Mathematical Foundations of Imaging, Tomography and Wavefield Inversion

Transcription

1 Mathematical Foundations of Imaging, Tomography and Wavefield Inversion Inverse problems are of interest and importance across many branches of physics, mathematics, engineering and medical imaging. In this text, the foundations of imaging and wavefield inversion are presented in a clear and systematic way. The necessary theory is gradually developed throughout the book, progressing from simple wave-equation-based models to vector wave models. By combining theory with numerous MATLAB-based examples, the author promotes a complete understanding of the material and establishes a basis for real-world applications. Key topics of discussion include the derivation of solutions to source radiation and scattering problems using Green-function techniques and eigenfunction expansions; the propagation and scattering of waves in homogeneous and inhomogeneous backgrounds; and the concepts of field time reversal and field back propagation and the key role that they play in imaging and inverse scattering. Bridging the gap between mathematics and physics, this multidisciplinary book will appeal to graduate students working in established areas of inverse scattering and to researchers developing new computational imaging modalities. Additional resources, including solutions to end-of-chapter problems and MATLAB codes for all the examples presented in the book, are available online at / Anthony J. Devaney is Distinguished Professor of Engineering at Northeastern University, Boston and he has worked in the general area of inverse problems for more than 40 years. He did his Ph.D. at the Institute of Optics at the University of Rochester and his thesis was supervised by Professor Emil Wolf. Professor Devaney has experience in geophysics inverse problems and inverse problems related to radar, optical and acoustic imaging. His patent on diffraction tomography (covered in Chapter 8 of this book) was selected as one of the top 50 patents from Schlumberger Doll Research (the principal basic research center of the Schlumberger Corporation) over its first 50 years of existence.

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3 Mathematical Foundations of Imaging, Tomography and Wavefield Inversion ANTHONY J. DEVANEY Northeastern University, Boston

4 CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York Information on this title: / c A. J. Devaney 2012 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2012 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data Devaney, Anthony J. Mathematical foundations of imaging, tomography and wavefield inversion / Anthony J. Devaney, Northeastern University, Boston. pages cm ISBN (hardback) 1. Wave equation. 2. Inverse problems (Differential equations) I. Title. QC W28D dc ISBN Hardback Additional resources for this publication at / Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

5 Preface page xv 1 Radiation and initial-value problems for the wave equation The radiation problem Fourier integral representations Green functions Retarded and advanced Green functions Frequency-domain Green functions Green-function solutions to the radiation problem The primary field solution Representation of the radiated field in terms of boundary values via the Kirchhoff Helmholtz theorem The interior field solution The initial-value problem for the wave equation Uniqueness Field back propagation Frequency-domain solution of the radiation problem The radiation pattern and the Sommerfeld radiation condition Radiated power and energy Non-radiating sources Non-radiating sources in the frequency domain A source decomposition theorem Essentially non-radiating sources The field uniqueness theorem Surface sources Non-radiating surface sources Active object cloaking 39 Further reading 40 Problems 41 2 Radiation and boundary-value problems in the frequency domain Frequency-domain formulation of the radiation problem Analytic-signal representation of time-domain fields The Helmholtz equation Lorentz dispersive medium 45 v

6 vi The Sommerfeld radiation condition in dispersive media Incoming- and conjugate-wave radiation conditions Green functions Green functions in two space dimensions Time-domain Green functions Key features of the time-domain Green functions Green-function solution of the radiation problem Solution of the radiation problem in two space dimensions The Kirchhoff Helmholtz representation of the radiated field The interior field solution and field back propagation Radiated power and energy Non-radiating and essentially non-radiating sources in dispersive media Non-radiating sources and the radiation pattern Essentially non-radiating sources Boundary-value problems for the Helmholtz equation The interior boundary-value problem The exterior boundary-value problem for closed boundaries The exterior boundary-value problem for open boundaries Symmetry of the Green functions The Rayleigh Sommerfeld boundary-value problem The Rayleigh Sommerfeld solution for two-dimensional wavefields Rayleigh Sommerfeld representation of the radiated field Solution of the RS problem using the Helmholtz identities Back propagation and the inverse RS boundary-value problem The inverse RS boundary-value problem Connection with wavefield time reversal Surface sources in dispersive media Non-radiating surface sources 82 Further reading 84 Problems 85 3 Eigenfunction expansions of solutions to the Helmholtz equation Separation of variables and the Sturm Liouville problem The Sturm Liouville problem Cartesian coordinates Homogeneous plane-wave expansions Plane-wave expansions that include inhomogeneous plane waves Plane-wave expansions involving evanescent plane waves Spherical coordinates Multipole expansions Multipole expansions of the Dirichlet and Neumann Green functions Plane-wave expansions of the multipole fields 110

7 vii 3.5 Circular cylindrical coordinates Two-dimensional wavefields Polar coordinates 113 Further reading 116 Problems Angular-spectrum and multipole expansions The Weyl expansion The angular-spectrum expansion for the conjugate-wave Green function The angular-spectrum expansion of the incoming-wave Green function Angle-variable forms of the Green-function expansions The angular-spectrum expansion of the radiated field The angle-variable form of the radiated field expansion The angular spectrum and radiation pattern The radiation pattern of a non-radiating source Forward and back propagation using the angular spectrum Back propagation from the radiation pattern Stabilized field back propagation and the inverse boundary-value problem Back propagation using the incoming-wave Green function Back propagation using the conjugate-wave Green function and field time reversal The angular-spectrum expansion of the scalar wavelet field Angular-spectrum expansions in two space dimensions The angular-spectrum expansion of the solution to the 2D radiation problem Two-dimensional forward and back propagation The angle-variable form of the 2D angular-spectrum expansion The Fresnel approximation and Fresnel transform The 3D Fresnel approximation and Fresnel transform The 2D Fresnel approximation Multipole expansions Multipole expansion of the radiated field Forward and back propagation using the multipole expansion Back propagation in the interior boundary-value problem Back propagation from the radiation pattern Multipole expansions of two-dimensional wavefields Connection between the angular-spectrum and multipole expansions Radiated energy out of plane and spherical boundaries Radiated energy into an infinite half-space Radiated energy from a spherical region 166 Further reading 167 Problems 167

8 viii 5 The inverse source problem The ISP for the wave equation The ISP integral equation The Porter Bojarski integral equation Time reversal and the back-propagated field The ISP in terms of Dirichlet or Neumann boundary-value data The ISP for surface sources The ISP for a planar surface source Solving the ISP integral equation Interpretation of the solution The ISP for 3D sources supported in plane-parallel slabs Solving for the source Limiting form as a surface source Time-reversal imaging for slab geometry The Hilbert-space formulation of the ISP The adjoint operator Singular value decomposition The range and null space of ˆT The least-squares pseudo-inverse Filtered back propagation and back-propagation imaging The antenna-synthesis problem Implementation of the SVD The solution to the far-field ISP The algorithm point-spread function Time reversal and back-propagation imaging Picard s condition and minimum-sized sources Antenna synthesis and the far-field ISP in 2D space Implementation of the SVD The solution to the 2D far-field ISP and algorithm PSF Two-dimensional scalar wavelet source The limited-view problem The 2D limited-view problem Computing the singular system 224 Further reading 225 Problems Scattering theory Potential scattering theory The Lippmann Schwinger equation The Lippmann Schwinger equation for the full Green function The formal solution to the LS equation Scattering from homogeneous penetrable objects Scattering from homogeneous spheres and cylinders Scattering from a homogeneous sphere 236

9 ix Scattering from a homogeneous cylinder Scattering from concentric cylinders The scattering amplitude The scattering amplitude in 2D space Reciprocity and translation theorems for the scattering amplitude Scattered field energy and the optical theorem Computing the scattered field from the scattering amplitude Field scattered by an arbitrary incident wave and the generalized scattering amplitude Computing the scattering amplitude from scattered field data over a plane Multipole expansion of the scattered field Multipole expansions of 2D scattered fields The transition operator The Lippmann Schwinger equation for the transition operator The Born series The Born approximation Incident plane waves The Born approximation for spherically and cylindrically symmetric scattering potentials Born scattering from homogeneous cylinders The error between the Born and exact scattering amplitudes Non-scattering potentials Non-scattering potentials within the Born approximation Incident plane waves The relationship between the two Born non-scattering potentials Almost-invisible weak scatterers Essentially non-scattering potentials The Rytov approximation The Ricatti equation for the complex phase of the field The Liouville Neumann expansion for the phase The Rytov approximation The short-wavelength limit The Rytov transformation A comparison of the Born and Rytov approximations The hybrid approximation Incident spherical waves and slant stacking Slant stacking from arbitrary surfaces Slant-stack computation of the scattering amplitude 282 Further reading 283 Problems Surface scattering and diffraction Formulation of the scattering problem for non-penetrable scatterers 286

10 x The scattering amplitude Liouville Neumann expansion Scattering from simple shapes Scattering from an infinite Dirichlet or Neumann plane Scattering from a sphere Scattering of a plane wave from a cylinder The physical-optics approximation Plane-wave incidence Simulation The Bojarski transformation and linearized inverse surface scattering The generalized Bojarski transformation Inverse scattering within the PO approximation Kirchhoff diffraction theory The Rayleigh Sommerfeld alternative to the Kirchhoff diffraction formula More general diffraction problems Algorithmic implementation of the diffraction formulas Inverse diffraction Inverse diffraction using back propagation The SVD formulation of the inverse diffraction problem The full data case The Slepian Pollak theory Determining the shape of a surface scatterer Surface reconstruction via back propagation The SVD approach to surface reconstruction 325 Further reading 331 Problems Classical inverse scattering and diffraction tomography Born inverse scattering from far-field data Born inversion from ideal data The filtered back-propagation algorithm Inverse scattering identity The FBP algorithm in two space dimensions Born inversion from limited and non-perfect data Non-perfect data Limited-data case I Limited-data case II Non-uniqueness and non-scattering scatterers Non-scattering potentials within the Born approximation Hilbert-space formulation of Born inverse scattering Adjoint and composite operators Singular value decomposition Solution to the inverse scattering problem 353

11 xi 8.5 Born inversion using non-plane-wave probes and arbitrary measurement surfaces Data collected on arbitrary surfaces Incident spherical waves Iterative algorithms Limited-data problems Incorporation of constraints Tomographic formulation of inverse scattering The Rytov approximation The short-wavelength limit of DT Computed tomography The projection-slice theorem The filtered back-projection algorithm Computed tomography of circularly symmetric objects Diffraction tomography Hybrid formulation Reduction to a set of 2D inverse scattering problems Diffraction tomography in two space dimensions The generalized projection-slice theorem The filtered back-propagation algorithm Diffraction tomography of circularly symmetric objects Simulations of DT with ideal Rytov data Three-dimensional diffraction tomography 381 Further reading 384 Problems Waves in inhomogeneous media Background-medium Green functions The reciprocity condition for the Green functions Plane-wave scattering states The radiation problem in non-uniform backgrounds The Green-function solution to the radiation problem The Kirchhoff Helmholtz representation of the radiated field The Porter Bojarski integral equation The radiation pattern of the field Generalized plane-wave expansions Generalized plane-wave expansions to the homogeneous Helmholtz equation in a non-uniform medium Generalized angular-spectrum expansions Angular-spectrum expansion of the radiated field in nonuniform media Non-radiating sources in non-uniform media Non-radiating sources and the radiation pattern The inverse source problem 404

