Applied and Numerical Harmonic Analysis

Series Editor John J. Benedetto University of Maryland

Editorial Advisory Board

Akram Aldroubi NIH, Biomedical Engineering/ Instrumentation

Ingrid Daubechies Princeton University

Christopher Heil Georgia Institute of Technology

James McClellan Georgia Institute of Technology

Michael Unser NIH, Biomedical Engineering/ Instrumentation

Victor Wickerhauser Washington University

Douglas Cochran Arizona State University

Hans G. Feichtinger University of Vienne

Murat Kunt Swiss Federal Institute of Technology, Lausanne

Wim Sweldens Lucent Technologies Bell Laboratories

Martin Vetterli Swiss Federal Institute of Technology, Lausanne

The Fourier Transform in Biomedical Engineering

Edited by

Terry M. Peters and Jackie Williams

With contributions from fason H. T. Bates

G. Bruce Pike Patrice Munger

Springer Science+Business Media, LLC

Terry M. Peters Advanced Imaging Research Group lP. Roberts Research Institute London ON N6A-5K8

Jackie Williams Department of Ophthalmology University ofWestern Ontario London ON N6A-5A5

Library of Congress Cataloging-in-Publication Data

The Fourier transform in biomedical engineering / Terry M. Peters, Jackie C. Williams (eds.): with contributions from Jason H.T. Bates, G. Bruce Pike, Patrice Munger.

p. cm. - (Applied and numerical harmonic analysis) Inc1udes bibliographical references and index. ISBN 978-1-4612-6849-9 ISBN 978-1-4612-0637-8 (eBook) DOI 10.1007/978-1-4612-0637-8 1. Fourier transformations. 2. Biomedical engineering-Mathematics.

1. Peters, T. M. II. Williams, Jackie c., 1954-III. Bates, Jason H. T. IV. Series.

[DNLM: 1. Fourier Analysi. 2. Biomedical Engineering. QA 403.5 F7751997] R857.F68F68 1998 6IO'.28-dc21 DNLM/DLC for Library of Congress

Printed on acid-free paper

© 1998 Springer Science+Business Media New York Originally published by Birkhăuser Boston in 1998 Softcover reprint of the hardcover 1 st edition 1998

Copyright is not claimed for works of U.S. Government employees.

98-30101 CIP

All rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner.

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ISBN 978-1-4612-6849-9 Typeset by Alden Bookset, Oxford, England

987 6 543 2 1

Contents List of Contributors . Dedication . . . . . . . . . Foreword by Robert Kearney Preface .......... .

1 Introduction to the Fourier Transform 1.1 Introduction....... . . . 1.2 Basic Functions . . . . . . . . 1.3 Sines, Cosines and Composite waves. 1.4 Orthogonality . . . . . . . . . . .

IX

Xl

X1l1

XV

1 1

2 3 4

1.5 Waves in time and space . . . . . . 7 1.6 Complex numbers. A Mathematical Tool 7 1.7 The Fourier transform . . . . . . . . . 11 1.8 Fourier transforms in the physical world: The Lens

as an FT computer. . . . 16 1.9 Blurring and convolution. 19

1.9.1 Blurring...... 19 1.9.2 Convolution. . . . 20

1.10 The "Point" or "Impulse" response function. 22 1.11 Band-limited functions 23 1.12 Summary. . . 23 1.13 Bibliography. . . . . 24

2 The I-D Fourier Transform 2.1 Introduction..... 2.2 Re-visiting the Fourier transform 2.3 The Sampling Theorem. 2.4 Aliasing ........... .

