Lifting and Filters II

Abstract

There are basically three forms for representing the building block in a DWT: The transform can be represented by a pair of filters (usually low pass and high pass filters) satisfying the perfect reconstruction conditions from Chap. 7, or it can be given as lifting steps, which are either given in the time domain as a set of equations, or in the frequency domain as a factored matrix of Laurent polynomials. The Daubechies 4 transform has been presented in all three forms in previous chapters, but so far we have only made casual attempts to convert between the various representations. When trying to do so, it turns out that only one conversion requires real work, namely conversion from filter to matrix and equation forms. In Chap. 7 we presented the theorem, which shows that it is always possible to do this conversion, but we did not show how to do it. This chapter is therefore dedicated to discussing the three basic forms of representation of the wavelet transform, as well as the conversions between them. In particular, we give a detailed proof of the `from filter to matrix/equation’ theorem stated in Chap. 7. The proof is a detailed and exemplified version of the proof found in I. Daubechies and W. Sweldens [7].