On the Effects of the Electromagnetic Source Modelling in the Iterative Multiscaling Method

UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38123 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

ON THE EFFECTS OF THE ELECTROMAGNETIC SOURCE

MODELLING IN THE ITERATIVE MULTISCALING METHOD

D. Franceschini, M. Donelli, and A. Massa

June 2007

in the Iterative Multis aling Method 2 3 4 5 6

Davide Fran es hini, Massimo Donelli,and Andrea Massa 7 8 9 10 11 12 13

Department of Information and Communi ation Te hnologies, 14

University of Trento,Via Sommarive 14,38050 Trento - Italy 15

Tel. +39 0461 882057,Fax +39 0461 882093 16

E-mail: andrea.massaing.unitn.it, {davide.fran es hini,massimo.donelli}dit.unitn.it 17

Web: http://www.eledia.ing.unitn.it 18

in the Iterative Multis aling Method

20

21

Davide Fran es hini, Massimo Donelli,and Andrea Massa 22

23

Abstra t 24

Thevalidationagainstexperimentaldataisafundamentalstepintheassessment of 25

theee tiveness of a mi rowave imaging algorithm. Itis aimed at pointing out the 26

limitationsofthe numeri al pro edureforasu essiveappli ation inareal environ-27

ment. Towards this end, this paper evaluatesthe re onstru tion apabilities of the 28

Iterative Multi-S aling Approa h (IMSA) when dealing withexperimental data by 29

onsidering dierent numeri al models of the illuminating setup. Infa t, sin e the 30

in ident ele tromagneti eld is usually olle ted in a limited set of measurement 31

points and inversion methods based on the use of the state equation require the 32

knowledge of the radiated eld in a ner grid of positions, an ee tive numeri al 33

pro edure for the synthesisof theele tromagneti sour e isgenerallyneeded. Con-34

sequently,theperforman esoftheinversionpro essmaybestrongly ae tedbythe

35

numeri almodeland,insu ha ase,agreat areshouldbedevotedtothiskeyissue

36

to guarantee suitableand reliable re onstru tions.

37

38

Keywords: 39

Mi rowaveImaging,InverseS attering,IterativeMulti-s aling Method,Sour eModeling. 40

Index Terms: 41

6982RadioS ien e: Tomographyandimaging;0629Ele tromagneti s: Inverses attering; 42

0669Ele tromagneti s: S atteringanddira tion. 43

Within the framework of the medi ine [Louis, 1992℄ and biomedi al engineering (see 45

for example [Liu et al., 2003℄ and the referen es ited therein), without forgetting the 46

industrial quality ontrol in industrial pro esses [Hoole et al., 1991℄ and the subsurfa e 47

sensing [Dubey et al., 1995; Daniels, 1996℄, many dierent appli ations require a non-48

invasivesensingofina essibleareas. Towardsthisend,mi rowaveimagingmethodologies 49

[Steinberg, 1991℄ have re ently gained a growing attention sin e they allow to retrieve 50

informationontheenvironmentprobedwithele tromagneti eldsby fullyexploitingthe 51

s attering phenomena [Colton and Kress, 1992℄. 52

Unfortunately, the inverse problem to be fa ed is intrinsi ally nonlinear, ill-posed, and 53

non-unique [Denisov, 1999℄. In parti ular, the ill-posedness and the non-uniqueness arise 54

fromthelimitedamountof information olle tableduringthea quisitionofthes attered 55

eld. Thenumberofindependents atteringdataislimited[Berteroetal.,1995;Bu iand 56

Fran es hetti, 1989℄ and they an onlybeused to retrieve anite numberof parameters 57

of the unknown ontrast fun tion. Tofully exploit su h aninformation and to a hieve a 58

suitableresolutiona ura y,severalmulti-resolutionstrategieshavebeenproposed[Miller 59

and Willsky, 1996a,1996b; Bu i et al.,2000a, 2000b;Baussard et al., 2004a,2004b℄. 60

The Iterative Multi-S aling Approa h belongs to this lass of algorithms [Caorsi et al., 61

2003℄. The unknown s atterers are iteratively re onstru ted by onsidering initially a 62

rough estimate of the diele tri distribution 1

and by enhan ing su essively the spatial 63

resolution in a set of regions-of-interest (RoIs) where the obje ts have been lo alized. 64

Su ha strategyis mathemati allyformulatedby dening asuitablemulti-resolution ost 65

fun tion whose global minimum is assumed as the estimated solution. The fun tional 66

is iteratively minimized by using a onjugate-gradient-based pro edure [Kleinman and 67

Van den Berg, 1992℄, but sto hasti [Massa, 2002℄ or hybrid algorithms an be suitably 68

applied. 69

In order to validate su h an approa h, the multi-resolution algorithm has been tested 70

against experimental data [Caorsi et al., 2004a℄ olle ted in a ontrolled environment 71

1

The IMSAisinitialized by onsidering thefreespa e distribution,then noa-priori information on the s enario under test is exploited. Moreover, the initialization of the intermediate steps is obtained from the re onstru tion of thepreviousstep with asimplemappingof the retrieved prolein the new dis retization oftheRoI.

indi ations and they modelanideal s enario. 73

In dealing with real data, one of the key issue is the modeling of the ele tromagneti 74

sour eoroftherelatedradiatedeld. Ingeneral,theele tromagneti eldemittedbythe 75

probing system is measured onlyin the observation domain. However, iterative methods 76

basedonDataandState equationsrequiretheknowledgeofthein identeld(i.e.,the 77

eld without the s atterers) generated from the sour e in the investigation domain. T o-78

wards this end, ana urate but simplymodel(i.e., requiringa reasonable omputational 79

burden) of the sour eshouldbe developed. Compli atednumeri almodelsa urately re-80

produ erealdata,buttheyaredi ulttobeimplementedstartingfromalimitednumber 81

of samples of the radiatedele tromagneti eld olle ted in a portion of the observation 82

domain. On the other hand, a rough model ould introdu e erroneous onstraints to 83

the re onstru tion pro ess. Nevertheless, whatever the sour e model, an ee tive inver-84

sion pro edure should be able to re onstru t the s atterer undertest with an a eptable 85

a ura y a ording toits robustness tothe noise. 86

In this framework, toassessthe ee tiveness and the robustness ofthe IMSA, the results 87

of aset of experiments, wheredierentmodels forapproximating the illuminatingsour e 88

are onsidered,will beshown. 89

The paper is organized as follows. In Se tion 2, the statement of the inverse problem 90

and the mathemati alformulationof the IMSA will bebriey resumed, while in Se tion 91

3 the numeri al models used to synthesize the probing ele tromagneti sour e will be 92

des ribed. A numeri al validation and an exhaustive analysis of the dependen e of the 93

re onstru tiona ura yonthemodelingoftheradiatedeldwillbe arriedoutinSe tion 94

4 by onsidering some experimental test ases. Finally, some on lusions will be drawn 95

(Se t. 5). 96

2 Mathemati al Formulation 97

The inversion pro edure willbeillustrated referringto atwo-dimensionalgeometry (Fig-98

ure 1). Let us onsider an investigation domain

D

I

, where an unknown s atterer is 99

supposed to be lo ated. The embeddingmedium is assumed lossless, non-magneti ,and 100

hara terized by a diele tri permittivity

ε

0

. Su h a s enario is illuminated by a set of 101

V

in ident mono hromati ele tromagneti elds

E

v

inc

(x, y)

