A Multi-Resolution Technique Based on Shape Optimization for the Reconstruction of Homogeneous Dielectric Objects

UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38123 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

A MULTI-RESOLUTION TECHNIQUE BASED ON SHAPE

OPTIMIZATION FOR THE RECONSTRUCTION OF

HOMOGENEOUS DIELECTRIC OBJECTS

M. Benedetti, D. Lesselier, M. Lambert, and A. Massa

January 2009

Shape Optimization for the Re onstru tion

of Homogeneous Diele tri Obje ts

M. Benedetti

1,2

, D. Lesselier

2

, M. Lambert

2

, and A. Massa

1

1

Departmentof InformationEngineeringandComputerS ien e,ELEDIAResear h Group,UniversityofTrento, ViaSommarive14,38050Trento-Italy,Tel. +390461 882057,Fax. +390461882093

2

DépartementdeRe her heenÉle tromagnétisme-LaboratoiredesSignauxet Systèmes,CNRS-SUPELEC Univ. ParisSud11,3rueJoliot-Curie,91192 Gif-sur-YvetteCEDEX,Fran e

E-mail: manuel.benedettidisi.unitn.it,lesselierlss.supele .fr,lambertlss.supele .fr, andrea.massaing.unitn.it

Abstra t. Intheframeworkofinverses atteringte hniques,thispaperpresentsthe

integration of a multi-resolution te hnique and the level-set method for qualitative mi rowaveimaging. Onone hand,in order to ee tively exploit thelimited amount of information olle table from s attering measurements, the iterative multi-s aling approa h (IMSA) is employed for enabling a detailed re onstru tion only where neededwithoutin reasingthenumberofunknowns. Ontheother hand,thea-priori information on the homogeneity of the unknown obje t is exploited by adopting a shape-based optimization and representing the support of the s atterer via a level set fun tion. Reliability and ee tiveness of the proposed strategy are assessed by pro essing both syntheti and experimental s attering data for simple and omplex geometries,aswell.

Key Words - Mi rowave Imaging, Inverse S attering, Level Sets, Iterative Multi-S alingApproa h,Homogeneous Diele tri S atterers.

1. Introdu tion

The non-invasive re onstru tion of position and shape of unknown targets is a topi of great interest in many appli ations, su h as non-destru tive evaluation and testing (NDE/NDT) for industrial monitoring and subsurfa e sensing [1℄. In this framework, many methodologies have been proposed based on x-rays [2℄, ultrasoni s [3℄, and eddy urrents [4℄. Furthermore, mi rowave imaging has been re ognized as a suitable methodology sin e [1℄[5℄: (a) ele tromagneti elds at mi rowave frequen ies an penetrate non-ideal ondu tor materials; (b) the eld s attered by the target is representative of itsinner stru ture and not only ofits boundary; ( )mi rowavesshow ahighsensibilitytothewater ontentofthe stru tureundertest; (d)mi rowavesensors anbeemployed withoutme hani al onta tswiththespe imen. Inaddition, ompared to x-ray and magneti resonan e, mi rowave-based approa hes minimize (or avoid) ollateralee ts in the spe imen undertest. Therefore, they an be safely employed in biomedi al imaging.

A further advan e in mi rowave non-invasive inspe tion is represented by inverse s atteringapproa hesaimedatre onstru tinga ompleteimageoftheregionundertest. Unfortunately,theunderlyingmathemati almodelis hara terizedbyseveraldrawba ks preventing their massiveemployment inNDE/NDT appli ations. Inparti ular, inverse s attering problems are intrinsi allyill-posed [6℄as well asnon-linear [7℄.

Sin e the ill-posednessis strongly relatedto the amountof olle table information andusually thenumberofindependentdataislowerthan thedimension ofthe solution spa e, multi-view/multi-illumination systems are generally adopted. However, it is well known that the olle table information is an upper-bounded quantity [8℄-[10℄. Consequently, it is ne essary to ee tively exploit the overall information ontained in the s attered eld samples fora hieving a satisfa tory re onstru tion.

