Conference rusure::math

    
    Any decent tutorial that will guide a person from the ground up.
    
    Anybody know of any real life code that does image compression
    and decompression.
    
    Darryl
    

    Re .0

    Thanks for those references.  It seems that wavelet theory is
    related to joint time-frequency representations such as radar
    ambiguity functions, Wigner transformations, short time Fourier
    transforms and the like.  I have some references on these which
    I can enter when I get home.

    Another application I'm familar with is loudspeaker and room
    acoustics; one wants a speaker to "speak" and then "shut up", without
    diffraction or other time smear to the signal, and without tonal
    colorations.  It is possible that these joint time/frequency
    representations can give perceptually relevant characterization of 
    speakers and listening rooms, or sound reinforcement systems.

    - Jim

	These are the topics on wavelets:

1272.0    KAZAN::CHALEAT        23-JUL-1990     2  wavelets
1269.0    MARX::ANDERSON        17-JUL-1990     2  waverettes (sp?)
955.0     PRSUD1::CHALEAT       21-OCT-1988     2  wavelet & time-frequency
							display

	Here are some more references from usenet sci.math:

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From: [email protected] (Robert J Frey)
Newsgroups: sci.math
Subject: References on Wavelet Theory
Keywords: wavelet theory, Fourier analysis
Message-ID: <[email protected]>
Date: 4 Sep 90 15:57:18 GMT
Organization: Kepler Financial Mgmt., Setuket, NY
Lines: 56
 
Below are summaries of the responses I received concerning wavelet theory. 
Thanks to those who responded.
 
The paper by Gilbert Strang in SIAM Review looks like the best place to start
reading.
 
------------------------------------------------------------------------------
From Knut Morken ([email protected]):
 
Daubechies I., 
Orthonormal bases of compactly supported wavelets.
Communications on Pure and Applied Mathematics, 41 (1988), 909-996.
 
Cavaretta A., Dahmen W., Michelli C., 
Stationary Subdivision, 
Preprint No. A-89-25, 
Freie Universitat Berlin, 
Fachbereich Mathematik: Serie A, Mathematik,
(to appear in Memoirs of the AMS.)
 
Dahmen W. and Michelli C., 
On Stationary Subdivision and the Construction of Compactly Supported 
Orthonormal Wavelets, 
Preprint No. A-90-6,
Freie Universitat Berlin, 
Fachbereich Mathematik: Serie A, Mathematik.
 
There was a paper in SIAM Review recently that, as the title says, gives
a brief introduction...
 
Strang G., 
Wavelets and dilation problems: a brief introduction, 
SIAM Review 31 (1989), 614-627.
 
------------------------------------------------------------------------------
From Andrew Mullhaupt ([email protected]):
 
Yves Meyer wrote a large number of papers on the wavelet theory
in the past  few years...Frazier and Jawerth had a different approach to 
atomic decompositions of function spaces which they worked out in a paper 
they published at a conference in Lund in 1984 (?)...
 
Jawerth did work out a "Fast" phi transform (as they called their 
decomposition) and has, I believe, used it in PDE applications and signal 
processing. Jawerth is at the Washington University in St. Louis, and although
he was visiting Berkeley (MSRI) for their year of analysis, Mike Frazier
is probably back at UNM by now. 
 
------------------------------------------------------------------------------
 
If more come in, I'll follow up.
 
-- 
Dr. Robert J Frey, Kepler Financial Management, Ltd.
[email protected] *or* [email protected]
voice: (516) 689-6300 * fax: (516) 751-8678

	*****	*****	*****	*****	*****	*****	*****	*****

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From: [email protected] (Dave Elliott)
Newsgroups: sci.math
Subject: Re: References on Wavelet Theory
Summary: Recent info
Keywords: wavelet theory, Fourier analysis
Message-ID: <[email protected]>
Date: 5 Sep 90 22:11:05 GMT
References: <[email protected]>
Reply-To: [email protected] (Dave Elliott)
Organization: Washington University, St. Louis MO
Lines: 16
 
Bjorn Jawerth is currently at the University of South Carolina, Columbia, SC;
I think Mike Frazier is there also this Fall. The Phi (Frazier-Jawerth)
transform and other aspects of wavelets *are* being investigated at Washington
University in Mathematics (Guido Weiss) and Electrical Engineering departments.
 
