Table of Contents

Copyright

Dedication

Preface

Series Editor’s Preface

Chapter 1: Introduction

1.1 Signals, Operators, and Imaging Systems

1.2 The Three Imaging Tasks

1.3 Examples of Optical Imaging

1.4 Imaging Tasks in Medical Imaging

Chapter 2: Operators and Functions

2.1 Classes of Imaging Operators

2.2 Continuous and Discrete Functions

Chapter 3: Vectors with Real-Valued Components

3.1 Scalar Products

3.2 Matrices

3.3 Vector Spaces

Chapter 4: Complex Numbers and Functions

4.1 Arithmetic of Complex Numbers

4.2 Graphical Representation of Complex Numbers

4.3 Complex Functions

4.4 Generalized Spatial Frequency – Negative Frequencies

4.5 Argand Diagrams of Complex-Valued Functions

Chapter 5: Complex-Valued Matrices and Systems

5.1 Vectors with Complex-Valued Components

5.2 Matrix Analogues of Shift-Invariant Systems

5.3 Matrix Formulation of Imaging Tasks

5.4 Continuous Analogues of Vector Operations

Chapter 6: 1-D Special Functions

6.1 Definitions of 1-D Special Functions

6.2 1-D Dirac Delta Function

6.3 1-D Complex-Valued Special Functions

6.4 1-D Stochastic Functions – Noise

6.5 Appendix A: Area of SINC[x] and SINC2[x]

6.6 Appendix B: Series Solutions for Bessel Functions J0[x] and J1[x]

Chapter 7: 2-D Special Functions

7.1 2-D Separable Functions

7.2 Definitions of 2-D Special Functions

7.3 2-D Dirac Delta Function and its Relatives

7.4 2-D Functions with Circular Symmetry

7.5 Complex-Valued 2-D Functions

7.6 Special Functions of Three (or More) Variables

Chapter 8: Linear Operators

8.1 Linear Operators

8.2 Shift-Invariant Operators

8.3 Linear Shift-Invariant (LSI) Operators

8.4 Calculating Convolutions

8.5 Properties of Convolutions

8.6 Autocorrelation

8.7 Crosscorrelation

8.8 2-D LSI Operations

8.9 Crosscorrelations of 2-D Functions

8.10 Autocorrelations of 2-D Functions

Chapter 9: Fourier Transforms of 1-D Functions

9.1 Transforms of Continuous-Domain Functions

9.2 Linear Combinations of Reference Functions

9.3 Complex-Valued Reference Functions

9.4 Transforms of Complex-Valued Functions

9.5 Fourier Analysis of Dirac Delta Functions

9.6 Inverse Fourier Transform

9.7 Fourier Transforms of 1-D Special Functions

9.8 Theorems of the Fourier Transform

9.9 Appendix: Spectrum of Gaussian via Path Integral

Chapter 10: Multidimensional Fourier Transforms

10.1 2-D Fourier Transforms

10.2 Spectra of Separable 2-D Functions

10.3 Theorems of 2-D Fourier Transforms

Chapter 11: Spectra of Circular Functions

11.1 The Hankel Transform

11.2 Inverse Hankel Transform

11.3 Theorems of Hankel Transforms

11.4 Hankel Transforms of Special Functions

11.5 Appendix: Derivations of Equations (11.12) and (11.14)

Chapter 12: The Radon Transform

12.1 Line-Integral Projections onto Radial Axes

12.2 Radon Transforms of Special Functions

12.3 Theorems of the Radon Transform

12.4 Inverse Radon Transform

12.5 Central-Slice Transform

12.6 Three Transforms of Four Functions

12.7 Fourier and Radon Transforms of Images

Chapter 13: Approximations to Fourier Transforms

13.1 Moment Theorem

13.2 1-D Spectra via Method of Stationary Phase

13.3 Central-Limit Theorem

13.4 Width Metrics and Uncertainty Relations

Chapter 14: Discrete Systems, Sampling, and Quantization

14.1 Ideal Sampling

14.2 Ideal Sampling of Special Functions

14.3 Interpolation of Sampled Functions

14.4 Whittaker–Shannon Sampling Theorem

14.5 Aliasing and Interpolation

14.6 “Prefiltering” to Prevent Aliasing

14.7 Realistic Sampling

14.8 Realistic Interpolation

14.9 Quantization

14.10 Discrete Convolution

Chapter 15: Discrete Fourier Transforms

15.1 Inverse of the Infinite-Support DFT

15.2 DFT over Finite Interval

15.3 Fourier Series Derived from Fourier Transform

15.4 Efficient Evaluation of the Finite DFT

15.5 Practical Considerations for DFT and FFT