,

v = 1, ..., V

, and the orre-102

sponding s attered elds

E

v

scatt



x

m

(v)

, y

m

(v)



,

v = 1, ..., V

, are available ( omputed asthe 103

dieren ebetween the eldwith

E

v

tot

and withoutthes atterer

E

v

inc

,

E

v

scatt

= E

tot

v

− E

inc

v

) 104

in

m

(v)

= 1, ..., M

(v)

,

v = 1, ..., V

,positionsbelongingtotheobservationdomain

D

M

. The 105

obje t is des ribed by a ontrast fun tion

τ (x, y) = ε

r

(x, y) − 1 − j

σ(x, y)

2πf ε

0

,

(x, y) ∈ D

I

, 106

ε

r

(x, y)

and

σ(x, y)

beingthediele tri permittivityandthe ele tri ondu tivity, respe -107

tively. 108

The arising s attering phenomena are mathemati ally des ribed through the well-known 109

Lippmann-S hwingerintegral equations [Colton and Kress, 1992℄: 110 111 112

E

v

scatt

(x

m(v)

, y

m(v)

) = k

2

0

R

D

I

G

2d

(x

m(v)

, y

m(v)

|x

′

_{, y}

′

_{)τ (x}

′

_{, y}

′

_)E

υ

tot

(x

′

, y

′

)dx

′

dy

′

, m

(v)

= 1, ..., M

(v)

(x

m

(v)

, y

m

(v)

) ∈ D

M

v = 1, ..., V

(1) (Data Equation) 113 114 115

E

v

inc

(x, y) = E

tot

v

(x, y) − k

2

0

Z

D

I

G

2d

(x, y|x

′

, y

′

)τ (x

′

, y

′

)E

tot

v

(x

′

, y

′

)dx

′

dy

′

(x, y) ∈ D

I

(2) (State Equation) 116 117

where

G

2d

denotes the Green fun tion of the ba kground medium[Jones, 1964℄. 118

Sin etheproblemasso iatedwith(??)isill-posed(see[Groets h,1993℄and[Vogel,2002℄) 119

the systemmatrixafterdis retizationoftheData Equation (a ordingtotheRi hmond's 120

pro edure[Ri hmond,1965℄)ishighlyill- onditioned,and,hen etheproblemisextremely 121

sensitive to the the noise. To remedy this ill- onditioning, a regularization is needed. 122

Thus, the problem is then reformulated in nding the unknown ontrast fun tion that 123

minimizesa suitable ost fun tiongenerally dened as follows 124

Φ {τ (x

n

, y) , E

tot

v

(x

n

, y

n

) ; n = 1, ..., N; v = 1, ..., V } =

=

P

V

v=1

P

M

_(v)

m

(v)

=1





E

v

scatt



x

m

(v)

, y

m

(v)



−

P

N

n=1

n

τ (x

n

, y

n

) E

tot

v

(x

n

, y

n

) G

ext

2d



A

n

, ρ

nm

(v)

o



2

+

P

V

v=1

P

N

n=1





E

v

inc

(x

n

, y

n

) −

h

E

v

tot

(x

n

, y

n

) −

P

N

u=1

{τ (x

u

, y

u

) E

tot

v

(x

u

, y

u

) G

int

2d

(A

u

, ρ

un

)}

i



2

where

G

int

2d

and

G

ext

2d

indi ate the dis retized forms of the internal and external Green's 125

operators [Colton and Kress, 1992℄,

ρ

nm

(v)

=

r

x

n

− x

m

(v)



2

+



y

n

− y

m

(v)



2

,

ρ

un

=

126

q

(x

u

− x

n

)

2

+ (y

u

− y

n

)

2

and

A

n

(

A

u

) is the area of the

n

-th (

u

-th) square dis retiza-127

tion domain. In parti ular,the rstterm of(??) enfor esdelitytothe s attereddata in 128

the observation domain (

E

v

scatt

(x

m

(v)

, y

m

(v)

)

,

(x

m

(v)

, y

m

(v)

) ∈ D

M

) and it amounts to the 129

residual errorwith respe t tothe s attered eld omputed fromthe Data Equation (??). 130

The se ond term is a regularization term equal to the residual error with respe t to the 131

in ident eld in the investigation domain (

E

v

inc

(x

n

, y

n

)

,

(x

n

, y

n

) ∈ D

I

) omputed from 132

the StateEquation (??). 133

However, due to the limitedamountof information ontent inthe input data [Bu i and 134

Fran es hetti, 1989℄, it would be problemati toparametrize the investigationdomain in 135

terms of a large number

N

of pixel values (in order to a hieve a satisfying resolution

136

levelinthere onstru ted image). Toover omethis drawba k, aninitialuniform( oarse)

137

dis retization is used and su essively an iterative parametrization of the test domain

138

allows to adaptively in rease the resolution level only in the region-of-interest of the

139

investigationareathusa hievingtherequiredre onstru tiona ura y[Caorsietal.,2003℄.

140

To retrieve the unknown s atterer (i.e., an obje t fun tion that better ts the problem 141

data, (

E

v

scatt

(x

m

(v)

, y

m

(v)

)

,

E

v

inc

(x, y)

),Eqs. (??)and(??)are dis retizeda ordingtothe 142

Ri hmond's pro edure [Ri hmond, 1965℄. Moreover, to better exploit the limited infor-143

mation ontent of the s attering data, an adaptive multi-resolution strategy is adopted 144

[Caorsi et al., 2003℄. 145

More in detail, su h an adaptive multi-resolution algorithm an be briey des ribed as

146

follows. Firstly, the IMSA onsiders (

i = 0

,

i

being the step index) an homogeneous

147

dis retization of the investigation domain with a number of dis retization domains

N

(0)

148

equal to the essential dimension of the s attered data and omputed a ording to the

149

riterion dened in [Isernia et al., 2001℄. Then, a  oarse re onstru tion of the investi-150

gation domain is yielded by minimizing (??) starting from the free-spa e onguration

151

[

τ (x

n

(0)

, y

n

(0)

) = 0.0

and

E

v

tot

(x

n

₍₀₎

, y

n

₍₀₎

) = E

inc

v

(x

n

₍₀₎

, y

n

₍₀₎

)

℄ in order toassess the robust-152

ness ofthe overallapproa hwithrespe t tothestartingguess inworst- ase. Afterthe 153

minimization, where a set of onjugate-gradient iterations (

k

being the iteration index)

(RoI),

D

O(i)

,

i = 0

, is dened. Su ha squared area is entered at 156

x

RoI

c

_(i)

=

x

RoI

re

(i)

+ x

RoI

im

(i)

2

, y

RoI

c

_(i)

=

y

RoI

re

(i)

+ y

RoI

im

(i)

2

(4) where

x

RoI

re

(i)

,

x

RoI

im

(i)

, y

RoI

re

(i)

and

y

RoI

im

(i)

are dened as

157 158

x

RoI

ℜ(i)

=

P

R

r=1

P

N(r)

n(r)=1

n

x

_n(r)

ℜ

h

τ



x

_n(r)

,y

_n(r)

io

P

N(r)

n(r)=1

n

ℜ

h

τ



x

_n(r)

,y

_n(r)

io

, R = i

(5) 159

y

_ℜ(i)

RoI

=

P

R

r=1

P

N

(r)

n

(r)

=1

n

y

n

(r)

ℜ

h

τ



x

n

(r)

, y

n

(r)

io

P

N

(r)

n

(r)

