4 December 2000 Wavelets and radial basis functions: a unifying perspective
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Abstract
Wavelets and radial basis functions (RBF) are two rather distinct ways of representing signals in terms of shifted basis functions. An essential aspect of RBF, which makes the method applicable to non-uniform grids, is that the basis functions, unlike wavelets, are non-local-in addition, they do not involve any scaling at all. Despite these fundamental differences, we show that the two types of representation are closely connected. We use the linear splines as motivating example. These can be constructed by using translates of the one-side ramp function, or, more conventionally, by using the shifts of a linear B-spline. This latter function, which is the prototypical example of a scaling function, can be obtained by localizing the one-side ramp function using finite differences. We then generalize the concept and identify 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, there exist a one-sided central basis function that spans the same multiresolution subspaces. The central property is that the multiresolution bases are generated by simple translation without any dilation.
© (2000) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
Michael A. Unser, Michael A. Unser, Thierry Blu, Thierry Blu, } "Wavelets and radial basis functions: a unifying perspective", Proc. SPIE 4119, Wavelet Applications in Signal and Image Processing VIII, (4 December 2000); doi: 10.1117/12.408636; https://doi.org/10.1117/12.408636
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