Biomedical Imaging GroupSTI
English only   BIG > Publications > Fractional Splines

 Home Page
 News & Events
 Tutorials and Reviews
 Download Algorithms

 All BibTeX References

Construction of Fractional Spline Wavelet Bases

M. Unser, T. Blu

Proceedings of the SPIE Conference on Mathematical Imaging: Wavelet Applications in Signal and Image Processing VII, Denver CO, USA, July 19-23, 1999, vol. 3813, pp. 422-431.

We extend Schoenberg's B-splines to all fractional degrees α > -1/2. These splines are constructed using linear combinations of the integer shifts of the power functions x+α(one-sided) or |x|*α(symmetric); in each case, they are α-Hölder continuous for α > 0. They satisfy most of the properties of the traditional B-splines; in particular, the Riesz basis condition and the two-scale relation, which makes them suitable for the construction of new families of wavelet bases. What is especially interesting from a wavelet perspective is that the fractional B-splines have a fractional order of approximation (α+1), while they reproduce the polynomials of degree [α]. We show how they yield continuous-order generalizations of the orthogonal Battle-Lemarié wavelets and of the semi-orthogonal B-spline wavelets. As α increases, these latter wavelets tend to be optimally localized in time and frequency in the sense specified by the uncertainty principle. The corresponding analysis wavelets also behave like fractional differentiators; they may therefore be used to whiten fractional Brownian motion processes.

AUTHOR="Unser, M. and Blu, T.",
TITLE="Construction of Fractional Spline Wavelet Bases",
BOOKTITLE="Proceedings of the {SPIE} Conference on Mathematical
        Imaging: {W}avelet Applications in Signal and Image Processing
address="Denver CO, USA",
month="July 19-23,",

© 1999 SPIE. 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 SPIE.
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.