The latest issue of IEEE Transactions on Information Theory has a survey
by Ingrid Daubechies, "The Wavelet Transform, Time-Frequency Localization
and Signal Analysis" (vol. 36, Sept. 1990, 961-1005) which may be the
most accessible article for most Net readers.
 
Otherwise I claim ignorance... and cannot correspond about wavelets.
 
                                David L. Elliott
				Dept. of Systems Science and Mathematics
                                Washington University, St. Louis, MO 63130
				[email protected]

	*****	*****	*****	*****	*****	*****	*****	*****

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From: [email protected] (Robert J Frey)
Newsgroups: sci.math
Subject: Wavelet References - II
Keywords: wavelets, Fourier analysis
Message-ID: <[email protected]>
Date: 7 Sep 90 14:01:56 GMT
Organization: Kepler Financial Mgmt., Setuket, NY
Lines: 65
 
 
Below are all of the references to wavelets received through Friday, September
7, 1990. Thanks to all who responded.
 
------------------------------------------------------------------------------
From Knut Morken ([email protected]):
 
	Daubechies I., 
	Orthonormal bases of compactly supported wavelets.
	Communications on Pure and Applied Mathematics, 41 (1988), 909-996.
 
	Cavaretta A., Dahmen W., Michelli C., 
	Stationary Subdivision, 
	Preprint No.  A-89-25, 
	Freie Universitat Berlin, Fachbereich Mathematik: Serie A, Mathematik,
	(to appear in Memoirs of the AMS).
 
	Dahmen W. and Michelli C., 
	On Stationary Subdivision and the Construction of Compactly Supported
		Orthonormal Wavelets, 
	Preprint No. A-90-6,
	Freie Universitat Berlin, Fachbereich Mathematik: Serie A, Mathematik.
 
	Strang G., 
	Wavelets and dilation problems: a brief introduction, 
	SIAM Review 31 (1989), 614-627.
 
------------------------------------------------------------------------------
From Andrew Mullhaupt ([email protected]):
 
	Yves Meyer wrote a large number of papers on the wavelet theory
in the past  few years...Frazier and Jawerth had a different approach to 
atomic decompositions of function spaces which they worked out in a paper 
they published at a conference in Lund in 1984 (?)...
 
	Jawerth did work out a "Fast" phi transform (as they called their 
decomposition) and has, I believe, used it in PDE applications and signal 
processing. Jawerth is at the Washington University in St. Louis, and although
he was visiting Berkeley (MSRI) for their year of analysis, Mike Frazier
is probably back at UNM by now. 
 
------------------------------------------------------------------------------
From Tom Menten ([email protected])
 
       Mallat,S.(1989) Multifrequency channel decompositions of images
       and wavelet models. IEEE transactions on Acoustics, Speech, and 
       Signal Processing, Vol 37, no 12, pp2091-2110
 
       Daubechies, Ingrid (1988) Time-frequency localization operators: 
       a geometric phase space approach.  IEEE Transactions on
       Information Theory v 34 n 4 pp605-612
 
       Daubechies, Ingrid (1986) Painless nonorthogonal expansions.
       J Math Phys vol 27, no 5 pp1271-83
 
       Daubechies, Ingrid (1988) Wavelet transform, time-frequency
       localization and signal analysis.  IEEE 1988 International
       Symposium on Information Theory.
 
------------------------------------------------------------------------------
 
-- 
Dr. Robert J Frey, Kepler Financial Management, Ltd.
[email protected] *or* [email protected]
voice: (516) 689-6300 * fax: (516) 751-8678

    I went to a talk called "Irregular Sampling and the Theory of Frames of
    Wavelets" givin by Professor John J. Benedetto.  This is interesting
    because irregular sampling may be desirable because of known properties
    of a signal.  Benedetto is a professor at the University of Maryland.
    
    Chuck

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From: [email protected] (Alan Algustyniak)
Newsgroups: sci.math
Subject: Wavelets
Message-ID: <[email protected]>
Date: 4 Oct 90 21:21:25 GMT
Sender: [email protected]
Reply-To: [email protected] (Alan Algustyniak)
Organization: SRI International, Menlo Park CA
Lines: 11
 
 
In the 24 August issue of SCIENCE, there is a news story about 'wavelets'.
 
