From Full Rank Subdivision Schemes to Multichannel Wavelets: A Constructive Approach

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


In this paper, we describe some recent results obtained in the context of vector subdivision schemes which possess the so-called full rank property. Such kind of schemes, in particular those which have an interpolatory nature, are connected to matrix refinable functions generating orthogonal multiresolution analyses for the space of vector-valued signals. Corresponding multichannel (matrix) wavelets can be defined and their construction in terms of a very efficient scheme is given. Some examples illustrate the nature of these matrix scaling functions/wavelets.


Full Rank Subdivision Scheme Common Zero Laurent Polynomial Matrix Completion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Bacchelli, M. Cotronei, T. Sauer, Wavelets for multichannel signals, Adv. Appl. Math., 29, 581–598 (2002)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    C. Conti, M. Cotronei, T. Sauer, Full rank interpolatory subdivision schemes: Kronecker, filters and multiresolution, J. Comput. Appl. Math., 233 (7), 1649–1659 (2010)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    C. Conti, M. Cotronei, T. Sauer, Full rank positive matrix symbols: interpolation and orthogonality, BIT, 48, 5–27 (2008)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    C. Conti, M. Cotronei, T. Sauer, Interpolatory vector subdivision schemes, in: A. Cohen, J. L. Merrien, L. L. Schumaker (eds.), Curves and Surfaces, Avignon 2006, Nashboro Press, (2007)Google Scholar
  5. 5.
    M. Cotronei, T. Sauer, Full rank filters and polynomial reproduction, Comm. Pure Appl. Anal., 6, 667–687 (2007)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    J.E. Fowler, L. Hua, Wavelet Transforms for Vector Fields Using Omnidirectionally Balanced Multiwavelets, IEEE Transactions on Signal Processing, 50, 3018–3027 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. S. Geronimo, D. P. Hardin, P. R. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory, 78 (3), 373–401 (1994)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    F.Keinert, Wavelets and Multiwavelets, Chapman & Hall/CRC, (2004)Google Scholar
  9. 9.
    J. Ježek and V. Kučera, Efficient algorithm for matrix spectral factorization, Automatica, 21, 663–669 (1985)MATHCrossRefGoogle Scholar
  10. 10.
    Lawton, W. M.; Micchelli, C. A., Bezout identities with inequality constraints. Vietnam J. Math. 28 (2), 97–126 (2000)MathSciNetMATHGoogle Scholar
  11. 11.
    C. A. Micchelli, T. Sauer, Regularity of multiwavelets, Adv. Comput. Math., 7 (4), 455–545 (1997)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    G. Strang, V. Strela, Short wavelets and matrix dilation equations, IEEE Trans. Signal Process., 43 (1), 108–115 (1995)CrossRefGoogle Scholar
  13. 13.
    X.G. Xia, Orthonormal matrix valued wavelets and matrix Karhunen–Loève expansion. In: A. Aldroubi, E. B. Lin (eds.) Wavelets, multiwavelets, and their applications, Contemporary Mathematics, 216, 159–175. Providence, RI: American Mathematical Society (1998)Google Scholar
  14. 14.
    X.G. Xia, B. Suter, Vector-valued wavelets and vector filter banks, IEEE Trans. Signal Process., 44, 508–518 (1996)CrossRefGoogle Scholar
  15. 15.
    A. T. Walden, A. Serroukh, Wavelet analysis of matrix-valued time-series, Proc. R. Soc. Lond. A, 458, 157–179 (2002)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Dipartimento di Energetica “Sergio Stecco”Università di FirenzeFirenzeItaly
  2. 2.DIMET, Università “Mediterranea” di Reggio CalabriaReggio CalabriaItaly

Personalised recommendations