Amara's
Wavelet
Please Note: This page is copyrighted material.
Wavelet Overview
The fundamental idea behind wavelets is to analyze according to
scale. Indeed, some researchers in the wavelet field feel that, by
using wavelets, one is adopting a whole new mindset or perspective
in processing data.
Wavelets are functions that satisfy certain mathematical requirements
and are used in representing data or other functions. This idea is
not new. Approximation using superposition of functions has existed
since the early 1800's, when Joseph Fourier discovered that he could
superpose sines and cosines to represent other functions. However,
in wavelet analysis, the scale that one uses in looking at data
plays a special role. Wavelet algorithms process data at different
scales or resolutions. If we look at a signal with alarge
"window," we would notice gross features. Similarly, if we look at
a signal with a small "window," we would notice small
discontinuities. The result in wavelet analysis is to "see the
forest and the trees."
Can you see why these features make wavelets interesting and useful?
For many decades, scientists have wanted more appropriate functions than the
the sines
and cosines which comprise the bases of Fourier analysis, to
approximate choppy signals. By their definition, these functions
are nonlocal (and stretch out to infinity), and therefore do a very
poor job in approximating sharp spikes. But with wavelet analysis,
we can use approximating functions that are contained neatly in
finite domains. Wavelets are wellsuited for approximating data with
sharp discontinuities.
The wavelet analysis procedure is to adopt a wavelet prototype
function, called an "analyzing wavelet" or "mother wavelet."
Temporal analysis is performed with a contracted, highfrequency
version of the prototype wavelet, while frequency analysis is
performed with a dilated, lowfrequency version of the prototype
wavelet. Because the original signal or function can be represented
in terms of a wavelet expansion (using coefficients in a linear
combination of the wavelet functions), data operations can be
performed using just the corresponding wavelet coefficients. And if
you further choose the best wavelets adapted to your data, or
truncate the coefficients below a threshold, your data is sparsely
represented. This "sparse coding" makes wavelets an excellent tool
in the field of data compression.
Other applied fields that are making use of wavelets are: astronomy,
acoustics, nuclear engineering, subband coding, signal and image
processing, neurophysiology, music, magnetic resonance imaging,
speech discrimination, optics, fractals, turbulence,
earthquakeprediction, radar, human vision, and pure mathematics
applications such as solving partial differential equations.
An Introduction to Wavelets Paper
The above is the first section from my paper: "An Introduction to Wavelets"
which was published in the IEEE Computational Sciences and
Engineering, Volume 2, Number 2, Summer 1995, pp 5061.
I want to:
Wavelet Tonebursts
Victor Wickerhauser has suggested that sound synthesis is a natural
use of wavelets. If one wishes to approximate the sound of a musical
instrument, the notes can be decomposed into its wavelet packet
coefficients. Reproducing the note would then require reloading
those coefficients into a wavelet packet generator and playing back
the result. Transient characteristics such as attack and decay can
be controlled separately (for example, with envelope generators), or
by using longer wave packets and encoding those properties, as well,
into each note.
Wickerhauser has done just this. He has created combinations of wave
packets that produces especially interesting sounds. In Fall 1994, he
kindly gave me one these wavelet tonebursts to use (many thanks!).
The sound of this particular toneburst is haunting. I found that I
couldn't get the sound out of my head for days after hearing it! If
your Web browser and/or WWW helper application supports .au sound
files, then hear this wavelet toneburst sound for yourself.
Press
to hear a wavelet toneburst (110 kB).
Panos Kudumakis in the Sound Engineering Research Group
at the King's College University of London has a wavelet toneburst program
that works interactively. His program uses random number wavelet packets
as a music synthesizer. More information about Kudumakis' program can be found by clicking
HERE.
Fourier Trivia
From "The Hartley Transform"
by Ronald N.