=1

n

ℜ

h

τ



x

n

(r)

, y

n

(r)

io

(6) 160

and its side

L

(i)

is dened asfollows

161 162

L

RoI

(i)

=

L

RoI

re

(i)

+ L

RoI

im

(i)

2

(7) 163 164

L

RoI

_ℜ(i)

= 2

P

R

r=1

P

N

(r)

n

(r)

=1







ρ

_n(r)c(i)

ℜ

h

τ



x

_n(r)

,y

_n(r)

i

max

_{n(r)=1,..,N(r)}

n

ℜ

h

τ



x

_n(r)

,y

_n(r)

io







P

R

r=1

P

N

(r)

n

(r)

=1







ℜ

h

τ



x

_n(r)

,y

_n(r)

i

max

_{n(r)=1,..,N(r)}

n

ℜ

h

τ



x

_n(r)

,y

_n(r)

io







(8) 165

where

ℜ

standsfortherealortheimaginarypartand

ρ

n

(r)

c

(i)

=

r

x

n

(r)

− x

RoI

c

(i)



2

+



y

n

(r)

− y

RoI

c

(i)



2

.

166

Su essively, the iterative pro ess starts (

i → i + 1

). A ording to the multi-resolution

167

strategy, an higher resolution level denoted by

R

(

R = i

) is adopted only for the RoI.

168

D

O(i−1)

is dis retized in

N

(i)

square sub-domain whi h number is always hosen equal to

169

the essential dimension of the s attered data [Bu i and Fran es hetti, 1989℄. A ner

170

obje tfun tionproleisthenretrieved,startingfromthe oarserre onstru tion a hieved

171

at the (i-1)-th step, by minimizing the multi-resolution ost fun tion,

Φ

(i)

, dened as

Φ

(i)















τ

(i)



_x

n

(r)

, y

n

(r)



, E

tot

v

(i)



x

n

(r)

, y

n

(r)



;

r = 1, ..., R = i;

n

(r)

= 1, ..., N

(r)

;

v = 1, ..., V















=

=

nP

V

v=1

P

M

(v)

m

_(v)

=1







E

scatt

v



x

m

(v)

, y

m

(v)



−

P

R

r=1

P

N

(r)

n

_(r)

=1

n

w



x

n

(r)

, y

n

(r)



τ

(i)



_x

n

(r)

, y

n

(r)



E

tot

v

(i)



x

n

(r)

, y

n

(r)



G

ext

2d



A

n

(r)

, ρ

n

(r)

m

(v)

o



2



+

nP

V

v=1

P

R

r=1

P

N

_(r)

n

_(r)

=1

n

w



x

n

(r)

, y

n

(r)

 

_E

v

inc



x

n

(r)

, y

n

(r)



−

h

E

tot

v

(i)



x

n

(r)

, y

n

(r)



−

P

N

(r)

u

(r)

=1

n

τ

(i)



_x

u

(r)

, y

u

(r)



E

tot

v

(i)



x

u

(r)

, y

u

(r)



G

int

2d



A

u

(r)

, ρ

u

(r)

n

(r)

oi



o

2



(9) where 174

w(x

n

(r)

, y

n

(r)

) =















0

if (x

n

_(r)

,

y

n

_(r)

) /

∈ D

O(i−1)

1

if (x

n

(r)

,

y

n

(r)

) ∈ D

O(i−1)

and

R

indi ates the resolution level and

D

O(i)

denotes the area of the RoI dened at

175

the i-th step of the iterative pro edure. It should be pointed out that the denition of

176

(??) requires not only the knowledge of the available s attered eld in the observation

177 domain [

E

v

scatt



x

m

(v)

, y

m

(v)



= E

v

tot



x

m

(v)

, y

m

(v)



− E

v

inc



x

m

(v)

, y

m

(v)



,



x

m

(v)

, y

m

(v)



∈

178

D

M

℄, but alsothat of the in ident eld in

D

O(i)

[

E

v

inc

(x

n

_(r)

, y

n

_(r)

)

,

(x

n

(r)

,

y

n

(r)

) ∈ D

O(i−1)

℄.

179

This latter information is generally not available from measurements [sin e, in general,

180

only the samples of

E

v

inc



x

m

(v)

, y

m

(v)



other than

E

v

tot



x

m

(v)

, y

m

(v)



are experimentally 181

measured℄, thereforeitshouldbesyntheti allygenerated by meansof asuitablemodelof

182

the ele tromagneti sour e.

183

The multi-steppro ess ontinues by omputing a new RoI a ording to(??)(??) and by

184

estimating anew diele tri distribution through theminimization ofthe updated version

185

of (??) untila "stationary re onstru tion"is rea hed [Caorsi et al.,2003℄ (

i = I

opt

) .

186

Su hapro edure an beextendedtomultiple-s atterersgeometriesby onsideringa

suit-187

able lustering pro edure [Caorsi etal.,2004b℄aimedatdeningthenumberofs atterers

188

Q

belonging to the investigation domain and the regions

D

(q)

O(i)

,

q = 1, ..., Q

, where the

189

syntheti zoomwillbe performed atea h step of the iterativepro ess.

Thein identelddataplaya ru ialroleintheimagingpro esssin etheknowledge/availability 192

of

E

v

inc

(x, y)

in the investigationdomainadds new information. Infa t, asit an be no-193

ti ed in the equation dening the multi-resolution ost fun tion (??), it allows to dene 194

another onstraint (??) for the problem solution then redu ing the ill-posedness of the 195

inverseproblem[BerteroandBo a i,1998℄sin esu haterm anbealso onsideredasa 196

sort ofregularizationterm. Clearly,anerroneousorimpre iseknowledgeofthe in ident 197

eld ould onsiderably ae t the reliability of the fun tional and onsequently of the 198

overall imaging pro ess sin e (??) ontrols the minimization pro edure. As a matter of 199

fa t, inmany pra ti alsituations, the in ident eld isonly availableatthe measurement 200

points belonging to the observation domain,

E

v

inc



x

m

(v)

, y

m

(v)



,



x

m

(v)

, y

m

(v)



∈ D

M

. 201

Su h a situation is ommonly en ountered when dealing with real data be ause of the 202

omplexityanddi ultiesin olle tingreliableandindependentmeasures inadensegrid 203

of points. Hen e, to fully exploit the knowledge of the in ident eld and before fa ing 204

with the data inversion, it is mandatory to develop a suitable model able to predi t the 205

in ident eld radiated by the a tual ele tromagneti sour e in the investigationdomain, 206

E

v

inc

(x, y)

,

(x, y) ∈ D

I

. Towards this aim, in the referen e literature (see [Belkebir and 207

Saillard, 2001℄ and the referen es ited therein), dierent solutions have been proposed. 208

They aremainlybasedonplaneor ylindri alwavesexpansions, sin efar-eld onditions 209

are usually satised. In this paper, su h models willbe analyzed and a new distributed 210

modelwill beproposed. More in detail, letus onsider 211

•

thePlane-Waves Model (PW-Model)wherethein identeldismodeledasthe 212

superposition of aset of

W

planewaves 213

E

υ

inc

(x, y) =

W

X

w=1

A

w

e

−jwk

0

(xcosθ

v

+ysinθ

v

)

(10)

θ

v

being the in ident angle,

k

0

the free-spa e propagation onstant, and

A

w

the 214

amplitude of

w

-th wave; 215

•

the Con entri -Cylindri al-Waves Model (CCW-Model) wherethe radiated 216

eld is represented through the superposition of ylindri alwaves a ording tothe 217

