# Contents

Wavelet analysis is a rather new, but meanwhile well established, technique for signal and image processing with various applications in other fields. For instance, the famous

JPEG2000 standard for image compression is based upon wavelets.

In this course we will learn the mathematical foundations of wavelet analysis which belong to the field of harmonic analysis. We will motivate wavelet analysis from the shortcomings of Fourier analysis with respect to time frequency representations of signals. Then we will study in detail the properties of the integral wavelet transform. The request for efficient evaluation of the wavelet transform leads to the concept of wavelet bases. Here, we will present the construction of orthogonal and bi-orthogonal wavelet systems. Finally, if time permits, some applications will be discussed: de-noising, image compression, etc.

# Notes

Multilevel representation by integral wavelet transform

Wavelet frames

Tight wavelet frames

Summary of Chapter 3.3

Fast wavelet algorithms

Perfect reconstruction filters

Stationary wavelet transform

Summary of Chapter 3.3.4

Daubechies wavelets

Lemma 3.30

3.6 Graphic iteration

3.7.1 Daubechies wavelets on [0,1]

# Other material

A direct way from integral to discrete WTs (original article)

Defect classification on specular surfaces using wavelets (original article)

# Slides

Scale(Frequency vs. Position on Treble Clef)

Inverse Fourier transform in the L^2 sense

Short time Fourier transform

Filter properties of integral wavelet transform

Example of integral wavelet transform

Multilevel decomposition

Approximation property 1

Approximation property 2

Cone of influence

Integral wavelet transform of a chirp

Distribution of phase space points for wavelet frames

Orthogonal wavelets (constructed in Chap. 3.3.4)

Bi-orthogonal wavelets

Wavelets on the interval

# Problem Sheets

Problem sheet 1 of October 28, 2013

Problem sheet 2 of November 04, 2013

Problem sheet 3 of November 11, 2013

Problem sheet 4 of November 18, 2013

Problem sheet 5 of November 25, 2013

Problem sheet 6 of December 02, 2013

Problem sheet 7 of December 09, 2013

Problem sheet 8 of December 16, 2013

Problem sheet 9 of December 23, 2013

Problem sheet 10 of January 13, 2014

Problem sheet 11 of January 20, 2014

Problem sheet 12 of January 27, 2014

Problem sheet 13 of February 03, 2014

Problem sheet 14 of February 10, 2014

# Course Administration and Mailing List

You can register online for the problem classes using the course administration website. Thereby, you subscribe to the mailing list which you can use to ask question of general interest and that is used to announce organizational issues.

# Literature

- Louis, Maass, Rieder: Wavelets - Theory and Applications, Wiley 1997
- Daubechies: Ten Lectures on Wavelets, SIAM 1993
- Mallat: A Wavelet Tour of Signal Processing, Academic Press 2008
- Stark: Wavelets and Signal Processing, Springer 2005