinov, M., Dyn, N., Levin, D.: Geometrically controlled 4-point interpolatory schemes. In: Advances in Multiresolution for Geometric Modelling, pp. 301–317. Springer, Berlin (2004) Google Scholar
  30. 30.
    Matei, B.: Smoothness characterization and stability in nonlinear multiscale framework: theoretical results. Asymptot. Anal. 41(3–4), 277–309 (2005) zbMATHMathSciNetGoogle Scholar
  31. 31.
    Oswald, P.: Smoothness of a nonlinear subdivision scheme. In: Cohen, A., Merrien, J.-L., Schumaker, L.L. (eds.) Curve and Surface Fitting: Saint-Malo 2002, pp. 323–332. Nashboro Press, Brentwood (2003) Google Scholar
  32. 32.
    Oswald, P.: Smoothness of nonlinear median-interpolation subdivision. Adv. Comput. Math. 20, 401–423 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Serna, S., Marquina, A.: Power ENO methods: a fifth-order accurate weighted power ENO method. J. Comput. Phys. 194, 632–658 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Ur Rahman, I., Drori, I., Stodden, V.C., Donoho, D.L., Schröder, P.: Multiscale representations for manifold-valued data. Multiscale Model. Simul. 4(4), 1201–1232 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Wallner, J.: Smoothness analysis of subdivision schemes by proximity. Constr. Approx. 24, 289–318 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Wallner, J., Dyn, N.: Convergence and C 1 analysis of subdivision schemes on manifolds by proximity. Comput. Aided Geom. Des. 22, 593–622 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Wallner, J., Nava Yazdani, E., Grohs, P.: Smoothness properties of Lie group subdivision schemes. Multiscale Model. Simul. 6, 493–505 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Xie, G., Yu, T.P.-Y.: Smoothness analysis of nonlinear subdivision schemes of homogeneous and affine invariant type. Constr. Approx. 22(2), 219–254 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Xie, G., Yu, T.P.-Y.: Smoothness equivalence properties of manifold-valued data subdivision schemes based on the projection approach. SIAM J. Numer. Anal. 45(3), 1200–1225 (2007) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Jacobs University, SESBremenGermany

Personalised recommendations