Group Theoretical Methods in Image Processing

Abstract

In this chapter we will introduce the concept of a representation of a group. The methods introduced here are of fundamental importance in the study of symmetries and they have been applied in such different fields as quantum mechanics and the theory of special functions. Apart from these "applications" the theory is also useful for the study of abstract groups since these abstract groups are linked via representations to well-known matrix groups. In the next section we will first introduce the algebraic concepts involved and then we will specialize the concepts to the case where the groups also have a topological structure. 3.1 Algebraic Representations Definition 3.1 Let G be any group, X a complex vector space and denote by GL(X) the group of invertible, linear mappings that carry X into itself. 1. Then we define a representation as a homomorphism T : G -+ GL(X); g ~-~ T(g). 2. X is called the representation space. 3. T(g) is called a representation operator. We call a representation in the sense of definition 3.1 an algebraic representation in con-trast to the continuous representations to be introduced in the next section 3.12 (see also [37] page 151). In the definition of continuous representations we will mainly re-quire that the operators T are continuous, a property which is of course coupled to the topological properties of the group and the vector space. The vector space is finite-dimensional in almost all applications of interest to us. In this case we define: Definition 3.2 1. If the representation space X is a finite-dimensional vector-space