Bracewell. (Oxford Press, 1st ed, 1986, page 6)
"When the FFT was brought into the limelight by Cooley and Tukey in 1965 it
had an enthusiastic reception in the populous world of electrical signal
analysis as the news spread via tutorial articles and special issues of
journals. This ferment occasioned mild surprise in the world of numerical
analysis, where related techniques were already known. Admirable sleuthing
by M.T. Heideman, C.S. Burrus, and D.H. Johnson (to appear in 'Archive for
History of the Exact Sciences') has now traced the origins of the method back
to a paper of C.F. Gauss (17771855) written in 1805, where he says,
'Experience will teach the user that this method will greatly lessen
the tedium of mechanical calculation.' "
"A fascinating sidelight of the historical investigation is that Gauss' fast
method for evaluating the sum of a Fourier series antedates the work on which
Fourier's fame is based. We should hasten to add that Gauss' paper was not
published until much later [Collected Works, Vol. 3 (Gottingen: Royal Society
of Sciences, 1876)], and we should remember that when Fourier introduced the
idea of representing an arbitrary periodic function as a trigonometric series
eminent mathematicians such as Lagrange resisted it."
Wavelet Digest
If you are interested in wavelets, you should subscribe to the Wavelet Digest. You'll
hear the latest announcements of available software, find out about errors in some
of the wavelet texts, find out about wavelet conferences, learn answers to questions
that you may have thought about, as well as ask questions of the experts that read it.
Subscribing and Submitting to the Wavelet Digest
Submissions:
Email to
wavelet@math.sc.edu
with "submit" as subject.
Subscriptions:
Email to
wavelet@math.sc.edu
with "subscribe" as subject.
To unsubscribe, email with "unsubscribe" followed by your email
address as subject. To change address, unsubscribe and resubscribe.
Preprints, references, and back issues can be obtained from their
information servers:
Wavelet Software
The amount of waveletsrelated software is multiplying. Many sources are on
Internet. If you are looking for papers and preprints, as well, browse around in some
of the Internet sites listed next. You may find papers in subdirectories named:
"/reports" or "/papers."
 "WaveLab" at Stanford University
 David Donoho and Iain Johnstone in the Stanford Statistics
Department, the Stanford graduate students Jonathan Buckheit and
Shaobing Chen, and Jeffrey Scargle at NASAAmes Research Center have
made publicly available: WaveLab .701, a library of Matlab routines for
wavelet analysis, waveletpacket analysis, cosinepacket analysis and
matching pursuit. The library, provided for Macintosh, UNIX and Windows
machines, is available free of charge over the Internet. WaveLab
currently has over 900 files consisting of scripts, Mfiles, MEXfiles,
datasets, self running demonstrations, and online documentation. It
been used in teaching courses in adapted wavelet analysis at Stanford
and at Berkeley, and is the basis for wavelet research by the authors.
IDL versions of many of these procedures are in progress (written by
myself).
WaveLab Matlab Software.
 "Wavelet Workbench" from Research Systems, Inc.
 Wavelet Workbench (WWB) is a library or "toolkit" written in IDL
that demonstrates wavelet concepts as well as providing wavelet
functions to manipulate data. This software is currently in beta
testing. This library can be run from a graphical user interface
("widget"), or it can be called from the IDL command level. Wavelet
Workbench currently consists of functions to perform scalegrams,
discrete wavelet transforms, multiresolution analysis, wavelet packet
analysis, wavelet transform compression, wavelet packet compression, and
denoising. To see a simple example with chirp functions in WWB, click on
the link below. If you wish to be a Beta Tester, that link will also
provide instructions for where to download the software.
Wavelet Workbench IDL Software.
 "MacWavelets" from Intergalactic Reality
 MacWavelets v1.00 is a free, simple, standalone, Macintosh program
for performing 1d discrete wavelet transforms and associated analyses.
