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Wavelet Digest, Vol. 4, Nr. 4.
Wavelet Digest Wednesday, April 12, 1995 Volume 4 : Issue 4
Today's Editor: Wim Sweldens
sweldens@math.scarolina.edu
Today's Topics:
1. Preprint: Adaptive hierarchical approximation on the sphere
2. Preprint: Characterization of signals by the ridges ...
3. Preprint: Wavelet correlations in hierarchical branching processes
4. Preprint: Preprints from the IRISA ftp host (A. Juditsky)
5. Preprint: Spherical wavelets: Texture processing
6. Preprint: CREW: Compression with Reversible Embedded Wavelets
7. Software: A fast matlab routine for calculating Daubechies filters
8. Software: Application of the Joint Time-Frequency Analyzer
9. Software: Discrete wavelet transform analysis (WPLW)
10. Meeting: Fractal Image Encoding and Analysis ASI (CFP)
11. Course: The 5th international summer school
12. Course: Mathematics and Physics of Wavelets with Applications
13. Answer: Wavelets and convolution (WD 4.2 #28)
14. Question: Derivatives of wavelets
15. Question: Identification of frequency modulation laws.
16. Question: Wavelets and speech recognition.
17. Question: Reference missing in Daubechies' Ten lectures ?
18. Question: Denoising Via Wavelet Techniques
19. Question: Example 3.2 in Chui's book.
20. Question: wavelets and space-time data.
21. Question: FWT from Numerical Recipes
22. Question: Backend Compression Methods for Wavelet Transformed Images ?
23. Question: self-similar sum of arbitrary function
24. Question: spectrum of wavelet opertors
25. Question: Structure detection, wavelets, preprint?
Submissions:
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Current number of subscribers: 4499
Calendar of events:
Jan-Apr : Spline Functions and the theory of Wavelets, Montreal WD 4.2 #14
Mar 7-Apr 18: Wavelets and Applications Course, Boston WD 4.3 #11
Apr 17-21: SPIE: Wavelet applications for dual use, Orlando WD 3.14 #5
May 17-20: Course: Wavelets: Principles, Applic. and Implement. WD 4.2 #17
May 22-24: Course: Fuzzy Logic, Chaos, and Neural Networks, UCLA WD 4.2 #15
May 30-Jun 3: Meeting of the Acoustical Society of America, DC WD 4.1 #7
*Jun 6-8 : Mathematics and Physics of Wavelets, College Park MD WD 4.4 #12
Jun 26-30: ANU Wavelets Workshop, Canberra, Australia WD 3.6 #6
Jul 3-7 : SIAM ICIAM 95, Hamburg, Germany WD 3.19 #15
*Jul 8-17: Fractal Image Encoding and Analysis, Trontheim, Norway WD 4.4 #10
Jul 13-14: SPIE: Mathematical Imaging: San Diego WD 4.1 #6
Jul 24-28: Wavelets in Electromagnetics PIERS, Seattle WD 3.18 #11
Jul 31-Aug 25: International Summer School, Jyvaskyla, Finland WD 4.3 #12
Aug 31-Sep 1: UK Symp. on Time-Freq. and Time-Scale, Warwick UK WD 4.3 #10
Sep 17-21: ASME Wavelets in Vibrations and Acoustics, Boston WD 3.17 #11
Dec 10-13: Neural Networks and Signal Processing, Nanjing, China WD 4.3 #9
--------------------------- Topic #1 -----------------------------------
From: froehlich@zib-berlin.de (Jochen Froehlich)
Subject: Preprint: Adaptive hierarchical approximation on the sphere
Title: An adaptive hierarchical approximation method on the sphere
using axisymmetric locally supported basis functions
Authors: R\"udiger Brand, Willi Freeden and Jochen Fr\"ohlich
Abstract:
Lacunary hierarchical approximation in a wavelet-type basis has to cope
with the discretization problem for the computation of the retained
amplitudes. Furthermore, in sperical geometry no uniform discrete
shift operator is available.
The paper describes a hierarchical approximation method which has been
constructed in the multiresolution spirit using non-orthogonal bump-like
basis functions. It involves discrete shift, dilation and scale separation
only in a qualitative way. The resulting scheme is appealing by its
simplicity and nevertheless successfull and efficient for real life
scattered data approximation on the sphere.
University Kaiserslautern, Math Dep., Preprint 130.
Konrad-Zuse-Zentrum Berlin, Preprint SC 95-2
anonymous ftp: ftp://ftp.ZIB-Berlin.DE/pub/zib-publications/reports/
www: http://www.zib-berlin.de/ZIBbib/Publications/
--------------------------- Topic #2 -----------------------------------
From: Rene Carmona <rcarmona@chelsea.math.uci.edu>
Subject: Preprint: Characterization of signals by the ridges ...
