Subspace based inversion methods for solving electromagnetic inverse scattering problems

Subspace-based Inversion Methods for Solving
Electromagnetic Inverse Scattering Problems
Zhong Yu
(M. Eng., B. Eng., Zhejiang University, China)
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY OF ENGINEERING
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2009
Acknowledgements
My deepest gratitude goes first and foremost to Dr. Chen Xudong, my supervisor,
for his constant warming encouragement and professional gu idance. Without his
consistent and illuminating instruction, this thesis could not have reached its present
form. I would like to thank the Nat io n al University of Singapore for providing
scholarsh ip to support me to pur su e my d octoral degree in electromagnetic inverse
problems. I also own my gratitude t o Prof. Ran Lixin from Zhejiang Univeristy,
who, as my supervisor when I was in Zhejiang University, introduced me into the
world-class electromagnetic re sear ch area. I would like to thank staff from Microwave
and RF research group in the Department of Electrical and Computer Engineering,
especially Prof. Leong Mook Seng, Prof. Li Le-wei, Prof. Ooi Ban Leong, Dr.
Koen Mouthaan, Mr. Sing Cheng Hiong, and Ms. Guo Lin for teaching me the
fundamentals of electrom agnetics and providing their kind assistance during my
doctoral study. I wou l d like to express my appreciation to my fellow team mates from
microwave research lab and MMIC lab, especially Krishna Agarwal, for her always
helpful discussion and her selflessness of maintaining the comput ing instruments,
Wang Ying, Zhang Yaqiong, Tang Xinyi, Nan Lan, Ch en Ying, and Zhong Zheng,
for t h ei r friendliness to share their most genuine happiness with me all t h ese year s.
Last but not least, I would like to present my heartfelt gratitude to my parents.
Without their decades’ support and sacrifice, I would not be able to pursue my
dream and reach the place where I am now. I would like to dedicate this thesis to
them, especiall y my father.

i
Contents
Acknowledgements i
Contents ii
Summary v
List of Figures vi
List of Acronyms ix
List of Publications xi
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Original contributions and overview of the thesis . . . . . . . . . . . . 6
2 Preliminaries 9
2.1 Forward problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Inversion methods for point-like scatterers . . . . . . . . . . . 17
2.2.2 Inversion methods for extende d scatterers . . . . . . . . . . . 21
ii
iii
3 Subspace-based inversion methods for small scatterers 26
3.1 A r obust non-iterative method for retrieving scattering strength . . . 27
3.1.1 The least squares retrieval method . . . . . . . . . . . . . . . 28
3.1.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 MUSIC imaging method for small anisotropic scatterers . . . . . . . . 33
3.2.1 Formulas for the forwar d problem of the multiple-scattering
small anisotropic spher es . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Inverse scattering problem . . . . . . . . . . . . . . . . . . . . 39
3.2.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . 43
3.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 MUSIC imaging method with enhanc ed resolution . . . . . . . . . . . 50

3.3.1 Forward scattering problem . . . . . . . . . . . . . . . . . . . 52
3.3.2 The MUSIC algorithm with enhanced resolution . . . . . . . . 53
3.3.3 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Subspace-based inversion methods for extended scatterers 64
4.1 SOM and nested SOM . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.1 The subspace-based optimizat i on method . . . . . . . . . . . . 65
4.1.2 The nested SOM . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Twofold SOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.1 The twofold SOM . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.2 Computational test . . . . . . . . . . . . . . . . . . . . . . . . 77
iv
4.2.3 Discussion and summary . . . . . . . . . . . . . . . . . . . . . 82
4.3 Improved SOM and its im plementation in three-dimensional i nverse
scattering problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3.1 Three-dimensional SOM . . . . . . . . . . . . . . . . . . . . . 85
4.3.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 90
4.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5 Conclusion 96
Bibliography 101
A Derivation of the Cramer-Rao bound 113
Summary
This thesis studies several methods for solving electromagnetic inverse scattering
problems, all of which are on the basis of the concept of subspace. The original
contrib u t i on s of this thesis can be cataloged into two folds: Firstly, we n ot only
apply the multiple signal classification (MUSIC) method to locate small anisotropic
scatterers, dimensions of which are much less than the wavelen gt h , but also pr o pose
a new MUSIC algorithm that improves resolution and in the meanwhile is able to
deal with small degenerate scatterers; Secondly, we propose a new series of subspace-
based optimization methods (SOM) to solve the inverse scattering problems for

