Filter design and comparison for two fast CWT algorithms

The continuous wavelet transform (CWT) is a powerful technique for signal analysis. Direct CWT computation by FFT requires O(N log₂ N) operations per scale, where N is the data length. The a trous algorithm and the Shensa (1992) algorithm are two fast methods to compute CWT recursively that require only O(N) operations per scale. Both of them can be described by the multiresolution analysis (MRA) structure but with different MRA filters. This paper proposes methods to design the MRA filters of the two algorithms to improve their accuracy on CWT computation. We begin with the formulation of the CWT computation error using the MRA structure. The MRA filters of the two algorithms are then designed to minimize the error. In either algorithm, both the lowpass and bandpass MRA filters can be optimized. The a trous algorithm has closed-form solutions for the two filters. The Shensa algorithm, on the other hand, has an analytic solution for the bandpass filter only. Finding the optimum lowpass filter requires a multidimensional numerical search. Simulation studies show that by using the proposed optimum filters, the Shensa algorithm, in general, outperforms the a trous algorithm