Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2013 Article ID 425184 8 pageshttpdxdoiorg1011552013425184

Research ArticleRecent Progress on the Factorization Method for ElectricalImpedance Tomography

Bastian Harrach

Department of Mathematics University of Wurzburg 97074 Wurzburg Germany

Correspondence should be addressed to Bastian Harrach bastianharrachuni-wuerzburgde

Received 12 November 2012 Accepted 15 March 2013

Academic Editor Bill Lionheart

Copyright copy 2013 Bastian Harrach This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The FactorizationMethod is a noniterative method to detect the shape and position of conductivity anomalies inside an objectThemethod was introduced by Kirsch for inverse scattering problems and extended to electrical impedance tomography (EIT) by Bruhland Hanke Since these pioneering works substantial progress has been made on the theoretical foundations of the method Thenecessary assumptions have been weakened and the proofs have been considerably simplified In this work we aim to summarizethis progress and present a state-of-the-art formulation of the Factorization Method for EIT with continuous data In particularwe formulate the method for general piecewise analytic conductivities and give short and self-contained proofs

1 Introduction

Electrical impedance tomography (EIT) aims to reconstructthe spatial conductivity distribution inside an imaging sub-ject Ω sube R119899 from current-voltage measurements on a partof its surface Σ sube 120597Ω Mathematically this leads to theproblem of recovering the coefficient 120590(119909) in the ellipticpartial differential equation

nabla sdot 120590nabla119906119892

120590= 0 in Ω 120590120597]119906

119892

120590|120597Ω

= 119892 on Σ

0 else(1)

fromknowledge of the correspondingNeumann-to-Dirichletoperator (NtD)

Λ (120590) 119892 997891997888rarr 119906119892

120590|Σ (2)

where 119906119892

120590is the solution of (1) We describe the precise

mathematical setting in Section 21In several applications EIT is used to determine the posi-

tion of conductivity changesThis includes anomaly detectionproblems where Λ(120590) is compared to a reference NtD Λ(120590

0)

in order to determine if and where 120590 differs from a knownbackground conductivity 120590

0 This problem also appears in

time-difference EIT where measurements at different timesare compared to monitor temporal conductivity changes

These applications lead to the shape reconstruction problemof determining the support of 120590 minus 120590

0from Λ(120590) and Λ(120590

0)

A prominent noniterative shape reconstruction methodis the Factorization Method It was introduced by Kirsch [1]for inverse scattering problems and extended to EIT by Bruhland Hanke [2ndash4] In its original form (cf [4]) the methodassumes that

1205900(119909) = 1 120590 (119909) = 1 + 120581 (119909) 120594

119863(119909) (3)

where119863 sube Ω is a union of separated smoothly bounded andsimply connected domains on which there is a conductivityjump of at least 120598 gt 0 that is

120581 (119909) ge 120598 forall119909 isin 119863 or 120581 (119909) le minus120598 forall119909 isin 119863 (4)

The method then characterizes the unknown shape 119863 by arange criterion For all unit vectors 119889 isin R119899 119889 = 1

119911 isin 119863 iff Φ119911119889|Σisin R (|Λ (120590) minus Λ (1)|

12

) (5)

whereΦ119911119889

is the so-called dipole function that is the solutionof

ΔΦ119911119889

= 119889 sdot nabla120575119911

in Ω 120597]Φ119911119889|120597Ω = 0 (6)

The range criterion (5) can be implemented numerically sothat each point 119911 isin Ω can be tested whether it belongs to theunknown inclusion or not

2 Computational and Mathematical Methods in Medicine

Substantial progress has been made on the FactorizationMethod since the original works of Kirsch Bruhl andHankeIn the following we restrict ourselves to progress in thecontext of EIT Overviews on the FM for EIT have beengiven by Hanke and Bruhl [5] in the book of Kirsch andGrinberg [6] and in a recent chapter of Hanke and Kirsch inScherzerrsquos Handbook of Mathematical Methods in Imaging[7]The FM for EIT has been treated as a special case of moregeneral elliptic problems by Kirsch [8] the author [9] andby Nachman Paivarinta and Teirila [10] A half-space settinghas been considered by Hanke and Schappel [11] Electrodemodels have been covered in the works of Bruhl LechleiterHakula Hanke Hyvonen and Pursiainen [5 12ndash15] The FMhas been extended to the complex conductivity case arisingin frequency-difference EIT by Seo Woo and the author[16 17] A priori separated indefinite inclusions have beentreated by Schmitt [18] and Schmitt and Kirsch discussedthe determination of the contrast level in the context of theFM for EIT in [19] Hyvonen and the author have removedthe assumptions on the inclusion contrast and boundaryregularity in [20] and [21] discusses the relation of the FMto localized potentials

In this work we aim to summarize the theoreticalprogress and present a state-of-the-art formulation of theFactorization Method for EIT with continuous data Inparticular wewill formulate themethod for general piecewiseanalytic conductivities and give short and self-containedproofs

2 Setting and Auxiliary Results

21The Setting We start by making the mathematical settingprecise Let Ω sube R119899 119899 ge 2 denote a bounded domainwith smooth boundary 120597Ω and outer normal vector ] LetΣ sube 120597Ω be an open part of the boundary 119871infin

+(Ω) denotes the

subspace of 119871infin(Ω)-functions with positive essential infima1198671

(Ω) and 1198712

(Σ) denote the spaces of1198671- and 1198712-functions

with vanishing integral mean on 120597Ω (resp Σ)For 120590 isin 119871

infin

+(Ω) and 119892 isin 119871

2

(Σ) there exists a unique

solution 119906119892

120590isin 119867

1

(Ω) of the elliptic partial differential

equation

nabla sdot 120590nabla119906119892

120590= 0 in Ω 120590120597]119906

119892

120590|120597Ω

= 119892 on Σ

0 else(7)

so that we can define the Neumann-to-Dirichlet operator(NtD)

Λ (120590) 1198712

(Σ) 997888rarr 119871

2

(Σ) 119892 997891997888rarr 119906

119892

120590|Σ (8)

where 119906119892120590isin 119867

1

(Ω) solves (7) Λ(120590) is a self-adjoint compact

linear operatorLet 120590

0isin 119871

infin

+(Ω) be piecewise analytic For each point 119911 isin

Ω that has a neighborhood in which 1205900is analytic and each

unit vector 119889 isin R119899 119889 = 1 let Φ119911119889

be the solution of

nabla sdot 1205900nablaΦ

119911119889= 119889 sdot nabla120575

119911in Ω 120590

0120597]Φ119911119889|120597Ω = 0 (9)

Φ119911119889

is called a dipole function

22 Auxiliary Results Our presentation of the FactorizationMethod in the next section relies on the following fourlemmas The first lemma is frequently called a monotonylemma since it shows that a larger conductivity leads to asmaller NtD More precisely it shows a relation betweenthe difference of two NtDs and the difference of the corre-sponding conductivities and the interior energy of an electricpotential The second lemma shows that this energy term isthe image of the adjoint of an auxiliary virtual measurementoperator that is defined on a subregion ofΩThe third lemmais a functional analytic relation between the norm of animage of an operator and the range of its adjoint Togetherwith the first two lemmas it implies that the range of theauxiliary virtual measurement operator can be calculatedfrom the NtDs Finally using the previous dipole functionsthe last lemma shows that the range of the auxiliary virtualmeasurement operator determines the region on which theyare defined

We start with the monotony lemma

Lemma 1 Let 1205901 1205900isin 119871

infin

+(Ω) Then for all 119892 isin 119871

2

(120597Ω)

intΩ(1205900minus 120590

1)1003816100381610038161003816nabla119906

0

1003816100381610038161003816

2d119909 leintΣ119892 (Λ

1minus Λ

0) 119892d119904 leint

Ω

1205900

1205901

(1205900minus 120590

1)1003816100381610038161003816nabla119906

0

1003816100381610038161003816

2d119909

(10)

where we abbreviated Λ119895= Λ(120590

119895) 119895 = 0 1 and 119906

0= 119906

119892

1205900

Proof The lemma seems to go back to Ikehata Kang Seo andSheen [22 23] cf also the similar arguments inKirsch [8] Ideet al [24] and in the works of Seo and the author [16 25] Forthe sake of completeness we copy the short proof from [25]For all 119892 isin 119871

2

(120597Ω) we have that

intΩ

1205901nabla119906

1sdot nabla119906

0d119909 = int

Σ

1198921199060d119904

= intΩ

1205900nabla119906

0sdot nabla119906

0d119909 = int

Σ

119892Λ0119892d119904

(11)

Hence from

intΩ

1205901nabla (119906

1minus 119906

0) sdot nabla (119906

1minus 119906

0) d119909

= intΩ

1205901

1003816100381610038161003816nabla11990611003816100381610038161003816

2d119909 minus intΩ

1205900

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

+ intΩ

(1205901minus 120590

0)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

(12)

we obtain that

intΣ

119892 (Λ1minus Λ

0) 119892d119904 = int

Ω

(1205900minus 120590

1)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

+ intΩ

1205901

1003816100381610038161003816nabla (1199061minus 119906

0)1003816100381610038161003816

2d119909(13)

which already yields the first asserted inequality

Computational and Mathematical Methods in Medicine 3

By interchanging 1205901and 120590

0 we conclude that

intΣ

119892 (Λ0minus Λ

1) 119892d119904

= intΩ

(1205901minus 120590

0)1003816100381610038161003816nabla1199061

1003816100381610038161003816

2d119909 + intΩ

1205900

1003816100381610038161003816nabla (1199060minus 119906

1)1003816100381610038161003816

2d119909

= intΩ

(1205901

10038161003816100381610038161003816100381610038161003816

nabla1199061minus1205900

1205901

nabla1199060

10038161003816100381610038161003816100381610038161003816

2

+ (1205900minus1205902

0

1205901

)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2

) d119909

(14)

and hence obtain the second inequality

Given a reference conductivity 1205900

isin 119871infin

+(Ω) and a

measurable subset119863 sube Ω we define the virtual measurementoperator 119871

119863by

119871119863 1198712

(119863)119899

997888rarr 1198712

(Σ) 119865 997891997888rarr V|

Σ (15)

where V isin 1198671

(Ω) solves

intΩ

1205900nablaV sdot nabla119908d119909 = int

119863

119865 sdot nabla119908d119909 forall119908 isin 1198671

(Ω) (16)

The energy term |nabla1199060|2 in Lemma 1 can be identified with the

norm of the adjoint of this virtual measurement operator

Lemma 2 The adjoint operator of 119871119863is given by

119871lowast

119863 1198712

(Σ) 997888rarr 119871

2

(119863)119899

119892 997891997888rarr nabla1199060|119863 (17)

where 1199060isin 119867

1

(Ω) solves

nabla sdot 1205900nabla119906

0= 0 119894119899 Ω 120590

0120597]1199060|120597Ω =

119892 119900119899 Σ

0 119890119897119904119890(18)

Proof For all 119892 isin 1198712

(Σ) and 119865 isin 119871

2

(119863)119899 we have that

int119863

(119871lowast

119863119892) sdot 119865 d119909 = int

Σ

119892 (119871119863119865) d119909 = int

Σ

119892V|Σd119909

= intΩ

1205900nabla119906

0sdot nablaV d119909 = int

119863

nabla1199060sdot 119865 d119909

(19)

which shows the assertion

The following functional analytic lemma uses bounds onthe image of an operator to characterize the range of its dualoperator

Lemma 3 Let 119883 and 119884 be real Hilbert spaces with innerproducts (sdot sdot)

119883and (sdot sdot)

119884 respectively Let 119860 isin L(119883 119884) and

1199091015840

isin 119883 Then

1199091015840

isin R (119860lowast

) iff exist119862 gt 0 10038161003816100381610038161003816(1199091015840

119909)119883

10038161003816100381610038161003816le 119862 119860119909 forall119909 isin 119883

(20)

In particular if1198831198841 and119884

2are three real Hilbert spaces119860

119894isin

L(119884119894 119883) 119894 = 1 2 and if there exists 119862 gt 0 with

1003817100381710038171003817119860lowast

11199091003817100381710038171003817 le 119862

1003817100381710038171003817119860lowast

21199091003817100381710038171003817 forall119909 isin 119883 (21)

thenR(1198601) sube R(119860

2)

Proof The assertion can be generalized to Banach spacesand in that context it is called the ldquo14th important propertyof Banach spacesrdquo in Bourbaki [26] For the sake of complete-ness we rewrite the proof from [27] to Hilbert spaces

If 1199091015840 isin R(119860lowast

) then there exists 1199101015840 isin 119884 such that 1199091015840 =119860lowast

1199101015840 Hence10038161003816100381610038161003816(1199091015840

119909)119883

10038161003816100381610038161003816=10038161003816100381610038161003816(119860

lowast

1199101015840

119909)119883

10038161003816100381610038161003816

=10038161003816100381610038161003816(1199101015840

119860119909)119884

10038161003816100381610038161003816le10038171003817100381710038171003817119910101584010038171003817100381710038171003817119860119909 forall119909 isin 119883

(22)

so that the assertion holds with 119862 = 1199101015840

Now let 1199091015840 isin 119883 be such that there exists 119862 gt 0 with

|(1199091015840

119909)119883| le 119862119860119909 for all 119909 isin 119883 We define

119891 (119910) = (1199091015840

119909)119883

for every 119910 = 119860119909 isin R (119860) (23)

Then 119891 is a well-defined continuous linear functional onR(119860) By setting it to zero on R(119860)

perp we can extend 119891 to acontinuous linear functional on 119884 Using the Riesz theoremit follows that there exists 1199101015840 isin 119884 with

(1199101015840

119910)119884

= 119891 (119910) forall119910 isin R (119860) (24)

Hence for all 119909 isin 119883 we have

(119860lowast

1199101015840

119909)119883

= (1199101015840

119860119909)119884

= 119891 (119860119909) = (1199091015840

119909)119883

(25)

so that 1199091015840 = 119860lowast

1199101015840

isin R(119860lowast

)

The last lemma shows that the range of the virtualmeasurement operator 119871

119863determines the region119863 onwhich

it is defined We state the lemma for a simple special casea generalized version of the lemma will be formulated inSection 32

Lemma 4 Let 1205900= 1 119863 sube Ω be open and 119863 sube Ω have a

connected complement Ω 119863Then for all unit vectors 119889 isin R119899 119889 = 1 and every point

119911 isin Ω 120597119863 it holds that

119911 isin 119863 iff Φ119911119889|Σisin R (119871

119863) (26)

Proof The proof is similar to the one of [21 Lemma 29]First let 119911 isin 119863 and 120598 gt 0 be such that 119861

120598(119911) sube 119863 We

choose

1198911isin 119867

1

(119861120598(119911)) with 119891

1|120597119861120598(119911)

= Φ119911119889|120597119861120598(119911)

1198912isin 119867

1

(119861120598(119911)) with Δ119891

2= 0

120597]1198912|120597119861120598(119911)= 120597]Φ119911119889|120597119861120598(119911)

(27)

and let 119865 isin 1198712

(119863)119899 be the zero continuation of nabla(119891

1minus 119891

2) to

119863Then the function

V = Φ119911119889

in Ω 119861120598(119911)

1198911

in 119861120598(119911)

(28)

4 Computational and Mathematical Methods in Medicine

fulfills V isin 1198671

(Ω) and for all 119908 isin 119867

1

(Ω)

intΩ

nablaV sdot nabla119908d119909

= intΩ119861120598(119911)

nablaΦ119911119889

sdot nabla119908d119909 + int119861120598(119911)

nabla1198911sdot nabla119908d119909

= minusint120597119861120598(119911)

120597]Φ119911119889 119908|120597119861120598(119911)d119904 + int

119861120598(119911)

nabla1198911sdot nabla119908d119909

= int119861120598(119911)

nabla (1198911minus 119891

2) sdot nabla119908d119909 = int

119863

119865 sdot nabla119908d119909

(29)

This shows thatΦ119911119889|Σ= V|

Σ= 119871

119863(119865) isin R(119871

119863)

Now let Φ119911119889|Σisin R(119871

119863) Let V isin 119867

1

(Ω) be the function

from the definition of 119871119863 Then

V|Σ= Φ

119911119889|Σ 120597]V|Σ = 0 = 120597]Φ119911119889|Σ (30)

so that it follows by unique continuation that V = Φ119911119889

in theconnected setΩ (119863 cup 119911)

If 119911 notin 119863 then 119889 sdot nabla120575119911notin 119867

minus2

(Ω 119863) and thus Φ119911119889

notin

1198712

(Ω 119863) which contradicts that V = Φ119911119889

in Ω (119863 cup 119911)Hence 119911 isin 119863

3 The Factorization Method

Nowwe will formulate the FactorizationMethod and charac-terize a region where a conductivity 120590 differs from a referenceconductivity 120590

0by a range criterion Before we turn to a

new general formulation of the method we first state it fora special case that is similar to the one that was treated in theoriginal works of Bruhl and Hanke [3 4]

31 The Factorization Method for a Simple Special Case

Theorem 5 Let 1205900= 1 and 120590 = 1 + 120594

119863 where 119863 sube Ω is an

open set so that 119863 sube Ω has a connected complement Ω 119863Then for all 119911 isin Ω 119911 notin 120597119863 and all dipole directions 119889 isin R119899119889 = 1

119911 isin 119863 iff Φ119911119889

1003816100381610038161003816Σisin R (|Λ (120590) minus Λ (1)|

12

) (31)

Proof Themonotony Lemma 1 yields that for all119892 isin 1198712

(120597Ω)

int119863

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 ge intΣ

119892 (Λ (1) minus Λ (120590)) 119892d119904 ge int119863

1

2

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(32)

Hence |Λ(120590) minusΛ(1)| = Λ(1) minusΛ(120590) and using Lemma 2 wecan restate this in the form

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

ge10038171003817100381710038171003817|Λ (120590) minus Λ (1)|

12

11989210038171003817100381710038171003817

2

ge1

2

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

(33)

Using the functional analytic Lemma 3 this implies that

R (119871119863) = R (|Λ (120590) minus Λ (1)|

12

) (34)

and thus the assertion follows from the relation between 119863

andR(119871119863) in Lemma 4

Obviously the same arguments can be used to treat thecase 120590(119909) = 1 + 120581(119909)120594

119863(119909) when there exists a conductivity

jump 120598 gt 0 so that either

120581 (119909) ge 120598 forall119909 isin 119863 or 120581 (119909) le minus120598 forall119909 isin 119863 (35)

32 The Factorization Method for the General PiecewiseAnalytic Case Now we drop the assumptions that the back-ground is constant that there is a clear conductivity jumpand that the complement of the inclusions is connected Wewill merely assume that the reference conductivity 120590

0is a

piecewise analytic function and that either 120590 minus 1205900ge 0 or

120590 minus 1205900le 0 Roughly speaking under this general assumption

the Factorization Method then characterizes the support of120590 minus 120590

0up to holes in the support that have no connections to

Σ For a precise formulation we use the concept of the innerand outer support from [28] that has been inspired by the useof the infinity support of Kusiak and Sylvester [29] see also[25 30]

Definition 6 A relatively open set 119880 sube Ω is called connectedto Σ if 119880 cap Ω is connected and 119880 cap Σ = 0

For a measurable function 120581 Ω rarr R we define

(a) the support supp(120581) as the complement (in Ω) of theunion of those relatively open119880 sube Ω for which 120581|

119880equiv

0(b) the inner support inn supp 120581 as the union of those

open sets 119880 sube Ω for which ess inf119909isin119880

|120581(119909)| gt 0(c) the outer support out

Σsupp 120581 as the complement (in

Ω) of the union of those relatively open 119880 sube Ω thatare connected to Σ and for which 120581|

119880equiv 0

The interior of a set119872 sube Ω is denoted by int119872 and its closure(with respect toR119899) by119872 If119872 is measurable we also define

(d) outΣ119872 = out

Σsupp120594

119872

It is easily checked that outΣ(supp 120581) = out

Σsupp 120581

With this concept we can extend the range characteriza-tion in Lemma 4 to a general setting (see also Remark 9 later)

Lemma 7 Let 1205900isin 119871

infin

+(Ω) be piecewise analytic Let 119863 sube Ω

be measurableThen for all unit vectors 119889 isin R119899 119889 = 1 and every point

119911 isin Ω that has a neighborhood in which 1205900is analytic

119911 isin int119863 implies Φ119911119889

1003816100381610038161003816Σisin R (119871

119863) (36)

and

Φ119911119889

1003816100381610038161003816Σisin R (119871

119863) implies 119911 isin out

Σ119863 (37)

Proof If 119911 isin int119863 then there exists a small ball 119861120598(119911) sube 119863

and the first assertion follows as in the proof of Lemma 4To show the second assertion letΦ