E

υ

inc

(x, y) =

W

X

w=−W

A

w

H

w

(2)

(k

0

ρ) e

jwφ

v

(11)

where

A

w

is an unknown oe ient,

H

(2)

w

indi ates the se ond kind

w

-th order 219

Hankel fun tion,

ρ

is the distan e between the observation point lo ated at

(x, y)

220

and the phase enter of the radiating system where the

w

-th line sour e is pla ed

221

and

φ

v

the orresponding angle;

222

•

the Distributed-Cylindri al-Waves Model (DCW-Model) where the a tual 223

sour e is repla ed with alinear array of equally-spa ed line-sour es, whi h radiates

224

an ele tri eld given by

225

E

υ

inc

(x, y) = −

k

2

0

8πf ε

0

W

X

w=1

A (x

w

, y

w

) H

0

(2)

(k

0

ρ

w

)

(12)

where

A(x

w

, y

w

)

is the unknown oe ientrelated tothe

w

-th elementand

ρ

w

the 226

distan e between the observation point and the

w

-th line sour e. 227

Su hmodelsare ompletelydenedwhenthesetofunknown oe ients,

A

w

or

A(x

w

, y

w

)

, 228

have been determined. Therefore, the solution of an inverse sour e problem, where the 229

known terms are the values of the in ident eld measured in the observation domain 230

E

v

inc



x

m

(v)

, y

m

(v)



, is required. More indetail, the followingsystem has tobe solved: 231

















































E

v

inc

(x

1

, y

1

)

...

...

E

v

inc

(x

m

_(v)

, y

m

_(v)

)

...

...

E

v

inc

(x

M

(v)

, y

M

(v)

)

















































=

















































G

11

...

G

1s

...

G

1S

...

...

...

...

...

...

...

...

...

...

G

m1

... G

ms

... G

mS

...

...

...

...

...

...

...

...

...

...

G

M1

... G

M s

... G

M S

































































































I

1

...

...

I

s

...

...

I

S

















































(13)

or ina more on iseform 232

where (a) for the PW-model

G

ms

= e

−jsk

0

d

m

,

d

m

= x

m

cosθ

v

+ y

m

sinθ

v

, and

I

s

=

233

A

s

,

s = 1, ..., S

,

S = W

; (b) for the CCW-model

G

ms

= H

(2)

s

(k

0

ρ

m

) e

jsφ

v

,

ρ

m

=

234

q

(x

m

− x

source

)

2

+ (y

m

− y

source

)

2

,

(x

source

, y

source

)

being the lo ation of the sour e, and 235

I

s

= A

s−1−W

,

s = 1, ..., S

,

S = 2W +1

;( )fortheDCW-model

G

ms

= −

k

2

₀

8πf ε

0

H

(2)

0

(k

0

ρ

ms

)

, 236

ρ

ms

=

q

(x

m

− x

s

)

2

+ (y

m

− y

s

)

2

, and

I

s

= A (x

s

, y

s

)

,

s = 1, ..., S

,

S = W

. 237

Unfortunately, (??)involvesthe limitationstypi al of an inverse-sour e problem(see for 238

example, [Devaney and Sherman, 1982℄). In parti ular,

[G]

is ill- onditioned and the 239

solution isusually non-stable and non-unique. Now, the problem of determining

[I]

from 240

the knowledge of the in ident eld an be re ast as the inversion of the linear operator 241

[G]

through the SVD-de omposition [Natterer, 1986℄ 242 243

[I] = [G]

+

[E]

(15) where 244

[G]

+

= [V] [Γ]

−1

[U]

∗

(16) and 245 246

[Γ]

−1

=

















1/γ

1

...

0

...

1/γ

s

...

0

...

1/γ

S

















(17)

Owing of the propertiesof

[G]

, the sequen e of singularvalues

{γ

s

}

S

s=1

willbe de reasing 247

and onvergenttozero. Consequently,thesolutionofequation(??)doesnot ontinuously 248

depend on problemdata and the unavoidable presen e of the noise, due to measurement 249

errors as well as to an ina urate model of the experimental setup, ould produ e an 250

unreliable sour e synthesis. 251

In the next se tion, anexhaustivenumeri alanalysiswill be arried out toassess the ro-252

bustness oftheIMSAagainsttheerrorinthein identelddataandtobetterunderstand 253

how and how mu h the model of the a tual ele tromagneti sour e ae ts the IMSA 254

performan es. 255

In this se tion,su hanassessmentwillbe performedby onsideringdierenttargetsand 257

starting from experimental data. The s attered data refers to the dataset available at 258

the Institute Fresnel - Marseille, Fran e. As des ribed in[Belkebir and Saillard, 2001; 259

Testorf and Fiddy, 2001; Marklein et al., 2001℄ and sket hed in Figure 2, the bistati 260

radar measurement system onsists of an emitting antenna pla ed at

r

s

= 720 ± 3mm

261

from the enter of the experimental setup and a re eiver whi h olle ts equally-spa ed

262 (

5

◦

)measurements of

E

v

tot



x

m

(v)

, y

m

(v)



and

E

v

inc



x

m

(v)

, y

m

(v)



ona ir ularinvestigation 263

domainofradius

r

m

= 760±3mm

. Notethepresen eofablind-se torof

θ

l

= 120

◦

around 264

the emittingantenna (Figure 2). The s atterers onsidered in the following experiments 265

are shown inFigs 2(a)-( )for referen e. 266

In the rst example [Fig. 2(a)℄, we will onsider the ir ular diele tri prole (

L

ref

=

267

30 mm

in diameter) positioned about

30 mm

from the enter of the experimental setup 268

(

x

c

ref

= 0.0, y

c

ref

= −30 mm

)and hara terizedbyahomogeneouspermittivity

ε

r

(x, y) =

269

3.0 ± 0.3

[

τ (x, y) = 2.0 ± 0.3

℄. The square investigation domain,

L

DI

= 30 cm

sided, 270

is partitioned in

N = 100

homogeneousdis retization domains and the re onstru tion is 271

performed by exploiting all the available measures (

M

(v)

= 49

,

v = 1, ..., V

) and views 272

(

V = 36

), but using mono-frequen y data (

f = 4 GHz

). 273

Theperforman esoftheIMSAintermsofquantitativeaswellasqualitativeimaginghave

274

beenassessed onsideringne essarilytheStateTerm

2

duringtheminimizationofthe ost

275

fun tion (??) and thusintrodu ingthe information- ontent of the in ident ele tri eld.

276

To do this, two simple models for the eld emitted by the probing antenna have been 277

preliminary taken into a ount. The rst one represents the radiated eld with a plane 278

wave(

W = S = 1

),theotherwitha ylindri alwave(

W = 0

,

S = 1

). Theamplitudesof 279

the modeledin identwavesare estimateda ordingtotheSVD-basedpro eduredetailed 280

in Se t. 3startingfromthe knowledgeof the values of the in identeld measured inthe 281

forwarddire tionand availabledire tly fromtheexperimentaldataset. They turn out to 282 be





A

(P W −M odel)

w=1







= 1.23

and





A

(CCW −M odel)

w=0







= 17.27

, respe tively. 283

In spite of the ina ura y inreprodu ing the values of the in ident eld olle ted at the 284

2

Some examples of algorithms employing only the Data Term an be found in the spe ial se tion

to lo alize the unknown targetwith asatisfa tory degree of a ura y as shown in Fig. 4

286

and onrmed by the geometri parameters reportedin Tab. I.

