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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley - IEEE
- Sprache: Englisch
A comprehensive and updated overview of the theory, algorithms and applications of for electromagnetic inverse scattering problems
- Offers the recent and most important advances in inverse scattering grounded in fundamental theory, algorithms and practical engineering applications
- Covers the latest, most relevant inverse scattering techniques like signal subspace methods, time reversal, linear sampling, qualitative methods, compressive sensing, and noniterative methods
- Emphasizes theory, mathematical derivation and physical insights of various inverse scattering problems
- Written by a leading expert in the field
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Seitenzahl: 565
Veröffentlichungsjahr: 2018
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Table of Contents
Cover
Title Page
Copyright
Dedication
Foreword
Preface
Chapter 1: Introduction
1.1 Introduction to Electromagnetic Inverse Scattering Problems
1.2 Forward Scattering Problems
1.3 Properties of Inverse Scattering Problems
1.4 Scope of the Book
References
Chapter 2: Fundamentals of Electromagnetic Wave Theory
2.1 Maxwell's Equations
2.2 General Description of a Scattering Problem
2.3 Duality Principle
2.4 Radiation in Free Space
2.5 Volume Integral Equations for Dielectric Scatterers
2.6 Surface Integral Equations for Perfectly Conducting Scatterers
2.7 Two-Dimensional Scattering Problems
2.8 Scattering by Small Scatterers
2.9 Scattering by Extended Scatterers
2.10 Far-Field Approximation
2.11 Reciprocity
2.12 Huygens' Principle and Extinction Theorem
References
Chapter 3: Time-Reversal Imaging
3.1 Time-Reversal Imaging for Active Sources
3.2 Time-Reversal Imaging for Passive Sources
3.3 Discussions
References
Chapter 4: Inverse Scattering Problems of Small Scatterers
4.1 Forward Problem: Foldy–Lax Equation
4.2 Uniqueness Theorem for the Inverse Problem
4.3 Numerical Methods
4.4 Inversion of a Vector Wave Equation
4.5 Discussions
References
Chapter 5: Linear Sampling Method
5.1 Outline of the Linear Sampling Method
5.2 Physical Interpretation
5.3 Multipole-Based Linear Sampling Method
5.4 Factorization Method
5.5 Discussions
References
Chapter 6: Reconstructing Dielectric Scatterers
6.1 Introduction
6.2 Noniterative Inversion Methods
6.3 Full-Wave Iterative Inversion Methods
6.4 Subspace-Based Optimization Method (SOM)
6.5 Discussions
References
Chapter 7: Reconstructing Perfect Electric Conductors
7.1 Introduction
7.2 Inversion Models Requiring Prior Information
7.3 Inversion Models Without Prior Information
7.4 Mixture of PEC and Dielectric Scatterers
7.5 Discussions
References
Chapter 8: Inversion for Phaseless Data
8.1 Introduction
8.2 Reconstructing Point-Like Scatterers by Subspace Methods
8.3 Reconstructing Point-Like Scatterers by Compressive Sensing
8.4 Reconstructing Extended Dielectric Scatterers
8.5 Discussions
References
Chapter 9: Inversion with an Inhomogeneous Background Medium
9.1 Introduction
9.2 Integral Equation Approach via Numerical Green's Function
9.3 Differential Equation Approach
9.4 Homogeneous Background Approach
9.5 Examples of Three-Dimensional Problems
9.6 Discussions
References
Chapter 10: Resolution of Computational Imaging
10.1 Diffraction-Limited Imaging System
10.2 Computational Imaging
10.3 Cramér–Rao Bound
10.4 Resolution under the Born Approximation
10.5 Discussions
10.6 Summary
References
Appendix A: Ill-Posed Problems and Regularization
A.1 Ill-Posed Problems
A.2 Regularization Theory
A.3 Regularization Schemes
A.4 Regularization Parameter Selection Methods
A.5 Discussions
Appendix B: Least Squares
B.1 Geometric Interpretation of Least Squares
B.2 Gradient of Squared Residuals
Appendix C: Conjugate Gradient Method
C.1 Solving General Minimization Problems
C.2 Solving Linear Equation Systems
Appendix D: Matrix-Vector Product by the FFT Procedure
D.1 One-Dimensional Case
D.2 Two-Dimensional Case
Appendix References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Introduction
Figure 1.1 Schematic diagram of inverse scattering problems.
Chapter 2: Fundamentals of Electromagnetic Wave Theory
Figure 2.1 Schematic diagram of a boundary between two different media. The infinitesimal loop and pillbox are used to derive boundary conditions.
Figure 2.2 Configuration for the derivation of Huygens' principle: The space encloses all electric current sources and its complementary space is a source-free region. The direction is from source region to source-free region .
Chapter 3: Time-Reversal Imaging
Figure 3.1 Basic principle of time-reversal imaging. (a) Imaging by a mirror; (b) the functionality of mirror can be considered as a device in the black box; and (c) the received signal is sent back along the original path to yield an image at the position of the original object.
Figure 3.2 Illustration of the time-reversal implementation steps. The TRM is a transmitter-receiver array. The first step consists of recording fields radiated by the source and the second step consists of transmitting time-reversed fields back through the same medium, which tend to focus at the original source position. Adapted from: Liu 2005, IEEE Trans.
Antennas Propag.
,
53
, 3058–3066. [39] Reproduced with permission of IEEE.
Figure 3.3 The spatial distribution of the TR field for 3D scalar wave case: (a) in the plane; (b) along the -axis. The values are normalized so that the peak value is 1.
Figure 3.4 The plot of the (normalized) : Top, middle, and bottom rows show the field distribution in the , , and planes, respectively; Left, middle, and right columns show the , , and components of the field, respectively.
Figure 3.5 The component of along the - and -axes.
Figure 3.6 The base-10 logarithm of the first 15 singular values of the MSR matrix. (a) A single scatterer under TM illumination; (b) a single scatterer under TE illumination; (c) two scatterers separated by under TM illumination; and (d) two scatterers separated by under TM illumination.
Figure 3.7 DORT imaging result for a single scatterer under TM illumination.
Figure 3.8 DORT imaging results for a single scatterer under TE illumination. Images (a) and (b) are obtained by the singular vectors corresponding to the first and the second singular values, respectively.
