Discrete Wavelet Transform Algorithms for Multi-Scale Analysis of Biomedical Signals

The discrete wavelet transform (DWT) algorithms have a firm position in multi-scale processing of biomedical signals, such as EMG and EEG. The DWT algorithms were initially based on the compactly supported conjugate quadrature filters (CQFs) (Smith & Barnwell, 1986; Daubechies, 1988). However, a drawback in CQFs is due to the nonlinear phase effects such as spatial dislocations in multi-scale analysis. This is avoided in biorthogonal discrete wavelet transform (BDWT) algorithms, where the scaling and wavelet filters are symmetric and linear phase. The biorthogonal filters are usually constructed by a ladder-type network called lifting scheme (Sweldens, 1988; ITU-T, 2000). Efficient lifting BDWT structures have been developed for microprocessor and VLSI environment (Olkkonen et al. 2005; Olkkonen & Olkkonen, 2008). Only integer register shifts and summations are needed for implementation of the analysis and synthesis filters. A severe obstacle in multi-scale DWT analysis is the dependence of the total energy of the wavelet coefficients in different scales on the fractional shifts of the analysed signal. If we have a discrete-time signal [ ] x n and the corresponding time shifted signal [ ] x n   , where [0,1]   , there occurs a notable difference in the energy of the wavelet coefficients as a function of the time shift. Kingsbury (2001) proposed a nearly shift invariant method, where the real and imaginary parts of the complex wavelet coefficients are approximately a Hilbert transform pair. The energy (absolute value) of the wavelet coefficients equals the envelope, which provides smoothness and approximate shift-invariance. Selesnick (2002) observed that using two parallel CQF banks, which are constructed so that the impulse responses of the scaling filters have half-sample delayed versions of each other: [ ] h n and [ 0.5] h n  , the corresponding wavelet bases are a Hilbert transform pair. Selesnick (2002) proposed a spectral factorization method based on the half delay all-pass Thiran filters for design of the scaling filters. However, the scaling filters do not owe coefficient symmetry and the nonlinearity interferes with the spatial timing in different scales and prevents accurate statistical correlations. In this book chapter we review the shift invariant DWT algorithms for multi-scale analysis of biomedical signals. We describe a dual-tree DWT, where two parallel CQF wavelet sequences form a Hilbert pair, which warrants the shift invariance. Next we review the construction of the shift invariant BDWT, which is based on the novel design of the Hilbert