12 xii General formulation Singular value decomposition The least-squares pseudo-inverse solution of the ISP Solution of the ISP for spherically symmetric backgrounds Solution of the ISP for a piecewise-constant spherically symmetric background Super-resolution Scattering in a non-uniform background medium The distorted-wave Born approximation The DWBA for a pair of concentric homogeneous cylinders Foldy Lax theory Inverse scattering within the DWBA The far-field limited-view ISCP Back-propagation imaging The limited-view problem in a homogeneous background The ISCP using data generated and acquired by sets of antennas 429 Further reading 432 Problems Time-reversal imaging for systems of discrete scatterers Experimental time-reversal imaging Experimental time-reversal imaging in non-uniform media Time-reversal imaging using a finite set of antennas Experimental time-reversal imaging Eigenvectors of the time-reversal matrix Focusing with the eigenvectors of the time-reversal matrix Computational time-reversal imaging Singular value decomposition of the multistatic data matrix DORT Time-reversal MUSIC Filtered DORT and multiple-frequency algorithms The inverse scattering problem 456 Further reading 457 Problems The electromagnetic field Maxwell equations Maxwell equations for a homogeneous isotropic medium Maxwell equations in the spatial frequency domain The Helmholtz theorem The EM radiation problem The dyadic Green function The radiation patterns The Kirchhoff Helmholtz representation of the radiated field 467

13 xiii 11.4 Angular-spectrum expansions of the radiated field The angle-variable form of the angular-spectrum expansion of the EM field Back propagation from the radiation patterns The Cartesian-variable form of the angular-spectrum expansion of the EM field Forward and back propagation from planar boundary-value data The Whittaker representation of the radiated fields Boundary-value problems and field back propagation using the Whittaker representation Debye representation and multipole expansions of radiated fields Multipole expansion of the radiated field Vector spherical-harmonic expansion of the radiation pattern The EM inverse source problem The EM ISP for sources supported in spherical regions Electromagnetic scattering theory The Lippmann Schwinger equations Electromagnetic scattering amplitudes angular-spectrum expansions The Born approximation Born scattering amplitudes Born inverse scattering 493 Further reading 494 Problems 494 Appendix A Proof of the scattering amplitude theorems 496 A.1 Proof of the reciprocity theorem 496 A.2 Proof of the translation theorem 497 A.3 Proof of the optical theorem 497 Appendix B Derivation of the generalized Weyl expansion 501 B.1 Off-shell Green functions and scattering wave states 501 B.2 Derivation of the generalized Weyl expansion 502 References 505 Index 515

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15 Preface xv I started this book roughly 20 years ago with the intention of producing a finished product within a year or so. But reality in the form of government research grants and publish or perish soon set in and so now, at long last, I have finally finished. The final product has of course changed significantly over these intervening years, both in content and in breadth. My original plan was to put together a six- or seven-chapter treatise on basic Fourier-based coherent imaging and diffraction tomography complete with Matlab codes implementing the imaging and inversion algorithms presented in the text. The current book certainly includes this material, but also includes a host of other material such as the chapter on time-reversal imaging and the four chapters on the propagation and scattering of waves in homogeneous and inhomogeneous backgrounds. More importantly, the Fourier-based inversion schemes originally used to develop much of coherent imaging and linearized inverse scattering (diffraction tomography) have been replaced by the much more powerful singular value decomposition (SVD). This approach allows virtually all of the linearized inverse problems associated with the wave and Helmholtz equation both in homogeneous and in inhomogeneous backgrounds to be treated in a uniform turn the crank manner. My work on imaging and wavefield inversion began as a graduate student under Professor Emil Wolf at the University of Rochester. Originally I had intended to pursue my Ph.D. in quantum optics, but had my plans changed significantly by an off-hand remark by Professor Wolf during one of our meetings. We were discussing the classical theory of imaging by lenses, at which point he asked the question what exactly is an image? The answer to that seemingly simple question set us off on a road that included non-radiating sources, non-scattering scatterers, and other bizarre objects that the mathematician would recognize as being members of the null space of the mapping from object to image. While the purely non-radiating sources and non-scattering scatterers are in the null space of the mapping from object to image, there are other strange objects that I have chosen to call essentially non-radiating sources (or scatterers). These objects are not in the null space but are very close to it, having the property that they only radiate (or scatter) evanescent waves outside of their support and are the cause for instability of inverse problems related to the wave and Helmholtz equations. I have tried to couple these physical interpretations of non-uniqueness and instability to the purely mathematical view of these properties throughout the book. Indeed, the melding of physics with mathematics is one of my major goals in this book. The general areas of imaging and inverse scattering are multidisciplinary in that they require a strong foundation in physics, mathematics, and signal processing. I have tried to include the necessary background in all three areas, but assume that the reader is already proficient in complex-variable theory and linear algebra at the senior