25 25 29 35 36

vi Contents

2.5 Convolution . 38 2.6 Digital Filtering 41 2.7 The Power Spectrum . 43 2.8 Deconvolution . 47 2.9 System Identification . 49 2.10 Summary. 51 2.11 Bibliography. 52

3 The 2-D Fourier Transform 53 3.1 Introduction . 53 3.2 Linear space-invariant systems in two dimensions 54 3.3 Ideal systems . 56 3.4 A simple X-ray imaging system . 59 3.5 Modulation Transfer Function (MTF). 65 3.6 Image processing . 70 3.7 Tomography . 73 3.8 Computed Tomography. 78 3.9 Summary. 87 3.10 Bibliography . 88

4 The Fourier Transform in Magnetic Resonance Imaging 89 4.1 Introduction . 89 4.2 The 2-D Fourier transform . 91 4.3 Magnetic Resonance Imaging . 91

4.3.1 Nuclear Magnetic Resonance. 91 4.3.2 Excitation, Evolution, and Detection . 95 4.3.3 The Received Signal: FIDs and Echos 97

4.4 MRI. 98 4.4.1 Localization: Magnetic Field Gradients. 98 4.4.2 The MRI Signal Equation . 100 4.4.3 2-D Spin-Warp Imaging. 103 4.4.4 Fourier Sampling: Resolution, Field-of-View,

and Aliasing . 106 4.4.5 2-D Multi-slice and 3-D Spin Warp Imaging 109 4.4.6 Alternate k-space Sampling Strategies. 113

4.5 Magnetic Resonance Spectroscopic Imaging . 118 4.5.1 Nuclear Magnetic Resonance Spectroscopy:

I-D. 118

Contents vii

4.5.2 Magnetic Resonance Spectroscopic Imaging: 2-D, 3-D, and 4-D . 119

4.6 Motion in MRI. 123 4.6.1 Phase Contrast Velocity Imaging. 124 4.6.2 Phase Contrast Angiography . 126

4.7 Conclusion. 127 4.8 Bibliography . 128

5 The Wavelet Transform 129 5.1 Introduction . 129

5.1.1 Frequency analysis: Fourier transform 130 5.2 Time-Frequencyanalysis . 131

5.2.1 Generalities . 131 5.2.2. How does time-frequency analysis work? 133 5.2.3 Windowed Fourier transform. 135 5.2.4 Wavelet transform . 140

5.3 Multiresolution Analysis 143 5.3.1 Scaling Functions 144 5.3.2 Definition . 148 5.3.3 Scaling Relation . 151 5.3.4 Relationship of mu1tiresolution analysis to

wavelets. 154 5.3.5 Multiresolution signal decomposition. 158 5.3.6 Digital filter interpretation . 160 5.3.7 Fast Wavelet Transform Algorithm. 164 5.3.8 Multidimensional Wavelet Transforms 164 5.3.9 Fourier vs. Wavelet Digital Signal Processing 169

5.4 Applications . 171 5.4.1 Image Compression 171 5.4.2 Irregular heart beat detection from EKG

signals. 172 5.5 Summary. 173 5.6 Bibliography . 173

6 The Discrete Fourier Transform and Fast Fourier Transform 174 6.1 Introduction . 174 6.2 From Continuous to Discrete. 174

viii Contents

6.2.1 The comb function . 175 6.2.2 Sampling 177 6.2.3 Interpreting DFT data in a cyclic buffer. 179

6.3 The Discrete Fourier Transform. 180 6.4 The Fast Fourier Transform 182

6.4.1 The DFT as a matrix equation . 184 6.4.2 Simplifying the transition matrix . 184 6.4.3 Signal-flow-graph notation. 186 6.4.4 The DFT expressed as a signal flow graph 186 6.4.5 Speed advantages of the FFT. 187

6.5 Caveats to using the DFTjFFT . 189 6.6 Conclusion. 193 6.7 Bibliography . 193

List of Contributors Terry M. Peters Advanced Imaging Research Group J.P. Roberts Research Institute London ON N6A-5K8

Jackie Williams Department of Ophthalmology University of Western Ontario London ON N6A-5A5

Jason H. T .Bates McGill University Meakins-Christie Laboratories Montreal, Quebec Canada, H2X 2P2

