Fourier Methods inImagingRoger L. Easton, Jr.Chester F. Carlson Center for Imaging ScienceRochester Institute of TechnologyRochester NY, USA

A John Wiley and Sons, Ltd, Publication

Fourier Methods in Imaging

Wiley-IS&T Series in Imaging Science and Technology

Series Editor:Michael A. Kriss

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Fourier Methods in ImagingRoger L. Easton, Jr.

Published in Association with the Society forImaging Science and Technology

Fourier Methods inImagingRoger L. Easton, Jr.Chester F. Carlson Center for Imaging ScienceRochester Institute of TechnologyRochester NY, USA

A John Wiley and Sons, Ltd, Publication

This edition first published 2010c© 2010 John Wiley & Sons, Ltd

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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 2010621.36’701515723–dc22

2010000343

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

ISBN: 9780470689837

Set in 9/11pt Times by Sunrise Setting Ltd, Torquay, UK.Printed in Singapore by Markono Print Media Pte Ltd.

To my parents and mystudents

Contents

Series Editor’s Preface xix

Preface xxiii

1 Introduction 11.1 Signals, Operators, and Imaging Systems . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The Imaging Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Three Imaging Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Examples of Optical Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Ray Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Wave Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.3 System Evaluation of Hubble Space Telescope . . . . . . . . . . . . 71.3.4 Imaging by Ground-Based Telescopes . . . . . . . . . . . . . . . . . 8

1.4 Imaging Tasks in Medical Imaging . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 Gamma-Ray Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.2 Radiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.3 Computed Tomographic Radiography . . . . . . . . . . . . . . . . . 13

2 Operators and Functions 152.1 Classes of Imaging Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Shift Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Continuous and Discrete Functions . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Functions with Continuous and Discrete Domains . . . . . . . . . . 172.2.3 Continuous and Discrete Ranges . . . . . . . . . . . . . . . . . . . 192.2.4 Discrete Domain and Range – “Digitized” Functions . . . . . . . . . 202.2.5 Periodic, Aperiodic, and Harmonic Functions . . . . . . . . . . . . . 212.2.6 Symmetry Properties of Functions . . . . . . . . . . . . . . . . . . . 26Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Vectors with Real-Valued Components 293.1 Scalar Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Scalar Product of Distinct Vectors . . . . . . . . . . . . . . . . . . . 323.1.2 Projection of One Vector onto Another . . . . . . . . . . . . . . . . . 34

3.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.1 Simultaneous Evaluation of Multiple Scalar Products . . . . . . . . . 343.2.2 Matrix–Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . 363.2.3 Square and Diagonal Matrices, Identity Matrix . . . . . . . . . . . . 37

viii Contents

3.2.4 Matrix Transposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.5 Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.1 Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.2 Vector Subspaces Associated with a System . . . . . . . . . . . . . 44Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Complex Numbers and Functions 514.1 Arithmetic of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.1 Equality of Two Complex Numbers . . . . . . . . . . . . . . . . . . . 524.1.2 Sum and Difference of Two Complex Numbers . . . . . . . . . . . . 524.1.3 Product of Two Complex Numbers . . . . . . . . . . . . . . . . . . . 534.1.4 Reciprocal of a Complex Number . . . . . . . . . . . . . . . . . . . 534.1.5 Ratio of Two Complex Numbers . . . . . . . . . . . . . . . . . . . . 53

4.2 Graphical Representation of Complex Numbers . . . . . . . . . . . . . . . . 534.3 Complex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.4 Generalized Spatial Frequency – Negative Frequencies . . . . . . . . . . . . 624.5 Argand Diagrams of Complex-Valued Functions . . . . . . . . . . . . . . . . 62

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Complex-Valued Matrices and Systems 655.1 Vectors with Complex-Valued Components . . . . . . . . . . . . . . . . . . . 65

5.1.1 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.1.2 Products of Complex-Valued Matrices and Vectors . . . . . . . . . . 67

5.2 Matrix Analogues of Shift-Invariant Systems . . . . . . . . . . . . . . . . . . 675.2.1 Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . . . . . . . 705.2.2 Projections onto Eigenvectors . . . . . . . . . . . . . . . . . . . . . 725.2.3 Diagonalization of a Circulant Matrix . . . . . . . . . . . . . . . . . . 755.2.4 Matrix Operators for Shift-Invariant Systems . . . . . . . . . . . . . 805.2.5 Alternative Ordering of Eigenvectors . . . . . . . . . . . . . . . . . . 84

5.3 Matrix Formulation of Imaging Tasks . . . . . . . . . . . . . . . . . . . . . . . 845.3.1 Inverse Imaging Problem . . . . . . . . . . . . . . . . . . . . . . . . 845.3.2 Solution of Inverse Problems via Diagonalization . . . . . . . . . . . 865.3.3 Matrix–Vector Formulation of System Analysis . . . . . . . . . . . . 87

5.4 Continuous Analogues of Vector Operations . . . . . . . . . . . . . . . . . . 885.4.1 Inner Product of Continuous Functions . . . . . . . . . . . . . . . . 885.4.2 Complete Sets of Basis Functions . . . . . . . . . . . . . . . . . . . 915.4.3 Orthonormal Basis Functions . . . . . . . . . . . . . . . . . . . . . . 925.4.4 Continuous Analogue of DFT . . . . . . . . . . . . . . . . . . . . . . 935.4.5 Eigenfunctions of Continuous Operators . . . . . . . . . . . . . . . . 93Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6 1-D Special Functions 976.1 Definitions of 1-D Special Functions . . . . . . . . . . . . . . . . . . . . . . . 98

6.1.1 Constant Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.1.2 Rectangle Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.1.3 Triangle Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.1.4 Signum Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.1.5 Step Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.1.6 Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.1.7 Sinusoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.1.8 SINC Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Contents ix

6.1.9 SINC2 Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.1.10 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.1.11 Quadratic-Phase Sinusoid – “Chirp” Function . . . . . . . . . . . . . 1156.1.12 Gaussian Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.1.13 “SuperGaussian” Function . . . . . . . . . . . . . . . . . . . . . . . 1196.1.14 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.1.15 Lorentzian Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.1.16 Thresholded Functions . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.2 1-D Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.2.1 1-D Dirac Delta Function Raised to a Power . . . . . . . . . . . . . 1316.2.2 Sifting Property of 1-D Dirac Delta Function . . . . . . . . . . . . . . 1326.2.3 Symmetric (Even) Pair of 1-D Dirac Delta Functions . . . . . . . . . 1336.2.4 Antisymmetric (Odd) Pair of 1-D Dirac Delta Functions . . . . . . . . 1346.2.5 COMB Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.2.6 Derivatives of 1-D Dirac Delta Function . . . . . . . . . . . . . . . . 1376.2.7 Dirac Delta Function with Functional Argument . . . . . . . . . . . . 139

6.3 1-D Complex-Valued Special Functions . . . . . . . . . . . . . . . . . . . . . 1426.3.1 Complex Linear-Phase Sinusoid . . . . . . . . . . . . . . . . . . . . 1436.3.2 Complex Quadratic-Phase Exponential Function . . . . . . . . . . . 1436.3.3 “Superchirp” Function . . . . . . . . . . . . . . . . . . . . . . . . . . 1456.3.4 Complex-Valued Lorentzian Function . . . . . . . . . . . . . . . . . 1476.3.5 Logarithm of the Complex Amplitude . . . . . . . . . . . . . . . . . . 149

