17 September 2005 Generalized biorthogonal Daubechies wavelets
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Proceedings Volume 5914, Wavelets XI; 59141X (2005) https://doi.org/10.1117/12.616536
Event: Optics and Photonics 2005, 2005, San Diego, California, United States
Abstract
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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Cedric Vonesch, Cedric Vonesch, Thierry Blu, Thierry Blu, Michael Unser, Michael Unser, "Generalized biorthogonal Daubechies wavelets", Proc. SPIE 5914, Wavelets XI, 59141X (17 September 2005); doi: 10.1117/12.616536; https://doi.org/10.1117/12.616536
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