Preprint Available by anonymous ftp from
chelsea.math.uci.edu
Title: Characterization of signals by the ridges of their wavelet transforms
Authors: Rene A. Carmona, Wen L. Hwang and Bruno Torresani
Abstract: Many procedures have been developed to characterize a signal
by some of the salient features of a specific transform, whether it is
the Fourier, the Gabor, the Wigner or more recently the wavelet
transform. Ridges in the modulus of the transform determine regions in
the transform domain with a high concentration of energy. For this
reason they are regarded as natural candidates for the
characterization and the reconstruction of the original signal. We
present a couple of new algorithmic procedures for the detection of
ridges in the modulus of the (continuous) wavelet transform of
one-dimensional signals. These detection procedures are shown to be
robust to additive noise. We also derive and test a new reconstruction
procedure. The latter uses only information of the restriction of the
wavelet transform to a sample of points from the ridge. This provides
with a very efficient way to code the information contained in the
signal.
--------------------------- Topic #3 -----------------------------------
From: 'Peter Lipa' <Peter.Lipa@oeaw.ac.at>
Subject: Preprint: Wavelet correlations in hierarchical branching processes
Title: Wavelet correlations in hierarchical branching processes
Authors: Martin Greiner, Jens Giesemann, Peter Lipa and Peter Carruthers
Comments: 31 pages, latex, 10 figures (available from authors).
Report-No: HEPHY-PUB 618/95, AZPH-TH/95-03
Abstract:
A study of correlations in tractable multiparticle cascade models in
terms of wavelets reveals many promising features. The selfsimilar
construction of the wavelet basis functions and their multiscale
localization properties provide a new approach to the statistical
analysis and analytical control of hierarchically organized branching
processes. The exact analytical solution of several discrete models
shows that the wavelet transformation supresses redundancy in the
correlation information. Wavelet correlations can be naturally
interpreted as correlations between structures (clumps) living on
different scales.
Note: This paper adresses to people interested in applications
of wavelets to stochastic (cascade) processes, statistical physics,
turbulence, high energy physics ....
The paper can be obtained electronically from the Los Alamos
preprint database: hep-ph@xxx.lanl.gov
paper: hep-ph/9503235 (no figures though)
The figures can be obtained in an uu-encoded ps-file (1.4 Mbyte !!)
via e-mail (or a hardcopy can be sent by snail-mail). Some
figures don't print on a NeXT machine.
Send e-mail request to:
lipa@hephy.oeaw.ac.at
Sincerely, Peter Lipa
--------------------------- Topic #4 -----------------------------------
From: Anatoli Iouditski <Anatoli.Iouditski@irisa.fr>
Subject: Preprint: Preprints from the IRISA ftp host (A. Juditsky)
title: Wavelet Estimators, Global Error Measures Revisited
author: Bernard Delyon, Anatoli Juditsky
Abstract: In the paper minimax rates of convergence for
wavelet estimators are studied. For the problems of density estimation and
non-parametric regression we establish upper bounds over a large range of
functional classes and global error measures.
The constructed estimate is simultaneously minimax (up to constant) for
all L_pi- error
measures, 0<pi<=infinity.
ftp: irisa.irisa.fr: /techreports/1993/PI-782.ps
title: Wavelet Estimators: Adapting to Unknown Smoothness
author: Anatoli Juditsky
Abstract: A wavelet thresholding algorithm is used to recover
a function of unknown smoothness from noisy data. It is known that
it can be
tuned to be minimax in order over a wide range of Besov-type
smoothness constraints and L_p-losses. We provide a method
to estimate an adaptive threshold parameter for each resolution level.
It is shown that the proposed algorithm is {\em adaptive in order}, i.e.
it attains the rate of convergence which is minimax up to a constant over
Besov regularity classes and L_p-error measures, 1<=p<= infinity.
The algorithm is computationally straightforward: the whole effort to compute
the threshold is order N log(N) for the sample size N.
ftp: irisa.irisa.fr: /techreports/1994/PI-815.ps
title: On the Computation of Wavelet Coefficients
author: Bernard Delyon, Anatoli Juditsky
Abstract: We consider fast algorithms of wavelet decomposition
of function f when
discrete observations of f (supp(f) belongs to [0,1]) are available.
The properties of the algorithms are studied
for three types of observation design: the regular design,
when the observations f(x_i) are
taken on the regular grid x_i=i/N, i=1, ..., N; the case of gittered regular
grid, when
it is only known that for all 0< i< N+1, i/N<= x_i<(i+1)/N;
the random design case: x_i, i=1, ..., N are
independent and identically distributed random variables on [0,1].
We show that these algorithms are in certain sense efficient when the accuracy
of approximation is concerned.
The proposed algorithms are computationally straightforward: the whole
effort to compute
the decomposition is order N for the sample size N.
ftp: irisa.irisa.fr: /techreports/1994/PI-856.ps
title: Computing wavelet density estimator for stochastic processes
author: Anatoli Juditsky, Frederique Leblanc
Abstract: Let (X_t) be a stictly stationary stochastic process (in continuous or discrete time). We are to estimate the density f of X_t on the basis
of discrete observations (X_i), i=1,..., N using a "linear" wavelet
estimator.
For the continuous time process (X_t), 0<= t<= T those observations
are the result of a regular discretization of
the continuous time trajectory.
We provide an adaptive version of the algorithm, which adapts automatically to
the regularity parameters of the density f. To compute the estimator we implement a fast algorithm (the whole effort to compute the estimate is order N log(N)
for the sample size N).
