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   -> Volume 3, Issue 9


Question: A proper grid for locating orthogonal wavelets ...
 
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Ali Akbar Safavi (akbar@ce.su.oz.au)
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PostPosted: Mon Dec 02, 2002 1:08 pm    
Subject: Question: A proper grid for locating orthogonal wavelets ...
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Question: A proper grid for locating orthogonal wavelets ...

Question: A proper grid for locating orthogonal wavelets of different
scales in a multi-resolution approximation :

Dear Wavelet-ers

I follow Mallat's work on " A Theory for Multi-Resolution Signal Representation:
The Wavelet Representation "
and publishied in : IEEE Trans. on pattern Analysis and... (1989).
I have a question and appreciate your comments.

My Question is:
If we construct a grid (whose interval in proportional to 2^(-j))
for locating wavelets at different scales,
then, should the wavelets at one scale be orthogonal to all the
wavelets of the other scales, in orther to have a multi-resolution
analysis without redundant information in our detail (the difference
between approximation at two different scales)?
(assumptions:
1. the orthogonality of translation of wavelets for each
particular scale is satisfied;
2. we only look at orthogonal wavelets;
3. the wavelet may be centered at t=0 or t=1/2; )

A negative answer accept the two following grids, while a positive answer accept
only the first grid,
(First) finest res. * * * *
* * *
* *
coarsest res. *

----------------------------------------------------------
(Second) finest res. * * * * *
* * *
coarsest res. * *


The grid that Daubechies uses in page 10-11 of "ten lectures on wavelets"
to give an example of multi-resolution with Haar wavelet is the same as
the first grid. Moreover, in that example she proves orthogonality of Haar
wavelets of all different scales.

Best regards

akbar safavi
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