287

As far as the single-plane-wave model is on erned, it should be pointed out that the 288

re onstru ted ontrast 3

is hara terized by an average value of the obje t fun tion equal

289

to

τ = 2.1

, then very lose to the a tual value of the real target. However, several

290

pixelsbelongingtotheareaof thereferen eprolepresent alargerobje tfun tionvalues

291

[

τ (x

n

, y

n

) = 2.5

℄and theretrievedobje t ontourdoesnot a uratelyreprodu ea ir ular

292

shape.

293

With respe t to the PW model, a better re onstru tion is obtained when a little more

294

omplexsour emodel(i.e.,thesingleCW-Model)isusedasit anbeobservedinFig. 4(b) 295

and inferred from the values of the error gures (whi h quantify the qualitative imaging 296

of the s atterer undertest) given in Tab. II and dened as follows 297

ρ

(q)

=

rh

x

(q)

c

_(Iopt)

− x

(q)

c

ref

i

2

+

h

y

c

(q)

_(Iopt)

− y

(q)

c

ref

i

2

λ

q = 1, ..., Q

(I

opt

)

(18) 298

∆

(q)

=











L

(q)

_(I

_opt

₎

− L

(q)

_ref







L

(q)

_ref







× 100 q = 1, ..., Q

(I

opt

)

(19) where the sub-s ript ref refers tothe a tualprole.

299

A ording to the indi ations drawn fromthese experiments,whi h point out that even a 300

rough representation of the in ident eld signi antly benets the inversion of the s at-301

tered eld data, the su essive pro edural step will be aimed at rening the numeri al 302

modelof the ele tromagneti sour e to further improve the ee tiveness of the retrieval 303

pro ess. However, it should be noti ed out that using a wrong, even though omplex, 304

model might a tually degrade the re onstru tion, thus great are is needed in dening 305

the most suitable omplex model. In order to point out su h a on ept, the problem 306

has beenstudied onsideringthe previouss atteringgeometry, butusingnumeri al mea-307

sured data with a ontrollable degree of noise. More in detail, the following analysis 308

has been arried out. Dierent ele tromagneti sour es have been onsidered to illumi-309

nate the s enarioundertest (i.e.,PW-Sour e, CCW-Sour e,and DCW-Sour e)and 310

3

domain

E

v

inc



x

m

(v)

, y

m

(v)



,



x

m

(v)

, y

m

(v)



∈ D

M

,varioussour emodels (i.e.,PW-Model, 312

CCW-Model, and DCW-Model) have been synthesized. Then, a noise hara terized 313

bya

SNR = 20 dB

hasbeensuperimposedtothedataandthere onstru tionpro esshas 314

been arried out starting from the dierent sour e models previously determined. The 315

obtainedresultsintermsofqualitative(??)-(??)andquantitativeerrorgures

ξ

(j)

dened 316 as 317

ξ

(j)

=

R

X

r=1

1

N

_(r)

(j)

N

_(r)

(j)

X

n

(r)

=1

(

τ (x

n

_(r)

, y

n

_(r)

) − τ

ref

(x

n

_(r)

, y

n

_(r)

)

τ

ref

_(x

n

(r)

, y

n

(r)

)

)

× 100

R = S

opt

(20) where

N

(j)

(r)

an range over the whole investigation domain (

j ⇒ tot

), or over the area 318

where the a tuals atterer is lo ated (

j ⇒ int

), or over the ba kground belonging tothe 319

investigationdomain(

j ⇒ ext

), arereportedinTab. III.Asexpe ted,the useofamodel 320

orresponding to the a tual sour e turns out to be the most suitable hoi e and more 321

omplex modeling ause larger errors. As an example, let us onsider the PW-sour e. 322

When theproleretrievalisperformedusingthePW-model thenthere onstru tion error

323

is equal to

ξ

tot

= 0.30

. Otherwise,

ξ

(DCW −M odel)

tot

= 13.30

and

ξ

(CCW −M odel)

tot

= 20.53

.

324

Similar on lusions hold true also for other illuminations and sour e models in terms of

325

quantitative error gures,as well.

326

Consequently,the more omplexsour e ongurations des ribed inSe tion3, whi h

on-327

siderthesuperpositionofplanewavesorof ylindri alwaves, havebeentakenintoa ount

328

in ordertodene the mostsuitablesour e model. Insu h aframeworksin e the

numeri-329

aldes ription ofthe a tualsour e inthe real measurementsetup isonlypartiallyornot

330

generally available, the optimalmodelhas tobe dened by lookingfor the most suitable

331

numberof the unknown sour e oe ients,

S

, and orrespondingvalues,

A

s

,

s = 1, ..., S

.

332

For ea h of the sour e models,

S

has been hosen by lookingfor the onguration that

333

provides a satisfa tory mat hing between measured and numeri ally- omputed values of

334

the in ident eld in the observation domain. Su h a mat hing has been evaluated by

335

omputing the following parameter

µ = (V M

(v)

)

−1

P

V

v=1

P

M

(v)

m

(v)

=1

h

Re

n

E

v

inc



x

m

(v)

, y

m

(v)

o

− Re

n

E

g

v

inc



x

m

(v)

, y

m

(v)

oi

2

+

h

Im

n

E

v

inc



x

m

_(v)

, y

m

_(v)

o

− Im

n

E

g

v

inc



x

m

_(v)

, y

m

_(v)

oi

2



1

2

(21) 338

where

Re {·}

and

Im {·}

stand for the real and imaginary part, respe tively, and the

339

super-s ript

e

indi ates anumeri ally-estimated quantity.

340

In Fig. 5, the behavior of the mat hing parameter is displayed for dierent sour e

341

models. As an be observed,

µ

redu es when

S

in reases. Thus, the optimal number of

342

sour e oe ients,

S

opt

,hasbeenheuristi ally-denedasthevaluebelongingtoastability

343

region. Consequently, the optimal values have been set to:

S

(P W −M odel)

opt

= 20

(where 344

µ ≃ 4 × 10

−4

)and

S

(CCW −M odel)

opt

= 11

(where

µ ≃ 10

−4

). Theamplitudesofthe weighting

345

sour e oe ients are shown in Fig. 6. The magnitudes of the CCW-Model oe ients

346

[Fig. 6(b)℄areverylargewhen omparedtothoseofthesinglePW-Model orsingle

CCW-347

Model. Asexpe ted,the orrespondingradiated-elddistributionsinsidetheinvestigation

348

domain

D

I

[Figs. 7( ),(d)℄ turn out to be una eptable (for omparison purposes, the

349

plot of the in ident ele tri eld omputed by means of the single CCW-Model is given

350

in Figs. 7(e),(f)). Moreover, Figs. 7(a),(b) show howeven the in identeld synthesized

351

bymeansofthe PW-Model presentsratherhighvalues withrespe ttothe distributionof

352

Figs. 7(e),(f). Sin ethein identeldistheguessvaluefortheoptimizationoftheinternal

353

eld, a ompletelywrongstartingdistributionmay onsiderablyae tthe wholeretrieval

354

pro edure. A ordingly, the adopted inversion strategy is not able to orre tly estimate

355

neitherthe shapenor thediele tri distributionofthe unknown s atterer (Fig. 8). As far

356

as the ase related to the PW-Model is on erned, it should be noted that the iterative

357

pro ess isstopped atthe fourth step(Tab. I)and the qualityof the re onstru ted prole

358

(Fig. 8(a))turns out to be strongly redu ed (if ompared to that of Fig. 4(a)) in terms

359

of qualitativeas wellas quantitativeimaging. Similarindi ations an be drawn fromthe

360

analysis of the retrieved distribution obtained with the CCW-Model. However, redu ing

361

the number ofterms inthe expansion ouldlead tobetter resultslike,forexample,those

362

presented in the spe ial se tion [Belkebir and Saillard, 2001℄ and those obtained in this

used in the proposed experiments.

