Group Algebras in Signal and Image Processing

Abstract

The linear and cyclic convolutions are computed as polynomial products. Group algebras are generalizations of polynomials and polynomial arithmetic based on algebraic groups. The multiplication in group algebra is called “group convolution.” The selection of coefficients from the cyclic convolution to generate output for the linear convolution can be extended to other functions called “value functions,” to allow for a wider variety of outputs. Group convolution of a signal requires selecting a single base group to work with. For images, group convolution is naturally based on the direct product of groups. Matrix representation of group algebra elements is discussed. The chapter discusses direct product group algebras. The ideas extend to convolution of data sets whose dimension is larger than two. Semidirect product group algebras that are used mainly in studying group algebras based on dihedral groups are discussed. The chapter also discusses inverting group algebra elements. Two methods for inverting elements are explained. In applications where nonzero and noninvertible group algebra elements may be a problem, a way to eliminate these elements via ideals and quotient rings is presented. Matrices whose elements are from group algebra are discussed. The chapter focuses on those matrices that have the convolution property—that is, those matrices for which the transform of a convolution is the product of the individual transforms. The chapter presents examples of the application of group algebras to edge detection in signals and images. The examples illustrate the use of direct product cyclic groups (the standard convolution case), direct product dihedral groups, and direct product quaternions. The chapter describes the process to implement group algebras in an object-oriented programming language.