G. Bruce Pike McGill University Montreal Nuerological Institute Montreal, QC Canada H3A-2B4

Patrice Munger McGill University Montreal Nuerological Institute Montreal, QC Canada H3A-2B4

DEDICATION

This book is dedicated to the memory of Professor Richard H T Bates, a remarkable engineering academic whose interests ranged from optical and radio astronomy, to crystallography, computed tomography, and physiological systems. Diverse as these activities were, they were connected by one theme, the Fourier Transform. In fact there was a rumour going around at one time that Richard actually spent part of his life in "Fourier-space".

For 25 years prior to his death in 1991, Richard was Professor of Electrical Engineering at the University of Canterbury, Christch­urch, New Zealand. He was the father of one of the contributors to this book (JHTB), the graduate advisor of another (TMP), and the inspiration to countless scientists and engineers the world over. In addition, the other contributors were graduate students of TMP, so this book it owes its existence to Richard by direct academic descent.

FOREWORD

In 1994, in my role as Technical Program Chair for the 17th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, I solicited proposals for mini-symposia to provide delegates with accessible summaries of important issues in research areas outside their particular specializations. Terry Peters and his colleagues submitted a proposal for a symposium on Fourier Trans­forms and Biomedical Engineering whose goal was "to demystify the Fourier transform and describe its practical application in biomedi­cal situations". This was to be achieved by presenting the concepts in straightforward, physical terms with examples drawn for the parti­cipants work in physiological signal analysis and medical imaging. The mini-symposia proved to be a great success and drew a large and appreciative audience. The only complaint being that the time allocated, 90 minutes, was not adequate to allow the participants to elaborate their ideas adequately. I understand that this feedback helped the authors to develop this book.

In his book on Fourier Analysis, T.W. Korner relates that Joseph Fourier submitted a memoir to the Academy of France in 1807 in which he derived the fundamental equations for heat conductance and solved them using a new expansion based on a trigonometric series. A commission of eminent mathematicians (including Laplace; Laplace; Lagrange and Poisson) examined the work and attacked it on two grounds. First, the mathematical validity of the new "Four­ier" series was questioned - Laplace apparently could not believe that cos(x) could possibly be expressed using a sine series. Secondly, the utility of the approach was questioned. Why should Fourier's expansion, even ifit were valid, be superior to expansions in terms of continued fractions, infinite products, power series?

Since that time, rigorous derivations have proved the validity of the Fourier expansion. Moreover with the advent of the digital computer and efficient computational algorithms" applications for Fourier methods have developed in all areas of science and technol­ogy. Nevertheless, many of those who use the Fourier transform are not aware of the method's theoretical foundation or equipped to appreciate the theoretical derivation that establishes it. Moreover,

XIV Foreword

the similarity of the different applications is often obscured by different terminology and symbols. Consequently, almost 200 years after its development there is still a need to demystify the Fourier transform and demonstrate its practical applications.

The authors of this book set out to meet this need for as wide an audience as possible by presenting their material in straightforward language and making extensive use of diagrams and "real world" examples. Writing such a book is a challenge; it is not easy to present difficult concepts in a straightforward manner while retaining their essence. I am delighted to see that the authors have met the challenge. They present their material in an easily understood manner without sacrificing accuracy. I found that the Chapters dealing with topics in my area of expertise provided me with new insights and a deeper understanding. Furthermore, chapters dealing with unfamiliar topics, such as MRI, proved to be accessible and easily understood; I came away with a much better understanding of how Fourier methods are used in imaging. I particularly enjoyed the manner in which the complexity of the discussion increased through the book starting with the "simple" I-D case, progressing to 2-D with CT, 3-D with MRI and concluding with 4-D (space and time) with the MRI and wavelet chapters. The final chapter, in keeping with an applications oriented book, provides many useful sugges­tions for the reader who wishes to apply the methods themselves.