15.6 FFTs of 2-D Arrays

15.7 Discrete Cosine Transform

Chapter 16: Magnitude Filtering

16.1 Classes of Filters

16.2 Eigenfunctions of Convolution

16.3 Power Transmission of Filters

16.4 Lowpass Filters

16.5 Highpass Filters

16.6 Bandpass Filters

16.7 Fourier Transform as a Bandpass Filter

16.8 Bandboost and Bandstop Filters

16.9 Wavelet Transform

Chapter 17: Allpass (Phase) Filters

17.1 Power-Series Expansion for Allpass Filters

17.2 Constant-Phase Allpass Filter

17.3 Linear-Phase Allpass Filter

17.4 Quadratic-Phase Filter

17.5 Allpass Filters with Higher-Order Phase

17.6 Allpass Random-Phase Filter

17.7 Relative Importance of Magnitude and Phase

17.8 Imaging of Phase Objects

17.9 Chirp Fourier Transform

Chapter 18: Magnitude–Phase Filters

18.1 Transfer Functions of Three Operations

18.2 Fourier Transform of Ramp Function

18.3 Causal Filters

18.4 Damped Harmonic Oscillator

18.5 Mixed Filters with Linear or Random Phase

18.6 Mixed Filter with Quadratic Phase

Chapter 19: Applications of Linear Filters

19.1 Linear Filters for the Imaging Tasks

19.2 Deconvolution – “Inverse Filtering”

19.3 Optimum Estimators for Signals in Noise

19.4 Detection of Known Signals – Matched Filter

19.5 Analogies of Inverse and Matched Filters

19.6 Approximations to Reciprocal Filters

19.7 Inverse Filtering of Shift-Variant Blur

Chapter 20: Filtering in Discrete Systems

20.1 Translation, Leakage, and Interpolation

20.2 Averaging Operators – Lowpass Filters

20.3 Differencing Operators – Highpass Filters

20.4 Discrete Sharpening Operators

20.5 2-D Gradient

20.6 Pattern Matching

20.7 Approximate Discrete Reciprocal Filters

Chapter 21: Optical Imaging in Monochromatic Light

21.1 Imaging Systems Based on Ray Optics Model

21.2 Mathematical Model of Light Propagation

21.3 Fraunhofer Diffraction

21.4 Imaging System based on Fraunhofer Diffraction

21.5 Transmissive Optical Elements

21.6 Monochromatic Optical Systems

21.7 Shift-Variant Imaging Systems

Chapter 22: Incoherent Optical Imaging Systems

22.1 Coherence

22.2 Polychromatic Source – Temporal Coherence

22.3 Imaging in Incoherent Light

22.4 System Function in Incoherent Light

Chapter 23: Holography

23.1 Fraunhofer Holography

23.2 Holography in Fresnel Diffraction Region

23.3 Computer-Generated Holography

23.4 Matched Filtering with Cell-Type CGH

23.5 Synthetic-Aperture Radar (SAR)

References

Index

This edition first published 2010

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Library of Congress Cataloging-in-Publication Data

Easton, Roger L. Jr.

Fourier methods in imaging / Roger L. Easton, Jr.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-68983-7 (cloth)

1. Image processing–Mathematics. 2. Fourier analysis. I. Title.

TA1637.E23 2010

621.36’701515723–dc22

2010000343

A catalogue record for this book is available from the British Library.

ISBN: 9780470689837

To my parents and my students

Preface

This book is intended to introduce the mathematical tools that can be applied to model and predict the action of imaging systems under some simplifying assumptions. A discussion of the mathematics used to model imaging systems encompasses such breadth of material that any single book that aspired to consider all aspects of the subject would be a massive tome. It should be made clear at the outset that this book intends no such pretense. Rather, its primary goal is to help readers develop an intuitive grasp of the most common mathematical methods that are useful for describing the action of general linear systems on signals of one or more spatial dimensions. In other words, the goal is to “develop images” of the mathematics. To assist in this development, many graphical and pictorial examples will be used for emphasis and to facilitate development of the readers’ intuition.

