Engineering Applications of the Motion-Group Fourier Transform

Summary

We review a number of engineering problems that can be posed or solved using Fourier transforms for the groups of rigid-body motions of the plane or three-dimensional space. Mathematically and computationally these problems can be divided into two classes: (1) physical problems that are described as degenerate diffusions on motion groups; (2) enumeration problems in which fast Fourier transforms are used to efficiently compute motion-group convolutions. We examine engineering problems including the analysis of noise in optical communication systems, the allowable positions and orientations reachable with a robot arm, and the statistical mechanics of polymer chains. In all of these cases, concepts from noncommutative harmonic analysis are put to use in addressing real-world problems, thus rendering them tractable.

1. Introduction

Noncommutative harmonic analysis is a beautiful and powerful area of pure mathematics that has connections to analysis, algebra, geometry, and the theory of algorithms. Unfortunately, it is also an area that is almost unknown to engineers. In our research group, we have addressed a number of seemingly intractable “real-world” engineering problems that are easily modeled and/or solved using techniques of noncommutative harmonic analysis. In particular, we have addressed physical/mechanical problems that are described well as functions or processes on the rotation and rigid-body-motion groups. The interactions and evolution of these functions are described using group-theoretic convolutions and diffusion equations, respectively. In this paper we provide a survey of some of these applications and show how computational harmonic analysis on motion groups is used.