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Factoring wavelet transforms into lifting steps (1998)
| Venue: | J. FOURIER ANAL. APPL |
| Citations: | 583 - 8 self |
BibTeX
@ARTICLE{Daubechies98factoringwavelet,
author = {Ingrid Daubechies and Wim Sweldens},
title = {Factoring wavelet transforms into lifting steps},
journal = {J. FOURIER ANAL. APPL},
year = {1998},
volume = {4},
pages = {247--269}
}
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Abstract
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 filtering 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 reduces 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.
Keyphrases
wavelet transforms lattice factorization electrical engineering community finite filter computational complexity linear system theory elementary matrix band subband polyphase matrix finite sequence self-contained derivation discrete wavelet transform basic principle non-unitary case simple filter-ing step subband filter euclidean algorithm ladder structure wavelet-like transform