--------------------------- Topic #5 -----------------------------------
From: Wim Sweldens, sweldens@math.scarolina.edu
Subject: Preprint: Spherical wavelets: Texture processing
Title: Spherical Wavelets: Texture Processing
Authors: Peter Schroeder and Wim Sweldens
Abstract:
Wavelets are a powerful tool for planar image processing. The
resulting algorithms are straightforward, fast, and efficient.
With the recently developed spherical wavelets this framework can
be transposed to spherical textures. We describe a class of
processing operators which are diagonal in the wavelet basis and
which can be used for smoothing and enhancement. Since the
wavelets (filters) are local in space and frequency, complex
localized constraints and spatially varying characteristics can be
incorporated easily. Examples from environment mapping and the
manipulation of topography/bathymetry data are given.
Anonymous ftp to ftp.math.scarolina.edu, file /pub/imi_95/imi95_4.ps
or imi95_4.ps.gz. The color images in this paper are available
separately in the files /pub/imi_95/imi95_4.pix/*.tif. Also, the
postscript file without the figures is available as imi95_4.nofig.ps.
--------------------------- Topic #6 -----------------------------------
From: schwartz@crc.ricoh.com "Edward L. Schwartz"
Subject: Preprint: CREW: Compression with Reversible Embedded Wavelets
Title: CREW: Compression with Reversible Embedded Wavelets
Authors: Ahmad Zandi, James D. Allen, Edward L. Schwartz, and Martin Boliek
RICOH California Research Center
Compression with reversible embedded wavelets (CREW) is a unified
lossless and lossy continuous-tone still image compression system. It
is a wavelet based compression scheme, that uses a "reversible"
approximation of one of the best wavelet filters. Reversible wavelets
are non-linear filters which implement exact-reconstruction systems
with minimal precision integer arithmetic.
CREW provides state of the art lossless compression of medical images
(> 8-bit deep), and state of the art lossy and lossless compression of
8-bit deep images with a single system. CREW has reasonable software
and hardware implementations. Applications for CREW include medical,
prepress, satellite and document image compression.
Available though WWW at
http://www.crc.ricoh.com/misc/crc-publications.html
Edward Schwartz
RICOH California Research Center
2882 Sand Hill Rd Suite 115
Menlo Park, CA 94025-7022
(415) 496-5712
(415) 854-8740 FAX
schwartz@crc.ricoh.com
--------------------------- Topic #7 -----------------------------------
From: Kevin Amaratunga <kevin@phaeton.mit.edu>
Subject: Software: A fast matlab routine for calculating Daubechies filters
Here is a matlab routine for computing Daubechies filter coefficients
which does the spectral factorization using cepstral analysis. This
method works well for long filter lengths and it tends to be faster
than methods which make use of root finding algorithms. With the
approximately 5.5 CPU seconds on a DECstation 5000/25. A quick check
shows that
sum(h(k)) = 2.00000000000044
sum(h(k)*g(k)) = 1.122455722880489e-23
sum(h(k)*h(k-2l)) = 1.871421275103674e-13 (Max over l, l != 0)
Note that cepstral aliasing due to the FFT is not a significant problem.
References:
Oppenheim and Schafer - Discrete-Time Signal Processing.
----
% function h = daub(Nh)
%
% Generate filter coefficients for the Daubechies orthogonal wavelets.
%
% h = filter coefficients of Daubechies orthonormal compactly supported
% wavelets
% Nh = length of filter.
function h = daub(Nh)
K = Nh/2;
L = Nh/2;
N = 512; % Use a 512 point FFT by default.
k = 0:N-1;
% Determine samples of the z transform of Mz (= Mz1 Mz2) on the unit circle.
% Mz2 = z.^L .* ((1 + z.^(-1)) / 2).^(2*L);
z = exp(j*2*pi*k/N);
tmp1 = (1 + z.^(-1)) / 2;
tmp2 = (1 - z.^(-1)) / 2;
Mz1 = zeros(1,N);
for l = 0:K-1
Mz1 = Mz1 + binomial(L+l-1,l) * (-z).^l .* tmp2.^(2*l);
end
Mz1 = 4 * Mz1;
% Mz1 has no zeros on the unit circle, so use the complex cepstrum to find
% its minimum phase spectral factor.
Mz1hat = log(Mz1);
m1hat = ifft(Mz1hat); % Real cepstrum of ifft(Mz1). (= cmplx
% cepstrum since Mz1 real, +ve.)
m1hat(N/2+1:N) = zeros(1,N/2); % Retain just the causal part.
m1hat(1) = m1hat(1) / 2; % Value at zero is shared between
% the causal and anticausal part.
G = exp(fft(m1hat,N)); % Min phase spectral factor of Mz1.
% Mz2 has zeros on the unit circle, but its minimum phase spectral factor
% is just tmp1.^L.
Hz = G .* tmp1.^L;
h = real(ifft(Hz));
h = h(1:Nh)';
John R. Williams and Kevin Amaratunga
Intelligent Engineering Systems Group Department of Civil Engineering
MIT Room 1-250 77 Massachusetts Avenue Cambridge, MA 02139.