Figure 3.9 DORT imaging results for two scatterers under TM illumination. The separation of the two scatterers is and their exact positions are marked by black dots. Images (a) and (b) are obtained by the singular vectors corresponding to the first and the second singular values, respectively.
Figure 3.10 DORT imaging results for two scatterers under TM illumination. The separation of the two scatterers is and their exact positions are marked by black dots. Images (a) and (b) are obtained by the singular vectors corresponding to the first and the second singular values, respectively.
Figure 3.11 DORT imaging result for a single scatterer under TM illumination, where transceivers cover only the angle .
Figure 3.12 DORT imaging results for a single scatterer under TE illumination, where transceivers cover only the angle . Images (a) and (b) are obtained by the singular vectors corresponding to the first and the second singular values, respectively.
Chapter 4: Inverse Scattering Problems of Small Scatterers
Figure 4.1 MUSIC method is applied to locate two point-like scatterers separated by . (a) and (b): The base 10 logarithm of the singular values of the MSR matrix for noise-free and SNR = 30 dB cases, respectively. (c) and (d): The base 10 logarithm of the pseudospectrum for noise-free and SNR = 30 dB cases, respectively.
Figure 4.2 MUSIC pseudospectrum for two identical cylinders with four different separation distances under four different noise levels. The MUSIC pseudospectrum is plotted for the line passing the two scatterers and it is normalized so that the maximum value in each subFigure is 1. The SNR levels are: No noise (solid line), 20 dB (dash line), 10 dB (dotted line), and 5 dB (dash-and-dot line).
Figure 4.3 Comparison of the result obtained by the least-squares retrieval method and that given by Marengo in [14]. The errors are averages over 1000 repetitions. The Cramer–Rao bound (CRB) of the estimation is also shown. Reproduced from Chen 2007,
J. Acoust. Soc. Am.
,
122
, 1325–1327, [15], with the permission of the Acoustical Society of America.
Figure 4.4 Logarithmic pseudospectrum for test positions in the plane for (a) and (b), and in the plane for (c) and (d). The MSR matrix is noise-free. The testing sources are: (a) an electric dipole oriented in the -axis (the position of the needle is correctly detected); (b) a magnetic dipole oriented in the -axis (the position of the needle is correctly detected); (c) an electric dipole oriented in the -axis (the position of the disk is correctly detected); and (d) a magnetic dipole oriented in the -axis (the position of the disk is correctly detected).
Source:
Chen 2008,
Journal of Physics: Conference Series
, 124, 012016. [39] Reproduced with permission of IOP Publishing.
Figure 4.5 Thirty largest singular values of the MSR matrix. (a) Noise-free and (b) 30dB Gaussian noise.
Source:
Chen 2008,
Journal of Physics: Conference Series
, 124, 012016. [39] Reproduced with permission of IOP Publishing.
Figure 4.6 Logarithmic pseudospectrum for test positions in the plane for (a) and (b), and in the plane for (c) and (d). The MSR matrix is contaminated with Gaussian noise, where SNR = 30 dB. The testing sources are: (a) an electric dipole oriented in the -axis (the position of the needle is not detected); (b) a magnetic dipole oriented in the -axis (the position of the needle is not detected); (c) an electric dipole oriented in the -axis (the position of the disk is not detected); and (d) a magnetic dipole oriented in the -axis (the position of the disk is not detected).
Figure 4.7 Logarithmic pseudospectrum for test positions in the plane for (a) and (b), and in the plane for (c) and (d). The MSR matrix is contaminated with Gaussian noise, where SNR = 30dB. The orientation of the test dipole is given by Eq. (4.35). (a) An electric test dipole (the position of the needle is correctly detected); (b) a magnetic test dipole (the position of the needle is not correctly detected); (c) an electric test dipole (the position of the disk is correctly detected); and (d) a magnetic test dipole (the position of the disk is correctly detected).
Figure 4.8 Singular values and the pseudospectrum obtained by the standard MUSIC algorithm in the noise free case. (a) The base 10 logarithm of the singular values of the MSR matrix (). (b), (c), and (d) are the base 10 logarithm of the pseudospectrum in plane obtained by the standard MUSIC algorithm with test dipoles in , , and directions, respectively.
Source:
Chen 2009,
Inverse Problems
, 25, 015008. [40] Reproduced with permission of IOP Publishing.
Figure 4.9 Pseudospectrum obtained by the proposed MUSIC algorithm in the noise free case. (a), (b), (c), and (d) are the base 10 logarithm of the pseudospectrum in the plane obtained by the proposed MUSIC algorithm corresponding to the , , , and cases, respectively.
Source:
Chen 2009,
Inverse Problems
, 25, 015008. [40] Reproduced with permission of IOP Publishing.
Figure 4.10 Singular values and the pseudospectrum obtained by the standard MUSIC algorithm in a noise-contaminated case (30dB). (a) The base 10 logarithm of the singular values of the MSR matrix (). (b), (c), and (d) are the pseudospectra in plane obtained by the standard MUSIC algorithm with test dipoles in , , and directions, respectively.
Source:
Chen 2009,
Inverse Problems
, 25, 015008. [40] Reproduced with permission of IOP Publishing.
Figure 4.11 Pseudospectra obtained by the proposed MUSIC algorithm in a noise-contaminated case (30dB). (a), (b), (c), (d), (e), and (f) are the pseudospectra in plane obtained by the proposed MUSIC algorithm corresponding to the , , , , , and cases, respectively.
Source:
Chen 2009,
Inverse Problems
, 25, 015008. [40] Reproduced with permission of IOP Publishing.
Chapter 5: Linear Sampling Method
Figure 5.1 Illustration of the current distribution for the sampling point and the proposed multipole-based interpretation. (a) Scatterer profile (relative permittivity); (b) Reconstruction LSM; (c) Current distribution (sampling point (0,0)); (d) Plot of for various
n
(sampling point (0,0)).
Source:
Agarwal 2010,
Opt. Express
,
18
, 6366–6381. [6] Reproduced with permission of The Optical Society.
Figure 5.2 Comparison of MLSM (
N
= 20) and MLSM (
N
= 1) for noise-free and noisy (10% additive Gaussian noise) scenarios. (a) Reconstruction based on
N
= 20 multipoles (noise-free); (b) Reconstruction based on
N
= 1 multipoles (noise-free); (c) Support estimated using (a) (noise-free); (d) Support estimated using (b) (noise-free); (e) Reconstruction based on
N
= 20 multipoles (10% noise); (f) Reconstruction based on
N
= 1 multipoles (10% noise); (g) Support estimated using (e) (10% noise); (h) Support estimated using (f) (10% noise).