extended scatterers, including th e nested SOM, twofold SOM, and improved SOM.
Based on the concept of subspace, we actually utilize the most stable part of the
measured scattered fields, thus, methods proposed in th i s thesis not only converge
fast but also are quite robust against noi se. Various numerical simulations have
been carried out and validate the proposed algorithms.
v
List of Figures
3.1 Comparison of the r esu l t obtained by least squares retrieval method
and that given i n [1] for the c ase that the scatterers have same scat-
tering strengths, τ
m
= 1, m = 1, 2, 3, 4. The er r o rs are averages over
1000 repetitio n s. The CRB of the estimation is also shown . . . . . . . 30
3.2 Normalized percentage of th e est i m at i on errors for the case that t he
scatterers have different scattering streng t h s, τ
m
= m, m = 1, 2, 3, 4.
The errors are averages over 1000 repetitions. The CRB of the esti-
mation is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Normalized percentage of th e est i m at i on errors for the case that t he
number of the transceivers is 31 and the scatterers have same scat-
tering strengths, τ
m
= 1, m = 1, 2, 3, 4. The er r o rs are averages over
1000 repetitio n s. The CRB of the estimation is also shown . . . . . . . 33
3.4 The definitio n of the rotation angles φ
n,m
, θ
n,m
and ϕ

n,m
, where e
(l)
n,m
is the l
th
electric (n = E) or magnetic (n = H) principle axis of the
m
th
scatterer, l = 1, 2, 3 and m = 1, 2, . . . , M. . . . . . . . . . . . . . 35
3.5 Pseudo-spectrum image and the accuracy of the retrieval of the scat-
tering strength tensors for two small anisotropic spheres located at
(0, 0) and (0, λ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6 Pseudo-spectrum image and the accuracy of the retrieval of the scat-
tering strength tensors for four small anisotropic spheres located at
(0, 0), (0, λ/12), (λ/12, 0) and (λ/12, λ/12). . . . . . . . . . . . . . . . 48
3.7 Singular values and pseudo-spectrum obtain ed by the standard MU-
SIC algorithm in noise free case. (a) The 10 base logarithm of the
singular values of the MSR matrix (j = 1, 2, . . . , 48). (b ) , (c) and (d)
are the 10 base logarithm of the pseudo-spectrum in y = x + 0.112λ
plane obtained by the standard MUSIC algorithm with test dipoles
in x, y and z directions, respectively. . . . . . . . . . . . . . . . . . . 57
vi
vii
3.8 Pseudo-spectrum obtained by the proposed MUSIC algorithm in noise
free case. (a), (b), (c) and (d) are the 10 base logarithm of the pseudo-
spectrum in y = x + 0.112λ plane obtained by the proposed MUSIC
algorithm corresponding to the L = 4, 5, 6 and 7 cases, respectivel y. . 58
3.9 Pseudo-spectrum obtai n ed by the proposed MUSIC algorithm in noise
free case when the test dipole i s constrained to be real . (a), (b), (c),

(d), (e) and (f) are the 10 base logarithm of the pseudo-spectrum in
y = x + 0.112λ plane obtained by the proposed MUSIC algorithm
corresponding to the L = 4, 5, 6, 7, 8 and 9 cases, respect i vely. . . . . 59
3.10 Si n g u l ar values a n d pseudo-spectrum obtained by the standard MU-
SIC algorithm in noise-contami n a te d case (30dB). (a) The 10 base
logarithm of the singular values of t h e MSR matrix (j = 1, 2, . . . , 48).
(b), (c) and (d) are the p seudo-spectrum in y = x + 0.112λ plane
obtained by the standard MUSIC al go r it hm with test dipoles in x, y
and z dir ect i ons, respectively. . . . . . . . . . . . . . . . . . . . . . . 60
3.11 Pseu d o -s pectrum obtained by the proposed MUSIC algorithm in noi se-
contamin a te d case ( 30 d B) . (a), (b), (c), (d), (e) and (f) are the
pseudo-spectrum i n y = x + 0.112λ plan e obtained by the proposed
MUSIC algorithm corresponding to the L = 4, 5, 6, 7, 8 and 9 cases,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.12 Pseu d o -s pectrum obtained by the proposed MUSIC algorithm in noi se-
contamin a te d case (30dB) when the test dipole is constrained to
be real. (a) , (b), (c), (d), (e) and (f) are the pseudo-spectrum in
y = x + 0.112λ plane obtained by the proposed MUSIC algorithm
corresponding to the L = 4, 5, 6, 7, 8 and 9 cases, respect i vely. . . . . 62
4.1 The original dielectric profile of the ’Austria’ structure. . . . . . . . . 71
4.2 Reconstruction r esu l t obtained using 20 × 20 mesh grid after 100
iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Reconstruction r esu l t obtained using 30 × 30 mesh grid after 100
iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4 Reconstruction r esu l t obtained using 30 × 30 mesh grid after 150
iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Reconstruction result obtained using 40 × 40 mesh grid after 1000
iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
viii
4.6 Reconstruction result obtained using 64 ×64 mesh grid after 5 iter a-

