## Factoring Wavelet Transforms into Lifting Steps

Publication ,  Journal Article
Daubechies, I; Sweldens, W
Published in: Journal of Fourier Analysis and Applications
January 1, 1998

This article 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 SL(n; R[z, z-1]) = E(z; z-1])); 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.

## Published In

Journal of Fourier Analysis and Applications

1069-5869

January 1, 1998

4

## Issue

3

• General Mathematics
• 4904 Pure mathematics
• 4901 Applied mathematics
• 0102 Applied Mathematics
• 0101 Pure Mathematics

### Citation

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Daubechies, I., & Sweldens, W. (1998). Factoring Wavelet Transforms into Lifting Steps. Journal of Fourier Analysis and Applications, 4(3). https://doi.org/10.1007/bf02476026
Daubechies, I., and W. Sweldens. “Factoring Wavelet Transforms into Lifting Steps.” Journal of Fourier Analysis and Applications 4, no. 3 (January 1, 1998). https://doi.org/10.1007/bf02476026.
Daubechies I, Sweldens W. Factoring Wavelet Transforms into Lifting Steps. Journal of Fourier Analysis and Applications. 1998 Jan 1;4(3).
Daubechies, I., and W. Sweldens. “Factoring Wavelet Transforms into Lifting Steps.” Journal of Fourier Analysis and Applications, vol. 4, no. 3, Jan. 1998. Scopus, doi:10.1007/bf02476026.
Daubechies I, Sweldens W. Factoring Wavelet Transforms into Lifting Steps. Journal of Fourier Analysis and Applications. 1998 Jan 1;4(3).

## Published In

Journal of Fourier Analysis and Applications

1069-5869

January 1, 1998

4

3