Towardsthisend,multi-resolutionstrategieshavebeenre entlyproposed. Theidea isthat ofusinganenhan edspatial resolutiononlyinthoseregionswherethe unknown s atterersarefoundtobelo ated. A ordingly,Milleretal. [11℄proposeda statisti ally-based method for determining the optimal resolution level, while Baussard et al. [12℄ developed astrategybased onsplinepyramidsforsub-surfa e imagingproblems. Asfor anexample on ernedwith qualitativemi rowaveimaging,Liet al. [13℄ implementeda multis alete hniquebasedonLinearSamplingMethod(LSM)toee tivelyre onstru t the ontourofthes atterers. Unlike[11℄-[13℄,the iterativemulti-s aleapproa h(IMSA)

developed by Caorsietal. [14℄performsamulti-step,multi-resolutioninversionpro ess inwhi htheratiobetween unknowns anddataiskeptsuitablylowand onstantatea h step of theinversion pro edure, thusredu ing the riskof o urren e of lo alminima[9℄ in the arisingoptimization problem.

On the other hand, the la k of informationae ting the inverse problemhas been addressed through the exploitation of the a-priori knowledge (when available) on the s enario under test by means of anee tive representation of the unknowns. As far as many NDE/NDT appli ations are on erned, the unknown defe t is hara terized by known ele tromagneti properties (i.e., diele tri permittivity and ondu tivity) and it lieswithinaknown hostregion. Undertheseassumptions,the imagingproblemredu es toashapeoptimizationproblemaimedatthesear h oflo ationandboundary ontours ofthedefe t. Parametri te hniquesaimedatrepresentingtheunknown obje tinterms ofdes riptiveparametersofreferen eshapes[15℄[16℄andmoresophisti atedapproa hes su h as evolutionary- ontrolled spline urves [17℄[18℄, shape gradients [19℄-[21℄ or level-sets [22℄-[30℄havethenbeen proposed. Asfaraslevel-set-based methodsare on erned, the homogeneousobje tisdened asthe zero levelof a ontinuousfun tion and,unlike pixel-based or parametri -based strategies, su h a des ription enables one to represent omplex shapes ina simple way.

Inordertoexploitboth theavailablea-priori knowledgeonthes enarioundertest (e.g.,the homogeneity ofthe s atterer)and theinformation ontentfromthe s attering measurements,thispaperproposestheintegrationoftheiterativemulti-s alingstrategy (IMSA) [14℄ and the level-set (LS)representation [23℄.

The paper is stru tured as follows. The integration between IMSA and LS is detailed in Se t. 2 dealing with a two-dimensional geometry. In Se tion 3, numeri al testing and experimentalvalidationare presented, a omparison with the standard LS implementationbeing made. Finally,some on lusions are drawn (Se t. 4).

2. Mathemati al Formulation

Let us onsider a ylindri al homogeneous non-magneti obje t with relative permittivity

ǫ

C

and ondu tivity

σ

C

that o upies a region

Υ

belonging to an investigationdomain

D

I

. Su h a s atterer is probed by a set of

V

transverse-magneti (

T M

)planewaves, withele tri elddire tedalongtheaxisofthe ylindri algeometry,

namely

ζ

v

_{(r) = ζ}

v

_(r)ˆz

(

v = 1, . . . , V

),

r = (x, y)

. The s attered eld,

ξ

v

_{(r) = ξ}

v

_(r)ˆz

,is olle ted at

M(v)

,

v = 1, ..., V

, measurement points

r

m

distributed in the observation domain

D

O

.

In order toele tromagneti ally des ribe the investigationdomain

D

I

,let usdene the ontrast fun tion

τ (r)

given by

τ (r) =







τ

C

0

r ∈ Υ

otherwise (1) where

τ

C

= (ǫ

C

− 1) − j

σ

C

2πf ε

0

,

f

being the frequen y of operation(the time dependen e

e

j2πf t

being implied).