Source:
Agarwal 2010,
Opt. Express
,
18
, 6366–6381. [6] Reproduced with permission of The Optical Society.
Figure 5.3 Cross checking the effect of reduction of multipoles. The values of are obtained for = 20 and = 1, respectively, and the difference between the two sets of is computed. The first, second, and third columns show this absolute value of the difference for = , 0, and 1, respectively.
Source:
Agarwal 2010,
Opt. Express
,
18
, 6366–6381. [6] Reproduced with permission of The Optical Society.
Figure 5.4 Comparison of LSM and MLSM. The results are obtained in the presence of noise. (a) Result of the conventional LSM for various values of α; (b) Plot of Error for various values of threshold.
Source:
Agarwal 2010,
Opt. Express
,
18
, 6366–6381. [6] Reproduced with permission of The Optical Society.
Figure 5.5 Examples of reconstruction of dielectric cylinders. The first column shows the scatterer profile (relative permittivity), the second column shows the reconstruction using the conventional LSM and the third column shows the reconstruction using the proposed MLSM. The results are obtained in the presence of noise. (a) Example 1: Austria profile [31]; (b) Example 2: Obstructed circular cylinders; (c) Example 3: Enclosed circular cylinders.
Source:
Agarwal 2010,
Opt. Express
,
18
, 6366–6381. [6] Reproduced with permission of The Optical Society.
Figure 5.6 Same as Figure 5.5 except that scatterers are perfectly conducting cylinders. (a) Example 1: Three cylinders; (b) Example 2: Nine circular cylinders.
Source:
Agarwal 2010,
Opt. Express
,
18
, 6366–6381. [6] Reproduced with permission of The Optical Society.
Chapter 6: Reconstructing Dielectric Scatterers
Figure 6.1 Inverse scattering experiment of the Austria profile: exact profile. The shaded bar shows the value of relative permittivity.
Figure 6.2 Application of noniterative inversion algorithms to the reconstruction of a weak-scattering “Austria” profile with . (a) BP method; (b) BA method; (c) EBA method; and (d) RA method.
Figure 6.3 The distribution of singular values of the operator , where the base 10 logarithm of the singular values is plotted.
Figure 6.4 BA reconstruction results as a function of regularization parameter : (a) ; (b) ; (c) ; (d) ; (e) ; and (f) .
Figure 6.5 The relative error of the BA reconstruction as a function of the regularization parameter .
Figure 6.6 Application of noniterative inversion algorithms to the reconstruction of “Austria” profile with . (a) BP method; (b) BA method; (c) EBA method; and (d) RA method. All methods fail to reconstruct a scatterer that is not weak-scattering.
Figure 6.7 The spectrum of the operator , where the base 10 logarithm of the singular values is plotted.
Source:
Chen 2010,
IEEE Trans. Geosci. Remote Sens.
,
48
, 42–49. [84] Reproduced with permission of IEEE.
Figure 6.8 The comparison of convergence trajectories in the first 50 iterations for different values of , where the base 10 logarithm of the objective function value is plotted.
Source:
Chen 2010,
IEEE Trans. Geosci. Remote Sens.
,
48
, 42–49. [84] Reproduced with permission of IEEE.
Figure 6.9 The comparison of trajectories of relative error in the first 50 iterations for different values of .
Figure 6.10 Reconstructed relative permittivity profiles at the 50th iteration for different values of . (a) . (b) . (c) . (d) . (e) . (f) .
Source:
Chen 2010,
IEEE Trans. Geosci. Remote Sens.
,
48
, 42–49. [84] Reproduced with permission of IEEE.
Figure 6.11 Reconstructed relative permittivity profiles at the 800th iteration for .
Source:
Chen 2010,
IEEE Trans. Geosci. Remote Sens.
,
48
, 42–49. [84] Reproduced with permission of IEEE.
Figure 6.12 Reconstructed relative permittivity profiles for at different noise levels. The optimization is terminated when there is no significant improvement in the objective function for two consecutive iterations, which are 59, 31, and 21, respectively, for the three noise levels. (a) noise; (b) noise; and (c) noise. Adapted from: Chen 2010,
IEEE Trans. Geosci. Remote Sens.
,
48
, 42–49. [84] Reproduced with permission of IEEE.
Figure 6.13 Illustration the concept of TSOM, where the special case , , and is considered. The subspace of is a straight line perpendicular to the plane. The intersection of two planes and is a straight line.
Figure 6.14 The left and right panels, respectively, show the real and imaginary parts of right singular vectors of corresponding to: (a) ; (b) ; (c) ; (d) ; and (e) order. Adapted from Xu, K. et al. (2014) Singular value decomposition of the current-to-field operator in solving inverse scattering problems,
IEEE Antennas and Propagation Society International Symposium
, Memphis, TN, 659–660. Reproduced with permission of IEEE.
Figure 6.15 The trajectories of the relative error of reconstruction for the NFFT-SOM and the Gs-SOM are compared.
Figure 6.16 Reconstruction results for the Gs-SOM (upper row) and the NFFT-SOM (lower row) at the 300th iteration for different choices of : for (a) and (d); for (b) and (e); for (c) and (f).
Figure 6.17 Comparisons of performances of the original NFFT-SOM and a gradual bases-expansion NFFT-SOM for , 30% noise, and . (a) Reconstruction results obtained by the original NFFT-SOM at the 400th iteration steps; (b) Reconstruction results obtained by the gradual bases-expansion NFFT-SOM at the 400th iteration steps; and (c) The trajectories of the relative errors of reconstruction for both inversion methods. In the legend, the suffix “g” denotes the “gradual bases expansion”.
Figure 6.18 Exact profile of the relative permittivity of digit patterns.
Source:
Pan 2009,
J. Opt. Soc. Am. A
,
26
, 1932–1937. [69] Reproduced with permission of The Optical Society.
Figure 6.19 Reconstructed profile of relative permittivity of digit patterns under 2D TE incidences.
Source:
Pan 2009,
J. Opt. Soc. Am. A
,
26
, 1932–1937. [69] Reproduced with permission of The Optical Society.
Figure 6.20 The scatterer is a coated cube with its inner edge length m and outer edge length m, The relative permittivity of the inner and outer layer is and , respectively.