With this software you can import and export 1d datasets, generate a
variety of test datasets, plot data, perform discrete and inverse
wavelet transforms using builtin and imported custom filters, display
wavelet coefficients (as 1d plots, scale density images, and as
scalegrams), manipulate datasets, sort, truncate, and unsort
coefficients, compress and expand data, remove noise from data, find
peaks in data, and display data statistics. All program operations are
available from the standard Macintosh user interface, including windows,
menus, and dialogs. No programming or scripting is required. Version
2.00 of the MacWavelets software will include the 2d discrete wavelet
transform, its inverse, and all associated compression, denoising, and
other functions. MacWavelets v2.00 will be available for downloading on
May 22, 1996.
MacWavelets
Software.
 "WaveLib" at INT, France
 WaveLib is a free, copyrighted (by INT) C library of wavelet
functions to generate wavelets, filters, perform wavelet transforms on
1D and 2D signals, calculate entropy, and perform thresholding, etc. In
addition, this package contains an interface with Matlab to display a
decomposition of a signal in the timefrequency plane as well as other
useful graphical wavelet displays. Some of the many new improvements
from last year are methods to perform a single/double quadtree
encryption, as well as fixing a number of bugs  especially the PC
version. Visit the Web site below. There is a form where you enter your
name, address, and email, and your interest in WaveLib. After you submit
the form, you will learn where to get the code, documentation, and will
read more details. Note: the WaveLib report is in French, but the
technical part describing the library is in English.
WaveLib at INT.
 "WavBox" Software by WavBox ToolSmiths
 A fullfeatured Matlab (GUI and commandline) toolbox by Carl Taswell for
performing wavelet transforms and adaptive wavelet packet
decompositions. WavBox contains a collection of wavelet transforms,
decompositions, and related functions that perform multiresolution
analyses of 1D multichannel signals and 2D images. The older
version 4.1 includes overscaled pyramid transforms, discrete wavelet
transforms, and adaptive wavelet and cosine packet decompositions by
best basis and matching pursuit as described by Mallat, Coifman,
Wickerhauser, and other authors, as well as Donoho and Johnstone's
wavelet shrinkage denoising methods. The new version 4.2 does the
above plus it implements Taswell's satisficing search algorithms for
the selection of nearbest basis decompositions with either additive
or nonadditive information costs. The new version also includes the
continuous wavelet transform valid for all wavelets including the
complex Morlet, real Gabor, and Mexican hat wavelets. Versions 13
are in the public domain. Versions later than this are
commercial.
Wavbox by WavBox ToolSmiths.
 Matlab "RiceWletTools"
 Free (but copyrighted) wavelet software for Matlab from the DSP group at Rice University.
RiceWletTools (RWT) is a collection of MATLAB Mfiles and MEXfiles implementing
wavelet and filter bank design and analysis. In addition to the design tools the toolbox provides
code for wavelet applications for both 1D and 2D denoising as well as code for processing of SAR
images.
Matlab RiceWletTools.
 Waterloo Maple Software Wavelet Share Libraries
 Free Maple wavelet libraries from the Maple share
library anonymous ftp sites. Files for calculating coefficients, values, graphs,
moments, and (anti)derivatives of Daubechies scaling
functions/wavelets, accuracy estimation and refinement of
approximations for the coefficients of Daubechies low pass filters,
calculating correct coefficients of Daubechies low pass filters, and
the 2p coefficients of the Daubechies Minium Phase filter using the
DeslauriersDubuc Lagrange interpolation method.
Maple Wavelet Software by anon ftp (Canada).
Maple Wavelet Software by anon ftp (Zurich).
 PVWave Signal Processing Toolkit (SPT)

The Signal Processing Toolkit (SPT) is an addon to PVWAVE
Advantage, almost entirely written in PVWAVE with the source
code supplied. The Toolkit concentrates on 1D signal processing, with
many of the functions extendable to 2D for image processing, etc. simply by
adding to the source code. In addition to the various functions
which cover areas such as: models and analysis, filter approximation and
realisation, transforms and spectrum analysis, statistical signal
processing, optimisation and convenience routines for polynomial
manipulation (of the transfer functions) and plotting functions, the wavelet
part of the product computes the wavelet transform of a data
sequence using compactly supported orthonormal wavelets (using a
quadrature mirror filters QMF) Functions
for computing and designing the QMF bank are supplied as well as a number
of examples.
