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   -> Volume 7, Issue 5


Question: Rediscovering Haar wavelets & wavelet packets
 
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"Dr. A. Avudainayagam" (avudai@acer.iitm.ernet.in)
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PostPosted: Thu Jan 01, 1970 12:59 am    
Subject: Question: Rediscovering Haar wavelets & wavelet packets
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#30 Question: Rediscovering Haar wavelets & wavelet packets

Recently Hugh Casement (Volume 7, Issue 3) attempted a numerical
experiment which resulted in the rediscovery of Haar wavelets! Here we
want to describe our similar efforts in meddling with Haar wavelets
which appears to give better(!) results when the aim is to make a
vector (representing a signal sampled at equal intervals of time)
sparse.
We simply swap the bottom n/2 rows and the top n/2 rows in an
nxn Haar matrix and call as H'.
_ _
| s -s 0 0 . . . . . . |
| 0 0 s -s . . . . . . |
| . . . . . . . . . . |
| . . . . . . . . . . |
| 0 0 . . . . . . s -s |
H'= | s s 0 0 . . . . . . | , s=1/sqrt(2)
| 0 0 s s . . . . . . |
| . . . . . . . . . . |
| . . . . . . . . . . |
| 0 0 0 0 . . . . s s |
- -
We apply H' in the same way as the Haar matrix on an nx1 vector which
means that we are transforming the details instead of the averages.
For any chosen threshold, H' makes the sampled signal more sparse than
the traditional Haar matrix H. For example: Consider the signal f(t)
= t squared, 0 <= t < 1 sampled at i/128, 0 <= i <= 127. Note that
for threshold value = 0.000001, the H'-transformed signal has 16
elements which may be ignored while the conventional H-transformed
signal has no element to be ignored. Similar results are obtained for
other signals given below. Threshold value = 0.000001

_____________________________________________________________________________
| | number of elements which can be |number of elements which can be |
| | ignored in H transformed signal |ignored in H' transformed signal |
|_____________________________________________________________________________|
| | | |
|t cubed | 1 | 9 |
| sin(t) | 0 | 8 |
| cos(t) | 0 | 12 |
| exp(t) | 0 | 8 |
------------------------------------------------------------------------------
A similar improvement(!) can also be acheived for compressing matrices (images).
Question: Does this mean H' compresses the signal better than H?
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