Source:
Zhong 2011,
IEEE Trans. Antennas Propag.
59
, 914–927. [93] Reproduced with permission of IEEE.
Figure 6.21 Reconstruction results for the coated cube by the FFT-TSOM. The first and second columns are the real and imaginary parts of the reconstruction result after 122 iterations. The first, second, and third rows correspond to the cross sections at m, m, and m, respectively.
Source:
Zhong 2011,
IEEE Trans. Antennas Propag.
59
, 914–927. [93] Reproduced with permission of IEEE.
Chapter 7: Reconstructing Perfect Electric Conductors
Figure 7.1 Reconstruction of a combination of closed-contour and line-shape PEC scatterers. (a) Exact contour, (b) reconstructed contour with noise-free data, and (c) reconstructed contour under 10% white Gaussian noise.
Source:
Ye 2011,
Inverse Problems
,
27
, 055011. [28] Reproduced with permission of IOP Publishing.
Figure 7.2 Reconstruction results of rotated line-shape PEC scatterer. (a) and (b) .
Source:
Shen 2013,
IEEE Trans. Antennas Propag.
,
61
, 4713–4721. [50] Reproduced with permission of IEEE.
Figure 7.3 Reconstruction results of multiple circular PEC scatterers.
Source:
Shen 2013,
IEEE Trans. Antennas Propag.
,
61
, 4713–4721. [50] Reproduced with permission of IEEE.
Figure 7.4 A ring dielectric scatterer and a square PEC scatterer: (a) original pattern and (b) reconstructed pattern.
Source:
Ye 2013,
IEEE Trans. Antennas Propag.
,
61
, 3774–3781. [35] Reproduced with permission of IEEE.
Figure 7.5 Two circular scatterers: One PEC scatterer and one lossy dielectric scatterer. (a) Original pattern of the real part of . (b) Original pattern of the imaginary part of . (c) Reconstructed pattern of the real part of . (d) Reconstructed pattern of the imaginary part of .
Source:
Ye 2013,
IEEE Trans. Antennas Propag.
,
61
, 3774–3781. [35] Reproduced with permission of IEEE.
Figure 7.6 Configuration of the scatterers that are used in the data set “FoamMetExt,” which is the experimental data collected by the Institut Fresnel. Adapted from: Geffrin 2005,
Inverse Problems
,
21
, S117, IOP Publishing. [36]
Figure 7.7 Frequency-hopping reconstruction at 2–12 GHz using the T-matrix Gs-SOM for the “FoamMetExt” experimental data. Real part of relative permittivity: (a) 2 GHz, (b) 4 GHz, (c) 6 GHz, (d) 8 GHz, (e) 10 GHz, and (f) 12 GHz.
Source:
Ye 2013,
IEEE Trans. Antennas Propag.
,
61
, 3774–3781. [35] Reproduced with permission of IEEE.
Chapter 8: Inversion for Phaseless Data
Figure 8.1 Singular values of the MSR matrix in the first (a) and the second (b) example.
Source:
Chen 2008,
J. Opt. Soc. Am. A
,
25
, 2018–2024. [21] Reproduced with permission of The Optical Society.
Figure 8.2 The pattern consisting of five point-like objects. (a) Exact pattern of relative permittivity. (b) Reconstructed pattern with no noise added into measurement. (c) Reconstructed pattern with Gaussian white noise added into measurement (SNR = 20 dB). (d) Reconstructed pattern with Gaussian white noise added into measurement (SNR = 10 dB).
Source:
Pan 2012,
IEEE Trans. Antennas Propag.
,
60
, 5472–5475. [31] Reproduced with permission of IEEE.
Figure 8.3 The pattern consisting of a circle and an annulus. (a) Exact relative permittivity. (b) Reconstructed relative permittivity with 31.6% Gaussian white noise.
Source:
Pan 2011,
IEEE Trans. Geosci. Remote Sens.
,
49
, 981–87. [34] Reproduced with permission of IEEE.
Chapter 9: Inversion with an Inhomogeneous Background Medium
Figure 9.1 The illustration of three kinds of boundaries for an inhomogeneous background ISP where some scatterers are located inside a square annulus background. The boundary B1 encloses all scatterers, i.e., anything that is different from the inhomogeneous background medium, the B2 encloses all inhomogeneities, i.e., the region exterior to B2 is homogenous, and the B3 requires reflectionlessness for all outgoing waves.
Figure 9.2 The configuration of the inhomogeneous background medium. It is a square wall with an outer side length and inner side length , and its refractive index is . The shaded bar represents the value of the square of refractive index, i.e., .
Source:
Chen 2010,
Inverse Problems
,
26
, 074007. [10] Reproduced with permission of IOP Publishing.
Figure 9.3 The configuration of the scatterer. (a) Exact profile, where three scatterers are present, i.e., an annulus, a square inside the wall, and a rectangle attached to the wall. (b) Reconstructed profile at the 50th iteration.
Source:
Chen 2010,
Inverse Problems
,
26
, 074007. [10] Reproduced with permission of IOP Publishing.
Figure 9.4 The FD scheme requires extension of the computational domain with one additional cell outward. The cells that are exterior to the boundary are the extended cells. Their centers are marked with crosses. The cells marked with solid dots are referred to as the boundary cells and the cells marked with circles are referred to as interior cells.
Figure 9.5 Reconstruction results by the finite-difference inversion method: Left and right panels show the real and imaginary parts of relative permittivity, respectively.
Source:
Zhong 2014, Differential equation based inversion method for solving inverse scattering problems,
Inverse Problems – from Theory to Applications (IPTA 2014)
, 90–94. [35] Reproduced with permission of IOP Publishing.
Figure 9.6 Experimental setup of the through-wall-imaging problem: (a) 2D schematic diagram of the experimental setup; (b) photograph of the experimental setup; (c) cylindrical scatterer is at the center; and (d) cylindrical scatterer is off center.
Source:
Meng 2016,
Electronics Letters
,
52
, 1933–1935. [23] Reproduced with permission of Institution of Engineering and Technology (IET).
Figure 9.7 Comparison of inversion results for the off-center cylinder. Inversion result by (a) OP-homo; (b) SOP-homo; (c) OP-inhomo; and (d) SOP-inhomo.
