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Factoring wavelet transforms into lifting steps

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Abstract

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 formulaSL(n;R[z, z−1])=E(n;R[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.

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Authors and Affiliations

  1. Program for Applied and Computational Mathematics, Princeton University, 08544, Princeton, NJ

    Ingrid Daubechies

  2. Bell Laboratories, Lucent Technologies, Rm. 2C-376, 600 Mountain Avenue, 07974, Murray Hill, NJ

    Wim Sweldens

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  1. Ingrid Daubechies
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  2. Wim Sweldens
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Communicated by John J. Benedetto

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Acknowledgements and Notes. Page 264.

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Daubechies, I., Sweldens, W. Factoring wavelet transforms into lifting steps. The Journal of Fourier Analysis and Applications 4, 247–269 (1998). https://doi.org/10.1007/BF02476026

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  • DOI: https://doi.org/10.1007/BF02476026

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