## Factoring wavelet transforms into lifting steps (1998)

Venue: | J. Fourier Anal. Appl |

Citations: | 434 - 7 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}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 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.