Automation of the lifting factorisation of wavelet transforms

Abstract

Wavelets are sets of basis functions used in the analysis of signals and images. In contrast to Fourier analysis, wavelets have both spatial and frequency localization, making them useful for the analysis of sharply-varying or non-periodic signals. The lifting scheme for finding the discrete wavelet transform was demonstrated by Daubechies and Sweldens (1996). In particular, they showed that this method depends on the factorization of polyphase matrices, whose entries are Laurent polynomials, using the Euclidean algorithm extended to Laurent polynomials. Such factorization is not unique and hence there are multiple factorizations of the polyphase matrix. In this paper we outline a Mathematica program that finds all factorizations of such matrices by automating the Euclidean algorithm for Laurent polynomials. Polynomial reduction using Gröbner bases was also incorporated into the program so as to reduce the number of wavelet filter coefficients appearing in a given expression through use of the relations they satisfy, thus permitting exact symbolic factorizations for any polyphase matrix.