The main emphasis of this book is the development of algorithms for processing multi-dimensional digital signals, and particularly, algorithms for multi-dimensional Fourier transforms in a form that is convenient for writing highly efficient code on a variety of vector and parallel computers. The rapidly increasing power of computing chips, the increased availability of vector and array processors, and the increasing size of the data sets to be analyzed make writing code that takes all the algorithmic possibilities into account and matches these to the target architecture a difficult task. By emphasizing the unified basis for the various approaches to multidimensional Fourier transforms, the book also clarifies how to exploit the differences in optimizing implementations. This book will be of interest not only to applied mathematicians and computer scientists, but also to seismologists, high-energy physicists, crystallographers, electrical engineers working on image processing, and others. Topics covered include: tensor products and the fast Fourier transform, one dimensional and multi-dimensional; finite Abelian groups and Fourier transforms; Cooley-Tukey and Good-Thomas algorithms; lines and planes; field algorithms; implementation on RISC and parallel architectures.