Amara's Wavelet Families Wavelet


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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 non-local (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 well-suited 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, high-frequency version of the prototype wavelet, while frequency analysis is performed with a dilated, low-frequency 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, sub-band coding, signal and image processing, neurophysiology, music, magnetic resonance imaging, speech discrimination, optics, fractals, turbulence, earthquake-prediction, 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 50-61.

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.

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 (1777-1855) 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:
E-mail to wavelet@math.sc.edu with "submit" as subject.

Subscriptions:
E-mail to wavelet@math.sc.edu with "subscribe" as subject. To unsubscribe, e-mail with "unsubscribe" followed by your e-mail 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 wavelets-related 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 NASA-Ames Research Center have made publicly available: WaveLab .701, a library of Matlab routines for wavelet analysis, wavelet-packet analysis, cosine-packet 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, M-files, MEX-files, datasets, self- running demonstrations, and on-line 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).
Red Button 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, multi-resolution 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.
Red Button Wavelet Workbench IDL Software.

"MacWavelets" from Intergalactic Reality
MacWavelets v1.00 is a free, simple, standalone, Macintosh program for performing 1-d discrete wavelet transforms and associated analyses. With this software you can import and export 1-d datasets, generate a variety of test datasets, plot data, perform discrete and inverse wavelet transforms using built-in and imported custom filters, display wavelet coefficients (as 1-d 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 2-d 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.
Red Button 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 time-frequency 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.
Red Button WaveLib at INT.

"WavBox" Software by WavBox ToolSmiths
A full-featured Matlab (GUI and command-line) 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 1-D multichannel signals and 2-D 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 near-best basis decompositions with either additive or non-additive 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 1-3 are in the public domain. Versions later than this are commercial.
Red Button Wavbox by WavBox ToolSmiths.

Matlab "Rice-Wlet-Tools"
Free (but copyrighted) wavelet software for Matlab from the DSP group at Rice University. Rice-Wlet-Tools (RWT) is a collection of MATLAB M-files and MEX-files 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.
Red Button Matlab Rice-Wlet-Tools.

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 Deslauriers-Dubuc Lagrange interpolation method.
Red Button Maple Wavelet Software by anon ftp (Canada).
Red Button Maple Wavelet Software by anon ftp (Zurich).

PV-Wave Signal Processing Toolkit (SPT)
The Signal Processing Toolkit (SPT) is an add-on to PV-WAVE Advantage, almost entirely written in PV-WAVE with the source code supplied. The Toolkit concentrates on 1-D signal processing, with many of the functions extendable to 2-D 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 ortho-normal wavelets (using a quadrature mirror filters- QMF) Functions for computing and designing the QMF bank are supplied as well as a number of examples.
Red Button PV-Wave 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.
Red Button Uvi_Wave Wavelet Software.
Red Button Uvi_Wave Wavelet Software by anon ftp.

Yale University
The Mathematics Department has made available wavelet software for de-noising, a wavelet packet library (written in C), and an educational package for X-windows.
Red Button Wavelet Software by anon ftp.

Khoros Wavelet and Compression Toolbox
One of the current efforts of the Computer Research and Applications Group (CIC/C-3) 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.
Red Button Khoros Wavelet and Compression Toolbox.

University of Missouri
Some more wavelets educational software can be found here.
Red Button 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.
Red Button 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 wavelet-related 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 shell-level 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 HP-UX), DOS, and (partially) Macintosh and should port to other systems with few problems.
Red Button Imager Wavelet Software Library.

MegaWave from Paris-IX 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.
Red Button MegaWave Wavelet Software by anon ftp.

W-Transform Matlab Toolbox
A toolbox to perform multiresolution analysis based on the W-transform is available. The W-transform 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 W-transform and a manual (108 Kb compressed PostScript) for the toolbox.
Red Button W-Transform Matlab Toolbox by anon ftp.

morletpackage
C and Matlab code to perform 1-D nonorthogonal discrete wavelet transforms, their inverses, and wavelet compression. It is set up to use Morlet, Daubechies and Tryme wavelets.
Red Button morletpackage and other wavelet pgms by anon ftp.

TimeStat Wavelet Application
Windows program that performs FFT, wavelet transforms (many bases) in an Excel-like spreadsheet environment.
Red Button 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.
Red Button Mathematica wavelet programs by anon ftp.

Aware's "WaveTool" Software
A general wavelet and multirate software toolbox for signal processing and scientific applications (commercial)
Red Button Aware's WaveTool Software.

Statsci's "S+ Wavelets" Software
S+WAVELETS toolkit includes an extensive set of over 500 analysis functions within an object-oriented 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 S-PLUS, which is based on the object-oriented S language developed at AT&T Bell Laboratories. (commercial)
Red Button 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 wavelet-based 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.
Red Button Institut fuer Algorithmen und Kognitive Systeme Wavelet Software.

"WaveThresh" Software from the University of Bristol
Wavethresh is an add-on package, from the people in the Department of Mathematics at the University of Bristol, for the statistical package S-PLUS. S-PLUS is based on the object-oriented S language developed at AT&T Bell Laboratories.
Red Button 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 Info-Mac Mac archive or one if its mirrors (below).
Red Button Wavelet Freeware Software.

"Wavelet Image Viewer" by Summus Ltd.
This is a commercial wavelet-based image compression plug-in for Netscape that claims to provide superior image quality, compression ratios, and speed. Visit their web page for a demo.
Red Button Wavelet Image Viewer Netscape Plug-in.