119911119889|Σisin R(119871

119863) and let

V isin 1198671

(Ω) be the function from the definition of 119871

119863 so that

(as in the proof of Lemma 4)

V|Σ= Φ

119911119889|Σ 120597]V|Σ = 0 = 120597]Φ119911119889|Σ (38)

Computational and Mathematical Methods in Medicine 5

Assume that 119911 notin outΣ119863 Then there exists a relatively

open 119880 sube Ω that is connected to Σ and contains 119911Hence by unique continuation it follows that V|

119880= Φ

119911119889|119880

and we obtain the same contradiction as in the proof ofLemma 4

Now we can formulate and prove the FactorizationMethod for general piecewise analytic conductivities

Theorem 8 Let 120590 isin 119871infin

+(Ω) and let 120590

0isin 119871

infin

+(Ω) be a

piecewise analytic function Let either

120590 (119909) ge 1205900(119909) forall119909 isin Ω or 120590 (119909) le 120590

0(119909) forall119909 isin Ω

(39)

Then for all 119911 isin Ω that have a neighborhood in which 1205900is

analytic as well as all unit vectors 119889 isin R119899 119889 = 1

119911isin innsupp (120590 minus 1205900) implies Φ

119911119889

1003816100381610038161003816Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

)

(40)

and

Φ119911119889

1003816100381610038161003816ΣisinR (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) implies 119911isin outΣsupp (120590 minus 120590

0)

(41)

Proof Let 119911 isin Ω have a neighborhood in which 1205900is analytic

and let 119889 isin R119899 be a unit vector with 119889 = 1 We onlyprove the assertions for 120590 ge 120590

0 The other case is completely

analogousFirst let 119911 isin inn supp (120590 minus 120590

0) Then there exists a small

ball 119861120598(119911) and 120575 gt 0 so that 120590 minus 120590

0ge 120575 on 119861

120598(119911) Using the

monotony Lemma 1 it follows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (1205900) minus Λ (120590)) 119892d119904

ge intΩ

1205900

120590(120590 minus 120590

0)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

ge 12057510038171003817100381710038171205900

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817100381710038171003817

1

120590

1003817100381710038171003817100381710038171003817119871infin(Ω)int119861120598(119911)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

ge 12057510038171003817100381710038171205900

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817100381710038171003817

1

120590

1003817100381710038171003817100381710038171003817119871infin(Ω)

10038171003817100381710038171003817119871lowast

119861120598(119911)11989210038171003817100381710038171003817

2

(42)

Using the functional analytic Lemma 3 we obtain that

R (119871119861120598(119911)

) sube R (1003816100381610038161003816Λ (120590) minus Λ (120590

0)1003816100381610038161003816

12

) (43)

and Lemma 7 yields that

Φ119911119889|Σisin R (119871

119861120598(119911)) sube R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) (44)

On the other hand with 119863 = supp(120590 minus 1205900) the monotony

Lemma 1 shows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (1205900) minus Λ (120590

1)) 119892d119904 le int

Ω

(1205901minus 120590

0)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

le10038171003817100381710038171205901 minus 120590

0

1003817100381710038171003817119871infin(Ω)int119863

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le10038171003817100381710038171205901 minus 120590

0

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

(45)

so that we obtain from the functional analytic Lemma 3

R (1003816100381610038161003816Λ (120590) minus Λ (120590

0)1003816100381610038161003816

12

) sube R (119871119863) (46)

Hence Lemma 7 yields that

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) sube R (119871119863) implies

119911 isin outΣ(supp (120590 minus 120590

0)) = out

Σsupp (120590 minus 120590

0)

(47)

Remark 9 Theorem 8 shows that the Factorization Methodis able to detect the support of a conductivity difference up tothe difference between the outer and the inner support thatis roughly speaking up to holes in the support that have noconnections to the boundary It leaves openwhether points insuch holes will fulfill the range criterion of the FactorizationMethod or not

A result of Hyvonen and the author [30 Lemma 25]shows that for every smooth domain119863with119863 sub Ω and everyunit vector 119889 isin R119899 119889 = 1

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) forall119911 isin 120597119863 (48)

implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) forall119911 isin 119863 (49)

In that sense we can expect that holes in the support willbe filled up and that the set detected by the FactorizationMethod is essentially the outer support of the conductivitydifference

33 The Factorization Method for the Indefinite Case It is along standing open theoretical problem whether the rangecriterion of the Factorization Method holds true withoutthe definiteness assumption that 120590 ge 120590

0on Ω or 120590 le

1205900on Ω However Grinberg Kirsch and Schmitt [18 31]

showed how to exclude a region 119864 sube Ω from Ω insuch a way that the Factorization Method only requires thedefiniteness assumption onΩ119864 In this subsection we showhow their idea can be incorporated into our formulation ofthe method

To point out the main idea we first formulate the resultfor a simple special case Let us stress that for 120590 = 1 + 120594

119863+ minus

(12)120594119863minus it is not known whether

119911 isin 119863+

cup 119863minus iff Φ

119911119889|Σisin R (|Λ (120590) minus Λ (1)|

12

) (50)

However we can still use the FactorizationMethod if we havesome a priori knowledge that separates 119863+ and 119863

minus Moreprecisely if we know a subset 119864 that contains 119863minus withoutintersecting 119863

+ then we can use the Factorization Methodto find119863+ cup 119864 (and thus119863+)

6 Computational and Mathematical Methods in Medicine

Theorem 10 Let 1205900= 1 and 120590 = 1 + 120594

119863+ minus (12)120594

119863minus where

119863+

119863minus

sube Ω are open Let 119864 sube Ω be an open set

(a) If 119863+ sube 119864 and 119863minus cup 119864 sube Ω has a connectedcomplement then for all 119911 isin Ω 119911 notin 120597(119863

minus

cup119864) and alldipole directions 119889 isin R119899 119889 = 1

119911 isin 119863minus

cup 119864 iff Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1) + 119871119864119871lowast

119864

1003816100381610038161003816

12

)

(51)

(b) If 119863minus sube 119864 and 119863+ cup 119864 sube Ω has a connectedcomplement then for all 119911 isin Ω 119911 notin 120597(119863

+

cup119864) and alldipole directions 119889 isin R119899 119889 = 1

119911 isin 119863+

cup 119864 iff Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1) minus 119871119864119871lowast

119864

1003816100381610038161003816

12

)

(52)

Proof Themonotony Lemma 1 yields that for all119892 isin 1198712

(120597Ω)

1

2int119863minus

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 minus int119863+

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le intΣ

119892 (Λ (120590) minus Λ (1)) 119892d119904

le int119863minus

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 minus1

2int119863+

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(53)

Since (cf Lemma 2)

intΣ

119892 (119871119864119871lowast

119864) 119892d119904 = 1003817100381710038171003817119871

lowast

1198641198921003817100381710038171003817

2

= int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 (54)

it follows for case (a) that

1

2int119863minuscup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 le intΣ

119892 (Λ (120590) minus Λ (1) + 2119871119864119871lowast

119864) 119892d119904

le int119863minuscup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909(55)

Using the functional analytic Lemma 3 this implies that

R (119871119863minuscup119864) = R (

1003816100381610038161003816Λ (120590) minus Λ (1) + 2119871119864119871lowast

119864

1003816100381610038161003816

12

) (56)

so that the assertion (a) follows from Lemma 4In case (b) we obtain that

3

2int119863+cup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 ge intΣ

119892 (Λ (1) minus Λ (120590) + 2119871119864119871lowast

119864) 119892d119904

ge1

2int119863+cup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909(57)

and the same arguments as above yield the assertion

We can also extend these ideas to the general setting ofSection 32

Theorem 11 Let 120590 isin 119871infin

+(Ω) and let 120590

0isin 119871

infin

+(Ω) be a

piecewise analytic function Let 119864 sube Ω be a measurable setChoose 120572 120573 isin R such that

120572 gt1003817100381710038171003817120590 minus 120590

0

1003817100381710038171003817119871infin(Ω) 120573 gt

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω) (58)

(a) If 120590 le 1205900on Ω 119864 then for all 119911 isin Ω that have a

neighborhood in which 1205900is analytic as well as all unit

vectors 119889 isin R119899 119889 = 1

119911 isin inn supp (120590 minus 1205900) cup 119864 implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590)minusΛ (1205900)+120572119871

119864119871lowast

119864

1003816100381610038161003816

12

)

(59)

and

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) implies

119911 isin outΣ(supp (120590 minus 120590

0) cup 119864)

(60)

(b) If 120590 ge 1205900on Ω 119864 then for all 119911 isin Ω that have a

neighborhood in which 1205900is analytic as well as all unit

vectors 119889 isin R119899 119889 = 1

119911 isin inn supp (120590 minus 1205900) cup 119864 implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) minus 120573119871

119864119871lowast

119864

1003816100381610038161003816

12

)

(61)

and

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) minus 120573119871

119864119871lowast

119864

1003816100381610038161003816

12

) implies

119911 isin outΣ(supp (120590 minus 120590

0) cup 119864)

(62)

Proof For every 119911 isin inn supp(120590minus1205900)with 119911 notin 119864 there exists

a small ball 119861120598(119911) and 120575 gt 0 so that 120590

0minus120590 ge 120575 on 119861

120598(119911) Using

the monotony Lemma 1 it follows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

ge intΩ

(1205900minus 120590

1)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

ge 120575int119861120598(119911)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909+(120572 minus1003817100381710038171003817120590 minus 120590

0

1003817100381710038171003817119871infin(Ω))int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(63)

Computational and Mathematical Methods in Medicine 7

As in the previous proofs we obtain from Lemmas 2 and 3that

R (1003816100381610038161003816Λ (120590) minus Λ (120590

0) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) supe R (119871119861120598(119911)cup119864

) (64)

so that the first implication of (a) follows from Lemma 7The monotony Lemma 1 also implies that

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

le intΩ

1205900

120590(1205900minus 120590)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω)intsupp(120590minus1205900)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

+ 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(65)

so that the second implication of (a) follows from Lemmas 23 and 7 Assertion (b) can be proven analogously

Remark 9 also applies to this case

4 Conclusions and Remarks

The Factorization Method can be used to detect regionsin which a conductivity differs from a known referenceconductivity In this work we summarized the progresson the methodrsquos theoretical foundation We formulated themethod for general piecewise analytic conductivities andgave comparatively simple and self-contained proofsWe alsoshowed how the idea of excluding a part of the imaging regioncan be incorporated into this formulation

The regularity assumptions can beweakened even furtherOur proofs only require unique continuation arguments forthe reference conductivity 120590

0and the existence of the dipole

functionsTwo major open theoretical questions still exist in the

context of the FactorizationMethodThe theoretical justifica-tion of the method requires a definiteness condition (on thewhole domain or after excluding an a priori known part of thedomain) It is unknownwhether themethodrsquos range criterionholds without such a definiteness conditionThe second openquestion concerns the numerical stability of the methodrsquosrange criterion So far there are no rigorous convergenceresults for numerical implementations of this range criterion(see however Lechleiter [32] for a first step in this direction)As a promising approach to overcome both problems wewould like to point out the recent work on monotony-basedmethods [28]

Acknowledgments

This paper was funded by the German Research Foundation(DFG) and the University of Wurzburg in the fundingprogramme Open Access Publishing

References

[1] A Kirsch ldquoCharacterization of the shape of a scattering obstacleusing the spectral data of the far field operatorrdquo Inverse Prob-lems vol 14 no 6 pp 1489ndash1512 1998

[2] M Bruhl Gebietserkennung in der elektrischen Impedanztomo-graphie [PhD thesis] Universitat Karlsruhe 1999

[3] M Bruhl and M Hanke ldquoNumerical implementation of twononiterative methods for locating inclusions by impedancetomographyrdquo Inverse Problems vol 16 no 4 pp 1029ndash10422000

[4] M Bruhl ldquoExplicit characterization of inclusions in electricalimpedance tomographyrdquo SIAM Journal on Mathematical Anal-ysis vol 32 no 6 pp 1327ndash1341 2001

[5] M Hanke and M Bruhl ldquoRecent progress in electricalimpedance tomographyrdquo Inverse Problems vol 19 no 6 ppS65ndashS90 2003

[6] A Kirsch andN GrinbergTheFactorizationMethod for InverseProblems vol 36 ofOxford Lecture Series inMathematics and ItsApplications Oxford University Press Oxford UK 2007

[7] M Hanke and A Kirsch ldquoSampling methodsrdquo in Handbook ofMathematical Models in Imaging O Scherzer Ed pp 501ndash550Springer 2011

[8] A Kirsch ldquoThe factorization method for a class of inverseelliptic problemsrdquo Mathematische Nachrichten vol 278 pp258ndash277 2005

[9] B Gebauer ldquoThe Factorization Method for real elliptic prob-lemsrdquo Zeitschrift fur Analysis und ihre Anwendung vol 25 no1 pp 81ndash102 2006

[10] A I Nachman L Paivarinta and A Teirila ldquoOn imagingobstacles inside inhomogeneous mediardquo Journal of FunctionalAnalysis vol 252 no 2 pp 490ndash516 2007

[11] M Hanke and B Schappel ldquoThe factorization method forelectrical impedance tomography in the half-spacerdquo SIAMJournal on Applied Mathematics vol 68 no 4 pp 907ndash9242008

[12] N Hyvonen ldquoComplete electrode model of electricalimpedance tomography approximation properties andcharacterization of inclusionsrdquo SIAM Journal on AppliedMathematics vol 64 pp 902ndash931 2004

[13] A Lechleiter N Hyvonen and H Hakula ldquoThe factorizationmethod applied to the complete electrode model of impedancetomographyrdquo SIAM Journal onAppliedMathematics vol 68 no4 pp 1097ndash1121 2008

[14] N Hyvonen H Hakula and S Pursiainen ldquoNumerical imple-mentation of the factorization method within the completeelectrode model of electrical impedance tomographyrdquo InverseProblems and Imaging vol 1 pp 299ndash317 2007

[15] HHakula andNHyvonen ldquoOn computation of test dipoles forfactorization methodrdquo BIT Numerical Mathematics vol 49 no1 pp 75ndash91 2009

[16] B Harrach and J K Seo ldquoDetecting inclusions in electri-cal impedance tomography without reference measurementsrdquoSIAM Journal on Applied Mathematics vol 69 no 6 pp 1662ndash1681 2009

[17] B Harrach J K Seo and E J Woo ldquoFactorization methodand its physical justification in frequency-difference electricalimpedance tomographyrdquo IEEE Transactions on Medical Imag-ing vol 29 no 11 pp 1918ndash1926 2010

[18] S Schmitt ldquoThe factorization method for EIT in the case ofmixed inclusionsrdquo Inverse Problems vol 25 no 6 Article ID065012 2009

8 Computational and Mathematical Methods in Medicine

[19] S Schmitt and A Kirsch ldquoA factorization scheme for determin-ing conductivity contrasts in impedance tomographyrdquo InverseProblems vol 27 Article ID 095005 2011

[20] B Gebauer and N Hyvonen ldquoFactorization method and irreg-ular inclusions in electrical impedance tomographyrdquo InverseProblems vol 23 no 5 pp 2159ndash2170 2007

[21] B Gebauer ldquoLocalized potentials in electrical impedancetomographyrdquo Inverse Problems and Imaging vol 2 pp 251ndash2692008

[22] H Kang J K Seo and D Sheen ldquoThe inverse conductivityproblem with one measurement Stability and estimation ofsizerdquo SIAM Journal on Mathematical Analysis vol 28 no 6 pp1389ndash1405 1997

[23] M Ikehata ldquoSize estimation of inclusionrdquo Journal of Inverse andIll-Posed Problems vol 6 no 2 pp 127ndash140 1998

[24] T Ide H Isozaki S Nakata S Siltanen and G UhlmannldquoProbing for electrical inclusions with complex sphericalwavesrdquo Communications on Pure and Applied Mathematics vol60 no 10 pp 1415ndash1442 2007

[25] B Harrach and J K Seo ldquoExact shape-reconstruction by one-step linearization in electrical impedance tomographyrdquo TheSIAM Journal onMathematical Analysis vol 42 no 4 pp 1505ndash1518 2010

[26] N Bourbaki Elements of Mathematics Topological VectorSpaces chapters 1ndash5 Springer Berlin Germany 2003

[27] F Fruhauf B Gebauer andO Scherzer ldquoDetecting interfaces ina parabolic-elliptic problem from surfacemeasurementsrdquo SIAMJournal on Numerical Analysis vol 45 no 2 pp 810ndash836 2007

[28] B Harrach and M Ullrich ldquoMonotony based shape recon-struction in electrical impedance tomographyrdquo submitted forpublication

[29] S Kusiak and J Sylvester ldquoThe scattering supportrdquoCommunica-tions on Pure and Applied Mathematics vol 56 no 11 pp 1525ndash1548 2003

[30] B Gebauer and N Hyvonen ldquoFactorization method andinclusions of mixed type in an inverse elliptic boundary valueproblemrdquo Inverse Problems and Imaging vol 2 pp 355ndash3722008

[31] N I Grinberg and A Kirsch ldquoThe factorization method forobstacles with a-priori separated sound-soft and sound-hardpartsrdquo Mathematics and Computers in Simulation vol 66 no4-5 pp 267ndash279 2004

[32] A Lechleiter ldquoA regularization technique for the factorizationmethodrdquo Inverse Problems vol 22 no 5 pp 1605ndash1625 2006

Submit your manuscripts athttpwwwhindawicom

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Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

2 Computational and Mathematical Methods in Medicine

Substantial progress has been made on the FactorizationMethod since the original works of Kirsch Bruhl andHankeIn the following we restrict ourselves to progress in thecontext of EIT Overviews on the FM for EIT have beengiven by Hanke and Bruhl [5] in the book of Kirsch andGrinberg [6] and in a recent chapter of Hanke and Kirsch inScherzerrsquos Handbook of Mathematical Methods in Imaging[7]The FM for EIT has been treated as a special case of moregeneral elliptic problems by Kirsch [8] the author [9] andby Nachman Paivarinta and Teirila [10] A half-space settinghas been considered by Hanke and Schappel [11] Electrodemodels have been covered in the works of Bruhl LechleiterHakula Hanke Hyvonen and Pursiainen [5 12ndash15] The FMhas been extended to the complex conductivity case arisingin frequency-difference EIT by Seo Woo and the author[16 17] A priori separated indefinite inclusions have beentreated by Schmitt [18] and Schmitt and Kirsch discussedthe determination of the contrast level in the context of theFM for EIT in [19] Hyvonen and the author have removedthe assumptions on the inclusion contrast and boundaryregularity in [20] and [21] discusses the relation of the FMto localized potentials

In this work we aim to summarize the theoreticalprogress and present a state-of-the-art formulation of theFactorization Method for EIT with continuous data Inparticular wewill formulate themethod for general piecewiseanalytic conductivities and give short and self-containedproofs

2 Setting and Auxiliary Results

21The Setting We start by making the mathematical settingprecise Let Ω sube R119899 119899 ge 2 denote a bounded domainwith smooth boundary 120597Ω and outer normal vector ] LetΣ sube 120597Ω be an open part of the boundary 119871infin

+(Ω) denotes the

subspace of 119871infin(Ω)-functions with positive essential infima1198671

(Ω) and 1198712

(Σ) denote the spaces of1198671- and 1198712-functions

with vanishing integral mean on 120597Ω (resp Σ)For 120590 isin 119871

infin

+(Ω) and 119892 isin 119871

2

(Σ) there exists a unique

solution 119906119892

120590isin 119867

1

(Ω) of the elliptic partial differential

equation

nabla sdot 120590nabla119906119892

120590= 0 in Ω 120590120597]119906

119892

120590|120597Ω

= 119892 on Σ

0 else(7)

so that we can define the Neumann-to-Dirichlet operator(NtD)

Λ (120590) 1198712

(Σ) 997888rarr 119871

2

(Σ) 119892 997891997888rarr 119906

119892

120590|Σ (8)

where 119906119892120590isin 119867

1

(Ω) solves (7) Λ(120590) is a self-adjoint compact

linear operatorLet 120590

0isin 119871

infin

+(Ω) be piecewise analytic For each point 119911 isin

Ω that has a neighborhood in which 1205900is analytic and each

unit vector 119889 isin R119899 119889 = 1 let Φ119911119889

be the solution of

nabla sdot 1205900nablaΦ

119911119889= 119889 sdot nabla120575

119911in Ω 120590

0120597]Φ119911119889|120597Ω = 0 (9)

