Table Of Content1846
Lecture Notes in Mathematics
Editors:
J.--M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
Habib Ammari
Hyeonbae Kang
Reconstruction of
Small Inhomogeneities
from Boundary
Measurements
123
Authors
HabibAmmari
CentredeMathe´matiquesApplique´es
E´colePolytechnique
UMRCNRS-7641
Palaiseau91128
France
e-mail:[email protected]
HyeonbaeKang
SchoolofMathematicalSciences
SeoulNationalUniversity
Seoul151-747
Korea
e-mail:[email protected]
LibraryofCongressControlNumber:2004108535
MathematicsSubjectClassification(2000):35R30,35B30
ISSN0075-8434
ISBN3-540-22483-1SpringerBerlinHeidelbergNewYork
DOI:10.1007/b98245
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Preface
Electrical impedance tomography (EIT) seeks to recover the electrical con-
ductivity distribution inside a body from measurements of current flows and
voltagesonits surface.Thevastandgrowingliteraturereflectsmanypossible
applications of EIT techniques, e.g., for medical diagnosis or nondestructive
evaluation of materials.
Since the underlying inverse problem is nonlinear and severely ill-posed,
general purpose EIT reconstruction techniques are likely to fail. Therefore it
is generally advisable to incorporate a-priori knowledge about the unknown
conductivity. One such type of knowledge could be that the body consists
of a smooth background containing a number of unknown small inclusions
with a significantly different conductivity. This situation arises for example
in breast cancer imaging or mine detection. In this case EIT seeks to recover
the unknown inclusions. Due to the smallness of the inclusions the associ-
ated voltage potentials measured on the surface of the body are very close
to the potentials corresponding to the medium without inclusion. So unless
one knows exactly what patterns to look for, noise will largely dominate the
information contained in the measured data. Furthermore, in applications it
is often not necessary to reconstruct the precise values of the conductivity or
geometry of the inclusions. The information of real interest is their positions
and size.
The main purpose of this book is to describe fresh and promising tech-
niquesforthereconstructionofsmallinclusionsfromboundarymeasurements
inareadableandinformativeform.Thesetechniquesrelyonaccurateasymp-
totic expansions of the boundary perturbations due to the presence of the
inclusions. The general approach we will take to derive these asymptotic ex-
pansionsisbasedonlayerpotentialtechniques.Thisallowsustohandleinclu-
sions with roughboundaries.In the course of deriving our asymptotic expan-
sions,weintroducenew conceptsofgeneralizedpolarizationtensors(GPT’s).
GPT’s contain significant information on the inclusion which will be investi-
gated. We then apply the asymptotic expansions for designing efficient direct
VI Preface
reconstruction algorithms to detect the location, size, and/or orientation of
the unknown inclusions.
This book would not have been possible without the collaborations and
the conversationswitha number ofoutstanding colleagues.We havenotonly
profitedfromgeneroussharingoftheirideas,insightsandenthusiasm,butalso
fromtheir friendship,support andencouragement.We feel specially indebted
to Gen Nakamura, Jin Keun Seo, Gunther Uhlmann and Michael Vogelius.
This book is dedicated to our friendship with them.
We also would like to thank our colleagues Mark Asch, Yves Capde-
boscq, and Darko Volkov for reading parts of the preliminary version of the
manuscript and our students Ekaterina Iakovleva, Eunjoo Kim, Mikyoung
Lim,KaoutherLouati,SofianeSoussi,KarimTouibi,andHabibZribiforpro-
viding us with numerical examples or/and carefully reading our manuscript.
During the preparation of this book we were supported by ACI Jeunes
Chercheurs (0693) from the French Ministry of Education and Scientific Re-
searchandKoreaScienceandEngineeringFoundationthroughthegrantR02-
2003-000-10012-0.