"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 hands-on 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).)
Red Button Wavelet Packet Laboratory.

"MIDAS" Astronomical Data Reduction Wavelet Transform
This is the online documentation from the "ESO-MIDAS User Guide Volume B: Data Reduction" for the Wavelet Transform functions built into the ESO-MIDAS software. ESO-MIDAS 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.
Red Button "MIDAS" Astronomical Data Reduction Wavelet Transform Documentation.
Red Button 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.
Red Button Wavelet Applications article via gopher.
(2)
G. Strang, "Wavelets," American Scientist, Vol. 82, 1994, pp. 250-255.
(3)
Rioul, Olivier and Martin Vetterli, "Wavelets and signal processing," IEEE Signal Processing Magazine, October 1991, p.14-38.
(4)
G. Strang, "Wavelet transforms versus Fourier transforms," Bull. (New Series) Amer. Math. Soc., Vol. 28, No. 2, 1993, pp. 288-305.
(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."
Red Button Wavelets Tutorial by Summus.
(6)
"The Engineer's Ultimate Guide to Wavelet Analysis" by Robi Polikar. ANOTHER Web-based wavelets tutorial. Excellent explanations and clear graphics. I put it here, even though it's not a "paper" or a "book."
Red Button 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. 74-81.
The code can be purchased separately on a MS-DOS or Macintosh diskette.
Red Button 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. 635-648.
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 self-extracting archive: WaveletCourse.ma.sea may be useful too.
Red Button Computers In Physics Mathematica Wavelet Code.
Red Button 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.)
Red Button 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. 674-693.
(11)
Meyer, Yves, "Review of two books on wavelets," Bull. (New Series) Amer. Math. Soc., Vol. 28, No. 2, 1993, pp. 350-360.
(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 0-7923-9536-0, 1995.
(14)
Crandall, Richard, Projects in Scientific Computation, NY, NY: Springer- Verlag, 1994, pp. 197-226.
This book has C and Mathematica code.
Red Button Projects in Scientific Computation.
(15)
Y Randy K. Young, Wavelet Theory and its Applications, Kluwer Academic Publishers, ISBN 0-7923-9271-X, 1993.
(16)
M. Vetterli and C. Herley, "Wavelets and Filter Banks: Theory and Design," IEEE Transactions on Signal Processing, Vol. 40, 1992, pp. 2207-2232.
(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.
Red Button Cohen and Chen DWT Fundaments.
Red Button 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 applications-oriented paper (light on the math, and heavy on the illustrations) about the Gabor wavelet. Topics include: joint space-frequency representations and wavelets, Gabor schemes of representation, vision modeling, image coding, enhancement and reconstruction, analysis and machine vision.
Red Button Representation with Gabor Wavelets Paper.
Red Button 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.
Red Button 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.
Red Button Fast Wavelet Transform Article by Mac Cody.
Red Button 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.
Red Button A Wavelet Analyzer Article by Mac Cody.
Red Button 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.
Red Button The Wavelet Packet Transform Article by Mac Cody.
Red Button 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 CBMS-NSF 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. 906-966.
(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[A-B].ps.Z". A nice tutorial (but not for kids!) available via anonymous ftp.
Red Button Vidakovic and Muller tutorial.
(28)
Martin Vetterli and Jelena Kovacevic, Wavelets and Subband Coding, Prentice-Hall, New Jersey, 1995.
(29)
Gilbert Strang and Truong Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996. The Table of Contents and Guide to the Book can be seen in their homepage:.
Red Button Wavelets and Filter Banks.
(30)
Ali N. Akansu and Richard A. Haddad, Multiresolution Signal Decomposition Transforms, Subbands, Wavelets, Academic Press, Inc., ISBN 0-12-047140-X.
(31)
Weiss, L. G., "Wavelets and Wideband Correlation Processing," IEEE Transactions on Signal Processing, January 1994, p13-32.
(32)
Argoul, F. et al., "Wavelet analysis of turbulence reveals the multifractal nature of the Richardson cascade," Nature, Vol. 338, 1989, pp. 51-53.
(33)
Farge, Marie, "Wavelet transforms and their application to turbulence," Ann. Rev. Fluid Mech. Vol. 24, 1989, pp.395-457.
(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.245-302.
(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. 173-205.
(36)
Wickerhauser, M.V., "Acoustic Signal Compression with Wave Packets," 1989. Available by anonymous ftp.
Red Button Acoustic Signal Compression with Wave Packets.
(37)
Kaiser,G., A Friendly Guide to Wavelets, Birkhauser, Boston, 1994.
(38)
John R. Smith and Shih-Fu Chang, "Frequency and Spatially Adaptive Wavelet Packets," Proceedings of the I.E.E.E. International Conference on Acoustics, Speech and Signal Processing (ICASSP-95), Birkhauser, Boston, 1994.
Red Button Frequency and Spatially Adaptive Wavelet Packets.
(39)
Jean-Luc Starck and Fionn Murtagh and Albert Bijaoui, "Image Restoration with Denoising Using Multi-Resolution," (unpublished?).
Red Button Image Restoration with Denoising.
(40)
Scargle, J. et al., "The Quasi-Periodic Oscillations and Very Low Frequency Noise of Scorpius X-1 as Transient Chaos: A Dripping Handrail?,"The Astrophysical Journal, Vol. 411, 1993, L91-L94.
(41)
J. Bradley, C. Brislawn, and T. Hopper, "The FBI Wavelet/Scalar Quantization Standard for Gray-scale Fingerprint Image Compression" Tech. Report LA-UR-93-1659 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



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