The s attering problem is des ribed by the well-known Lippmann-S hwinger integral equations

ξ

v

(r

m

) =



_2π

λ



2

Z

D

I

τ (r

′

_{) E}

v

(r

′

_{) G}

2D

(r

m

, r

′

) dr

′

,

r

m

∈ D

O

(2)

ζ

v

(r) = E

v

(r) −



_2π

λ



2

Z

D

I

τ (r

′

) E

v

(r

′

) G

2D

(r, r

′

) dr

′

, r

∈ D

I

(3)

where

λ

is the ba kground wavelength,

E

v

is the total ele tri eld, and

G

2D

(r, r

′

_{) =}

−

j

4

H

(2)

0



2π

λ

kr − r

′

_k



isthefree-spa e two-dimensionalGreen'sfun tion,

H

(2)

0

being the se ond-kind, zeroth-order Hankel fun tion.

In order to retrieve the unknown position and shape of the target

Υ

by step-by-stepenhan ingthespatialresolutiononlyinthatregion, alledregion-of-interest(

RoI

),

R ∈ D

I

, where the s atterer is lo ated [14℄, the following iterative pro edure of

S

max

steps is arriedout.

With referen e to Fig. 1(a) and to the blo k diagram displayed in Fig. 2, at the rst step (

s = 1

,

s

being the step number) a trial shape

Υ

s

= Υ

1

, belonging to

D

I

, is hosen and the region of interest

R

s

[

R

s=1

= D

I

℄ is partitioned into

N

IM SA

equal square sub-domains,where

N

IM SA

depends onthe degrees offreedomofthe problemat hand and it is omputed a ording to the guidelines suggested in [9℄.

In addition, the level set fun tion

φ

s

is initialized by means of a signed distan e fun tion dened as follows [23℄[25℄:

φ

s

(r) =







−

min

b=1,...,B

s

kr − r

b

k

if

τ (r) = τ

C

min

b=1,...,B

s

kr − r

b

k

if

τ (r) = 0

(4)

where

r

b

= (x

b

, y

b

)

is the

b

-thborder- ell (

b = 1, . . . , B

s

) of

Υ

s=1

.

•

Problem Unknown Representation - The ontrast fun tion is represented in

terms of the levelset fun tionas follows

e

τ

k

s

(r) =

s

X

i=1

N

IM SA

_X

n

i

=1

τ

k

i

B



r

n

i



r

∈ D

I

(5)

where the index

k

s

indi ates the

k

-th iteration at the

s

-th step [

k

s

= 1, ..., k

opt

s

℄,

B



r

n

i



isare tangularbasisfun tionwhosesupportisthe

n

-thsub-domainatthe

i

-th resolution level [

n

i

= 1, ..., N

IM SA

,

i = 1, ..., s

℄, and the oe ient

τ

k

i

is given by

τ

k

i

=







τ

C

if

Ψ

k

i



r

n

i



≤ 0

0

otherwise (6) letting

Ψ

k

i



r

n

i



=











φ

k

i



r

n

i



if

i = s

φ

_k

opt

i



r

n

i



if

(i < s)

and



r

n

i

∈ R

i



(7) with

i = 1, ..., s

asin (5).

•

Field Distribution Updating - On e

τ

e

k

s

(r)

has been estimated, the ele tri eld

E

v

k

s

(r)

is numeri ally omputed a ording toa point-mat hing version of the Methodof Moments (MoM)[31℄ as

e

E

v

k

i



r

n

i



=

P

N

IM SA

p

i

=1

ζ

v



_r

p

i

 h

1 −

τ

e

k

i



r

p

i



G

2D



r

n

i

, r

p

i

i

−1

,

r

n

i

, r

p

i

∈ D

I

n

i

= 1, ..., N

IM SA

.

