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   -> Volume 2, Issue 6


Response: Implementing C. Chui's semi-orthogonal wavelets (WD 2.5, #9)
 
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Michael Unser, National Institutes of Health
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PostPosted: Fri Nov 29, 2002 2:30 pm    
Subject: Response: Implementing C. Chui's semi-orthogonal wavelets (WD 2.5, #9)
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Response: Implementing C. Chui's semi-orthogonal wavelets (WD 2.5, #9)

Response to Topic #9, Wavelet digest Volume 2 : Issue 5

I can give you some advice of the implementation of the B-spline wavelets
which are the same as the Chui-Wang semi-orthogonal spline wavelets. We
have had quite some experience with those wavelets (having constructed them
independently) and we have successfully applied them to image processing
(texture analysis, in particular).

The implementation rule that insures reversibility is simple : you should
use boundary conditions that are compatible between the different scales
and residual signals. The safer approach that works for all wavelet types
is to use a simple periodic signal extension.

In image processing, one usually prefers periodic symmetrization which
avoids border artifacts. This approach certainly works for B-spline
wavelets of odd degree (which are symmetrical), as well as for the
Battle-Lemarie wavelets. Implementation details (including how to handle
the boundaries) can be found in [1], and [2] (for a tabulated filter-bank
example).

Unfortunately, this approach does not work directly for the
anti-symmetrical B-spline wavelets of even degree (or odd order, depending
how they are defined) described by Chui and Wang [3]. The problem is that
filtering a symmetrical signal (at the boundary) with an anti-symmetrical
template yields an anti-symmetric signal; in other words, for the wavelet
transform to be reversible, you would have to use some form of
anti-symmetric boundary conditions for the residues. I let you work out the
details.

If you want to avoid these problems just stick with symmetrical wavelets !
I can send you reprints of [1] and [2] if you give me your postal address.

References :
[1] M. Unser, A. Aldroubi and M. Eden, "A family of polynomial spline
wavelet transforms", Signal Processing, vol. 30, pp. 141-162, January 1993.
[2] M. Unser, A. Aldroubi and M. Eden, "On the asymptotic convergence of
B-spline wavelets to Gabor functions", IEEE Trans. Information Theory, vol.
38, pp. 864-872, March 1992.
[3] C.K. Chui and J.Z. Wang, "On compactly supported spline wavelets and a
duality principle", Trans. Amer. Math. Soc., vol. 330, pp. 903-915, 1992.


Michael Unser

BEIP, Bldg 13, Room 3W13
National Institutes of Health
Bethesda, MD 20892, USA

Email : Unser@helix.nih.gov
All times are GMT + 1 Hour
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