tions with result in Fig. 4.2 as the initial guess. . . . . . . . . . . . . 72
4.7 Singular values of
G
S
and G
D
. . . . . . . . . . . . . . . . . . . . . . 7 6
4.8 The recalcul at ed objective function values whi le choosing M
0
= 50, 100, 200
and 500. The original SOM’s curve i s also presented. . . . . . . . . . 78
4.9 Reconstruction results using M
0
= 50, 100 , 200 and 500 with 10%
additive noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.10 Recon s t r u ct i on result after 5 iterations using M
0
= 500 with the
initial guess generated by using M
0
= 50 after 30 iteration s with 10%
additive noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.11 Recon s t r u ct i on results in the first and second step of TSOM on scat-
tering data with 30% additive noise. . . . . . . . . . . . . . . . . . . . 80
4.12 Recon s t r u ct i on results in the first and second step of TSOM on scat-
tering data with 50% additive noise. . . . . . . . . . . . . . . . . . . . 81
4.13 Recon s t r u ct i on results of inhomogeneous scatterers in the first and
second step of TSOM on scattering data wi th 10% additive noise. . . 82
4.14 A coated cube with its inner edge length a = 0.6λ and outer edge
length b = 1.6λ. The relative permittivity of the in n e r cube is ǫ

r1
=
1.6 and the relative permittivity of the outer layer is ǫ
r2
= 1.3. . . . . 90
4.15 Si n g u l ar values of
G
3D
S
. . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.16 The objective function values withi n 60 iterations when L = 1, 5, 10, 30
and 60. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.17 The real part of the retrieval result of the dielectric profile for the
domain of inter est after 60 it er at i ons. The real par t of the rela t ive
permittivity of the inner cube and outer layer are 1.6 and 1.3, respec-
tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.18 The imaginar y part of the retrieval result of the dielectric profile for
the domain of interest after 60 iterations. The imaginary part of the
relative permittivity of the inner cube and ou t er layer are both 0. . . 94
List of Acronyms
2D Two-dimens io n al
3D Three-dimensional
AD Analog to digital
CDM Coupled dipol e method
CG Conjugate gradient
CRB Cramer-Rao bound
CSI Contrast source inversion
DDA Discrete dipol e approximation
DOA Direction of arrival
DORT Decomposition of the time reversal operator

EFIE Electric field integral equation
EISP-ES Electromagnetic inverse scattering problems for extended scatterers
EISP-PLS Electromagnetic inverse scattering problems for point - l i ke scatterers
FFT Fast Fou r i er transform
LM Levenberg-Marquardt
MOM Method of moment
MSE Multiple scattering effect
MSR Multi-static response
MUSIC Multiple signal classificati on
ix
x
SNR Signal to noise ratio
SOM Subspace-based optimization method
SVD singular value deco m position
TM Transverse magnetic
TSOM Twofold subspace-based optimization method
TSVD Truncat ed singular value decomposition
List of Publications
1. Th e content in Chapter 3, Sectio n 1 has been published as
Xudong Chen and Yu Zhong, “A robust noniterative method for obtain-
ing scattering strengths of multiply scattering point targets,” J. Acous. Soc.
Am., Vol. 122, pp. 1325 -1 327 , 2007.
2. Th e content in Chapter 3, Sectio n 2 has been published as
Yu Zhong and Xudong Chen, “MUSIC imaging and electromagnetic inverse
scattering of multiple-scattering small anisotropic sp h er es ,” IEEE Trans. An-
tenna and Propag., Vol. 5 5, pp. 3542-3549, 2007.
3. Th e content in Chapter 3, Sectio n 3 has been published as
Xudong Chen and Yu Zhong, “MUSIC electromagnetic imaging with en-
hanced resolution for small inclusions,” I nver se Problems, Vol.25, ID:0150 08
(12pp), 2009.