(8)

•

Cost Fun tion Evaluation - Starting from the total ele tri eld distribution

(8), the re onstru ted s attered eld

ξ

e

v

k

s

(r

m

)

at the

m

-th measurement point,

m = 1, ..., M(v)

, isupdated by solving the following equation

e

ξ

v

k

s

(r

m

) =

s

X

i=1

N

IM SA

_X

n

i

=1

e

τ

k

i



r

n

i



e

E

v

k

i



r

n

i



G

2D



r

m

, r

n

i



(9)

and the t between measured and re onstru ted data is evaluated by the multi-resolution ost fun tion

Θ

dened as

Θ {φ

k

s

} =

P

V

v=1

P

M(v)

m=1





 e

ξ

v

k

s

(r

m

) − ξ

v

k

s

(r

m

)







2

P

V

v=1

P

M(v)

m=1







ξ

v

k

s

(r

m

)







2

.

(10)

•

MinimizationStopping -Theiterativepro essstops[i.e.,

k

opt

s

= k

s

and

τ

e

opt

s

=

τ

e

k

s

℄ when: (a)aset of onditionsonthestabilityof there onstru tionholds trueor(b) when the maximum number of iterations is rea hed [

k

s

= K

max

℄ or ( ) when the

valueofthe ostfun tionissmallerthanaxedthreshold

γ

th

. Asfarasthestability of the re onstru tion is on erned [ ondition (a)℄,the rst orresponding stopping riterion is satised when, for a xed number of iterations,

K

τ

, the maximum number of pixelswhi h varytheir value is smallerthan auser dened threshold

γ

τ

a ording to the relationship

max

j=1,...,K

τ







N

IM SA

_X

n

s

=1

|

τ

e

k

s

(r

n

s

) −

τ

e

k

s

−j

(r

n

s

)|

τ

C







< γ

τ

· N

IM SA

.

(11)

These ond riterion, aboutthestabilityofthere onstru tion,issatisedwhenthe ost fun tion be omes stationarywithin a windowof

K

Θ

iterationsas follows:

1

K

Θ

K

Θ

X

j=1

Θ {φ

k

s

} − Θ {φ

k

s

−j

}

Θ {φ

k

s

}

< γ

Θ

.

(12)

K

Θ

beingaxed numberofiterationsand

γ

Θ

beinguser-denedthresholds;. When

the iterative pro ess stops, the solution

τ

e

opt

s

at the

s

-th step is sele ted as the one represented by the best level set fun tion

φ

opt

s

dened as

φ

opt

s

=

arg

h

min

h=1,...,k

opt

s

(Θ {φ

h

})

i

.

(13)

•

Iteration Update - The iterationindex is updated [

k

s

→ k

s

+ 1

℄;

•

Level Set Update - Thelevelsetis updateda ordingtothe following

Hamilton-Ja obi relationship

φ

k

s

(r

n

s

) = φ

k

s

−1

(r

n

s

) − ∆t

s

V

k

s

−1

(r

n

s

) H {φ

k

s

−1

(r

n

s

)}

(14)

where

H {·}

isthe Hamiltonianoperator[32℄[33℄ given as

H

2

_{φ

k

s

(r

n

s

)} =











































































max

2

n

_D

x−

k

s

; 0

o

+

min

2

n

_D

x+

k

s

; 0

o

+

+

max

2

n

_D

y−

k

s

; 0

o

+

min

2

n

_D

y+

k

s

; 0

o

if

V

k(s)



r

n(s)



≥ 0

min

2

n

_D

x−

k

s

; 0

o

+

max

2

n

_D

x+

k

s

; 0

o

+

+

min

2

n

_D

y−

k

s

; 0

o

+

max

2

n

_D

y+

k

s

; 0

o

otherwise (15) and

D

x±

k

s

=

±φ

_ks

(x

ns±1

,y

ns

)∓φ

ks

(x

ns

,y

ns

)

l

s

,

D

y±

k

s

=

±φ

_ks

(x

ns

,y

ns±1

)∓φ

ks

(x

ns

,y

ns

)

l

s

.

