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Wavelet question.
 
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Ming-Haw Yaou, National Chiao Tung University Taiwan.
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PostPosted: Mon Aug 17, 1992 11:18 pm    
Subject: Wavelet question.
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Wavelet question.

In the wavelet digest vol.1 nr.1 (Topic #6), I asked a question
about orthonormal wavelet transform.

It is very kind of many readers to email me their constructive
advice. I also have to apologize for that some misunderstanding was
happened due to my unclear expression in the problem. To specify my question
more clearly, I correct some mistakes in the properties (a) and (b) and express
them in explicit equation.

------------------ The rewritten properties (a) and (b) ------------------

(a) Both Pmn(x) and Qmn(x) are orthonormal bases. This can be written as
< P(x-n) , P(x-m) > = delta(m,n).
< Q(x-n) , Q(x-m) > = delta(m,n).
Pmn(x) and Qmn(x) are mutually orthogonal (under the same value of m).
This can be expressed as
< P(x-n) , Q(x-m) > = 0.
(b) Denote the second derivatives of P(x) and Q(x) as P"(x) and Q"(x).
P"(x) and Q"(x) satisfy
< P"(x-n) , Q"(x-m) > = 0.

(c) Further constraints (for avoiding trivial solution) :
The regions of support of P(x) and Q(x) should not be non-overlapped.
Also the regions of support of P(x) and Q(x) should be greater than 1,
i.e., in the interval [p,q] and |p-q| > 1.

Where operation < a , b > stands for the inner product of continuous functions
a with b. Symbol delta(m,n) stands for the Kronecker delta function.
Variables m and n are integers and x are real.
The constraints in property(a) are well known as the necessary conditions in
the construction of dyadic orthonormal wavelet bases. The property(b) came
from my current study ( I wish to solve curve fitting problem from wavelet
theory).

PS:
It is known that the property(a) can be indirectly solved by
transferring the constraints to the filters of an associated QMF banks.
And P(x) and Q(x) can be iteratively calculated from the filters. I try to
transfer the additional property(b) to the filters. But It seems not to be
very easy.

I will very appreciate if any answer or advice is sent from you.
Please e-mail to : u7811837@cc.nctu.edu.tw

Thank in advance, Ming-Haw Yaou.
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