4. Th e content in Chapter 4, Sectio n 1 has been accepted as
Yu Zhong and Xudong Chen, “A Nested Subspace-Based Algorit h m for Solv-
ing Inverse Scattering Problem,” International Conference on Inverse Prob-
lems, Wuhan, China, accepted.
5. Th e content in Chapter 4, Sectio n 2 has been published as
Yu Zhong and Xudo n g Chen , “Twofold su b space-based optimization meth od
for solving inverse scattering problems,” Inverse Problems, Vol 25, ID:085003
xi
xii
(11pp), 2009.
6. Th e content in Chapter 4, Sectio n 3 has been accepted as
Yu Zhong, Xudong Chen, and Krishna Agarwal, “An improved sub sp ac e-
based opt i m iz at i on method and its im p l em entation in solving three-dimensional
inverse problems,” IEEE Trans. Geosci. Remote Sens., accepted, 2010.
In addition, there are three coauthored papers the content of which are not
included in this thesis:
1. Li n fa n g Shen, Xudong Chen, Yu Zhong and Krishna Aga r wal, “The effect of
absorption on terahertz surface plasmon polaritons propagating along period-
ically co rr ugated metal wires,” Physical Review B, Vol. 77, pp. 075408(1-7),
2008.
2. Li Pan, Krishna Agarwal, Yu Zhong, Swee Ping Yeo, and Xudong Chen,
“Subspace-based opt i m iz at i on method for reconstructin g extended scatterers:
Transverse Electric case,” J. Optic. Soc. Am. A, Vol. 26, pp. 1932-1937,
2009.
3. Kri shna Agarwal, Xudong Chen, and Yu Zhong, “A multipole-expansion
based linear sampling method for solving inverse scatter ing pr ob l em s, ” Optics
Express, Vol. 18, pp. 6366-6381, 2010.
4. Xu d ong Chen and Yu Zhong, “Influence of multiple scattering on the reso-
lution in inverse scattering,” J. Optic. Soc. Am. A, Vol. 27, pp. 245-250,
2010.

5. Xiu z hu Ye, Xudong Chen, Yu Zhong, and Krishna Agarwal, “Subspace-
based optimization method for reconstructing perfectly electric conductors,”
Progress in Electromagnetic Research, Vol. 100, pp. 119-128, 2010.
6. Tianjian Lu, Krishna Agarwal, Yu Zhong, and Xudong Chen , “Through-
wall imaging: application of subspace- based optimizatio n meth od,” Progress
in Electromagnetic Research, Vo l. 102, pp. 351-366, 2010
Chapter 1
Introduction
This thesis deals with inversion methods for solving electromagnetic i nverse scatter-
ing problems, by which we use electromagneti c wave to probe the location, shape,
and physical characteristics of scattere rs . Methods studied in the thesis include
those for small scatterers, dimensions of whi ch are much smaller than the wave-
length of the illumination, and those for extended scat t er e rs , dimension s of which
are comparabl e with the wavelength of the illumination. In this introductory chap-
ter, a brief survey of th e topic is given, followed by original contributions of this
thesis and the structure of the thesis.
1.1 Background
When talking about inverse problem, one has to mention its counterpart, the for-
ward problem. We are usually used to describe a natu r al phenomenon by using a
physical model. For instance, we use Coulomb’s law to precisely describe the interac-
tion between two small charges, i.e., after knowing the quantities of two charges and
the distan ce between them, we can calculate the force on th e two ch a r ges. We may
1
2
define this prob l em as forward problem. Then, its counterpart, the inverse problem,
could be, knowing the force on the two charges and the quantities of charges, we
want to find out the distance between the two small charges. From this example,
we may summarize that inverse problem and forward problem actually are a pair
of problems regarding the sam e model but the input (con d i t io n ) and output (so lu-
tion) of these two types of problems are somewhat reversed [2] [3], e.g., in above

