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   -> Volume 1, Issue 2

Preprints available
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Akram Aldroubi

PostPosted: Thu Nov 28, 2002 10:46 am    
Subject: Preprints available
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Preprints available

The following two preprints are available:

1. Families of multiresolution and wavelet spaces with optimal properties

Akram Aldroubi, Michael Unser


Under suitable conditions, if the scaling functions $varphi_1$ and
$varphi_2$ generate the multiresolutions $V_{(j)}(varphi_1)$ and
$V_{(j)}(varphi_2 )$ , then their convolution
$varphi_1*varphi_2$ also generates a multiresolution
$V_{(j)}(varphi_1*varphi_2)$. Moreover, if p is an appropriate
convolution operator from $l_2$ into itself and if $varphi$ is a
scaling function generating the multiresolution $V_{(j)}(varphi)$,
then $p*varphi$ is a scaling function generating the same
multiresolution $V_{(j)}(varphi)=V_{(j)}(p*varphi)$. Using these
two properties, we group the scaling and wavelet functions into
equivalent classes and consider various equivalent basis functions
of the associated function spaces. We use the n-fold convolution
product to construct sequences of multiresolution and wavelet
spaces $V_{(j)}(varphi^n)$ and $W_{(j)}(varphi^n)$ with increasing
regularity. We discuss the link between multiresolution analysis
and Shannon's sampling theory. We then show that the interpolating
and orthogonal pre- and post-filters associated with the
multiresolution sequence $V_{(0)}(varphi^n)$ asymptotically
converge to the ideal lowpass filter of Shannon. We also prove that
the filters associated with the sequence of wavelet spaces
$W_{(0)}(varphi^n)$ converge to the ideal bandpass filter. Finally,
we construct the basic-wavelet sequences $psi_b^n$ and show that
they tend to Gabor functions. This provides wavelets that are nearly
time-frequency optimal. The theory is illustrated with the example
of polynomial splines.

2. Discrete spline filters for multiresolutions and wavelets of $l_2$

Akram Aldroubi, Michael Unser, Murray Eden


We consider the problem of approximation by B-spline functions in the
context of the discrete sequence-space $l_2$ instead of the usual space
$L_2$. This setting is natural for digital signal/image processing, and for
numerical analysis. To this end, we use sampled B-splines to define a
family of approximation spaces $SS^n_m subset l_2$ that we partition
into sets of multiresolution and wavelet spaces of $l_2$. We show that
the least square approximation in $SS^n_m$ of a finite energy signal is
obtained using translation-invariant filters. We study the asymptotic
properties of these filters and provide the link with Shannon's sampling
procedure. We derive and compare two pyramidal representations of
signals; the $l_2$-optimal and the stepwise $l_2$-optimal pyramids. The
advantage of the latter pyramid is that it can be computed by the
repetitive application of a single procedure. Finally, we derive a step by
step discrete wavelet transform of $l_2$ that is based on the stepwise
optimal representation of signals. As an application we implement and
compare these representations with the Gaussian/Laplacian pyramid
which is widely used in computer vision.

For a copy of these preprints, send an e-mail to

Note from the editor: There is a list of other papers by these authors
together with the abstracts available by anonymous ftp.
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