Balancing Multiwavelets using Groebner Bases and Relinearization Techniques

Jerome Lebrun
Department of Communication Systems
Swiss Federal Institute of Technology - Lausanne
lcavwww.epfl.ch/~lebrun

Email:  Jerome.Lebrun@epfl.ch

Abstract:

Wavelets and filter banks have become useful in digital signal processing in
part because of their ability to represent piecewise smooth signals with
relative efficiency.  For such signals, the discrete wavelet transform (DWT)
developed as a main tool for signal compression (JPEG 2000), fast algorithms,
and signal estimation and modeling (noise suppression and image segmentation,
etc). The DWT is usually implemented as an iterated digital filter bank tree,
so the design of a wavelet transform amounts to the design of a filter bank.

While the spectral factorization approach is the most convenient method to
construct the classic Daubechies wavelets (and the corresponding digital
filters), it is not applicable anymore to most of the other wavelet design
problems where additional constraints are imposed.  A typical case comes with
the construction of multiwavelets (corresponding to filter banks with relaxed
requirements on their time-invariance).  Multiwavelets are a natural
generalization of wavelets where one allows the associated multiresolution
analysis to be generated by more than one scaling function so as to overcome
the limitation preventing the construction of orthogonal wavelets with compact
support and symmetries.  Conditions of balancing are then introduced in the
design so as to ensure that the multiwavelets behave like bona-fide wavelets up
to a given order of approximation.  These conditions and stronger conditions of
interpolation leading to multiCoiflets will be extensively detailed.

Besides, although the spectral factorization approach can not be used anymore,
as for many of these design problems, the design equations can be written as a
multivariate polynomial system of equations.   Accordingly, Groebner algorithms
offer a way to obtain solutions in these cases.  At the same time, even though
the computation of a Groebner basis is the crucial point in our approach, one
should not forget that it is only the first step in the solving process.
Methods to implement change of ordering of the Groebner basis, and alternative
approaches like relinearization techniques leading to triangular systems and
rational univariate representation of the system are also key tools.  Some of
these methods will be discussed.