Fast Algorithms for Running Wavelet Analyses

We present a general framework for the design and efficient implementation of various types of running (or over-sampled) wavelet transforms (RWT) using polynomial splines. Unlike previous techniques, the proposed algorithms are not necessarily restricted to scales that are powers of two; yet they all achieve the lowest possible complexity : 0(N) per scale, where N is signal length. In particular, we propose a new algorithm that can handle any integer dilation factor and use wavelets with a variety of shapes (including Mexican-Hat and cosine-Gabor). A similar technique is also developed for the computation of Gabor-like complex RWTs. We also indicate how the localization of the analysis templates (real or complex B-spline wavelets) can be improved arbitrarily (up to the limit specified by the uncertainty principle) by increasing the order of the splines. These algorithms are then applied to the analysis of EEG signals and yield several orders of magnitude speed improvement over a standard implementation.