Quaternion Fourier Transforms for Signal and Image Processing E-Book

Todd A. Ell

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 207

Veröffentlichungsjahr: 2014

Beschreibung

Based on updates to signal and image processing technology made in the last two decades, this text examines the most recent research results pertaining to Quaternion Fourier Transforms. QFT is a central component of processing color images and complex valued signals. The book's attention to mathematical concepts, imaging applications, and Matlab compatibility render it an irreplaceable resource for students, scientists, researchers, and engineers.

Leseprobe

Contents

Nomenclature

Preface

Introduction

I.1. Fourier analysis

I.2. Quaternions

I.3. Quaternion Fourier transforms

I.4. Signal and image processing

I.5. Other hypercomplex algebras

I.6. Practical application

I.7. Overview of the remaining chapters

1 Quaternion Algebra

1.1. Definitions

1.2. Properties

1.3. Exponential and logarithm of a quaternion

1.4. Representations

1.5. Powers of a quaternion

1.6. Subfields

2 Geometric Applications

2.1. Euclidean geometry (3D and 4D)

2.2. Spherical geometry

2.3. Projective space (3D)

3 Quaternion Fourier Transforms

3.1. 1D quaternion Fourier transforms

3.2. 2D quaternion Fourier transforms

3.3. Computational aspects

4 Signal And Image Processing

4.1. Generalized convolution

4.2. Generalized correlation

4.3. Instantaneous phase and amplitude of complex signals

Bibliography

Index

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK

www.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.wiley.com

© ISTE Ltd 2014The rights of Todd A. Ell, Nicolas Le Bihan and Stephen J. Sangwine to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2014934161

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISSN 2051-2481 (Print)ISSN 2051-249X (Online)ISBN 978-1-84821-478-1

Nomenclature

NOTE.– The complex root of –1 which is usually denoted i, or, in engineering texts j, is denoted throughout this book by a capital letter I, in order to avoid any confusion with the first of the three quaternion roots of –1, all three of which are denoted throughout in bold font like this: i, j, k.

Preface

This book aims to present the state of the art, together with the most recent research results in the use of quaternion Fourier transforms (QFTs) 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 some applications to signals and images. The book, together with the MATLAB® toolbox developed by the authors, [SAN 13b] allows the readers to make use of the presented concepts and experiment with them in practice through the examples provided.

Todd A. ELLMinneapolis, MN, USA

Nicolas le BIHANMelbourne, Australia

Stephen J. SANGWINEColchester, UK

April 2014

Introduction

This book covers a topic that combines two branches of mathematical theory to provide practical tools for the analysis and processing of signals (or images) with three- or four-dimensional samples (or pixels). The two branches of mathematics are not recent developments, but their combination has occurred only within the last 25–30 years, and mostly since just before the millennium.

I.1. Fourier analysis

Fourier analysis was, in 1822, with Joseph Fourier’s development of techniques, the first to analyze mathematical functions into sinusoidal components. In signal and image processing, Fourier’s ideas underpin the two fundamental representations of a signal: one in the time (or image) domain where the signal (or image) is represented by samples (or pixels) with amplitudes and the other in the frequency domain where the signal (or image) is represented by sinusoidal frequency components, each with an amplitude and a phase. Mathematically, these concepts are not limited to time and frequency: one can use Fourier analysis on a function of any variable, resulting in a representation in terms of sinusoidal functions of that variable. However, this book is concerned with signal and image processing, and we will therefore use the terms time and frequency rather than more general concepts. It should be understood throughout that when we talk of images, the concept of time is replaced by the two spatial coordinates that define pixel position within an image.

Today, Fourier analysis is classically taught to mathematicians, scientists and engineers in several related ways, each applicable to a specific subset of mathematical functions or signals: