Generalized coiflets with nonzero-centered vanishing moments

We show that imposing a certain number of vanishing moments on a scaling function (e.g., coiflets) leads to fairly small phase distortion on its associated filter bank in the neighborhood of DC. However, the phase distortion at the other frequencies can be much larger. We design a new class of real-valued, compactly supported, orthonormal, and nearly symmetric wavelets (we call them generalized coiflets) with a number of nonzero-centered vanishing moments equally distributed on scaling function and wavelet. Such a generalization of the original coiflets offers one more free parameter, the mean of the scaling function, in designing filter banks. Since this parameter uniquely characterizes the first several moments of the scaling function, it is related to the phase response of the lowpass filter at low frequencies. We search for the optimal parameter to minimize the maximum phase distortion of the filter bank over the lowpass half-band. Also, we are able to construct nearly odd-symmetric generalized coiflets, whose associated lowpass filters are surprisingly similar to those of some biorthogonal spline wavelets. These new wavelets can be useful in a broad range of signal and image processing applications because they provide a better tradeoff between the two desirable but conflicting properties of the compactly supported and real-valued wavelets, i.e., orthonormality versus symmetry, than the original coiflets