Overview

The course provides a detailed and mathematically precise introduction to Fourier, Wavelet and multiresolution analysis from a computational point of view. This includes algorithmical aspects, complexity analysis, and exemplary applications relevant to scientific and visual computing.

Goals and Objectives

The course is algorithmically oriented aiming to enable the students to develop principled computational methods for problems related to Fourier, Wavelet and multiresolution analysis.

Required Knowledge

The course will assume solid knowledge (calculus and linear algebra) such as taught in undergraduate mathematics courses or in AMCS 101, 131, and 151.

Assignments and Evaluation

There will be a problem set assigned each week to prepare for the final exam. This homework track is mostly theoretical, but it will include smaller programming tasks along the way. The students may collaborate on the assignments. Grading policy: 100% final exam.

Syllabus

1. Function Spaces and Fourier Series

2. Continuous-time Fourier Transform (CTFT)

3. Laplace Transform and Bromwich Integral

4. Discrete-time Fourier Transform (DTFT)

5. Fast Fourier Transform (FFT) and the Cooley-Tukey FFT Algorithm

6. Rader's FFT Algorithm

7. Chirp Z-transform (CZT) and Bluestein's Algorithm

8. Windowed Fourier Transform (WFT) and Heisenberg’s Uncertainty Principle

9. Discrete Cosine Transform (DCT)

10. Wavelet Functions

11. Haar's Theorem

12. Continuous-time Wavelet Transform (CTWT)

13. Discrete-time Wavelet Transform (DTWT)

14. Mallat's Multiresolution Analysis (MRA)

15. Fast Wavelet Transform (FWT)

Literature

• J. C. Goswami and A. K. Chan

  Fundamentals of Wavelets: Theory, Algorithms, and Applications

  Wiley, 2011

• G. Kaiser

  A Friendly Guide to Wavelets

  Birkhäuser, 2011

• K. P. Ramachandran, K. I. Resmi, and N. G. Soman

  Insight into Wavelets: From Theory to Practice

  PHI, 2010

• D. K. Ruch and P. J. Van Fleet

  Wavelet Theory: An Elementary Approach with Applications

  Wiley, 2009

• E. J. Stollnitz, A. D. DeRose, and D. H. Salesin

  Wavelets for Computer Graphics: Theory and Applications

  Morgan Kaufmann, 1996

Instructor

Prof. Dr. Dominik L. Michels

Assistant

Dr. Dmitry A. Lyakhov

Class Schedule

1:15 PM – 2:45 PM | Sun Wed | 2021-01-23 – 2021-05-18 | Bldg 1, R 2107