6.4 1-D Stochastic Functions – Noise . . . . . . . . . . . . . . . . . . . . . . . . 1496.4.1 Moments of Probability Distributions . . . . . . . . . . . . . . . . . . 1516.4.2 Discrete Probability Laws . . . . . . . . . . . . . . . . . . . . . . . . 1526.4.3 Continuous Probability Distributions . . . . . . . . . . . . . . . . . . 1566.4.4 Signal-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 1606.4.5 Example: Variance of a Sinusoid . . . . . . . . . . . . . . . . . . . . 1616.4.6 Example: Variance of a Square Wave . . . . . . . . . . . . . . . . . 1616.4.7 Approximations to SNR . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.5 Appendix A: Area of SINC[x] and SINC2[x] . . . . . . . . . . . . . . . . . . . 1626.6 Appendix B: Series Solutions for Bessel Functions J0[x] and J1[x] . . . . . . 166

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7 2-D Special Functions 1717.1 2-D Separable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.1.1 Rotations of 2-D Separable Functions . . . . . . . . . . . . . . . . . 1727.1.2 Rotated Coordinates as Scalar Products . . . . . . . . . . . . . . . 172

7.2 Definitions of 2-D Special Functions . . . . . . . . . . . . . . . . . . . . . . . 1747.2.1 2-D Constant Function . . . . . . . . . . . . . . . . . . . . . . . . . 1747.2.2 Rectangle Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1757.2.3 Triangle Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1767.2.4 2-D Signum and STEP Functions . . . . . . . . . . . . . . . . . . . 1767.2.5 2-D SINC Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1787.2.6 SINC2 Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1787.2.7 2-D Gaussian Function . . . . . . . . . . . . . . . . . . . . . . . . . 1807.2.8 2-D Sinusoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7.3 2-D Dirac Delta Function and its Relatives . . . . . . . . . . . . . . . . . . . . 1827.3.1 2-D Dirac Delta Function in Cartesian Coordinates . . . . . . . . . . 1837.3.2 2-D Dirac Delta Function in Polar Coordinates . . . . . . . . . . . . 1847.3.3 2-D Separable COMB Function . . . . . . . . . . . . . . . . . . . . . 186

x Contents

7.3.4 2-D Line Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . 1877.3.5 2-D “Cross” Function . . . . . . . . . . . . . . . . . . . . . . . . . . 1947.3.6 “Corral” Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.4 2-D Functions with Circular Symmetry . . . . . . . . . . . . . . . . . . . . . . 1957.4.1 Cylinder (Circle) Function . . . . . . . . . . . . . . . . . . . . . . . . 1967.4.2 Circularly Symmetric Gaussian Function . . . . . . . . . . . . . . . 1977.4.3 Circularly Symmetric Bessel Function of Zero Order . . . . . . . . . 1977.4.4 Besinc or Sombrero Function . . . . . . . . . . . . . . . . . . . . . . 2007.4.5 Circular Triangle Function . . . . . . . . . . . . . . . . . . . . . . . . 2017.4.6 Ring Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

7.5 Complex-Valued 2-D Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2047.5.1 Complex 2-D Sinusoid . . . . . . . . . . . . . . . . . . . . . . . . . . 2047.5.2 Complex Quadratic-Phase Sinusoid . . . . . . . . . . . . . . . . . . 205

7.6 Special Functions of Three (or More) Variables . . . . . . . . . . . . . . . . . 205Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

8 Linear Operators 2078.1 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2088.2 Shift-Invariant Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2138.3 Linear Shift-Invariant (LSI) Operators . . . . . . . . . . . . . . . . . . . . . . 216

8.3.1 Linear Shift-Variant Operators . . . . . . . . . . . . . . . . . . . . . 2218.4 Calculating Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.4.1 Examples of Convolutions . . . . . . . . . . . . . . . . . . . . . . . . 2238.5 Properties of Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

8.5.1 Region of Support of Convolutions . . . . . . . . . . . . . . . . . . . 2258.5.2 Area of a Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . 2258.5.3 Convolution of Scaled Functions . . . . . . . . . . . . . . . . . . . . 226

8.6 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2268.6.1 Autocorrelation of Stochastic Functions . . . . . . . . . . . . . . . . 2288.6.2 Autocovariance of Stochastic Functions . . . . . . . . . . . . . . . . 229

8.7 Crosscorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2298.8 2-D LSI Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

8.8.1 Line-Spread and Edge-Spread Functions . . . . . . . . . . . . . . . 2338.9 Crosscorrelations of 2-D Functions . . . . . . . . . . . . . . . . . . . . . . . . 2348.10 Autocorrelations of 2-D Functions . . . . . . . . . . . . . . . . . . . . . . . . 235

8.10.1 Autocorrelation of the Cylinder Function . . . . . . . . . . . . . . . . 236Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

9 Fourier Transforms of 1-D Functions 2399.1 Transforms of Continuous-Domain Functions . . . . . . . . . . . . . . . . . . 239

9.1.1 Example 1: Input and Reference Functions are Even Sinusoids . . . 2429.1.2 Example 2: Even Sinusoid Input, Odd Sinusoid Reference . . . . . 2459.1.3 Example 3: Odd Sinusoid Input, Even Sinusoid Reference . . . . . 2469.1.4 Example 4: Odd Sinusoid Input and Reference . . . . . . . . . . . . 247

9.2 Linear Combinations of Reference Functions . . . . . . . . . . . . . . . . . . 2509.2.1 Hartley Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2519.2.2 Examples of the Hartley Transform . . . . . . . . . . . . . . . . . . . 2519.2.3 Inverse of the Hartley Transform . . . . . . . . . . . . . . . . . . . . 252

9.3 Complex-Valued Reference Functions . . . . . . . . . . . . . . . . . . . . . . 2549.4 Transforms of Complex-Valued Functions . . . . . . . . . . . . . . . . . . . . 2569.5 Fourier Analysis of Dirac Delta Functions . . . . . . . . . . . . . . . . . . . . 2599.6 Inverse Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Contents xi

9.7 Fourier Transforms of 1-D Special Functions . . . . . . . . . . . . . . . . . . 2639.7.1 Fourier Transform of δ[x] . . . . . . . . . . . . . . . . . . . . . . . . 2649.7.2 Fourier Transform of Rectangle . . . . . . . . . . . . . . . . . . . . . 2649.7.3 Fourier Transforms of Sinusoids . . . . . . . . . . . . . . . . . . . . 2669.7.4 Fourier Transform of Signum and Step . . . . . . . . . . . . . . . . . 2689.7.5 Fourier Transform of Exponential . . . . . . . . . . . . . . . . . . . . 2709.7.6 Fourier Transform of Gaussian . . . . . . . . . . . . . . . . . . . . . 2759.7.7 Fourier Transforms of Chirp Functions . . . . . . . . . . . . . . . . . 2769.7.8 Fourier Transform of COMB Function . . . . . . . . . . . . . . . . . . 279