Paris and Seoul, Habib Ammari
May 2004 Hyeonbae Kang
Contents
1 Introduction............................................... 1
Part I Detection of Small Conductivity Inclusions
2 Transmission Problem .................................... 11
2.1 Some Notations and Preliminaries ......................... 11
2.2 Layer Potentials for the Laplacian ......................... 14
2.3 Neumann and Dirichlet Functions ......................... 30
2.4 Representation Formula .................................. 34
2.5 Energy Identities ........................................ 39
3 Generalized Polarization Tensors .......................... 41
3.1 Definition .............................................. 42
3.2 Uniqueness Result ....................................... 45
3.3 Symmetry and Positivity of GPT’s ........................ 47
3.4 Bounds for the PolarizationTensor of Po´lya–Szego¨ .......... 49
3.5 Estimates of the Weighted Volume and the Center of Mass ... 52
3.6 PolarizationTensors of Multiple Inclusions.................. 56
3.6.1 Definition ........................................ 57
3.6.2 Properties........................................ 58
3.6.3 Representation by Equivalent Ellipses................ 62
4 Derivation of the Full Asymptotic Formula ................ 65
4.1 Energy Estimates........................................ 67
4.2 Asymptotic Expansion ................................... 71
4.3 Closely Spaced Small Inclusions ........................... 76
5 Detection of Inclusions .................................... 79
5.1 Constant Current Projection Algorithm .................... 80
5.2 Quadratic Algorithm..................................... 85
VIII Contents
5.3 Least-Squares Algorithm ................................. 88
5.4 Variational Algorithm.................................... 89
5.5 Linear Sampling Method ................................. 91
5.6 Lipschitz-Continuous Dependence and Moments Estimations.. 97
5.6.1 Lipschitz-Continuous Dependence ................... 97
5.6.2 Moments Estimations .............................100
Part II Detection of Small Elastic Inclusions
6 Transmission Problem for Elastostatics ....................109
6.1 Layer Potentials for the Lam´e System......................109
6.2 Kelvin Matrix Under Unitary Transforms...................113
6.3 Transmission Problem....................................115
6.4 Complex Representation of Displacement Vectors............123
7 Elastic Moment Tensor ....................................129
7.1 Asymptotic Expansion in Free Space.......................129
7.2 Properties of EMT’s .....................................132
7.3 EMT’s Under Linear Transforms ..........................140
7.4 EMT’s for Ellipses.......................................143
7.5 EMT’s for Elliptic Holes and Hard Ellipses .................147
8 Derivation of Full Asymptotic Expansions ................151
8.1 Full Asymptotic Expansions ..............................151
9 Detection of Inclusions ....................................159
9.1 Detection of EMT’s......................................159
9.2 Representation of the EMT’s by Ellipses ...................163
9.3 Detection of the Location.................................164
9.4 Numerical Results .......................................167
Part III Detection of Small Electromagnetic Inclusions
10 Well-Posedness ............................................179
10.1 Existence and Uniqueness of a Solution ....................179
11 Representation of Solutions................................185
11.1 Preliminary Results......................................185
11.2 Representation Formulae .................................188
12 Derivation of Asymptotic Formulae........................197
12.1 Asymptotic Expansion ...................................197
Contents IX
13 Reconstruction Algorithms ................................207
13.1 Asymptotic Expansion ...................................207
13.2 Reconstruction of Multiple Inclusions ......................209
13.2.1 The Fourier Transform Algorithm ...................209
13.2.2 The MUSIC Algorithm.............................213
A Appendices ................................................215
A.1 Theorem of Coifman, McIntosh, and Meyer .................215
A.2 Continuity Method ......................................216
A.3 Collectively Compact Operators ..........................217
A.4 Uniqueness for the Inverse Conductivity Problem............217
A.4.1 Uniqueness With Many Measurements ...............217
A.4.2 Uniqueness of Disks With One Measurement..........220
References.....................................................223
Index..........................................................237
1
Introduction
Electrical Impedance Tomography (EIT) is designed to produce, efficiently
and accurately, images of the conductivity distribution inside a body from
measurements of current flows and voltages on the body’s surface. Due to
several merits of EIT such as safety, low cost, real time monitoring, EIT has
received considerable attention for the last two decades; see for instance the
review papers [255, 256, 78, 58] and the extensive list of references therein.
The vast and growing literature reflects many possible applications of EIT,
e.g., for medical diagnosisornondestructive evaluationofmaterials.In medi-
calapplications,EITcouldpotentiallybe usedfor monitoringlungproblems,
noninvasive monitoring of heartfunction and blood flow, screening for breast
andprostatecancer,andimprovingelectrocardiogramsandelectroencephalo-
grams [78, 37, 82, 83, 163, 164, 143].
