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   -> Volume 6, Issue 11


Answer: Wavelets and convolution theorem (WD 6.10 #26)
 
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Andrew Dorrell (andrewd@ee.uts.edu.au)
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PostPosted: Wed Oct 01, 1997 5:02 pm    
Subject: Answer: Wavelets and convolution theorem (WD 6.10 #26)
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#21 Answer: Wavelets and convolution theorem (WD 6.10 #26)

My experience is that there is noting particularly nice about wavelet
domain convolution (as there are a lot of cross terms---especially in
ND). I have been doing a bit of it but find that any computational
advantage really comes from (compressing) the data, not the operator.
I have not however played around with many different bases or operator
classes. Would love to benefit from any insights that you gain.

Gerald Kaiser has however identified a class of convolutions which can
be implemented as multiplications in the scale domain:

@article{Kai95,
author = {Gerald Kaiser},
title = {Wavelet filtering with the {M}ellin transform},
journal= "Applied Math. Letters",
year = 1995,
note = {preprint}
}

@inproceedings{Kai96,
author = {Gerald Kaiser},
title = {Wavelet filtering in the scale domain},
booktitle= "Wavelet Applications in Signal and Image Processing",
volume = 2825,
organization = "SPIE",
address = "Denver, Colarado",
month = aug,
year = 1996
}

A number of authors have reported interest in what I think of as the
"half mapping" of convolution to the wavelet domain. That is, absorb
the convolution operator into either the analysis or synthesis filter
bank. Refs include:

@article{Vai93,
title = {Orthonormal and Biorthonormal Filter Banks
as Convolvers, and Convolutional Coding Gain},
author = {P. P. Vaidyanathan},
journal = ieee:tsp,
volume = 41, number = 6, month = jun, year = 1993}

@unpublished{BeyPPa,
author = {G. Beylkin and B. Torr'{e}sani},
title = {Implementation of Operators via Filter Banks,
Autocorrelation Shell and {H}ardy Wavelets},
note =
{url{ftp://amath-ftp.colarado.edu/pub/wavelets/papers/filterbanks_hardy.ps.Z}}
}

You may also be interested in checking out the perspective expressed in
@unpublished{Lin94,
author = {Alan R. Lindsey},
title = {The Non-Existence of a Wavelet Function Admitting a Wavelet
Transform Convolution Theorem of the Fourier Type},
note =
{url{ftp://voyager.cns.ohiou.edu/pub/papers/lindsey/wvltconv.ps}},
month = aug, year = 1994
}

---
Note from the editor:
The issue of wavelets and convolution has come up earlier
in the Wavelet Digest. Other answers you can find at

http://www.wavelet.org/wavelet/digest_04/digest_04.03.html#14
and
http://www.wavelet.org/wavelet/digest_04/digest_04.04.html#13
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