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Institutes
Institute for Applied and Numerical Mathematics
Research Group 3: Scientific Computing
Wavelets

Wavelets (Winter Semester 2013/14)

Schedule
Lecture: Tuesday 9:45-11:15 1C-04
Wednesday 11:30-13:00 1C-03
Problem class: Thursday 15:45-17:15 Z 2
Lecturers
Lecturer Prof. Dr. Andreas Rieder
Office hours: Until further notice only on appointment.
Room 3.040 Kollegiengebäude Mathematik (20.30)
Email: andreas.rieder(at)kit.edu
Problem classes Dr. Tim Kreutzmann
Office hours:
Room Kollegiengebäude Mathematik (20.30)
Email:

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.
barb_matlab.png
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