Would someone please post references to this 'theory'?
 
IMHO, the report in SCIENCE is so un-informative, yet so laudatory, as to
be downright sadistic.
 
	Alan
 
SCIENCE, Vol.249, pp.858-859

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From: [email protected] (David Heisterberg)
Newsgroups: sci.math
Subject: Re: Wavelets
Message-ID: <[email protected]>
Date: 5 Oct 90 13:48:38 GMT
References: <[email protected]>
Sender: [email protected]
Organization: Ohio SuperComputer Center, Columbus, OH, USA
Lines: 14
 
In article <[email protected]> [email protected] (Alan Algustyniak) writes:
>
>In the 24 August issue of SCIENCE, there is a news story about 'wavelets'.
>Would someone please post references to this 'theory'?
 
For about two days (before someone else reserved it) I checked out a
fairly recent book, "Wavelets, Time - Frequency Methods, and Phase
Space" is a reasonable approximation to the title.  The editor I don't
recall; it is primarily conference proceedings.  The first two papers
are a general introduction to the topic.
--
David J. Heisterberg		[email protected]		And you all know
The Ohio Supercomputer Center	[email protected]	security Is mortals'
Columbus, Ohio  43212		ohstpy::djh		chiefest enemy.

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From: [email protected] (Arne Jensen)
Newsgroups: sci.math
Subject: Re: Wavelets
Message-ID: <[email protected]>
Date: 11 Oct 90 09:34:53 GMT
References: <[email protected]>
Organization: CS and Math, University of Aalborg, Denmark
Lines: 22
 
In article <[email protected]> [email protected] (Alan Algustyniak) writes:
>
>In the 24 August issue of SCIENCE, there is a news story about 'wavelets'.
>
>Would someone please post references to this 'theory'?
 
For the mathematical theory I recommend the following reference:
 
   Yves Meyer: Ondelettes et Operateurs 
   Tome I: Ondelettes
   Hermann 1990
   ISBN 2 7056 6125 0
 
   Arne Jensen
   Department of Mathematics and Computer Science
   Institute for Electronic Systems
   Aalborg University
   Fr. Bajers Vej 7
   DK-9220 Aalborg 0
   Denmark
 
   [email protected]

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From: [email protected] (Ghanashyam Joshi)
Newsgroups: sci.math
Subject: Wavelet bibliography, UPTODATE
Message-ID: <[email protected]>
Date: 13 Dec 90 02:27:34 GMT
Organization: North Dakota State University, Fargo
Lines: 118
 
Here is the long awaited (I guess judging from the e-mails received from all
over the world) Bibliography on Wavelet theory and applications. Thanks 
to many contributors to this list, I have done every attempt to include every
relevant reference. Please let me know of any ommisions of repetitions or
enhancements to this list. Please e-mail to 
		[email protected]
Thanks again to many contributors, I wish I can thank them individually!
-------------------------------------------------------------------------------
::::: Wavelet Bibliography :::::
 
Cavaretta A., Dahmen W., Michelli C.
Stationary Subdivision
Fachbereich Mathematik: Serie A, Mathematik,
   Freie Universitat Berlin, 
   Preprint No. A-89-25, 
   (to appear in Memoirs of the AMS.)
 
Dahmen W. and Michelli C., 
On Stationary Subdivision and the Construction of 
   Compactly Supported Orthonormal Wavelets, 
Fachbereich Mathematik: Serie A, Mathematik,
   Freie Universitat Berlin, 
   Preprint No. A-90-6,
 
Daubechies I., 
Orthonormal bases of compactly supported wavelets.
Communications on Pure and Applied Mathematics, 
   41 (1988), 909-996.
 