Φ119911119889

is called a dipole function

22 Auxiliary Results Our presentation of the FactorizationMethod in the next section relies on the following fourlemmas The first lemma is frequently called a monotonylemma since it shows that a larger conductivity leads to asmaller NtD More precisely it shows a relation betweenthe difference of two NtDs and the difference of the corre-sponding conductivities and the interior energy of an electricpotential The second lemma shows that this energy term isthe image of the adjoint of an auxiliary virtual measurementoperator that is defined on a subregion ofΩThe third lemmais a functional analytic relation between the norm of animage of an operator and the range of its adjoint Togetherwith the first two lemmas it implies that the range of theauxiliary virtual measurement operator can be calculatedfrom the NtDs Finally using the previous dipole functionsthe last lemma shows that the range of the auxiliary virtualmeasurement operator determines the region on which theyare defined

We start with the monotony lemma

Lemma 1 Let 1205901 1205900isin 119871

infin

+(Ω) Then for all 119892 isin 119871

2

(120597Ω)

intΩ(1205900minus 120590

1)1003816100381610038161003816nabla119906

0

1003816100381610038161003816

2d119909 leintΣ119892 (Λ

1minus Λ

0) 119892d119904 leint

Ω

1205900

1205901

(1205900minus 120590

1)1003816100381610038161003816nabla119906

0

1003816100381610038161003816

2d119909

(10)

where we abbreviated Λ119895= Λ(120590

119895) 119895 = 0 1 and 119906

0= 119906

119892

1205900

Proof The lemma seems to go back to Ikehata Kang Seo andSheen [22 23] cf also the similar arguments inKirsch [8] Ideet al [24] and in the works of Seo and the author [16 25] Forthe sake of completeness we copy the short proof from [25]For all 119892 isin 119871

2

(120597Ω) we have that

intΩ

1205901nabla119906

1sdot nabla119906

0d119909 = int

Σ

1198921199060d119904

= intΩ

1205900nabla119906

0sdot nabla119906

0d119909 = int

Σ

119892Λ0119892d119904

(11)

Hence from

intΩ

1205901nabla (119906

1minus 119906

0) sdot nabla (119906

1minus 119906

0) d119909

= intΩ

1205901

1003816100381610038161003816nabla11990611003816100381610038161003816

2d119909 minus intΩ

1205900

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

+ intΩ

(1205901minus 120590

0)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

(12)

we obtain that

intΣ

119892 (Λ1minus Λ

0) 119892d119904 = int

Ω

(1205900minus 120590

1)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

+ intΩ

1205901

1003816100381610038161003816nabla (1199061minus 119906

0)1003816100381610038161003816

2d119909(13)

which already yields the first asserted inequality

Computational and Mathematical Methods in Medicine 3

By interchanging 1205901and 120590

0 we conclude that

intΣ

119892 (Λ0minus Λ

1) 119892d119904

= intΩ

(1205901minus 120590

0)1003816100381610038161003816nabla1199061

1003816100381610038161003816

2d119909 + intΩ

1205900

1003816100381610038161003816nabla (1199060minus 119906

1)1003816100381610038161003816

2d119909

= intΩ

(1205901

10038161003816100381610038161003816100381610038161003816

nabla1199061minus1205900

1205901

nabla1199060

10038161003816100381610038161003816100381610038161003816

2

+ (1205900minus1205902

0

1205901

)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2

) d119909

(14)

and hence obtain the second inequality

Given a reference conductivity 1205900

isin 119871infin

+(Ω) and a

measurable subset119863 sube Ω we define the virtual measurementoperator 119871

119863by

119871119863 1198712

(119863)119899

997888rarr 1198712

(Σ) 119865 997891997888rarr V|

Σ (15)

where V isin 1198671

(Ω) solves

intΩ

1205900nablaV sdot nabla119908d119909 = int

119863

119865 sdot nabla119908d119909 forall119908 isin 1198671

(Ω) (16)

The energy term |nabla1199060|2 in Lemma 1 can be identified with the

norm of the adjoint of this virtual measurement operator

Lemma 2 The adjoint operator of 119871119863is given by

119871lowast

119863 1198712

(Σ) 997888rarr 119871

2

(119863)119899

119892 997891997888rarr nabla1199060|119863 (17)

where 1199060isin 119867

1

(Ω) solves

nabla sdot 1205900nabla119906

0= 0 119894119899 Ω 120590

0120597]1199060|120597Ω =

119892 119900119899 Σ

0 119890119897119904119890(18)

Proof For all 119892 isin 1198712

(Σ) and 119865 isin 119871

2

(119863)119899 we have that

int119863

(119871lowast

119863119892) sdot 119865 d119909 = int

Σ

119892 (119871119863119865) d119909 = int

Σ

119892V|Σd119909

= intΩ

1205900nabla119906

0sdot nablaV d119909 = int

119863

nabla1199060sdot 119865 d119909

(19)

which shows the assertion

The following functional analytic lemma uses bounds onthe image of an operator to characterize the range of its dualoperator

Lemma 3 Let 119883 and 119884 be real Hilbert spaces with innerproducts (sdot sdot)

119883and (sdot sdot)

119884 respectively Let 119860 isin L(119883 119884) and

1199091015840

isin 119883 Then

1199091015840

isin R (119860lowast

) iff exist119862 gt 0 10038161003816100381610038161003816(1199091015840

119909)119883

10038161003816100381610038161003816le 119862 119860119909 forall119909 isin 119883

(20)

In particular if1198831198841 and119884

2are three real Hilbert spaces119860

119894isin

L(119884119894 119883) 119894 = 1 2 and if there exists 119862 gt 0 with

1003817100381710038171003817119860lowast

11199091003817100381710038171003817 le 119862

1003817100381710038171003817119860lowast

21199091003817100381710038171003817 forall119909 isin 119883 (21)

thenR(1198601) sube R(119860

2)

Proof The assertion can be generalized to Banach spacesand in that context it is called the ldquo14th important propertyof Banach spacesrdquo in Bourbaki [26] For the sake of complete-ness we rewrite the proof from [27] to Hilbert spaces

If 1199091015840 isin R(119860lowast

) then there exists 1199101015840 isin 119884 such that 1199091015840 =119860lowast

1199101015840 Hence10038161003816100381610038161003816(1199091015840

119909)119883

10038161003816100381610038161003816=10038161003816100381610038161003816(119860

lowast

1199101015840

119909)119883

10038161003816100381610038161003816

=10038161003816100381610038161003816(1199101015840

119860119909)119884

10038161003816100381610038161003816le10038171003817100381710038171003817119910101584010038171003817100381710038171003817119860119909 forall119909 isin 119883

(22)

so that the assertion holds with 119862 = 1199101015840

Now let 1199091015840 isin 119883 be such that there exists 119862 gt 0 with

|(1199091015840

119909)119883| le 119862119860119909 for all 119909 isin 119883 We define

119891 (119910) = (1199091015840

119909)119883

for every 119910 = 119860119909 isin R (119860) (23)

Then 119891 is a well-defined continuous linear functional onR(119860) By setting it to zero on R(119860)

perp we can extend 119891 to acontinuous linear functional on 119884 Using the Riesz theoremit follows that there exists 1199101015840 isin 119884 with

(1199101015840

119910)119884

= 119891 (119910) forall119910 isin R (119860) (24)

Hence for all 119909 isin 119883 we have

(119860lowast

1199101015840

119909)119883

= (1199101015840

119860119909)119884

= 119891 (119860119909) = (1199091015840

119909)119883

(25)

so that 1199091015840 = 119860lowast

1199101015840

isin R(119860lowast

)

The last lemma shows that the range of the virtualmeasurement operator 119871

119863determines the region119863 onwhich

it is defined We state the lemma for a simple special casea generalized version of the lemma will be formulated inSection 32

Lemma 4 Let 1205900= 1 119863 sube Ω be open and 119863 sube Ω have a

connected complement Ω 119863Then for all unit vectors 119889 isin R119899 119889 = 1 and every point

119911 isin Ω 120597119863 it holds that

119911 isin 119863 iff Φ119911119889|Σisin R (119871

119863) (26)

Proof The proof is similar to the one of [21 Lemma 29]First let 119911 isin 119863 and 120598 gt 0 be such that 119861

120598(119911) sube 119863 We

choose

1198911isin 119867

1

(119861120598(119911)) with 119891

1|120597119861120598(119911)

= Φ119911119889|120597119861120598(119911)

1198912isin 119867

1

(119861120598(119911)) with Δ119891

2= 0

120597]1198912|120597119861120598(119911)= 120597]Φ119911119889|120597119861120598(119911)

(27)

and let 119865 isin 1198712

(119863)119899 be the zero continuation of nabla(119891

1minus 119891

2) to

119863Then the function

V = Φ119911119889

in Ω 119861120598(119911)

1198911

in 119861120598(119911)

(28)

4 Computational and Mathematical Methods in Medicine

fulfills V isin 1198671

(Ω) and for all 119908 isin 119867

1

(Ω)

intΩ

nablaV sdot nabla119908d119909

= intΩ119861120598(119911)

nablaΦ119911119889

sdot nabla119908d119909 + int119861120598(119911)

nabla1198911sdot nabla119908d119909

= minusint120597119861120598(119911)

120597]Φ119911119889 119908|120597119861120598(119911)d119904 + int

119861120598(119911)

nabla1198911sdot nabla119908d119909

= int119861120598(119911)

nabla (1198911minus 119891

2) sdot nabla119908d119909 = int

119863

119865 sdot nabla119908d119909

(29)

This shows thatΦ119911119889|Σ= V|

Σ= 119871

119863(119865) isin R(119871

119863)

Now let Φ119911119889|Σisin R(119871

119863) Let V isin 119867

1

(Ω) be the function

from the definition of 119871119863 Then

V|Σ= Φ

119911119889|Σ 120597]V|Σ = 0 = 120597]Φ119911119889|Σ (30)

so that it follows by unique continuation that V = Φ119911119889

in theconnected setΩ (119863 cup 119911)

If 119911 notin 119863 then 119889 sdot nabla120575119911notin 119867

minus2

(Ω 119863) and thus Φ119911119889

notin

1198712

(Ω 119863) which contradicts that V = Φ119911119889

in Ω (119863 cup 119911)Hence 119911 isin 119863

3 The Factorization Method

Nowwe will formulate the FactorizationMethod and charac-terize a region where a conductivity 120590 differs from a referenceconductivity 120590

0by a range criterion Before we turn to a

new general formulation of the method we first state it fora special case that is similar to the one that was treated in theoriginal works of Bruhl and Hanke [3 4]

31 The Factorization Method for a Simple Special Case

Theorem 5 Let 1205900= 1 and 120590 = 1 + 120594

119863 where 119863 sube Ω is an

open set so that 119863 sube Ω has a connected complement Ω 119863Then for all 119911 isin Ω 119911 notin 120597119863 and all dipole directions 119889 isin R119899119889 = 1

119911 isin 119863 iff Φ119911119889

1003816100381610038161003816Σisin R (|Λ (120590) minus Λ (1)|

12

) (31)

Proof Themonotony Lemma 1 yields that for all119892 isin 1198712

(120597Ω)

int119863

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 ge intΣ

119892 (Λ (1) minus Λ (120590)) 119892d119904 ge int119863

1

2

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(32)

Hence |Λ(120590) minusΛ(1)| = Λ(1) minusΛ(120590) and using Lemma 2 wecan restate this in the form

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

ge10038171003817100381710038171003817|Λ (120590) minus Λ (1)|

12

11989210038171003817100381710038171003817

2

ge1

2

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

(33)

Using the functional analytic Lemma 3 this implies that

R (119871119863) = R (|Λ (120590) minus Λ (1)|

12

) (34)

and thus the assertion follows from the relation between 119863

andR(119871119863) in Lemma 4

Obviously the same arguments can be used to treat thecase 120590(119909) = 1 + 120581(119909)120594

119863(119909) when there exists a conductivity

jump 120598 gt 0 so that either

120581 (119909) ge 120598 forall119909 isin 119863 or 120581 (119909) le minus120598 forall119909 isin 119863 (35)

32 The Factorization Method for the General PiecewiseAnalytic Case Now we drop the assumptions that the back-ground is constant that there is a clear conductivity jumpand that the complement of the inclusions is connected Wewill merely assume that the reference conductivity 120590

0is a

piecewise analytic function and that either 120590 minus 1205900ge 0 or

120590 minus 1205900le 0 Roughly speaking under this general assumption

the Factorization Method then characterizes the support of120590 minus 120590

0up to holes in the support that have no connections to

Σ For a precise formulation we use the concept of the innerand outer support from [28] that has been inspired by the useof the infinity support of Kusiak and Sylvester [29] see also[25 30]

Definition 6 A relatively open set 119880 sube Ω is called connectedto Σ if 119880 cap Ω is connected and 119880 cap Σ = 0

For a measurable function 120581 Ω rarr R we define

(a) the support supp(120581) as the complement (in Ω) of theunion of those relatively open119880 sube Ω for which 120581|

119880equiv

0(b) the inner support inn supp 120581 as the union of those

open sets 119880 sube Ω for which ess inf119909isin119880

|120581(119909)| gt 0(c) the outer support out

Σsupp 120581 as the complement (in

Ω) of the union of those relatively open 119880 sube Ω thatare connected to Σ and for which 120581|

119880equiv 0

The interior of a set119872 sube Ω is denoted by int119872 and its closure(with respect toR119899) by119872 If119872 is measurable we also define

(d) outΣ119872 = out

Σsupp120594

119872

It is easily checked that outΣ(supp 120581) = out

Σsupp 120581

With this concept we can extend the range characteriza-tion in Lemma 4 to a general setting (see also Remark 9 later)

Lemma 7 Let 1205900isin 119871

infin

+(Ω) be piecewise analytic Let 119863 sube Ω

be measurableThen for all unit vectors 119889 isin R119899 119889 = 1 and every point

119911 isin Ω that has a neighborhood in which 1205900is analytic

119911 isin int119863 implies Φ119911119889

1003816100381610038161003816Σisin R (119871

119863) (36)

and

Φ119911119889

1003816100381610038161003816Σisin R (119871

119863) implies 119911 isin out

Σ119863 (37)

Proof If 119911 isin int119863 then there exists a small ball 119861120598(119911) sube 119863

and the first assertion follows as in the proof of Lemma 4To show the second assertion letΦ

119911119889|Σisin R(119871

119863) and let

V isin 1198671

(Ω) be the function from the definition of 119871

119863 so that

(as in the proof of Lemma 4)

V|Σ= Φ

119911119889|Σ 120597]V|Σ = 0 = 120597]Φ119911119889|Σ (38)

Computational and Mathematical Methods in Medicine 5

Assume that 119911 notin outΣ119863 Then there exists a relatively

open 119880 sube Ω that is connected to Σ and contains 119911Hence by unique continuation it follows that V|

119880= Φ

119911119889|119880

and we obtain the same contradiction as in the proof ofLemma 4

Now we can formulate and prove the FactorizationMethod for general piecewise analytic conductivities

Theorem 8 Let 120590 isin 119871infin

+(Ω) and let 120590

0isin 119871

infin

+(Ω) be a

piecewise analytic function Let either

120590 (119909) ge 1205900(119909) forall119909 isin Ω or 120590 (119909) le 120590

0(119909) forall119909 isin Ω

(39)

Then for all 119911 isin Ω that have a neighborhood in which 1205900is

analytic as well as all unit vectors 119889 isin R119899 119889 = 1

119911isin innsupp (120590 minus 1205900) implies Φ

119911119889

1003816100381610038161003816Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

)

(40)

and

Φ119911119889

1003816100381610038161003816ΣisinR (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) implies 119911isin outΣsupp (120590 minus 120590

0)

(41)

Proof Let 119911 isin Ω have a neighborhood in which 1205900is analytic

and let 119889 isin R119899 be a unit vector with 119889 = 1 We onlyprove the assertions for 120590 ge 120590

0 The other case is completely

analogousFirst let 119911 isin inn supp (120590 minus 120590

0) Then there exists a small

ball 119861120598(119911) and 120575 gt 0 so that 120590 minus 120590

0ge 120575 on 119861

120598(119911) Using the

monotony Lemma 1 it follows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (1205900) minus Λ (120590)) 119892d119904

ge intΩ

1205900

120590(120590 minus 120590

0)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

ge 12057510038171003817100381710038171205900

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817100381710038171003817

1

120590

1003817100381710038171003817100381710038171003817119871infin(Ω)int119861120598(119911)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

ge 12057510038171003817100381710038171205900

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817100381710038171003817

1

120590

1003817100381710038171003817100381710038171003817119871infin(Ω)

10038171003817100381710038171003817119871lowast

119861120598(119911)11989210038171003817100381710038171003817

2

(42)

Using the functional analytic Lemma 3 we obtain that

R (119871119861120598(119911)

) sube R (1003816100381610038161003816Λ (120590) minus Λ (120590

0)1003816100381610038161003816

12

) (43)

and Lemma 7 yields that

Φ119911119889|Σisin R (119871

119861120598(119911)) sube R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) (44)

On the other hand with 119863 = supp(120590 minus 1205900) the monotony

Lemma 1 shows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (1205900) minus Λ (120590

1)) 119892d119904 le int

Ω

(1205901minus 120590

0)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

le10038171003817100381710038171205901 minus 120590

0

1003817100381710038171003817119871infin(Ω)int119863

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le10038171003817100381710038171205901 minus 120590

0

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

(45)

so that we obtain from the functional analytic Lemma 3

R (1003816100381610038161003816Λ (120590) minus Λ (120590

0)1003816100381610038161003816

12

) sube R (119871119863) (46)

Hence Lemma 7 yields that

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) sube R (119871119863) implies

119911 isin outΣ(supp (120590 minus 120590

0)) = out

Σsupp (120590 minus 120590

0)

(47)

Remark 9 Theorem 8 shows that the Factorization Methodis able to detect the support of a conductivity difference up tothe difference between the outer and the inner support thatis roughly speaking up to holes in the support that have noconnections to the boundary It leaves openwhether points insuch holes will fulfill the range criterion of the FactorizationMethod or not

A result of Hyvonen and the author [30 Lemma 25]shows that for every smooth domain119863with119863 sub Ω and everyunit vector 119889 isin R119899 119889 = 1

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) forall119911 isin 120597119863 (48)

implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) forall119911 isin 119863 (49)

In that sense we can expect that holes in the support willbe filled up and that the set detected by the FactorizationMethod is essentially the outer support of the conductivitydifference

33 The Factorization Method for the Indefinite Case It is along standing open theoretical problem whether the rangecriterion of the Factorization Method holds true withoutthe definiteness assumption that 120590 ge 120590

0on Ω or 120590 le

1205900on Ω However Grinberg Kirsch and Schmitt [18 31]

showed how to exclude a region 119864 sube Ω from Ω insuch a way that the Factorization Method only requires thedefiniteness assumption onΩ119864 In this subsection we showhow their idea can be incorporated into our formulation ofthe method

To point out the main idea we first formulate the resultfor a simple special case Let us stress that for 120590 = 1 + 120594

119863+ minus

(12)120594119863minus it is not known whether

119911 isin 119863+

cup 119863minus iff Φ

119911119889|Σisin R (|Λ (120590) minus Λ (1)|

12

) (50)

However we can still use the FactorizationMethod if we havesome a priori knowledge that separates 119863+ and 119863

minus Moreprecisely if we know a subset 119864 that contains 119863minus withoutintersecting 119863

+ then we can use the Factorization Methodto find119863+ cup 119864 (and thus119863+)

6 Computational and Mathematical Methods in Medicine

Theorem 10 Let 1205900= 1 and 120590 = 1 + 120594

119863+ minus (12)120594

119863minus where

119863+

119863minus

sube Ω are open Let 119864 sube Ω be an open set

(a) If 119863+ sube 119864 and 119863minus cup 119864 sube Ω has a connectedcomplement then for all 119911 isin Ω 119911 notin 120597(119863

minus

cup119864) and alldipole directions 119889 isin R119899 119889 = 1

119911 isin 119863minus

cup 119864 iff Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1) + 119871119864119871lowast

119864

1003816100381610038161003816

12

)

(51)

(b) If 119863minus sube 119864 and 119863+ cup 119864 sube Ω has a connectedcomplement then for all 119911 isin Ω 119911 notin 120597(119863

+

cup119864) and alldipole directions 119889 isin R119899 119889 = 1

119911 isin 119863+

cup 119864 iff Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1) minus 119871119864119871lowast

119864

1003816100381610038161003816

12

)

(52)

Proof Themonotony Lemma 1 yields that for all119892 isin 1198712

(120597Ω)

1

2int119863minus

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 minus int119863+

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le intΣ

119892 (Λ (120590) minus Λ (1)) 119892d119904

le int119863minus

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 minus1

2int119863+

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(53)

Since (cf Lemma 2)

intΣ

119892 (119871119864119871lowast

119864) 119892d119904 = 1003817100381710038171003817119871

lowast

1198641198921003817100381710038171003817

2

= int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 (54)

it follows for case (a) that

1

2int119863minuscup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 le intΣ

119892 (Λ (120590) minus Λ (1) + 2119871119864119871lowast

119864) 119892d119904

le int119863minuscup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909(55)

