Factoring wavelet transforms into lifting steps-英文文献.pdf

FACTORING WAVELET TRANSFORMS INTO LIFTING STEPS INGRID DAUBECHIES AND WIM SWELDENS September 1996, revised November 1997 AB STRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filter- ing steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formula ); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically re- duces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers. 1. INTRODUCTION Over the last decade several constructions of compactly supported wavelets originated both from mathematical analysis and the signal processing community. The roots of critically sampled wavelet transforms are actually older than