9.8 Theorems of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 2809.8.1 Multiplication by Constant . . . . . . . . . . . . . . . . . . . . . . . . 2819.8.2 Addition Theorem (Linearity) . . . . . . . . . . . . . . . . . . . . . . 2819.8.3 Fourier Transform of a Fourier Transform . . . . . . . . . . . . . . . 2819.8.4 Central-Ordinate Theorem . . . . . . . . . . . . . . . . . . . . . . . 2849.8.5 Scaling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2849.8.6 Shift Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2879.8.7 Filter Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2899.8.8 Modulation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2959.8.9 Derivative Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 2979.8.10 Fourier Transform of Complex Conjugate . . . . . . . . . . . . . . . 2989.8.11 Fourier Transform of Crosscorrelation . . . . . . . . . . . . . . . . . 2999.8.12 Fourier Transform of Autocorrelation . . . . . . . . . . . . . . . . . . 3029.8.13 Rayleigh’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3029.8.14 Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3049.8.15 Fourier Transform of Periodic Function . . . . . . . . . . . . . . . . . 3069.8.16 Spectrum of Sampled Function . . . . . . . . . . . . . . . . . . . . . 3079.8.17 Spectrum of Discrete Periodic Function . . . . . . . . . . . . . . . . 3089.8.18 Spectra of Stochastic Signals . . . . . . . . . . . . . . . . . . . . . . 3089.8.19 Effect of Nonlinear Operations of Spectra . . . . . . . . . . . . . . . 310

9.9 Appendix: Spectrum of Gaussian via Path Integral . . . . . . . . . . . . . . . 320Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

10 Multidimensional Fourier Transforms 32510.1 2-D Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

10.1.1 2-D Fourier Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . 32610.2 Spectra of Separable 2-D Functions . . . . . . . . . . . . . . . . . . . . . . . 327

10.2.1 Fourier Transforms of Separable Functions . . . . . . . . . . . . . . 32810.2.2 Fourier Transform of δ[x, y] . . . . . . . . . . . . . . . . . . . . . . . 32810.2.3 Fourier Transform of δ[x − x0, y − y0] . . . . . . . . . . . . . . . . . 33010.2.4 Fourier Transform of RECT[x, y] . . . . . . . . . . . . . . . . . . . . 33210.2.5 Fourier Transform of TRI[x, y] . . . . . . . . . . . . . . . . . . . . . 33210.2.6 Fourier Transform of GAUS[x, y] . . . . . . . . . . . . . . . . . . . . 33210.2.7 Fourier Transform of STEP[x] · STEP[y] . . . . . . . . . . . . . . . . . 33410.2.8 Theorems of Spectra of Separable Functions . . . . . . . . . . . . . 33410.2.9 Superpositions of 2-D Separable Functions . . . . . . . . . . . . . . 335

10.3 Theorems of 2-D Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . 33510.3.1 2-D “Transform-of-a-Transform” Theorem . . . . . . . . . . . . . . . 33610.3.2 2-D Scaling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 33610.3.3 2-D Shift Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33610.3.4 2-D Filter Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33710.3.5 2-D Derivative Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 338

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10.3.6 Spectra of Rotated 2-D Functions . . . . . . . . . . . . . . . . . . . 34010.3.7 Transforms of 2-D Line Delta and Cross Functions . . . . . . . . . . 341Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

11 Spectra of Circular Functions 34711.1 The Hankel Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

11.1.1 Hankel Transform of Dirac Delta Function . . . . . . . . . . . . . . . 35111.2 Inverse Hankel Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35311.3 Theorems of Hankel Transforms . . . . . . . . . . . . . . . . . . . . . . . . . 354

11.3.1 Scaling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35411.3.2 Shift Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35411.3.3 Central-Ordinate Theorem . . . . . . . . . . . . . . . . . . . . . . . 35411.3.4 Filter and Crosscorrelation Theorems . . . . . . . . . . . . . . . . . 35511.3.5 “Transform-of-a-Transform” Theorem . . . . . . . . . . . . . . . . . . 35511.3.6 Derivative Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 35511.3.7 Laplacian of Circularly Symmetric Function . . . . . . . . . . . . . . 356

11.4 Hankel Transforms of Special Functions . . . . . . . . . . . . . . . . . . . . . 35611.4.1 Hankel Transform of J0(2πrρ0) . . . . . . . . . . . . . . . . . . . . . 35611.4.2 Hankel Transform of CYL(r) . . . . . . . . . . . . . . . . . . . . . . . 35811.4.3 Hankel Transform of r−1 . . . . . . . . . . . . . . . . . . . . . . . . 36011.4.4 Hankel Transforms from 2-D Fourier Transforms . . . . . . . . . . . 36111.4.5 Hankel Transform of r2 GAUS(r) . . . . . . . . . . . . . . . . . . . . 36311.4.6 Hankel Transform of CTRI(r) . . . . . . . . . . . . . . . . . . . . . . 364

11.5 Appendix: Derivations of Equations (11.12) and (11.14) . . . . . . . . . . . . 365Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

12 The Radon Transform 37112.1 Line-Integral Projections onto Radial Axes . . . . . . . . . . . . . . . . . . . 371

12.1.1 Radon Transform of Dirac Delta Function . . . . . . . . . . . . . . . 37712.1.2 Radon Transform of Arbitrary Function . . . . . . . . . . . . . . . . . 379

12.2 Radon Transforms of Special Functions . . . . . . . . . . . . . . . . . . . . . 38012.2.1 Cylinder Function CYL(r) . . . . . . . . . . . . . . . . . . . . . . . . 38012.2.2 Ring Delta Function δ(r − r0) . . . . . . . . . . . . . . . . . . . . . . 38212.2.3 Rectangle Function RECT[x, y] . . . . . . . . . . . . . . . . . . . . . 38412.2.4 Corral Function COR[x, y] . . . . . . . . . . . . . . . . . . . . . . . . 385

12.3 Theorems of the Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . 38712.3.1 Radon Transform of a Superposition . . . . . . . . . . . . . . . . . . 38712.3.2 Radon Transform of Scaled Function . . . . . . . . . . . . . . . . . 38812.3.3 Radon Transform of Translated Function . . . . . . . . . . . . . . . . 38912.3.4 Central-Slice Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 38912.3.5 Filter Theorem of the Radon Transform . . . . . . . . . . . . . . . . 390

12.4 Inverse Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39112.4.1 Recovery of Dirac Delta Function from Projections . . . . . . . . . . 39212.4.2 Summation of Projections over Azimuths . . . . . . . . . . . . . . . 398

12.5 Central-Slice Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40212.5.1 Radial “Slices” of f [x, y] . . . . . . . . . . . . . . . . . . . . . . . . 40212.5.2 Central-Slice Transforms of Special Functions . . . . . . . . . . . . 40312.5.3 Inverse Central-Slice Transform . . . . . . . . . . . . . . . . . . . . 409

12.6 Three Transforms of Four Functions . . . . . . . . . . . . . . . . . . . . . . . 41012.7 Fourier and Radon Transforms of Images . . . . . . . . . . . . . . . . . . . . 419

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

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13 Approximations to Fourier Transforms 42113.1 Moment Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