Using the functional analytic Lemma 3 this implies that

R (119871119863minuscup119864) = R (

1003816100381610038161003816Λ (120590) minus Λ (1) + 2119871119864119871lowast

119864

1003816100381610038161003816

12

) (56)

so that the assertion (a) follows from Lemma 4In case (b) we obtain that

3

2int119863+cup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 ge intΣ

119892 (Λ (1) minus Λ (120590) + 2119871119864119871lowast

119864) 119892d119904

ge1

2int119863+cup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909(57)

and the same arguments as above yield the assertion

We can also extend these ideas to the general setting ofSection 32

Theorem 11 Let 120590 isin 119871infin

+(Ω) and let 120590

0isin 119871

infin

+(Ω) be a

piecewise analytic function Let 119864 sube Ω be a measurable setChoose 120572 120573 isin R such that

120572 gt1003817100381710038171003817120590 minus 120590

0

1003817100381710038171003817119871infin(Ω) 120573 gt

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω) (58)

(a) If 120590 le 1205900on Ω 119864 then for all 119911 isin Ω that have a

neighborhood in which 1205900is analytic as well as all unit

vectors 119889 isin R119899 119889 = 1

119911 isin inn supp (120590 minus 1205900) cup 119864 implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590)minusΛ (1205900)+120572119871

119864119871lowast

119864

1003816100381610038161003816

12

)

(59)

and

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) implies

119911 isin outΣ(supp (120590 minus 120590

0) cup 119864)

(60)

(b) If 120590 ge 1205900on Ω 119864 then for all 119911 isin Ω that have a

neighborhood in which 1205900is analytic as well as all unit

vectors 119889 isin R119899 119889 = 1

119911 isin inn supp (120590 minus 1205900) cup 119864 implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) minus 120573119871

119864119871lowast

119864

1003816100381610038161003816

12

)

(61)

and

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) minus 120573119871

119864119871lowast

119864

1003816100381610038161003816

12

) implies

119911 isin outΣ(supp (120590 minus 120590

0) cup 119864)

(62)

Proof For every 119911 isin inn supp(120590minus1205900)with 119911 notin 119864 there exists

a small ball 119861120598(119911) and 120575 gt 0 so that 120590

0minus120590 ge 120575 on 119861

120598(119911) Using

the monotony Lemma 1 it follows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

ge intΩ

(1205900minus 120590

1)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

ge 120575int119861120598(119911)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909+(120572 minus1003817100381710038171003817120590 minus 120590

0

1003817100381710038171003817119871infin(Ω))int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(63)

Computational and Mathematical Methods in Medicine 7

As in the previous proofs we obtain from Lemmas 2 and 3that

R (1003816100381610038161003816Λ (120590) minus Λ (120590

0) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) supe R (119871119861120598(119911)cup119864

) (64)

so that the first implication of (a) follows from Lemma 7The monotony Lemma 1 also implies that

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

le intΩ

1205900

120590(1205900minus 120590)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω)intsupp(120590minus1205900)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

+ 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(65)

so that the second implication of (a) follows from Lemmas 23 and 7 Assertion (b) can be proven analogously

Remark 9 also applies to this case

4 Conclusions and Remarks

The Factorization Method can be used to detect regionsin which a conductivity differs from a known referenceconductivity In this work we summarized the progresson the methodrsquos theoretical foundation We formulated themethod for general piecewise analytic conductivities andgave comparatively simple and self-contained proofsWe alsoshowed how the idea of excluding a part of the imaging regioncan be incorporated into this formulation

The regularity assumptions can beweakened even furtherOur proofs only require unique continuation arguments forthe reference conductivity 120590

0and the existence of the dipole

functionsTwo major open theoretical questions still exist in the

context of the FactorizationMethodThe theoretical justifica-tion of the method requires a definiteness condition (on thewhole domain or after excluding an a priori known part of thedomain) It is unknownwhether themethodrsquos range criterionholds without such a definiteness conditionThe second openquestion concerns the numerical stability of the methodrsquosrange criterion So far there are no rigorous convergenceresults for numerical implementations of this range criterion(see however Lechleiter [32] for a first step in this direction)As a promising approach to overcome both problems wewould like to point out the recent work on monotony-basedmethods [28]

Acknowledgments

This paper was funded by the German Research Foundation(DFG) and the University of Wurzburg in the fundingprogramme Open Access Publishing

References

[1] A Kirsch ldquoCharacterization of the shape of a scattering obstacleusing the spectral data of the far field operatorrdquo Inverse Prob-lems vol 14 no 6 pp 1489ndash1512 1998

[2] M Bruhl Gebietserkennung in der elektrischen Impedanztomo-graphie [PhD thesis] Universitat Karlsruhe 1999

[3] M Bruhl and M Hanke ldquoNumerical implementation of twononiterative methods for locating inclusions by impedancetomographyrdquo Inverse Problems vol 16 no 4 pp 1029ndash10422000

[4] M Bruhl ldquoExplicit characterization of inclusions in electricalimpedance tomographyrdquo SIAM Journal on Mathematical Anal-ysis vol 32 no 6 pp 1327ndash1341 2001

[5] M Hanke and M Bruhl ldquoRecent progress in electricalimpedance tomographyrdquo Inverse Problems vol 19 no 6 ppS65ndashS90 2003

[6] A Kirsch andN GrinbergTheFactorizationMethod for InverseProblems vol 36 ofOxford Lecture Series inMathematics and ItsApplications Oxford University Press Oxford UK 2007

[7] M Hanke and A Kirsch ldquoSampling methodsrdquo in Handbook ofMathematical Models in Imaging O Scherzer Ed pp 501ndash550Springer 2011

[8] A Kirsch ldquoThe factorization method for a class of inverseelliptic problemsrdquo Mathematische Nachrichten vol 278 pp258ndash277 2005

[9] B Gebauer ldquoThe Factorization Method for real elliptic prob-lemsrdquo Zeitschrift fur Analysis und ihre Anwendung vol 25 no1 pp 81ndash102 2006

[10] A I Nachman L Paivarinta and A Teirila ldquoOn imagingobstacles inside inhomogeneous mediardquo Journal of FunctionalAnalysis vol 252 no 2 pp 490ndash516 2007

[11] M Hanke and B Schappel ldquoThe factorization method forelectrical impedance tomography in the half-spacerdquo SIAMJournal on Applied Mathematics vol 68 no 4 pp 907ndash9242008

[12] N Hyvonen ldquoComplete electrode model of electricalimpedance tomography approximation properties andcharacterization of inclusionsrdquo SIAM Journal on AppliedMathematics vol 64 pp 902ndash931 2004

[13] A Lechleiter N Hyvonen and H Hakula ldquoThe factorizationmethod applied to the complete electrode model of impedancetomographyrdquo SIAM Journal onAppliedMathematics vol 68 no4 pp 1097ndash1121 2008

[14] N Hyvonen H Hakula and S Pursiainen ldquoNumerical imple-mentation of the factorization method within the completeelectrode model of electrical impedance tomographyrdquo InverseProblems and Imaging vol 1 pp 299ndash317 2007

[15] HHakula andNHyvonen ldquoOn computation of test dipoles forfactorization methodrdquo BIT Numerical Mathematics vol 49 no1 pp 75ndash91 2009

[16] B Harrach and J K Seo ldquoDetecting inclusions in electri-cal impedance tomography without reference measurementsrdquoSIAM Journal on Applied Mathematics vol 69 no 6 pp 1662ndash1681 2009

[17] B Harrach J K Seo and E J Woo ldquoFactorization methodand its physical justification in frequency-difference electricalimpedance tomographyrdquo IEEE Transactions on Medical Imag-ing vol 29 no 11 pp 1918ndash1926 2010

[18] S Schmitt ldquoThe factorization method for EIT in the case ofmixed inclusionsrdquo Inverse Problems vol 25 no 6 Article ID065012 2009

8 Computational and Mathematical Methods in Medicine

[19] S Schmitt and A Kirsch ldquoA factorization scheme for determin-ing conductivity contrasts in impedance tomographyrdquo InverseProblems vol 27 Article ID 095005 2011

[20] B Gebauer and N Hyvonen ldquoFactorization method and irreg-ular inclusions in electrical impedance tomographyrdquo InverseProblems vol 23 no 5 pp 2159ndash2170 2007

[21] B Gebauer ldquoLocalized potentials in electrical impedancetomographyrdquo Inverse Problems and Imaging vol 2 pp 251ndash2692008

[22] H Kang J K Seo and D Sheen ldquoThe inverse conductivityproblem with one measurement Stability and estimation ofsizerdquo SIAM Journal on Mathematical Analysis vol 28 no 6 pp1389ndash1405 1997

[23] M Ikehata ldquoSize estimation of inclusionrdquo Journal of Inverse andIll-Posed Problems vol 6 no 2 pp 127ndash140 1998

[24] T Ide H Isozaki S Nakata S Siltanen and G UhlmannldquoProbing for electrical inclusions with complex sphericalwavesrdquo Communications on Pure and Applied Mathematics vol60 no 10 pp 1415ndash1442 2007

[25] B Harrach and J K Seo ldquoExact shape-reconstruction by one-step linearization in electrical impedance tomographyrdquo TheSIAM Journal onMathematical Analysis vol 42 no 4 pp 1505ndash1518 2010

[26] N Bourbaki Elements of Mathematics Topological VectorSpaces chapters 1ndash5 Springer Berlin Germany 2003

[27] F Fruhauf B Gebauer andO Scherzer ldquoDetecting interfaces ina parabolic-elliptic problem from surfacemeasurementsrdquo SIAMJournal on Numerical Analysis vol 45 no 2 pp 810ndash836 2007

[28] B Harrach and M Ullrich ldquoMonotony based shape recon-struction in electrical impedance tomographyrdquo submitted forpublication

[29] S Kusiak and J Sylvester ldquoThe scattering supportrdquoCommunica-tions on Pure and Applied Mathematics vol 56 no 11 pp 1525ndash1548 2003

[30] B Gebauer and N Hyvonen ldquoFactorization method andinclusions of mixed type in an inverse elliptic boundary valueproblemrdquo Inverse Problems and Imaging vol 2 pp 355ndash3722008

[31] N I Grinberg and A Kirsch ldquoThe factorization method forobstacles with a-priori separated sound-soft and sound-hardpartsrdquo Mathematics and Computers in Simulation vol 66 no4-5 pp 267ndash279 2004

[32] A Lechleiter ldquoA regularization technique for the factorizationmethodrdquo Inverse Problems vol 22 no 5 pp 1605ndash1625 2006

Submit your manuscripts athttpwwwhindawicom

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Oxidative Medicine and Cellular Longevity

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PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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ObesityJournal of

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Research and TreatmentAIDS

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 3

By interchanging 1205901and 120590

0 we conclude that

intΣ

119892 (Λ0minus Λ

1) 119892d119904

= intΩ

(1205901minus 120590

0)1003816100381610038161003816nabla1199061

1003816100381610038161003816

2d119909 + intΩ

1205900

1003816100381610038161003816nabla (1199060minus 119906

1)1003816100381610038161003816

2d119909

= intΩ

(1205901

10038161003816100381610038161003816100381610038161003816

nabla1199061minus1205900

1205901

nabla1199060

10038161003816100381610038161003816100381610038161003816

2

+ (1205900minus1205902

0

1205901

)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2

) d119909

(14)

and hence obtain the second inequality

Given a reference conductivity 1205900

isin 119871infin

+(Ω) and a

measurable subset119863 sube Ω we define the virtual measurementoperator 119871

119863by

119871119863 1198712

(119863)119899

997888rarr 1198712

(Σ) 119865 997891997888rarr V|

Σ (15)

where V isin 1198671

(Ω) solves

intΩ

1205900nablaV sdot nabla119908d119909 = int

119863

119865 sdot nabla119908d119909 forall119908 isin 1198671

(Ω) (16)

The energy term |nabla1199060|2 in Lemma 1 can be identified with the

norm of the adjoint of this virtual measurement operator

Lemma 2 The adjoint operator of 119871119863is given by

119871lowast

119863 1198712

(Σ) 997888rarr 119871

2

(119863)119899

119892 997891997888rarr nabla1199060|119863 (17)

where 1199060isin 119867

1

(Ω) solves

nabla sdot 1205900nabla119906

0= 0 119894119899 Ω 120590

0120597]1199060|120597Ω =

119892 119900119899 Σ

0 119890119897119904119890(18)

Proof For all 119892 isin 1198712

(Σ) and 119865 isin 119871

2

(119863)119899 we have that

int119863

(119871lowast

119863119892) sdot 119865 d119909 = int

Σ

119892 (119871119863119865) d119909 = int

Σ

119892V|Σd119909

= intΩ

1205900nabla119906

0sdot nablaV d119909 = int

119863

nabla1199060sdot 119865 d119909

(19)

which shows the assertion

The following functional analytic lemma uses bounds onthe image of an operator to characterize the range of its dualoperator

Lemma 3 Let 119883 and 119884 be real Hilbert spaces with innerproducts (sdot sdot)

119883and (sdot sdot)

119884 respectively Let 119860 isin L(119883 119884) and

1199091015840

isin 119883 Then

1199091015840

isin R (119860lowast

) iff exist119862 gt 0 10038161003816100381610038161003816(1199091015840

119909)119883

10038161003816100381610038161003816le 119862 119860119909 forall119909 isin 119883

(20)

In particular if1198831198841 and119884

2are three real Hilbert spaces119860

119894isin

L(119884119894 119883) 119894 = 1 2 and if there exists 119862 gt 0 with

1003817100381710038171003817119860lowast

11199091003817100381710038171003817 le 119862

1003817100381710038171003817119860lowast

21199091003817100381710038171003817 forall119909 isin 119883 (21)

thenR(1198601) sube R(119860

2)

Proof The assertion can be generalized to Banach spacesand in that context it is called the ldquo14th important propertyof Banach spacesrdquo in Bourbaki [26] For the sake of complete-ness we rewrite the proof from [27] to Hilbert spaces

If 1199091015840 isin R(119860lowast

) then there exists 1199101015840 isin 119884 such that 1199091015840 =119860lowast

1199101015840 Hence10038161003816100381610038161003816(1199091015840

119909)119883

10038161003816100381610038161003816=10038161003816100381610038161003816(119860

lowast

1199101015840

119909)119883

10038161003816100381610038161003816

=10038161003816100381610038161003816(1199101015840

119860119909)119884

10038161003816100381610038161003816le10038171003817100381710038171003817119910101584010038171003817100381710038171003817119860119909 forall119909 isin 119883

(22)

so that the assertion holds with 119862 = 1199101015840

Now let 1199091015840 isin 119883 be such that there exists 119862 gt 0 with

|(1199091015840

119909)119883| le 119862119860119909 for all 119909 isin 119883 We define

119891 (119910) = (1199091015840

119909)119883

for every 119910 = 119860119909 isin R (119860) (23)

Then 119891 is a well-defined continuous linear functional onR(119860) By setting it to zero on R(119860)

perp we can extend 119891 to acontinuous linear functional on 119884 Using the Riesz theoremit follows that there exists 1199101015840 isin 119884 with

(1199101015840

119910)119884

= 119891 (119910) forall119910 isin R (119860) (24)

Hence for all 119909 isin 119883 we have

(119860lowast

1199101015840

119909)119883

= (1199101015840

119860119909)119884

= 119891 (119860119909) = (1199091015840

119909)119883

(25)

so that 1199091015840 = 119860lowast

1199101015840

isin R(119860lowast

)

The last lemma shows that the range of the virtualmeasurement operator 119871

119863determines the region119863 onwhich

it is defined We state the lemma for a simple special casea generalized version of the lemma will be formulated inSection 32

Lemma 4 Let 1205900= 1 119863 sube Ω be open and 119863 sube Ω have a

connected complement Ω 119863Then for all unit vectors 119889 isin R119899 119889 = 1 and every point

119911 isin Ω 120597119863 it holds that

119911 isin 119863 iff Φ119911119889|Σisin R (119871

119863) (26)

Proof The proof is similar to the one of [21 Lemma 29]First let 119911 isin 119863 and 120598 gt 0 be such that 119861

120598(119911) sube 119863 We

choose

1198911isin 119867

1

(119861120598(119911)) with 119891

1|120597119861120598(119911)

= Φ119911119889|120597119861120598(119911)

1198912isin 119867

1

(119861120598(119911)) with Δ119891

2= 0

120597]1198912|120597119861120598(119911)= 120597]Φ119911119889|120597119861120598(119911)

(27)

and let 119865 isin 1198712

(119863)119899 be the zero continuation of nabla(119891

1minus 119891

2) to

119863Then the function

V = Φ119911119889

in Ω 119861120598(119911)

1198911

in 119861120598(119911)

(28)

4 Computational and Mathematical Methods in Medicine

fulfills V isin 1198671

(Ω) and for all 119908 isin 119867

1

(Ω)

intΩ

nablaV sdot nabla119908d119909

= intΩ119861120598(119911)

nablaΦ119911119889

sdot nabla119908d119909 + int119861120598(119911)

nabla1198911sdot nabla119908d119909

= minusint120597119861120598(119911)

120597]Φ119911119889 119908|120597119861120598(119911)d119904 + int

119861120598(119911)

nabla1198911sdot nabla119908d119909

= int119861120598(119911)

nabla (1198911minus 119891

2) sdot nabla119908d119909 = int

119863

119865 sdot nabla119908d119909

(29)

This shows thatΦ119911119889|Σ= V|

Σ= 119871

119863(119865) isin R(119871

119863)

Now let Φ119911119889|Σisin R(119871

119863) Let V isin 119867

1

(Ω) be the function

from the definition of 119871119863 Then

V|Σ= Φ

119911119889|Σ 120597]V|Σ = 0 = 120597]Φ119911119889|Σ (30)

so that it follows by unique continuation that V = Φ119911119889

in theconnected setΩ (119863 cup 119911)

If 119911 notin 119863 then 119889 sdot nabla120575119911notin 119867

minus2

(Ω 119863) and thus Φ119911119889

notin

1198712

(Ω 119863) which contradicts that V = Φ119911119889

in Ω (119863 cup 119911)Hence 119911 isin 119863

3 The Factorization Method

Nowwe will formulate the FactorizationMethod and charac-terize a region where a conductivity 120590 differs from a referenceconductivity 120590

0by a range criterion Before we turn to a

new general formulation of the method we first state it fora special case that is similar to the one that was treated in theoriginal works of Bruhl and Hanke [3 4]

31 The Factorization Method for a Simple Special Case

Theorem 5 Let 1205900= 1 and 120590 = 1 + 120594

119863 where 119863 sube Ω is an

open set so that 119863 sube Ω has a connected complement Ω 119863Then for all 119911 isin Ω 119911 notin 120597119863 and all dipole directions 119889 isin R119899119889 = 1

119911 isin 119863 iff Φ119911119889

1003816100381610038161003816Σisin R (|Λ (120590) minus Λ (1)|

12

) (31)

Proof Themonotony Lemma 1 yields that for all119892 isin 1198712

(120597Ω)

int119863

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 ge intΣ

119892 (Λ (1) minus Λ (120590)) 119892d119904 ge int119863

1

2

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(32)

Hence |Λ(120590) minusΛ(1)| = Λ(1) minusΛ(120590) and using Lemma 2 wecan restate this in the form

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

ge10038171003817100381710038171003817|Λ (120590) minus Λ (1)|

12

11989210038171003817100381710038171003817

2

ge1

2

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

(33)

Using the functional analytic Lemma 3 this implies that

R (119871119863) = R (|Λ (120590) minus Λ (1)|

12

) (34)

and thus the assertion follows from the relation between 119863

andR(119871119863) in Lemma 4

Obviously the same arguments can be used to treat thecase 120590(119909) = 1 + 120581(119909)120594

119863(119909) when there exists a conductivity

jump 120598 gt 0 so that either

120581 (119909) ge 120598 forall119909 isin 119863 or 120581 (119909) le minus120598 forall119909 isin 119863 (35)

32 The Factorization Method for the General PiecewiseAnalytic Case Now we drop the assumptions that the back-ground is constant that there is a clear conductivity jumpand that the complement of the inclusions is connected Wewill merely assume that the reference conductivity 120590

0is a

piecewise analytic function and that either 120590 minus 1205900ge 0 or

120590 minus 1205900le 0 Roughly speaking under this general assumption

the Factorization Method then characterizes the support of120590 minus 120590

0up to holes in the support that have no connections to

Σ For a precise formulation we use the concept of the innerand outer support from [28] that has been inspired by the useof the infinity support of Kusiak and Sylvester [29] see also[25 30]