PVWave Home Page.
 "Uvi_Wave" at Universidad de Vigo, Spain
 Wavelet software for Matlab and Khoros to help research and educational
professionals. The software includes the Discrete Wavelet Transform, the
Wavelet Transform, the Inverse Discrete Wavelet Transform, Scale Functions,
Wavelet Functions, Multiresolution analysis, Non subsampled filter banks,
can be useful for singularity detection, Wavelet design, and some demos and
utilities for subband managing and viewing. A new Matlab version (2.0) was just
released.
Uvi_Wave Wavelet Software.
Uvi_Wave Wavelet Software by anon ftp.
 Yale University
 The Mathematics Department has made available wavelet software for denoising,
a wavelet packet library (written in C), and an educational package for Xwindows.
Wavelet Software by anon ftp.
 Khoros Wavelet and Compression Toolbox
 One of the current efforts of the Computer Research and
Applications Group (CIC/C3) at Los Alamos National Laboratory is to
develop a Khoros toolbox that implements the discrete wavelet
transformation (DWT), Compression algorithms and associated support
routines. The wavelet toolbox provides programs for both 1D and 2D
wavelet transforms.
Khoros Wavelet and Compression Toolbox.
 University of Missouri
 Some more wavelets educational software can be found here.
Wavelet Software by anon ftp.
 Dept. of Mathematics, University of South Carolina
 Some more wavelets software can be found here. Read the README files first.
Some Wavelet Software at U of S Carolina via Gopher.
 "Wavelet at Imager" University of British Columbia
 The Imager
Wavelet Library, "wvlt", is a small library of waveletrelated functions
in C that perform forward and inverse transforms and refinement.
Support for 15 popular wavelet bases is included, with the ability to add
more. The package also includes source for three shelllevel programs to do
wavelet processing on ASCII files and PPM images with some demo
scripts. (The demos require "gnuplot" and "perl" to be installed on
your system.) The code has been compiled and tested under various
UNIX flavors (AIX, SunOS, IRIX, and HPUX), DOS, and (partially)
Macintosh and should port to other systems with few problems.
Imager Wavelet Software Library.
 MegaWave from ParisIX Dauphine University's Ceremade Laboratory
 Two software environments (MegaWave1 and MegaWave2) containing C programs to
implement wavelet transforms, anisotropic diffusion and segmentation, the
the AMSS model (Affine Morphological Scale Space), and snakes. Read the README
for more details.
MegaWave Wavelet Software by anon ftp.
 WTransform Matlab Toolbox
 A toolbox to perform multiresolution analysis based on the
Wtransform is available. The Wtransform is a class of discrete
transforms that treats signal endpoints differently than usual and
allows signals of any length to be handled efficiently. In addition
to the toolbox (310 Kb tarred/compressed) there is a paper (590 Kb
compressed PostScript) describing the Wtransform and a manual (108
Kb compressed PostScript) for the toolbox.
WTransform Matlab Toolbox by anon ftp.
 morletpackage
 C and Matlab code to perform 1D nonorthogonal discrete
wavelet transforms, their inverses, and wavelet compression. It is set up to use Morlet,
Daubechies and Tryme wavelets.
morletpackage and other wavelet pgms by anon ftp.
 TimeStat Wavelet Application

Windows program that performs FFT, wavelet transforms (many bases) in an
Excellike spreadsheet environment.
TimeStat program by anon ftp.
 Mathematica wavelet programs
 This directory contains a series of Mathematica programs
designed to display the features and properties of various types of
wavelets. There are also PostScript files documenting the programs
as well as some additional documents about wavelets.
Mathematica wavelet programs by anon ftp.
 Aware's "WaveTool" Software
 A general wavelet and multirate software toolbox for signal processing and
scientific applications (commercial)
Aware's WaveTool Software.
