### Academic Press, 1998

## A Wavelet Tutorial

Adapted for the Web by F.
CHAPLAIS

###

for those who hate
preambles### Warning

This presentation is inspired from S.G. Mallat's book and does not
pretend to reflect it exactly. It is concerned with the following
topics:

- Fourier analysis (chapter 2)
- time-frequency analysis (chapter 4, except for the
quadratic energy distributions))
- frames (chapter 5)
- singularity analysis and reconstruction (chapter 6 except
for the multifractals)
- wavelet bases and filter banks (chapter 7)

The following topics from the book are not covered here:

- chapter 3 about discrete signals (except for the FFT and
convolutions algorithms, which are briefly described)
- chapter 8 on wavelet packets and local cosine bases
- chapter 9 on approximation
- chapter 10 about estimation (which is being revised by S.G.
Mallat for the French edition and the second US edition)
- chapter 11 on compression and coding (hope to do it
someday)

### Proposed Tours

Four tours are proposed, corresponding to four
different topics. These tours are linked to each other sequentially.
Many links allow navigation from one to topic to another for a
nonlinear browsing.

### Index

For direct access, here is a list of links that point to the main
topics:

Fourier
Transform

Instantaneous
Frequency of an Analytic Signal

Time-Frequency
Localization

Windowed
Fourier Transform and Wavelet Transform

Frames and
Riesz Bases

Windowed
Fourier Frames and Wavelet Frames

Multiresolution
Approximations

Wavelet
Bases

Filter
Banks

Regularity
Analysis of a Signal

Detection
of Singularities

Reconstruction
from Dyadic Maxima

Edge
Detection and Image Reconstruction

Algorithms:
Fast
Fourier Transforms and Convolutions

Fast
Windowed Fourier Transform

Fast
Wavelet Transform

Fast
Dyadic Wavelet Transform

Decomposition
and Reconstruction over Orthonormal Wavelet Bases

### Numerical Computations

All numerical figures in the book have been computed using
Wavelab, a freeware Matlab Toolbox, available at

http://www-stat.stanford.edu/~wavelab/

Uvi Wave is another
freeware Matlab Toolbox.

I have just finished a Matlab Wavelet Tutorial.

###

### Updates:

June 4, 1999: Full Strang
and Fix conditions added, with proof!

May 18, 2001:
added mathematical transition
from filter banks to multiresolution analysis

January 2008: improved previous connection between discrete and continuous wavelets

Last update: January 31, 2008

Feedback is welcome.