Skip to main content
×
Home
    • Aa
    • Aa

A fast algorithm for constructing orthogonal multiwavelets

  • Yang Shouzhi (a1)
Abstract
Abstract

Multiwavelets possess some nice features that uniwavelets do not. A consequence of this is that multiwavelets provide interesting applications in signal processing as well as in other fields. As is well known, there are perfect construction formulas for the orthogonal uniwavelet. However, a good formula with a similar structure for multiwavelets does not exist. In particular, there are no effective methods for the construction of multiwavelets with a dilation factor a (a ≥ 2, aZ). In this paper, a procedure for constructing compactly supported orthonormal multiscaling functions is first given. Based on the constructed multiscaling functions, we then propose a method of constructing multiwavelets, which is similar to that for constructing uniwavelets. In addition, a fast numerical algorithm for computing multiwavelets is given. Compared with traditional approaches, the algorithm is not only faster, but also computationally more efficient. In particular, the function values of several points are obtained simultaneously by using our algorithm once. Finally, we give three examples illustrating how to use our method to construct multiwavelets.

    • Send article to Kindle

      To send this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A fast algorithm for constructing orthogonal multiwavelets
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      A fast algorithm for constructing orthogonal multiwavelets
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      A fast algorithm for constructing orthogonal multiwavelets
      Available formats
      ×
Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]A. C. Cabrelli and L. M. Gordillo , “Existence of multiwavelets in Rn”, Proc. Amer. Math. Soc. 130 (2002) 14131424.

[2]C. K. Chui and J. Lian , “A study on orthonormal multiwavelets”, Appl. Numer Math. 20 (1996) 273298.

[3]C. K. Chui and J. Z. Wang , “A cardinal spline approach to wavelets”, Proc. Amer Math. Soc. 113 (1991) 785793.

[4]I. Daubechies , Ten lectures on wavelets (SIAM, Philadelphia, PA, 1992).

[5]I. Daubechies , “Orthonormal bases of compactly supported wavelets”, Comm. Pure Appl. Math. 41 (1998) 909996.

[6]I. Daubechies and J. C. Lagarias , “Two-scale difference equations. I Existence and global regularity of solutions”, SIAM J. Math. Anal. 22 (1991) 13881410.

[7]G. C. Donovan , J. Geronimo and D. P. Hardin , “Construction of orthogonal wavelets using fractal interpolation functions”, SIAM J. Math. Anal. 27 (1996) 11581192.

[8]G. C. Donovan , J. Geronimo and D. P. Hardin , “Intertwining multiresolution analysis and construction of piecewise polynomial wavelets”, SIAM J. Math. Anal. 27 (1996) 17911815.

[10]J. Geronimo , D. P. Hardin and P. Massopust , “Fractal functions and wavelet expansions based on several scaling functions”, J. Approx. Theory 78 (1998) 373401.

[11]T. N. T. Goodman , S. L. Lee and W. S. Tang , “Wavelets in wandering subspaces”, Trans. Amer. Math. Soc. 338 (1993) 639654.

[12]B. Kessler , “A construction of compactly supported biorthogonal scaling vectors and multiwavelets on R2”, J. Approx. Theory 117 (2002) 229254.

[13]P. Massopust , D. Ruch and P. Van Fleet , “On the support properties of scaling vectors”, Appl. Comput. Harmon. Anal. 3 (1996) 229238.

[14]I. Selesnick , “Interpolating multiwavelet bases and the sampling theorem”, IEEE Trans. Signal Process. 47 (1999) 16151621.

[16]G. Strang and V. Strela , “Orthogonal multiwavelets with vanishing moments”, J. Opt. Eng. 33 (1994) 21042107.

[17]V. Strela , P. N. Heller , G. Strang , P. Topiwala and C. Heil , “The application of multiwavelet filter banks to image processing”, IEEE Trans. Image Process. 8 (1999) 548563.

[18]C. J. Tymczak , A. M. N. Niklasson and H. Röder , “Separable and nonseparable multiwavelets in multiple dimensions”, J. Comput. Phys. 175 (2002) 363397.

[19]D. Zhou , “Interpolatory orthogonal multiwavelets and refinable functions”, IEEE Trans. Signal Process. 50 (2002) 520527.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax