An innovative multi-source strategy for enhancing the reconstruction capabilities of inverse scattering techniques

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UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38123 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

AN INNOVATIVE MULTI-SOURCE STRATEGY FOR ENHANCING

THE RECONSTRUCTION CAPABILITIES OF INVERSE

SCATTERING TECHNIQUES

F. Caramanica, and G. Oliveri

January 2011

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Re onstru tion Capabilities of Inverse S attering T

e h-niques

Federi o Caramani aand Gia omo Oliveri

Department of Information and Communi ation Te hnology

University of Trento-Via Sommarive,14 - 38050Trento - ITALY

Tel.: +39 0461 882057;Fax: +39 0461 882093

E-mail: federi o. aramani adisi.unitn.it,gia omo.oliveridisi.unitn.it

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Re onstru tion Capabilities of Inverse S attering T

e h-niques

Federi o Caramani aand Gia omo Oliveri

Abstra t

A tivemi rowaveimagingte hniquesareaimedatre onstru tinganunknownregion

undertestbymeansofsuitableinversionalgorithmsstartingfromthemeasurement

of thes attered ele tromagneti eld. Withinsu ha framework,this paperfo uses

onaninnovativestrategy thatfullyexploitsthe informationarisingfromthe

illumi-nation ofthe investigationdomain withdierent ongurations aswell asradiation

patterns of the probing sour es. The proposed approa h an be easily integrated

with multiview te hniques and, unlike multifrequen y methods, it doesnot require

additive a-priori information on the diele tri nature of the s atterer under test.

A large numberof numeri al simulations onrmsthe ee tiveness of theinversion

strategy as well as its robustness with respe t to noise on data. Moreover, the

re-sultsofa omparative study withsingle-sour e methodologies furtherpointout the

advantagesand potentialities ofthenew approa h.

Keywords:

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In the last years, the s ienti ommunity has addressed a growing interest to the

de-te tion and imaging of unknown obje ts lo ated in ina essible domains by means of

ele tromagneti elds at mi rowave frequen ies. As a matter of fa t, the propagation of

anele tromagneti waveinthe mi rowaverange issigni antly ae tedbythe

hara ter-isti s of the medium. Therefore, itisprotable toexploit su h aphenomenon in orderto

senseanunknowns enarioinanon-invasivefashion. Towardsthis end,several resear hes

havebeen pursued inthe frameworkof subsurfa e monitoring[1℄[2℄, non-destru tive

eval-uation and testing [3℄[4℄[5℄, and biomedi al diagnosti s [6℄[7℄[8℄[9℄.

Whatever the appli ation, a mi rowave imaging setup onsists of a probing sour e that

sensesanina essibleinvestigationdomainandaset ofre eivers olle tingsamplesofthe

ele tromagneti elds atteredbythestru tureundertest. Afterthemeasurementphase,

a post-pro essing of the olle ted data is performed to provide a faithful re onstru tion

of the s enario under test. Su h a retrieval pro ess presents some intrinsi drawba ks

[10℄[11℄, whi h make the inversion of the s attering data hard to ope with. Firstly, if

a omplete and quantitative re onstru tion of the ele tromagneti properties is desired,

multiple s attering ee ts annot be negle ted and a full non-linear model should be

onsidered. Moreover, the ill-posedness and the ill- onditioning of the problem [12℄ are

key-issuestobe arefullyaddressed. They areduetothe la kofinformation omingfrom

measured s attering data. During the imaging pro ess, a huge amount of parameters

has toberetrieved startingfrom a limitednumberof independent measurements. Thus,

if neither a-priori information are available nor other physi al onstraints are imposed,

there isthe need to olle tother informationby meansof suitablete hniques besidesthe

use of ee tiveretrievalte hniques [13℄[14℄[15℄[16℄[17℄[18℄[19℄.

Withinsu haframework,dierentstrategiesaimedatin reasingthe information ontent

of s atteringdatahavebeen proposed intherelatedliterature. Letus onsider the

multi-view strategy proposed in [20℄ where the s atterer is illuminated from dierent angular

dire tions in order to give an overview of the s enario under test. As determined in

[21℄, su h a te hnique allows asigni ant in reasing of independent s attering data with

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Anotherwidely-used ountermeasureresortstoamulti-frequen yapproa h[22℄[23℄[24℄[25℄[26℄

orafrequen y-hoppings heme[27℄[28℄. Asfarasthe olle tableinformationis on erned,

the number of independent data ertainly in reases sin e dierent and omplementary

s attering ee ts are ex ited by a set of in ident ele tromagneti elds at dierent

fre-quen ies. However,tofullyexploitsu hanenhan ementofthe knowledgeonthes enario

at hand, some a-priori assumptions have to be done about the dispersion model of the

diele tri hara teristi sof the s atterer [23℄.

The informative ontent hasbeenalsoenhan edbyexploitingthe so- alleda quired

in-formation duringtheinversion[29℄[30℄orusingdierentpolarizations[31℄[32℄[33℄[34℄[35℄[36℄[37℄.

In this paper, an innovative methodology aimed at in reasing the amount of s attering

data(avoidingfurthera-prioriassumptionsontheinvestigationdomain)isproposed. The

multi-sour e (MS) approa h supposes the investigation domain illuminated by dierent

probing sour es, ea h of them with a proper (and dierent) radiationpattern, to indu e

dierent s attering intera tions able to show dierent aspe ts of the s atterer under

test. Integrated with a multi-view strategy, the exploitation of the sour e diversity

(through the denition of a suitablemulti-sour e/multi-view ost fun tion) enlargesin a

non-negligible fashion the number of retrievable unknowns by enhan ing the robustness

of the imaging pro ess against the noise and the stability of the inversion pro edure as

wellasthe re onstru tiona ura y. Thearisingredu tionof theratiobetween dimension

of the spa eofthe unknownsand that ofdataalsoimpliesade reased sensitivity tofalse

solutions [38℄ leadingto amore tra table optimization problem.

In the following, after the mathemati al formulation of inverse s attering intera tions

arising in the mi rowave imaging pro ess, the multi-sour e te hnique is introdu ed and

des ribed (Se t 2). Se tion 3 is devoted to the numeri al assessment of the ee tiveness

androbustnessoftheproposedstrategy. Towardsthisaim,sele tednumeri alexperiments

on erned with layered as well as omplex s atterers willbe dis ussed. To point out the

enhan ed re onstru tion a ura y, the results of a omparative study will be presented

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Let us onsider a two-dimensional s enario for mi rowave imaging of ylindri al bodies,

ˆz

being the symmetry axis. With referen e to Figure 1, the ele tromagneti properties of the investigation domain

D

I

are des ribed through the unknown ontrast fun tion

τ (x, y) = ε

r

(x, y) − 1 − j

σ(x,y)

_{2πf ε}

₀

,

(x, y) ∈ D

I

. The ba kground is known and it is assumed to be homogeneousand hara terized by

τ

0

.

Toimagethe s enarioundertestandre onstru t thediele tri prole

τ (x, y)

,the investi-gation region is sensed with

S

ele tromagneti sour es radiatingdierent beampatterns. Moreover, ea h sour e su essively probes

D

I

from

V

dierent angular positions

θ

v

a - ording to a multi-view approa h [20℄. The radiated in ident ele tri eld is denoted by

E

inc

v,s

(x, y)ˆz

,

v = 1, ..., V

,

s = 1, ..., S

.

The ee ts of the intera tions between in ident elds and s atterers are revealed by

ol-le tingasetof measurementsofthe s atteredele tri eld. Atea hsensor lo atedwithin

the observation domain

D

O

, the following set of samples is available:

E

v,s

scatt

(x

v

m

, y

m

v

)ˆz

,

v = 1, ..., V

,

m

v

_{= 1, .., M}

v

,

s = 1, ..., S

. Analyti ally,the relationbetween s atterers and diused eld an beexpressed through the integral Data equation [10℄

E

scatt

v,s

(x

v

m

, y

v

m

) = j

k

2

0

4 Z

D

I

τ (x

′

, y

′

)E

tot

v,s

(x

′

, y

′

)G

2D

(x

v

m

, y

v

m

|x

′

, y

′

)dx

′

dy

′

(x

v

m

, y

v

m

) ∈ D

O

(1)

where

k

0

is the ba kground wavenumber and

G

2D

is the two-dimensional free-spa e Green's fun tion [39℄. Analogously, the known in ident eld

E

v,s

inc

(x, y)ˆz

in

D

I

is related to the s atterers properties asfollows (State integral equation)

E

inc

v,s

(x, y) = E

v,s

tot

(x, y) − j

k

2

0

4 Z

D

I

τ (x

′

, y

′

)E

tot

v,s

(x

′

, y

′

)G

2D

(x, y |x

′

, y

′

)dx

′

dy

′

(x, y) ∈ D

I

(2)

Unfortunately, su h a des ription is mathemati ally tra table only after a suitable

dis- retization. A ording to the Ri hmond's pro edure [40℄, by onsidering

N

re tangular basis fun tion

Ω

n

(x, y) =











1 (x, y) ∈ D

I

(n)

0 (x, y) /

∈ D

_I

(n)

(3)

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(

D

(n)

I

being the

n

-th partition of the investigation domain), the re onstru tion pro ess moves towards the retrieval of the dis retized representation of the unknowns

τ (x, y) =

N

X

n=1

τ

n

Ω

n

(x, y)

E

tot

v,s

(x, y) =

N

X

n=1

E

n

v,s

Ω

n

(x, y).

Thus,theproblemunknownsturnouttobe

τ

n

,

E

v,s

n

,

n = 1, ..., N

,

v = 1, ..., V

,

s = 1, ..., S

. The numberof the expansion oe ients denes the dimension of the unknown-spa e

U

. Typi aldrawba ksof the inverse s atteringproblemathand are itsill-posednessand the

ill- onditioning[12℄, bothduetothelimitedamountoftheavailableand olle table

infor-mation,thatmakethe inversionofs atteringdatainstableandina uratewithoutproper

ountermeasures. Consequently, the inverseproblemhas tobe arefullymanaged by

pro-perlydening aleast-squaresolutionand asuitableregularizationstrategy. Towards this

end,awidelyadoptedte hnique onsistsinimposingasetof onstraintsrelatedtoinverse

s attering data or to the a-priori knowledge to be satised in a least-square fashion by

minimizinga suitable ost fun tion.

It is wellknown that to ome to a well-posed and well- onditionedproblem, a ne essary

ondition (although not su ient) is that

U

be less than the essential dimension of the s attering data

I

or the number of arising independent onstraints

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. Therefore,

mana-gings atteringdata(and orresponding onstraints)aswellastheexploitationofa-priori

informationisakey issue,sin e itstronglyae ts theoverall inversionpro edureandthe

possibility of obtaining a faithful re onstru tion of the a tual prole. To ee tively

ad-dress su h an issue, an innovative approa h that fully exploits s attering data from a

multi-sour esystem isformulatedinthe following.

Let us onsider a standard multi-view arrangement where the

s

-th sour e probes the investigationdomain. Asfarasthe ttingwiththe arisings atteringdata [

E

s,v

scatt

(x

v

m

, y

m

v

)

and

E

s,v

inc

(x

n

, y

n

)

℄is on erned,thesolutionisrequestedtosatisfythefollowing onstraints

φ

(s)

_Data

{τ

n

, E

n

v,s

} =

C

_Data

(s)

{τ

n

, E

n

v,s

}

c

(s)

_Data

= 0

φ

(s)

State

{τ

n

, E

n

v,s

} =

C

_State

(s)

{τ

n

, E

n

v,s

}

c

(s)

_State

= 0

(4)

C

_Data

(s)

{τ

n

, E

n

v,s

} =

V

X

v=1

M

v

X

m

v

₌₁

E

scatt

s,v

(x

v

m

, y

m

v

) −

N

X

n=1

τ

n

E

n

v,s

G

v

2D

(x

v

m

, y

v

m

|x

n

, y

n

)

2

(5)

(1)

Infa t,ea hlinear onstraint anbeequivalentlyseenasaredu tionofthenumberofindependent unknowns.

(9)

C

_State

(s)

{τ

n

, E

n

v,s

} =

V

X

v=1

N

X

n=1

E

inc

s,v

(x

n

, y

n

) −

"

E

n

v,s

−

N

X

p=1

τ

p

E

p

v,s

G

v

2D

(x

n

, y

n

|x

p

, y

p

)

#

2

(6) where

c

(s)

data

=

P

V

v=1

P

M

v

m

v

₌₁

|E

s,v

scatt

(x

v

m

, y

m

v

)|

2

,

c

(s)

State

=

P

V

v=1

P

N

n=1

|E

s,v

inc

(x

n

, y

n

)|

2

, and

G

v

_2D

(x

r

, y

r

|x

t

, y

t

) =











(j/2)

h

πk

0 a

r

H

1 (2)

(k

0 a

r

) − 2j

i

if r = t and (x

r

, y

r

) ∈ D

I

(jπk

0 a

r

/2)H

(2)

0 (k

0 ρ

v

rt

)J

1 (k

0 a

r

)

if r 6= t or (x

r

, y

r

) /

∈ D

I

(7)

ρ

v

rt

=

q

(x

v

r

− x

t

)

2 + (y

r

v

− y

t

)

2

;

H

(2)

0 ( . )

and

H

(1)

0 ( . )

are the rst and se ond kind

0

-th

order Hankel fun tion, respe tively;

J

1 ( . )

is the Bessel fun tion;

k

0 =

2π

λ

0

(

λ

0

being the

ba kground wavelength) and

a

r

=

q

D

_I

(n)

π

.

Be ause of the properties of the s attered eld, the amount of independent information

arising from su h a (

s

-th) multi-view experiment (and olle table from the

P

V

v=1

M

v

measurementsin

D

O

) is not larger than a xed threshold

I

[21℄. Therefore, the solution

{τ

n

, E

n

v,s

}

that best ts (4) annot be faithfully retrieved without redu ing the spatial resolution a ura y

N

until

U

is lowerthan su h athreshold.

Asamatteroffa t,theinformationrelatedtoasingle-sour e(SS)multi-viewexperiment

might be insu ient to guarantee satisfa tory re onstru tions and dierent strategies

have to be investigated to in rease the information ontent of s attering data. A widely

adoptedte hniqueistheso- alledmulti-frequen yapproa h[22℄[23℄[25℄that ertainly an

improvethe re onstru tion a ura y of the imaging pro ess by in reasing the number of

independentdata, but generallyitrequiressome assumptionson thediele tri properties

ofthe s enarioundertest. Indeed, even thoughtheobje tfun tiondoesnot hangewhen

hanging the sour e-position (or illumination), it varies with frequen y. Thus, without

somea-priori information,thesolutionofdierentinverses atteringproblemsisne essary

with a large in reasing of the omputationalload and omputer memory requirements.

Within the same framework, the multi-sour e te hnique also is aimed at enlarging the

amount of informative s attering-data in order to enhan e the re onstru tion a ura y.

However, whilethe ontrastfun tion ofthe multi-frequen y approa h isafun tion ofthe

frequen y [

τ

n

∼ τ

n

(f )

℄, in su h a ase, the unknown oe ients

τ

n

are independent on the sour e model [

τ

n

≁ τ

n

(s)

℄. The underlying idea of the multi-sour e approa h lies in the assumption (preliminarily veried in [41℄) that dierent ele tromagneti sour es

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sin edierentintera tionso urbetweens atterersandin identelds. Consequently,the

measurementsofthearisings atteredeldsallowonetokeepdierentand omplementary

informationthatit ouldbeprotabletoexploitforimprovingthea ura yoftheimaging

pro ess. To fully exploit su h information, a suitable ombination of the onstraints in

(4) on erned with ea h sour e shouldbe onsidered. Towards this end, letus denethe

multi-sour e ost fun tion

Φ

M S

Φ

M S

{τ

n

, E

n

v,s

} = Φ

Data

M S

{τ

n

, E

n

v,s

} + Φ

State

M S

{τ

n

, E

n

v,s

}

(8) where

Φ

Data

M S

{τ

n

, E

n

v,s

} =

P

S

s=1

C

(s)

Data

{τ

n

, E

n

v,s

}

P

S

s=1

c

(s)

Data

(9)

Φ

State

M S

{τ

n

, E

n

v,s

} =

P

S

s=1

C

(s)

State

{τ

n

, E

n

v,s

}

P

S

s=1

c

(s)

State

(10)

to be minimized. Sin e the multi-sour e te hnique does not depend on the minimization

algorithm,a tuallytheoptimizationpro edure onstitutesabla kbox intheoverall

sys-tem. Then,a deterministi optimizerbased on the alternating dire tionimpli it method

[42℄ is used

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(see Appendix A). Unlike the modied gradient method [49℄, where the

unknown elds and ontrast are updated simultaneously, but a ording to the ontrast

sour e inversion method [50℄,

τ

k

= {[τ

n

]

k

; n = 1, ..., N}

and

E

k

= {[E

v,s

n

]

k

; n = 1, ..., N;

v = 1, ..., V ; s = 1, ..., S}

are iteratively re onstru ted (

k

being the iterationnumber) by alternativelyupdatingthetwo sequen es. Theminimizationalgorithmisstopped whena

maximumnumberofiterations,

K

(i.e.,

k ≤ K

),orathresholdonthe ostfun tionvalue,

δ

(i.e.,

Φ

(k)

M S

= Φ

M S

{τ

k

, E

k

} ≤ δ

), or the value of the ost fun tion remains unaltered in a xed per entage of the total amount of the minimization-algorithm iterations(i.e.,

˛

K

w

Φ

(k)

M S

−

P

_Kw

h=1

Φ

(h)

M S

˛

Φ

(s)

n

_τ

(s)

k

,E

(s)

k

o

≤ ς

,

K

w

being an integer number).

