Flexible Wavelet Transforms Using Lifting

Roger L. Claypoole, Jr. and Richard G. Baraniuk, Rice University Summary We introduce and discuss biorthogonal wavelet transforms using the lifting construction. The lifting construction exploits a spatial{domain, prediction{error interpretation of the wavelet transform and provides a powerful framework for designing customized transforms. We discuss the application of lifting to adaptive and non{linear transforms, transforms of non{uniformly sampled data, and related issues. Introduction Many applications (compression, analysis, denoising, etc.) bene t from signal representation with as few coe cients as possible. We also wish to characterize a signal as a series of course approximations, with sets of ner and ner details. The Discrete Wavelet Transform (DWT) provides such a representation. The DWT represents a real-valued discrete-time signal in terms of shifts and dilations of a lowpass scaling function and a bandpass wavelet function [2]. The DWT decomposition is multiscale: it consists of a set of scaling coe cients c0[n], which represent coarse signal information at scale j = 0, and a set of wavelet coe cients dj [n], which represent detail information at scales j = 1; 2; : : : ; J . The forward DWT has an e cient implementation in terms of a recursive multirate lterbank based around a lowpass lter h and highpass lter g [12, pp. 302{332]. The inverse DWT employs an inverse lterbank with lowpass lter eh and highpass lter eg, as shown in Figure 1 For special choices of h, g, eh, and eg, the underlying wavelet and scaling functions form a biorthogonal wavelet basis [2].