 Statsci's "S+ Wavelets" Software
 S+WAVELETS toolkit includes an extensive set of over 500 analysis functions within an
objectoriented environment. Contains functios such as the discrete wavelet transform,
wavelet optimal signal estimation, wavelet packet analysis, local cosine analysis, "best basis" selection,
matching pursuit analysis, robust wavelets analysis, and more.
S+WAVELETS operates in conjunction with SPLUS, which is based on the objectoriented S language
developed at AT&T Bell Laboratories. (commercial)
Statsci's S+ Wavelets Software.
 Wavelet Software from the Institut fuer Algorithmen und Kognitive Systeme, Karlsruhe, Germany
 The software offerred here includes "wvfloat," an interactive tool to
visualize waveletbased decompositions of 2D grayscale images in "pgm"
format, "xmorlet," which demonstrates the continous 1D wavelet transform
for sound files using the Morlet wavelet, and "wavelook," a tool for
experimenting with the parametrization of the 1D orthogonal compactly
supported wavelets according to D. Pollen.
Institut fuer Algorithmen und Kognitive Systeme Wavelet Software.
 "WaveThresh" Software from the University of Bristol
 Wavethresh is an addon package, from the people in the Department of Mathematics at
the University of Bristol, for the statistical package SPLUS. SPLUS is based on
the objectoriented S language developed at AT&T Bell Laboratories.
U of Bristol's WaveThresh Software.
 "Wavelet" Freeware Mac Software from Darryl Spencer
 This little Macintosh program demonstrates lossy image compression using
wavelets. Wavelet lets the observer decide how much information to eliminate
in an image via a scroll bar, and then rapidly calculates what the picture will look
like compared to the original. You can find this program at the Sumex InfoMac Mac
archive or one if its mirrors (below).
Wavelet Freeware Software.
 "Wavelet Image Viewer" by Summus Ltd.
 This is a commercial waveletbased image compression plugin for
Netscape that claims to provide superior image quality, compression
ratios, and speed. Visit their web page for a demo.
Wavelet Image Viewer Netscape Plugin.
 "Wavelet Packet Laboratory for Windows" by Digital Diagnostics Corporation & Yale University.
 The Wavelet Packet Laboratory for Windows is an interactive software
tool for the Microsoft Windows operating environment that allows you to
explore the properties of the Wavelet Packet and Local Trigonometric
Transforms by performing adapted waveform analysis on digital signals.
This package includes a users' manual and program PC diskette that
allows handson signal analysis. Publisher/Price: AK Peters  1994 3.5"
Disk & Manual $300.00 (My guess is that this software is closely
associated with M.V. Wickerhauser's wavelet packets research and his
book: Adapted Wavelet Analysis from Theory to Software listed
below in the Beginners Bibliography (AG).)
Wavelet Packet Laboratory.
 "MIDAS" Astronomical Data Reduction Wavelet Transform
 This is the online documentation from the "ESOMIDAS User Guide
Volume B: Data Reduction" for the Wavelet Transform functions built into
the ESOMIDAS software. ESOMIDAS is the acronym for the European Southern Observatory  Munich Image Data
Analysis System which is developed and maintained by the European Southern Observatory.
The MIDAS system provides general tools for image processing and data reduction with emphasis
on astronomical applications including imaging and special reduction packages for ESO
instrumentation at La Silla. I attach below also another piece of text from
MIDAS documentation at a different site that describes how to compute convolutions
from continuous wavelet transforms.
"MIDAS" Astronomical Data Reduction Wavelet Transform Documentation.
More "MIDAS" Documentation Describing Convolutions and the CWT.
Wavelet Beginners Bibliography, Some with Code
We all have to
start somewhere.. The following references are selected for ease of
understanding by someone unfamiliar with the signal processing
literature. The easiest are listed first. (Thanks to Gerard Middleton
for the initial list!)
 (1)
 Barry A. Cipra, "Wavelet Applications Come to the Fore,"
SIAM News (Mathematics that Counts), 11/93.
Wavelet Applications article via gopher.
