Course syllabus for Wavelet analysis

The course syllabus contains changes
See changes

Course syllabus adopted 2020-03-12 by Head of Programme (or corresponding).

Overview

  • Swedish nameWaveletanalys
  • CodeTMA462
  • Credits7.5 Credits
  • OwnerMPENM
  • Education cycleSecond-cycle
  • Main field of studyMathematics
  • DepartmentMATHEMATICAL SCIENCES
  • GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail

Course round 1

The course round is cancelled. This course round is planned to be given every other year. For further questions, please contact the director of studies
  • Teaching language English
  • Application code 20129
  • Open for exchange studentsYes

Credit distribution

0101 Examination 7.5 c
Grading: TH		 7.5 c

In programmes

Examiner

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Eligibility

General entry requirements for Master's level (second cycle)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Specific entry requirements

English 6 (or by other approved means with the equivalent proficiency level)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Course specific prerequisites

Basic Fourier analysis

Aim

Fourier analysis (frequency analysis) is an indispensable tool for deterministic and statistical signal analysis and processing (and in the theory of partial differential equations). Presently, the so-called wavelet transform is widely used as a complement to traditional Fourier transforms. The aim of the course is, in part, to describel how these transforms are used in practice, e.g., for 'sampling' of signals, in antenna theory, in geometrical optics, in computer tomography, in probability theory, and also the 'fast' transforms which are now executed by computers in this context, for example in image processing.

Learning outcomes (after completion of the course the student should be able to)

Identify problems that can be solved using discrete Fourier and wavelet trasforms and to pick up the relevant type of transform for the particular model problem and data set.

Sample signals and perform image compression and image processing writing and implementing Fourier/wavelet codes.

Read and pick up adequate information from the research papers in the field.


Content

The course has three main ingredients: Discrete (and continuous) Fourier transforms, distribution theory/generalized functions and wavelet transforms. The transition between these parts are smooth and well motivated. In this regard the concept of fix point of Fourier transforms and Fourier transform of tempered distributions are linked, the signal sampling links dicrete Fourier to Haar wavelets.

We introduce multi resolution analysis, construct the wavelet and scaling functions, and study dual and bi-orthogonal wavelet bases

Some advanced Fourier related transforms in higher dimensions (as Hankel, Abel. Hilbert and Radon transforms) as well as higher (two in our case) dimensional wavelets are also coved.

 

Organisation

The Class meets 4 times/week (8 hourse) in 7 weeks. 5 hourse/week is devoted covering the theory and
3 hours/week is for exercises. TA has office hours for computer support avialable weeks 2-7.

As my office hours: You may drop in any time netween 11 and 13.

Literature

All electronic and available through the course web page:

1. Bergh, Notes on Generalized Functions and Fourier Transforms, compendium, Chalmers and Göteborg University.

2. Problems from Bergh, Ekstedt, Lindberg: Wavelets , Studentlitteratur. The book is not available to purchase, but it is posted in the course site.

  • 3. Bracewell The Fourier Transform and its Applications . Parts of chapters 1-14 are included in the course. However, lecture notes will be made available, and the book is not essential for the course.

  • 4. My Primary Lecture notes (5  chapter 86 pp) can be downloaded from the course site.  

    • Examination including compulsory elements

    Written examination and exercises handed in, partly based on computer calculations.

    The course syllabus contains changes

    • Changes to course rounds:
      • 2020-03-12: Cancelled Changed to cancelled by UOL
        [Course round 1] Cancelled