∆t

s

is the time-step hosen as

∆t

s

= ∆t

1

l

s

l

1

with

∆t

1

to be set heuristi allya ording to the literature [23℄,

l

s

being the ell-side atthe

s

-thresolution level.

V

k

fun tion omputedfollowingthe guidelines suggested in[23℄ by solving theadjoint problemof (8) inorder todetermine the adjointeld

F

v

k

s

. A ordingly,

V

k

s

(r

n

s

) = −ℜ

( P

V

v=1

τ

C

E

v

ks

(

r

ns

)

F

ks

v

(

r

ns

)

P

V

v=1

P

M(v)

m=1

|

ξ

v

ks

(

r

m

)|

2

)

,

n

s

= 1, ..., N

IM SA

(16)

where

ℜ

standsfor the real part.

When the

s

-th minimization pro ess terminates, the ontrast fun tion is updated [

τ

e

opt

s

(r)=

τ

e

k

s−1

(r)

,

r ∈ D

I

(5)℄ aswell asthe

RoI

[

R

s

→ R

s−1

℄. Todo so, the following operations are arriedout:

•

Computation of the Bary enter of the RoI - the enter of

R

s

of oordinates

(

x

e

c

s

,

y

e

s

c

)

is determinedby omputing the enter of mass of the re onstru ted shapes as follows [14℄ [Fig. 1(b)℄

e

x

c

s

=

R

D

I

x

τ

e

opt

s

(r) B (r) dx dy

R

D

I

τ

e

opt

s

(r) B (r) dx dy

(17)

e

y

c

s

=

R

D

I

y

τ

e

opt

s

(r) B (r) dx dy

R

D

I

τ

e

opt

s

(r) B (r) dx dy

;

(18)

•

Estimation of the Size of the RoI -theside

L

s

of

R

s

is omputedbyevaluating

the maximum of the distan e

δ

c

(r) =

q

(x −

x

e

c

s

)

2

+ (y −

y

e

c

s

)

2

in order to en lose the s atterer, namely

e

L

s

=

max

r

(

2 ×

τ

e

opt

s

(r)

τ

C

δ

c

(r)

)

.

(19)

On e the

RoI

has been identied, the level of resolution is enhan ed [

k

s

→ k

s−1

℄ only in this region by dis retizing

R

s

into

N

IM SA

sub-domains [Fig. 1( )℄ and by repeating the minimization pro ess untilthe syntheti zoombe omes stationary(

s = s

opt

), i.e.,

(

|Q

s−1

− Q

s

|

|Q

s−1

|

× 100

)

< γ

Q

,

Q =

x

e

c

,

y

e

c

,

L

e

(20)

γ

Q

being a threshold set as in [14℄, oruntil a maximum number of steps (

s

opt

= S

max

)

is rea hed.

Atthe end ofthe multi-steppro ess (

s = s

opt

),the problemsolutionis obtainedas

e

τ

opt



_r

n

i



=

τ

e

opt

s



r

n

i



,

n

i

= 1, ..., N

IM SA

,

i = 1, ..., s

opt

.

3. Numeri al Validation

In order to assess the ee tiveness of the IMSA-LS approa h, a sele ted set of representativeresults on ernedwithbothsyntheti and experimentaldataispresented herein. Theperforman esa hievedareevaluatedbymeansofthefollowingerrorgures:

•

Lo alization Error

δ

δ|

_p

=

r

e

x

c

s

|

p

− x

c

|

p



2

−



y

e

c

s

|

p

− y

c

|

p



2

λ

× 100

(21) where

r

c

|

p

=



x

c

_|

p

, y

c

_|

p



is the enter of the

p

-th true s atterer,

p = 1, ..., P

,

P

being the number of obje ts. The average lo alizationerror

< δ >

is dened as

< δ >=

1

P

P

X

p=1

δ|

_p

.

(22)

•

Area EstimationError

∆

∆ =





I

X

i=1

1

N

IM SA

N

IM SA

_X

n

i

=1

N

n

i





× 100

(23) where

N

n

i

isequal to

1

if

τ

e

opt



_r

n

i



= τ



r

n

i



and

0

otherwise.