example, the dist an ce between the two charges is the input of the forward problem
but the output of the inverse problem. After giving this initial impression of th e
inverse problem, let us focus on the topic of this thesis, the electromagneti c inverse
scattering problem. It is the same procedure as above that, when we are talking
about the inverse problem, we need to refer to its counter p ar t . Th e electromagnetic
forward scattering problem, or the name we usually use, the electromagnetic scat-
tering problem, is the problem to find out electromagnetic scattered fields generated
by some obstacles or some inhomogeneous media. The inputs (cond i t i on s ) are the
electromagnetic i n ci d ent wave and the physical properties of scatterers, such as the
geometric distribution, the permittivity, and permeability of every scatterer. One
of the most famous electromagnetic scattering problem is the Rayleigh scattering
problem, which exp l ai n s why the color of the sky is blue [4]. If we reverse the
solution and the conditions of the electromagnetic forward scattering problem, we
have the electromagnetic inverse scattering problem, i.e., by given the knowledge of
scattered fields, the inci d e n ces and maybe some other a priori informa ti o n , we need
to find out the geometric dist r i b u t i on, like the shape and the location, and physical
parameters, like permittivity and ( or ) permeability, of scatterers [5].
Electromagnetic inverse scatterin g problems have played a central role in many
important civil and military applications in our daily life, such as in radar , non-
destruction detection, medical examinati on , cell-level imaging, semiconductor flaw
detection, etc Due to all these important applications, it is essential to deve lo p
precise and fast methods to solve various inverse scattering probl em s. However,
3
because of the intrin si c nonlinearity and (or) ill-posedness, the inverse scatt e ri ng
problems usually can only be solved within some precision and the computational
cost is usually quite large.
The electromagn et i c inverse scattering problems studied in this thesis can be
divided into two types: electromagnetic inverse scattering problems for point-like
scatterers (EISP-PLS), the dimensions of which are much smaller than the wave-
length o f the illuminating waves; el ect r om a gn et i c inverse scattering probl em s for

extended scatterers (EISP-ES), the dimensions of which are comparable with the
wavelength of the illuminating waves. These two types of problems are both nonlin-
ear, but are fundame ntally different from each other from the view of whether the
problem is properly posed. In Hadam ar d ’ s sense [6], a problem is properly posed,
or well-posed, if
• The solution of the problem exists;
• The solution of the problem is uniqu e;
• The solution of the problem is stable.
If the number of the detectors an d the number of the incidences are both larger
than the number of the induced independent se con dary point sources inside those
point-like scatterers, there is a well-determined gap between the large singular values
and small singular values of the multi-st at i c response (MSR) matrix for these small
scatterers. Such a characteristic insures an injectivity between the scattered fields
and the induced secondary sources [7], and thus it is easy to construct a stable
inversion from the scattered fields to t h e induced secondar y sources. In this sense,
according to above definition, the EISP-PLS are well-posed . If any of the above
three co n d i t i on s cannot be fulfi l l ed , the problem is improperly posed, or ill-posed.
The EISP-ES are ill-posed due to its com p a ct scattering operator that maps the
induced current inside scatterers to the measured scattered fi el d s [8, 9, 5]. Methods
4
for solving EISP-PLS and EISP-ES are quite diff er ent, since one needs to deal with
the stability problem when solving the latter one.
Because EISP-PLS are properly posed, methods for solving th ese problems only
need to address the non l i n ea r ity characteristic and try to give a more precise solution,
or in other word, try to obtain a better resolution under the condition of noise level.
Traditi onal methods, like beamforming method, do not have good resolving ability,
thus researchers turn to some methods that are based on the spectral information of
the measured scattering data, such as the two that have been intensively discussed
recently, including th e Decomposition of the Time Reversal Operator (DORT, a
French acronym) [10–14] and Multiple Signal Classification (MUSIC) [11, 15–17, 7]

