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   -> Volume 10, Issue 4

Preprint: Wavelets, Fractals and Radial Basis Functions
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Thierry BLU (

PostPosted: Mon Nov 19, 2001 2:53 pm    
Subject: Preprint: Wavelets, Fractals and Radial Basis Functions
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#8 Preprint: Wavelets, Fractals and Radial Basis Functions

The following paper has recently been accepted for publication in the
IEEE Transactions on Signal Processing. In particular, we give an
explicit time-domain representation of dyadic wavelets

Wavelets, Fractals and Radial Basis Functions

Thierry BLU and Michael UNSER
(Swiss Federal Institute of Technology, Lausanne)


Wavelets and radial basis functions (RBF) lead to two distinct ways of
representing signals in terms of shifted basis functions. RBFs, unlike
wavelets, are non-local and do not involve any scaling, which makes them
applicable to non-uniform grids. Despite these fundamental differences, we
show that the two types of representation are closely linked together...
through fractals. First, we identify and characterize the whole class of
self-similar radial basis functions that can be localized to yield
conventional multiresolution wavelet bases. Conversely, we prove that, for
any compactly supported scaling function phi, there exists a one-sided
central basis function ho that spans the same multiresolution subspaces.
The central property is that the multiresolution bases are generated by
simple translations of ho, without any dilation. We also present an
explicit time-domain representation of a scaling function as a sum of
harmonic splines. The leading term in the decomposition corresponds to the
fractional splines; a recent, continuous-order generalization of the
polynomial splines.
Thierry BLU

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