(617) 253 7657
--------------------------- Topic #8 -----------------------------------
From: Shie Qian <qian@eagle.natinst.com>
Subject: Software: Application of the Joint Time-Frequency Analyzer
Application of Time-Frequency Representation for Inverse Synthetic
Aperture Radar
The response to National InstrumentsU official release, last spring,
of the Joint Time-Frequency Analyzer (JTFA) has been tremendous.
Hundreds of engineers and scientists around the world now use
JTFA in such diverse areas as acoustics, aeronautics, biomedical
engineering, communications, economics, and seismology. As we
continue to improve the JTFA product (e.g., adding data acquisition
functions), we will post some successful application examples,
under the permission of authors and related institutions, on the
National Instruments ftp site.
ftp.natinst.com
/support/support_notes/PostScript
or /support/support_notes/mac-viewable
/support/support_notes/windows_viewable
The first paper posted, ISSSE95.ps, discusses the application for
the inverse synthetic aperture radar. This will be an invited paper
for the International Symposium on Signals, Systems, and
Electronic T95.
--------------------------- Topic #9 -----------------------------------
From: ANANDKUMAR <akumar@hns.com>
Subject: Software: Discrete wavelet transform analysis (WPLW)
A Nice Software package for Discrete Wavelet Transform Analysis (WPLW)
I am a researcher, trying to apply Wavelets to compress ECG data. For the past
6 months, I have been working with a package called WPLW (Wavelet Packet
Laboratory for Windows) to do my research. I do very strongly recommend this
tool for researchers who do not want to write their own code for wavelet
analysis. The WPLW is very user friendly, and runs under windows. It has a
library of Daubechies, Coifman, and several localized trignometric functions.
It is very simple to use, yet very powerful, in that the user can verify
multiple transcriptions and the choose the best basis, automatically. (the
best basis is chosen based on minimum entropy). The tool gives you the wavelet
packet and wavelet transform analysis. I have had very good success with this
tool, as I can very quickly do multiple wavelet filter analysis, and the
choose the best one that fits the signal of interest.
For those of you that are interested in more information, you could contact
the distributor
A K Peters, Ltd.
289 Linden St.
Wellesley, MA 02181-5910
(617) 235-2404
The best part of it all, is the outstanding technical support for this tool,
provided by Digital Diagnostic Co. in CT. It continues to be a pleasure
dealing with the folks at DDC, who support WPLW.
Sincerely
ANANDKUMAR J
HUGHES NETWORK SYSTEMS
GERMANTOWN, MD 20876
--------------------------- Topic #10 -----------------------------------
From: Frederic Gilbert <Frederic.Gilbert@inria.fr>
Subject: Meeting : Fractal Image Encoding and Analysis ASI (CFP)
CALL FOR PARTICIPATION
Fractal Image Encoding and Analysis
a NATO Advanced Study Institute
July 8-17, 1995
at the
Reso Royal Garden Hotel
Trondheim, Norway
Objectives
The Advanced Study Institute (ASI) is an expository meeting in which invited
speakers present new advances in fractal image encoding and analysis. The
meeting has two main objectives: First, the ASI will focus on presenting
material that is not taught elsewhere and that is not available from any one
source. Researchers with fractal image analysis and encoding backgrounds will
demonstrate their techniques to each other in order to advance both fields.
Workshops in both areas will discuss specific implementation issues and generate
a list of interesting unsolved problems.
Scope
Topics to be covered include, but are not limited to:
o Fractal Analysis and Modeling of 2D data.
o Multifractal image segmentation
o Encoding methods: VQ, classification, breaking the time complexity, search
strategies.
o Decoding methods: pyramid representation, pixel chasing, finite step
decoders, matrix inversion.
o Video and Color encoding.
o Analysis and Implementation Workshops.
Structure/Participation
Four invited talks will be given each day, followed by one or two shorter
selected talks by participants in the institute. A poster session will be held
during the breaks and in the evenings. The institute will also include two
workshops, one on image encoding and one on image analysis.
Participants wishing to present talks or posters should e-mail, fax, or mail an
extended abstract (2-3 pages) describing their proposed lecture/poster-session
to the ASI (see contact information) before April 15, 1995. NATO asks that
participants stay for the full duration of the ASI.
Registration Fees
before April 15, 1995 after April 15, 1995
advanced registration rates regular rates
regular | US$439.00 | US$589.00
student ** | US$219.00 | US$399.00
** proof of status required
To register, complete the registration form available from the
http://inls.ucsd.edu/y/ASI/ WWW page, and return it by mail or fax to
(619) 534-7664 with your payment (currently payment can only be made by check or
money order made payable to "NATO ASI"; credit card payments will be available
soon).
Preregister by April 15 to take advantage of advanced registration rates.
Registration received after April 15, 1995 will be charged at normal rates. NATO
limits the size of the institute to a maximum of 80 people. On-site registration
will not be possible.
Invited Speakers
o THE LIST IS IN THE PROCESS OF BEING FINALIZED.
Proceedings
Talks given by the invited speakers will be collected in a NATO ASI series book
published by Springer Verlag. Participants in the institute will receive a free
copy when the book is published.
Sponsors
o NATO
o Rockwell International.
o Institute for Nonlinear Science, University of California, San Diego.