methods. Both methods are based on the singular value decomposition of the so-
called multi-st a t ic response (MSR) matrix. The main difference between these two
methods is that the DORT m et hod needs scatterers are well-resolved in order to lo-
cate them, while MUSIC method is not constrained by such condition. Thus, MUSIC
method could obtain a better resolution than the DORT method does. However,
if one uses the DORT technique in a wide-band scenario, he can locate small scat-
terers embedded in inhomogeneous background, which is the time-reversal mirror
techniqu e [18]. Despite its good resolving ability, MUSIC method was mainly used
in solving direction of arrival (DOA) problem in signal processing society [19–26] and
was only recentl y introduced into acoustical society for solvi n g acoustic inverse scat-
tering problems. The transplant of MUSIC method from acoustic inverse scattering
problem to electro m agnetic inverse scattering problem is not so straightforward due
to electromagnetic wave’s polarization characteristic, which may also supply some
margin to further develop the m et h od. After obt ai n i n g the location of point-like
scatterers, scattering str en g t h s of scatterers need to b e retrieved either by iterative
method [15] or non-iterative method [1].
For EISP-ES, as aforementioned, due to the intrinsic ill-posedness, it is difficult
5
to solve them. Research wor kers in mathemati cal society, physical society, and elec-
trical engineering society have devoted themsel ves in developing more stable and
more efficient solvers for decades. Chronologically, methods for one-dimensional
EISP-ES were first developed, followed by the methods for two-dimensional and
three-dimensional EISP-ES thanks to the amelioration of the fast computational
techniqu es and high performance computing equipments. Methods that have been
intensively discussed and used in practical scenarios could be cast into two large
groups: methods based on the integral equation solver of Maxwell equations and
methods bas ed on the differential equation solver of Maxwell equations. These two
kinds of methods have thei r own advantages and disadvantages due to the forward
problem solver they adopt, e.g., the numbe r of unknowns in methods based on dif-
ferential equat io n solver is usually larger than the one in methods based on integral

equation solver, but the form er does not need the explicit expression of Green’s func-
tion while the latter needs [27]. In this thesis, the methods based on integral equa-
tion solver are investigated. Until nowadays, methods tha t are based on the integral
equation solver of Maxwell equations mainly include the Born iterative method [28],
distorted Born iterative method [29–32], modified gradient method [33], an d con-
trast source invers io n (CSI) method [34]. They use the integral equation solution
of Ma xwell equations to set up an objective function that measures the mismat ch
of the scattering dat a an d (or) the mismatch of the ind u ced cu r r ent sources inside
the domain of interest. By minimizing such an objective function, the spatial dis-
tribution of permittivity and (or) permeability of scat t er er s could be obtained. Due
to aforementioned intrinsic nonlinearity and ill-posedness of this problem, iterative
optimization strategy and regularization are necessary. Usually, when the num-
ber of unknowns of the problem becomes large, the optimization usually converges
quite slow and thus the com p u t at i onal burd en dramatically increases. Besides these
methods, there are some other methods that have been discussed in applied math-
ematical society, such as linear sampling method, factorization method, level set
method, etc. [3 5, 36]. The linear samp l ing method and factorization method be-
6
long to the quantitative method that has quite low computati on cost . However, as
mention ed by some researchers, they have difficulty in reconstructing the g eom et r i c
shape of scatterer that is not simply connected, such as an annular object [37].
The subject of this thesis i s in two fo lds: First, to investigate MUSIC methods
for solving electromagnetic inverse scattering problems for point-like sca t te r er s, so
as to obtain a better resolution; Second, to investigate methods for solving elec-
tromagnetic inverse scatterin g problems for extended scatterers, which makes the
optimization converge fast er and obtain satisfactory reconstruction results, so as to
decrease the whole computational cost of the solver.
1.2 Original contributions and overview of the the-
sis
The original contributions of t h e thesis consist two parts: methods for solving EIS P-

PLS and methods for solving EIS P- ES , both of which are on the basis of the concept
of subsp ace. The subspace-based methods for solvi n g EISP-PLS are introduced in
Chapter 3, while those for EISP-ES are introduced in Chapter 4. Before these two
chapter, in Chapter 2, some preliminaries are given, and the thesis is summarized
in Chapter 5. Following are the detailed construction of this thesis.
Chapter 2 r ev ie ws a forward problem model that is usually used in solving
inverse scattering problems, and several inversion techniques that are closely related
to our works. For the forward problem model, the coupled dipole method (CDM)
is derived. This method is on the basis of electric field integral equation (EFIE).
The singularity of the original EFIE is rigor o u sl y discussed. In the second part of
Chapter 2, several inversion techniques are discussed, for both point-like scatterers
and extended scatterers. First, the MUSIC that is used in solving acoustic inverse
7
scattering problems and its preliminary usage in solving electromagnetic inverse
scattering pr o b l em s for point-like scatterers is presented. Second, the inversion
techniqu e based on EFIE for solving t h e electromagneti c inverse scattering pr o b l em s
for extended scatterers are discussed, such as Born iterative metho d , distorted Born
iterative method, contrast sour c e inversion m e thod, and some other methods that
treat the whole inverse scattering problem as two separate physical pr ocesses: the
process of scattering from the induced secondary sources inside scatterers and the
process of inducing those secondary sources.
In Chapter 3, the applica ti o n and extension of MUSIC method in solving the
electromagnetic inverse scattering problems for point-li ke scatterers are investigated.
First, based on th e app l i cat i on in solving ac ou st i c inverse scattering problems, a
new non-iterative retrieval met hod is proposed to re t ri eve the scattering strengths
after obtaining the locations of scatterers using MUSIC method. Second, MUSIC
method i s app l i ed to locate small anisotropic scatterer s and the non- i t er at i ve re-
trieval method is extended to restore the scatterin g strength tensors of anisotrop i c
scatterers. Further, by utili zi n g th e stab le su b sp a ce of the MSR matrix an d th e
polarization char a ct er is t ic of the electromagnetic wave, a new MUS IC met h od is