As far as the numeri al experiments are on erned, the re onstru tions have been performedbyblurringthe s atteringdatawithanadditiveGaussiannoise hara terized by a signal-to-noise-ratio(

SNR

)

SNR = 10

log

P

V

v=1

P

M

(v)

m=1

|ξ

v

(r

m

)|

2

P

V

v=1

P

M(v)

m=1

|µ

v,m

|

2

(24)

µ

v,m

being a omplex Gaussian randomvariable with zero mean value.

3.1. Syntheti Data - Cir ular Cylinder

3.1.1. Preliminary Validation In the rst experiment, a lossless ir ular o- entered

s atterer of known permittivity

ǫ

C

= 1.8

and radius

ρ = λ/4

is lo ated in a square investigationdomain of side

L

D

= λ

[23℄.

V = 10 T M

plane waves are impingingfrom the dire tions

θ

v

= 2π (v − 1)/ V

,

v = 1, ..., V

, and the s attering measurements are olle ted at

M = 10

re eivers uniformlydistributed ona ir leof radius

ρ

O

= λ

.

As faras the initializationofthe IMSA-LS algorithmis on erned,the initialtrial obje t

Υ

1

is a disk with radius

λ/4

and entered in

D

I

. The initial value of the time step is set to

∆t

1

= 10

−2

stopping riteria,the following ongurationof parametershas been sele ted a ording to a preliminary alibration dealing with simple known s atterers and noiseless data:

S

max

= 4

(maximum number of steps),

γ

e

x

c

= γ

e

y

c

= 0.01

and

γ

e

L

_{= 0.05}

(multi-step pro ess thresholds),

K

max

= 500

(maximum number of optimization iterations),

γ

Θ

= 0.2

and

γ

τ

= 0.02

(optimization thresholds),

K

Θ

= K

τ

= 0.15 K

max

(stability

ounters), and

γ

th

= 10

−5

(threshold on the ost fun tion).

Figure 3 shows samples of re onstru tions with the IMSA-LS. At the rst step [Fig. 3(a) -

s = 1

℄, the s atterer is orre tly lo ated, but its shape is only roughly estimated. Thankstothemulti-resolutionrepresentation,thequalitativeimagingofthe s attererisimproved inthenext step[Fig. 3(b)-

s = s

opt

= 2

℄as onrmedbytheerror indexesinTab. 1. For omparisonpurposes,theproleretrievedbythesingle-resolution method [23℄ (indi ated in the following as Bare-LS), when

D

I

has been dis retized in

N

Bare

= 31 × 31

equalsub-domains, isshown [Fig. 3( )℄. In general,the dis retization

of the Bare-LS has been hosen in order to a hieve in the whole investigation domain a re onstru tion with the same level of spatial resolution obtained by the IMSA-LS in the

RoI

at

s = s

opt

.

Although the nal re onstru tions [Figs. 3(b)( )℄ a hieved by the two approa hes aresimilarandquite losetothetrues atterersampledatthespatialresolutionof Bare-LS [Fig. 3(d)℄ and IMSA-LS [Fig. 3(b)℄, the IMSA-LS more faithfully retrieves the symmetryofthea tualobje t,eventhoughthere onstru tionerrorappearstobelarger thanthe one of the Bare-LS (Fig. 4). During the iterativepro edure, the ost fun tion

Θ

opt

= Θ {φ

opt

s

}

isinitially hara terizedbyamonotoni allyde reasingbehavior. Then,

Θ

opt

⌋

_{IM SA}

be omesstationaryuntilthe stopping riteriondened by relationships(11)

and (12) is satised (Fig. 4 -

s = 1

). Then, after the update of the eld distribution indu ing the error spike when

s = s

opt

= 2

and

k

s

= 1

,

Θ

opt

⌋

IM SA

settles to a value of

6.28 × 10

−4

whi hisoftheorder oftheBare-LS error(

Θ

opt

⌋

Bare

= 1.42 × 10

−4

). Su ha slightdieren ebetween

Θ

opt

⌋

IM SA

and

Θ

opt

⌋

Bare

dependsonthedierentdis retization [i.e., the basis fun tions

B



r

_n(i=2)