Source:
Meng 2016,
Electronics Letters
,
52
, 1933–1935. [23] Reproduced with permission of Institution of Engineering and Technology (IET).
Figure 9.8 Schematic diagram of a confocal optical microscopy system. The modeling consists of three subsystems: focusing of incident light, interaction of focal field with the object structures, and the detection of the scattered light. OL, Objective lens; TL, Tube lens; BS, Beam splitter; PH, pinhole; PMT, Photomultiplier tube. The inset shows the focal spot and computational domain (the line box with the arrow) used in the subsystem II. The laser light of 405 nm wavelength is focused by objective lens and the intensity is recorded by a PMT with a pinhole of 20 m diameter. The sample is expanded and shown on the left-hand side.
Source:
Chen 2016,
Optica
,
3
, 1339–1347. [28] Reproduced with permission of The Optical Society.
Figure 9.9 Inverse reconstruction using four-square and four-disk patterns with 160 nm center-to-center and 40 nm edge-to-edge distances: (a) and (b) SEM images of four-square and four-disk patterns; (c) and (d) The simulated images of four-square and four-disk patterns using the proposed optical model; (e) and (f) The calibrated images from experimental images of four-square and four-disk patterns using the CLSM setup; (g) and (h) Inverse reconstruction images based on images in (e) and (f) using the proposed inversion approach with 56 iterations. Scale bars in (a) and (b) are 80 nm.
Source:
Chen 2016,
Optica
,
3
, 1339–1347. [28] Reproduced with permission of The Optical Society.
Figure 9.10 A schematic of the MIM. GHz voltage modulation is delivered to the tip of a metallic probe. When the tip of the probe is brought close to and scanned across the surface of a sample, variations of tip-sample-ground admittance are recorded, the real and imaginary parts of which are denoted as MIM-Re and MIM-Im signals, respectively.
Source:
Wei 2016,
IEEE Trans. Microw. Theory Techn.
,
64
, 1402–1408. [31] Reproduced with permission of IEEE.
Figure 9.11 (a) A three-dimensional sample with a “51” shape perturbation presented. The substrate provides a background relative permittivity and the perturbation has ; (b) top view of exact distribution of relative permittivity in (a); (c) the simulated MIM signal, where 5% white Gaussian noise is added; and (d) reconstruction of relative permittivity from the signal in (c).
Figure 9.12 (a) Three-dimensional view of a two-layer medium with , , , and . Top view of the exact distribution of relative permittivity for the (b) bottom layer and (c) top layer. (d) The simulated received MIM with 1% additive white Gaussian noise. Reconstructed distribution of relative permittivity for the (e) bottom layer and (f) top layer.
Chapter 10: Resolution of Computational Imaging
Figure 10.1 A single point source, located at the origin, is mapped to a spreading spot in a traditional optical microscopy. (a) A schematic diagram of the lens system, where the dashed box might be composed of several lenses, with various distances between them. (b) Distribution of intensity in the image plane.
Figure 10.2 The CRBs of and for (a) a single incidence and (b) 10 incidences. The wavelength is equal to 1.
Figure 10.3 (a) Ewald's sphere of , associated with the incident wavevector . (b) Ewald's limiting sphere . It is the envelope of the spheres , , …, associated with all possible wavevectors , , …of the incident fields. Adapted from: Born and Wolf 1999, Figure 13.4, p. 701. [10] Reproduced with permission of Cambridge University Press.
Figure 10.4 Inverse experiment of the Austria profile: exact profile.
Figure 10.5 Reconstruction results of an Austria profile for (a) 1.01, (b) 1.05, (c) 1.1, (d) 1.3, (e) 2, and (f) 3.5 with , respectively.
Figure 10.6 Reconstruction results of an Austria profile for (a) 1.01, (b) 1.05, (c) 1.1, (d) 1.3, (e) 2, and (f) 3.5 with , respectively.
List of Tables
Chapter 7: Reconstructing Perfect Electric Conductors
Table 7.1 The small-term () asymptotic expansions of and for both PEC and dielectric small circular scatterers.
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Ye 2013,
IEEE Trans. Antennas Propag.
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, 3774–3781. [35] Reproduced with permission of IEEE
Computational Methods for Electromagnetic Inverse Scattering
Xudong Chen
National University of Singapore
This edition first published 2018
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To Lin, Yuexin, Yide, and my parents.
Foreword
I am thankful to Dr. Xudong Chen for asking me to write a Foreword to his book on Computational Methods for Electromagnetic Inverse Scattering. This book comes at an opportune time as the field of inverse scattering has been studied for several decades now. I feel that this field is about to enter a new era, just as the field of artificial intelligence has evolved in the last three decades. To recount the history of artificial intelligence briefly, it started out as a field in computer science to emulate human intelligence with computers. However, to emulate human intelligence with the computers of three decades ago was a tall order. Very quickly, the field evolved to a less ambitious goal of developing expert systems to replace humans. Expert systems found applications in many machines that can perform quasi-intelligent menial tasks for humans. When the field of artificial neural networks was conceived, it again aroused much excitement in the computer science community: it portended great potential for machines to emulate the inner workings of the human brain. However, the excitement period subsided gradually, as many of the algorithms were too slow, and it was too difficult and time consuming to train neural networks of high complexity. Nevertheless, neural networks re-emerged later in the new field of machine learning. This was especially significant when machines were trained to beat humans in a game as complicated as the ancient oriental board game go in Japanese, or weiqi (weichi in Wade–Giles phonetics) in Chinese.
Three main reasons precipitate this breakthrough in artificial intelligence: (1) Computers have become at least 10 million times faster in the last three decades. (2) Computer memories are a lot cheaper compared to three decades ago, due to the compounding effect of Moore's Law. (3) Algorithms for information propagation through neural nets have become cleverer and faster.
Inverse scattering is facing the same juncture at this point as it shares many similar features with artificial intelligence; for instance, one of the bottle-necks of the inverse scattering algorithm is its computational cost or labor. But after several decades, computer technologies have grown a lot more powerful and cheaper. The clever use of modern computer technologies in massively parallel computations, the use of a priori data in inverse scattering and imaging, and the development of compressive sensing knowledge can be the game changers in this field. Moreover, the dogged pursuit of more efficient inverse scattering algorithms by many researchers makes the time ripe for this field to undergo a major revolution, as has been witnessed in the field of artificial intelligence.
