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   -> Volume 6, Issue 1


Answer: Wavelet smoothing (WD 5.10 #22)
 
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Francesco P. Lovergine (loverg@ercole.iesi.ba.cnr.it)
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PostPosted: Mon Dec 09, 1996 9:41 am    
Subject: Answer: Wavelet smoothing (WD 5.10 #22)
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#16 Answer: Wavelet smoothing (WD 5.10 #22)

> From: Christian Oehreneder <coehrene@geoinfo.tuwien.ac.at>
> Subject: #22 Question: I) smoothing on the interval / II) an integral
>
> Dear wavelet-colleagues!
>
> I need help on tow wavelet-related problems.
>
> I) I need an efficient smoothing algorithm for functions defined on an
> interval in R or R^2. One but not the only way would be a wavelet
> basis for those domains. So far I have used the Lifting-Scheme, but I
> want to test others too. What kind of solutions do exist on this
> problem? Has anybody got computer-code, so that I can try it on my
> problem?
>

Wavelet smoothing is not the only scheme of smoothing, of course. A
different class of smoother are based on regularization theory a la
Thikonov. See Tomaso Poggio and many others for applications in image
processing and computer vision. These method are roughly based on the
definition of an energy functional to minimize, which measures both
data compatibility of the final solution and its smoothing. The
resulting problem is well-posed and is expressed in terms of a PDE
problem. Classical models used are the membrane and thin-plate model.
More complex scheme (with non-convex functionals) can be defined to
integrate discontinuities in the solution. This scheme are not
well-posed, generally. For numerical methods see D.Terzopoulos and
many others.

Some references:

author = "T. Poggio and V. Torre and C. Koch",
title = "Computational Vision and Regularization Theory",
journal = "Nature",
volume = 317,
year = 1985,
pages = "314--319"

@article{bib:Terzopoulos86b,
author="D. Terzopoulos",
title="Regularization of Inverse Visual Problems Involving
Discontinuities",
journal=PAMI, year=1986, volume=8, number=4, month=jul, pages={413--424}
}

@article{bib:Terzopoulos88,
author="D. Terzopoulos",
title="The Computation of Visible Surface Representation",
journal=PAMI, volume=10, number=4, month=jul, pages={417--438},
year=1988
}

@book{bib:Tikhonov77,
author="A.N. Tikhonov and V.Y. Arsenin",
title="Solutions of Ill-posed Problems",
publisher="W.H. Winston ed.", year=1977
}

f.lovergine@iesi.ba.cnr.it
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