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   -> Volume 7, Issue 4

Thesis: Factoring Wavelet Transforms
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Paul Abbott (

PostPosted: Wed Mar 18, 1998 9:40 am    
Subject: Thesis: Factoring Wavelet Transforms
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#5 Thesis: Factoring Wavelet Transforms

Factoring Wavelet Transforms

Mark Maslen <>

The discrete wavelet transform can be carried out using a sequence of
simple operations called lifting steps. The theory of lifting may be
developed by considering the polyphase representations of the filters
for a given basis, defined in terms of Laurent polynomials. The
polyphase matrix is constructed from these polynomials, and this
matrix is used to carry out the wavelet transform. Using the Euclidean
algorithm for Laurent polynomials, the polyphase matrix may be
factorised, giving rise to the sequence of lifting steps. An important
algebraic result is that the quotients determined from this algorithm
are not uniquely defined, so there are several possible
factorisations. This allows a choice of optimal factorisation for a
given purpose. In this project, a program was developed to find all
factorisations of a given filter. The inverse of the wavelet
transform, when expressed in terms of lifting steps is simple to find,
and this too has been automated. It is demonstrated that for large
datasets, the lifting implementation is significantly faster than the
traditional method of conducting the wavelet transform via quadrature
mirror filters.
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