A second goal of this book is to develop a consistent mathematical formalism for characterizing imaging systems. This effort requires derivation of equations used to describe both the action of the imaging system and its effect on the quality of the output image. Success in meeting the first objective of developing intuition should facilitate the achievement of the second goal. In the course of this discussion, we will derive representations of images that are defined over both continuous and discrete domains and for continuous and discrete ranges. These same representations may be used to describe imaging systems as well. Representations that are defined over a continuous domain are convenient for describing realistic objects, imaging systems, and the resulting images. Representations in discrete coordinates (i.e., using sampled functions) are essential for modeling general objects, images, and systems in a computer. Discrete images and systems are conveniently represented as vectors and matrices.

Authors of technical books at this level must always be cognizant of the different levels of preparation by the readers for the subject. Though the treatment may target readers with a “median” background, there always will be some spread about that median.

The contents of the book can be roughly grouped into five parts. Some chapters (and even some sections) may be skipped by many readers depending on their particular needs. The particular order of consideration of topics was chosen with some care to ensure a sequential discussion. That said, the choice also reflects the particular biases of the author to some extent.

After the introduction, the first part of the book (Chapters 2–5) attempts to address the inevitable variation in reader experience and preparation. It introduces the basic mathematical concepts of linear algebra for vectors and functions that are necessary for understanding the subsequent discussions. These include complex-valued functions, vector spaces, and inner products of vectors and functions and are intended to provide broad and less than rigorous reviews for readers who have already encountered mathematical discussions of these topics in previous studies, such as in quantum mechanics. This discussion is similar in tone to treatments of these subjects presented by Image Reconstruction in Radiology, by Anthony J. Parker (1990), and many readers will likely be able to skim (or even skip) some or all of it. Readers desiring or requiring a deeper discussion should consult some of the standard texts, such as Linear Algebra and its Applications, by Gilbert Strang (2005), and Advanced Mathematical Models for Engineering and Science Students, by Geoffrey Stephenson and Paul M. Radmore (1990).

The second part (Chapters 6–13) defines a set of “special” functions and describes the mathematical operations and transformations of continuous functions that are useful for describing imaging systems. The Fourier transforms of 1-D and 2-D functions are considered in detail, and the Radon transform is introduced. The last chapter in this part considers approximations of the Fourier transform and figures of merit that are useful metrics of the representations in the two domains. Note that other sources exist for discussions of the special functions, including Linear Systems, Fourier Transforms, and Optics, by Jack Gaskill (1978), and The Fourier Transform and its Applications, by Ronald N. Bracewell (1986).

The third part spans Chapters 14 and 15, and considers the Fourier transform of discrete functions. The importance of this discussion cannot be overstated, as many (if not most) applications require operations with sampled functions. The fourth part (Chapters 16–20) considers the description of imaging systems as linear “filters”, and applies the mathematical tools to solve specific imaging tasks. In particular, Chapter 20 considers the application of linear filters to discrete functions. The fifth part considers in the remaining chapters the application of linear systems to model optical imaging systems, including holography.

The selection of parts depends on the needs of the readers. Many may find the first part to be a review of concepts that were considered in other venues. For these readers, a logical progression would include skimming the first part, more careful study of the second and third parts, and then selection of the appropriate topics from the fourth and fifth parts.

Two software programs used to create the examples in this book are available online for free. The original DOS program, “signals”, creates and processes 1-D functions. It was originally written for classroom demonstrations. The program may be downloaded from
http://www.cis.rit.edu/resources/software/index.html.
This program runs directly on a Windows PC or may also be used in Linux and the Macintosh OS in the DOSBox environment (http://www.dosbox.com). As part of her senior research project in 2009, Juliet Bernstein wrote the second program “SignalShow” in Java. It creates and processes both 1-D and 2-D functions. It is available from the website http://www.signalshow.com.

