Introduction to wavelets
Wavelet theory is a part of time frequency analysis which allows the study of the local properties of functions by convolving them with shrinking windows and thus describing the structure of functions on various scales, such as fractals for example.
Nowadays, wavelets are widely used in applied sciences (signal and image analysis, data compression, filter banks) as well as in “pure” disciplines (singular operators, functional spaces).
The goal of the course is to give an introduction to the subject in order to provide the students with tools for conducting their own research using the wavelet techniques.
Nowadays, wavelets are widely used in applied sciences (signal and image analysis, data compression, filter banks) as well as in “pure” disciplines (singular operators, functional spaces).
The goal of the course is to give an introduction to the subject in order to provide the students with tools for conducting their own research using the wavelet techniques.
Lecturer
Date
19th September ~ 14th December, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 09:50 - 11:25 | A3-2-303 | ZOOM 07 | 559 700 6085 | BIMSA |
Prerequisite
Basics of Fourier analysis and Hilbert spaces, however I’ll remind the main notions and facts as soon as needed.
Syllabus
Introduction
- Basic facts from Fourier analysis
- Uncertainty principles
- Ideas of time frequency analysis
- Various types of time-frequency representations
Continuous wavelets transform
- Definition. Reconstruction formula, energy preservation
- Hilbert spaces with reproducing kernel
- Digression: Space of bandlimited signals, Shannon sampling
- Lipschitz regularity
- Local regularity measurements via the Wavelet transform
- Detection of singularities
- Wavelet maxima for images
- Self-similarity. Multifractals.
Discrete wavelets
- Motivation, examples
- Multiresolutional analysis
- Conjugate mirror filters
- Construction of wavelets
- Choosing the filter
- Compactly supported wavelets
- 2D wavelets, image compression
- Wavelets on an interval
Functional spaces
- Survey: Functional spaces, Unconditional basis
- Survey: Tools from real analysis: maximal functions, Marzinkevich interpolation theorem, Calderon- Zygmund operators, Khinchin inequality ( I plan to formulate the main results without proofs, yet if time permitted more details will be given).
- Wavelet basis in L^p spaces
- Wavelet basis in Sobolev spaces
- Basic facts from Fourier analysis
- Uncertainty principles
- Ideas of time frequency analysis
- Various types of time-frequency representations
Continuous wavelets transform
- Definition. Reconstruction formula, energy preservation
- Hilbert spaces with reproducing kernel
- Digression: Space of bandlimited signals, Shannon sampling
- Lipschitz regularity
- Local regularity measurements via the Wavelet transform
- Detection of singularities
- Wavelet maxima for images
- Self-similarity. Multifractals.
Discrete wavelets
- Motivation, examples
- Multiresolutional analysis
- Conjugate mirror filters
- Construction of wavelets
- Choosing the filter
- Compactly supported wavelets
- 2D wavelets, image compression
- Wavelets on an interval
Functional spaces
- Survey: Functional spaces, Unconditional basis
- Survey: Tools from real analysis: maximal functions, Marzinkevich interpolation theorem, Calderon- Zygmund operators, Khinchin inequality ( I plan to formulate the main results without proofs, yet if time permitted more details will be given).
- Wavelet basis in L^p spaces
- Wavelet basis in Sobolev spaces
Reference
We will not follow one textbook. However, the main textbooks are:
1. Mallat, Stéphane, A wavelet tour of signal processing. The sparse way. Elsevier/Academic Press, Amsterdam, 2009.
2. Daubechies, Ingrid, Ten lectures on wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
3. Meyer, Yves, Wavelets and operators. Cambridge Studies in Advanced Mathematics, 37. Cambridge University Press, Cambridge, 1992.
1. Mallat, Stéphane, A wavelet tour of signal processing. The sparse way. Elsevier/Academic Press, Amsterdam, 2009.
2. Daubechies, Ingrid, Ten lectures on wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
3. Meyer, Yves, Wavelets and operators. Cambridge Studies in Advanced Mathematics, 37. Cambridge University Press, Cambridge, 1992.
Audience
Advanced Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Yurii Lyubarskii received PhD degree in mathematics in 1974 in the Institute for Low Temperature Physics and Engineering, Ukrainian Academy of Sciences and Doctor degree in St. Petersburg branch of Russian mathematical institute in 1990. He taught at the Norwegian University of Science and Technology and also at St. Petersburg State University. Scientific interests of Yu. Lyubarskii include complex and harmonic analysis and applications to the signal analysis.