13.1.1 First Moment – Centroid . . . . . . . . . . . . . . . . . . . . . . . . . 42413.1.2 Second Moment – Moment of Inertia . . . . . . . . . . . . . . . . . 42413.1.3 Central Moments – Variance . . . . . . . . . . . . . . . . . . . . . . 42513.1.4 Evaluation of 1-D Spectra from Moments . . . . . . . . . . . . . . . 42713.1.5 Spectra of 1-D Superchirps via Moments . . . . . . . . . . . . . . . 43113.1.6 2-D Moment Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 43313.1.7 Moments of Circularly Symmetric Functions . . . . . . . . . . . . . 435

13.2 1-D Spectra via Method of Stationary Phase . . . . . . . . . . . . . . . . . . 43613.2.1 Examples of Spectra via Stationary Phase . . . . . . . . . . . . . . 440

13.3 Central-Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45213.4 Width Metrics and Uncertainty Relations . . . . . . . . . . . . . . . . . . . . 454

13.4.1 Equivalent Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45413.4.2 Uncertainty Relation for Equivalent Width . . . . . . . . . . . . . . . 45513.4.3 Variance as a Measure of Width . . . . . . . . . . . . . . . . . . . . 455Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

14 Discrete Systems, Sampling, and Quantization 45914.1 Ideal Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

14.1.1 Ideal Sampling of 2-D Functions . . . . . . . . . . . . . . . . . . . . 46114.1.2 Is Sampling a Linear Operation? . . . . . . . . . . . . . . . . . . . . 46214.1.3 Is the Sampling Operation Shift Invariant? . . . . . . . . . . . . . . . 46214.1.4 Aliasing Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46514.1.5 Operations Similar to Ideal Sampling . . . . . . . . . . . . . . . . . 467

14.2 Ideal Sampling of Special Functions . . . . . . . . . . . . . . . . . . . . . . . 46714.2.1 Ideal Sampling of δ[x] and COMB[x] . . . . . . . . . . . . . . . . . . 470

14.3 Interpolation of Sampled Functions . . . . . . . . . . . . . . . . . . . . . . . 47214.3.1 Examples of Interpolation . . . . . . . . . . . . . . . . . . . . . . . . 478

14.4 Whittaker–Shannon Sampling Theorem . . . . . . . . . . . . . . . . . . . . . 47914.5 Aliasing and Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

14.5.1 Frequency Recovered from Aliased Samples . . . . . . . . . . . . . 48014.5.2 “Unwrapping” the Phase of Sampled Functions . . . . . . . . . . . . 482

14.6 “Prefiltering” to Prevent Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . 48314.6.1 Prefiltered Images Recovered from Samples . . . . . . . . . . . . . 48414.6.2 Sampling and Reconstruction of Audio Signals . . . . . . . . . . . . 485

14.7 Realistic Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48614.8 Realistic Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

14.8.1 Ideal Interpolator for Compact Functions . . . . . . . . . . . . . . . 49114.8.2 Finite-Support Interpolators in Space Domain . . . . . . . . . . . . . 49114.8.3 Realistic Frequency-Domain Interpolators . . . . . . . . . . . . . . . 495

14.9 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50014.9.1 Quantization “Noise” . . . . . . . . . . . . . . . . . . . . . . . . . . . 50314.9.2 SNR of Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 50514.9.3 Quantizers with Memory – “Error Diffusion” . . . . . . . . . . . . . . 507

14.10 Discrete Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

15 Discrete Fourier Transforms 51115.1 Inverse of the Infinite-Support DFT . . . . . . . . . . . . . . . . . . . . . . . . 51315.2 DFT over Finite Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

15.2.1 Finite DFT of f [x] = 1[x] . . . . . . . . . . . . . . . . . . . . . . . . 522

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15.2.2 Scale Factor in DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 52415.2.3 Finite DFT of Discrete Dirac Delta Function . . . . . . . . . . . . . . 52615.2.4 Summary of Finite DFT . . . . . . . . . . . . . . . . . . . . . . . . . 526

15.3 Fourier Series Derived from Fourier Transform . . . . . . . . . . . . . . . . . 52715.4 Efficient Evaluation of the Finite DFT . . . . . . . . . . . . . . . . . . . . . . . 529

15.4.1 DFT of Two Samples – The “Butterfly” . . . . . . . . . . . . . . . . . 53015.4.2 DFT of Three Samples . . . . . . . . . . . . . . . . . . . . . . . . . 53115.4.3 DFT of Four Samples . . . . . . . . . . . . . . . . . . . . . . . . . . 53215.4.4 DFT of Six Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . 53215.4.5 DFT of Eight Samples . . . . . . . . . . . . . . . . . . . . . . . . . . 53315.4.6 Complex Matrix for Computing 1-D DFT . . . . . . . . . . . . . . . . 534

15.5 Practical Considerations for DFT and FFT . . . . . . . . . . . . . . . . . . . . 53415.5.1 Computational Intensity . . . . . . . . . . . . . . . . . . . . . . . . . 53415.5.2 “Centered” versus “Uncentered” Arrays . . . . . . . . . . . . . . . . 53615.5.3 Units of Measure in the Two Domains . . . . . . . . . . . . . . . . . 53815.5.4 Ensuring Periodicity of Arrays – Data “Windows” . . . . . . . . . . . 53915.5.5 A Garden of 1-D FFT Windows . . . . . . . . . . . . . . . . . . . . . 54515.5.6 Undersampling and Aliasing . . . . . . . . . . . . . . . . . . . . . . 55115.5.7 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55415.5.8 Zero Padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55415.5.9 Discrete Convolution and the Filter Theorem . . . . . . . . . . . . . 55515.5.10 Discrete Transforms of Quantized Functions . . . . . . . . . . . . . 55915.5.11 Parseval’s Theorem for DFT . . . . . . . . . . . . . . . . . . . . . . 56015.5.12 Scaling Theorem for Sampled Functions . . . . . . . . . . . . . . . 562

15.6 FFTs of 2-D Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56315.6.1 Interpretation of 2-D FFTs . . . . . . . . . . . . . . . . . . . . . . . 56415.6.2 2-D Hann Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

15.7 Discrete Cosine Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

16 Magnitude Filtering 57316.1 Classes of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574

16.1.1 Magnitude Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57416.1.2 Phase (“Allpass”) Filters . . . . . . . . . . . . . . . . . . . . . . . . . 575

16.2 Eigenfunctions of Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . 57616.3 Power Transmission of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 57716.4 Lowpass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

16.4.1 1-D Test Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58116.4.2 Ideal 1-D Lowpass Filter . . . . . . . . . . . . . . . . . . . . . . . . 58116.4.3 1-D Uniform Averager . . . . . . . . . . . . . . . . . . . . . . . . . . 58116.4.4 2-D Lowpass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

16.5 Highpass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58516.5.1 Ideal 1-D Highpass Filter . . . . . . . . . . . . . . . . . . . . . . . . 58516.5.2 1-D Differentiators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58616.5.3 2-D Differentiators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58716.5.4 High-Frequency Boost Filters – Image Sharpeners . . . . . . . . . . 588

16.6 Bandpass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58916.7 Fourier Transform as a Bandpass Filter . . . . . . . . . . . . . . . . . . . . . 59416.8 Bandboost and Bandstop Filters . . . . . . . . . . . . . . . . . . . . . . . . . 59616.9 Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