Another reason that this field has become very interesting is that it is a field that is highly inter-disciplinary, drawing upon knowledge from mathematics, wave physics, and signal processing, as well as computer science. The confluence of various forms of knowledge and their judicious synergy are important to stimulate the next generation of technology that can follow from inverse scattering: for instance, in various forms of imaging, detection, and identification applications. This book will become an excellent resource for researchers and students who wish to learn the relevant knowledge needed for studying inverse scattering and related topics. Dr. Chen has started from the fundamentals of electromagnetic scattering theory and guides the readers slowly into the advanced form of scattering and inverse scattering theory. He also gives comprehensive coverage of the major inverse scattering techniques, plus pertinent signal processing methods. It is pleasing to see that both perfect electric conductor inversion and dielectric object inversion are discussed, as well as the complicated case when the background is inhomogeneous. Small-scatterer inversion is discussed alongside with large-scatterer inversion. The issue of phaseless imaging (or reconstruction) as well as imaging with phase information have been discussed. Phase imaging has been done at microwave frequency but is becoming increasingly popular at optical frequency as optical measurements become more precise. The manner the book is organized makes this knowledge accessible to researchers who are not in mainstream electromagnetic physics. Also, topics are added to ease the learning of computational mathematics and signal processing.
In summary, Dr. Chen should be lauded for spending the effort to write this book, which will become an important resource for researchers and students in this field.
Weng Cho ChewPurdue University
September 2017
Preface
This book is dedicated to presenting computational methods for solving electromagnetic inverse scattering problems. The intended audience includes graduate students and researchers in electrical engineering and physical sciences who are interested in inverse scattering and related imaging or who may encounter this subject in their work. Researchers in applied mathematics might also find the book useful.
There are two main reasons that motivated me to write this monograph. First, despite the fact that a rapidly expanding number of research articles on inverse scattering have been published, thanks to its wide range of real-world applications as well as the availability of powerful and cheaper computational resources, few research textbooks have been written on the subject. In particular, there has not yet been a book dedicated to solving electromagnetic inverse scattering problems without making linearization approximations. The lack of a suitable reference book has been an inconvenience for many researchers who are either in this area or are interested in entering into this subject. Second, although progress in the research into inverse scattering would not be possible without the confluence of various forms of knowledge, researchers in the engineering community usually have little knowledge on the theories and tools that have been developed in the applied mathematical community. Although there are excellent textbooks on the topic in applied mathematics, these books are usually inaccessible to engineering readers due to a lack of sufficient training in mathematics.
Based on my research experiences in the subject during 2006–2016, I wrote this monograph, keeping in mind these two concerns. The book mainly addresses inverting exact wave equations, without making linearization approximations, which results in a highly nonlinear problem. The book is written in such a way that it presents the following features:
1.
Most of the major inversion algorithms are reviewed and, in particular, their strengths and weakness are discussed, as well as their relationships to other algorithms.
2.
Important mathematical concepts, such as existence, uniqueness, and stability, are introduced. A general introduction to ill-posed problems and regularization is provided in the Appendix. Some inversion algorithms that prevail in the applied mathematical community are also introduced, such as the well-established linear sampling method. All these mathematical topics are presented in a way accessible to engineering readers.
3.
The book is highly oriented to the practical implementation of algorithms. The details of solving the forward problem and the implementation steps of individual inversion algorithms are presented such that readers can practice them without a long learning curve. Along the same pragmatic direction, several important tools are provided in Appendices.
To summarize, the book presents inverse scattering for an engineering audience in a well-balanced way; that is, emphasizing pragmatism of computational methods but still with the right formal rigor.
Keeping in mind that the research into the inverse problem requires a deep or fairly good understanding of the corresponding forward problem, I always hesitate to directly apply a general optimization method to a high-dimensional nonlinear problem, where the original forward problem is iteratively evaluated. I am convinced that insights and intuitions, no matter whether they are mathematical, physical, or engineering, potentially help us to solve the problem in a more efficient and elegant way. In inverse scattering problems, induced source plays an essential role. The analysis of induced source, such as its degrees of freedom, multipole expansion, Fourier series, and expansion with respect to singular vectors, provides deep insights into solving inverse scattering problems, which is demonstrated throughout this book.
Supplementary materials, such as the MATLAB m-files used to generate many of the examples and figures, can be found on my personal website. These materials help readers make rapid progress in learning the subject and comparing the various solution methods.
I am indebted to my Ph.D. supervisor Professor Jin Au Kong who taught me electromagnetic wave theory and to my Masters supervisors Professor Guangzheng Ni and Professor Shiyou Yang who introduced me to the field of optimization and taught me the importance of physical insight. Their passion and enthusiasm in teaching greatly influenced my view on education. I am very grateful to Professor Weng Cho Chew who was so generous in writing the Foreword to the book and provided me with valuable suggestions on my writing. The depth and width of his knowledge, as well as his interest in learning whenever and wherever possible, have deeply impressed and influenced me. I would like to thank my close collaborators Dr. Dominique Lesselier, Professor Colin Sheppard, Professor Lixin Ran, and Professor Zhi-Xun Shen, together with whom I worked on various inverse problems and imaging projects. I appreciate my friendship with many mathematicians; in particular, Professor Gunther Uhlmann, Professor Jun Zou, Professor Hongkai Zhao, Professor Jenn-Nan Wang, and Professor Gen Nakamura, who have helped me in various ways, taught me mathematics, and influenced my style of research.
I have been very fortunate to work with brilliant Ph.D. students and postdoctoral fellows on this subject, in particular, Yu Zhong, Krishna Agarwal, Li Pan, Xiuzhu Ye, Rencheng Song, Rui Chen, and Zhun Wei. Dr. Zhong and Dr. Agarwal, my first two Ph.D. students, started working on inverse scattering almost at the same time as I did. I cherish the time and effort we spent together in embarking on a new journey in inverse scattering. Special thanks go to Dr. Wei and Dr. Chen who generated many of the figures and provided a lot of editorial assistance to the book. I would also like to thank Dr. Maokun Li, who read most of chapters and provided many suggestions for improvements.
Finally, I am deeply grateful to my wife, Lin, my children, Yuexin and Yide, and my parents, for their tremendous support, patience and love during this project.