Scientific and Organizing Committee
o Director:
o Yuval Fisher
o Co-directors:
o Jacques Levy-Vehel
o Geir Oien
o Claude Tricot
Contact Information
NATO Fractal Image Encoding and Analysis ASI
Institute for Nonlinear Science 0402
University of California, San Diego
9500 Gilman Drive
La Jolla, CA 92093-0402
USA
Phone: (619) 534-5599 Fax: (619) 534-7664
E-mail: asi@poincare.ucsd.edu WWW: http://inls.ucsd.edu/y/ASI/
--------------------------- Topic #11 -----------------------------------
From: Laura Varpula <iss5@tukki.jyu.fi>
Subject: Course: The 5th international summer school
The University of Jyvaskyla, Finland, is organizing
THE 5th INTERNATIONAL SUMMER SCHOOL
31 Jul - 25 Aug 1995
What is the Summer School?
The aim of the International Summer School is to
offer advanced courses in various topics to both
undergraduate and graduate students.
What is the programme of the Summer School?
The programme of the 5th International Summer
School consists of courses on the following topics:
mathematics
applied mathematics
computer science
statistics
biology
chemistry
physics
How to apply for the Summer School?
To apply for the International Summer School
please fill in the application form
(http://www.math.jyu.fi/summerschool.html)
and send it to the organizers by 31 March 1995.
What are the costs?
There is no tuition fee for the Summer School.
Housing and living expenses and travel costs
are the responsibility of each participant.
For further information please contact
The 5th International Summer School
Faculty of Mathematics and Natural Sciences
University of Jyvaskyla
P.O. Box 35
FIN-40351 Jyvaskyla
FINLAND
Phone: +358 41 602 205
Fax: +358 41 602 201
E-mail and WWW addresses:
iss5@tukki.jyu.fi
http://www.math.jyu.fi/summerschool.html
Please find enclosed more detailed information about the courses
of the Wavelet-topic in the Summer School.
MA 1 Introduction to wavelets,
30 h, 31 July - 11 August
The topics of the course are: Outline of basic Fou-
rier analysis, multiresolution approximation, or-
thogonal wavelet bases, wavelet frames, biortho-
gonal bases, applications to various fields includ-
ing functional analysis. The students are assumed
to know the first principles of Lebesgue integration
and basic functional analysis.
Course book: Y. Meyer, "Wavelets and Operators".
Dr. D.L. Salinger, Univ. of Leeds, UK.
MA 2 Signal processing and multifractal analysis
with wavelets, 30 h, 7 - 18 August
The topics of the course are: Signal and image
processing with wavelets and other time-frequency
algorithms, optimal denoising, wavelets and multi-
fractal analysis. The basic understanding of wave-
lets is assumed.
Course books: Y. Meyer, "Wavelets, Algorithms
and Applications" (SIAM 1992); P. Flandrin,
"Temps-frequence" (Hermes 1993).
Prof Y. Meyer, Universite Paris-Dauphine and the
French Academy of Sciences, France.
MA 3 Applications of wavelets to analysis,
30 h, 10 - 22 August
The topics of the course are: Regularity analysis
through wavelets: global analysis related to har-
monic analysis (Sobolev spaces, Holder spaces,
weighted Lebesgue spaces) and microlocal analysis
(with applications to multifractal analysis, partial
differential equations and turbulence). Basic know-
ledge of Lebesgue integration and basic Fourier
and functional analysis is required.
Course book: Y. Meyer, "Wavelets and Operators".
Also of interest: I. Daubechies, "Ten Lectures on
Wavelets" (SIAM, 1992); J.P. Kahane and P.G.
Lemarie-Rieusset, "Fourier Series and Wavelets"
(Gordon and Breach, to appear).
Prof. P.G. Lemarie-Rieusset, Universite Paris-Sud,
France.
--------------------------- Topic #12 -----------------------------------
From: "Jenkins, James W." <JenkiJW1@subtech1.spacenet.jhuapl.edu>
Subject: Course: Mathematics and Physics of Wavelets with Applications
Offered by The Applied Technology Institute
COURSE TITLE:
The Mathematics and Physics of Wavelets,
with Applications to Remote Sensing
INSTRUCTOR:
Gerald Kaiser, Mathematical Sciences Department,
University of Massachusetts at Lowell
TIME & PLACE:
June 6-8, 1995 in College Park, MD (301) 345 6700
DESCRIPTION: 12 lectures, each 50 minutes,
followed by 10 minutes for questions. The instructor
will also be available for extended discussions. A mix
of university, goverment and industry participants are expected.
LECTURES:
1. Continuous time-frequency analysis
2. Continuous time-scale (wavelet) analysis
3. Time-frequency sampling theorems
(Discrete windowed Fourier transforms)
4. Time-scale sampling theorems
(Discrete wavelet transforms)
5. Multiresolution analysis and subband filtering
6. Daubechies' wavelet bases
7. Radar and sonar wideband ambiguity functions:
A canonical application of time-scale analysis
8. Radar and sonar narrowband ambiguity functions:
A canonical application of time-frequency analysis
9. Physical (acoustic and electromagnetic) wavelets:
Localized solutions of wave and Maxwell equations
10. The scattering of physical wavelets,
based on conformal translformations
11. Emission and reception of physical wavelets:
A variational approach to remote sensing
12. Other applications.
Participants will be given a set of lecture notes
and a copy of the instructor's recent book
"A Friendly Guide to Wavelets" (Birkhauser, 1994).
