An introduction will be given to the
theory of Wavelets and Fourier transforms, and
their applications.
The course will be divided
in two parts.
The first part is an introduction to Fourier
analysis, including the discrete Fourier transform, the fast Fourier
transform and applications such as computerized tomography.
In the second part the focus will be on wavelets
including the Haar wavelets, multi-resolution analysis, Daubechies
wavelets, discrete wavelet transforms and applications to signal
processing and data compression.
AIM
The student will understand the basic theory and is familiar with the
advantages and disadvantage of Fourier and Wavelet transforms. He/she
can read the technical literature in which these transforms play a
crucial role. In practical situations, he/she will know when these
transforms will be useful and he/she can identify the most appropriate
transform. He/she can analyze and solve practical problems using these
transforms and can employ standard software (based on fast Fourier and
fast wavelet transform).
EDUCATIONAL FORM
Mixture of standard lectures, exercise class and computer laboratory.
Location:Buys Ballot Lab, room 071
(BBL071).
Princetonplein 5
(Google maps),
De Uithof, 3584 CC Utrecht
Time: 15 lectures on the Mondays, 10:00-12:45 Period: 3 + 4
February 3, 2014 (week 6) - May 26, 2014 (week 22)
There will be no lecture in the weeks 17 (April 21 [Easter Monday])
and
19 (May 5 [Liberation Day]).
COURSE MATERIAL
Course material will be provided during the course
or can be downloaded from this page (see below).
Fourier Theory
Schedule
The schedule below for lectures to come is from the previous course.
We will roughly follow this schedule in the present course.
Schedule as well as transparencies and handouts will be updated
when updates are available (as a rule, before the lecture).
A printed version of the Lecture Notes will be on sale
from Aes-kwadraat.
For Lecture Notes, Transparencies, etc., see the
Documentation page
(limited access).
Number of Chapters and exercises refer to the Lecture Notes.
Lecture 1. Discussed in class:
Introduction (see transparancies), Chapter 1, up to Paragraph 1.8,
exercise 1.3 of the Lecture Notes plus the exercises on transparencies.
Assignments: Ex. 1.8, 1.13 and 1.16.
Please, read the Paragraphs 1.9 and 1.10 at home.
Lecture 2. Discussed in class:
Chapter 2,
plus the exercises on transparencies.
Assignments: Ex. 2.6 and 2.8; visit the website on `Hearing Fourier Series'
mentioned on the
Fourier Documentation
page (limited access) .
Lecture 3. Discussed in class:
Chapter 3, 3.1-3.16, Chapter 7, 7.1-7.2, Exercises 3.2 and 1.8.
Assignments: Ex. 3.3, 3.4 (Hint: use the results of Ex. 3.2), and 3.9
(for 3.9, it might be helpful to read section 2.6 first).
Lecture 4. Discussed in class: Exercises 3.3, 3.9, 3.11,
Chapter 3.16-3.17.
Section 7.D
Assignments: Ex. 3.16 and 3.18.
Lecture 5. First computer session (see “Computer session
I”
at the end of the Lecture Notes). In room BBL103
Lecture 6. Discussed in class: Chapter 4 and 5.
Assignments: Ex. 5.3 and 5.5 (a)-(d).
The schedule below is from the previous course.
We will roughly follow this schedule in the present course.
Schedule as well as transparencies and handouts will be updated
when updates are available (as a rule, before the lecture).
For Transparencies, exercises, etc., see the
documentation page
(limited access)
Number of an Exercise refer to the an exercise in the exercise set.
Lecture 10. Introduction wavelets.
- We discussed the first 113 transparencies
Assignments: Ex. 1.1 and 1.4
Lecture 11. Wavelets, filters and block Toeplitz matrices.
- We discussed the transparencies 115-246
Assignments: Ex. 3.2 and 3.3
Lecture 12. Splines, approximation quality.
- We discussed the transparencies 247-351
Assignments: Ex. 4.4
Lecture 13. Conditioning, Existence, smoothness, orthogonality.
- We discussed the transparencies 352-436
Assignments: Ex. 5.1 and 5.3 (you may use Riesz' Lemma of Ex. 5.2).
Computer sessions on wavelets.
To do: follow the instructions in the text, make notes and discuss these notes with
Arno or Gerard.
Exercises by Arno Swart based on material from
Jeroen de Waard, Stefan van der Baan, Martijn Pistorius, Sander de Putter
Write a report on one of the following subjects.
You can select a practical subject,
or a pure theoretical one, or a literature study.
Before you start working on one of these subjects,
talk to the teacher first.
One student per report.
Computerized Tomography To do: Write a computer program (either C or Matlab)
and a report on Computerized Tomography. In this report, you should give
theoretical background and discuss the effect of discretizations
(and rounding errors) on the quality of back transform.
Use the questions in the text
(ps.gz)
for inspiration. See also the documentation
page.
There is a small Matlab program, Images.m (tar.gz*), that may help you
to generate image matrices.
Run this program in Matlab 6 (or higher). This program may be helpful
if you are programming in Matlab but also if you program in C or C++.
Files to get started:
Matlab files
(tar.gz*) (for matlab programmers).
This package contains stripped versions of a working package.
For comparison convenience,
pictures as produced by this operational version have been
displayed here.
An introduction JPEG-2000-like
compression using MATLAB and its wavelet toolbox.
To do: Write a report; go to the Wavelet
& FFT resources
page for instructions.
Exercise by Arno Swart
Theoretical subject.
Write a paper on Theorem 2.4 of the Lecture notes on Fourier Theory.
The report should contain a proof (the proof is scattered among a number of
exercises in the Lecture notes [like Ex.2.2--2.24 and Ex.6.17-6.20]),
but also motivations, explanations, examples, background information.
The presentation of the arguments should be well structured
(lemmas, theorems, etc.) and motivated.
Literature study. Write a
reportor design a Matlab session on a subject from applications
in which Fourier theory and/or wavelet transform plays a crucial role.
The application should require some extension to theory as presented in the
course (‘new’ aspects).
The report should be a small appendix to the lecture notes
(5 to 10 pages. Chapter 9 in the Lecture Notes forms an example). In
particular, the notation should match the one of the Lecture Notes. It
should not double results from the lecture notes:
in such a case, a cross-reference to the appropriate result in the Lecture Notes
suffices. Provide the background information from the application as
concise as possible, elaborate on ‘new’ aspects form
Fourier Theory/ wavelets. The focus should be on
subjects from the course. If the discussion on certain aspects
is going to be to technical, consider
to put these details in an exercise (separate from the report,
also hand in the solution to
these exercises).
For the Matlab session, provide some background material, write
the exercises (questions), and the Matlab code (on a separate paper,
give the solutions).
Note. One author per report, but here is a possibility for
a joint project. One person can work on the report, another can design
a Matlab session that matches the report.
Examples.
MRI
MP3 compression
Signal analysis in Geophysics
...
It is appreciated if you come up with a suggestion
by yourself.
