Factoring wavelet transforms into lifting steps |
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Authors: | Ingrid Daubechies Wim Sweldens |
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Institution: | (1) Program for Applied and Computational Mathematics, Princeton University, 08544 Princeton, NJ;(2) Bell Laboratories, Lucent Technologies, Rm. 2C-376, 600 Mountain Avenue, 07974 Murray Hill, NJ |
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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;Rz, z−1])=E(n;Rz, 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.
Research Tutorial
Acknowledgements and Notes. Page 264. |
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Keywords: | 42C15 42C05 19-02 |
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