365

The obtained dis ouraging results an be properly motivated by observing the

singular-366

valuesspe trum(Fig. 9)andby omputingthe onditionnumber

η

ofthelinearmatrix

op-367

erator

[G]

(denedasfollows

η =

max

p

{σ

p

}

min

p

{σ

p

}

),whi h learly pointoutanintrinsi instability

368

of the system and the ill- onditioningof the problem. Inmore detail, the ill- onditioning

369

index turns out to be equal to

η

(P W −M odel)

_{= 41.07}

and to

η

(CCW −M odel)

_{= 5.62 × 10}

7

, 370 respe tively. 371

Apossiblesolutionforsuitablydeningthesour emodeland, onsequently,forimproving

372

the resolution a ura y of the retrieval pro ess (alternative to employ a trun ated-SVD

373

regularizationalgorithmassuggestedbythestep-likebehaviorofthesingular-values

spe -374

trum), isto dene aspatially-distributed line-sour e modelas des ribed in Se t. 3.

375

A ordingtothepro edurefor hoosingthenumberaswellasthemagnitudeofthesour e

376

weights previously des ribed, a reasonable onguration is

S

(DCW −M odel)

opt

= 15

(Fig. 5)

377

with the oe ients distributed as shown in Fig. 10(a). For ompleteness, in order to

378

givean idea of the tting between measured and omputed data, Figs. 10(b)-( )display

379

the values of the amplitude and phase of the radiated-eld omputed in the observation

380

domain. Moreover, Fig. 11 gives a gray-level representation of the in ident ele tri eld

381

synthesized in the investigationdomain.

382

The use of su h a model for the in ident eld allows a signi ant improvement in the

383

re onstru tion. Su haresult anbeappre iatedinFig. 12wherethegray-level

represen-384

tation of the obje tfun tion isgiven. In parti ular, forthis representative onguration,

385

also the intermediate re onstru tions [Figs. 12(a)-( )℄ of the multi-s aling pro ess are

386

reported in order to show how the prole improves during the iterative pro edure. As

387

it an be noti ed, even though the omputationaldomain isnot nely dis retized atthe

388

st step [Fig. 12(a)℄,the IMSA iterativelyin reases the resolution inthe RoI inorder to

389

obtain an a urate dis retization at the onvergen e step [Fig. 12( )℄ where a

meaning-390

ful prole is obtained. As a matter of fa t, the lo alization as well as the dimensioning

391

error of the onvergen e step [Fig 12( )℄redu es with respe t to the other sour emodels

392

(

ρ

(DCW −M odel)

_{= 0.045λ}

0

,

∆

(DCW −M odel)

_{≈ 9}

-Tab. II)and thehomogeneityofthe a tual

su h anapproa hwith respe t tothe othersour e-synthesis modalities is on erned,it is

395

mainly motivated by the faithful and stable reprodu tion [Figs. 10(b)-( )℄ of the a tual

396

values of the eld measured inthe observation domain.

397

To further assess the robustness and the ee tiveness of the IMSA, by validating the

398

radiated-eldsynthesisaswell,these ondexample onsidersamultiple-s attererss enario

399

(twodielTM_8f.exp -[Belkebir andSaillard,2001℄). Under thesame assumptionsofthe

400

previousexampleintermsofmeasures,radiationfrequen y,andviewsaswellasextension

401

and partitioning ofthe investigationdomain,two diele tri (

τ

(q)

_{= 2.0 ± 0.3}

,

q = 1, ..., Q

,

402

Q = 2

) ir ular(

L

(q)

ref

= 30 mm

indiameter) ylinders are pla ed

90 mm

from ea h other

403

[Fig. 2(b)℄.

404

Fig. 13 shows the results of the re onstru tion pro ess in orresponden e with dierent

405

sour e models. As an be seen, whatever the stable sour e synthesis the two targets

406

are orre tly lo ated and dimensioned with a satisfa tory a ura y. Certainly, the more

407

sophisti atedsynthesis approa h(DCW-Model -

S = 15

)allows toobtain abetter

re on-408

stru tion as onrmed by the geometri parameters of the retrieved proles resumed in

409

Tab. IV. In order toshow the apabilities of the IMSA in estimating the lossless nature

410

ofthe diele tri s atterers, there onstru tion orrespondingtothe DCW-Model hasbeen

411

run using a blind inversion s heme, that is without a-priori informationof its

hara ter-412

isti s. Su h assumption does not exploit the alternative denition of the solution spa e,

413

whi h allows to re onstru t only the real part of the obje t fun tion. A ordingly, Fig.

414

13(d) points out that the minimum of the imaginary part of the obje t fun tion is

0.08

415

( orresponding to

σ = 1.78 × 10

−3 S

m

).

416

Finally, inorder to omplete the validationof the approa h,the lastexample deals with

417

a metalli stru ture. The s atterer is an U-shaped metalli ylinder [Fig. 2( )℄ and the

418

re onstru tion is performed starting from the omplete data olle tion of the dataset

419

uTM_shaped.exp [Belkebir and Saillard, 2001℄at the working frequen y of

f = 4 GHz

.

420

A ordingtothestrategyproposedin[VandenBergetal.,1995℄,onlytheimaginarypart

421

of the obje t fun tion has been retrieved onsidering a lower bound in the re onstru ted

422

ontrast and if at some iterationthe estimated

Im {τ (x, y)}

is lowerthan

τ

max

Im

= −15.0

,

then the ontrast is repla ed by

τ

max

Im

. As a result, the imaginary part of the retrieved

424

proleinthe ongurationwith theDCW-Model for thesynthesis ofthe radiatedeld, is

425

depi ted inFig. 14. Atthe onvergen e step(

I

opt

= 4

),the re onstru tion learly reveals

426

that we are dealing with a U-shaped target. The outer and the inner ontour of the U

427

are well reprodu ed (even though little artifa ts appear) onrming the ee tiveness of

428

approa h inshaping and lo ating diele tri aswell metalli s atterers.