 (2)
 G. Strang, "Wavelets," American Scientist, Vol. 82, 1994, pp.
250255.
 (3)
 Rioul, Olivier and Martin Vetterli, "Wavelets and signal processing,"
IEEE Signal Processing Magazine, October 1991, p.1438.
 (4)
 G. Strang, "Wavelet transforms versus Fourier transforms,"
Bull. (New Series) Amer. Math. Soc., Vol. 28, No. 2, 1993,
pp. 288305.
 (5)
 "Wavelets" by Summus. I liked this WWW wavelets tutorial so much that I
decided to put it here, even though it's not a "paper" or a "book."
Wavelets Tutorial by Summus.
 (6)
 "The Engineer's Ultimate Guide to Wavelet Analysis" by Robi Polikar.
ANOTHER Webbased wavelets tutorial. Excellent explanations and clear
graphics. I put it here, even though it's not a "paper" or a "book."
Engineer's Guide to Wavelet Analysis by Polikar.
 (7)
 Robinson, Sam L. and Ryczek, Peter F., "Wavelets,"
The Mathematica Journal, Volume 5, Issue 1, Winter 1995, pp. 7481.
The code can be purchased separately on a MSDOS or Macintosh diskette.
The Mathematica Journal.
 (8)
 Rowe, Alistair C.H. and Abbott, Paul C., "Daubechies Wavelets and
Mathematica," Computers in Physics, Volume 9, Issue 6,
November/December 1995, pp. 635648.
This article is very similar in
content to the above The Mathematica Journal article. (Both are very
nice articles.) The code can be downloaded via anonymous ftp. Download:
WaveletTransform.m. The Mac selfextracting archive: WaveletCourse.ma.sea
may be useful too.
Computers In Physics Mathematica Wavelet Code.
Related Daubechies Mathematica Package at Colorado School of Mines.
 (9)
 Press, William; Teukolsky, Saul; Vetterling, William; Flannery, Brian 1992.
Numerical Recipes in Fortran (or C), NY, NY: Cambridge University Press.
The wavelet routines can be freely copied
and redistributed. (UNLIKE much of the rest of the Numerical Recipes code.)
Numerical Recipes Wavelets Software.
 (10)
 S. G. Mallat, "A Theory for Multiresolution Signal Decomposition: The
Wavelet Representation," IEEE Transactions on Pattern Analysis and
Machine Intelligence, Vol II, 7, 1989, pp. 674693.
 (11)
 Meyer, Yves, "Review of two books on wavelets,"
Bull. (New Series) Amer. Math. Soc., Vol. 28, No. 2, 1993,
pp. 350360.
 (12)
 Y. Meyer, Wavelets: Algorithms and Applications, Society for
Industrial and Applied Mathematics, Philadelphia, 1993.
 (13)
 Y T Chan, Wavelet Basics, Kluwer Academic Publishers,
ISBN 0792395360, 1995.
 (14)
 Crandall, Richard, Projects in Scientific Computation, NY, NY: Springer
Verlag, 1994, pp. 197226.
This book has C and Mathematica code.
Projects in Scientific Computation.
 (15)
 Y Randy K. Young, Wavelet Theory and its Applications, Kluwer Academic Publishers,
ISBN 079239271X, 1993.
 (16)
 M. Vetterli and C. Herley, "Wavelets and Filter Banks: Theory
and Design," IEEE Transactions on Signal Processing, Vol. 40,
1992, pp. 22072232.
 (17)
 Cohen, Jack and Chen, Tong 1993;
"Fundamentals of the Discrete Wavelet Transform for Seismic Data Processing," unpublished.
An excellent tutorial about the wavelet transform, including 2D wavelets. See also their
other equally excellent wavelet papers at their site.
Cohen and Chen DWT Fundaments.
Other Wavelet Articles and Mathematica Software.