Xudong Chen
September 2017, Singapore
Chapter 1Introduction
The purpose of this chapter is to provide an overview of the book. First, the concept of electromagnetic inverse scattering problems (ISPs) is introduced, which is followed by their scientific and real-world applications. Second, we address the forward scattering problem, also known as the direct problem. Third, the fundamental properties of electromagnetic ISPs, including the existence, uniqueness, and stability of the solution, are presented. The inherent nonlinearity of ISPs is emphasized and the classification of ISPs is discussed. Finally, the scope of the book is specified. The topics covered by the remaining chapters are overviewed, which is followed by extension of the methods presented in the book to other areas. Other related topics that are not covered by the book are briefly mentioned.
1.1 Introduction to Electromagnetic Inverse Scattering Problems
The electromagnetic scattering problem deals with determining the scattered field generated by a given scatterer when it is illuminated by incoming electromagnetic waves. This is also called the forward or direct problem. The opposite of the forward problem is called the inverse problem. Electromagnetic inverse scattering is concerned with determining the nature of an unknown scatterer, such as its shape, position, and material, from knowledge about measured scattered fields.
Figure 1.1 shows a schematic diagram of inverse scattering problems. An unknown scatterer is located in the domain , referred to as the domain of interest (DOI), and is illuminated by incoming waves generated by transmitters labelled Tx1, Tx2, . For each illumination, the scattered fields are measured by an array of receivers labelled Rx1, Rx2, . The goal of the inverse scattering problem is to determine the shape, position, and material of the scatterer from the measured scattered fields.
Figure 1.1 Schematic diagram of inverse scattering problems.
Using electromagnetic waves to probe obscured or remote regions, the imaging techniques based on electromagnetic ISPs are suitable for a wide range of applications. For example, in nondestructive evaluation (NDE), the ISP has been applied to detection of possible cracks in civil and industrial structures [1–4]. In geography, this is used in remote detection of subsurface inclusions, such as detecting unexploded ordnance and mines [5, 6]. In the oil industry, it is used for oil and gas exploration [7]. In medicine, it is used for the detection of the early stages of breast cancer [8–12]. In security checks, it is applied to concealed weapon detection [13]. It can also be used for material characterization, such as the determination of constituents and evaluation of porosity [14]. Some real-world applications of inverse scattering in the microwave range can be found in chapter 10 of [15]. In physical science, the interpretation of Rutherford's gold foil experiment, which discovered the atomic nucleus, is also an inverse scattering problem.
From this short and incomplete list, it is apparent that the scope of electromagnetic ISP is extensive and its applications are diverse and important. Nevertheless, compared with its increasing importance, research in inverse scattering technique is still in the nascent stage. The purpose of this book is to introduce several computational methods for solving electromagnetic ISPs. Before discussing the inverse problem, we have to give the rudiments of the corresponding forward problem, which is the topic of the next section.
1.2 Forward Scattering Problems
Electromagnetic scattering theory is based on Maxwell's equations. Maxwell's equations are four partial differential equations that describe the electric and magnetic fields arising from distributions of electric charges and currents. Electromagnetic scattering occurs when scatterers are illuminated by a radiation source. The perturbation field due to the presence of scatterers is referred to as the scattered field; that is, the scattered field is the difference between the fields with and without the scatterers. Since the scattering problem is formulated in an unbounded domain, the boundary condition at infinity is called the radiation boundary condition, which requires the scattered field to be a local plane wave that propagates outward.
Broadly speaking, scatterers can be categorized into two types: Penetrable and impenetrable scatterers. For penetrable scatterers, the wave field is not zero inside the scatterers and satisfies the wave equation that depends on the constitutive parameters of the scatterer. At the interface between a penetrable scatterer and the background medium, continuity of certain components of electric and magnetic fields should be satisfied. For an impenetrable scatterer, the wave field is zero inside the scatterer and the total field satisfies a certain boundary condition, such as the Dirichlet (or sound-soft [16]) boundary condition, Neumann (or sound-hard [16]) boundary condition, or the impedance boundary condition. In this book, scatterers made of nonmagnetic dielectric material are penetrable scatterers, and scatterers made of perfect electric conductors (PEC) are chosen for impenetrable scatterers. In solving ISPs, the values of permittivity of dielectric scatterers have to be reconstructed, whereas the boundary of PEC scatterers has to be determined. In the applied mathematical community, scattering problems involving penetrable and impenetrable scatterers are often referred to as the medium and obstacle problem, respectively [16, 17].
This book deals with time-harmonic waves; that is, monochromatic waves. We do not specify any particular frequency range; for example, radio frequency, microwave, millimeter wave, or optical wave. Instead, we are interested in expressing dimensions and positions in terms of wavelength. The mathematical methods, both theoretical and numerical ones, used to investigate the forward and inverse scattering problems depend heavily on the operating frequency of the wave. For scatterers whose dimensions are much larger than the wavelength, the mathematical methods used to study their scattering phenomena are very different from those used for scatterers whose dimensions are much smaller than, or comparable to, the wavelength. This book is primarily concerned with the forward and inverse scattering problems associated with the scatterers whose dimensions are much smaller than, or comparable to, the wavelength.
The theories, formulations, and computational methods for the (forward) scattering problem are provided in Chapter 2.
1.3 Properties of Inverse Scattering Problems
Following the definition by Hadamard [18], a problem is well posed if its solution exists, is unique, and depends continuously on data. If one of these conditions is not satisfied, the problem is ill- or improperly posed. It is obvious that the first two properties, that is, the existence and uniqueness, should be discussed when the data is noise-free. Otherwise, for example, for a given set of measurement data that are contaminated with noise (such as measurement error and background noise), if there is no candidate acting as an input to the problem that produces an output exactly matching the measured data, then the solution to the problem does not exist. The last property, referred to as continuity (or stability), essentially means that a small perturbation of the data results in a small perturbation of the solution. Mathematical techniques known as regularization methods have been developed to construct a stable approximate solution of an ill-posed problem. More details on ill-posedness and regularization can be found in Appendix A.
For electromagnetic inverse scattering problems, we will address the following questions: the existence, uniqueness, and stability of the solution, the inherent nonlinearity, and classifications.