16.9.1 Tiling of Frequency Domain with Orthogonal Wavelets . . . . . . . . 600

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16.9.2 Example of Wavelet Decomposition . . . . . . . . . . . . . . . . . . 602Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602

17 Allpass (Phase) Filters 60317.1 Power-Series Expansion for Allpass Filters . . . . . . . . . . . . . . . . . . . 60417.2 Constant-Phase Allpass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 60517.3 Linear-Phase Allpass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60617.4 Quadratic-Phase Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

17.4.1 Impulse Response and Transfer Function . . . . . . . . . . . . . . . 60817.4.2 Scaling of Quadratic-Phase Transfer Function . . . . . . . . . . . . 61217.4.3 Limiting Behavior of the Quadratic-Phase Allpass Filter . . . . . . . 61517.4.4 Impulse Response of Allpass Filters of Order 0, 1, 2 . . . . . . . . . 615

17.5 Allpass Filters with Higher-Order Phase . . . . . . . . . . . . . . . . . . . . . 61517.5.1 Odd-Order Allpass Filters with n≥ 3 . . . . . . . . . . . . . . . . . . 61817.5.2 Even-Order Allpass Filters with n≥ 4 . . . . . . . . . . . . . . . . . 619

17.6 Allpass Random-Phase Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 61917.6.1 Information Recovery after Random-Phase Filtering . . . . . . . . . 626

17.7 Relative Importance of Magnitude and Phase . . . . . . . . . . . . . . . . . . 62617.8 Imaging of Phase Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62817.9 Chirp Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632

17.9.1 1-D “M–C–M” Chirp Fourier Transform . . . . . . . . . . . . . . . . . 63217.9.2 1-D “C–M–C” Chirp Fourier Transform . . . . . . . . . . . . . . . . . 63417.9.3 M–C–M and C–M–C with Opposite-Sign Chirps . . . . . . . . . . . 63717.9.4 2-D Chirp Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 63817.9.5 Optical Correlator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63817.9.6 Optical Chirp Fourier Transformer . . . . . . . . . . . . . . . . . . . 641Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

18 Magnitude–Phase Filters 64718.1 Transfer Functions of Three Operations . . . . . . . . . . . . . . . . . . . . . 648

18.1.1 Identity Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64818.1.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64818.1.3 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

18.2 Fourier Transform of Ramp Function . . . . . . . . . . . . . . . . . . . . . . . 65318.3 Causal Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65418.4 Damped Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 65818.5 Mixed Filters with Linear or Random Phase . . . . . . . . . . . . . . . . . . . 66118.6 Mixed Filter with Quadratic Phase . . . . . . . . . . . . . . . . . . . . . . . . 661

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666

19 Applications of Linear Filters 66719.1 Linear Filters for the Imaging Tasks . . . . . . . . . . . . . . . . . . . . . . . 66719.2 Deconvolution – “Inverse Filtering” . . . . . . . . . . . . . . . . . . . . . . . . 669

19.2.1 Conditions for Exact Recovery via Inverse Filtering . . . . . . . . . . 67119.2.2 Inverse Filter for Uniform Averager . . . . . . . . . . . . . . . . . . . 67219.2.3 Inverse Filter for Ideal Lowpass Filter . . . . . . . . . . . . . . . . . 67519.2.4 Inverse Filter for Decaying Exponential . . . . . . . . . . . . . . . . 678

19.3 Optimum Estimators for Signals in Noise . . . . . . . . . . . . . . . . . . . . 67919.3.1 Wiener Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68019.3.2 Wiener Filter Example . . . . . . . . . . . . . . . . . . . . . . . . . . 68819.3.3 Wiener–Helstrom Filter . . . . . . . . . . . . . . . . . . . . . . . . . 689

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19.3.4 Wiener–Helstrom Filter Example . . . . . . . . . . . . . . . . . . . . 69319.3.5 Constrained Least-Squares Filter . . . . . . . . . . . . . . . . . . . 695

19.4 Detection of Known Signals – Matched Filter . . . . . . . . . . . . . . . . . . 69619.4.1 Inputs for Matched Filters . . . . . . . . . . . . . . . . . . . . . . . . 701

19.5 Analogies of Inverse and Matched Filters . . . . . . . . . . . . . . . . . . . . 70319.5.1 Wiener and Wiener–Helstrom “Matched” Filter . . . . . . . . . . . . 706

19.6 Approximations to Reciprocal Filters . . . . . . . . . . . . . . . . . . . . . . . 70819.6.1 Small-Order Approximations of Reciprocal Filters . . . . . . . . . . . 71119.6.2 Examples of Approximate Reciprocal Filters . . . . . . . . . . . . . 713

19.7 Inverse Filtering of Shift-Variant Blur . . . . . . . . . . . . . . . . . . . . . . . 719Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720

20 Filtering in Discrete Systems 72320.1 Translation, Leakage, and Interpolation . . . . . . . . . . . . . . . . . . . . . 724

20.1.1 1-D Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72420.1.2 2-D Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726

20.2 Averaging Operators – Lowpass Filters . . . . . . . . . . . . . . . . . . . . . 72820.2.1 1-D Averagers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72820.2.2 2-D Averagers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 730

20.3 Differencing Operators – Highpass Filters . . . . . . . . . . . . . . . . . . . . 73120.3.1 1-D Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73120.3.2 2-D Derivative Operators . . . . . . . . . . . . . . . . . . . . . . . . 73220.3.3 1-D Antisymmetric Differentiation Kernel . . . . . . . . . . . . . . . 73420.3.4 Second Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73420.3.5 2-D Second Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 73620.3.6 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737

20.4 Discrete Sharpening Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 74020.4.1 1-D Sharpeners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74020.4.2 2-D Sharpening Operators . . . . . . . . . . . . . . . . . . . . . . . 742

20.5 2-D Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74320.6 Pattern Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744

20.6.1 Normalization of Contrast of Detected Features . . . . . . . . . . . 74720.6.2 Amplified Discrete Matched Filters . . . . . . . . . . . . . . . . . . . 748

20.7 Approximate Discrete Reciprocal Filters . . . . . . . . . . . . . . . . . . . . . 74920.7.1 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751

21 Optical Imaging in Monochromatic Light 75321.1 Imaging Systems Based on Ray Optics Model . . . . . . . . . . . . . . . . . 754

21.1.1 Seemingly “Plausible” Models of Light in Imaging . . . . . . . . . . . 75421.1.2 Imaging Systems Based on Ray “Selection” by Absorption . . . . . 75821.1.3 Imaging System that Selects and Reflects Rays . . . . . . . . . . . 76021.1.4 Imaging Systems Based on Refracting Rays . . . . . . . . . . . . . 76121.1.5 Model of Imaging Systems . . . . . . . . . . . . . . . . . . . . . . . 761

21.2 Mathematical Model of Light Propagation . . . . . . . . . . . . . . . . . . . . 76221.2.1 Wave Description of Light . . . . . . . . . . . . . . . . . . . . . . . . 76221.2.2 Irradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76521.2.3 Propagation of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . 76521.2.4 Examples of Fresnel Diffraction . . . . . . . . . . . . . . . . . . . . . 772

21.3 Fraunhofer Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78321.3.1 Examples of Fraunhofer Diffraction . . . . . . . . . . . . . . . . . . . 785

