Biomedical Imaging GroupSTI
English only   BIG > Publications > Fractional Differentiators

 Home Page
 News & Events
 Tutorials and Reviews
 Download Algorithms

 All BibTeX References

Semi-Orthogonal Wavelets That Behave like Fractional Differentiators

D. Van De Ville, T. Blu, B. Forster, M. Unser

Proceedings of the SPIE Optics and Photonics 2005 Conference on Mathematical Methods: Wavelet XI, San Diego CA, USA, July 31-August 3, 2005, vol. 5914, pp. 59140C-1/59140C-8.

The approximate behavior of wavelets as differential operators is often considered as one of their most fundamental properties. In this paper, we investigate how we can further improve on the wavelet's behavior as differentiator. In particular, we propose semi-orthogonal differential wavelets. The semi-orthogonality condition ensures that wavelet spaces are mutually orthogonal. The operator, hidden within the wavelet, can be chosen as a generalized differential operator ∂τγ, for a γ-th order derivative with shift τ. Both order of derivation and shift can be chosen fractional. Our design leads us naturally to select the fractional B-splines as scaling functions. By putting the differential wavelet in the perspective of a derivative of a smoothing function, we find that signal singularities are compactly characterized by at most two local extrema of the wavelet coefficients in each subband. This property could be beneficial for signal analysis using wavelet bases. We show that this wavelet transform can be efficiently implemented using FFTs.

AUTHOR="Van De Ville, D. and Blu, T. and Forster, B. and Unser, M.",
TITLE="Semi-Orthogonal Wavelets That Behave like Fractional
BOOKTITLE="Proceedings of the {SPIE} Conference on Mathematical Imaging:
        {W}avelet {XI}",
address="San Diego CA, USA",
month="July 31-August 3,",

© 2005 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.