Definition 6 A relatively open set 119880 sube Ω is called connectedto Σ if 119880 cap Ω is connected and 119880 cap Σ = 0

For a measurable function 120581 Ω rarr R we define

(a) the support supp(120581) as the complement (in Ω) of theunion of those relatively open119880 sube Ω for which 120581|

119880equiv

0(b) the inner support inn supp 120581 as the union of those

open sets 119880 sube Ω for which ess inf119909isin119880

|120581(119909)| gt 0(c) the outer support out

Σsupp 120581 as the complement (in

Ω) of the union of those relatively open 119880 sube Ω thatare connected to Σ and for which 120581|

119880equiv 0

The interior of a set119872 sube Ω is denoted by int119872 and its closure(with respect toR119899) by119872 If119872 is measurable we also define

(d) outΣ119872 = out

Σsupp120594

119872

It is easily checked that outΣ(supp 120581) = out

Σsupp 120581

With this concept we can extend the range characteriza-tion in Lemma 4 to a general setting (see also Remark 9 later)

Lemma 7 Let 1205900isin 119871

infin

+(Ω) be piecewise analytic Let 119863 sube Ω

be measurableThen for all unit vectors 119889 isin R119899 119889 = 1 and every point

119911 isin Ω that has a neighborhood in which 1205900is analytic

119911 isin int119863 implies Φ119911119889

1003816100381610038161003816Σisin R (119871

119863) (36)

and

Φ119911119889

1003816100381610038161003816Σisin R (119871

119863) implies 119911 isin out

Σ119863 (37)

Proof If 119911 isin int119863 then there exists a small ball 119861120598(119911) sube 119863

and the first assertion follows as in the proof of Lemma 4To show the second assertion letΦ

119911119889|Σisin R(119871

119863) and let

V isin 1198671

(Ω) be the function from the definition of 119871

119863 so that

(as in the proof of Lemma 4)

V|Σ= Φ

119911119889|Σ 120597]V|Σ = 0 = 120597]Φ119911119889|Σ (38)

Computational and Mathematical Methods in Medicine 5

Assume that 119911 notin outΣ119863 Then there exists a relatively

open 119880 sube Ω that is connected to Σ and contains 119911Hence by unique continuation it follows that V|

119880= Φ

119911119889|119880

and we obtain the same contradiction as in the proof ofLemma 4

Now we can formulate and prove the FactorizationMethod for general piecewise analytic conductivities

Theorem 8 Let 120590 isin 119871infin

+(Ω) and let 120590

0isin 119871

infin

+(Ω) be a

piecewise analytic function Let either

120590 (119909) ge 1205900(119909) forall119909 isin Ω or 120590 (119909) le 120590

0(119909) forall119909 isin Ω

(39)

Then for all 119911 isin Ω that have a neighborhood in which 1205900is

analytic as well as all unit vectors 119889 isin R119899 119889 = 1

119911isin innsupp (120590 minus 1205900) implies Φ

119911119889

1003816100381610038161003816Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

)

(40)

and

Φ119911119889

1003816100381610038161003816ΣisinR (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) implies 119911isin outΣsupp (120590 minus 120590

0)

(41)

Proof Let 119911 isin Ω have a neighborhood in which 1205900is analytic

and let 119889 isin R119899 be a unit vector with 119889 = 1 We onlyprove the assertions for 120590 ge 120590

0 The other case is completely

analogousFirst let 119911 isin inn supp (120590 minus 120590

0) Then there exists a small

ball 119861120598(119911) and 120575 gt 0 so that 120590 minus 120590

0ge 120575 on 119861

120598(119911) Using the

monotony Lemma 1 it follows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (1205900) minus Λ (120590)) 119892d119904

ge intΩ

1205900

120590(120590 minus 120590

0)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

ge 12057510038171003817100381710038171205900

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817100381710038171003817

1

120590

1003817100381710038171003817100381710038171003817119871infin(Ω)int119861120598(119911)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

ge 12057510038171003817100381710038171205900

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817100381710038171003817

1

120590

1003817100381710038171003817100381710038171003817119871infin(Ω)

10038171003817100381710038171003817119871lowast

119861120598(119911)11989210038171003817100381710038171003817

2

(42)

Using the functional analytic Lemma 3 we obtain that

R (119871119861120598(119911)

) sube R (1003816100381610038161003816Λ (120590) minus Λ (120590

0)1003816100381610038161003816

12

) (43)

and Lemma 7 yields that

Φ119911119889|Σisin R (119871

119861120598(119911)) sube R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) (44)

On the other hand with 119863 = supp(120590 minus 1205900) the monotony

Lemma 1 shows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (1205900) minus Λ (120590

1)) 119892d119904 le int

Ω

(1205901minus 120590

0)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

le10038171003817100381710038171205901 minus 120590

0

1003817100381710038171003817119871infin(Ω)int119863

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le10038171003817100381710038171205901 minus 120590

0

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

(45)

so that we obtain from the functional analytic Lemma 3

R (1003816100381610038161003816Λ (120590) minus Λ (120590

0)1003816100381610038161003816

12

) sube R (119871119863) (46)

Hence Lemma 7 yields that

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) sube R (119871119863) implies

119911 isin outΣ(supp (120590 minus 120590

0)) = out

Σsupp (120590 minus 120590

0)

(47)

Remark 9 Theorem 8 shows that the Factorization Methodis able to detect the support of a conductivity difference up tothe difference between the outer and the inner support thatis roughly speaking up to holes in the support that have noconnections to the boundary It leaves openwhether points insuch holes will fulfill the range criterion of the FactorizationMethod or not

A result of Hyvonen and the author [30 Lemma 25]shows that for every smooth domain119863with119863 sub Ω and everyunit vector 119889 isin R119899 119889 = 1

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) forall119911 isin 120597119863 (48)

implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) forall119911 isin 119863 (49)

In that sense we can expect that holes in the support willbe filled up and that the set detected by the FactorizationMethod is essentially the outer support of the conductivitydifference

33 The Factorization Method for the Indefinite Case It is along standing open theoretical problem whether the rangecriterion of the Factorization Method holds true withoutthe definiteness assumption that 120590 ge 120590

0on Ω or 120590 le

1205900on Ω However Grinberg Kirsch and Schmitt [18 31]

showed how to exclude a region 119864 sube Ω from Ω insuch a way that the Factorization Method only requires thedefiniteness assumption onΩ119864 In this subsection we showhow their idea can be incorporated into our formulation ofthe method

To point out the main idea we first formulate the resultfor a simple special case Let us stress that for 120590 = 1 + 120594

119863+ minus

(12)120594119863minus it is not known whether

119911 isin 119863+

cup 119863minus iff Φ

119911119889|Σisin R (|Λ (120590) minus Λ (1)|

12

) (50)

However we can still use the FactorizationMethod if we havesome a priori knowledge that separates 119863+ and 119863

minus Moreprecisely if we know a subset 119864 that contains 119863minus withoutintersecting 119863

+ then we can use the Factorization Methodto find119863+ cup 119864 (and thus119863+)

6 Computational and Mathematical Methods in Medicine

Theorem 10 Let 1205900= 1 and 120590 = 1 + 120594

119863+ minus (12)120594

119863minus where

119863+

119863minus

sube Ω are open Let 119864 sube Ω be an open set

(a) If 119863+ sube 119864 and 119863minus cup 119864 sube Ω has a connectedcomplement then for all 119911 isin Ω 119911 notin 120597(119863

minus

cup119864) and alldipole directions 119889 isin R119899 119889 = 1

119911 isin 119863minus

cup 119864 iff Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1) + 119871119864119871lowast

119864

1003816100381610038161003816

12

)

(51)

(b) If 119863minus sube 119864 and 119863+ cup 119864 sube Ω has a connectedcomplement then for all 119911 isin Ω 119911 notin 120597(119863

+

cup119864) and alldipole directions 119889 isin R119899 119889 = 1

119911 isin 119863+

cup 119864 iff Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1) minus 119871119864119871lowast

119864

1003816100381610038161003816

12

)

(52)

Proof Themonotony Lemma 1 yields that for all119892 isin 1198712

(120597Ω)

1

2int119863minus

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 minus int119863+

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le intΣ

119892 (Λ (120590) minus Λ (1)) 119892d119904

le int119863minus

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 minus1

2int119863+

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(53)

Since (cf Lemma 2)

intΣ

119892 (119871119864119871lowast

119864) 119892d119904 = 1003817100381710038171003817119871

lowast

1198641198921003817100381710038171003817

2

= int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 (54)

it follows for case (a) that

1

2int119863minuscup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 le intΣ

119892 (Λ (120590) minus Λ (1) + 2119871119864119871lowast

119864) 119892d119904

le int119863minuscup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909(55)

Using the functional analytic Lemma 3 this implies that

R (119871119863minuscup119864) = R (

1003816100381610038161003816Λ (120590) minus Λ (1) + 2119871119864119871lowast

119864

1003816100381610038161003816

12

) (56)

so that the assertion (a) follows from Lemma 4In case (b) we obtain that

3

2int119863+cup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 ge intΣ

119892 (Λ (1) minus Λ (120590) + 2119871119864119871lowast

119864) 119892d119904

ge1

2int119863+cup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909(57)

and the same arguments as above yield the assertion

We can also extend these ideas to the general setting ofSection 32

Theorem 11 Let 120590 isin 119871infin

+(Ω) and let 120590

0isin 119871

infin

+(Ω) be a

piecewise analytic function Let 119864 sube Ω be a measurable setChoose 120572 120573 isin R such that

120572 gt1003817100381710038171003817120590 minus 120590

0

1003817100381710038171003817119871infin(Ω) 120573 gt

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω) (58)

(a) If 120590 le 1205900on Ω 119864 then for all 119911 isin Ω that have a

neighborhood in which 1205900is analytic as well as all unit

vectors 119889 isin R119899 119889 = 1

119911 isin inn supp (120590 minus 1205900) cup 119864 implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590)minusΛ (1205900)+120572119871

119864119871lowast

119864

1003816100381610038161003816

12

)

(59)

and

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) implies

119911 isin outΣ(supp (120590 minus 120590

0) cup 119864)

(60)

(b) If 120590 ge 1205900on Ω 119864 then for all 119911 isin Ω that have a

neighborhood in which 1205900is analytic as well as all unit

vectors 119889 isin R119899 119889 = 1

119911 isin inn supp (120590 minus 1205900) cup 119864 implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) minus 120573119871

119864119871lowast

119864

1003816100381610038161003816

12

)

(61)

and

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) minus 120573119871

119864119871lowast

119864

1003816100381610038161003816

12

) implies

119911 isin outΣ(supp (120590 minus 120590

0) cup 119864)

(62)

Proof For every 119911 isin inn supp(120590minus1205900)with 119911 notin 119864 there exists

a small ball 119861120598(119911) and 120575 gt 0 so that 120590

0minus120590 ge 120575 on 119861

120598(119911) Using

the monotony Lemma 1 it follows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

ge intΩ

(1205900minus 120590

1)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

ge 120575int119861120598(119911)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909+(120572 minus1003817100381710038171003817120590 minus 120590

0

1003817100381710038171003817119871infin(Ω))int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(63)

Computational and Mathematical Methods in Medicine 7

As in the previous proofs we obtain from Lemmas 2 and 3that

R (1003816100381610038161003816Λ (120590) minus Λ (120590

0) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) supe R (119871119861120598(119911)cup119864

) (64)

so that the first implication of (a) follows from Lemma 7The monotony Lemma 1 also implies that

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

le intΩ

1205900

120590(1205900minus 120590)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω)intsupp(120590minus1205900)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

+ 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(65)

so that the second implication of (a) follows from Lemmas 23 and 7 Assertion (b) can be proven analogously

Remark 9 also applies to this case

4 Conclusions and Remarks

The Factorization Method can be used to detect regionsin which a conductivity differs from a known referenceconductivity In this work we summarized the progresson the methodrsquos theoretical foundation We formulated themethod for general piecewise analytic conductivities andgave comparatively simple and self-contained proofsWe alsoshowed how the idea of excluding a part of the imaging regioncan be incorporated into this formulation

The regularity assumptions can beweakened even furtherOur proofs only require unique continuation arguments forthe reference conductivity 120590

0and the existence of the dipole

functionsTwo major open theoretical questions still exist in the

context of the FactorizationMethodThe theoretical justifica-tion of the method requires a definiteness condition (on thewhole domain or after excluding an a priori known part of thedomain) It is unknownwhether themethodrsquos range criterionholds without such a definiteness conditionThe second openquestion concerns the numerical stability of the methodrsquosrange criterion So far there are no rigorous convergenceresults for numerical implementations of this range criterion(see however Lechleiter [32] for a first step in this direction)As a promising approach to overcome both problems wewould like to point out the recent work on monotony-basedmethods [28]

Acknowledgments

This paper was funded by the German Research Foundation(DFG) and the University of Wurzburg in the fundingprogramme Open Access Publishing

References

[1] A Kirsch ldquoCharacterization of the shape of a scattering obstacleusing the spectral data of the far field operatorrdquo Inverse Prob-lems vol 14 no 6 pp 1489ndash1512 1998

[2] M Bruhl Gebietserkennung in der elektrischen Impedanztomo-graphie [PhD thesis] Universitat Karlsruhe 1999

[3] M Bruhl and M Hanke ldquoNumerical implementation of twononiterative methods for locating inclusions by impedancetomographyrdquo Inverse Problems vol 16 no 4 pp 1029ndash10422000

[4] M Bruhl ldquoExplicit characterization of inclusions in electricalimpedance tomographyrdquo SIAM Journal on Mathematical Anal-ysis vol 32 no 6 pp 1327ndash1341 2001

[5] M Hanke and M Bruhl ldquoRecent progress in electricalimpedance tomographyrdquo Inverse Problems vol 19 no 6 ppS65ndashS90 2003

[6] A Kirsch andN GrinbergTheFactorizationMethod for InverseProblems vol 36 ofOxford Lecture Series inMathematics and ItsApplications Oxford University Press Oxford UK 2007

[7] M Hanke and A Kirsch ldquoSampling methodsrdquo in Handbook ofMathematical Models in Imaging O Scherzer Ed pp 501ndash550Springer 2011

[8] A Kirsch ldquoThe factorization method for a class of inverseelliptic problemsrdquo Mathematische Nachrichten vol 278 pp258ndash277 2005

[9] B Gebauer ldquoThe Factorization Method for real elliptic prob-lemsrdquo Zeitschrift fur Analysis und ihre Anwendung vol 25 no1 pp 81ndash102 2006

[10] A I Nachman L Paivarinta and A Teirila ldquoOn imagingobstacles inside inhomogeneous mediardquo Journal of FunctionalAnalysis vol 252 no 2 pp 490ndash516 2007

[11] M Hanke and B Schappel ldquoThe factorization method forelectrical impedance tomography in the half-spacerdquo SIAMJournal on Applied Mathematics vol 68 no 4 pp 907ndash9242008

[12] N Hyvonen ldquoComplete electrode model of electricalimpedance tomography approximation properties andcharacterization of inclusionsrdquo SIAM Journal on AppliedMathematics vol 64 pp 902ndash931 2004

[13] A Lechleiter N Hyvonen and H Hakula ldquoThe factorizationmethod applied to the complete electrode model of impedancetomographyrdquo SIAM Journal onAppliedMathematics vol 68 no4 pp 1097ndash1121 2008

[14] N Hyvonen H Hakula and S Pursiainen ldquoNumerical imple-mentation of the factorization method within the completeelectrode model of electrical impedance tomographyrdquo InverseProblems and Imaging vol 1 pp 299ndash317 2007

[15] HHakula andNHyvonen ldquoOn computation of test dipoles forfactorization methodrdquo BIT Numerical Mathematics vol 49 no1 pp 75ndash91 2009

[16] B Harrach and J K Seo ldquoDetecting inclusions in electri-cal impedance tomography without reference measurementsrdquoSIAM Journal on Applied Mathematics vol 69 no 6 pp 1662ndash1681 2009

[17] B Harrach J K Seo and E J Woo ldquoFactorization methodand its physical justification in frequency-difference electricalimpedance tomographyrdquo IEEE Transactions on Medical Imag-ing vol 29 no 11 pp 1918ndash1926 2010

[18] S Schmitt ldquoThe factorization method for EIT in the case ofmixed inclusionsrdquo Inverse Problems vol 25 no 6 Article ID065012 2009

8 Computational and Mathematical Methods in Medicine

[19] S Schmitt and A Kirsch ldquoA factorization scheme for determin-ing conductivity contrasts in impedance tomographyrdquo InverseProblems vol 27 Article ID 095005 2011

[20] B Gebauer and N Hyvonen ldquoFactorization method and irreg-ular inclusions in electrical impedance tomographyrdquo InverseProblems vol 23 no 5 pp 2159ndash2170 2007

[21] B Gebauer ldquoLocalized potentials in electrical impedancetomographyrdquo Inverse Problems and Imaging vol 2 pp 251ndash2692008

[22] H Kang J K Seo and D Sheen ldquoThe inverse conductivityproblem with one measurement Stability and estimation ofsizerdquo SIAM Journal on Mathematical Analysis vol 28 no 6 pp1389ndash1405 1997

[23] M Ikehata ldquoSize estimation of inclusionrdquo Journal of Inverse andIll-Posed Problems vol 6 no 2 pp 127ndash140 1998

[24] T Ide H Isozaki S Nakata S Siltanen and G UhlmannldquoProbing for electrical inclusions with complex sphericalwavesrdquo Communications on Pure and Applied Mathematics vol60 no 10 pp 1415ndash1442 2007

[25] B Harrach and J K Seo ldquoExact shape-reconstruction by one-step linearization in electrical impedance tomographyrdquo TheSIAM Journal onMathematical Analysis vol 42 no 4 pp 1505ndash1518 2010

[26] N Bourbaki Elements of Mathematics Topological VectorSpaces chapters 1ndash5 Springer Berlin Germany 2003

[27] F Fruhauf B Gebauer andO Scherzer ldquoDetecting interfaces ina parabolic-elliptic problem from surfacemeasurementsrdquo SIAMJournal on Numerical Analysis vol 45 no 2 pp 810ndash836 2007

[28] B Harrach and M Ullrich ldquoMonotony based shape recon-struction in electrical impedance tomographyrdquo submitted forpublication

[29] S Kusiak and J Sylvester ldquoThe scattering supportrdquoCommunica-tions on Pure and Applied Mathematics vol 56 no 11 pp 1525ndash1548 2003

[30] B Gebauer and N Hyvonen ldquoFactorization method andinclusions of mixed type in an inverse elliptic boundary valueproblemrdquo Inverse Problems and Imaging vol 2 pp 355ndash3722008

[31] N I Grinberg and A Kirsch ldquoThe factorization method forobstacles with a-priori separated sound-soft and sound-hardpartsrdquo Mathematics and Computers in Simulation vol 66 no4-5 pp 267ndash279 2004

[32] A Lechleiter ldquoA regularization technique for the factorizationmethodrdquo Inverse Problems vol 22 no 5 pp 1605ndash1625 2006

Submit your manuscripts athttpwwwhindawicom

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Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

4 Computational and Mathematical Methods in Medicine

fulfills V isin 1198671

(Ω) and for all 119908 isin 119867

1

(Ω)

intΩ

nablaV sdot nabla119908d119909

= intΩ119861120598(119911)

nablaΦ119911119889

sdot nabla119908d119909 + int119861120598(119911)

nabla1198911sdot nabla119908d119909

= minusint120597119861120598(119911)

120597]Φ119911119889 119908|120597119861120598(119911)d119904 + int

119861120598(119911)

nabla1198911sdot nabla119908d119909

= int119861120598(119911)

nabla (1198911minus 119891

2) sdot nabla119908d119909 = int

119863

119865 sdot nabla119908d119909

(29)

This shows thatΦ119911119889|Σ= V|

Σ= 119871

119863(119865) isin R(119871

119863)

Now let Φ119911119889|Σisin R(119871

119863) Let V isin 119867

1

(Ω) be the function

from the definition of 119871119863 Then

V|Σ= Φ

119911119889|Σ 120597]V|Σ = 0 = 120597]Φ119911119889|Σ (30)

so that it follows by unique continuation that V = Φ119911119889

in theconnected setΩ (119863 cup 119911)

If 119911 notin 119863 then 119889 sdot nabla120575119911notin 119867

minus2

(Ω 119863) and thus Φ119911119889

notin

1198712

(Ω 119863) which contradicts that V = Φ119911119889

in Ω (119863 cup 119911)Hence 119911 isin 119863

3 The Factorization Method

Nowwe will formulate the FactorizationMethod and charac-terize a region where a conductivity 120590 differs from a referenceconductivity 120590

0by a range criterion Before we turn to a

new general formulation of the method we first state it fora special case that is similar to the one that was treated in theoriginal works of Bruhl and Hanke [3 4]

