Generalized Daubechies Wavelets

We present a generalization of the Daubechies wavelet family. The context is that of a non-stationary multiresolution analysis—i.e., a sequence of embedded approximation spaces generated by scaling functions that are not necessarily dilates of one another. The constraints that we impose on these scaling functions are: (1) orthogonality with respect to translation, (2) reproduction of a given set of exponential polynomials, and (3) minimal support. These design requirements lead to the construction of a general family of compactly-supported, orthonormal wavelet-like bases of $L _{ 2 }$ . If the exponential parameters are all zero, then one recovers Daubechies wavelets, which are orthogonal to the polynomials of degree (N − 1) where N is the order (vanishing-moment property). A fast filterbank implementation of the generalized wavelet transform follows naturally; it is similar to Mallat's algorithm, except that the filters are now scale-dependent. The new transforms offer increased flexibility and are tunable to the spectral characteristics of a wide class of signals.