EPFL | Biomedical Imaging Group | Fractional Wavelets

Fractional Wavelets: Properties and Applications

M. Unser, T. Blu

Proceedings of the SIAM Conference on Imaging Science (IS'02), Boston MA, USA, March 4-6, 2002, Session MS1, pp. 33.

We introduce the concept of fractional wavelets which extends the conventional theory to non-integer orders. This allows for the construction of new wavelet bases that are indexed by a continuously-varying order parameter, as opposed to an integer. An essential feature of the method is to gain control over the key wavelet properties (regularity, time-frequency localization, etc…). Practically, this translates into the fact that all important wavelet parameters are adjustable in a continuous fashion so that the new basis functions can be fine-tuned for the application at hand. We present some specific examples of wavelets (fractional splines) and investigate the main implications of the fractional order property. In particular, we prove that these wavelets essentially behave like fractional derivative operators which makes them good candidates for the analysis and synthesis of fractal-like processes. We also consider non-separable extensions to quincunx lattices which are well suited for image processing. Finally, we deal with the practical aspect of the evaluation of these transforms and present a fast implementation based on the FFT.

@INPROCEEDINGS(http://bigwww.epfl.ch/publications/unser0202.html,
AUTHOR="Unser, M. and Blu, T.",
TITLE="Fractional Wavelets: {P}roperties and Applications",
BOOKTITLE="Proceedings of the {SIAM} Conference on Imaging Science
	({IS'02})",
YEAR="2002",
editor="",
volume="MS1",
series="",
pages="33",
address="Boston MA, USA",
month="March 4-6,",
organization="",
publisher="",
note="")

© 2002 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.