Moreover, to redu e the o urren e of false solutions, the re onstru tion approa h has

been integrated into a multi-resolutionmethodology (IMSA) [51℄[52℄[53℄ insteadof using

(2)

Morere entoptimizationapproa hestostandard2Dinverses atteringproblemsdevelopedatthe

ELEDIALabhavebeendes ribedelsewhere[43℄[44℄[45℄[46℄andtheirintegration,aswellastheintegration

ofother state-of-the-artsto hasti minimizationte hniques [47℄[48℄, intothemulti-sour ete hniquewill

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multi-step(

i

beingthestepindex)syntheti zoomoftheregion-of-interest(RoI)wherethe s atterer isestimated, isdependent neitherontheminimizationapproa h[29℄[54℄[55℄nor

on theimagingsystem [56℄[57℄ and s atteringdata [58℄. Thus, it an beeasilyintegrated

intothe multi-sour e imaging system by enhan ing its omputational ee tiveness.

3 Numeri al Analysis

Inthis se tion,the ee tivenessoftheMSapproa hisevaluatedby onsideringasele ted

and representative set of numeri alexperiments.

As a rst test ase(Test Case1),letus onsider the referen eprole shown inFig. 2(a)

where a two-dimensional investigation domain of side

L

D

I

= 3.0 λ

0

ontains a square stratied s atterer

L

ext

= 0.9 λ

0

-sided entered at

x

ref

c

= −0.6 λ

0

,

y

ref

c

= 0.6 λ

0

. The diele tri permittivityoftheinnerregion(

L

int

= 0.3 λ

0

)isequalto

ε

int

r

= 3.0

(

τ

int

(x, y) =

2.0

),whiletheouterlayerhasa ontrastfun tion

τ

ext

(x, y) = 0.5

. Su ha ongurationhas been su essively illuminated by

V = 4

dierent dire tions and the measures have been olle ted at

M

v

_{= 26}

,

v = 1, ..., V

equally-spa ed points lying on a ir ular observation domain of radius

r

O

= 2.2 λ

0

. The s attering data have been syntheti ally-generated by onsidering the followingele tromagneti sour e models:

•

PlaneWave(PW)model hara terizedbyanin identeld

E

v,s

inc

(x, y)|

s=1

= E

0 e

jkr

ˆz

,

r = x cosθ

v

_{+ y sinθ}

v

;

•

Isotropi Line (IL)sour e lo ated at the point(

x

p

= r

p

cosθ

v

,

y

p

= r

p

sinθ

v

), where

r

p

= 2.4 λ

0

and radiatingaeld equal to

E

v,s

inc

(x, y)|

s=2

= −I

0 k

2

0 8πf ε

0 H

₀

(2)

(k

0 ρ

p

)ˆz

(11)

ρ

p

being the distan e between

(x

p

, y

p

)

and

(x, y)

;

I

0 =

q

8P

0 ηk

0

,

P

0 = 1

mW

m

being the

radiatedpower for unitlength and

η

the intrinsi impedan e of the ba kground;

•

Dire tive Line (DL) sour e modeled by using an expression similar to the Silver's equation [59℄

E

v,s

inc

(x, y)|

s=3

= −I

0 k

2

0 8πf ε

0

(12)

where

B(ψ) =

psin

3 _(ψ)

if

0 ≤ ψ ≤ π

and

B(ψ) = 0

otherwise,

ψ

being the polar angle in a oordinate system entered at

(x

p

, y

p

)

;

•

Composite Sour e(CS)radiatinga eld obtained as follows

E

v

inc

(x, y) =

S

X

s=1

E

v,s

inc

(x, y) S = 3

(13)

Moreover, in order to simulate a real environment, a Gaussian noise hara terized by a

SNR = 20 dB

has been addedto s attered eld data.

As far as the IMSA approa h is on erned, the following parametri onguration has

been used:

K = 2000

,

δ = 10

−5

,

ς = 10

−6

, and

K

w

=

K

10

. Furthermore, in order to test

the approa hin worst ase onditions, the ba kground has been hosen as initialguess

for the unknown ontrast.

To give some quantitative indi ations on the re onstru tion a ura y of the retrieval

pro ess, the following error gureswillbeused

χ

j

=

1 N

(j)

N

(j)

X

n=1

τ (x

n

, y

n

) − τ

ref

(x

n

, y

n

)

τ

ref

_(x

n

, y

n

)

× 100

(14) where

N

(j)

anrangeoverthewholeinvestigationdomain(

j ⇒ tot

),overtheareaa tually o upied by the s atter (

j ⇒ int

),or overthe ba kground aroundthe obje ts(

j ⇒ ext

). On the other hand,

ξ

( enter lo ation error) and

Λ

(shaping error) will estimate the a ura y of the qualitativeimaging

ξ =

r

h

x

c

− x

ref

c

i

2 +

h

y

c

− y

c

ref

i

2 λ

0

(15)

Λ =

(

R − R

ref

R

ref

)

× 100

(16)

where the super-s ript ref refers to the a tual prole,

R

being the radius of the RoI where the s atterer issupposed tobelo ated.

