Publications [#287148] of Ingrid Daubechies

Papers Published

1. Daubechies, I; Sweldens, W, Factoring Wavelet Transforms into Lifting Steps, Journal of Fourier Analysis and Applications, vol. 4 no. 3 (January, 1998), pp. x1-268 [doi]
   (last updated on 2024/04/19)

   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 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.