EPFL
 Biomedical Imaging GroupSTI
EPFL
  Publications
English only   BIG > Publications > Biorthogonal Wavelets


 CONTENTS
 Home Page
 News & Events
 People
 Publications
 Tutorials and Reviews
 Research
 Demos
 Download Algorithms

 DOWNLOAD
 PDF
 Postscript
 All BibTeX References

Approximation Power of Biorthogonal Wavelet Expansions

M. Unser

IEEE Transactions on Signal Processing, vol. 44, no. 3, pp. 519-527, March 1996.



This paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions. The Strang-Fix conditions imply that the error for an orthogonal wavelet approximation at scale a = 2-i globally decays as aN, where N is the order of the transform. This is why, for a given number of scales, higher order wavelet transforms usually result in better signal approximations. We prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone. We also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero. These results strengthen Sweldens earlier analysis and confirm that the approximation power of biorthogonal and (semi-)orthogonal wavelet expansions is essentially the same. Finally, we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines. We also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error.


@ARTICLE(http://bigwww.epfl.ch/publications/unser9604.html,
AUTHOR="Unser, M.",
TITLE="Approximation Power of Biorthogonal Wavelet Expansions",
JOURNAL="{IEEE} Transactions on Signal Processing",
YEAR="1996",
volume="44",
number="3",
pages="519--527",
month="March",
note="")

© 1996 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from IEEE.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.