Daubechies, I.
The Wavelet Transform, Time-Frequency Localization 
   and Signal Analysis
IEEE Transactions on Information Theory vol. 36, 
   Sept. 1990, 961-1005
 
Daubechies, I.
Time-frequency localization operators: a geometric 
   phase space approach
IEEE Transactions on Information Theory, vol. 34, no. 4, 1988
 
Daubechies, I. Grossman A. Meyer Y.
Painless nonorthogonal expansions.
J Math Phys, vol 27, no 5, 1986, pp 1271-1283
 
Mallat S.
A theory for multiresolution signal decomposition - the wavelet
   representation.
IEEE Transactions on pattern analysis and machine intelligence, 
1989 jul, v11 n7:674-693.
 
Mallat,S.(1989) Multifrequency channel decompositions of images
and wavelet models. IEEE transactions on Acoustics, Speech, and 
Signal Processing, Vol 37, no 12, pp2091-2110
 
Mallat S. (1989), Multiresolution Approximations and Wavelet Orthonormal
Bases of L2(R), Trans. Amer. Math. Soc. 315(1), pp 69-87
 
Strang G., 
Wavelets and dilation problems: a brief introduction, 
SIAM Review 31 (1989), 614-627.
 
Yves Meyer: Ondelettes et Operateurs 
Tome I: Ondelettes
Hermann 1990
ISBN 2 7056 6125 0
 
Huntsberger T.L. and B.A.
Hypercube algorithm for Image Decomposition and Analysis in the Wavelet
Representation, Proceedings of the fifth 
distributed memory conference, South Carolina, pp 171-175, 1990.
 
Heil C.E. and Walnut D.F.
Continuous and discrete wavelet transforms, SIAM Review 31 (1989), pp 628-666
 
Alex Pentland
Fast Surface Estimation Using Wavelet bases, MIT Media Lab Vision and Modeling 
Group Technical Report No. 142, June 1990
 
Berkhout A. G.
Least-squares inverse filtering and wavelet deconvolution, Geophysics, 1977,
Vol. 42, pp 1369-83
 
 
Grossman A. and Morlet J.
Decomposition of Hardy functions into square integrable wavelets of constant    shape, SIAM J. Math. Anal., 15(4), 1984, pp 723-736
 
Grossman A and Morlet J and Paul T
Transforms associated to square integrable group representations. I. General
results, J. Math. Phys. 26 (10), October 1985 pp 2473-2478
 
Grossman A, Morlet J and Paul T
Transforms associated to square integrable group representations II: Examples
Ann. Inst. Henri Poincare'-Physique the'orique, Vol 45, No. 3, 1986, pp293-309
 
Wornell G. W. and Openheim A. V., Fractal Signal Modeling and Processing using
wavelets, Research Laboratory of electronics, Massachusetts Institute of Tech.
 
Daubechies I and Theirry Paul
Time-frequency localisation operators--a geometric phase space approach: II
The use of dilations, Inverse Problems 4 (1988) 661-680
 
Holschneider Matthias
Wavelet analysis on the circle, J. Math. Phys. 31 (1), January 1990, pp 39-44
 
Arneodo A, Argoul F, Elezgaray J and Grasseau G
Wavelet transform analysis of fractals: Applications to nonequilibrium phase
transitions, Center de Rescherche Paul Pascal, Domaine Universitaire, 33405 
Talence Cedex, France
 
Argoul F, Arneodo A, Elezgaray J, Grasseau G
Wavelet Transform of fractal aggregates, Aug 29, 1988
-------------------------------------------------------------------------
 
Ghanshyam Joshi
ME-EM Department
Michigan Technological University
Houghton, Michigan-49931
USA

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From: [email protected] (Darrin Wahl)
Newsgroups: sci.math
Subject: Wavelet Theory Bibliography (Long)
Keywords: wavelets, oh-so-hot topics
Message-ID: <[email protected]>
Date: 28 Jan 91 17:11:20 GMT
Sender: [email protected]
Reply-To: [email protected] (Darrin Wahl)
Organization: Science Aplications Int'l Corp.
Lines: 259
 
 
Here is a list of the references I have received on wavelet theory.
Note that some references are repeated.
 
 
From: Charles Thayer <[email protected]>
 
===== Don Lancaster, Radio Electronics, Feb '91 pp 77 =====
Beresford, Smith
Dispersive noise removal in t-x space
Geophysics USA, vol 53 no 3 pp 346-58.
 
Brown C.
New wave number crunching.
E.E. Times, 5 Nov 1990, pp 31-34.
 
