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> Volume 6, Issue 2
Preprint: Preprints from Akram Aldroubi on multiwavelets.

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Akram Aldroubi (aldroubi@helix.nih.gov) Guest

Posted: Wed Jan 29, 1997 1:57 pm Subject: Preprint: Preprints from Akram Aldroubi on multiwavelets.




#3 Preprint: Preprints from Akram Aldroubi on multiwavelets.
New preprints available
1) Title: Oblique and hierarchical multiwavelet bases
Author: Akram Aldroubi
Abstract: We develop the theory of oblique multiwavelet bases, which
encompasses the orthogonal, semiorthogonal and biorthogonal cases, and
we circumvent the noncommutativity problems that arise in the
construction of multiwavelets. Oblique multiwavelets preserve the
advantages of orthogonal and biorthogonal wavelets and enhance the
flexibility of the theory to accommodate a wider variety of wavelet
bases. For example, for a given multiresolution, we can construct
supercompact wavelets for which the support is half the size of the
shortest orthogonal, semiorthogonal, or biorthogonal wavelet. The
theory also produces the htype, piecewise linear hierarchical bases
used in finite element methods, and it allows us to construct new
htype, smooth hierarchical bases, as well as htype hierarchical
bases that use several template functions. For the hierarchical
bases, and for all other types of oblique wavelets, the expansion of a
function can still be implemented with a perfect reconstruction filter
bank. We illustrate the results using the Haar scaling function and
the CohenDaubechiesPlonka multiscaling function. We also construct a
supercompact spline uniwavelet of order 3 and a hierarchical basis
that is based on the Hermit cubic spline, and explicitly give the
coefficients of the corresponding filter bank.
A copy can be downloaded from
http://aa2mac.ncrr.nih.gov/preprints.html
or Email to aldroubi@helix.nih.gov
2) Title:Prefiltering for the initialization of multiwavelet
transforms
Authors: Michael J. Vrhel, Akram Aldroubi
Abstract:We introduce a new method for initializing the multiwavelet
decomposition algorithm. The approach assumes that the input signal
is contained within some welldefined subspace of $L_2$ (e.g. space of
bandlimited functions). The initialization algorithm is the
orthogonal projection of the input signal into the space defined by
the multiscaling function. Unlike an interpolation approach, the
projection method will always have a solution. We provide examples and
implementation details.
A copy can be downloaded from
http://aa2mac.ncrr.nih.gov/preprints.html
or Email to aldroubi@helix.nih.gov
3) Title: Construction of biorthogonal wavelets starting from any two
multiresolutions
Authors: A. Aldroubi, P. Abry and M. Unser
Abstract: Starting from any two given multiresolution approximations
${V^1_j}_{j in Z}$ and ${V^2_j}_{j in Z}$, we construct
biorthogonal wavelet bases that are associated with this chosen pair
of multiresolutions. Thus, our construction method takes a point of
view opposite to the one of CohenDaubechiesFeauveau which starts
from a wellchosen pair of biorthogonal discrete filters. In our
construction, the necessary and sufficient condition is the
nonperpendicularity of the multiresolutions. This condition can be
checked explicitly by a simple formula that computes the angle between
the spaces. Thus, by selecting different sets of nonperpendicular
multiresolutions, we generate different biorthogonal wavelet bases
with various desired shapes and properties. For biorthogonal wavelets
generated by this method, we derive the formulae that give the filter
bank coefficients needed in the Mallattype
decomposition/reconstruction algorithms.
A copy can be downloaded from
http://aa2mac.ncrr.nih.gov/preprints.html
or Email to aldroubi@helix.nih.gov
4) Title:Complete iterative reconstruction algorithms for irregularly
sampled data in splinelike spaces
Authors: Akram Aldroubi and Hans Feichtinger
Abstract: We prove that the exact reconstruction of a function $fv$
from its samples $fv (x_i)$ on any "sufficiently dense" sampling set
$X_{i in ind subset RR^n}$, where $ind$ is a countable indexing
set, can be obtained for a large class of splinelike spaces that
belong to $Lp (RR^n)$. Moreover, The reconstruction can be
implemented using fast algorithms. Since, a special case is the space
of bandlimited functions, our result generalizes the classical
Shannonwhittacker sampling theorem on regular sampling and the
PaleyWiener theorem on nonuniform sampling.
A copy can be downloaded from
http://aa2mac.ncrr.nih.gov/preprints.html
or Email to aldroubi@helix.nih.gov
5) Title: Exact iterative reconstruction algorithm for multivariate
irregularly sampled functions in splinelike spaces: The $Lp$theory
Authors: Akram Aldroubi and Hans Feichtinger
Abstract: We prove that the exact reconstruction of a function $fv$
from its samples $fv (x_i)$ on any "sufficiently dense" sampling set
${x_i}_{iin Lambda}$ can be obtained, as long as $fv$ is known to
belong to a large class of splinelike spaces in $Lp (RR^n)$.
Moreover, the reconstruction can be implemented using fast algorithms.
Since a limiting case is the space of bandlimited functions, our result
generalizes the classical ShannonWhittacker sampling theorem on regular
sampling and the PaleyWiener theorem on nonuniform sampling.
A copy can be downloaded from
http://aa2mac.ncrr.nih.gov/preprints.html
or Email to aldroubi@helix.nih.gov
Thanks
Akram Aldroubi
BEIP/NIH
Building 13/3N17
13 South DR MSC 5766
Bethesda MD 208925766
USA
http://aa2mac.ncrr.nih.gov 





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