429

5 Con lusions 430

The Iterative Multi-S aling Approa h has been tested against experimentally-a quired 431

data byfo usingtheattentiononitsrobustness asregardsdierentmathemati almodels 432

usedtosynthesizethein identele tri eld. Theee tivenessoftheiterativeminimization

433

of the ost fun tional in re onstru ting unknowns s atterers presents a ertaindegree of

434

sensitivity to the model of the in ident eld used to formalize the onstraint stated by

435

the StateEquation. By onsideringamore omplexapproximation model(DCM-Model),

436

satisfa tory lo alizations and re onstru tions have been arried out by indi ating the

437

positiveee t of asuitable synthesis methodology on the inversion pro ess. 438

However, even thoughana urate approximationmodelgenerallymightresultina more 439

a urate re onstru tion, whi h omplex model is more appropriate for the in ident eld 440

may depend on the measurement setup, espe ially the mi rowave sour e onguration. 441

For example, for simple plane-wave in ident eld, using the PW-model might redu e 442

artifa ts whi h result from measurement noise. So future investigations are needed by 443

onsideringotherexperimentaldatasets( urrently not-available,but underdevelopment) 444

to generalize the on lusions of su han analysis. 445

Moreover,theresultsofthenumeri alanalysis arriedoutinthepaperandthe omparison

446

with there onstru tions obtainedinthe relatedliteraturesuggestthatimproved imaging

447

te hniques (e. g., multi-frequen y te hniques) or additional regularization terms may

448

probablydiminishtheimpa tofthein identeldmodel. Sin ethispointhasnot dire tly

449

investigated other resear hes will be aimed at further improving the ee tiveness of the

450

IMSA by onsidering multi-frequen y strategies, further regularization terms and more

451

ee tiveoptimizationalgorithmsfortheminimizationofthemulti-resolution ostfun tion

455

Baussard, A., E. L. Miller, and D. Lesselier(2004a),Adaptivemultis aleapproa h

456

for2Dmi rowavetomography,URSI - InternationalSymposium onEle tromagneti

The-457

ory,Pisa, Italia,pp. 1092-1094.

458

Baussard, A., E. L. Miller, and D. Lesselier (2004b), Adaptive multis ale

re on-459

stru tion of buriedobje ts, Inverse Problems,20,S1-S15.

460

Belkebir, K., and M. Saillard (2001), Spe ial se tion: Testing Inversion Algorithms 461

against Experimental Data, InverseProblems, 17,1565-1702. 462

Bertero, M., C. De Mol, and E. R. Pike (1995), "Linear inverse problems with 463

dis retedata. I: Generalformulationand singularsystem analysis,"Inverse Problems, 1, 464

301-330. 465

Bertero, M., and P. Bo a i (1998), Introdu tion to Inverse Problem in Imaging, 466

IoP Publishing, Philadelphia. 467

Bu i, O. M., and G. Fran es hetti(1989),On the DegreesofFreedomofS attered 468

Fields, IEEE Trans. AntennasPropagat.,37, 918-926. 469

Bu i, O. M., L. Cro o, T. Isernia, and V. Pas azio (2000a), Wavelets in non-470

linear inverse s attering, Pro . IEEE Geos ien e and Remote Sensing Symp., IGARSS-471

2000, 7, 3130-3132. 472

Bu i, O. M., L. Cro o, and T. Isernia (2000b), An adaptive wavelet-based ap-473

proa h for non destru tive evaluation appli ations, Pro . IEEE Antennas Propagation 474

Symp., APS-2000, 3,1756-1759. 475

Caorsi, S., M. Donelli, D. Fran es hini, and A. Massa (2003), A new methodol-476

ogy based on an iterative multi-s aling for mi rowave imaging, IEEE Trans. Mi rowave 477

Theory Te h., 51,1162-1173. 478

Caorsi,S., M.Donelli,andA.Massa(2004a),Analysisofthestabilityandrobustness

479

ofthe iterativemultis alingapproa hformi rowaveimagingappli ations,Radio S i.,39,

480

1-17.

481

Caorsi, S., M. Donelli, and A. Massa (2004b),Lo ation, dete tion, and imaging of 482

multiples atterersbymeansoftheiterativemultis alingmethod,IEEETrans. Mi rowave 483

Colton, D., and R. Krees (1992), Inverse a ousti s and ele tromagneti s attering 485

theory, Berlin, Germany: Springer-Verlag. 486

Daniels, D. J. (1996), Surfa e penetrating radar, IEE Ele tron. Comm. Eng. J., 8, 487

165-182. 488

Denisov, A. M. (1999), Elements of theory of inverse problems, Utre ht, The Nether-489

lands: VSP. 490

Devaney, A. J., and G. C. Sherman (1982), Nonuniqueness in inverse sour e and 491

s attering problems, IEEE Trans. Antennas Propagat., 33, 1034-1037. 492

Dubey,A.C.(1995),Dete tionte hnologyforminesandmineliketargets,Eds. Orlando, 493

FL. 494

Groets h, C. W. (1993),Inverse Problems in Mathemati al S ien es,Wiesbaden, Ger-495

many: Vieweg. 496

Hoole, S. R. H. (1991),Inverse problemmethodologyand nite elementsinthe identi-497

 ations of ra ks, sour es, materials,and their geometry inina essible lo ations,IEEE 498

Trans. Magn., 27, 3433-3443. 499

Isernia, T., V. Pas azio, and R. Pierri (2001), On the lo al minima problem in a 500

tomographi imagingte hnique, IEEE Trans. Geos i. Remote Sensing, 39, 1596-1607. 501

Jones, D. S. (1964),The Theory of Ele tromagnetism,Oxford,U.K.: PergamonPress. 502

Kleinman, R. E., and P. M. Van den Berg(1992), A modiedgradient method for 503

two-dimensionalproblems in tomography, J. Comput. Appl. Math., 42,17-35. 504

Liu, Q. H., Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A. Ybarra, L. W. 505

Nolte, and W. T. Joines (2003),A tive mi rowaveimaging 1-2D forward and inverse 506

s attering methods, IEEE Trans. Mi rowave Theory Te h., 50, 123-133. 507

Louis, K. (1992), Medi al imaging: state of the art and future development, Inverse 508

Problems,8, 709-738. 509

Marklein, R., K. Balasubramanian, A. Quing,and K. J. Lagenberg(2001), Lin-510

ear andnonlinear iteratives alarinversionof multi-frequen ymulit-bistati experimental 511

ele tromagneti s attering data, Inverse Problems, 17,1565-1702. 512

Massa, A.(2002),Geneti algorithmbased te hniques for2Dmi rowaveinverse s atter-513

Pandalai,Transworld Resear h Network Press, Trivandrum, India. 515

Miller, E. L., and A. S. Willsky (1996a), A multis ale, statisti ally based inversion 516

s heme for linearized inverse s attering problems, IEEE Trans. Geos i. Remote Sensing, 517

34, 346-357. 518

Miller, E.L., andA.S.Willsky(1996b),Wavelet-basedmethodsfornonlinearinverse 519

s attering problem using the extended Born approximation, Radio S i., 31,51-65. 520

Natterer, F. (1986), Numeri al treatment of ill-posed problems, in Inverse Problems, 521

Ed. G.Talenti, Le ture Notes in Mathemati s, p. 1225. 522

Ri hmond, J. H. (1965), S attering by a diele tri ylinder of arbitrary ross se tion 523

shape, IEEE Trans. Antennas Propagat., 13, 334-341. 524

Steinberg, B. D. (1991), Mi rowave imaging te hniques, New York: Wiley. 525

Testorf,M., and M.Fiddy (2001),Imagingfromreals atteredelddatausingalinear 526

spe tral estimation te hniques, Inverse Problems, 17,1565-1702. 527

Vogel, C. R. (2002), Computational Methods for Inverse Problems, Philadelphia, PA: 528

SIAM. 529

•

Figure 1. Geometry of the problem.

532

•

Figure 2. Numeri al Experiments: (a) o- entered homogeneous ir ular

ylin-533

der (Real dataset Marseille [Belkebir and Saillard, 2001℄ - dielTM_de 8f.exp),

534

(b)twohomogeneous ir ular ylinders(RealdatasetMarseille [Belkebirand

Sail-535

lard,2001℄-twodielTM_8f.exp),and( )U-shapedmetalli ylinder(Realdataset

536

Marseille [Belkebir and Saillard,2001℄ - uTM_shaped.exp).