Cipra B.
A New wave in applied mathematics.
Science 24 Aug 1990, pp 858-859.
 
Daugman J.
Complete discrete 2-D Gabor transforms.
IEEE Trans Acoustics & Speech, vol 36 no 7 pp 1169-79.
 
Daugman J.
Entropy reduction and decorrelation in visual coding.
IEEE Trans. Biomedical Engineering, Vol 36 no 1 pp 107-114.
 
Daugman J.
Non-orthogonal wavelet representations 
	in relaxation networks.
New Developments in Neural Computing, pp 233-50.
 
Jurkevics A.
A critique of seismic deconvolution methods.
Geophysics, vol 49 no 12 pp 2109-16.
 
Lewis A.
Vdeo compression using 3D wavelet transforms.
Electronic Letters, vol 26 no 6 pp 396-8.
 
Mallat S.
A theory for multiresolution signal decomposition.
IEEE Transactions on Machine Intelligence, v11-7 pp 674-93.
 
Persoglia S.
Adaptive deconvolution by lattice filters
Bulletin of Geophysics Theory, vol 27 no 107 pp 169-83.
 
Robinson E.
Statistical pulse compression.
IEEE Proceedings, vol 72 no 10 pp 1276-89.
 
Tuteur F.
Wavelet transformation in signal detection.
ICASSP Speech Conference 88, vol 3 pp 1435-8.
8th IFAC/IFORS Symposium, vol 2 pp 1061-5.
 
 
 
**************************************************************
 
 
::::: Wavelet Bibliography :::::
compiled originally by: [email protected].
 
Argoul F, Arneodo A, Elezgaray J, Grasseau G
Wavelet Transform of fractal aggregates, Aug 29, 1988.
Centre de Rescherche Paul Pascal, Domaine Universitaire, 
	33405 Talence Cedex, France.
 
Arneodo A, Argoul F, Elezgaray J, and Grasseau G
Wavelet transform analysis of fractals: 
	Applications to nonequilibrium phase transitions.
Centre de Rescherche Paul Pascal, Domaine Universitaire, 
	33405 Talence Cedex, France.
 
Beresford, Smith
Dispersive noise removal in t-x space
Geophysics USA, vol 53 no 3 pp 346-58.
 
Berkhout A. G.
Least-squares inverse filtering and wavelet deconvolution, Geophysics, 1977,
Vol. 42, pp 1369-83
 
Brown C.
New wave number crunching.
E.E. Times, 5 Nov 1990, pp 31-34.
 
Cavaretta A., Dahmen W., Michelli C.
Stationary Subdivision
Fachbereich Mathematik: Serie A, Mathematik,
   Freie Universitat Berlin, 
   Preprint No. A-89-25, 
   (to appear in Memoirs of the AMS.)
 
Cipra B.
A New wave in applied mathematics.
Science 24 Aug 1990, pp 858-859.
 
Dahmen W. and Michelli C., 
On Stationary Subdivision and the Construction of 
   Compactly Supported Orthonormal Wavelets, 
Fachbereich Mathematik: Serie A, Mathematik,
   Freie Universitat Berlin, 
   Preprint No. A-90-6,
 
Daubechies I.
Orthonormal bases of compactly supported wavelets.
Communications on Pure and Applied Mathematics, 
   41 (1988), 909-996.
 
Daubechies I. Grossman A. Meyer Y.
Painless nonorthogonal expansions.
J Math Phys, vol 27, no 5, 1986, pp 1271-1283
 
Daubechies I.
Time-frequency localization operators: a geometric 
   phase space approach
IEEE Transactions on Information Theory, vol. 34, no. 4, 1988
 
Daubechies I. Theirry P.
Time-frequency localisation operators: a geometric 
	phase space approach: II The use of dilations
Inverse Problems 4 (1988) 661-680.
 
Daubechies, I.
The Wavelet Transform, Time-Frequency Localization 
   and Signal Analysis
IEEE Transactions on Information Theory vol. 36, 
   Sept. 1990, 961-1005
 
Daugman J.
Complete discrete 2-D Gabor transforms.
IEEE Trans Acoustics & Speech, vol 36 no 7 pp 1169-79.
 