31 The Factorization Method for a Simple Special Case

Theorem 5 Let 1205900= 1 and 120590 = 1 + 120594

119863 where 119863 sube Ω is an

open set so that 119863 sube Ω has a connected complement Ω 119863Then for all 119911 isin Ω 119911 notin 120597119863 and all dipole directions 119889 isin R119899119889 = 1

119911 isin 119863 iff Φ119911119889

1003816100381610038161003816Σisin R (|Λ (120590) minus Λ (1)|

12

) (31)

Proof Themonotony Lemma 1 yields that for all119892 isin 1198712

(120597Ω)

int119863

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 ge intΣ

119892 (Λ (1) minus Λ (120590)) 119892d119904 ge int119863

1

2

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(32)

Hence |Λ(120590) minusΛ(1)| = Λ(1) minusΛ(120590) and using Lemma 2 wecan restate this in the form

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

ge10038171003817100381710038171003817|Λ (120590) minus Λ (1)|

12

11989210038171003817100381710038171003817

2

ge1

2

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

(33)

Using the functional analytic Lemma 3 this implies that

R (119871119863) = R (|Λ (120590) minus Λ (1)|

12

) (34)

and thus the assertion follows from the relation between 119863

andR(119871119863) in Lemma 4

Obviously the same arguments can be used to treat thecase 120590(119909) = 1 + 120581(119909)120594

119863(119909) when there exists a conductivity

jump 120598 gt 0 so that either

120581 (119909) ge 120598 forall119909 isin 119863 or 120581 (119909) le minus120598 forall119909 isin 119863 (35)

32 The Factorization Method for the General PiecewiseAnalytic Case Now we drop the assumptions that the back-ground is constant that there is a clear conductivity jumpand that the complement of the inclusions is connected Wewill merely assume that the reference conductivity 120590

0is a

piecewise analytic function and that either 120590 minus 1205900ge 0 or

120590 minus 1205900le 0 Roughly speaking under this general assumption

the Factorization Method then characterizes the support of120590 minus 120590

0up to holes in the support that have no connections to

Σ For a precise formulation we use the concept of the innerand outer support from [28] that has been inspired by the useof the infinity support of Kusiak and Sylvester [29] see also[25 30]

Definition 6 A relatively open set 119880 sube Ω is called connectedto Σ if 119880 cap Ω is connected and 119880 cap Σ = 0

For a measurable function 120581 Ω rarr R we define

(a) the support supp(120581) as the complement (in Ω) of theunion of those relatively open119880 sube Ω for which 120581|

119880equiv

0(b) the inner support inn supp 120581 as the union of those

open sets 119880 sube Ω for which ess inf119909isin119880

|120581(119909)| gt 0(c) the outer support out

Σsupp 120581 as the complement (in

Ω) of the union of those relatively open 119880 sube Ω thatare connected to Σ and for which 120581|

119880equiv 0

The interior of a set119872 sube Ω is denoted by int119872 and its closure(with respect toR119899) by119872 If119872 is measurable we also define

(d) outΣ119872 = out

Σsupp120594

119872

It is easily checked that outΣ(supp 120581) = out

Σsupp 120581

With this concept we can extend the range characteriza-tion in Lemma 4 to a general setting (see also Remark 9 later)

Lemma 7 Let 1205900isin 119871

infin

+(Ω) be piecewise analytic Let 119863 sube Ω

be measurableThen for all unit vectors 119889 isin R119899 119889 = 1 and every point

119911 isin Ω that has a neighborhood in which 1205900is analytic

119911 isin int119863 implies Φ119911119889

1003816100381610038161003816Σisin R (119871

119863) (36)

and

Φ119911119889

1003816100381610038161003816Σisin R (119871

119863) implies 119911 isin out

Σ119863 (37)

Proof If 119911 isin int119863 then there exists a small ball 119861120598(119911) sube 119863

and the first assertion follows as in the proof of Lemma 4To show the second assertion letΦ

119911119889|Σisin R(119871

119863) and let

V isin 1198671

(Ω) be the function from the definition of 119871

119863 so that

(as in the proof of Lemma 4)

V|Σ= Φ

119911119889|Σ 120597]V|Σ = 0 = 120597]Φ119911119889|Σ (38)

Computational and Mathematical Methods in Medicine 5

Assume that 119911 notin outΣ119863 Then there exists a relatively

open 119880 sube Ω that is connected to Σ and contains 119911Hence by unique continuation it follows that V|

119880= Φ

119911119889|119880

and we obtain the same contradiction as in the proof ofLemma 4

Now we can formulate and prove the FactorizationMethod for general piecewise analytic conductivities

Theorem 8 Let 120590 isin 119871infin

+(Ω) and let 120590

0isin 119871

infin

+(Ω) be a

piecewise analytic function Let either

120590 (119909) ge 1205900(119909) forall119909 isin Ω or 120590 (119909) le 120590

0(119909) forall119909 isin Ω

(39)

Then for all 119911 isin Ω that have a neighborhood in which 1205900is

analytic as well as all unit vectors 119889 isin R119899 119889 = 1

119911isin innsupp (120590 minus 1205900) implies Φ

119911119889

1003816100381610038161003816Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

)

(40)

and

Φ119911119889

1003816100381610038161003816ΣisinR (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) implies 119911isin outΣsupp (120590 minus 120590

0)

(41)

Proof Let 119911 isin Ω have a neighborhood in which 1205900is analytic

and let 119889 isin R119899 be a unit vector with 119889 = 1 We onlyprove the assertions for 120590 ge 120590

0 The other case is completely

analogousFirst let 119911 isin inn supp (120590 minus 120590

0) Then there exists a small

ball 119861120598(119911) and 120575 gt 0 so that 120590 minus 120590

0ge 120575 on 119861

120598(119911) Using the

monotony Lemma 1 it follows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (1205900) minus Λ (120590)) 119892d119904

ge intΩ

1205900

120590(120590 minus 120590

0)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

ge 12057510038171003817100381710038171205900

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817100381710038171003817

1

120590

1003817100381710038171003817100381710038171003817119871infin(Ω)int119861120598(119911)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

ge 12057510038171003817100381710038171205900

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817100381710038171003817

1

120590

1003817100381710038171003817100381710038171003817119871infin(Ω)

10038171003817100381710038171003817119871lowast

119861120598(119911)11989210038171003817100381710038171003817

2

(42)

Using the functional analytic Lemma 3 we obtain that

R (119871119861120598(119911)

) sube R (1003816100381610038161003816Λ (120590) minus Λ (120590

0)1003816100381610038161003816

12

) (43)

and Lemma 7 yields that

Φ119911119889|Σisin R (119871

119861120598(119911)) sube R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) (44)

On the other hand with 119863 = supp(120590 minus 1205900) the monotony

Lemma 1 shows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (1205900) minus Λ (120590

1)) 119892d119904 le int

Ω

(1205901minus 120590

0)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

le10038171003817100381710038171205901 minus 120590

0

1003817100381710038171003817119871infin(Ω)int119863

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le10038171003817100381710038171205901 minus 120590

0

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

(45)

so that we obtain from the functional analytic Lemma 3

R (1003816100381610038161003816Λ (120590) minus Λ (120590

0)1003816100381610038161003816

12

) sube R (119871119863) (46)

Hence Lemma 7 yields that

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) sube R (119871119863) implies

119911 isin outΣ(supp (120590 minus 120590

0)) = out

Σsupp (120590 minus 120590

0)

(47)

Remark 9 Theorem 8 shows that the Factorization Methodis able to detect the support of a conductivity difference up tothe difference between the outer and the inner support thatis roughly speaking up to holes in the support that have noconnections to the boundary It leaves openwhether points insuch holes will fulfill the range criterion of the FactorizationMethod or not

A result of Hyvonen and the author [30 Lemma 25]shows that for every smooth domain119863with119863 sub Ω and everyunit vector 119889 isin R119899 119889 = 1

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) forall119911 isin 120597119863 (48)

implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) forall119911 isin 119863 (49)

In that sense we can expect that holes in the support willbe filled up and that the set detected by the FactorizationMethod is essentially the outer support of the conductivitydifference

33 The Factorization Method for the Indefinite Case It is along standing open theoretical problem whether the rangecriterion of the Factorization Method holds true withoutthe definiteness assumption that 120590 ge 120590

0on Ω or 120590 le

1205900on Ω However Grinberg Kirsch and Schmitt [18 31]

showed how to exclude a region 119864 sube Ω from Ω insuch a way that the Factorization Method only requires thedefiniteness assumption onΩ119864 In this subsection we showhow their idea can be incorporated into our formulation ofthe method

To point out the main idea we first formulate the resultfor a simple special case Let us stress that for 120590 = 1 + 120594

119863+ minus

(12)120594119863minus it is not known whether

119911 isin 119863+

cup 119863minus iff Φ

119911119889|Σisin R (|Λ (120590) minus Λ (1)|

12

) (50)

However we can still use the FactorizationMethod if we havesome a priori knowledge that separates 119863+ and 119863

minus Moreprecisely if we know a subset 119864 that contains 119863minus withoutintersecting 119863

+ then we can use the Factorization Methodto find119863+ cup 119864 (and thus119863+)

6 Computational and Mathematical Methods in Medicine

Theorem 10 Let 1205900= 1 and 120590 = 1 + 120594

119863+ minus (12)120594

119863minus where

119863+

119863minus

sube Ω are open Let 119864 sube Ω be an open set

(a) If 119863+ sube 119864 and 119863minus cup 119864 sube Ω has a connectedcomplement then for all 119911 isin Ω 119911 notin 120597(119863

minus

cup119864) and alldipole directions 119889 isin R119899 119889 = 1

119911 isin 119863minus

cup 119864 iff Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1) + 119871119864119871lowast

119864

1003816100381610038161003816

12

)

(51)

(b) If 119863minus sube 119864 and 119863+ cup 119864 sube Ω has a connectedcomplement then for all 119911 isin Ω 119911 notin 120597(119863

+

cup119864) and alldipole directions 119889 isin R119899 119889 = 1

119911 isin 119863+

cup 119864 iff Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1) minus 119871119864119871lowast

119864

1003816100381610038161003816

12

)

(52)

Proof Themonotony Lemma 1 yields that for all119892 isin 1198712

(120597Ω)

1

2int119863minus

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 minus int119863+

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le intΣ

119892 (Λ (120590) minus Λ (1)) 119892d119904

le int119863minus

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 minus1

2int119863+

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(53)

Since (cf Lemma 2)

intΣ

119892 (119871119864119871lowast

119864) 119892d119904 = 1003817100381710038171003817119871

lowast

1198641198921003817100381710038171003817

2

= int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 (54)

it follows for case (a) that

1

2int119863minuscup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 le intΣ

119892 (Λ (120590) minus Λ (1) + 2119871119864119871lowast

119864) 119892d119904

le int119863minuscup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909(55)

Using the functional analytic Lemma 3 this implies that

R (119871119863minuscup119864) = R (

1003816100381610038161003816Λ (120590) minus Λ (1) + 2119871119864119871lowast

119864

1003816100381610038161003816

12

) (56)

so that the assertion (a) follows from Lemma 4In case (b) we obtain that

3

2int119863+cup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 ge intΣ

119892 (Λ (1) minus Λ (120590) + 2119871119864119871lowast

119864) 119892d119904

ge1

2int119863+cup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909(57)

and the same arguments as above yield the assertion

We can also extend these ideas to the general setting ofSection 32

Theorem 11 Let 120590 isin 119871infin

+(Ω) and let 120590

0isin 119871

infin

+(Ω) be a

piecewise analytic function Let 119864 sube Ω be a measurable setChoose 120572 120573 isin R such that

120572 gt1003817100381710038171003817120590 minus 120590

0

1003817100381710038171003817119871infin(Ω) 120573 gt

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω) (58)

(a) If 120590 le 1205900on Ω 119864 then for all 119911 isin Ω that have a

neighborhood in which 1205900is analytic as well as all unit

vectors 119889 isin R119899 119889 = 1

119911 isin inn supp (120590 minus 1205900) cup 119864 implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590)minusΛ (1205900)+120572119871

119864119871lowast

119864

1003816100381610038161003816

12

)

(59)

and

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) implies

119911 isin outΣ(supp (120590 minus 120590

0) cup 119864)

(60)

(b) If 120590 ge 1205900on Ω 119864 then for all 119911 isin Ω that have a

neighborhood in which 1205900is analytic as well as all unit

vectors 119889 isin R119899 119889 = 1

119911 isin inn supp (120590 minus 1205900) cup 119864 implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) minus 120573119871

119864119871lowast

119864

1003816100381610038161003816

12

)

(61)

and

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) minus 120573119871

119864119871lowast

119864

1003816100381610038161003816

12

) implies

119911 isin outΣ(supp (120590 minus 120590

0) cup 119864)

(62)

Proof For every 119911 isin inn supp(120590minus1205900)with 119911 notin 119864 there exists

a small ball 119861120598(119911) and 120575 gt 0 so that 120590

0minus120590 ge 120575 on 119861

120598(119911) Using

the monotony Lemma 1 it follows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

ge intΩ

(1205900minus 120590

1)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

ge 120575int119861120598(119911)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909+(120572 minus1003817100381710038171003817120590 minus 120590

0

1003817100381710038171003817119871infin(Ω))int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(63)

Computational and Mathematical Methods in Medicine 7

As in the previous proofs we obtain from Lemmas 2 and 3that

R (1003816100381610038161003816Λ (120590) minus Λ (120590

0) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) supe R (119871119861120598(119911)cup119864

) (64)

so that the first implication of (a) follows from Lemma 7The monotony Lemma 1 also implies that

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

le intΩ

1205900

120590(1205900minus 120590)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω)intsupp(120590minus1205900)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

+ 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(65)

so that the second implication of (a) follows from Lemmas 23 and 7 Assertion (b) can be proven analogously

Remark 9 also applies to this case

4 Conclusions and Remarks

The Factorization Method can be used to detect regionsin which a conductivity differs from a known referenceconductivity In this work we summarized the progresson the methodrsquos theoretical foundation We formulated themethod for general piecewise analytic conductivities andgave comparatively simple and self-contained proofsWe alsoshowed how the idea of excluding a part of the imaging regioncan be incorporated into this formulation

The regularity assumptions can beweakened even furtherOur proofs only require unique continuation arguments forthe reference conductivity 120590

0and the existence of the dipole

functionsTwo major open theoretical questions still exist in the

context of the FactorizationMethodThe theoretical justifica-tion of the method requires a definiteness condition (on thewhole domain or after excluding an a priori known part of thedomain) It is unknownwhether themethodrsquos range criterionholds without such a definiteness conditionThe second openquestion concerns the numerical stability of the methodrsquosrange criterion So far there are no rigorous convergenceresults for numerical implementations of this range criterion(see however Lechleiter [32] for a first step in this direction)As a promising approach to overcome both problems wewould like to point out the recent work on monotony-basedmethods [28]

Acknowledgments

This paper was funded by the German Research Foundation(DFG) and the University of Wurzburg in the fundingprogramme Open Access Publishing

References

[1] A Kirsch ldquoCharacterization of the shape of a scattering obstacleusing the spectral data of the far field operatorrdquo Inverse Prob-lems vol 14 no 6 pp 1489ndash1512 1998

[2] M Bruhl Gebietserkennung in der elektrischen Impedanztomo-graphie [PhD thesis] Universitat Karlsruhe 1999

[3] M Bruhl and M Hanke ldquoNumerical implementation of twononiterative methods for locating inclusions by impedancetomographyrdquo Inverse Problems vol 16 no 4 pp 1029ndash10422000

[4] M Bruhl ldquoExplicit characterization of inclusions in electricalimpedance tomographyrdquo SIAM Journal on Mathematical Anal-ysis vol 32 no 6 pp 1327ndash1341 2001

[5] M Hanke and M Bruhl ldquoRecent progress in electricalimpedance tomographyrdquo Inverse Problems vol 19 no 6 ppS65ndashS90 2003

[6] A Kirsch andN GrinbergTheFactorizationMethod for InverseProblems vol 36 ofOxford Lecture Series inMathematics and ItsApplications Oxford University Press Oxford UK 2007

[7] M Hanke and A Kirsch ldquoSampling methodsrdquo in Handbook ofMathematical Models in Imaging O Scherzer Ed pp 501ndash550Springer 2011

[8] A Kirsch ldquoThe factorization method for a class of inverseelliptic problemsrdquo Mathematische Nachrichten vol 278 pp258ndash277 2005

[9] B Gebauer ldquoThe Factorization Method for real elliptic prob-lemsrdquo Zeitschrift fur Analysis und ihre Anwendung vol 25 no1 pp 81ndash102 2006

[10] A I Nachman L Paivarinta and A Teirila ldquoOn imagingobstacles inside inhomogeneous mediardquo Journal of FunctionalAnalysis vol 252 no 2 pp 490ndash516 2007

[11] M Hanke and B Schappel ldquoThe factorization method forelectrical impedance tomography in the half-spacerdquo SIAMJournal on Applied Mathematics vol 68 no 4 pp 907ndash9242008

[12] N Hyvonen ldquoComplete electrode model of electricalimpedance tomography approximation properties andcharacterization of inclusionsrdquo SIAM Journal on AppliedMathematics vol 64 pp 902ndash931 2004

[13] A Lechleiter N Hyvonen and H Hakula ldquoThe factorizationmethod applied to the complete electrode model of impedancetomographyrdquo SIAM Journal onAppliedMathematics vol 68 no4 pp 1097ndash1121 2008

[14] N Hyvonen H Hakula and S Pursiainen ldquoNumerical imple-mentation of the factorization method within the completeelectrode model of electrical impedance tomographyrdquo InverseProblems and Imaging vol 1 pp 299ndash317 2007

[15] HHakula andNHyvonen ldquoOn computation of test dipoles forfactorization methodrdquo BIT Numerical Mathematics vol 49 no1 pp 75ndash91 2009

[16] B Harrach and J K Seo ldquoDetecting inclusions in electri-cal impedance tomography without reference measurementsrdquoSIAM Journal on Applied Mathematics vol 69 no 6 pp 1662ndash1681 2009

[17] B Harrach J K Seo and E J Woo ldquoFactorization methodand its physical justification in frequency-difference electricalimpedance tomographyrdquo IEEE Transactions on Medical Imag-ing vol 29 no 11 pp 1918ndash1926 2010

[18] S Schmitt ldquoThe factorization method for EIT in the case ofmixed inclusionsrdquo Inverse Problems vol 25 no 6 Article ID065012 2009

8 Computational and Mathematical Methods in Medicine

[19] S Schmitt and A Kirsch ldquoA factorization scheme for determin-ing conductivity contrasts in impedance tomographyrdquo InverseProblems vol 27 Article ID 095005 2011

[20] B Gebauer and N Hyvonen ldquoFactorization method and irreg-ular inclusions in electrical impedance tomographyrdquo InverseProblems vol 23 no 5 pp 2159ndash2170 2007

[21] B Gebauer ldquoLocalized potentials in electrical impedancetomographyrdquo Inverse Problems and Imaging vol 2 pp 251ndash2692008

[22] H Kang J K Seo and D Sheen ldquoThe inverse conductivityproblem with one measurement Stability and estimation ofsizerdquo SIAM Journal on Mathematical Analysis vol 28 no 6 pp1389ndash1405 1997

[23] M Ikehata ldquoSize estimation of inclusionrdquo Journal of Inverse andIll-Posed Problems vol 6 no 2 pp 127ndash140 1998

[24] T Ide H Isozaki S Nakata S Siltanen and G UhlmannldquoProbing for electrical inclusions with complex sphericalwavesrdquo Communications on Pure and Applied Mathematics vol60 no 10 pp 1415ndash1442 2007

[25] B Harrach and J K Seo ldquoExact shape-reconstruction by one-step linearization in electrical impedance tomographyrdquo TheSIAM Journal onMathematical Analysis vol 42 no 4 pp 1505ndash1518 2010

[26] N Bourbaki Elements of Mathematics Topological VectorSpaces chapters 1ndash5 Springer Berlin Germany 2003

[27] F Fruhauf B Gebauer andO Scherzer ldquoDetecting interfaces ina parabolic-elliptic problem from surfacemeasurementsrdquo SIAMJournal on Numerical Analysis vol 45 no 2 pp 810ndash836 2007

[28] B Harrach and M Ullrich ldquoMonotony based shape recon-struction in electrical impedance tomographyrdquo submitted forpublication

[29] S Kusiak and J Sylvester ldquoThe scattering supportrdquoCommunica-tions on Pure and Applied Mathematics vol 56 no 11 pp 1525ndash1548 2003

[30] B Gebauer and N Hyvonen ldquoFactorization method andinclusions of mixed type in an inverse elliptic boundary valueproblemrdquo Inverse Problems and Imaging vol 2 pp 355ndash3722008

