EPFL
 Biomedical Imaging GroupSTI
EPFL
  Publications
English only   BIG > Publications > Fractional Splines


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

 DOWNLOAD
 PDF
 Postscript
 All BibTeX References

Fractional Splines and Wavelets

M. Unser, T. Blu

SIAM Review, vol. 42, no. 1, pp. 43-67, March 2000.



unser9901fig01.gif

We extend Schoenberg's family of polynomial splines with uniform knots to all fractional degrees α > -1. These splines, which involve linear combinations of the one-sided power functions x+α = max(0, x)α, belong to L1 and are α-Hölder continuous for α > 0. We construct the corresponding B-splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains. We show that these functions satisfy most of the properties of the traditional B-splines, including the convolution property, and a generalized fractional differentiation rule that involves finite differences only. We characterize the decay of the B-splines which are not compactly supported for non-integral α's. Their most astonishing feature (in reference to the Strang-Fix theory) is that they have a fractional order of approximation α + 1 while they reproduce the polynomials of degree [α]. For α > 1/2, they satisfy all the requirements for a multiresolution analysis of L2 (Riesz bounds, two scale relation) and may therefore be used to build new families of wavelet bases with a continuously-varying order parameter. Our construction also yields symmetrized fractional B-splines which provide the connection with Duchon's general theory of radial (m,s)-splines (including thin-plate splines). In particular, we show that the symmetric version of our splines can be obtained as solution of a variational problem involving the norm of a fractional derivative. (Front cover).


@ARTICLE(http://bigwww.epfl.ch/publications/unser9901.html,
AUTHOR="Unser, M. and Blu, T.",
TITLE="Fractional Splines and Wavelets",
JOURNAL="{SIAM} Review",
YEAR="2000",
volume="42",
number="1",
pages="43--67",
month="March",
note="")

© 2000 SIAM. 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 SIAM.
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.