,

n(i) = 1, ..., N

IM SA

are not the same as those of

Bare-LS℄, but it does not ae t the re onstru tion in terms of both lo alization and area estimation, sin e

δ⌋

IM SA−LS

< δ⌋

Bare−LS

and

∆⌋

IM SA−LS

< ∆⌋

Bare−LS

(Tab. 1). Fig. 4 alsoshows that the multi-step multi-resolutionstrategy is hara terized by a lower omputationalburden be ause of the smallernumber of iterationsfor rea hing the onvergen e (Fig. 4 -

k

tot

⌋

IM SA

= 125

vs.

k

tot

⌋

Bare

= 177

, being

k

tot

the total numberofiterationsdenedas

k

tot

=

P

s

opt

redu ed number of oating-pointoperations. As a matterof fa t,sin e the omplexity of the LS-based algorithms is of the order of

O (2 × η

3

₎

,

η = N

IM SA

, N

Bare

(i.e., the solutionof twodire t problems is ne essary for omputing anestimate of the s attered eld and for updating the velo ity ve tor), the omputational ost of the IMSA-LS at ea hiteration istwoorders in magnitudesmaller than that of the Bare-LS.

3.1.2. Noisy Data As for the stability of the proposed approa h, Figure 5 shows

the re onstru tions with the IMSA-LS [Figs. 5(a)( )(e)℄ ompared to those of the Bare-LS [Figs. 5(b)(d)(f)℄ with dierent levels of additive noise on the s attered data [

SNR = 20 dB

(top);

SNR = 10 dB

(middle);

SNR = 5 dB

(bottom)℄. As expe ted, when the

SNR

de reases, the performan es worsen. However, as outlined by the behavior of the error gures in Tab. 2, blurred data and/or noisy onditions ae t moreevidentlythe Bare implementationthan themulti-resolutionapproa h. For ompleteness, the behavior of

Θ

opt

⌋

IM SA

versus the iteration index is reported in Fig. 6 for dierent levels of

SNR

. As it an be noti ed, the value of the error atthe end of the iterative pro edure de reases as the

SNR

in reases.

In the se ond experiment, the same ir ular s atterer, but entered at a dierent position within a larger investigation square of side

L

D

= 2λ

(

ρ

O

= 2λ

), has been re onstru ted. A ording to [9℄,

M = 20

;

v = 1, ..., V

re eivers and

V = 20

views are onsidered and

D

I

isdis retized in

N

IM SA

= 13 × 13

pixels.

Figure 7(a) shows the re onstru tion obtained at the onvergen e (

s

opt

= 3

) by IMSA-LS when

SNR = 5 dB

. The result rea hed by the Bare-LS (

N

BARE

= 47 × 47

) is reported in Fig. 7(b) as well. As it an be noti ed, the multi-resolution inversion is hara terizedby abetterestimationof the obje t enter and shapeas onrmed by the values of

δ

and

∆

(

δ⌋

IM SA−LS

= 0.59

vs.

δ⌋

Bare−LS

= 2.72

and

∆⌋

IM SA−LS

= 0.48

vs.

∆⌋

_Bare−LS

= 0.64

). As for the omputationalload,the same on lusions fromprevious

experiments hold true.

3.2. Syntheti Data - Re tangular S atterer

These ondtest asedealswith amore omplexs attering onguration. Are tangular o- entered s atterer (

L = 0.27λ

and

W = 0.13λ

) hara terized by a diele tri permittivity

ǫ

C

= 1.8

islo atedwithinaninvestigationdomainof

L

D

= 3λ

asindi ated bythe reddashedlineinFig. 8. Insu ha ase,theimagingsetupismadeup of

V = 30

sour es and

M = 30

measurement points for ea h view

v

[9℄.