[31] N I Grinberg and A Kirsch ldquoThe factorization method forobstacles with a-priori separated sound-soft and sound-hardpartsrdquo Mathematics and Computers in Simulation vol 66 no4-5 pp 267ndash279 2004

[32] A Lechleiter ldquoA regularization technique for the factorizationmethodrdquo Inverse Problems vol 22 no 5 pp 1605ndash1625 2006

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

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Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

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Diabetes ResearchJournal of

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Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 5

Assume that 119911 notin outΣ119863 Then there exists a relatively

open 119880 sube Ω that is connected to Σ and contains 119911Hence by unique continuation it follows that V|

119880= Φ

119911119889|119880

and we obtain the same contradiction as in the proof ofLemma 4

Now we can formulate and prove the FactorizationMethod for general piecewise analytic conductivities

Theorem 8 Let 120590 isin 119871infin

+(Ω) and let 120590

0isin 119871

infin

+(Ω) be a

piecewise analytic function Let either

120590 (119909) ge 1205900(119909) forall119909 isin Ω or 120590 (119909) le 120590

0(119909) forall119909 isin Ω

(39)

Then for all 119911 isin Ω that have a neighborhood in which 1205900is

analytic as well as all unit vectors 119889 isin R119899 119889 = 1

119911isin innsupp (120590 minus 1205900) implies Φ

119911119889

1003816100381610038161003816Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

)

(40)

and

Φ119911119889

1003816100381610038161003816ΣisinR (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) implies 119911isin outΣsupp (120590 minus 120590

0)

(41)

Proof Let 119911 isin Ω have a neighborhood in which 1205900is analytic

and let 119889 isin R119899 be a unit vector with 119889 = 1 We onlyprove the assertions for 120590 ge 120590

0 The other case is completely

analogousFirst let 119911 isin inn supp (120590 minus 120590

0) Then there exists a small

ball 119861120598(119911) and 120575 gt 0 so that 120590 minus 120590

0ge 120575 on 119861

120598(119911) Using the

monotony Lemma 1 it follows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (1205900) minus Λ (120590)) 119892d119904

ge intΩ

1205900

120590(120590 minus 120590

0)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

ge 12057510038171003817100381710038171205900

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817100381710038171003817

1

120590

1003817100381710038171003817100381710038171003817119871infin(Ω)int119861120598(119911)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

ge 12057510038171003817100381710038171205900

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817100381710038171003817

1

120590

1003817100381710038171003817100381710038171003817119871infin(Ω)

10038171003817100381710038171003817119871lowast

119861120598(119911)11989210038171003817100381710038171003817

2

(42)

Using the functional analytic Lemma 3 we obtain that

R (119871119861120598(119911)

) sube R (1003816100381610038161003816Λ (120590) minus Λ (120590

0)1003816100381610038161003816

12

) (43)

and Lemma 7 yields that

Φ119911119889|Σisin R (119871

119861120598(119911)) sube R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) (44)

On the other hand with 119863 = supp(120590 minus 1205900) the monotony

Lemma 1 shows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (1205900) minus Λ (120590

1)) 119892d119904 le int

Ω

(1205901minus 120590

0)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909

le10038171003817100381710038171205901 minus 120590

0

1003817100381710038171003817119871infin(Ω)int119863

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le10038171003817100381710038171205901 minus 120590

0

1003817100381710038171003817119871infin(Ω)

1003817100381710038171003817119871lowast

1198631198921003817100381710038171003817

2

(45)

so that we obtain from the functional analytic Lemma 3

R (1003816100381610038161003816Λ (120590) minus Λ (120590

0)1003816100381610038161003816

12

) sube R (119871119863) (46)

Hence Lemma 7 yields that

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) sube R (119871119863) implies

119911 isin outΣ(supp (120590 minus 120590

0)) = out

Σsupp (120590 minus 120590

0)

(47)

Remark 9 Theorem 8 shows that the Factorization Methodis able to detect the support of a conductivity difference up tothe difference between the outer and the inner support thatis roughly speaking up to holes in the support that have noconnections to the boundary It leaves openwhether points insuch holes will fulfill the range criterion of the FactorizationMethod or not

A result of Hyvonen and the author [30 Lemma 25]shows that for every smooth domain119863with119863 sub Ω and everyunit vector 119889 isin R119899 119889 = 1

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) forall119911 isin 120597119863 (48)

implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900)1003816100381610038161003816

12

) forall119911 isin 119863 (49)

In that sense we can expect that holes in the support willbe filled up and that the set detected by the FactorizationMethod is essentially the outer support of the conductivitydifference

33 The Factorization Method for the Indefinite Case It is along standing open theoretical problem whether the rangecriterion of the Factorization Method holds true withoutthe definiteness assumption that 120590 ge 120590

0on Ω or 120590 le

1205900on Ω However Grinberg Kirsch and Schmitt [18 31]

showed how to exclude a region 119864 sube Ω from Ω insuch a way that the Factorization Method only requires thedefiniteness assumption onΩ119864 In this subsection we showhow their idea can be incorporated into our formulation ofthe method

To point out the main idea we first formulate the resultfor a simple special case Let us stress that for 120590 = 1 + 120594

119863+ minus

(12)120594119863minus it is not known whether

119911 isin 119863+

cup 119863minus iff Φ

119911119889|Σisin R (|Λ (120590) minus Λ (1)|

12

) (50)

However we can still use the FactorizationMethod if we havesome a priori knowledge that separates 119863+ and 119863

minus Moreprecisely if we know a subset 119864 that contains 119863minus withoutintersecting 119863

+ then we can use the Factorization Methodto find119863+ cup 119864 (and thus119863+)

6 Computational and Mathematical Methods in Medicine

Theorem 10 Let 1205900= 1 and 120590 = 1 + 120594

119863+ minus (12)120594

119863minus where

119863+

119863minus

sube Ω are open Let 119864 sube Ω be an open set

(a) If 119863+ sube 119864 and 119863minus cup 119864 sube Ω has a connectedcomplement then for all 119911 isin Ω 119911 notin 120597(119863

minus

cup119864) and alldipole directions 119889 isin R119899 119889 = 1

119911 isin 119863minus

cup 119864 iff Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1) + 119871119864119871lowast

119864

1003816100381610038161003816

12

)

(51)

(b) If 119863minus sube 119864 and 119863+ cup 119864 sube Ω has a connectedcomplement then for all 119911 isin Ω 119911 notin 120597(119863

+

cup119864) and alldipole directions 119889 isin R119899 119889 = 1

119911 isin 119863+

cup 119864 iff Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1) minus 119871119864119871lowast

119864

1003816100381610038161003816

12

)

(52)

Proof Themonotony Lemma 1 yields that for all119892 isin 1198712

(120597Ω)

1

2int119863minus

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 minus int119863+

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le intΣ

119892 (Λ (120590) minus Λ (1)) 119892d119904

le int119863minus

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 minus1

2int119863+

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(53)

Since (cf Lemma 2)

intΣ

119892 (119871119864119871lowast

119864) 119892d119904 = 1003817100381710038171003817119871

lowast

1198641198921003817100381710038171003817

2

= int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 (54)

it follows for case (a) that

1

2int119863minuscup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 le intΣ

119892 (Λ (120590) minus Λ (1) + 2119871119864119871lowast

119864) 119892d119904

le int119863minuscup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909(55)

Using the functional analytic Lemma 3 this implies that

R (119871119863minuscup119864) = R (

1003816100381610038161003816Λ (120590) minus Λ (1) + 2119871119864119871lowast

119864

1003816100381610038161003816

12

) (56)

so that the assertion (a) follows from Lemma 4In case (b) we obtain that

3

2int119863+cup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 ge intΣ

119892 (Λ (1) minus Λ (120590) + 2119871119864119871lowast

119864) 119892d119904

ge1

2int119863+cup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909(57)

and the same arguments as above yield the assertion

We can also extend these ideas to the general setting ofSection 32

Theorem 11 Let 120590 isin 119871infin

+(Ω) and let 120590

0isin 119871

infin

+(Ω) be a

piecewise analytic function Let 119864 sube Ω be a measurable setChoose 120572 120573 isin R such that

120572 gt1003817100381710038171003817120590 minus 120590

0

1003817100381710038171003817119871infin(Ω) 120573 gt

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω) (58)

(a) If 120590 le 1205900on Ω 119864 then for all 119911 isin Ω that have a

neighborhood in which 1205900is analytic as well as all unit

vectors 119889 isin R119899 119889 = 1

119911 isin inn supp (120590 minus 1205900) cup 119864 implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590)minusΛ (1205900)+120572119871

119864119871lowast

119864

1003816100381610038161003816

12

)

(59)

and

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) implies

119911 isin outΣ(supp (120590 minus 120590

0) cup 119864)

(60)

(b) If 120590 ge 1205900on Ω 119864 then for all 119911 isin Ω that have a

neighborhood in which 1205900is analytic as well as all unit

vectors 119889 isin R119899 119889 = 1

119911 isin inn supp (120590 minus 1205900) cup 119864 implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) minus 120573119871

119864119871lowast

119864

1003816100381610038161003816

12

)

(61)

and

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) minus 120573119871

119864119871lowast

119864

1003816100381610038161003816

12

) implies

119911 isin outΣ(supp (120590 minus 120590

0) cup 119864)

(62)

Proof For every 119911 isin inn supp(120590minus1205900)with 119911 notin 119864 there exists

a small ball 119861120598(119911) and 120575 gt 0 so that 120590

0minus120590 ge 120575 on 119861

120598(119911) Using

the monotony Lemma 1 it follows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

ge intΩ

(1205900minus 120590

1)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

ge 120575int119861120598(119911)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909+(120572 minus1003817100381710038171003817120590 minus 120590

0

1003817100381710038171003817119871infin(Ω))int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(63)

Computational and Mathematical Methods in Medicine 7

As in the previous proofs we obtain from Lemmas 2 and 3that

R (1003816100381610038161003816Λ (120590) minus Λ (120590

0) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) supe R (119871119861120598(119911)cup119864

) (64)

so that the first implication of (a) follows from Lemma 7The monotony Lemma 1 also implies that

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

le intΩ

1205900

120590(1205900minus 120590)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω)intsupp(120590minus1205900)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

+ 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(65)

so that the second implication of (a) follows from Lemmas 23 and 7 Assertion (b) can be proven analogously

Remark 9 also applies to this case

4 Conclusions and Remarks

The Factorization Method can be used to detect regionsin which a conductivity differs from a known referenceconductivity In this work we summarized the progresson the methodrsquos theoretical foundation We formulated themethod for general piecewise analytic conductivities andgave comparatively simple and self-contained proofsWe alsoshowed how the idea of excluding a part of the imaging regioncan be incorporated into this formulation

The regularity assumptions can beweakened even furtherOur proofs only require unique continuation arguments forthe reference conductivity 120590

0and the existence of the dipole

functionsTwo major open theoretical questions still exist in the

context of the FactorizationMethodThe theoretical justifica-tion of the method requires a definiteness condition (on thewhole domain or after excluding an a priori known part of thedomain) It is unknownwhether themethodrsquos range criterionholds without such a definiteness conditionThe second openquestion concerns the numerical stability of the methodrsquosrange criterion So far there are no rigorous convergenceresults for numerical implementations of this range criterion(see however Lechleiter [32] for a first step in this direction)As a promising approach to overcome both problems wewould like to point out the recent work on monotony-basedmethods [28]

Acknowledgments

This paper was funded by the German Research Foundation(DFG) and the University of Wurzburg in the fundingprogramme Open Access Publishing

References

[1] A Kirsch ldquoCharacterization of the shape of a scattering obstacleusing the spectral data of the far field operatorrdquo Inverse Prob-lems vol 14 no 6 pp 1489ndash1512 1998

[2] M Bruhl Gebietserkennung in der elektrischen Impedanztomo-graphie [PhD thesis] Universitat Karlsruhe 1999

[3] M Bruhl and M Hanke ldquoNumerical implementation of twononiterative methods for locating inclusions by impedancetomographyrdquo Inverse Problems vol 16 no 4 pp 1029ndash10422000

[4] M Bruhl ldquoExplicit characterization of inclusions in electricalimpedance tomographyrdquo SIAM Journal on Mathematical Anal-ysis vol 32 no 6 pp 1327ndash1341 2001

[5] M Hanke and M Bruhl ldquoRecent progress in electricalimpedance tomographyrdquo Inverse Problems vol 19 no 6 ppS65ndashS90 2003

[6] A Kirsch andN GrinbergTheFactorizationMethod for InverseProblems vol 36 ofOxford Lecture Series inMathematics and ItsApplications Oxford University Press Oxford UK 2007

[7] M Hanke and A Kirsch ldquoSampling methodsrdquo in Handbook ofMathematical Models in Imaging O Scherzer Ed pp 501ndash550Springer 2011

[8] A Kirsch ldquoThe factorization method for a class of inverseelliptic problemsrdquo Mathematische Nachrichten vol 278 pp258ndash277 2005

[9] B Gebauer ldquoThe Factorization Method for real elliptic prob-lemsrdquo Zeitschrift fur Analysis und ihre Anwendung vol 25 no1 pp 81ndash102 2006

[10] A I Nachman L Paivarinta and A Teirila ldquoOn imagingobstacles inside inhomogeneous mediardquo Journal of FunctionalAnalysis vol 252 no 2 pp 490ndash516 2007

[11] M Hanke and B Schappel ldquoThe factorization method forelectrical impedance tomography in the half-spacerdquo SIAMJournal on Applied Mathematics vol 68 no 4 pp 907ndash9242008

[12] N Hyvonen ldquoComplete electrode model of electricalimpedance tomography approximation properties andcharacterization of inclusionsrdquo SIAM Journal on AppliedMathematics vol 64 pp 902ndash931 2004

[13] A Lechleiter N Hyvonen and H Hakula ldquoThe factorizationmethod applied to the complete electrode model of impedancetomographyrdquo SIAM Journal onAppliedMathematics vol 68 no4 pp 1097ndash1121 2008

[14] N Hyvonen H Hakula and S Pursiainen ldquoNumerical imple-mentation of the factorization method within the completeelectrode model of electrical impedance tomographyrdquo InverseProblems and Imaging vol 1 pp 299ndash317 2007

[15] HHakula andNHyvonen ldquoOn computation of test dipoles forfactorization methodrdquo BIT Numerical Mathematics vol 49 no1 pp 75ndash91 2009

[16] B Harrach and J K Seo ldquoDetecting inclusions in electri-cal impedance tomography without reference measurementsrdquoSIAM Journal on Applied Mathematics vol 69 no 6 pp 1662ndash1681 2009

[17] B Harrach J K Seo and E J Woo ldquoFactorization methodand its physical justification in frequency-difference electricalimpedance tomographyrdquo IEEE Transactions on Medical Imag-ing vol 29 no 11 pp 1918ndash1926 2010

[18] S Schmitt ldquoThe factorization method for EIT in the case ofmixed inclusionsrdquo Inverse Problems vol 25 no 6 Article ID065012 2009

8 Computational and Mathematical Methods in Medicine

[19] S Schmitt and A Kirsch ldquoA factorization scheme for determin-ing conductivity contrasts in impedance tomographyrdquo InverseProblems vol 27 Article ID 095005 2011

[20] B Gebauer and N Hyvonen ldquoFactorization method and irreg-ular inclusions in electrical impedance tomographyrdquo InverseProblems vol 23 no 5 pp 2159ndash2170 2007

[21] B Gebauer ldquoLocalized potentials in electrical impedancetomographyrdquo Inverse Problems and Imaging vol 2 pp 251ndash2692008

[22] H Kang J K Seo and D Sheen ldquoThe inverse conductivityproblem with one measurement Stability and estimation ofsizerdquo SIAM Journal on Mathematical Analysis vol 28 no 6 pp1389ndash1405 1997

[23] M Ikehata ldquoSize estimation of inclusionrdquo Journal of Inverse andIll-Posed Problems vol 6 no 2 pp 127ndash140 1998

[24] T Ide H Isozaki S Nakata S Siltanen and G UhlmannldquoProbing for electrical inclusions with complex sphericalwavesrdquo Communications on Pure and Applied Mathematics vol60 no 10 pp 1415ndash1442 2007

[25] B Harrach and J K Seo ldquoExact shape-reconstruction by one-step linearization in electrical impedance tomographyrdquo TheSIAM Journal onMathematical Analysis vol 42 no 4 pp 1505ndash1518 2010

[26] N Bourbaki Elements of Mathematics Topological VectorSpaces chapters 1ndash5 Springer Berlin Germany 2003

[27] F Fruhauf B Gebauer andO Scherzer ldquoDetecting interfaces ina parabolic-elliptic problem from surfacemeasurementsrdquo SIAMJournal on Numerical Analysis vol 45 no 2 pp 810ndash836 2007

[28] B Harrach and M Ullrich ldquoMonotony based shape recon-struction in electrical impedance tomographyrdquo submitted forpublication

[29] S Kusiak and J Sylvester ldquoThe scattering supportrdquoCommunica-tions on Pure and Applied Mathematics vol 56 no 11 pp 1525ndash1548 2003

[30] B Gebauer and N Hyvonen ldquoFactorization method andinclusions of mixed type in an inverse elliptic boundary valueproblemrdquo Inverse Problems and Imaging vol 2 pp 355ndash3722008

[31] N I Grinberg and A Kirsch ldquoThe factorization method forobstacles with a-priori separated sound-soft and sound-hardpartsrdquo Mathematics and Computers in Simulation vol 66 no4-5 pp 267ndash279 2004

[32] A Lechleiter ldquoA regularization technique for the factorizationmethodrdquo Inverse Problems vol 22 no 5 pp 1605ndash1625 2006

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

6 Computational and Mathematical Methods in Medicine

Theorem 10 Let 1205900= 1 and 120590 = 1 + 120594

119863+ minus (12)120594

119863minus where

119863+

119863minus

sube Ω are open Let 119864 sube Ω be an open set

(a) If 119863+ sube 119864 and 119863minus cup 119864 sube Ω has a connectedcomplement then for all 119911 isin Ω 119911 notin 120597(119863

minus

cup119864) and alldipole directions 119889 isin R119899 119889 = 1

119911 isin 119863minus

cup 119864 iff Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1) + 119871119864119871lowast

119864

1003816100381610038161003816

12

)

(51)

(b) If 119863minus sube 119864 and 119863+ cup 119864 sube Ω has a connectedcomplement then for all 119911 isin Ω 119911 notin 120597(119863

+

cup119864) and alldipole directions 119889 isin R119899 119889 = 1

119911 isin 119863+

cup 119864 iff Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1) minus 119871119864119871lowast

119864

1003816100381610038161003816

12

)

(52)

Proof Themonotony Lemma 1 yields that for all119892 isin 1198712

(120597Ω)

1

2int119863minus

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 minus int119863+

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le intΣ

119892 (Λ (120590) minus Λ (1)) 119892d119904

le int119863minus

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 minus1

2int119863+

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(53)

Since (cf Lemma 2)

intΣ

119892 (119871119864119871lowast

119864) 119892d119904 = 1003817100381710038171003817119871

lowast

1198641198921003817100381710038171003817

2

= int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 (54)

it follows for case (a) that

1

2int119863minuscup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 le intΣ

119892 (Λ (120590) minus Λ (1) + 2119871119864119871lowast

119864) 119892d119904

le int119863minuscup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909(55)

Using the functional analytic Lemma 3 this implies that

R (119871119863minuscup119864) = R (

1003816100381610038161003816Λ (120590) minus Λ (1) + 2119871119864119871lowast

119864

1003816100381610038161003816

12

) (56)

so that the assertion (a) follows from Lemma 4In case (b) we obtain that

3

2int119863+cup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 ge intΣ

119892 (Λ (1) minus Λ (120590) + 2119871119864119871lowast

119864) 119892d119904

ge1

2int119863+cup119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909(57)

and the same arguments as above yield the assertion

We can also extend these ideas to the general setting ofSection 32

Theorem 11 Let 120590 isin 119871infin

+(Ω) and let 120590

0isin 119871

infin

+(Ω) be a

piecewise analytic function Let 119864 sube Ω be a measurable setChoose 120572 120573 isin R such that

120572 gt1003817100381710038171003817120590 minus 120590

0

1003817100381710038171003817119871infin(Ω) 120573 gt

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω) (58)

(a) If 120590 le 1205900on Ω 119864 then for all 119911 isin Ω that have a

neighborhood in which 1205900is analytic as well as all unit

vectors 119889 isin R119899 119889 = 1

119911 isin inn supp (120590 minus 1205900) cup 119864 implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590)minusΛ (1205900)+120572119871

119864119871lowast

119864

1003816100381610038161003816

12

)

