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Foundations of Computational Mathematics

, Volume 11, Issue 6, pp 617–656 | Cite as

Normal Multi-scale Transforms for Curves

  • S. Harizanov
  • P. Oswald
  • T. Shingel
Article

Abstract

Extending upon Daubechies et al. (Constr. Approx. 20:399–463, 2004) and Runborg (Multiscale Methods in Science and Engineering, pp. 205–224, 2005), we provide the theoretical analysis of normal multi-scale transforms for curves with general linear predictor S, and a more flexible choice of normal directions. The main parameters influencing the asymptotic properties (convergence, decay estimates for detail coefficients, smoothness of normal re-parametrization) of this transform are the smoothness of the curve, the smoothness of S, and its order of exact polynomial reproduction. Our results give another indication why approximating S may not be the first choice in compression applications of normal multi-scale transforms.

Keywords

Nonlinear geometric multi-scale transforms Approximating subdivision schemes Lipschitz smoothness Curve representation Detail decay estimate 

Mathematics Subject Classification (2000)

65D17 65D15 65T60 26A16 

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References

  1. 1.
    R. Baraniuk, M. Janssen, S. Lavu, Multiscale approximation of piecewise smooth two-dimensional functions using normal triangulated meshes, Appl. Comput. Harmon. Anal. 19, 92–130 (2005). MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    M.A. Berger, Y. Wang, Bounded semigroups of matrices, Linear Algebra Appl. 166, 21–27 (1992). MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    P. Binev, N. Dyn, R.A. DeVore, N. Dyn, Adaptive approximation of curves, in Approximation Theory (Acad. Publ. House, Sofia, 2004), pp. 43–57. Google Scholar
  4. 4.
    A.S. Cavaretta, W. Dahmen, C.A. Micchelli, Stationary Subdivision, Memoirs AMS, vol. 93 (Am. Math. Soc., Providence, 1991). Google Scholar
  5. 5.
    I. Daubechies, O. Runborg, W. Sweldens, Normal multiresolution approximation of curves, Constr. Approx. 20, 399–463 (2004). MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    N. Dyn, M.S. Floater, K. Hormann, A C 2 four-point subdivision scheme with fourth order accuracy and its extensions, in Mathematical Methods for Curves and Surfaces, ed. by M. Dæhlen, K. Mørken, L.L. Schumaker (Nashboro Press, Brentwood, 2005), pp. 145–156. Google Scholar
  7. 7.
    N. Dyn, K. Hormann, M.A. Sabin, Z. Shen, Polynomial reproduction by symmetric subdivision schemes, J. Approx. Theory 155, 28–42 (2008). MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    N. Dyn, D. Levin, Subdivision schemes in geometric modelling, Acta Numer. 11, 73–144 (2002). MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    I. Friedel, A. Khodakovski, P. Schröder, Variational normal meshes, ACM Trans. Graph. 23, 1061–1073 (2004). CrossRefGoogle Scholar
  10. 10.
    P. Grohs, A general proximity analysis of nonlinear subdivision schemes, SIAM J. Math. Anal. 42, 729–750 (2010). MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    I. Guskov, K. Vidimce, W. Sweldens, P. Schröder, Normal meshes, in Computer Graphics (SIGGRAPH’00: Proceedings), ed. by K. Akeley (ACM, New York, 2000), pp. 95–102. Google Scholar
  12. 12.
    A. Khodakovsky, I. Guskov, Compression of normal meshes, in Geometric Modeling for Scientific Visualization (Springer, Berlin, 2003), pp. 189–207. Google Scholar
  13. 13.
    J.M. Lane, R.F. Riesenfeld, A theoretical development for the computer generation and display of piecewise polynomial surfaces, IEEE Trans. Pattern Anal. Mach. Intell. 2, 35–46 (1980). zbMATHCrossRefGoogle Scholar
  14. 14.
    S. Lavu, H. Choi, R. Baraniuk, Geometry compression of normal meshes using rate-distortion algorithms, in Proceedings of the Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, ed. by L. Kobbelt, P. Schröder, H. Hoppe (Eurographics Association, Aire-la-Ville, 2003), pp. 52–61. Google Scholar
  15. 15.
    O. Runborg, Introduction to normal multiresolution analysis, in Multiscale Methods in Science and Engineering, ed. by B. Engquist, P. Lötstedt, O. Runborg. Lecture Notes in Computational Science and Engineering, vol. 44 (Springer, Heidelberg, 2005), pp. 205–224. CrossRefGoogle Scholar
  16. 16.
    O. Runborg, Fast interface tracking via a multiresolution representation of curves and surfaces, Commun. Math. Sci. 7, 365–389 (2009). MathSciNetzbMATHGoogle Scholar

Copyright information

© SFoCM 2011

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität KaiserslauternKaiserslauternGermany
  2. 2.School of Engineering and ScienceJacobs UniversityBremenGermany
  3. 3.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

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