In the rst experiment, the inversion pro ess has been arried out by onsidering simple

single-sour earrangements(i.e.,PW-model,IL-model,andDL-model)inordertogenerate

some referen e ases for evaluating the ee tiveness of the MS approa h. By applying

(13)

pro edure) in

N = 49

square sub-domains and the obtained results are shown in Figs. 2(b)-(d) with a grey-level representation

(3)

. As it an be observed, whatever the simple

single-sour e illumination, the retrieved proles slightly relate to the a tual stratied

onguration (Fig. 2). Although no-artifa ts are present in the re onstru tion and the

s atterer is roughly lo ated in the orre t area of the investigation domain (

ξ ≃ 0.30

-Tab. I), signi ant errors turn out both in the shaping and in the dete tion of the

two-layer stru ture as onrmed by the values of the error gures (

Λ > 38.0

and

χ

j

> 21.0

,

j ⇒ tot, int, ext

- Tab. I). Moreover, it is worth noting that the best re onstru tion is yielded with the simplest sour e-model(i.e., the plane-wave illumination). Su h a result

seems toindi atethat,ingeneral,thereisnot adire t relationbetween omplexityofthe

illuminationmodelingand a hievable re onstru tion a ura y.

To further onrm su h a on ept, let us onsider the retrieved prole when the

so- alled omposite sour e (CS) is used. In this situation, the ele tromagneti sour e has

been obtained asthe superposition of the in ident eldsradiated by the SS-models. The

obtained image[Fig. 2(e)℄turnsout tobeeven worse thanthatof theSS-modelswithan

in reasingofthequalitative(

ξ = 1.38

and

Λ = 67.11

)aswellasquantitative(

χ

int

= 51.77

) errors.

By leaving aside the study of the optimal illuminationfor the problem at hand (whose

analysis is beyond the s ope of this paper), the IMSA-MS strategy has been taken into

a ount. The re onstru tion turns out signi antly improved both pi torially[Fig. 2(f)℄

andintermsoferrorgures. Inparti ular,itshouldbeobservedthatthelayeredstru ture

is learly distinguishableand the s atterer turns out better lo alized(

ξ

(DL)

_{≃ 2.4 ξ}

(M S)

) and dimensioned (

Λ

(M S)

_{= 24.00}

vs.

Λ

(IL)

_{= 38.67}

).

Su h a result has been a hieved by minimizingthe ost fun tion (8) as shown in Figure

3 where the two terms of the fun tionalare given, as well. When the IMSA-MS method

is adopted, the minimization rea hes a lower value of the ost fun tion thus allowing a

more a urate data-inversion.

In the se ond experiment, the numberof views has been in reased from

V = 4

to

V = 6

in order to assess the ee tiveness of the proposed approa h in dierent illumination

(3)

Pleasenotethatthebla kpixelinthelowerrightborderisusedforreferen eandthedashedline indi ates theregiono upiedbythea tuals atterer.

(14)

with a de rease of the error values (

ξ

,

Λ

, and

χ

tot

) of about one order in magnitude ompared to the

V = 4

ondition (as an example,

χ

(M S)

_tot

˛

V

=4

χ

(M S)

_tot

˛

V

=6

≃ 9.0

). Whatever strategy, the two-layers prole is learly dete ted. However, the IMSA-MS further onrms its

potentialities by over oming other SS-strategies (

χ

(IL)

tot

≃ 3.25 χ

(M S)

tot

,

χ

(DL)

int

≃ 1.50 χ

(M S)

int

, and

χ

(DL)

ext

χ

(M S)

_ext

≃ 1.0 × 10

2

)andby a hievingafaithfulimageoftheoriginalprole[Fig. 4(e)℄.

Although su h experiments indi ate that it proves onveniently in terms of a hievable

resolution a ura y to use the multi-sour e strategy, it is needed to evaluate the arising

omputational burden, as well. Towards this end, Figure 5 gives an overview of the

omputational s enario by resuming the two experiments with dierent illuminations.

Moreindetail,the followingrepresentativeparametersareshown: the numberofproblem

unknowns

U

[Fig. 5(a)℄,thenumberofstepsoftheIMSA

I

opt

[Fig. 5(b)℄,thetotalnumber of iterationsof the minimization pro edure

K

tot

=

P

I

opt

i=1

k

(i)

conv

(

k

(i)

conv

beingthe number of

iterationsneededtoa hievethe onvergen e atthe

i

-thstepofthemulti-s alingpro ess) [Fig. 5( )℄,and the iterationtime

t

k

[Fig. 5(d)℄. As it an be observed, even thoughthe numberof unknowns triples, the multi-s alingpro ess isterminated atthe same number

of steps (

I

(M S)

opt

V

=4

= 2

and

I

(M S)

opt

V

=6

= 3

) as for SS-approa hes and, generally, with

a lower

K

tot

. Therefore, the expe ted in reasing of the iteration-time does not impa t so-signi antly and it does not prevent the feasibility of the proposed approa h. As a

matter of fa t, the problem at hand is still omputationally tra table (thanks to the

integration with the IMSA). Moreover, the trade-o between in reased omputational

osts and enhan ed re onstru tion a ura yseems to be infavor of the IMSA-MS.