I believe this book will be a useful addition to the library of anyone working with Fourier transforms.

Robert Kearney, Ph.D., Eng., Department of Biomedical Engineering,

McGill University,

August 1997.

PREFACE

A number of years ago, one of us (TMP) gave an examination to aspiring MRI technologists. One of the questions, asked for a simple definition of the Fourier transform. Among the responses, two stood out. "I'm not really sure, but it must be terribly important since the instructor gets so excited about it," and "It's like a prism that separates light into its component colors." This second answer gave us hope and convinced us that topics like Fourier theory could indeed be presented to the non mathematician, using meaningful examples, without confusing the issue by over-reliance on mathe­matics. Thus was sown the seed for this book.

The concept of the Fourier transform was first suggested in 1810 by Jean Baptiste Joseph Fourier, when he published a paper on heat conduction. Fourier was a brilliant man of many talents, and he led a fascinating life. He was born in Auxerre, France, in 1768, the son of a tailor. His rather humble birth precluded him from his ambition to become an artillery officer, but he was able to go to school and he became an engineer, teacher, and administrator. Fourier lived in turbulent times and he became embroiled in the French Revolution, and went from the extremes of almost being sent to the guillotine, to being an important man in Napoleon's regime. He rose to become the Prefect of the department of Isere, and was made a Baron by Napoleon in 1808. His fortunes fluctuated along with Napoleon's, but it was during one of the periods when he was out of favor that he had time to work on the problem of heat conduction, which led to the development of what became known as the Fourier Series Analysis, which later became generalized as the Fourier transform.

Fourier had wide mathematical interests and in l8lO he won a competition to develop equations that would describe how heat is diffused. In his analysis of this subject, he discovered that using sums of trigonometric functions (sines and cosines) to represent more general functions, provided a powerful analytical tool that enabled him to find the solution to complex heat conduction problems. His methods became generally adopted when it was realized that such a decomposition of signals or waveforms into these simpler (basis)

xvi Preface

functions had a much wider applicability in areas other than heat conduction.

The idea for this book arose from a workshop on Fourier Transforms in Biomedical Engineering, presented at the IEEE Engineering in Medicine and Biology Society Meeting, held in Montreal, Quebec, September, 1995. Usually, engineers and physi­cists think and write using complicated mathematical equations, and most text books on Fourier transforms incorporate this mathema­tical approach. Although the workshop was attended by members of the biomedical engineering community, who understood many of the basic concepts, many in the audience were from a broad spectrum within the biomedical engineering community, and we felt that the material needed to be presented in straightforward language.

There are many people working in biomedical fields, who do not necessarily have an extensive mathematical background, but who would benefit from an understanding of Fourier transforms. We wanted to produce a book that would be comprehensible without extensive reliance on mathematics. A few years ago the editors produced a (relatively!) simple monograph on k-space in Magnetic Resonance Imaging, which also had arisen from a workshop for MRI technologists. Having one author with a background in psychology and limited mathematical training helped to make the explanations less technical, and we have endeavored to keep the same approach in this book. Despite our best efforts it seems inevitable that the content will seem insultingly simplified to some, while others may struggle with some of the concepts. The mathe­matics have not been ignored, but those who are not as comfortable deciphering equations should not be put off by their presence in this book. They have been included to make the text more intelligible for those who are used to working with them, but where possible the concepts are also explained simply for those who aren't. Throughout the book diagrams have been included to reinforce the text, and practical examples from commonly used applications are described to show how the Fourier transform works in the real world.

Over the last century the Fourier transform has become one of the most useful analytical tools available to the applied mathema­tician, physicist, and engineer. In other disciplines related to biome-

Preface xvii

dieine, the Fourier transform has become invaluable, not only for the analysis of data, but also as a means of describing the physical mechanism of collecting and reconstructing data. It finds application in such diverse fields as radio-astronomy, crystallography, spectro­photometry, music and many others, not to mention the medical image processing and signal analysis techniques discussed in this book.