proposed to improve the resolving abili ty, which is also able to deal with small de-
generate scatterer whose scattering s tr e n gt h tensor is rank deficient or almost rank
deficient .
In Chapter 4, based on the concept of subsp ace, several meth ods for solv ing
electromagnetic inverse problem s for extended scatterers are proposed, which could
converge fast to satisfactory resul t s. To begin with, the sketch of the su b space-
based optimization meth od (S O M) is presented, and a multilevel scheme of applying
the method is followed. Based on the SOM, an even faster convergent method ,
the twofold SOM, is proposed for solving the two-d i m ensional inverse scattering
problems. Before ending this chapter, a new current construction method is used
8
to decrease the computational cost of the SOM and thus such an improved SOM
is able to solve three-dimensiona l el ect r om a gn et i c inverse scattering proble m s for
extended scatterers.
Finally, in Chapter 5, su m m a r i zat i on of this thesis is presented, as well as dis-
cussions of some aspects of the future work t h a t m ay further improve the solver of
three-dimensional electromagnetic inverse scattering problems.
Chapter 2
Preliminaries
As mentioned in the introduction chapter, when talking about inverse problem, one
has to first mention its counterpart , the forward probl em . In our topic, the forward
problem is th e electromagnetic scattering problem, which has been stud i ed for a
long time. Thus, in the first part of this chapter, a forward problem solver based on
the integral equat i on sol u t i on of the Maxwell equations is presented. Such forward
problem solver is used in solving the inverse scattering problems in the rest part of
the thesis. After introducing the forward problem solver, those methods mentioned
in the previou s chapter for solving the electromagnetic inverse scattering problems
for both point-like scatterers and extended scatterers will be introduced.
2.1 Forward problem
As we know, the Maxwell equations consist of four equations, Faraday’s law, Am-

pere’s law, Gauss’ law for magnetic fields and Gauss’ law for electric fi el d s, as
∇ × E = −
∂
B
∂t
, (2.1a)
9
10
∇ ×
H =
∂
D
∂t
+
J, (2.1b)
∇ ·
B = 0, (2.1c)
and
∇ ·
D = ρ, (2.1d)
respectively, with
E the electric field strength, H the magnetic field stren g t h , D the
electric displacement,
B the magnetic flux density, J the electric current density, and
ρ the electric charge density [4, 38]. These four eq u at i ons describe the el ect r om ag -
netic wave’s behavior , i.e., the interaction between the electric fields and magnetic
fields. Besides, since the electromagnetic wave propagates in some medium, the in-
teraction between the wave and the medium is governed by the constitutive relations
as
D = ǫ · E, (2.2a)

B = µ ·H, (2.2b)
where
ǫ = diag [ǫ
1
, ǫ
2
, ǫ
3
] is the permittivity tensor and µ = diag [µ
1
, µ
2
, µ
3
] is the
permeability tensor of the medium in which the wave propagates. We may catalo g
various types of media into different types according to the different relationship
between the pr i n ci pal elements of the two t en sor s . When all principal elements in
the permittivity tensor (permeability tensor) are the same, the medium is called
electrically (magnetica ll y ) isotropic medium. When any of the principal element
of the per m i t t i vi ty tensor (permeab i l ity tensor) differs from the rest, the medium
is called electrically (magnetically) anisotropic medium. Such as the background
medium that we usually assume is the air with ǫ = diag [ǫ
0
, ǫ
0
, ǫ
0
] = Iǫ
0

