Designing Multiresolution Analysis-type Wavelets and Their Fast Algorithms

Often, the Dyadic Wavelet Transform is performed and implemented with the Daubechies wavelets, the Battle-Lemarie wavelets, or the splines wavelets, whereas in continuous-time wavelet decomposition a much larger variety of mother wavelets is used. Maintaining the dyadic time-frequency sampling and the recursive pyramidal computational structure, we present various methods for constructing wavelets ψ_wanted, with some desired shape and properties and which are associated with semi-orthogonal multiresolution analyses. We explain in detail how to design any desired wavelet, starting from any given multiresolution analysis. We also explicitly derive the formulae of the filter bank structure that implements the designed wavelet. We illustrate these wavelet design techniques with examples that we have programmed with Matlab routines.