The advantages of the MS over SS-strategies in terms of re onstru tion a ura y are

further pointed out and emphasized when more omplex geometries are at hand. In

these situations, the enhan ement allowed by the proposed strategy turnsout to beeven

more signi ant than in Test Case 1. To show su h a behavior, the se ond test (Test

Case 2) deals with the asymmetri prole shown in Figure 6(a) and hara terized by

the followingdiele tri /geometri parameters:

x

ref

c

= −0.392 λ

0

,

y

ref

c

= 0.374 λ

0

,

H

ext

=

1.05 λ

0

,

t

arm

= 0.15 λ

0

(thi kness of the arms),

d

arm

= 0.3λ

0

(distan e between arms), and

τ = 2.0

. The s atterer has been su essively illuminated by

V = 6

dire tions and a

(15)

noise of

SNR = 20 dB

hasbeen addedtotheelddata. Unfortunately,the IMSA-ILand the IMSA-CS single-sour e approa hes ompletelyfail inre onstru ting the shape of the

obje tundertestasit an beseeninFig. 6( )and Fig. 6(e),respe tively,andquantied

in Tab. III (e.g.,

χ

(IL)

tot

= 17.67

and

χ

(CS)

tot

= 28.47

;

Λ

(IL)

_{= 11.60}

and

Λ

(CS)

_{= 65.0}

). As

far as the IMSA-PW is on erned, the upper arm is lost while the lowerone is orre tly

dete ted [Fig. 6(b)℄. A breaking improvement of the image quality arises when the

DL-SS illumination is used [Fig. 6(d)℄. In su h a ase, while the s atterer annot be

identied exa tly, the retrieval pro edure onverges to a stru ture that o upies a large

subsetofthe trueobje t. However, on eagaintheMSstrategy[Fig. 6(f)℄allowsabetter

re onstru tion (

χ

(DL)

tot

≃ 1.7 χ

(M S)

tot

,

χ

(DL)

ext

≃ 2.0 χ

(M S)

tot

,and

ξ

(DL)

_{≃ 10 ξ}

(M S)

). Asa matter

of fa t,the nal re onstru tion isessentially identi altothat one would a hieve withthe

la k of edge-preserving [60℄ orbinary-regularization te hniques [61℄.

4 Con lusions

A nonlinear multisour e strategy for the quantitativeimaging of unknown s atterers has

been presented. The approa h is aimed at in reasing the re onstru tion a ura y by

en-larging the non-redundant information on the s enario under test through an ee tive

exploitationof dierentele tromagneti intera tionsbetween variousprobingsour esand

s atterers. The method has been developed by exploiting and extending the multiview

te hnique to the ase of multisour e data through the denition of a suitable ost

fun -tion to be minimized. Notwithstanding its simpli ity, the proposed strategy turned out

to be ee tive in re overing various permittivity proles from simple shapes up to

om-plex ongurations. As far as the in rement of the omputational load is on erned,

the integration of the multisour e approa h into an iterative multi-s alingpro edure

al-lowedasustainable overhead. Moreover, itisworth pointing outthatthe dierentsour e

ontributions an be pro essed almost in an independent fashion so that a parallel

im-plementation [62℄ would be very easy. This task together with the use of more ee tive

(16)

In order to apply the onjugate-gradient minimization pro edure, the omputation of

∇Φ

M S

given by

∇Φ

M S

= ∇Φ

Data

M S

+ ∇Φ

State

M S

(17)

is derived. The omputation of partial derivatives of

Φ

Data

M S

and

Φ

State

M S

with respe t to the variables

Re {τ

n

}

,

Im {τ

n

}

,

Re {E

v,s

n

}

, and

Im {E

v,s

n

}

is required (where

Re { }

and

Im { }

indi atereal and imaginary part,respe tively).

Firstly, letus give some preliminarydenitions useful for the following

Ψ

(s)

_Data

{τ

n

, E

n

v,s

} = E

s,v

scatt

(x

v

m

, y

v

m

) − Θ

(s)

Data

{τ

n

, E

n

v,s

}

(18)

Ψ

(s)

_State

{τ

n

, E

n

v,s

} = E

s,v

inc

(x

n

, y

n

) − Θ

(s)

State

{τ

n

, E

n

v,s

}

(19) where

Θ

(s)

Data

{τ

n

, E

n

v,s

} =

P

N

n=1

τ

n

E

n

v,s

G

v

2D

(x

m

v

, y

m

v

|x

n

, y

n

)

and

Θ

(s)

State

{τ

n

, E

n

v,s

} = E

n

v,s

−

P

N

p=1

τ

p

E

p

v,s

G

v

2D

(x

n

, y

n

|x

p

, y

p

)

.

As farasthe Data term is on erned, bymeans ofsimple mathemati almanipulations,

the following expressions for the partial derivativesof

Ψ

(s)

Data

are obtained

∂Ψ

(s)

_Data

∂ {Re (τ

n

)}

= −

∂Re

n

Θ

(s)

Data

{τ

n

, E

n

v,s

}

o

∂ {Re (τ

n

)}

− j

∂Im

n

Θ

(s)

Data

{τ

n

, E

n

v,s

}

o

∂ {Re (τ

n

)}

(20)

∂Ψ

(s)

_Data

∂ {Im (τ

n

)}

= −

∂Re

n

Θ

(s)

_Data

{τ

n

, E

n

v,s

}

o

∂ {Im (τ

n

)}

− j

∂Im

n

Θ

(s)

_Data

{τ

n

, E

n

v,s

}

o

∂ {Im (τ

n

)}

(21) where

∂

n

Re

Θ

(s)

_Data

o

∂ {Re (τ

n

)}

=

π

2 k

0 ρ

n

J

1 (k

0 ρ

n

) [Re {E

v,s

n

} Y

0 (k

0 d

mn

) − Im {E

n

v,s

} J

0 (k

0 d

mn

)]

(22)

∂

n

Re

Θ

(s)

_Data

o

∂ {Im (τ

n

)}

= −

π

2 k

0 ρ

n

J

1 (k

0 ρ

n

) [Im {E

v,s

n

} Y

0 (k

0 d

mn

) + Re {E

n

v,s

} J

0 (k

0 d

mn

)]

(23)

∂

n

Im

Θ

(s)

