Fast Quasi-Continuous Wavelet Algorithms for Analysis and Synthesis of One-Dimensional Signals

The wavelet transform is a widely used time-frequency tool for signal processing. However, with some rare exceptions, its use in signal processing is limited to discrete-time critically sampled transforms, which are particular cases of subband coding. On the other hand, interest in continuous wavelet analyses has been repeatedly demonstrated in the literature. However, implementation challenges limit their practical uses: continuous analyses are time consuming, and current fast algorithms are often restricted to particular generating analysis wavelets; syntheses are even more time consuming and usually too approximate. This paper formalizes the quasi-continuous wavelet transform. A review of algorithms proposed in the literature is presented. Thereafter, a fast quasi-continuous algorithm for analysis is proposed using filter banks. It is valid for almost any generating analysis wavelet. Another version that minimizes the redundancies between subbands is also presented. Different synthesis algorithms are described with filter bank implementations. Different methods answer different needs. The “fair synthesis” algorithm gives the same weight to each point of the time-scale plane. It is important for selective reconstructions of portions of this plane. On the other hand, the “closest takes most” method allows a hierarchical approach. Finally, the direct summation method often gives fair approximations. The proposed algorithms are compatible with hybrid wavelet transforms where, for example, the wavelet can change from one scale level to another.