Biomedical Imaging GroupSTI
English only   BIG > Publications > Fractional Splines

 Home Page
 News & Events
 Tutorials and Reviews
 Download Algorithms

 PDF not available
 PS not available
 All BibTeX References

Fractional Splines and Wavelets: From Theory to Applications

M. Unser, T. Blu

Joint IDR-IMA Workshop: Ideal Data Representation, Minneapolis MN, USA, April 9-13, 2001.

In the first part, we present the theory of fractional splines; an extension of the polynomial splines for non-integer degrees. Their basic constituents are piecewise power functions of degree α. The corresponding B-splines are obtained through a localization process similar to the classical one, replacing finite differences by fractional differences. We show that the fractional B-splines share virtually all the properties of the classical B-splines, including the two-scale relation, and can therefore be used to define new wavelet bases with a continuously varying order parameter. We discuss some of their remarkable properties; in particular, the fact that the fractional spline wavelets behave like fractional derivatives of order α + 1.

In the second part, we turn to applications. We first describe a fast implementation of the fractional wavelet transform, which is essential to make the method practical. We then present an application of fractional splines to tomographic reconstruction where we take advantage of explicit formulas for computing the fractional derivatives of splines. We also make the connection with ridgelets. Finally, we consider the use of fractional wavelets for the detection and localization of brain activation in fMRI sequences. Here, we take advantage of the continuously varying order parameter which allows us to fine-tune the localization properties of the basis functions.

AUTHOR="Unser, M. and Blu, T.",
TITLE="Fractional Splines and Wavelets: {F}rom Theory to
BOOKTITLE="Joint {IDR-IMA} Workshop: {I}deal Data Representation",
address="Minneapolis MN, USA",
month="April 9-13,",

© 2001 IMA. 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 IMA.
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.