 (18)
 Navarro, Rafael; Tabernero, Antonio; and Cristobal, Gabriel, 1995; "Image
Representation with Gabor Wavelets and its Applications," to be
published in Advances in Imaging and Electron Physics. An
excellent applicationsoriented paper (light on the math, and heavy on
the illustrations) about the Gabor wavelet. Topics include: joint
spacefrequency representations and wavelets, Gabor schemes of
representation, vision modeling, image coding, enhancement and
reconstruction, analysis and machine vision.
Representation with Gabor Wavelets Paper.
Figures for Representation with Gabor Wavelets Paper.
 (19)
 Various Wavelet Researchers;
"Wavelets and Their Applications in Computer Graphics," Wavelets course
organized by Alain Fournier for SIGRAPHs '94 and '95. They are in the
form of a (UNIX) compressed file 3.4MB in size that uncompresses to an
11.0MB PostScript file. The printed document is 239 pages long. These
notes are provided through the courtesy of Dr. Fournier.
SIGRAPH Computer Graphics Wavelet Course.
 (20)
 Cody, Mac A., "The Fast Wavelet Transform," Dr.
Dobb's Journal, April 1992.
The C code and article can be found at Cody Associates Home Page.
Fast Wavelet Transform Article by Mac Cody.
Fast Wavelet Transform Code by Mac Cody.
 (21)
 Cody, Mac A., "A Wavelet Analyzer," Dr.
Dobb's Journal, April 1993.
The C code and article can be found at Cody Associates Home Page.
A Wavelet Analyzer Article by Mac Cody.
A Wavelet Analyzer Code by Mac Cody.
 (22)
 Cody, Mac A., "The Wavelet Packet Transform," Dr.
Dobb's Journal, April 1994.
The C code and article can be found at Cody Associates Home Page.
The Wavelet Packet Transform Article by Mac Cody.
The Wavelet Packet Transform Code by Mac Cody.
 (23)
 Daubechies, Ingrid, Ten Lectures on Wavelets, Society for
Industrial and Applied Mathematics Press, vol. 61 of CBMSNSF Regional
Conference Series in Applied Mathematics, Philadelphia, 1992.
 (24)
 Wickerhauser Victor, Adapted Wavelet Analysis from Theory to
Software, AK Peters, Boston, 1994.
This book has C code. (There are some inconsistencies
between the code in the text and that on the diskette. Has
anyone else noticed this? AG)
 (25)
 Daubechies, Ingrid, "Orthonormal Bases of Compactly Supported Wavelets,"
Comm. Pure Appl. Math., Vol 41, 1988, pp. 906966.
 (26)
 Newland, D.E., An Introduction to Random Vibrations, Spectral and
Wavelet Analysis, New York, John Wiley, 1993.
The third edition of the book has a few Matlab scripts.
A somewhat more extensive collection is available from him at:
Lynxvale Limited, 20 Trumpington Street, Cambridge CB2 1QA U.K. (cost US $100)
 (27)
 Vidakovic, Brani, and Muller, Peter, "Wavelets for Kids," unpublished,
"wav4kids[AB].ps.Z".
A nice tutorial (but not for kids!) available via anonymous ftp.
Vidakovic and Muller tutorial.
 (28)
 Martin Vetterli and Jelena Kovacevic, Wavelets and Subband Coding,
PrenticeHall, New Jersey, 1995.
 (29)
 Gilbert Strang and Truong Nguyen, Wavelets and Filter Banks,
WellesleyCambridge Press, 1996.
The Table of Contents and Guide to the Book can be seen in their homepage:.
Wavelets and Filter Banks.
 (30)
 Ali N. Akansu and Richard A. Haddad, Multiresolution Signal Decomposition
Transforms, Subbands, Wavelets,
Academic Press, Inc., ISBN 012047140X.
 (31)
 Weiss, L. G., "Wavelets and Wideband Correlation Processing,"
IEEE Transactions on Signal Processing, January 1994, p1332.
 (32)
 Argoul, F. et al., "Wavelet analysis of turbulence reveals the
multifractal nature of the Richardson cascade,"
Nature, Vol. 338, 1989, pp. 5153.
