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   -> Volume 7, Issue 3


Question: Reinvented wavelets?
 
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Hugh Casement (hughcasement@compuserve.com)
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PostPosted: Mon Mar 16, 1998 6:14 am    
Subject: Question: Reinvented wavelets?
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#26 Question: Reinvented wavelets?

Dear Waveleteers / Chers Ondeletteurs / Liebe Wellchen-Leutchen,

Forgive a stupid question from a physicist who left the academic world 25=

years ago.

I recently tried the following experiment. I took a 2x2 block of pixels
with intensities
-------
| a | b |
|---|---|
| c | d |
-------
and applied the transformation
A = 0.5 (a + b + c + d)
B = 0.5 (a + b - c - d)
C = 0.5 (a - b + c - d)
D = 0.5 (a - b - c + d)

The inverse transformation is precisely the same, swapping upper and
lowe r case. I repeated this for all 2x2 blocks in an image, creating
four new arrays each a quarter the size. The B, C, and D arrays
contained many zeros, so would be amenable to run-length encoding. I
repeated the process on the A (sum) array, and then on its sum array,
and so on. Theoretically one cou ld take a 1024 x 768 array and
reduce it progressively to 4 x 3, though I suspect it's more
reasonable to break off the process rather sooner.

What I'd like to know is: have I reinvented wavelets? The process
seems to have little in common with the abstruse mathematics appearing
in articles on the subject in learned journals. Looking at it from a
musician's poin t of view one could say that, whereas a Fourier
transform reduces a wavefor m to its partials (arithmetic progression
of frequencies), my method reduce s it to octaves (geometric
progression). Assuming this is a kind of primitive wavelet transform,
perhaps someone could suggest a more refined method that gives better
image compression without being complicated to implement.

Hugh Casement <hughcasement@compuserve.com>

Note from the editor: You have rediscovered Haar wavelets. For a good
on-line tutorial on how to go beyond Haar look at Amara's web page:
http://www.amara.com/current/wavelet.html
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