Daugman J.
Entropy reduction and decorrelation in visual coding.
IEEE Trans. Biomedical Engineering, Vol 36 no 1 pp 107-114.
 
Daugman J.
Non-orthogonal wavelet representations 
	in relaxation networks.
New Developments in Neural Computing, pp 233-50.
 
Grossman A. Morlet J.
Decomposition of Hardy functions into square 
	integrable wavelets of constant shape.
SIAM J. Math. Anal., 15(4), 1984, pp 723-736.
 
Grossman A. Morlet J. Paul T.
Transforms associated to square integrable 
	group representations. I. General results.
J. Math. Phys. 26 (10), October 1985 pp 2473-2478.
 
Grossman A. Morlet J. Paul T.
Transforms associated to square integrable 
	group representations II: Examples.
Ann. Inst. Henri Poincare'-Physique the'orique, 
	Vol 45, No. 3, 1986, pp293-309.
 
 
Heil C.E. and Walnut D.F.
Continuous and discrete wavelet transforms.
SIAM Review 31 (1989), pp 628-666.
 
Holschneider Matthias
Wavelet analysis on the circle.
J. Math. Phys. 31 (1), January 1990, pp 39-44.
 
Huntsberger T.L. and B.A.
Hypercube algorithm for Image Decomposition 
	and Analysis in the Wavelet Representation.
Proceedings of the fifth distributed memory conference, 
	South Carolina, pp 171-175, 1990.
 
Jurkevics A.
A critique of seismic deconvolution methods.
Geophysics, vol 49 no 12 pp 2109-16.
 
Lewis A.
Vdeo compression using 3D wavelet transforms.
Electronic Letters, vol 26 no 6 pp 396-8.
 
Mallat S.
A theory for multiresolution signal decomposition - the wavelet
   representation.
IEEE Transactions on pattern analysis and machine intelligence, 
1989 jul, v11 n7:674-693.
 
Mallat,S.(1989) Multifrequency channel decompositions of images
and wavelet models. IEEE transactions on Acoustics, Speech, and 
Signal Processing, Vol 37, no 12, pp2091-2110
 
Mallat S. (1989), Multiresolution Approximations and Wavelet Orthonormal
Bases of L2(R), Trans. Amer. Math. Soc. 315(1), pp 69-87
 
Pentland, Alex
Fast Surface Estimation Using Wavelet bases.
MIT Media Lab Vision and Modeling Group 
	Technical Report No. 142, June 1990.
 
Persoglia S.
Adaptive deconvolution by lattice filters
Bulletin of Geophysics Theory, vol 27 no 107 pp 169-83.
 
Robinson E.
Statistical pulse compression.
IEEE Proceedings, vol 72 no 10 pp 1276-89.
 
Strang G., 
Wavelets and dilation problems: a brief introduction, 
SIAM Review 31 (1989), 614-627.
 
Tuteur F.
Wavelet transformation in signal detection.
ICASSP Speech Conference 88, vol 3 pp 1435-8.
8th IFAC/IFORS Symposium, vol 2 pp 1061-5.
 
Wornell G. W., Openheim A. V.
Fractal Signal Modeling and Processing using
	wavelets.
Research Laboratory of electronics, MIT.
 
Yves Meyer: Ondelettes et Operateurs 
Tome I: Ondelettes
Hermann 1990
ISBN 2 7056 6125 0
 
***************************************************************
 
From 				Jordi Sod
				[email protected]
 
For the mathematical theory I recommend the following reference:
 
   Yves Meyer: Ondelettes et Operateurs 
   Tome I: Ondelettes
   Hermann 1990
   ISBN 2 7056 6125 0
 
   Arne Jensen
   Department of Mathematics and Computer Science
   Institute for Electronic Systems
   Aalborg University
   Fr. Bajers Vej 7
   DK-9220 Aalborg 0
   Denmark
 
   [email protected]
 
 
--								             --
Darrin Wahl                                                 [email protected]
Science Applications International, Corp.
10260 Campus Point Dr.
San Diego, CA  92121		
 
Only a fool would express these opinions!