(59)

and

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) implies

119911 isin outΣ(supp (120590 minus 120590

0) cup 119864)

(60)

(b) If 120590 ge 1205900on Ω 119864 then for all 119911 isin Ω that have a

neighborhood in which 1205900is analytic as well as all unit

vectors 119889 isin R119899 119889 = 1

119911 isin inn supp (120590 minus 1205900) cup 119864 implies

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) minus 120573119871

119864119871lowast

119864

1003816100381610038161003816

12

)

(61)

and

Φ119911119889|Σisin R (

1003816100381610038161003816Λ (120590) minus Λ (1205900) minus 120573119871

119864119871lowast

119864

1003816100381610038161003816

12

) implies

119911 isin outΣ(supp (120590 minus 120590

0) cup 119864)

(62)

Proof For every 119911 isin inn supp(120590minus1205900)with 119911 notin 119864 there exists

a small ball 119861120598(119911) and 120575 gt 0 so that 120590

0minus120590 ge 120575 on 119861

120598(119911) Using

the monotony Lemma 1 it follows that for all 119892 isin 1198712

(120597Ω)

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

ge intΩ

(1205900minus 120590

1)1003816100381610038161003816nabla1199060

1003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

ge 120575int119861120598(119911)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909+(120572 minus1003817100381710038171003817120590 minus 120590

0

1003817100381710038171003817119871infin(Ω))int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(63)

Computational and Mathematical Methods in Medicine 7

As in the previous proofs we obtain from Lemmas 2 and 3that

R (1003816100381610038161003816Λ (120590) minus Λ (120590

0) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) supe R (119871119861120598(119911)cup119864

) (64)

so that the first implication of (a) follows from Lemma 7The monotony Lemma 1 also implies that

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

le intΩ

1205900

120590(1205900minus 120590)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω)intsupp(120590minus1205900)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

+ 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(65)

so that the second implication of (a) follows from Lemmas 23 and 7 Assertion (b) can be proven analogously

Remark 9 also applies to this case

4 Conclusions and Remarks

The Factorization Method can be used to detect regionsin which a conductivity differs from a known referenceconductivity In this work we summarized the progresson the methodrsquos theoretical foundation We formulated themethod for general piecewise analytic conductivities andgave comparatively simple and self-contained proofsWe alsoshowed how the idea of excluding a part of the imaging regioncan be incorporated into this formulation

The regularity assumptions can beweakened even furtherOur proofs only require unique continuation arguments forthe reference conductivity 120590

0and the existence of the dipole

functionsTwo major open theoretical questions still exist in the

context of the FactorizationMethodThe theoretical justifica-tion of the method requires a definiteness condition (on thewhole domain or after excluding an a priori known part of thedomain) It is unknownwhether themethodrsquos range criterionholds without such a definiteness conditionThe second openquestion concerns the numerical stability of the methodrsquosrange criterion So far there are no rigorous convergenceresults for numerical implementations of this range criterion(see however Lechleiter [32] for a first step in this direction)As a promising approach to overcome both problems wewould like to point out the recent work on monotony-basedmethods [28]

Acknowledgments

This paper was funded by the German Research Foundation(DFG) and the University of Wurzburg in the fundingprogramme Open Access Publishing

References

[1] A Kirsch ldquoCharacterization of the shape of a scattering obstacleusing the spectral data of the far field operatorrdquo Inverse Prob-lems vol 14 no 6 pp 1489ndash1512 1998

[2] M Bruhl Gebietserkennung in der elektrischen Impedanztomo-graphie [PhD thesis] Universitat Karlsruhe 1999

[3] M Bruhl and M Hanke ldquoNumerical implementation of twononiterative methods for locating inclusions by impedancetomographyrdquo Inverse Problems vol 16 no 4 pp 1029ndash10422000

[4] M Bruhl ldquoExplicit characterization of inclusions in electricalimpedance tomographyrdquo SIAM Journal on Mathematical Anal-ysis vol 32 no 6 pp 1327ndash1341 2001

[5] M Hanke and M Bruhl ldquoRecent progress in electricalimpedance tomographyrdquo Inverse Problems vol 19 no 6 ppS65ndashS90 2003

[6] A Kirsch andN GrinbergTheFactorizationMethod for InverseProblems vol 36 ofOxford Lecture Series inMathematics and ItsApplications Oxford University Press Oxford UK 2007

[7] M Hanke and A Kirsch ldquoSampling methodsrdquo in Handbook ofMathematical Models in Imaging O Scherzer Ed pp 501ndash550Springer 2011

[8] A Kirsch ldquoThe factorization method for a class of inverseelliptic problemsrdquo Mathematische Nachrichten vol 278 pp258ndash277 2005

[9] B Gebauer ldquoThe Factorization Method for real elliptic prob-lemsrdquo Zeitschrift fur Analysis und ihre Anwendung vol 25 no1 pp 81ndash102 2006

[10] A I Nachman L Paivarinta and A Teirila ldquoOn imagingobstacles inside inhomogeneous mediardquo Journal of FunctionalAnalysis vol 252 no 2 pp 490ndash516 2007

[11] M Hanke and B Schappel ldquoThe factorization method forelectrical impedance tomography in the half-spacerdquo SIAMJournal on Applied Mathematics vol 68 no 4 pp 907ndash9242008

[12] N Hyvonen ldquoComplete electrode model of electricalimpedance tomography approximation properties andcharacterization of inclusionsrdquo SIAM Journal on AppliedMathematics vol 64 pp 902ndash931 2004

[13] A Lechleiter N Hyvonen and H Hakula ldquoThe factorizationmethod applied to the complete electrode model of impedancetomographyrdquo SIAM Journal onAppliedMathematics vol 68 no4 pp 1097ndash1121 2008

[14] N Hyvonen H Hakula and S Pursiainen ldquoNumerical imple-mentation of the factorization method within the completeelectrode model of electrical impedance tomographyrdquo InverseProblems and Imaging vol 1 pp 299ndash317 2007

[15] HHakula andNHyvonen ldquoOn computation of test dipoles forfactorization methodrdquo BIT Numerical Mathematics vol 49 no1 pp 75ndash91 2009

[16] B Harrach and J K Seo ldquoDetecting inclusions in electri-cal impedance tomography without reference measurementsrdquoSIAM Journal on Applied Mathematics vol 69 no 6 pp 1662ndash1681 2009

[17] B Harrach J K Seo and E J Woo ldquoFactorization methodand its physical justification in frequency-difference electricalimpedance tomographyrdquo IEEE Transactions on Medical Imag-ing vol 29 no 11 pp 1918ndash1926 2010

[18] S Schmitt ldquoThe factorization method for EIT in the case ofmixed inclusionsrdquo Inverse Problems vol 25 no 6 Article ID065012 2009

8 Computational and Mathematical Methods in Medicine

[19] S Schmitt and A Kirsch ldquoA factorization scheme for determin-ing conductivity contrasts in impedance tomographyrdquo InverseProblems vol 27 Article ID 095005 2011

[20] B Gebauer and N Hyvonen ldquoFactorization method and irreg-ular inclusions in electrical impedance tomographyrdquo InverseProblems vol 23 no 5 pp 2159ndash2170 2007

[21] B Gebauer ldquoLocalized potentials in electrical impedancetomographyrdquo Inverse Problems and Imaging vol 2 pp 251ndash2692008

[22] H Kang J K Seo and D Sheen ldquoThe inverse conductivityproblem with one measurement Stability and estimation ofsizerdquo SIAM Journal on Mathematical Analysis vol 28 no 6 pp1389ndash1405 1997

[23] M Ikehata ldquoSize estimation of inclusionrdquo Journal of Inverse andIll-Posed Problems vol 6 no 2 pp 127ndash140 1998

[24] T Ide H Isozaki S Nakata S Siltanen and G UhlmannldquoProbing for electrical inclusions with complex sphericalwavesrdquo Communications on Pure and Applied Mathematics vol60 no 10 pp 1415ndash1442 2007

[25] B Harrach and J K Seo ldquoExact shape-reconstruction by one-step linearization in electrical impedance tomographyrdquo TheSIAM Journal onMathematical Analysis vol 42 no 4 pp 1505ndash1518 2010

[26] N Bourbaki Elements of Mathematics Topological VectorSpaces chapters 1ndash5 Springer Berlin Germany 2003

[27] F Fruhauf B Gebauer andO Scherzer ldquoDetecting interfaces ina parabolic-elliptic problem from surfacemeasurementsrdquo SIAMJournal on Numerical Analysis vol 45 no 2 pp 810ndash836 2007

[28] B Harrach and M Ullrich ldquoMonotony based shape recon-struction in electrical impedance tomographyrdquo submitted forpublication

[29] S Kusiak and J Sylvester ldquoThe scattering supportrdquoCommunica-tions on Pure and Applied Mathematics vol 56 no 11 pp 1525ndash1548 2003

[30] B Gebauer and N Hyvonen ldquoFactorization method andinclusions of mixed type in an inverse elliptic boundary valueproblemrdquo Inverse Problems and Imaging vol 2 pp 355ndash3722008

[31] N I Grinberg and A Kirsch ldquoThe factorization method forobstacles with a-priori separated sound-soft and sound-hardpartsrdquo Mathematics and Computers in Simulation vol 66 no4-5 pp 267ndash279 2004

[32] A Lechleiter ldquoA regularization technique for the factorizationmethodrdquo Inverse Problems vol 22 no 5 pp 1605ndash1625 2006

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Computational and Mathematical Methods in Medicine 7

As in the previous proofs we obtain from Lemmas 2 and 3that

R (1003816100381610038161003816Λ (120590) minus Λ (120590

0) + 120572119871

119864119871lowast

119864

1003816100381610038161003816

12

) supe R (119871119861120598(119911)cup119864

) (64)

so that the first implication of (a) follows from Lemma 7The monotony Lemma 1 also implies that

intΣ

119892 (Λ (120590) minus Λ (1205900) + 120572119871

119864119871lowast

119864) 119892d119904

le intΩ

1205900

120590(1205900minus 120590)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909 + 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

le

1003817100381710038171003817100381710038171003817

1205900

120590(1205900minus 120590)

1003817100381710038171003817100381710038171003817119871infin(Ω)intsupp(120590minus1205900)

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

+ 120572int119864

1003816100381610038161003816nabla11990601003816100381610038161003816

2d119909

(65)

so that the second implication of (a) follows from Lemmas 23 and 7 Assertion (b) can be proven analogously

Remark 9 also applies to this case

4 Conclusions and Remarks

The Factorization Method can be used to detect regionsin which a conductivity differs from a known referenceconductivity In this work we summarized the progresson the methodrsquos theoretical foundation We formulated themethod for general piecewise analytic conductivities andgave comparatively simple and self-contained proofsWe alsoshowed how the idea of excluding a part of the imaging regioncan be incorporated into this formulation

The regularity assumptions can beweakened even furtherOur proofs only require unique continuation arguments forthe reference conductivity 120590

0and the existence of the dipole

functionsTwo major open theoretical questions still exist in the

context of the FactorizationMethodThe theoretical justifica-tion of the method requires a definiteness condition (on thewhole domain or after excluding an a priori known part of thedomain) It is unknownwhether themethodrsquos range criterionholds without such a definiteness conditionThe second openquestion concerns the numerical stability of the methodrsquosrange criterion So far there are no rigorous convergenceresults for numerical implementations of this range criterion(see however Lechleiter [32] for a first step in this direction)As a promising approach to overcome both problems wewould like to point out the recent work on monotony-basedmethods [28]

Acknowledgments

This paper was funded by the German Research Foundation(DFG) and the University of Wurzburg in the fundingprogramme Open Access Publishing

References

[1] A Kirsch ldquoCharacterization of the shape of a scattering obstacleusing the spectral data of the far field operatorrdquo Inverse Prob-lems vol 14 no 6 pp 1489ndash1512 1998

[2] M Bruhl Gebietserkennung in der elektrischen Impedanztomo-graphie [PhD thesis] Universitat Karlsruhe 1999

[3] M Bruhl and M Hanke ldquoNumerical implementation of twononiterative methods for locating inclusions by impedancetomographyrdquo Inverse Problems vol 16 no 4 pp 1029ndash10422000

[4] M Bruhl ldquoExplicit characterization of inclusions in electricalimpedance tomographyrdquo SIAM Journal on Mathematical Anal-ysis vol 32 no 6 pp 1327ndash1341 2001

[5] M Hanke and M Bruhl ldquoRecent progress in electricalimpedance tomographyrdquo Inverse Problems vol 19 no 6 ppS65ndashS90 2003

[6] A Kirsch andN GrinbergTheFactorizationMethod for InverseProblems vol 36 ofOxford Lecture Series inMathematics and ItsApplications Oxford University Press Oxford UK 2007

[7] M Hanke and A Kirsch ldquoSampling methodsrdquo in Handbook ofMathematical Models in Imaging O Scherzer Ed pp 501ndash550Springer 2011

[8] A Kirsch ldquoThe factorization method for a class of inverseelliptic problemsrdquo Mathematische Nachrichten vol 278 pp258ndash277 2005

[9] B Gebauer ldquoThe Factorization Method for real elliptic prob-lemsrdquo Zeitschrift fur Analysis und ihre Anwendung vol 25 no1 pp 81ndash102 2006

[10] A I Nachman L Paivarinta and A Teirila ldquoOn imagingobstacles inside inhomogeneous mediardquo Journal of FunctionalAnalysis vol 252 no 2 pp 490ndash516 2007

[11] M Hanke and B Schappel ldquoThe factorization method forelectrical impedance tomography in the half-spacerdquo SIAMJournal on Applied Mathematics vol 68 no 4 pp 907ndash9242008

[12] N Hyvonen ldquoComplete electrode model of electricalimpedance tomography approximation properties andcharacterization of inclusionsrdquo SIAM Journal on AppliedMathematics vol 64 pp 902ndash931 2004

[13] A Lechleiter N Hyvonen and H Hakula ldquoThe factorizationmethod applied to the complete electrode model of impedancetomographyrdquo SIAM Journal onAppliedMathematics vol 68 no4 pp 1097ndash1121 2008

[14] N Hyvonen H Hakula and S Pursiainen ldquoNumerical imple-mentation of the factorization method within the completeelectrode model of electrical impedance tomographyrdquo InverseProblems and Imaging vol 1 pp 299ndash317 2007

[15] HHakula andNHyvonen ldquoOn computation of test dipoles forfactorization methodrdquo BIT Numerical Mathematics vol 49 no1 pp 75ndash91 2009

[16] B Harrach and J K Seo ldquoDetecting inclusions in electri-cal impedance tomography without reference measurementsrdquoSIAM Journal on Applied Mathematics vol 69 no 6 pp 1662ndash1681 2009

[17] B Harrach J K Seo and E J Woo ldquoFactorization methodand its physical justification in frequency-difference electricalimpedance tomographyrdquo IEEE Transactions on Medical Imag-ing vol 29 no 11 pp 1918ndash1926 2010

[18] S Schmitt ldquoThe factorization method for EIT in the case ofmixed inclusionsrdquo Inverse Problems vol 25 no 6 Article ID065012 2009

8 Computational and Mathematical Methods in Medicine

[19] S Schmitt and A Kirsch ldquoA factorization scheme for determin-ing conductivity contrasts in impedance tomographyrdquo InverseProblems vol 27 Article ID 095005 2011

[20] B Gebauer and N Hyvonen ldquoFactorization method and irreg-ular inclusions in electrical impedance tomographyrdquo InverseProblems vol 23 no 5 pp 2159ndash2170 2007

[21] B Gebauer ldquoLocalized potentials in electrical impedancetomographyrdquo Inverse Problems and Imaging vol 2 pp 251ndash2692008

[22] H Kang J K Seo and D Sheen ldquoThe inverse conductivityproblem with one measurement Stability and estimation ofsizerdquo SIAM Journal on Mathematical Analysis vol 28 no 6 pp1389ndash1405 1997

[23] M Ikehata ldquoSize estimation of inclusionrdquo Journal of Inverse andIll-Posed Problems vol 6 no 2 pp 127ndash140 1998

[24] T Ide H Isozaki S Nakata S Siltanen and G UhlmannldquoProbing for electrical inclusions with complex sphericalwavesrdquo Communications on Pure and Applied Mathematics vol60 no 10 pp 1415ndash1442 2007

[25] B Harrach and J K Seo ldquoExact shape-reconstruction by one-step linearization in electrical impedance tomographyrdquo TheSIAM Journal onMathematical Analysis vol 42 no 4 pp 1505ndash1518 2010

[26] N Bourbaki Elements of Mathematics Topological VectorSpaces chapters 1ndash5 Springer Berlin Germany 2003

[27] F Fruhauf B Gebauer andO Scherzer ldquoDetecting interfaces ina parabolic-elliptic problem from surfacemeasurementsrdquo SIAMJournal on Numerical Analysis vol 45 no 2 pp 810ndash836 2007

[28] B Harrach and M Ullrich ldquoMonotony based shape recon-struction in electrical impedance tomographyrdquo submitted forpublication

[29] S Kusiak and J Sylvester ldquoThe scattering supportrdquoCommunica-tions on Pure and Applied Mathematics vol 56 no 11 pp 1525ndash1548 2003

[30] B Gebauer and N Hyvonen ldquoFactorization method andinclusions of mixed type in an inverse elliptic boundary valueproblemrdquo Inverse Problems and Imaging vol 2 pp 355ndash3722008

[31] N I Grinberg and A Kirsch ldquoThe factorization method forobstacles with a-priori separated sound-soft and sound-hardpartsrdquo Mathematics and Computers in Simulation vol 66 no4-5 pp 267ndash279 2004

[32] A Lechleiter ldquoA regularization technique for the factorizationmethodrdquo Inverse Problems vol 22 no 5 pp 1605ndash1625 2006

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

8 Computational and Mathematical Methods in Medicine

[19] S Schmitt and A Kirsch ldquoA factorization scheme for determin-ing conductivity contrasts in impedance tomographyrdquo InverseProblems vol 27 Article ID 095005 2011

[20] B Gebauer and N Hyvonen ldquoFactorization method and irreg-ular inclusions in electrical impedance tomographyrdquo InverseProblems vol 23 no 5 pp 2159ndash2170 2007

[21] B Gebauer ldquoLocalized potentials in electrical impedancetomographyrdquo Inverse Problems and Imaging vol 2 pp 251ndash2692008

[22] H Kang J K Seo and D Sheen ldquoThe inverse conductivityproblem with one measurement Stability and estimation ofsizerdquo SIAM Journal on Mathematical Analysis vol 28 no 6 pp1389ndash1405 1997

[23] M Ikehata ldquoSize estimation of inclusionrdquo Journal of Inverse andIll-Posed Problems vol 6 no 2 pp 127ndash140 1998

[24] T Ide H Isozaki S Nakata S Siltanen and G UhlmannldquoProbing for electrical inclusions with complex sphericalwavesrdquo Communications on Pure and Applied Mathematics vol60 no 10 pp 1415ndash1442 2007

[25] B Harrach and J K Seo ldquoExact shape-reconstruction by one-step linearization in electrical impedance tomographyrdquo TheSIAM Journal onMathematical Analysis vol 42 no 4 pp 1505ndash1518 2010

[26] N Bourbaki Elements of Mathematics Topological VectorSpaces chapters 1ndash5 Springer Berlin Germany 2003

[27] F Fruhauf B Gebauer andO Scherzer ldquoDetecting interfaces ina parabolic-elliptic problem from surfacemeasurementsrdquo SIAMJournal on Numerical Analysis vol 45 no 2 pp 810ndash836 2007

[28] B Harrach and M Ullrich ldquoMonotony based shape recon-struction in electrical impedance tomographyrdquo submitted forpublication

[29] S Kusiak and J Sylvester ldquoThe scattering supportrdquoCommunica-tions on Pure and Applied Mathematics vol 56 no 11 pp 1525ndash1548 2003

[30] B Gebauer and N Hyvonen ldquoFactorization method andinclusions of mixed type in an inverse elliptic boundary valueproblemrdquo Inverse Problems and Imaging vol 2 pp 355ndash3722008

[31] N I Grinberg and A Kirsch ldquoThe factorization method forobstacles with a-priori separated sound-soft and sound-hardpartsrdquo Mathematics and Computers in Simulation vol 66 no4-5 pp 267ndash279 2004

[32] A Lechleiter ldquoA regularization technique for the factorizationmethodrdquo Inverse Problems vol 22 no 5 pp 1605ndash1625 2006

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MEDIATORSINFLAMMATION

of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Behavioural Neurology

EndocrinologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Disease Markers

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

OncologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Oxidative Medicine and Cellular Longevity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational and Mathematical Methods in Medicine

OphthalmologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Diabetes ResearchJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Research and TreatmentAIDS

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Gastroenterology Research and Practice

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom