Wavelets

Course manual 2023/2024

Course content

Theoretical aspects: The course contains an abstract mathematical introduction on the topic of wavelets and their general construction using the concept of multiresolution analysis. Based on two concrete examples, being the classical Haar wavelet and the less know and somewhat less trivial Stroemberg wavelet, the more abstract concept of multiresolution analysis is introduced, and together with it the scaling function. It is shown how wavelets can be constructed from the multiresolution analysis, and hence from the scaling function. Apart from revisiting the Haar- and Stroemberg wavelets, this also leads to several types of wavelets, such as periodic wavelets, Meyer's wavelets, spline wavelets, and Daubechies' compactly supported wavelets. Special attention is paid to their decay, their vanishing moments and their smoothness.

Practical aspects:  Apart form the theoretical aspects, we will pay attention to applications of wavelets in both linear and nonlinear approximation theory and signal processing. In particular we discuss the wavelet transform, the discrete wavelet transform and the fast discrete wavelet transform, and their implementation.

Historical aspects: We will also spend some time on the historical development and the status of wavelet research in mathematics, highlighted by the Abel Prize awarded to Yves Meyer in 2017, Ingrid Daubechies' membership of the Royal Netherlands Academy of Arts and Sciences, and the Fast Fourier Transform being named in the Top 10 algorithms in the 20th century.

Context:  The vast topic of Wavelets and Multiresolution Analysis is situated both in abstract and in applied analysis. Their existence and mathematical properties are worth studying on their own account, nonetheless they find applications in many fields, particularly in signal processing (wavelets were used in the recent detection of gravitational waves). Wavelets are well-suited to analyze irregular and non-smooth signals (such as from geophysics, finance, etc).  Wavelets are also used in ingenious approximation methods for solutions of PDEs. 

Study materials

Literature

• A Mathematical Introduction to Wavelets, by P. Wojtaszczyk. London Mathematical Society. Student Texts 37.

• A First Course in Wavelets with Fourier Analysis, by Albert Boggess

• Semiclassical Analysis, byM. Zworski

• Introduction to the Theory of Distributions, by Friedlander and Joshi

Software

• Matlab, Python

Other

• Hand-outs by the lecturer, slides.

Objectives

• student understands the definition and properties of Schwartz functions and tempered distributions
• student knows and understands the definition of Fourier transform on tempered distributions
• student knows and understands the definition of wavelet and some basic examples of wavelets
• student can prove basic properties of the Haar wavelet and other types of wavelets
• student knows and understands the definition of a multiresolution analysis
• student understands the role of Fourier transforms in the analysis of wavelets
• student knows and understands the definition of scaling function
• student understands a typical construction procedure for wavelets
• student understands the relevance of wavelet properties as decay, vanishing moments, smoothness
• student understands the concept of best N-term approximation and the role of wavelets therein
• student knows and understands the wavelet transform and its fast version
• student has seen applications of wavelets in numerical analysis and signal processing

Teaching methods

• Lecture
• Self-study
• Computer lab session/practical training
• Presentation/symposium

The course is a lecture course. However, in weeks 4 and 12, students are asked to work on assignments that they should hand in.

Learning activities

Attendance

This programme does not have requirements concerning attendance (TER-B).

Assessment

The final individual exam counts for 70%, whereas each of the assessments counts for 15%. The score for the individual exam should be at least 5 out of 10. In the event of a resit, the results for the assessments will still be valid, with the same weights. In particular, you can only resit the exam. There is no possibility to replace an assessment. The minimum score for each assessment and for the exam is 1 out of 10.

Inspection of assessed work

Inspection of assessed work is possible on appointment with the lecturer.

Assignments

Assignment 1 will be due on Oct 20th. It will be 5 questions. This will be 15% of your grade.

Assignment 2 will be due on Dec 10th. It will have 4 questions. This will be 15% of your grade.

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Timetable

The schedule for this course is published on DataNose.

Contact information

Coordinator

• L.L. Tzou