Accession Number:

ADA433614

Title:

Wavelets, Fractals, and Radial Basis Functions

Descriptive Note:

Conference paper

Corporate Author:

SWISS FEDERAL INST OF TECHNOLOGY LAUSANNE (SWITZERLAND)

Personal Author(s):

Report Date:

2005-01-07

Pagination or Media Count:

12.0

Abstract:

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 phichi, there exists a one-sided central basis function rhochi that spans the same multiresolution subspaces. The central property is that the multiresolution bases are generated by simple translation of rho 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.

Subject Categories:

  • Numerical Mathematics
  • Theoretical Mathematics

Distribution Statement:

APPROVED FOR PUBLIC RELEASE