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Akram Aldroubi Guest

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




Preprints available
The following two preprints are available:
1. Families of multiresolution and wavelet spaces with optimal properties
Akram Aldroubi, Michael Unser
Abstract:
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 nfold 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 postfilters 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 basicwavelet sequences $psi_b^n$ and show that
they tend to Gabor functions. This provides wavelets that are nearly
timefrequency 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
Abstract:
We consider the problem of approximation by Bspline functions in the
context of the discrete sequencespace $l_2$ 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 Bsplines 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 translationinvariant 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 email to aldroubi@helix.nih.gov.
Note from the editor: There is a list of other papers by these authors
together with the abstracts available by anonymous ftp.
Connect to maxwell.math.sc.edu and retrieve the file
/pub/wavelet/abstracts/aldroubi.txt. 





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