_Data

o

∂ {Re (τ

n

)}

=

π

2 k

0 ρ

n

J

1 (k

0 ρ

n

) [Re {E

v,s

n

} J

0 (k

0 d

mn

) + Im {E

n

v,s

} Y

0 (k

0 d

mn

)]

(24)

(17)

∂

n

Im

Θ

(s)

_Data

o

∂ {Im (τ

n

)}

=

π

2 k

0 ρ

n

J

1 (k

0 ρ

n

) [−Im {E

v,s

n

} J

0 (k

0 d

mn

) + Re {E

n

v,s

} Y

0 (k

0 d

mn

)]

(25) being

ρ

n

=

L

n

√

_π

,

d

mn

=

q

(x

n

− x

m

)

2 − (y

n

− y

m

)

2

and

∂Ψ

(s)

_Data

∂ {Re (E

n

v,s

)}

= −

∂Re

n

Θ

(s)

_Data

{τ

n

, E

n

v,s

}

o

∂ {Re (E

n

v,s

)}

− j

∂Im

n

Θ

(s)

_Data

{τ

n

, E

n

v,s

}

o

∂ {Re (E

n

v,s

)}

(26)

∂Ψ

(s)

Data

∂ {Im (E

n

v,s

)}

= −

∂Re

n

Θ

(s)

_Data

{τ

n

, E

n

v,s

}

o

∂ {Im (τ

n

)}

− j

∂Im

n

Θ

(s)

_Data

{τ

n

, E

n

v,s

}

o

∂ {Im (τ

n

)}

(27) where

∂

n

Re

Θ

(s)

_Data

o

∂ {Re (E

n

v,s

)}

=

π

2 k

0 ρ

n

J

1 (k

0 ρ

n

)

σ

n

2πf ε

0 J

0 (k

0 d

mn

) + (ε

r

n

− 1) Y

0 (k

0 d

mn

)

(28)

∂

n

Re

Θ

(s)

_Data

o

∂ {Im (E

n

v,s

)}

=

π

2 k

0 ρ

n

J

1 (k

0 ρ

n

)

σ

n

2πf ε

0 Y

0 (k

0 d

mn

) − (ε

r

n

− 1) J

0 (k

0 d

mn

)

(29)

∂

n

Im

Θ

(s)

_Data

o

∂ {Re (E

n

v,s

)}

=

π

2 k

0 ρ

n

J

1 (k

0 ρ

n

)

(ε

r

n

− 1) J

0 (k

0 d

mn

) −

σ

n

2πf ε

0 Y

0 (k

0 d

mn

)

(30)

∂

n

Im

Θ

(s)

Data

o

∂ {Im (E

n

v,s

)}

=

π

2 k

0 ρ

n

J

1 (k

0 ρ

n

)

σ

n

2πf ε

0 Y

0 (k

0 d

mn

) − (ε

r

n

− 1) J

0 (k

0 d

mn

)

(31)

A ordingly, the partial derivativesof

Φ

Data

M S

are given by

∂Φ

Data

M S

∂

_{Re(τ

n

)}

=

1 P

S

s=1

P

V

v=1

P

M

m=1

|

E

s,v

scatt

(x

v

m

,y

m

v

)

|

2 P

M

m=1

2Re

n

Ψ

(s)

_Data

{τ

n

, E

n

v,s

}

o

∂Re

n

Ψ

(s)

_Data

{

τ

n

,E

v,s

n

}

o

∂

_{Re(τ

n

)}

+

2Im

n

Ψ

(s)

_Data

{τ

n

, E

n

v,s

}

o

∂Im

n

Ψ

(s)

_Data

{

τ

n

,E

n

v,s

}

o

∂

_{Re(τ

n

)}

(32)

∂Φ

Data

M S

∂

{Im(τ

n

)}

=

1 P

S

s=1

P

V

v=1

P

M

m=1

|

E

s,v

scatt

(x

v

m

,y

m

v

)

|

2 P

M

m=1

2Re

n

Ψ

(s)

Data

{τ

n

, E

n

v,s

}

o

∂Re

n

Ψ

(s)

_Data

{

τ

n

,E

n

v,s

}

o

∂

{Im(τ

n

)}

+

2Im

n

Ψ

(s)

Data

{τ

n

, E

n

v,s

}

o

∂Im

n

Ψ

(s)

_Data

{

τ

n

,E

n

v,s

}

o

∂

{Im(τ

n

)}

(33)

(18)

∂Φ

Data

M S

∂

{

Re

(

E

v,s

n

)}

=

1 P

S

s=1

P

V

v=1

P

M

m=1

|

E

s,v

scatt

(x

v

m

,y

m

v

)

|

2 P

M

m=1

2Re

n

Ψ

(s)

Data

{τ

n

, E

n

v,s

}

o

∂Re

n

Ψ

(s)

_Data

{

τ

n

,E

v,s

n

}

o

∂

{

Re

(

E

n

v,s

)}

+

2Im

n

Ψ

(s)

_Data

{τ

n

, E

n

v,s

}

o

∂Im

n

Ψ

(s)

_Data

{

τ

n

,E

v,s

n

}

o

∂

{

Re

(

E

n

v,s

)}

(34)

∂Φ

Data

M S

∂

{

Im

(

E

v,s

n

)}

=

1 P

S

s=1

P

V

v=1

P

M

m=1

|

E

s,v

scatt

(x

v

m

,y

m

v

)

|

2 P

M

m=1

2Re

n

Ψ

(s)

_Data

{τ

n

, E

n

v,s

}

o

∂Re

n

Ψ

(s)

_Data

{

τ

n

,E

v,s

n

}

o

∂

{

Im

(

E

n

v,s

)}

+

2Im

n

Ψ

(s)

_Data

{τ

n

, E

n

v,s

}

o

∂Im

n

Ψ

(s)