537

•

Figure 3. Comparisonsbetween the in ident eld measured in

D

M

and the values

538

synthesized by means of the PW-Model ((a) amplitude and (b) phase), and

CCW-539

Model (( ) amplitudeand (d) phase).

540

•

Figure 4. Re onstru tions of an o- entered homogeneous ir ular ylinder (Real

541

dataset Marseille [Belkebir and Saillard, 2001℄ - dielTM_de 8f.exp) a hieved

542

at the onvergen e step of the inversion pro edure by modeling the radiated eld

543

through (a)the single PW-Model and (b) the single CCW-Model.

544

•

Figure 5. Fitting between omputed and measured values of the radiated eld

545

in the observation domain versus various numbers of sour e oe ients,

S

,and for

546

dierentsour e models.

547

•

Figure 6. Behaviorofweighting sour e oe ientsasafun tionofthe index

w

for

548

(a) the PW-Model (

S = 20

) and for(b) the CCW-Model (

S = 11

).

549

•

Figure7. Plotsoftheradiatedelds(

V = 1

) omputedbymeansofthePW-Model

550

(

S = 20

) (amplitude (a) and phase (b) distributions), the CCW-Model (

S = 11

)

551

(amplitude ( ) and phase (d) distributions), and the single CCW-Model (

S = 1

)

552

(amplitude (e) and phase (f) distributions).

553

•

Figure 8. Re onstru tions of an o- entered homogeneous ir ular ylinder (Real

554

dataset Marseille [Belkebir and Saillard, 2001℄ - dielTM_de 8f.exp) a hieved

555

at the onvergen e step of the inversion pro edure by modeling the radiated eld

556

through (a)the PW-Model (

S = 20

) and (b) the CCW-Model (

S = 11

).

•

Figure 9. Normalizedbehavior of the singularvalues of

[G]

for (a) the PW-Model

558

(

S = 20

) and for (b)the CCW-Model (

S = 11

).

559

•

Figure 10. Radiated-eld modeling: DCW-Model (

S = 15

). (a) Behavior of

560

weighting sour e oe ientsasa fun tionof the index

w

. Comparison between the

561

in identeld measuredin

D

M

andthe numeri ally- omputedvalues ((b) amplitude

562

and ( ) phase).

563

•

Figure 11. Plots of the radiated eld (

V = 1

) omputed by means of the

DCW-564

Model (

S = 15

) (amplitude(e)and phase(f) distributions).

565

•

Figure 12. Re onstru tion of ano- entered homogeneous ir ular ylinder(Real

566

dataset Marseille [Belkebir and Saillard, 2001℄ - dielTM_de 8f.exp)a hieved at

567

(a) i=1, (b) i=2and ( ) at the onvergen e step (i=3) of the inversion pro edure

568

by modelingthe radiated eld through the DCW-Model (

S = 15

).

569

•

Figure 13. Re onstru tions of two homogeneous ir ular ylinders (Real dataset

570

Marseille [Belkebir andSaillard,2001℄-twodielTM_8f.exp)a hievedatthe

on-571

vergen e step ofthe inversionpro edurebymodeling theradiatedeld through (a)

572

the single PW-Model, (b) the single CCW-Model and the DCW-Model (

S = 15

)

573

[( ) real part and (d)imaginary part℄.

574

•

Figure 14. Re onstru tion of an U-shaped metalli ylinder (Real dataset

Mar-575

seille [Belkebir and Saillard, 2001℄ - uTM_shaped.exp) a hieved at the

onver-576

gen e step of the inversion pro edure by modeling the radiated eld through the

577

DCW-Model (

S = 15

).

•

Table I. Re onstru tion of an o- entered homogeneous ir ular ylinder (Real 580

datasetMarseille [Belkebir and Saillard,2001℄ -dielTM_de 8f.exp)-Estimated 581

geometri al parameters. 582

•

Table II. Re onstru tion of an o- entered homogeneous ir ular ylinder (Real 583

dataset Marseille [Belkebir and Saillard, 2001℄ - dielTM_de 8f.exp)- Error g-584

ures. 585

•

TableIII.Re onstru tionofano- enteredhomogeneous ir ular ylinder(

SNR =

586

20 dB

) for dierent illuminationsand onsidering variousele tromagneti sour es -587

Quantitativeerror gures [(a)

ξ

tot

, (b)

ξ

int

and ( )

ξ

ext

℄.

588

•

Table IV. Re onstru tion of two homogeneous ir ular ylinders (Real dataset 589

Marseille [Belkebirand Saillard,2001℄-twodielTM_8f.exp)-Estimated geomet-590 ri al parameters (

d

(I

opt

)

=

rn

x

(1)

c

_(Iopt)

− x

(2)

c

_(Iopt)

o

2

+

n

y

(1)

c

_(Iopt)

− y

c

(2)

_(Iopt)

o

2

). 591 592

594 595 596

(x , y )

_m

_m

D

_M

D

_I

φ

_v

θ

inc

x

y

Observation Domain

Investigation Domain

597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612

Figure 1 - D. Fran es hini et al., On the Ee ts of the Ele tromagneti ...

(x , y )

m

m

r

_m

r

_s

x

y

30 mm

θ

_l

(a)

(x , y )

m

m

r

_m

r

_s

x

y

θ

l

90 mm

(b)

(x , y )

_m

_m

r

_m

r

_s

x

y

θ

_l

40 mm

50 mm

80 mm

( ) 614 615 616 617

Figure 2 - D. Fran es hini et al., On the Ee ts of the Ele tromagneti ...

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0

5

10

15

20

25

30

35

40

45

50

|E

v

inc

(x

m

,y

m

)|,v=1

m

PW-Model (S=1)

PW-Model (S=20)

Actual Values

(a)

-4

-3

-2

-1

0

1

2

3

4

0

5

10

15

20

25

30

35

40

45

50

Phase[E

v

inc

(x

m

,y

m

)],v=1

m

PW-Model (S=1)

PW-Model (S=20)

Actual Values

(b) 619 620 621 622 623 624

Figure 3(I) - D. Fran es hiniet al., On the Ee ts of the Ele tromagneti ...

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

5

10

15

20

25

30

35

40

45

50

|E

v

inc

(x

m

,y

m

)|,v=1

m

CCW-Model (S=1)

CCW-Model (S=11)

Actual Values

( )

-4

-3

-2

-1

0

1

2

3

4

0

5

10

15

20

25

30

35

40

45

50

Phase[E

v

inc

(x

m

,y

m

)],v=1

m

CCW-Model (S=1)

CCW-Model (S=11)

Actual Values

(d) 626 627 628 629 630 631

Figure 3(II) - D. Fran es hiniet al., On the Ee ts of the Ele tromagneti ...

x

y

2.6

Re {τ (x, y)}

0.0

(a)

x

y

2.6

Re {τ (x, y)}

0.0

(b) 634 635 636 637 638

Figure 4 - D. Fran es hini et al., On the Ee ts of the Ele tromagneti ...

641 642 643 644

0

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0

5

10

15

20

25

µ

S

DCW-Model

PW-Model

CCW-Model

645 646 647 648 649 650 651 652 653 654 655 656 657 658 659

Figure 5 - D. Fran es hini et al., On the Ee ts of the Ele tromagneti ...