I must thank many individuals who have participated in the writing of this book. Many students have provided inspiration and impetus to the process. Of particular note, I acknowledge John Knapp, Derek Walvoord, Ranjit Bhaskar, Ted Tantalo, David Wilbur, Anthony Calabria, Gary Hoffmann, Sharon Cady, Sally Robinson, Scott Brown, Kate Johnson, Alec Greenfield, Alvin Spivey, Noah Block, and Katie Hoheusle. Harry Barrett, Kyle Myers, Fenella France, and Jack Gaskill inspired by their examples. Special thanks to Juliet Bernstein for creating the computer-generated holograms.

Several colleagues also contributed in positive ways to the preparation of this text, including Keith Knox, Zoran Ninkov, Elliott Horch, William Cirillo, Ed Przybylowicz, Rodney Shaw, Jeff Pelz, Jon Arney, William A. Christens-Barry, Mike Toth, P. R. Mukund, Ajay Pasupuleti, and Reiner Eschbach. A few “colleagues”, especially some other faculty, who made negative contributions will not be mentioned by name.

I would also like to thank some others who provided personal inspiration over the last 30+ years of my professional life. Among those who inspired the work are Harry Barrett, William Noel, Sue Chan, Andrea Zizzi, Fenella France, Catherine Carlson, and Judith Knight, as well my parents Roger and Barbara Easton.

Roger L. Easton, Jr.

Rochester, New York

September 2009

Series Editor’s Preface

Science, like life, is full of unintended consequences and unanticipated benefits. For example, by 1893 M. J. Hadamard had developed a set of functions represented as matrices that could “break down” the nature of a “signal” into components taken from a specific set of “basis” functions, which had complicated waveforms but simple binary values of “1” or “−1”. The amplitudes of these functions defined the signal in terms of the basis functions. By the 1930s this set of functions had been codified into the Walsh–Hadamard transform, which was formed from a complete and orthogonal set of basis functions (in one or two dimensions). While the Walsh–Hadamard transform was of great mathematical interest, it did not have a lot of practical value until the age of the computer and its first major impact may have been when it was used to convert raw image data from deep-space probes into a series of bits that represented the “1s” and “−1s” of the basis functions. These binary signals were ideal for transmission from deep space and the image was easily reconstructed on Earth using the inverse transform. A second example of the unanticipated benefits is more relevant to this the 10th offering of the Wiley/IS&T Series on Imaging Science and Technology: Fourier Methods in Imaging by Dr. Roger L. Easton, Jr. In the early 1800s many of the world greatest physicists and chemists were focusing on the nature of heat, heat conduction, and steam engines and were creating the foundations of classical thermodynamics. One of theses scientists was Joseph Fourier. Fourier focused on solving the most basic nature of how heat (and temperature) moved through solids and this resulted in his work entitled The Analytical Theory of Heat. His solutions resulted in unique series and integrals using sine and cosine functions to provide the final solution to the heat conduction, over time and space, for a given system (with a well-defined geometry). This expansion into harmonic functions came to be known as the Fourier series, Fourier integral, or more simply the Fourier transform. Fourier analysis is used today in all fields of science and engineering.

Fourier also served as Secretary of the Institut d’Egypte in 1798–1801 during Napoleon’s expedition to that country. The most important artifact found during this mission was the Rosetta Stone, which included copies of the same text in Greek, demotic, and Egyptian hieroglyphics. Several years later, Fourier encouraged the young Jean-Fran¸cois Champollion to work on translating the writings, and Champollion discovered the secret in 1822. Thomas Young, another famous scientist with significant contributions in optics, had also searched for the solution to this enigma. This aspect of Fourier’s career and that of Dr. Easton will show an interesting similarity as noted below.

