EPFL | Biomedical Imaging Group | Daubechies Wavelets

Generalized Biorthogonal Daubechies Wavelets

C. Vonesch, T. Blu, 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. 59141X-1–59141X-6.

We propose a generalization of the Cohen-Daubechies-Feauveau (CDF) and 9⁄7 biorthogonal wavelet families. This is done within the framework of non-stationary multiresolution analysis, which involves a sequence of embedded approximation spaces generated by scaling functions that are not necessarily dilates of one another. We consider a dual pair of such multiresolutions, where the scaling functions at a given scale are mutually biorthogonal with respect to translation. Also, they must have the shortest-possible support while reproducing a given set of exponential polynomials. This constitutes a generalization of the standard polynomial reproduction property.

The corresponding refinement filters are derived from the ones that were studied by Dyn et al. in the framework of non-stationary subdivision schemes. By using different factorizations of these filters, we obtain a general family of compactly supported dual wavelet bases of L₂. In particular, if the exponential parameters are all zero, one retrieves the standard CDF B-spline wavelets and the 9⁄7 wavelets. Our generalized description yields equivalent constructions for E-spline wavelets.

A fast filterbank implementation of the corresponding wavelet transform follows naturally; it is similar to Mallat's algorithm, except that the filters are now scale-dependent. This new scheme offers high flexibility and is tunable to the spectral characteristics of a wide class of signals. In particular, it is possible to obtain symmetric basis functions that are well-suited for image processing.

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