Group theoretical methods and their applications
In the simplest form the Fourier transform decomposes a function on the circle into a linear combination of sin/cos functions. Two natural questions are the following: (A) What is so special about the trigonometric functions and (B) what is the corresponding Fourier transform on other domains like the line, the sphere or the cone?
A modern interpretation of Fourier analysis shows that the answers to both questions can be found in the symmetry properties of the domains on which these functions are defined.
Reiner Lenz / Michael Felsberg / Klas Norberg
The course is open to students enrolled in a Ph.D. program at
Linkoping University/ISY. External participants are welcome in
accordance to the usual LIU practice.
If you have interest to participate, please register by sending an email to the instructor
PrerequisitesLinear Algebra is a must and Fourier transforms and PDE's a bonus
Course outline (tentative)
In the course we will introduce the basic concepts from group theory needed to model these symmetries and we will use them to describe the corresponding generalizations of the Fourier transform.
Two results that can be derived in this framework are (A) the construction of invariants/covariants and (B) a characterization of principal component analysis of stochastic processes which are stationary in a group-theoretical sense.
Groups that are relevant to signal processing and computer vision include (ordered in increasing complexity)
- Finite groups like permutations and transformations on grids
- The 2-D rotation group
- Translation groups
- The 3-D rotation group
- The Lorentz-groups
The Weyl-Heisenberg Group
Topics covered are:
- Basic facts from group theory
- Group representations
- Linear operators with symmetry properties (Schur’s Lemma)
- Basic properties of Lie groups and Lie algebras
- Illustration of the usage of group characters
The content of the course will also, to some degree, depend on the background and the interests of the participants
The course will consist of a series of lectures and project work
The mathematical content of the course is contained in:
Fässler & Stiefel: Group theoretical methods and their applications http://link.springer.com/book/10.1007/978-1-4612-0395-7/page/1
Selected parts of this book will be complemented with articles that describe applications of the theory to problems from signal processing and computer vision
Last updated: 2014-03-18