and µ =
diag [ µ
0
, µ
0
, µ
0
] =
Iµ
0
, meaning that the air is isotropic medium, where ǫ
0
≈ 8.85 ×
10
−12
and µ
0
= 4π ×10
−7
. Because of t h ese two constit u t i ve relat i ons, one sees that
when the electromagnetic wave encounters different media, the propagatio n of th e
wave will be changed. The most usual behaviors of electromagnetic wave describing
11
such changes include reflection, refraction, diffraction, and scattering. Now, we can
derive the integral solution of the Maxwell equations.
First, we need a solution of the scalar Helmholtz equation in source region. For
a continuous source ρ(
r) in region D, it has been rigorously p r oven that the solution
of the scalar Helmholtz equation ∇
2

φ(
r) + k
2
0
φ(r) = −ρ(r) is
φ(
r) = φ
0
(r) + lim
δ→0

D−V
δ
g(
r, r
′
)ρ(r
′
)dV (r
′
), (2.3)
for
r ∈ D [39], where g( r, r
′
) =
exp(ik
0
R)
4πR
with R = |

r − r
′
| is the scalar G r een ’ s
function in three-dimensional situation , φ
0
is the unperturbed solution , and V
δ
is
the princi p al volume containing
r with surface S
V
δ
. Nex t, followi n g the procedure
in [40], we can obtain the integral equation solution of the Maxwell equation for
inhomogeneous medium by using (2.3). Now we assume that al l inhomogeneous
regions are included in d o m ai n D with surface S
D
. From Maxwell equations and
the two constitutive relations, we have
∇
2

E + (ǫ
r
−
I) ·E

+ k
2
0


E + (ǫ
r
−
I) ·E

= −∇ ×∇ ×

(ǫ
r
− I) · E

− iωµ
0
∇ ×

µ
r
− I

· H

, (2.4)
where k
0
= ω
√
ǫ
0
µ

0
is the wavenumber of the b a ckground medium, ǫ
r
= ǫ/ǫ
0
is
the rel at i ve permittivity tensor,
µ
r
= µ/µ
0
is th e relative permeab i l i ty tensor,
and
I is a 3 by 3 identity tensor. This equat i on is valid due to these conditions
∇·

E + (ǫ
r
− I) · E

= ∇·D/ǫ
0
= 0 and ∇·B = 0. If we consider the right-hand
side of the Eq. (2.4) as the secondary sources, this equation act ually consists of
three scalar Helmholtz equations. Thus, by (2.3), its so lution is
12
E(r) = E
inc
(r) − (ǫ
r

− I) ·E(r)
+ lim
δ→0

D−V
δ
g(
r, r
′
)

∇
′
× ∇
′
×

(ǫ
r
− I) · E(r
′
)

+ iωµ
0
∇
′
×

µ

r
− I

· H(r
′
)

dV (r
′
). (2.5)
By using identity g(∇
′
×
A) = ∇
′
×(gA)−∇
′
g×A and

v
∇
′
×AdV (r
′
) =

s
ˆn×AdS,
the above equation could be writ t en as
E(r) = E

inc
(r) − (ǫ
r
− I) · E
−lim
δ→0

D−V
δ
∇
′
g(
r, r
′
) ×∇
′
×

(ǫ
r
− I) · E

dV (r
′
)
− iωµ
0
lim
δ→0


D−V
δ
∇
′
g(
r, r
′
) ×

µ
r
− I

· H(r
′
)

dV (r
′
)
+ lim
δ→0

S
D
+S
V
δ
ˆn ×


g(
r, r
′
)∇
′
×

(ǫ
r
− I) ·E

dS(r
′
)
+iωµ
0
lim
δ→0

S
D
+S
V
δ
ˆn ×

g(
r, r
′
)


µ
r
− I

· H(r
′
)

dS(r
′
), (2.6)
where ˆn = ˆn
D
is th e unit vector outward normal to S
D
if r
′
∈ S
D
, and ˆn = −ˆn
V
δ
is the unit vector inward norm al to S
V
δ
if
r
′
∈ S

V
δ
. The limit of t h e fifth term on
the right-hand side is zero, since on S
D
we have
ǫ
r
= I and on S
V
δ
when δ → 0 the
scalar Green’s function g behaves as 1/R but the surface integration acts as R
2
as
R → 0. So does the sixt h term. Now only the third and the fourth terms need to
be address ed , and the details of which can be found in [40]. Ultimately, we arrive
at this useful expression