However,insensitivity of boundary measurements to any change of inner-
body conductivity values has hampered EIT from providing accurate static
conductivityimages[4].Inpracticecapturedcurrent-to-voltagepairsmustbe
limited by the number of electrodes attached on the surface of the body that
confine the resolution of the image [153, 101]. We can definitely increase the
resolution of the conductivity image by increasing the number of electrodes.
However, it should be noticed that beyond a certain level, increasing the
number of electrodes may nothelp in producing a better image for the inner-
regionofthebodyifwetakeintoaccounttheinevitablenoiseinmeasurements
and the inherent insensitivity mentioned before.
In its most general form EIT is severely ill-posed and nonlinear. These
major and fundamental difficulties can be understood by means of a mean
value type theorem in elliptic partial differential equations. The value of the
voltagepotentialateachpointinsidetheregioncanbeexpressedasaweighted
average of its neighborhood potential where the weight is determined by the
conductivity distribution. In this weighted averaging way, the conductivity
distribution is conveyed to the boundary potential. Therefore, the boundary
data is entangled in the global structure of the conductivity distribution in
a highly nonlinear way. This is the main obstacle to finding non-iterative
H.AmmariandH.Kang:LNM1846, pp.1–4,2004.
(cid:1)c Springer-VerlagBerlinHeidelberg2004
2 1 Introduction
reconstruction algorithms with limited data. If, however, in advance we have
additional structural information about the medium, then we may be able to
determine specific features about the conductivity distribution with a good
resolution. One such type of knowledge could be that the body consists of a
smooth backgroundcontaining a number of unknown small inclusions with a
significantlydifferentconductivity.Thissituationarisesforexampleinbreast
andprostatecancerimaging[37,82,83,163,164,26,243]orminedetection.In
this case EIT seeks to recover the unknown inclusions. Due to the smallness
of the inclusions the associated voltage potentials measured on the surface
of the body are very close to the potentials corresponding to the medium
without inclusions. So unless one knows exactly what patterns to look for,
noise will largely dominate the information contained in the measured data.
Furthermore,inapplicationsitisoftennotnecessarytoreconstructtheprecise
values of the conductivity or the geometry of the inclusions. The information
of real interest is their positions and size.
Takingadvantageofthesmallnessoftheinclusions,manypromisingrecon-
struction techniques have been designed since the pioneering works by Fried-
manandVogelius [120,121,123].It turns outthat the method ofasymptotic
expansions of small volume inclusions provides a useful framework to recon-
struct the location and geometric features of the inclusions in a stable way,
even for moderately noisy data [73, 53, 28, 64, 191, 39, 261].
In this book we have made an attempt to describe these new and promis-
ing techniques for the reconstruction of small inclusions from boundary mea-
surements in a readable and informative form. As we said, these techniques
rely on accurate asymptotic expansions of the boundary perturbations due
to the presence of the inclusions. The general approach we will take in this
book is as follows. Based on layer potential techniques and decomposition
formulae like the one due to Kang and Seo in [168] for the conductivity
problem, we first derive complete asymptotic expansions. This allows us to
handle inclusions with rough boundaries and those with extreme conductiv-
ities. In the course of deriving our asymptotic expansions, we introduce new
concepts of generalized polarization tensors (GPT’s). These concepts gen-
eralize those of classical Po´lya–Szego¨ polarization tensors which have been
extensively studied in the literature by many authors for various purposes
[72, 24, 73, 212, 104, 105, 200, 198, 123, 180, 94, 186, 231, 232, 241, 95]. The
GPT’s appear naturally in higher-orderasymptotics of the steady-state volt-
agepotentialsunderthe perturbationofconductorbyconductivityinclusions
ofsmalldiameter.GPT’scontainsignificantinformationontheinclusionthat
will be investigated. We then apply the asymptotic expansions for designing
efficient direct reconstruction algorithms to detect the location, size, and/or
orientation of the unknown inclusions.
The book is intended to be self-contained. However, a certain familiarity
with layer potential techniques is required. The book is divided into three
parts that can be read independently.
