​​

​​​​AMCS 394: Fourier and Wavelet Theory (Spring 2018)

Overview

The course provides are 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. This homework track is mostly theoretical, but it will include smaller programming tasks along the way. The students may collaborate on the assignments provided each student writes up his or her own solutions and clearly lists the names of all the students in the group (grading policy: 50% homework assignments and 50% 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, M.Sc.

Assistant

Dr. Dmitry A. Lyakhov, M.Sc.

Class Schedule

TBA