EPFL | Biomedical Imaging Group | Generalized Daubechies

Generalized Daubechies Wavelet Families

C. Vonesch, T. Blu, M. Unser

IEEE Transactions on Signal Processing, vol. 55, no. 9, pp. 4415–4429, September 2007.

We present a generalization of the orthonormal Daubechies wavelets and of their related biorthogonal flavors (Cohen-Daubechies-Feauveau, 9⁄7). Our fundamental constraint is that the scaling functions should reproduce a predefined set of exponential polynomials. This allows one to tune the corresponding wavelet transform to a specific class of signals, thereby ensuring good approximation and sparsity properties. The main difference with the classical construction of Daubechies et al. is that the multiresolution spaces are derived from scale-dependent generating functions. However, from an algorithmic standpoint, Mallat's Fast Wavelet Transform algorithm can still be applied; the only adaptation consists in using scale-dependent filter banks. Finite support ensures the same computational efficiency as in the classical case. We characterize the scaling and wavelet filters, construct them and show several examples of the associated functions. We prove that these functions are square-integrable and that they converge to their classical counterparts of the corresponding order.

@ARTICLE(http://bigwww.epfl.ch/publications/vonesch0702.html,
AUTHOR="Vonesch, C. and Blu, T. and Unser, M.",
TITLE="Generalized {D}aubechies Wavelet Families",
JOURNAL="{IEEE} Transactions on Signal Processing",
YEAR="2007",
volume="55",
number="9",
pages="4415--4429",
month="September",
note="")

© 2007 IEEE. 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 IEEE. 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.