Contents xvii

21.4 Imaging System based on Fraunhofer Diffraction . . . . . . . . . . . . . . . . 79021.5 Transmissive Optical Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 792

21.5.1 Optical Elements with Constant or Linear Phase . . . . . . . . . . . 79321.5.2 Lenses with Spherical Surfaces . . . . . . . . . . . . . . . . . . . . 794

21.6 Monochromatic Optical Systems . . . . . . . . . . . . . . . . . . . . . . . . . 79621.6.1 Single Positive Lens with z1 � 0 . . . . . . . . . . . . . . . . . . . . 79621.6.2 Single-Lens System, Fresnel Description of Both Propagations . . . 79921.6.3 Amplitude Distribution at Image Point . . . . . . . . . . . . . . . . . 80321.6.4 Shift-Invariant Description of Optical Imaging . . . . . . . . . . . . . 80621.6.5 Examples of Single-Lens Imaging Systems . . . . . . . . . . . . . . 807

21.7 Shift-Variant Imaging Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 81121.7.1 Response of System at “Nonimage” Point . . . . . . . . . . . . . . . 81121.7.2 Chirp Fourier Transform and Fraunhofer Diffraction . . . . . . . . . . 816Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819

22 Incoherent Optical Imaging Systems 82322.1 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823

22.1.1 Optical Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 82322.1.2 Spatial Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828

22.2 Polychromatic Source – Temporal Coherence . . . . . . . . . . . . . . . . . . 83822.2.1 Coherence Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 842

22.3 Imaging in Incoherent Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84222.4 System Function in Incoherent Light . . . . . . . . . . . . . . . . . . . . . . . 845

22.4.1 Incoherent MTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84622.4.2 Comparison of Coherent and Incoherent Imaging . . . . . . . . . . 847Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853

23 Holography 85523.1 Fraunhofer Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856

23.1.1 Two Points: Object and Reference . . . . . . . . . . . . . . . . . . . 85623.1.2 Multiple Object Points . . . . . . . . . . . . . . . . . . . . . . . . . . 86223.1.3 Fraunhofer Hologram of Extended Object . . . . . . . . . . . . . . . 86423.1.4 Nonlinear Fraunhofer Hologram of Extended Object . . . . . . . . . 866

23.2 Holography in Fresnel Diffraction Region . . . . . . . . . . . . . . . . . . . . 86723.2.1 Object and Reference Sources in Same Plane . . . . . . . . . . . . 86823.2.2 Reconstruction of Virtual Image from Hologram with

Compact Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87223.2.3 Reconstruction of Real Image: z2 > 0 . . . . . . . . . . . . . . . . . 87223.2.4 Object and Reference Sources in Different Planes . . . . . . . . . . 87323.2.5 Reconstruction of Point Object . . . . . . . . . . . . . . . . . . . . . 87823.2.6 Extended Object and Planar Reference Wave . . . . . . . . . . . . 88223.2.7 Interpretation of Fresnel Hologram as Lens . . . . . . . . . . . . . . 88323.2.8 Reconstruction of Real Image of 3-D Extended Object . . . . . . . . 885

23.3 Computer-Generated Holography . . . . . . . . . . . . . . . . . . . . . . . . 88523.3.1 CGH in the Fraunhofer Diffraction Region . . . . . . . . . . . . . . . 88623.3.2 Examples of Cell CGHs . . . . . . . . . . . . . . . . . . . . . . . . . 89023.3.3 2-D Lohmann Holograms . . . . . . . . . . . . . . . . . . . . . . . . 89423.3.4 Error-Diffused Quantization . . . . . . . . . . . . . . . . . . . . . . . 895

23.4 Matched Filtering with Cell-Type CGH . . . . . . . . . . . . . . . . . . . . . . 89823.5 Synthetic-Aperture Radar (SAR) . . . . . . . . . . . . . . . . . . . . . . . . . 900

23.5.1 Range Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90423.5.2 Azimuthal Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 906

xviii Contents

23.5.3 SAR System Architecture . . . . . . . . . . . . . . . . . . . . . . . . 907Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914

References 917

Index 921

Series Editor’s Preface

Science, like life, is full of unintended consequences and unanticipated benefits. For example, by 1893M. 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 hadcomplicated waveforms but simple binary values of “1” or “−1”. The amplitudes of these functionsdefined the signal in terms of the basis functions. By the 1930s this set of functions had been codifiedinto the Walsh–Hadamard transform, which was formed from a complete and orthogonal set of basisfunctions (in one or two dimensions). While the Walsh–Hadamard transform was of great mathematicalinterest, it did not have a lot of practical value until the age of the computer and its first major impact mayhave been when it was used to convert raw image data from deep-space probes into a series of bits thatrepresented the “1s” and “−1s” of the basis functions. These binary signals were ideal for transmissionfrom deep space and the image was easily reconstructed on Earth using the inverse transform. A secondexample of the unanticipated benefits is more relevant to this the 10th offering of the Wiley/IS&TSeries on Imaging Science and Technology: Fourier Methods in Imaging by Dr. Roger L. Easton, Jr. Inthe 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. Oneof 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 ofHeat. His solutions resulted in unique series and integrals using sine and cosine functions to providethe final solution to the heat conduction, over time and space, for a given system (with a well-definedgeometry). This expansion into harmonic functions came to be known as the Fourier series, Fourierintegral, or more simply the Fourier transform. Fourier analysis is used today in all fields of science andengineering.

Fourier also served as Secretary of the Institut d’Egypte in 1798–1801 during Napoleon’s expeditionto that country. The most important artifact found during this mission was the Rosetta Stone, whichincluded copies of the same text in Greek, demotic, and Egyptian hieroglyphics. Several years later,Fourier encouraged the young Jean-François Champollion to work on translating the writings, andChampollion discovered the secret in 1822. Thomas Young, another famous scientist with significantcontributions in optics, had also searched for the solution to this enigma. This aspect of Fourier’s careerand 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 interpretedby means of the visual system as an image. This is the natural spatial domain representation of theimage and can be used to understand and alter the image as one wishes. However, there is an alternatemathematical representation of this image that can also be used to understand and alter the image. Thisalternate representation is the spatial frequency domain or the Fourier transform of the image. Considerthe following set of operations. Find the average value of the image. Subtract the average value of theimage from each point in the image resulting in a new image that now has both negative and positivevalues (light has no negative values, so this is just a mathematical abstraction of the image). Nowconstruct a large (really infinite) set of two-dimensional patterns that are made up of sine and cosinefunctions. They will also have negative and positive values. Take each of these patterns and multiply

xx Series Editor’s Preface

them point by point with the average adjusted image and then sum all the values. This sum is thenthe coefficient (amplitude) for the given pattern used; this can be thought of as the projection of eachpattern onto the average adjusted image. Repeat for all the patterns (basis functions). The resulting setof coefficients, along with the average value, now represent the alternate mathematical representationof the image, its Fourier transform. One can reconstruct the image by taking all the basis functionsand 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 spatialfrequency domain, one can operate on the coefficients to alter the nature of the image and then follow theabove process to reconstruct the altered image. How all this can be done mathematically is presented inthis excellent and precise text. One other example of the Fourier transform is useful to establish a morecomplete frame of reference for this text. Using basic wave optic reconstruction it is possible to showthat the focal plane of a lens (not the image plane) contains the Fourier transform of the image. Thus onecan perform operations on the image (in real time) by placing active devices (that alter the image andre-emit light) or passive devices (that just attenuate the light) in the focal plane of the lens and then usingan identical lens to perform the inverse Fourier transform to get the altered image. Hence we see howFourier’s quest to understand how heat and temperature flow through solids has led, unintentionally, tosuch a vast and rich branch of imaging science.

