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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Fourier Methods in Imaging introduces the mathematical tools for modeling linear imaging systems to predict the action of the system or for solving for the input. The chapters are grouped into five sections, the first introduces the imaging “tasks” (direct, inverse, and system analysis), the basic concepts of linear algebra for vectors and functions, including complex-valued vectors, and inner products of vectors and functions. The second section defines "special" functions, mathematical operations, and transformations that are useful for describing imaging systems. Among these are the Fourier transforms of 1-D and 2-D function, and the Hankel and Radon transforms. This section also considers approximations of the Fourier transform. The third and fourth sections examine the discrete Fourier transform and the description of imaging systems as linear "filters", including the inverse, matched, Wiener and Wiener-Helstrom filters. The final section examines applications of linear system models to optical imaging systems, including holography.
- Provides a unified mathematical description of imaging systems.
- Develops a consistent mathematical formalism for characterizing imaging systems.
- Helps the reader develop an intuitive grasp of the most common mathematical methods, useful for describing the action of general linear systems on signals of one or more spatial dimensions.
- Offers parallel descriptions of continuous and discrete cases.
- Includes many graphical and pictorial examples to illustrate the concepts.
This book helps students develop an understanding of mathematical tools for describing general one- and two-dimensional linear imaging systems, and will also serve as a reference for engineers and scientists
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Seitenzahl: 1504
Veröffentlichungsjahr: 2010
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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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Cover Art based on images of the Archimedes Palimpsest, © Copyright Owner of the Archimedes Palimpsest, Licensed for use under Creative Commons Attribution 3.0 Unported Access Rights.
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 itsApplications, by Gilbert Strang (2005), and Advanced Mathematical Models forEngineering 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 fromhttp://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 ImagingMagazine, 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 FourierMethods 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
