Advances in Computational Mathematics

, Volume 41, Issue 3, pp 709–726 | Cite as

An anisotropic directional subdivision and multiresolution scheme

  • Mariantonia Cotronei
  • Daniele Ghisi
  • Milvia Rossini
  • Tomas Sauer
Article

Abstract

In order to handle directional singularities, standard wavelet approaches have been extended to the concept of discrete shearlets in Kutyniok and Sauer (SIAM J. Math. Anal. 41, 1436–1471, 2009). One disadvantage of this extension, however, is the relatively large determinant of the scaling matrices used there which results in a substantial data complexity. This motivates the question whether some of the features of the discrete shearlets can also be obtained by means of different geometries. In this paper, we give a positive answer by presenting a different approach, based on a matrix with small determinant which therefore offers a larger recursion depth for the same amount of data.

Keywords

Subdivision Filterbank Multiple multiresolution analysis 

Mathematics Subject Classification (2000)

65T60 41A05 42C15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Conti, C., Cotronei, M., Sauer, T.: Full rank interpolatory subdivision: a first encounter with the multivariate realm. J. Approx. Theor. 162, 559–575 (2010)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Conti, C., Hormann, K.: Polynomial reproduction for univariate subdivision schemes of any arity. J. Approx. Theory 163(4), 413–437 (2011)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Derado, J.: Multivariate refinable interpolating functions. Appl. Comp. Harm. Anal. 7, 165–183 (1999)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Hamming, R. W.: Digital filters, Prentice-Hall (1989)Google Scholar
  5. 5.
    Häuser, S.: Fast Finite Shearlet Transform: a tutorial, University of Kaiserslautern, http://www.mathematik.uni-kl.de/imagepro/members/haeuser/ffst/
  6. 6.
    Hutchinson, J. E.: Fractals and self similarity. Indiana Univ. Math. J. 30, 713–747 (1981)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Kovačević, J., Vetterli, M.: Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R n. IEEE Trans. Inform. Th. 38(2), 533–555 (1992)CrossRefGoogle Scholar
  8. 8.
    Kutyniok, G., Labate, D. (eds.): Shearlets. Springer (2011)Google Scholar
  9. 9.
    Kutyniok, G., Lim, W.Q., Reisenhofer, R.: ShearLab 3D - Faithful digital shearlet transform with compactly supported shearlets, preprint available at arXiv:1402.5670 (2014)
  10. 10.
    Kutyniok, G., Sauer, T.: Adaptive directional subdivision schemes and shearlet multiresolution analysis. SIAM J. Math. Anal. 41, 1436–1471 (2009)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Kutyniok, G., Shahram, M., Zhuang, X.: ShearLab: A rational design of a digital parabolic scaling algorithm. SIAM J. Imaging Sci. 5(4), 1291–1332 (2012)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Latour, V., Müller, J., Nickel, W.: Stationary subdivision for general scaling matrices. Math. Z. 227, 645–661 (1998)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Marcus, M., Minc, H.: A survey of matrix theory and matrix inequalities, Prindle, Weber & Schmidt (1969)Google Scholar
  14. 14.
    Park, H., Woodburn, C.: An algorithmic proof of Suslin’s stability theorem for polynomial rings. J. Algebra 178, 277–298 (1995)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Sauer, T.: Stationary vector subdivision – quotient ideals, differences and approximation power. Rev. R. Acad. Cien. Serie A. Mat. 96, 257–277 (2002)MATHGoogle Scholar
  16. 16.
    Sauer, T.: Multiple subdivision. In: Boissonnat, J.-D., et al. (eds.) Curves and Surfaces 2011, Lecture Notes in Computer Science 6920, pp 612–628. Springer, Berlin (2011)Google Scholar
  17. 17.
    Sauer, T.: Shearlet Multiresolution and Multiple Refinement. In: Kutyniok, G., Labate, D. (eds.) Shearlets, Springer (2011)Google Scholar
  18. 18.
    Strang, G., Nguyen, T.: Wavelets and Filter Banks, Wellesley–Cambridge Press (1996)Google Scholar
  19. 19.
    Vetterli, M., Kovačević, J.: Wavelets and subband coding, Prentice Hall (1995)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Mariantonia Cotronei
    • 1
  • Daniele Ghisi
    • 2
  • Milvia Rossini
    • 2
  • Tomas Sauer
    • 3
  1. 1.DIIESUniversità Mediterranea di Reggio CalabriaReggio CalabriaItaly
  2. 2.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Milano-BicoccaMilanoItaly
  3. 3.Lehrstuhl für Mathematik mit Schwerpunkt Digitale SignalverarbeitungUniversität PassauPassauGermany

Personalised recommendations