Fourier Methods in Imaging provides the reader with a complete and coherent view of operatingon images in the spatial frequency domain and how these operations relate to methods in the spatialdomain, but may be easier to implement or more flexible in achieving a given goal. The first part ofthe text provides a clear review and exposition of the mathematical nature of linear system analysisfor images and carefully considers the impact of sampling (moving from the continuous domain toa 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 clearexposition of Hankel transforms (Fourier transforms in circular coordinates) and the Radon transformthat forms the basis of many medical imaging systems like X-ray tomography. Dr. Easton then providesa comprehensive review of discrete transforms (used on all computers and in digital signal processingsystems embedded in digital cameras and other digital imaging devices). These discrete transformsare the equivalent of the more general continuous transforms, but used on sampled images. Once themathematical methods have been clearly explained, Dr. Easton uses the mathematics to implement aseries of filtering applications in the spatial frequency domain, which are equivalent to more complexand 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 adigitally encoded image like that from a camera or scanner.

Fourier Methods in Imaging represents an outstanding and practical review of operating in the spatialfrequency domain for both “live optical” images and captured digital images. Every scientist or engineerworking in modern imaging systems will find this to be an indispensable reference, one that sits on hisor 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 SciencesCenter in 1986. He joined the Carlson Center for Imaging Science at The Rochester Institute ofTechnology (RIT) upon graduation and has been an integral part of the Center as both an outstandinginstructor and researcher. He received the Professor Raymond C. Bowman Award for undergraduateteaching in Imaging Science from the Society for Imaging Science and Technology in 1997. Over hisyears at RIT he has concentrated on the mathematical treatment of linear imaging systems and onthe experimental image processing techniques of “real” and captured images. In addition to his basicresearch in image science, Dr. Easton has been part of a team of scientific and historical scholars whohave focused on the preservation and reconstruction of ancient manuscripts, including the Dead SeaScrolls and the Archimedes Palimpsest. In this small way, he shares one of Fourier’s own interests inthe meaning of historical artifacts. This work resulted in his winning of the Archie Mahan Prize fromthe Optical Society of America, for the article “Imaging and the Dead Sea Scrolls” in 1988, and the

Series Editor’s Preface xxi

2003 Imaging Solution of the Year Award, by Advanced Imaging Magazine, for “Multispectral Imagingof the Archimedes Palimpsest”, January 2003.

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

MICHAEL A. KRISS

Formerly of the Eastman KodakResearch Laboratories

and the University of Rochester

Preface

This book is intended to introduce the mathematical tools that can be applied to model and predict theaction of imaging systems under some simplifying assumptions. A discussion of the mathematics usedto model imaging systems encompasses such breadth of material that any single book that aspired toconsider all aspects of the subject would be a massive tome. It should be made clear at the outset thatthis book intends no such pretense. Rather, its primary goal is to help readers develop an intuitive graspof the most common mathematical methods that are useful for describing the action of general linearsystems 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 usedfor emphasis and to facilitate development of the readers’ intuition.

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

Authors of technical books at this level must always be cognizant of the different levels ofpreparation 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 somesections) may be skipped by many readers depending on their particular needs. The particular orderof consideration of topics was chosen with some care to ensure a sequential discussion. That said, thechoice 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 inevitablevariation in reader experience and preparation. It introduces the basic mathematical concepts of linearalgebra for vectors and functions that are necessary for understanding the subsequent discussions. Theseinclude complex-valued functions, vector spaces, and inner products of vectors and functions and areintended to provide broad and less than rigorous reviews for readers who have already encounteredmathematical discussions of these topics in previous studies, such as in quantum mechanics. Thisdiscussion is similar in tone to treatments of these subjects presented by Image Reconstruction inRadiology, by Anthony J. Parker (1990), and many readers will likely be able to skim (or evenskip) some or all of it. Readers desiring or requiring a deeper discussion should consult some of thestandard texts, such as Linear Algebra and its Applications, by Gilbert Strang (2005), and AdvancedMathematical Models for Engineering and Science Students, by Geoffrey Stephenson and Paul M.Radmore (1990).

xxiv Preface

The second part (Chapters 6–13) defines a set of “special” functions and describes the mathematicaloperations 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 isintroduced. The last chapter in this part considers approximations of the Fourier transform and figuresof merit that are useful metrics of the representations in the two domains. Note that other sources existfor discussions of the special functions, including Linear Systems, Fourier Transforms, and Optics, byJack 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 requireoperations with sampled functions. The fourth part (Chapters 16–20) considers the description ofimaging 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 partconsiders 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 areview of concepts that were considered in other venues. For these readers, a logical progression wouldinclude skimming the first part, more careful study of the second and third parts, and then selection ofthe appropriate topics from the fourth and fifth parts.

Two software programs used to create the examples in this book are available online for free. Theoriginal DOS program, “signals”, creates and processes 1-D functions. It was originally written forclassroom 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 theDOSBox environment (http://www.dosbox.com). As part of her senior research project in 2009, JulietBernstein wrote the second program “SignalShow” in Java. It creates and processes both 1-D and 2-Dfunctions. 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 haveprovided inspiration and impetus to the process. Of particular note, I acknowledge John Knapp, DerekWalvoord, 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 KatieHoheusle. 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 KeithKnox, Zoran Ninkov, Elliott Horch, William Cirillo, Ed Przybylowicz, Rodney Shaw, Jeff Pelz, JonArney, 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 bementioned by name.

I would also like to thank some others who provided personal inspiration over the last 30+ years ofmy 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 andBarbara Easton.

ROGER L. EASTON, JR.Rochester, New York

September 2009

1

Introduction

1.1 Signals, Operators, and Imaging SystemsAs 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 oftenelectromagnetic radiation with units of, say, watts per unit area, while the output may be a transparentnegative 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 tomathematically describe the properties of imaging systems, and it is often convenient to use the modelof a system as a chain of links.

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

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., photonsto 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. Sometimesit is useful to model the imaging system with a second stage of energy propagation after collection, aswhen evaluating optical imaging systems composed of a single thin lens.