How does Fourier analysis impact on imaging? Consider the following two cases. When one “looks” at a recorded scene, one sees a continuous two-dimensional array of light values, which is interpreted by means of the visual system as an image. This is the natural spatial domain representation of the image and can be used to understand and alter the image as one wishes. However, there is an alternate mathematical representation of this image that can also be used to understand and alter the image. This alternate representation is the spatial frequency domain or the Fourier transform of the image. Consider the following set of operations. Find the average value of the image. Subtract the average value of the image from each point in the image resulting in a new image that now has both negative and positive values (light has no negative values, so this is just a mathematical abstraction of the image). Now construct a large (really infinite) set of two-dimensional patterns that are made up of sine and cosine functions. They will also have negative and positive values. Take each of these patterns and multiply them point by point with the average adjusted image and then sum all the values. This sum is then the coefficient (amplitude) for the given pattern used; this can be thought of as the projection of each pattern onto the average adjusted image. Repeat for all the patterns (basis functions). The resulting set of coefficients, along with the average value, now represent the alternate mathematical representation of the image, its Fourier transform. One can reconstruct the image by taking all the basis functions and multiplying by the appropriate coefficient, summing point by point, and adding the average value; this is equivalent to the inverse Fourier transform. Once the image has been transformed to the spatial frequency domain, one can operate on the coefficients to alter the nature of the image and then follow the above process to reconstruct the altered image. How all this can be done mathematically is presented in this excellent and precise text. One other example of the Fourier transform is useful to establish a more complete frame of reference for this text. Using basic wave optic reconstruction it is possible to show that the focal plane of a lens (not the image plane) contains the Fourier transform of the image. Thus one can perform operations on the image (in real time) by placing active devices (that alter the image and re-emit light) or passive devices (that just attenuate the light) in the focal plane of the lens and then using an identical lens to perform the inverse Fourier transform to get the altered image. Hence we see how Fourier’s quest to understand how heat and temperature flow through solids has led, unintentionally, to such a vast and rich branch of imaging science.

Fourier Methods in Imaging provides the reader with a complete and coherent view of operating on images in the spatial frequency domain and how these operations relate to methods in the spatial domain, but may be easier to implement or more flexible in achieving a given goal. The first part of the text provides a clear review and exposition of the mathematical nature of linear system analysis for images and carefully considers the impact of sampling (moving from the continuous domain to a discrete domain) that is found in most digital imaging systems. Dr. Easton provides a host of often-used functions (SINC functions, triangle functions, etc.) in one-dimensional and two-dimensional cases, each of which are encountered in many practical image processing applications. He also provides a clear exposition of Hankel transforms (Fourier transforms in circular coordinates) and the Radon transform that forms the basis of many medical imaging systems like X-ray tomography. Dr. Easton then provides a comprehensive review of discrete transforms (used on all computers and in digital signal processing systems embedded in digital cameras and other digital imaging devices). These discrete transforms are the equivalent of the more general continuous transforms, but used on sampled images. Once the mathematical methods have been clearly explained, Dr. Easton uses the mathematics to implement a series of filtering applications in the spatial frequency domain, which are equivalent to more complex and harder to implement operations in the spatial domain. The topics of operating on coherent and non-coherent light and holography round out the text; these operations are on the actual image rather than a digitally encoded image like that from a camera or scanner.

Fourier Methods in Imaging represents an outstanding and practical review of operating in the spatial frequency domain for both “live optical” images and captured digital images. Every scientist or engineer working in modern imaging systems will find this to be an indispensable reference, one that sits on his or her desk and will be used time and time again.

Dr. Easton received his Ph.D. in Optical Science from the University of Arizona’s Optical Sciences Center in 1986. He joined the Carlson Center for Imaging Science at The Rochester Institute of Technology (RIT) upon graduation and has been an integral part of the Center as both an outstanding instructor and researcher. He received the Professor Raymond C. Bowman Award for undergraduate teaching in Imaging Science from the Society for Imaging Science and Technology in 1997. Over his years at RIT he has concentrated on the mathematical treatment of linear imaging systems and on the experimental image processing techniques of “real” and captured images. In addition to his basic research in image science, Dr. Easton has been part of a team of scientific and historical scholars who have focused on the preservation and reconstruction of ancient manuscripts, including the Dead Sea Scrolls and the Archimedes Palimpsest. In this small way, he shares one of Fourier’s own interests in the meaning of historical artifacts. This work resulted in his winning of the Archie Mahan Prize from the Optical Society of America, for the article “Imaging and the Dead Sea Scrolls” in 1988, and the 2003 Imaging Solution of the Year Award, by Advanced Imaging Magazine, for “Multispectral Imaging of the Archimedes Palimpsest”, January 2003.

