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Factoring wavelet transforms into lifting steps
 J. Fourier Anal. Appl
, 1998
"... ABSTRACT. This paper 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 dec ..."
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Cited by 434 (7 self)
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ABSTRACT. This paper 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); 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. 1.
Learning to Swim in a Sea of Wavelets
 Bulletin of the Belgian Mathematical Society  Simon Stevin
, 1994
"... We give some introductory notes about wavelets, motivating and deriving the basic relations that are used in this context. These notes should be considered as in introduction to the literature. They are far from complete but we hope it can motivate some readers to get involved with a quite interesti ..."
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Cited by 5 (0 self)
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We give some introductory notes about wavelets, motivating and deriving the basic relations that are used in this context. These notes should be considered as in introduction to the literature. They are far from complete but we hope it can motivate some readers to get involved with a quite interesting piece of mathematics which is the result of a lucky mariage between the results of the signal processing community and results in multiresolution analysis. We try to give answers to the questions: What are wavelets? What is their relation to Fourier analysis? Where do the scaling function and the wavelet function come from? Why can they be useful? What is a wavelet transform? Where and how are they applied? Contents 1 History 2 Motivation 3 Discrete versus continuous wavelet transforms 4 Multiresolution analysis 5 The father function or scaling function 6 Solution of the dilation equation 7 Interpretation of multiresolution 8 The mother function 9 More properties of the scaling function ...