BIOMEDICAL IMAGING GROUP (BIG)
Laboratoire d'imagerie biomédicale (LIB)

Wavelets, Fractals, and Radial Basis Functions

T. Blu, M. Unser

IEEE-SPS best paper award, IEEE Transactions on Signal Processing, vol. 50, no. 3, pp. 543-553, March 2002.

Wavelets and radial basis functions (RBFs) lead to two distinct ways of representing signals in terms of shifted basis functions. RBFs, unlike wavelets, are nonlocal and do not involve any scaling, which makes them applicable to nonuniform 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 φ(x), there exists a one-sided central basis function ρ+(x) that spans the same multiresolution subspaces. The central property is that the multiresolution bases are generated by simple translation of ρ+ 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.

IEEE Signal Processing Society's 2003 Best Paper Award 

@ARTICLE(http://bigwww.epfl.ch/publications/blu0201.html,
AUTHOR="Blu, T. and Unser, M.",
TITLE="Wavelets, Fractals, and Radial Basis Functions",
JOURNAL="{IEEE} Transactions on Signal Processing",
YEAR="2002",
volume="50",
number="3",
pages="543--553",
month="March",
note="{IEEE-SPS} best paper award")

© 2002 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.