On a personal note, I had the pleasure of working with Dr. Easton while I was with the Eastman Kodak Research Laboratories and at the University of Rochester’s Center for Electronic Imaging Systems, a joint effort with RIT, from 1986 through 1999. Dr. Easton is a truly dedicated teacher with proven experience in finding experimental solutions to complex imaging problems. As such, his offering of Fourier Methods in Imaging reflects his deep understanding of imaging problems, applications, and solutions. I will be proud to have this text on my desk and I highly recommend it to all working in the field of imaging science and technology.

Michael A. Kriss

Formerly of the Eastman Kodak

Research Laboratories

and the University of Rochester

1

Introduction

1.1 Signals, Operators, and Imaging Systems

As a simple definition, we may consider an imaging system to map the distribution of the input “object” to a “similar” distribution at the output “image” (where the meaning of “similar” is to be determined). Often the input and output amplitudes are represented in different units. For example, the input is often electromagnetic radiation with units of, say, watts per unit area, while the output may be a transparent negative emulsion measured in dimensionless units of “density” or “transmittance”. In other words, the system often changes the form of the energy; it is a “transducer”. The goal of this book is to mathematically describe the properties of imaging systems, and it is often convenient to use the model of a system as a chain of links.

1.1.1 The Imaging Chain

An imaging system is often modeled as a “chain” of links that transfer information in the form of energy from an input (the object) to the output (the image) of the system. Many schemes of links in an imaging system are plausible, depending on details, but one eight-link model is appropriate in many imaging systems:

1. The source of energy (usually in the form of electromagnetic radiation).

2. The object to be imaged, which interacts with the energy from the source by reflection, refraction, absorption, scattering, and/or other mechanism.

3. Propagation of the energy to the imaging system.

4. Energy collection (often using an optical system composed of lenses and/or mirrors).

5. Sensing or detection by a transducer (converts incident energy to a measurable form, e.g., photons to electrons).

6. Image processing, including data compression (if any).

7. Storage and/or transmission (if any).

8. Display.

The source and object often are one and the same, e.g., radiating objects such as stars. Sometimes it is useful to model the imaging system with a second stage of energy propagation after collection, as when evaluating optical imaging systems composed of a single thin lens.

In this book, we consider a simplified picture of the imaging chain that combines the source, object, and energy propagation into one entity that we will call the “input function” (or just the “input”); it usually is specified by a single-valued physical quantity f that varies over continuous coordinates in space, and perhaps in time t and color, specified by wavelength λ (or equivalently by temporal frequency ν):

1.1a 1.1

Though these quantities have explicit spectral and temporal coordinates in addition to the spatial dimensions, the signals considered in this book most often will be functions of one or two spatial coordinates only and are specified by f[x] or f[x, y]. In other words, the input and output signals will be constant in time and wavelength. The number of coordinates necessary to specify the function is the dimensionality, and the set of all possible such coordinates defines its domain. We will use the shorthand notation of “1-D” and “2-D” for one- and two-dimensional signals, respectively; 2-D signals such as f[x, y] are of greatest interest in imaging applications, but the study of 1-D signals will be considered in depth in this book as well. This is because 1-D systems often are easy to visualize and the results may be directly transferable to problems with higher dimensionality.

The output of the imaging system also will usually be specified by a single-valued physical quantity that will be denoted by g. Though the domain of the image g may be identical to that of the object f, it is more common to require different coordinates, and these will be denoted, when necessary, by primed coordinates. In many cases, the output will be a function of two spatial coordinates with no dependence on wavelength:

1.1b 1.2

In imaging applications, the numerical value assigned to the dependent variable of the input or output signals in Equation (1.1) represents a measurable physical quantity. Based on the familiarity of optical imaging, a common descriptive name for f is the “brightness” of the scene, though this terminology is not used in some subdisciplines of optics, such as radiometry. Regardless of the nomenclature, the appropriate quantity (i.e., the units of ) depends on the particular imaging system. In optical imaging, the relevant quantity is the at each coordinate: the time average of the square of the magnitude of the electric field. In X-ray or gamma-ray imaging, the measured quantity is the number of quanta incident on the sensor as a function of spatial location. In acoustic imaging (sonar or ultrasound), the acoustic power radiated, transmitted, or reflected by an object is the quantity of record.

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