OF SIGNAL PROCESSING

BY STÉPHANE MALLAT

Adapted for the Web by F. CHAPLAIS

for those who hate preambles

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)

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.

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

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.

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.