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From: [email protected] (Amol Joshi)
Newsgroups: sci.math,sci.engr
Subject: Re: Wavelet Theory
Message-ID: <[email protected]>
Date: 28 Jan 91 20:58:27 GMT
References: <[email protected]> <[email protected]>
Organization: none
Lines: 31
Xref: ryn.mro4.dec.com sci.math:13744 sci.engr:591
 
>
>I recently read a small article in Scientific American about a type of
>high frequency transform based on "wavelets".  It is supposed to be helpful
>in distinguishing the onset of high frequency components of signals, such
>as vowels in speech processing.
>Does anyone have any info and/or references on this subject?  Thanks
>in advance.
>Darrin Wahl                                                 [email protected]
 
I will cite one ref. You can get others from it:
 
Orthonormal Bases of Compactly Supported Wavelets
Ingrid Daubechies
Commun. on Pure and Applied Math., Vol. XLI 909-996 (1988).
 
The basic idea behind the wavelets is simple.
In 1986, Meyer (a French mathematician) showed that a simple construction
can yield an orthonormal basis of an L2 space. Moreover, the basis
functions have small essential supports in *both* time and frequency
domains. This is an improvement over the Fourier transform where the
basis functions (the complex exponentials) are perfectly concentrated
in the frequency domain but spread all over in the time domain.
 
Hope this helps.
Amol
 
-- 
------------------------------------------------------------
Amol Joshi                         | [email protected]
Department of Chemical Engineering |
Washington University in St. Louis.|

Article        17409
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From: [email protected] (Penelope E. Smith)
Newsgroups: sci.math
Subject: RE: WAVELETS: Anyone Know Anything
Message-ID: <[email protected]>
Date: 21 Jun 91 22:48:14 GMT
Lines: 8
 
Hi, Ihave just joint authored a handbook on wavelets for the military
which is a clear introduction to this field in the english language
suitable for engineers. It can be obtained via military sources or by
contacting Fastman Inc 1-215-691-2577. We also have a program for sale
called FASTWAVE which impliments in a very friendly way this and
several other time-frequency transforms using a fastwavelet
algorithim similar to that of coiffman. (Penny Smith)[email protected]
or [email protected].

Has anybody found an equivalent to the FFT for wavelets?
If so, does anybody at DEC have a copy of the software?

	Just a fast FFT is needed !
		... plus some good display tools.

    


	The wavelet transform applied to a signal s(t) is defined by :


                         /
                c(a,b) = | s(t) . g((t-a)/b) . dt
                         / 

	where

                        -2   jwt   -t^2/2
                g(t) = a . e   . e

                with   w  = about 5.5

	is the analysing wavelet.

	The wavelet coefficients c(a,b) ,which are complex numbers,
	can be displayed as 2D time/frequency pictures with 'b' as the
	time coordinate and '1/a' on the frequency direction, one picture
	for the modulus	and another for the phase.

	If we choose a value for 'a', the equation is a convolution so
	the fourier transform of c() is the product of the respective
	fourier transforms of s() and of g().

	It appears (thanks to computers, but I can't prove it ) that the
	fourier transform of g() is a pure gaussian curve !

	Result is that this wavelet transform is nothing else than
	a filtering (in the frequency domain) by a sliding gaussian
	window, the width increasing with the center frequency.	

	A simple way to check the interest of the transform can be
	demonstrated using a pure sine waveform with just one point
	corrupted. 

	The  effect is the mainly noticeable on the modulus but very
	light on the phase of the transforms. 

	In the litterature I have seen phase pictures showing sharp
	evidence of such minor discontinuity in the signal. This is
	explained by the (artificial but usual) modulo 2pi expression
	of the phase. When one looks at the phase plot, it is not
	to find any good reason to keep this modulo 2pi representation.


	program flow:
		compute S = direct fft of signal s(t)
		for a = ...
			{
			compute gauss curv G(a)
			compute C = complex product S*G
			compute c = reverse fft of C	
			compute modulus & phase
			display and/or store
			}

	When I will get some time to cleanup my stuff I will try to
	provide	the source of a demo program.

	Marcel

Re: .13

>When I will get some time to cleanup my stuff I will try to
>provide the source of a demo program.


Thanks, that would be very helpful.  I'm not very comfortable with the theory
behind this stuff, and looking at an implementation often helps.

James