To register or to receive a full course catalog
phone the Applied Technology Institute at
(410) 531-6034 or Fax (410) 531-1013.
Enrollment is limited.
For more information about the course content,
contact kaiserg@woods.uml.edu .
--------------------------- Topic #13 -----------------------------------
From: tsuhan@big.att.com (Tsuhan Chen)
Subject: Answer: Wavelets and convolution (WD 4.2 #28)
Dear Elharti Abdelmoula,
In response to your question posted in the Feb 1995 issue of the
Wavelet Digest, a wavelet convolution theorem was presented in
(See pp. 2018-2019):
Tsuhan Chen and P. P. Vaidyanathan, "Vector space framework for
unification of one- and multidimensional filter bank theory," IEEE
Trans. on Signal Processing, vol. 42, no. 8, pp. 2006-2021, August
1994.
This wavelet convolution theorem is similar to, although not exactly
the same as, that of the DFT/FFT type.
Best regards,
Tsuhan Chen
AT&T Bell Laboratories
Visual Communications Research Department
101 Crawfords Corner Road, 4C528
Holmdel, NJ 07733-3030
Tel: (908) 949-2708
Email: tsuhan@research.att.com
--------------------------- Topic #14 -----------------------------------
From: Tony Cai <tcai@compstat.wharton.upenn.edu>
Subject: Question: Derivatives of wavelets
I am studying function estimation with discrete observations contaminated by
random noise. In my work, I need to know the derivative functions of the
father wavelet and mother wavelet (of Daubechies' o.n. family and the Coiflets)
. I am looking for a fast algorithm to compute the derivative of the mother
wavelet. I wish you can help me. Any help is appreciated.
Tony Cai
tcai@compstat.wharton.upenn.edu
--------------------------- Topic #15 -----------------------------------
From: "SELEGHIM Paulo 148957" <SELEGHIM@dtp.cea.fr>
Subject: Question: Identification of frequency modulation laws.
I am looking for references on the identification of frequency modulation
laws from the wavelet transform and particularly on the application of the
stationary phase method. Keywords are for example instantaneous frequency,
ridge and skeleton.
I intend to apply these techniques in the study of two-phase flow.
Thanks in advance
Paulo Seleghim Junior
Centre d'Etudes Nucleaires de Grenoble
DTP/STI-LEF 17, rue des Martyrs 38054 Grenoble Cedex 9 France
EM : seleghim@dtp.cea.fr
--------------------------- Topic #16 -----------------------------------
From: URBINI@aquila.infn.it
Subject: Question: Wavelets and speech recognition.
Hello,
I'm looking for wavelets and speech recognition.
I wish info for software (C, C++) or tools for Matlab (for Windows).
I'm a student on the University of L' Aquila - Italy.
Thanks in advance and sorry for my english...
Paolo Urbini
Urbini@aquila.infn.it
--------------------------- Topic #17 -----------------------------------
From: andrewd@ee.uts.edu.au (Andrew Dorrell)
Subject: Question: Reference missing in Daubechies' Ten lectures ?
Please,
Is there a reference missing from Daubechies "Ten Lectures in Wavelets"
Chapter 8?
Page 282 reads "A first application to image analysis of these biorthogonal
bases associated to the laplacian pyramid is given in Antonini et al. (1990)."
The references section contains two references bearing Antonini's name but
both are 1991 and neither seem directly related. Does anyone know what
this reference should be? Will post responses of course.
Many thanks,
Note from the editor: My guess is that this is a typo and that it should
be (1991). Cf. the comment at the start of the bibliography.
Mr Andrew Dorrell
School of Electrical Engineering, University of Technology, Sydney
Phone: 61 2 330 2395
Fax: 61 2 330 2435
email: andrewd@ee.uts.edu.au OR dorrell@ihf.uts.edu.au
--------------------------- Topic #18 -----------------------------------
From: el_williams@ccmail.pnl.gov
Subject: Question: Denoising Via Wavelet Techniques
Comment:
Software:
I am an avid user of Wavelet Packet Laboratory for Windows.
This is a commercially available package sold via Digital Diagnostic
Corporation in Hamden, CT. The software has allowed me to extensive
research in signal processing of Mass Spectrometry Data.
Question:
Need Resources on Denoising Techniques:
The following is my current perception.
Wavelet Techniques for denoising:
I. What is Denoising?
A. Coherent structure extraction
1. Let's view our signal of interest as being composed of
a noisy incoherent signal plus an information signal. We would
really like to capture or extract the information signal and
discard the noise.
2. The task is to choose a basis in which to represent the
information signal. The basis function must be chosen such that the
correlation between it and the information signal is
maximized. This will then leave the noisy signal not well correlated
with the basis function. Of course now this noisy signal is left
susceptible to elimination via some non-linear
statistical filter.