Fourier Methods in Imaging Roger L. Easton, Jr.c© 2010 John Wiley & Sons, Ltd

2 Fourier Methods in Imaging

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 coordinatesin space, and perhaps in time t and color, specified by wavelength λ (or equivalently by temporalfrequency ν):

Input to imaging system = f [x, y, z, t, λ] or f [x, y, z, t, ν] (1.1a)

Though these quantities have explicit spectral and temporal coordinates in addition to the spatialdimensions, the signals considered in this book most often will be functions of one or two spatialcoordinates only and are specified by f [x] or f [x, y]. In other words, the input and output signalswill be constant in time and wavelength. The number of coordinates necessary to specify the functionis the dimensionality, and the set of all possible such coordinates defines its domain. We will use theshorthand notation of “1-D” and “2-D” for one- and two-dimensional signals, respectively; 2-D signalssuch as f [x, y] are of greatest interest in imaging applications, but the study of 1-D signals will beconsidered in depth in this book as well. This is because 1-D systems often are easy to visualize and theresults 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 quantitythat will be denoted by g. Though the domain of the image g may be identical to that of the object f , itis more common to require different coordinates, and these will be denoted, when necessary, by primedcoordinates. In many cases, the output will be a function of two spatial coordinates with no dependenceon wavelength:

Output of imaging system = g[x, y, z, t, λ] → g[x, y] (1.1b)

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

Now consider some important examples in a bit more detail. For example, optical images maybe generated by light that is coherent or incoherent. At this point, we can think of coherent light ascomposed of a single wavelength λ and incoherent light as composed of many wavelengths, such asnatural light. The relevant quantity in optical imaging with coherent illumination is the complex-valuedamplitude of the electric field (including both its magnitude and the phase). The appropriate measuredquantity in incoherent (natural) light is the time average of the square of the magnitude of the complex-valued amplitude of the electromagnetic field; this is the irradiance and may be denoted by 〈|f |2〉. Instill other applications, the physical quantity represented by f may have a very different form. Thoughthe signal and the system may have different forms, most of the principles discussed in this book willbe applicable to some extent in all imaging situations.

The description of the imaging system requires a mathematical model of its action upon the inputfunction f to generate the output g. This action will be denoted by an operator represented as anupper-case script character, such as O{f [x, y, . . .]}. The operator symbolizes the mathematical rulethat assigns a particular output amplitude g to every location in its domain. In many cases, it will bepossible to describe the action of the system as the combination of a specific function associated withthe imaging system (the “system function”) and a particular mathematical operation (e.g., multiplicationor integration).

It is obvious that the output image generally is affected both by the mathematical form of the specificinput object and by the characteristics of the system. The functional expression for a common type ofsimple image would be:

O{f [x, y, z, t, λ]} = g[x′, y′] (1.2)

Introduction 3

g

Figure 1.1 Schematic of an imaging system that acts on a time-varying input with three spatialdimensions and color, f [x, y, z, t, λ] to produce a 2-D monochrome (gray-scale) image g[x′, y′].

The schematic of the imaging process is shown in Figure 1.1. The spatial domain of the output imageis often different from that of the input object, hence the use of primed characters in Equation (1.2). Inrealistic situations, the amplitude g also is affected by other parameters, such as the time, the exposuretime �t , and the spectral response of the sensor. In Equation (1.2), the effects of these additionalparameters could be considered to be implicit in the system operatorO.

1.2 The Three Imaging TasksIn many imaging applications, input objects and output images are functions of spatial dimensions only.Examples of mathematical relations for 1-D and 2-D systems are:

O{f [x]} = g[x′] (1.3a)

O{f [x, y]} = g[x′, y′] (1.3b)

Simply put, the imaging chain relates three “entities”: the input object, the action of the imaging system,and the output image. These three entities are denoted by the symbols f , O{ }, and g, respectively.A simple description of an imaging “task” is the process of specifying one of the three entities fromknowledge of the other two. Three cases are evident:

1. The forward or direct problem: to find the mathematical expression for the image g[x′, . . .]given complete knowledge of the input object f [x, . . .] and the systemO.

2. The inverse problem: to evaluate the input f [x, . . .] from the measured image g[x′, . . .] and thesystemO.

3. The system analysis problem: to determine the action of the operatorO from the input f [x, . . .]and the image g[x′, . . .] (the solution is often very similar in form to that of the inverse problem).

The solution of the direct task is often rather easy, while the others may be difficult or evenmathematically impossible. Other and more complicated variants of these imaging problems arecommon, including the cases where knowledge of the entities (f , g, and/or O) may be incompleteor contaminated by random noise. Some of the variants of the imaging problem will be considered inlater chapters.

The additional complexity of the more general imaging system model is perhaps evident just fromobservation of the form of the general 1-D imaging relation in Equation (1.3a); the operator O mustbe a function of x and x′ because it relates the object to the image. The most general form of O

4 Fourier Methods in Imaging

in Equation (1.3a) may modify the “brightness” f and/or the “location” x of all or part of the inputsignal by rearrangement, amplification, attenuation, or removal in an arbitrary fashion. For example, theimage amplitude g at a specific location could be derived from the input amplitude at the correspondinglocation, from that at a different location, or from amplitudes at multiple locations in the input f [x, . . .].The functional form of the relationship between f [x, . . .] and g[x′, . . .] may be linear or nonlinear,deterministic or random.

Though it is desirable to mathematically represent the action of system operators so that they areboth concise and generally applicable, these two characteristics usually are mutually exclusive. In otherwords, a general system operator appropriate for the imaging task likely is impossible to specify in aconcise mathematical notation.

Perhaps these examples give the readers some flavor of the difficulties to be attacked when specifyingthe action of an imaging system that is more general than the usual simple cases.

1.3 Examples of Optical Imaging

At this point we introduce a few simplified examples of optical and medical imaging systems to illustratethe imaging “tasks” and the mathematical concepts introduced in this book.

1.3.1 Ray Optics

Solution of a particular imaging task demands that the available “entities” f , g, andO be representedor modeled as mathematical expressions, which are then manipulated to derive an expression forthe unknown entity. To illustrate the concept, consider the particularly simple, yet still very useful,mathematical model of optical imaging from introductory optics. A point source of energy emitsgeometrical “rays” of light that propagate in straight lines to infinity in all directions. The imaging“system” is an optical “element” that interacts with any ray it intercepts. The interaction mechanism isa physical process (usually refraction or reflection) that “diverts” the ray from its original direction. Inthis example, the optical element is a single thin lens located at a distance z1 from the source. If thediameter of the lens is infinite, then all rays that move at all from left to right will intercept the lensand be diverted. Such a system may be described by the single parameter, the “focal length”, whichdetermines the “power” of the system to redirect rays. We will denote the focal length by the bold-faced roman “f” to distinguish it from the italic character “f ” that will be used to represent the inputamplitude. In the example of Figure 1.2, f is positive and the system redirects the light rays that emergefrom the same object point so that they converge to an image point located at some distance z2 fromthe lens. The mathematical descriptions relevant to the input “object”, the imaging system, and outputimage are respectively the distance z1 from the object to the lens, the focal length f, and the distance z2from the lens to the image point. The relationship of these three distances is the mathematical model ofthe imaging system, which is most commonly presented in the form:

1

z1+ 1

z2= 1

f(1.4a)

This simple model of the optical system is “perfect” in the sense that all light rays emerging from asingle source point of infinitesimal size at the object are assumed to converge to a single infinitesimalarea about the location: the point “image”. The relation of Equation (1.4a) may be rearranged into formsfor each of the three tasks by placing the known parameters on one side of the equation and the unknownvalue on the other. The relations for the three tasks are trivial to derive from Equation (1.4a):

1. Direct task: given the object distance z1 and the parameter f of the imaging system, find theoutput image distance z2. The mathematical expression to be solved is a simple rearrangementof Equation (1.4a) with the two known quantities on the left-hand side and the unknown on the