_Data

{

τ

n

,E

n

v,s

}

o

∂

{

Im

(

E

n

v,s

)}

(35)

Analogously,itispossible toevaluatethe partialderivativesof

Φ

State

M S

,whi hare given by

∂Φ

State

M S

∂

_{Re(τ

n

)}

=

1 P

S

s=1

P

V

v=1

P

N

p=1

|

E

s,v

inc

(x

p

,y

p

)

|

2 n

P

N

p=1

πk

o

ρ

n

h

Re

n

Ψ

(s)

_State

τ

p

, E

p

v,s

o

Im E

v,s

p

I

np

− Re

E

p

v,s

R

np

− Im

n

Ψ

(s)

State

τ

p

, E

p

v,s

o

Re E

v,s

p

I

np

− Re

E

p

v,s

R

np

io

(36)

∂Φ

State

M S

∂

_{Im(τ

n

)}

=

1 P

S

s=1

P

V

v=1

P

N

p=1

|

E

s,v

inc

(x

p

,y

p

)

|

2 n

P

N

p=1

πk

o

ρ

n

h

Re

n

Ψ

(s)

_State

τ

p

, E

p

v,s

o

Im E

v,s

p

R

np

+ Re

E

p

v,s

I

np

+ Im

n

Ψ

(s)

_State

τ

p

, E

p

v,s

o

Im E

v,s

p

I

np

− Re

E

p

v,s

R

np

io

(37)

∂Φ

State

M S

∂

{

Re

(

E

n

v,s

)}

=

1 P

S

s=1

P

V

v=1

P

N

p=1

|

E

s,v

inc

(x

p

,y

p

)

|

2 n

P

N

p=1

πk

o

ρ

n

h

Re

n

Ψ

(s)

_State

τ

p

, E

p

v,s

o

n

−

2δ

np

πk

0 ρ

n

− ε

r

p

− 1 R

np

−

σ

p

2πf ε

0 I

np

o

+ Im

n

Ψ

(s)

State

τ

p

, E

p

v,s

o n

ε

r

p

− 1 I

np

+

σ

p

2πf ε

0 R

np

oio

(38)

∂Φ

State

M S

∂

{

Im

(

E

v,s

n

)}

=

1 P

S

s=1

P

V

v=1

P

N

p=1

|

E

s,v

inc

(x

p

,y

p

)

|

2 n

P

N

p=1

πk

o

ρ

n

h

Re

n

Ψ

(s)

_State

τ

p

, E

p

v,s

o

n

−

σ

p

2πf ε

0 R

np

+ ε

r

p

− 1 I

np

o

− Im

n

Ψ

(s)

_State

τ

p

, E

p

v,s

o n

2δ

np

πk

0 ρ

n

+ ε

r

p

− 1 R

np

+

σ

p

2πf ε

0 I

np

oio

(39) where

R

np

=











J

1 (k

0 ρ

n

) Y

0 (k

0 d

np

) if n = p

Y

1 (k

0 ρ

n

) if n 6= p

(40)

I

np

=











J

1 (k

0 ρ

n

) J

0 (k

0 d

np

) if n = p

J

1 (k

0 ρ

n

) −

_πk

4 ₀

_ρ

_n

if n 6= p

(41)

(19)

and

δ

np

= 1

if

n = p

,

δ

np

= 0

otherwise. Finally,thearray

∇Φ

M S

=

[∇Φ

M S

]

τ

n

, [∇Φ

M S

]

E

v,s

n

; n = 1, ..., N; v = 1, ..., V ; s = 1, ..., S

an be omputed as

[∇Φ

M S

]

_τ

_n

=

∂Φ

Data

M S

∂ {Re (τ

n

)}

+

∂Φ

State

M S

∂ {Re (τ

n

)}

+ j

∂Φ

Data

M S

∂ {Im (τ

n

)}

+

∂Φ

State

M S

∂ {Im (τ

n

)}

(42)

[∇Φ

M S

]

_E

v,s

n

=

∂Φ

Data

M S

∂ {Re (E

n

v,s

)}

+

∂Φ

State

M S

∂ {Re (E

n

v,s

)}

+ j

∂Φ

Data

M S

∂ {Im (E

n

v,s

)}

+

∂Φ

State

M S

∂ {Im (E

n

v,s

)}

(43)

(20)

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•

Figure 1. Geometry of the multi-sour e/multi-viewimagingsystem.

•

Figure 2. Multi-layered diele tri prole (

τ

ext

= 2.0

,

τ

int

= 0.5

). Referen e dis-tribution (a). Re onstru ted diele tri distribution by using (b) IMSA-PW, ( )

IMSA-IL,(d)IMSA-DL,(e)IMSA-CS,and(f)IMSA-MS (

SNR = 20 dB

-

V = 4

).

• τ

ext

= 2.0

,

τ

int

= 0.5

-

SNR = 20 dB

,

V = 4

). Behavior of the (a) ost fun tion and related (b) data and ( ) state terms during the iterativepro ess.

• τ

ext

= 2.0

,

τ

int

= 0.5

-

SNR = 20 dB

,

V = 6

). Re onstru teddiele tri distributionbyusing(b)IMSA-PW,( )IMSA-IL, (d) IMSA-DL,(e) IMSA-CS, and (f) IMSA-MS.

• τ

ext

= 2.0

,

τ

int

= 0.5

-

SNR = 20 dB

). Evaluationofthe omputationalburden/ omplexity: (a)problemdimension

U

, (b) number of IMSA steps

I

opt

, ( ) total number of iterations

K

tot

, and (d) iteration time

t

k

[msec]

.

•

Figure 6. Complex diele tri prole (

τ = 2.0

). Referen edistribution (a). Re on-stru teddiele tri distributionbyusing(b)IMSA-PW,( )IMSA-IL,(d)IMSA-DL,