3. There is a library of Wavelet bases that one can
choose from. How does one determine which basis to choose? One
method is to perform an exhaustive search of the library
selecting the basis that
has the least amount of cost. By cost we mean, which basis
represents our signal in the most efficient way. So we just simply
select a basis in which are signal has a minimum entropy. Now, the
work has been done for us with commercial software packages; all
that we have to do is to choose the minimum best basis from the set of
best basis derived from the library. The best basis gives the
smallest entropy expansion of our signal. We can also view this as
the smallest measure of distance between our signal and its
orthogonal decomposition on some subspace.
4. Now that we have a nice representation of our signal
along the best-basis, select the coefficients above a threshold say,
t (like above 0.1 %), an eliminate the others. We then reconstruct
the signal from these coefficients and save it.
Ah ha! There may be coherent structures still hiding in the residual
(i.e. buried in the noise if you will). How do we know when to
stop looking for hidden structures in the residual!!?? Well once
again we need some sort of threshold to compare against So that we
continue to test for information in the residual
until we reach a point where the presumed extracted coherent part is
greater than some threshold. We can
also just choose to iterate a fixed number of times (i.e. the old
trial by error method). Now unless you have develop some model for
which you think you understand the nature of this noise, then it is
time for human interpretation. (i.e. are
you happy with the results that you see? Do the results make sense?
Are the results useful?). I guess what I am trying to emphasize is
that in order to calculate "signal-to-noise" ratios one must be able
to measure the noise. Everyone likes to assume that the noise is
white noise or guassian. Sometimes the assumption is a good
approximation. Anyway enough babbling, just be careful on how you
interpret your results. This is not magic you have to be
interactive!!!!!
5. Finally, I would to describe another denoising technique using
wavelets. This one comes to us via Stanford. A technique known as
Wavelet Shrinkage was developed Donoho and Johnstone etc... at
Stanford University. Those guys are very good statisticians and you
know statisticians are always looking for robustness in both
parametric and non-parametric data analysis. These guys view
denoising as non-parametric statistical estimation and smoothing,
whereas I view these operations as nonlinear filter theory. Same
thing right!!???? Well this method starts with the idea that we want
to estimate our signal of interest, however, it is contaminated by
iid zero mean guassian noise. Then they use the additive noise model.
Now the next step is to model your signal of interest if you have a
lot of apriori (or any at all) information or you have a nice
phenomenological model of your signal (i.e. you what's going on with
the physics/chemistry of your process).
The latter technique is a parametric approach. If on the other hand
you don't know much about your signal then nonparametric methods
should be employed. Let's get back to wavelet shrinkage. Well
manually it is very simple;since noise effects all wavelet
coefficients by shrinking the coefficients toward
zero we can reduce the noise and simultaneously preserve the features
of our signal. (i.e. blurring of any sharp feature is minimize).
Now we simply do the inverse transform and viola!!!! We have a
denoised signal. I hope didn't lead you on by saying that this was
easy. The hard part is to select a wavelet basis and come up with a
statistically based thresholding rule. The thresholding rule is the
means by which you decide how much to shrink the wavelet coefficients.
There are few thresholding rules developed by Donoho and his crew.
Such as , thresholding based on SURE (Stein Unbiased Risk Estimator),
minimax criteria, etc... There are details also concerning
the type of Shrinkage function. (i.e. hard shrinkage vs. soft
shrinkage).
Thank You for your help in advance,
Ernest L. Williams Jr.
Battelle Pacific Northwest Labs
Applied Physics Center
(509) 375-3930
e-mail el_williams@pnl.gov
--------------------------- Topic #19 -----------------------------------
From: keith.weintraub@tf-mail.citicorp.com (Keith Weintraub - dpt12)
Subject: Question: Example 3.2 in Chui's book.
I'm a newcomer to wavelets and I am trying to learn by reading "An
Introduction to Wavelets" by Charles K. Chui (1992). I am stuck on the proof
of example 3.2 on page 54. I believe the result (which can be arrived at by
computing and inspecting the Fourier transforms) but the proof as written in
the book escapes me. Any ideas?
Also, does anyone know of an errata list for the book?
Thanks
Keith Weintraub (KW) -- Citibank Global Derivatives
399 Park 7/2, NY, NY 10043 keithw@Citicorp.COM uunet!ccorp!keithw
212-291-5827
--------------------------- Topic #20 -----------------------------------
From: baert@astr.ucl.ac.be (Etienne Baert)
Subject: Question: wavelets and space-time data.
I am looking for some informations on the feasibility of using wavelets
transform for space-time data analysis.
Are they anyone working on this area or aware of people who are working
in this area ? Any suggestions on whether it is feasible or issues
involved will be appreciated.
Please write to me at : baert@astr.ucl.ac.be
Thanking you in advance.
Baert Etienne
Institute of Astronomy and Geophysics Georges Lemaitre
Universite Catholique de Louvain
Louvain-la-Neuve (Belgium)
--------------------------- Topic #21 -----------------------------------
From: PSOMMA@chiostro.univr.it
Subject: Question: FWT from Numerical Recipes
We are using an adapted version of the FWT from "Numerical
Recipes in C". We tried many other Daubechies's wavelets, but we had
problems with the biorthogonal wavelets, in particular when the
number of coefficents is odd. In this case, how should be the wrap-
around of the matrix? And its inverse?