 (33)
 Farge, Marie, "Wavelet transforms and their application to turbulence,"
Ann. Rev. Fluid Mech. Vol. 24, 1989, pp.395457.
 (34)
 Muzy, J.F., E. Bacry and A. Arneodo, "The multifractal formalism revisited
with wavelets,"
Internatl. Jour. Bifurcation Chaos, Vol. 4, No. 2, 1994, pp.245302.
 (35)
 D. Donoho, "Nonlinear Wavelet Methods for Recovery of Signals,
Densities, and Spectra from Indirect and Noisy Data," Different
Perspectives on Wavelets, Proceeding of Symposia in Applied
Mathematics, Vol 47, I. Daubechies ed. Amer. Math. Soc., Providence,
R.I., 1993, pp. 173205.
 (36)
 Wickerhauser, M.V., "Acoustic Signal Compression with Wave
Packets," 1989. Available by anonymous ftp.
Acoustic Signal Compression with Wave
Packets.
 (37)
 Kaiser,G., A Friendly Guide to Wavelets, Birkhauser, Boston, 1994.
 (38)
 John R. Smith and ShihFu Chang, "Frequency and Spatially Adaptive Wavelet Packets,"
Proceedings of the I.E.E.E. International Conference on Acoustics, Speech and Signal
Processing (ICASSP95), Birkhauser, Boston, 1994.
Frequency and Spatially Adaptive Wavelet Packets.
 (39)
 JeanLuc Starck and Fionn Murtagh and Albert Bijaoui, "Image Restoration with Denoising
Using MultiResolution," (unpublished?).
Image Restoration with Denoising.
 (40)
 Scargle, J. et al., "The QuasiPeriodic Oscillations and
Very Low Frequency Noise of Scorpius X1 as Transient Chaos: A Dripping
Handrail?,"The Astrophysical Journal, Vol. 411, 1993, L91L94.
 (41)
 J. Bradley, C. Brislawn, and T. Hopper, "The FBI Wavelet/Scalar
Quantization Standard for Grayscale Fingerprint Image Compression"
Tech. Report LAUR931659 Los Alamos Nat'l Lab, Los Alamos, N.M.
1993. See also C. Brislawn's WWW site below.
Wavelet Bibliographies on WWW
Wavelet WWW Sites
 Wavelet Resources (Dept of Mathematics, Salzburg University).
 UC Berkeley Wavelet Group.
 Steve Baum's Wavelet Resources.
 Wavelet Movies, Papers, and Software (Yale University).
 Wavelet Image Compression Example.
 Chris Brislawn's Fingerprint WSQ Compression Info.
 Wavelets in Statistics.
 Multiscale EdgeDetection with Wavelets.
 Dept. of Applied Science at Lawrence Livermore Wavelet Page.
 CMRG Wavelet Page (Dept. of Engr., U of Aberdeen, Scotland).
 Wavelets at PhysNum (Lab of Nuclear Physics, U of Montreal).
 The Wavelet Project at Intelligent Engineering Systems Laboratory, MIT.
 Nonlinear Waveletbased Processing at Rice.
 Wavelet Group at Karlsruhe, Germany.
 Mac A. Cody Associates Home Page.
 Multirate Signal Processing Page at Univ of WI, Madison.
 Berndtgen's Wavelets and Cluster Analysis.
 Wavelet Methods for Radiance Computations at Stanford.
 Jo Yew's Wavelet and Image Compression Page.
 Wavelet NetCare at Washington UniversitySt. Louis.
 Aware's Wavelet Transform Processor Chip.
 Wavelet Seismic Inversion Lab (Co. School of Mines).
 Peter De Gersem's Wavelet/Music Bookmarks.
 Ronald Spencer  Gabor Wavelets.
 Multiscale Shape Representation.
 (NICE) Wavelet Packet Image Compression Demo by Smith and Chang, Columbia U.

"Why Wavelets" Explanations by S. Santini, UCSD.
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