Math @ Duke

Publications [#287148] of Ingrid Daubechies
Papers Published
 Daubechies, I; Sweldens, W, Factoring Wavelet Transforms into Lifting Steps,
Journal of Fourier Analysis and Applications, vol. 4 no. 3
(December, 1998),
pp. x1268
(last updated on 2019/05/25)
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 wellknown to algebraists (and expressed by the formula SL(n; R[z, z1]) = E(z; z1])); it is also used in linear systems theory in the electrical engineering community. We present here a selfcontained 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., nonunitary 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 waveletlike transform that maps integers to integers.


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