16 xvi Preface undergraduate/first-year graduate level and has at least a rudimentary familiarity with the wave and Helmholtz equations in a homogeneous medium such as free space. I have also tried to emphasize the underlying physics of the various topics covered in the book but, unfortunately, at the expense of mathematical rigor. This is especially true in the development of time-independent scattering theory in Chapters 6 and 9, which follow the purely formal approach used in non-relativistic quantum scattering (collision) theory. The vast majority of the book treats scalar wave theory, with only the last chapter devoted to vector waves in the form of the electromagnetic (EM) field. The reasons for this are that all of the essential ingredients of coherent imaging and inverse scattering are already contained in the scalar theory and that the vector theory, at least for the EM field, can be reduced to three or fewer coupled scalar wave problems. Indeed, by using the so-called Whittaker or Debye representations presented in Chapter 11, EM inverse source and scattering problems for planar or spherical geometries can be reduced to two uncoupled scalar wave problems that are treated exactly in the manner presented in earlier chapters of the book. I have also, for the most part, restricted the treatment of the various inverse problems to linearized formulations of the corresponding forward problems. The exceptions to this are the inverse source problem which, by its nature, is a linear problem and one of the formulations of inverse scattering from conducting surfaces in Chapter 7. The goal of this book is to present the mathematical (and physical) foundations of imaging and wavefield inversion rather than to push specific inversion schemes or algorithms or to present detailed results of the use of such algorithms on real data. To this end, I have concentrated on simple yet representative Matlab-based examples that are easily understood and directly related to the theoretical development presented in the book. The myriad details that attend any actual application of these algorithms to real data are not presented. Such details include the methods required to retrieve the phase of an optical field in an optical-imaging or inverse-scattering algorithm and the need to align, usually through the use of digital filters, the outputs from antenna or transducer arrays in ultrasound or EM inverse-scattering or time-reversal imaging experiments. Finally, a word about the references cited in the book. Originally I intended to include as complete a list as possible of the majority of papers and books by workers in the general field of inverse scattering and wavefield inversion. I soon found the list growing beyond bound and was forced to limit the list to those references that I felt to be directly related to the material presented in the book. The book is mostly about linearized formulations of inverse scattering and, thus, I have left out an enormous number of references, especially within the mathematics community, to exact non-linear approaches to inverse scattering. I have also left out virtually all references to applications since the book is about the underlying theory of linearized inverse scattering and is not concerned with applications of this theory in various fields such as optics, acoustics, etc. I apologize to the many researchers who may feel slighted by not being included in the bibliography or not being suitably referenced. I would like to thank my former professor and good friend and colleague over the past (can it be 40?) years Emil Wolf. Much of the material in the book can be traced back to my Ph.D. thesis and to joint papers by Emil and myself. I would also like to thank my colleague of many years standing Dr. George Sherman and the dozens of current and

17 xvii Preface former students and colleagues who collaborated on the development and application of the material presented in the book. Special thanks go to my former friend and colleague Alan Witten, who died unexpectedly in Alan, who was professor of geophysics at the University of Oklahoma, used acoustic diffraction tomography to help find and unearth seismosaurus, the longest dinosaur yet discovered (see NY Times New X-Ray Technique Helps Dinosaur Hunters, Science Section, Dec. 12, 1989), and whose work was, at least partially, the motivation for the opening scenes in the original Jurassic Park movie. I would also like to thank Dr. Arje Nachman of the AFOSR and Dr. Richard Albanese, director of the mathematical products division at the Brooks Air Force Base in San Antonio, for financial and inspirational support over the past 20 years. Finally, I must thank Simon Capelin and the wonderful staff at Cambridge University Press. Simon first met me about the book in 1990 in my company office in downtown Boston to discuss the project that I promised would be finished in less than a year.