A WAVELET TOUR
OF SIGNAL PROCESSING
BY STÉPHANE MALLAT

Academic Press, 1998

A Wavelet Tutorial
Adapted for the Web by F. CHAPLAIS

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.

Frequency Analysis

From dyadic wavelets to filters

From filters to dyadic wavelets

Regularity Analysis

Frames

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