Chapter 1 introduces the basic concepts of waves and sinusoids (sines and cosines), and how they relate to natural phenomena. In nature, waves occur both as functions of time (e.g. a musical tone), or of space (ripples on a pond). This chapter serves as a general overview of the Fourier transform, as well as an introduction to the notation used. It also explains the concept of complex numbers and the rationale behind their use in this area. A discussion of how complicated signals can be built up from the sums of simpler "basis" functions is also presented. The optical lens is used both as an example of a device that "calculates" a Fourier transform as well as being a 2-dimensional linear system. Following the discussion of image reconstruction from Fourier components, the concepts of point response functions and image blurring are introduced.

The second chapter is concerned with the processing of 1-dimensional signals. The concept of linear systems is discussed, and why linearity is such an important issue in Fourier transforms. This chapter builds upon the earlier material dealing with breaking down signals into component sinusoids, and introduces the concepts of sampling and aliasing. The chapter concludes with a number of examples demonstrating the usefulness of Fourier processing of digitized physiological signals.

Chapter 3, "The 2-D Fourier Transform" extends the concepts from Chapter 2 concerning I-dimensional signals into the 2-dimensional realm. This chapter demonstrates how images can be built up from composites of sine waves, and conversely, how complex images may be decomposed into their individual wave components. It uses a simple X-ray imaging system as a example and explains how imaging systems may be optimized using the Modulation Transfer Function. The chapter concludes with a description of Computed Tomography (CT) , a technology that owes its existence to Fourier theory, including a discussion of how

XVlll Preface

the data are acquired and processed using Fourier techniques to form images.

From the general discussion of 2-D Fourier transform, Chapter 4 moves into the field of Magnetic Resonance Imaging (MRI), which is one of the most important practical applications of 2- and 3-dimensional Fourier processing. While not pretending to be an exhaustive treatment of the physics of MRI, it nevertheless gives the reader an introduction to some of the physical principles, as a basis for understanding the relationship between MRI and Fourier transforms discussed later in the chapter. The chapter describes the fundamental role of the Fourier transform in the operation of an MRI machine. Moreover, it reinforces the discussions of previous chapters by emphasizing the physical phenomena of extracting spatial frequency components of 3-D anatomical images, through the use of magnetic field gradients.

Chapter 5 is concerned with Wavelet (literally "little waves") analysis, which is a more recent signal processing tool. One of the limitations of Fourier theory is that the component sinusoids are infinite in extent, even when the structures they represent are small and/or isolated. The concept of time-frequency analysis, or the representation of signals of different frequencies or scales at different times, is introduced in this chapter. Wavelet analysis is an extension of the concepts of Fourier theory, to allow for compact, rather than extended basis functions. This chapter is the most mathematical in the book, and may be more difficult conceptually than the others. However, to aid an understanding of the concepts that are presented mathematically, analogies are drawn to the process of listening to music, and how the appreciation of a musical piece can be described in terms of its transitory characteristics, rather than a series of sinusoidal signals. The development of the theory is followed by illustrations of the use of Wavelets in image and signal analysis.

Throughout the book, we discuss the use of the discrete Fourier (DFT) and the fast Fourier transform (FFT) algorithm as a means of computing it. Chapter 6 deals with how we make the transition from the continuous to the discrete world, in order that we can use digital computers to perform the calculations on signals and images. In addition, the consequences of sampling are considered, along with a description of the FFT algorithm, whose increase in computational

Preface xix

speed makes the practical calculation of Fourier transforms possible. It also presents some practical advice on the use of the FFT.

We sincerely wish to thank the contributors to this book, who are good friends as well as colleagues, for all the hard work they put into this book. They all responded with great patience to the demands made by us during the writing process. It has been a pleasure working with them.

Terry Peters and Jackie Williams,

July 1997.