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   -> Volume 6, Issue 9


Answer: Shiftability (WD 6.8 #25)
 
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andrewd@eng.uts.edu.au (Andrew Dorrell)
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PostPosted: Wed Aug 13, 1997 7:41 am    
Subject: Answer: Shiftability (WD 6.8 #25)
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#13 Answer: Shiftability (WD 6.8 #25)

<From the Question>

There they have over come an old problem of the shift invariance of
the wavelet transform by not down sampling while taking the wavelet
transform.They have showed with examples that it works.

I would like to know why this works and will it always work? If you
have any suggestions or any reference materials,can you please
forward them to my address.

Yes it will always work. The reason is simply that shift variance is
introduced because Nyquist sampling is violated in each of the
(wavelet decomposed) subbands - wavelet are not ideal filters after
all! By not downsampling the problem is avioded. There are in fact
many informative articles on this subject. Daubechies talks about
creating tight frames in chapter 3 of her definative book. Although
not explicit, the motivation is shift invariance. Chui has also
published a number of papers on maintaining tight frames at certain
oversampling rates. A well explained work. The 'a-trous' algorithm
is in fact a fast implementation of a wavelet transform with no
downsampling - used because of its shift invariance. Eero Simoncelli
(look up his web page) also has a number of papers specifically
addressing the issue of shiftability and developing jointly shiftabl,
and overcomplete, wavelet tight frames. More recently a number of
authors have examined other methods of achieving shiftability - some
are as simple as deciding, on a level by level basis, whether to
retain odd or even samples. Anyway the motivation for using these
other thechniques in preference to the a-trous (not downsampled)
wavelet transform is thatthey retain some of the compactness
properties, and reduced computational complexity of orthogonal
tranasforms.

Whether some are applicable may still depend on the data you are working
with.

Mr Andrew Dorrell
School of Electrical Engineering
University of Technology, Sydney

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