Quaternion Fourier Transforms for Signal and Image Processing

FOCUS Series in Digital Signal and Image Processing

Quaternion Fourier Transforms for Signal and Image Processing

Todd A. Ell, UTC Aerospace Systems, Burnsville, MN, USA
Nicolas Le Bihan, CNRS - GIPSA-Lab, Grenoble, France
Stephen J. Sangwine, University of Essex, Colchester, UK

ISBN : 9781848214781

Publication Date : May 2014

Hardcover 160 pp

75.00 USD



This book presents the state of the art, together with the most recent research results, in the use of Quaternion Fourier Transforms (QFT) for the processing of color images and complex valued signals. It is based on the work of the authors in this area since the 1990s and presents the mathematical concepts, computational issues and applications on images and signals. The book, together with the MATLAB toolbox developed by two of the authors (QTFM, http://qtfm.sourceforge.net/), allows the reader to make use of the presented concepts and experiment with them in practice through the examples provided in the book.
Following the Introduction, Chapter 1 introduces the quaternion algebra H and presents some properties which will be of use in the subsequent chapters. Chapter 2 gives an overview of the geometric transformations which can be represented using quaternions. Chapter 3 provides the definition and properties of QFT. The signals and images considered are those with vector-valued samples/pixels.
The fourth and final chapter is dedicated to the illustration of the use of QFT to process color images and complex improper signals. The concepts presented in this chapter are illustrated on simulated and real images and signals.


1. Quaternion Algebra.
2. Geometric Applications.
3. Quaternion Fourier Transforms.
4. Signal and Image Processing.

About the authors/editors

Todd A. Ell is an Engineering Fellow at UTC Aerospace Systems, Burnsville, MN, USA. He is also a Visiting Fellow at the University of Essex, Colchester, UK. His interests include the study and application of hypercomplex algebras to dynamic systems analysis.
Nicolas Le Bihan is currently a Chargé de Recherche at the CNRS working in the Department of Images & Signals (DIS) at the GIPSA-Lab in Grenoble, France. His research interests include polarized signal processing, statistical signal processing on groups and manifolds, geometric (Berry) phases, random processes on non-commutative algebraic structures and applications in physics and geophysics.
Stephen J. Sangwine is a Senior Lecturer with the School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK. His interests include linear vector filtering and transforms of vector signals and images, color image processing, and digital hardware design.