We will be grateful to anyone that can help us.
Paolo Sommaruga, Stefano Lonardi
email: psomma@chiostro.univr.it
Universita di Verona
ITALY
--------------------------- Topic #22 -----------------------------------
From: sev@gdstech.grumman.com (Sev Binello)
Subject: Question: Backend Compression Methods for Wavelet Transformed Images ?
Hi,
I am experimenting wavelet compression on IR, radar and visual images.
I have heard of some large compression ratios available via wavelet
compression, however, I do not seem to be able to attain such ratios
( i.e often hear of 100 : 1). Are there benchmarks available anywhere?
Are descriptions of the techniques used to compress the images
available?
Basically, I perform a wavelet transform on an image, and
iteratively transform the low-low pass image a variable number of times.
Then scale the coeeficients back to pixel values and then use various back
end compression methods on the transformed and scaled images.
I have tried both lossy (Vq, Adpcm) and lossless (LZW, ARIITHMETIC)
techniques, in addition to cascading the two. The compression ratios
I get are usually on a par with JPEG if not a little lower.
Comments and Suggestions Appreciated
Sev
--------------------------- Topic #23 -----------------------------------
From: Amy Caplan <amyc@well.sf.ca.us>
Subject: Question: self-similar sum of arbitrary function
I am curious about under what conditions the sum of a translated,
scaled "basis" function g() will tend to generate a statistically
self-similar function, as k->inf:
f(t) = sum_k w[k]*g(s[k]*t-o[k]),
where o[k] translates and s[k] and w[k] stretch and scale the function g().
Does anyone have any guidance to offer about how to reason about this?
Suppose s[k] are all 1. It seems obvious to me (thinking 'spectrally')
that no possible w[k] will generate a self-similar image
(unless g() already has a 1/f type spectrum. let's assume it does not).
Allow s[k] to change: increasing s[k] will shrink g() and create a
proportional bandwidth increase; the peak frequency amplitude will
also go down so as to maintain constant "energy".
If you plot the spectrum of different g() with s() taking on many values
it looks like the envelope of these spectra are already 1/f.
But, what is the distribution of s[k] needed to make this true?
Intuitively, it seems to me that
- the positioning o[k] does not affect the spectrum and can be random.
- if w[k] = 1 and s[k] are chosen to be uniformly distributed values,
then the spectrum might be 1/f. The function will be 'wierd' however--
isolated copies of highly-shrunken g() will poke up in random places.
- if s[k] are chosen proportionally, e.g. twice as many s[k]=2 as s[k]=1
and the corresponding w[k] falloff correspondingly somehow,
a standard fractal construction will result?
- If so, what should the w[k] be? (1/s[k]??), *and how do you conclude this?*
Thanks for any guidance (or discussion!)
--------------------------- Topic #24 -----------------------------------
From: Jean-Philippe Brunet <brunet@Think.COM>
Subject: Question: spectrum of wavelet opertors
What is known about the spectrum of wavelet operators?
The spectrum of the Fourier transform operator is uniformely distributed on
the unit circle. I am curious as to what is the distribution of eigenvalues
for a wavelet transform operator, in particular whether it would have a
fractal character. I tried a few keyword searches in the Wavelet digest but
I thought I would ask you directly.
Regards,
Jean-Philippe brunet
Thinking Machines Corporation
245 First Street
Cambridge, MA 02142
I computed the eigenvalues of the
discrete wavelet transform matrix (DAUB4) of moderate size (N=1000). The
distribution on the unit circle is non uniform, but that's all I can really
say.
If one were to relax the orthogonality constraint then the
eigenvalues will spread over the complex plane, perhaps showing interesting
(fractal?) patterns (but are there non othogonal wavelet operators of
practical interest?). Before I spend some time on this I was curious of what
had been done.
--------------------------- Topic #25 -----------------------------------
From: berg@pool.informatik.rwth-aachen.de (Stephen R. van den Berg)
Subject: Question: Structure detection, wavelets, preprint?
I am reading an abstracts pamflet here, it contains the following
reference to an article that is yet to be published:
E. Lega, H. Scholl, J.-M. Alimi, A. Bijaoui and P. Bury
A parallel algorithm for structure detection based on wavelet and
segmentation analysis
Departerment C.E.R.G.A., Observatoire de la Cote d'Azur, B.P.229,
06304 Nice, France Departerment G.D.Cassini., Observatoire de la Cote
d'Azur, B.P.229, 06304 Nice, France Laboratoire d'Astrophysique
Extragalactique et de Cosmologie, U.R.A. CNRS 173, Observatoire de
Paris-Meudon, 92195 Meudon, France
The abstract begins with (no, I'm not going to type it in completely :-):
We present a parallel algorithm which allows to recognize rapidly
structures in a 3-dimensional set of discrete data points resulting from
numerical experiments, and to study their morphological properties.
...
And now for the question: does anyone have an email address of one of the
authors? Or an ftp address/url for the paper?
Any pointers are appreciated.
Sincerely, berg@pool.informatik.rwth-aachen.de
Stephen R. van den Berg (AKA BuGless).
-------------------- End of Wavelet Digest -----------------------------