For electromagnetic inverse scattering problems, the question about existence is trivially confirmative since the measured scattering data for an inverse scattering problem must be generated by a certain scatterer and obviously this scatterer is an automatic solution to the inverse scattering problem. Turning to the question of uniqueness, [19] and Section 7.1 of [16] proved the uniqueness theorem under certain conditions for dielectric and PEC scatterers, respectively. The conclusion for dielectric scatterers is that, under certain conditions, for a fixed wavenumber and all directions of incidence and all polarizations of the electric field, the knowledge of the electric far field pattern for all angles uniquely determines permittivity. The conclusions for PEC scatterers are that (1), for a fixed wavenumber, the electric far field patterns for all incidence direction and all polarizations uniquely determine the PEC scatterer; and (2), for one fixed incidence direction and polarization, the electric far field pattern for all wavenumbers contained in some interval uniquely determines the PEC scatterer.
It is important to note that this book concentrates mainly on computational methods that solve inverse scattering problems with a unique solution. Inverse scattering problems that do not have a unique solution are not considered in this book. In fact, the conditions of non-uniqueness are rather stringent, and thus in practice such inverse scattering problems are not often encountered. For example, for anisotropic scatterers, if the permittivity and permeability are allowed to be zero or infinite, then it is possible to have infinite solutions to the inverse scattering problem. One of the applications of such non-uniqueness is invisibility and cloaking, and the idea of designing such kinds of anisotropic scatterers is referred to as transformation optics [20, 21]. In addition, many inverse scattering methods cannot work reliably when a penetrable isotropic scatterer does not scatter off a certain incidence wave for certain wavenumbers. In the mathematical community, such a wavenumber is referred to as the transmission eigenvalue of the interior transmission problem [22]. This book considers only scattering problems where all discrete physical and numerical resonance frequencies are avoided [23].
Next, we turn to the question of stability. Inverse scattering problems involving dielectric or PEC scatterers cannot be stably solved. In fact, even if the amount of data collected is sufficient to guarantee uniqueness, the unknown parameters (either the boundaries for PEC scatterers or the values of permittivity for dielectric scatterers) do not usually depend on the measured data in a stable way (mathematically referred to as continuous). An obvious question to ask is how large the error of the solution could be in the worst case if the error in the measured data is at most . For an ill-posed problem, the error in solution could be arbitrarily large, which means instability. In order to recover some kind of stability, we need to restrict the space of admissible unknowns by assuming that they satisfy a priori conditions, such as some kind of smoothness, sparseness, or non-negative constraint. With this a priori information, it is possible to prove that the unknowns depend in a continuous way on the measured data. Determining the modulus of this continuity is referred to as the stability estimate (Section 2.2 of [24]). For electromagnetic inverse scattering problems, it has been proven that the stability is of a logarithmic type [25, 26]. Roughly speaking, if the error in the measured data is at most , then the error of solution in the worst case is on the order of (where ). By the L'Hôpital's rule, as approaches zero, we see that a small error in measured data leads to a much larger error in the solution.
In addition to instability, the second main difficulty of inverse scattering problems is the fact that inverse problems are nonlinear, even if the corresponding forward problems are linear ones. The inverse scattering problem deals with the relationship between scattered field and scatterer's parameter, whereas the forward scattering problem deals with the relationship between the incident and scattered fields. The nonlinearity of the inverse problem is obvious due to the fact that the scattered field will not be doubled when the scatterer's permittivity is doubled. The nonlinearity is due to the multiple scattering effect that physically exists. In addition, the nonlinearity is not a convex one. The intrinsic nonlinearity of the inverse scattering problem makes the development of effective algorithms a difficult task because a solution procedure can get trapped in false solutions that are in fact different from the exact one.
Since multiple scattering effects physically exist, any imaging algorithm that ignores multiple scattering effects will cause an error, which is hard or impossible to remove using simple post-processing methods. Thus, numerical reconstruction that takes multiple scattering effects into account is expected to be one of the main research directions for the inverse problem community in the near future. Usually, the nonlinear problem is solved by casting it into an optimization problem, where the mismatch between the measured and predicted data is minimized by adjusting the unknowns that are used for prediction. The bottleneck of reconstruction algorithms that take multiple scattering effects into account lies in the high computational cost for solving the associated forward problem conducted at each iteration of the optimization process. However, the enormous increase in computing power of modern computers and the development of powerful inversion algorithms are expected to make it possible to reconstruct objects within the framework of multiple scattering in many real-world applications in the near future.
Electromagnetic inverse scattering problems can be classified by different criteria. Depending on the material properties of scatterers, the ISPs can be classified into penetrable and impenetrable ones. Depending on the size of scatterers, in comparison with the wavelength, the ISPs can be classified into two types involving small scatterers (also known as point-like scatterers) and extended scatterers. Depending on the property of background medium, classification can be homogenous and inhomogeneous background ISPs. In terms of availability of phase information of measured scattered fields, classification can be ISPs with phaseless data or phase-available data. From the angle of inversion algorithms, classification can be linear and nonlinear, iterative and noniterative, or quantitative and qualitative. When multiple scattering is taken into account, the ISP is inherently nonlinear, and consequently any inversion algorithm that directly solves the nonlinear problem is called a full-wave nonlinear algorithm. Under some conditions, such as scatterers being weak in scattering, the nonlinear problem can be well approximated by a linear one, and the accompanying algorithms are linear ones. Certain inversion algorithms are able to provide accurate or good approximate solutions with one or only a few steps of manipulation, whereas some algorithms provide final reconstruction results by iteratively minimizing the mismatch between measured and predicted data. In practical applications, the objectives of ISPs can be very different. If the values of permittivity are needed, then quantitative inversion algorithms should be adopted. If only approximate information on the shape, position, and size of scatterer is needed, then qualitative inversion algorithms suffice.
1.4 Scope of the Book
The main purpose of this book is to introduce computational methods for solving electromagnetic inverse scattering problems. The remaining chapters cover the following topics regarding forward and inverse electromagnetic scattering problems.
Chapter 2 gives the rudiments of the theory behind the forward scattering problem. The fundamentals of electromagnetic wave theory are reviewed, and equations are developed for scattering problems involving dielectric scatterers and perfect electric conductors. The chapter aims to provide readers with some quick practice in solving forward scattering problems.
Chapters 3–5 present qualitative inversion algorithms, which include time-reversal imaging, the multiple signal classification (MUSIC) method, and the linear sampling method. These inversion algorithms do not provide the values of permittivity of scatterers, but instead provide indicators that show the possibility of the existence of scatterers at particular spatial points from which the size, shape, and position of scatterers can be inferred. In particular, Chapter 4
