Lifting and Wavelets: A Factorization Method

We begin with the concept of a discrete wavelet transformation. We begin with a scaling function satisfying a certain number of properties to be able to implement the wavelet transformation. We will then require translates of the scaling function to obey the same set of properties and this set will form what is called a Multiresolution Analysis. We then switch the ideas to the Fourier transform domain were we get equivalent results. However, instead of choosing a scaling function with coefficients to satisfy a set of properties, we will work backwards and construct a scaling function based on having a set of coefficients satisfy the necessary properties. This method of construction will lead us to some classic wavelet transformations known as the Haar and Daubechies wavelets. We then introduce the lifting method for wavelet transformations. Unlike the discrete wavelet transform, the lifting scheme divides the signal into even and odd components and performs both a prediction and update step. We will then show through implementation how this method is computationally more efficient. Furthermore, as shown in Daubechies and Sweldens [2], we will prove that any discrete wavelet transformation can be decomposed into lifting steps. This decomposition corresponds to a factorization of the associated polyphase matrix of a wavelet transform. This factorization is implemented with the help of the Euclidean algorithm, with the focus on the Laurent polynomials to represent our filters. This paper will then conclude with some classical wavelet transformations and compare the results to using the lifting method through implementation on Mathematica