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Advances in Multiresolution for Geometric Modelling pp 369–390Cite as

Periodic and Spline Multiresolution Analysis and the Lifting Scheme

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Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Summary

The lifting scheme is a well-known general framework for the construction of wavelets, especially in finitedimensional settings. After a short introduction about the basics of lifting, we discuss how wavelet constructions, in two specific finite settings, can be related to the lifting approach. These examples concern, on the one hand, polynomial splines and, on the other, the Fourier approach for translation-invariant spaces of periodic functions.

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References

  1. Carnicer, J.M., Dahmen, W., Pena, J.M.: Local decomposition of refinable spaces and wavelets. Appl. Comput. Harmon. Anal., 3, 127–153 (1996).

    Article  MathSciNet  Google Scholar 

  2. Chui, C.K., Mhaskar, H.N.: On trigonometric wavelets. Constr. Approx., 9, 167–190 (1993).

    Article  MathSciNet  Google Scholar 

  3. Cohen, A., Daubechies, I., Feauveau, J.-C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math., 45, 485–560 (1992).

    MathSciNet  Google Scholar 

  4. Dahmen, W., Kunoth, A., Urban, K.: Biorthogonal spline wavelets on the interval — stability and moment conditions. Appl. Comput. Harmon. Anal., 6, 132–196 (1999).

    Article  MathSciNet  Google Scholar 

  5. Dahmen, W., Micchelli, C.A.: Banded matrices with banded inverses. II: Locally finite decomposition of spline spaces. Constr. Approx., 9, 263–281 (1993).

    Article  MathSciNet  Google Scholar 

  6. Dahmen, W., Schneider, R.: Wavelets on manifolds. I: Construction and domain decomposition. SIAM J. Math. Anal., 31, 184–230 (1999).

    Article  MathSciNet  Google Scholar 

  7. Daubechies, I., Sweldens, W.: Factoring wavelet transforms into lifting steps. J. Fourier Anal. Appl., 4, 247–269 (1998).

    MathSciNet  Google Scholar 

  8. Koh, Y.W., Lee, S.L., Tan, H.H. Periodic orthogonal splines and wavelets. Appl. Comput. Harmonic Anal., 2, 201–218 (1995).

    Article  MathSciNet  Google Scholar 

  9. Lyche, T., Mørken, K., Quak, E.: Theory and algorithms for nonuniform spline wavelets. In: Dyn, N., Leviatan, D., Levin, D., and Pinkus, A., (eds.), Multivariate approximation and applications, Cambridge University Press, 152–187 (2001).

    Google Scholar 

  10. Masson, R.: Biorthogonal spline wavelets on the interval for the resolution of boundary problems. Math. Models Methods Appl. Sci., 6, 749–791 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  11. Narcowich, F.J., Ward, J.D.: Wavelets associated with periodic basis functions. Appl. Comput. Harmon. Anal., 3, 40–56 (1996).

    Article  MathSciNet  Google Scholar 

  12. Plonka, G., Tasche, M.: A unified approach to periodic wavelets. In: Chui, C.K., Montefusco, L., and Puccio, L. (eds.), Wavelets: Theory, Algorithms, and Applications, Academic Press, New York, 137–151 (1994).

    Google Scholar 

  13. Plonka, G., Tasche, M.: On the computation of periodic spline wavelets. Appl. Comput. Harmon. Anal., 2, 1–14 (1995).

    Article  MathSciNet  Google Scholar 

  14. Prestin, J., Quak, E.: Trigonometric interpolation and wavelet decompositions. Num. Alg., 9, 293–317 (1995).

    Article  MathSciNet  Google Scholar 

  15. Quak, E.: Nonuniform B-splines and B-wavelets. In: Iske, A., Quak, E., Floater M. S., (eds.), Tutorials on Multiresolution in Geometric Modelling, Springer, 101–146 (2002).

    Google Scholar 

  16. Schneider, R.: Multiskalenund Wavelet-Matrixkompression. Analysisbasierte Methoden zur effizienten Lösung großer vollbesetzter Gleichungssysteme. Advances in Numerical Mathematics. B. G. Teubner Stuttgart (1998).

    Google Scholar 

  17. Selig, K.: Periodische Wavelet-Packets und eine gradoptimale Schauderbasis. PhD thesis, Universität Rostock, Germany, Shaker Verlag, Aachen (1998).

    Google Scholar 

  18. Sweldens, W., Schröder, P.: Building your own wavelets at home. Wavelets in Computer Graphics, ACM SIGGRAPH Course Notes (1996).

    Google Scholar 

  19. Sweldens, W.: The lifting scheme: A custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal., 3, 186–200 (1996).

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Lübeck, Germany

    Jürgen Prestin

  2. Department of Applied Mathematics, SINTEF ICT, Oslo, Norway

    Ewald Quak

Authors
  1. Jürgen Prestin
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  2. Ewald Quak
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Editor information

Editors and Affiliations

  1. Computer Laboratory, University of Cambridge, William Gates Building, 15 J. J. Thomson Avenue, Cambridge, CB3 0FD, UK

    Neil A. Dodgson

  2. Computer Science Department, Oslo University, P. O. Box 1053, 0316, Blindern, Oslo, Norway

    Michael S. Floater

  3. Numerical Geometry Ltd., 26 Abbey Lane, Lode, Cambridge, CB5 9EP, UK

    Malcolm A. Sabin

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© 2005 Springer-Verlag Berlin Heidelberg

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Prestin, J., Quak, E. (2005). Periodic and Spline Multiresolution Analysis and the Lifting Scheme. In: Dodgson, N.A., Floater, M.S., Sabin, M.A. (eds) Advances in Multiresolution for Geometric Modelling. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26808-1_21

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