AMCS 237: Fourier and Wavelet Theory (Fall 2023)
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

Function Spaces and Fourier Series

Continuoustime Fourier Transform (CTFT)

Laplace Transform and Bromwich Integral

Discretetime Fourier Transform (DTFT)

Fast Fourier Transform (FFT) and the CooleyTukey FFT Algorithm

Rader's FFT Algorithm

Chirp Ztransform (CZT) and Bluestein's Algorithm

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

Discrete Cosine Transform (DCT)

Wavelet Functions

Haar's Theorem

Continuoustime Wavelet Transform (CTWT)

Discretetime Wavelet Transform (DTWT)

Mallat's Multiresolution Analysis (MRA)

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
8:30 AM – 10:00 AM  Sun Wed  20230827 – 20231206  Bldg 1, R 2107