D

I

is partitioned into

N

IM SA

= 19 × 19

sub-domains (while

N

Bare

= 33 × 33

)and

∆t

1

isset to

0.06

.

3.2.1. Validation of the Stopping Criteria Before dis ussing the re onstru tion

apabilities,let usshowa result on erned with the behaviorof the proposedapproa h when varying the user-dened thresholds (

γ

Θ

,

γ

τ

,

γ

e

x

c

,

γ

e

y

c

,

γ

e

L

)of the stopping riteria. Figure8 displays the re onstru tions a hieved by using the sets of parameters given in Tab. 3[

Γ

1

- Fig. 8(a);

Γ

2

-Fig. 8(b);

Γ

3

-Fig. 8( );

Γ

4

-Fig. 8(d)℄while thebehaviors of the ost fun tion are depi ted in Fig. 9. As it an be noti ed, the total number of iterations

k

tot

in reases as the values of the thresholds

γ

Θ

and

γ

τ

de rease. However, in spite of a larger

k

tot

, using lowerthreshold values does not providebetter results,as shown by the omparison between settings

Γ

2

and

Γ

4

[Figs. 8(b)-(d),and Fig. 9℄. The sets of parameters hara terized by

γ

Θ

= 0.2

and

γ

τ

= 0.02

provide a good trade-o between the arising omputational burden and the quality of the re onstru tions. As far as the stopping riterion of the multi-resolution pro edure is on erned, Figure 9 also shows two dierent behaviors of the ost fun tion when using

Γ

2

and

Γ

3

(letting

γ

Θ

= 0.2

and

γ

τ

= 0.02

). In parti ular, the proposed approa h stops at

s

opt

= 3

,

insteadof

s

opt

= 4

,whenin reasing by adegreeofmagnitudethe values of

γ

e

x

c

,

γ

e

y

c

,and

γ

_L

_e

. Although with a heavier omputational burden, the hoi e

γ

e

x

c

= γ

_e

_y

c

= 0.01

and

γ

_L

_e

= 0.05

results more ee tive [see Fig. 8(b)vs. Fig. 8( )℄.

3.2.2. Noisy Data Figures 10-12 and Table 4 show the results from the omparative

study arriedoutin orresponden ewithdierentvaluesofsignal-to-noiseratio[

SNR =

20 dB

- Fig. 10(a) vs. Fig. 10(b);

SNR = 10 dB

- Fig. 10( ) vs. Fig. 10(d);

SNR =

5 dB

- Fig. 10(e) vs. Fig. 10(f)℄. They further onrm the reliability and e ien y

of the multi-resolution strategy in terms of qualitative re onstru tion errors (Fig. 11), espe iallywhen the noise levelgrows. In parti ular,the Bare implementationdoes not yield either the position or the shape of the re tangular s atterer when

SNR = 5 dB

, whereas the IMSA-LS properly retrieves both the bary enter and the ontour of the target. As forthe omputational ost, itshouldbenoti edthatalthoughtheIMSA-LS requires a greater number of iterationsfor rea hing the onvergen e (Fig. 12,Tab. 4), the total amountof omplex oating-pointoperations,

f

pos

= O (2 × η

3

_{) × k}

tot

, usually results smaller(Tab. 4).

3.3. Numeri alData - Hollow Cylinder

The third test ase is on erned with the inversion of the data s attered by a higher permittivity (

ǫ

C

= 2.5

) o- entered ylindri al ring, letting

L

D

= 3λ

. The external radius of the ring is

ρ

ext

=

2

3

λ

, and the internal one is

ρ

int

=

λ

3

. By assuming the same arrangement of emitters and re eivers as inSe tion 3.2, the investigation domain isdis retized with

N

IM SA

= 19 × 19

and

N

Bare

= 35 × 35

square ells forthe IMSA-LS and the Bare-LS,respe tively. Moreover,

∆t

1

is initializedto

0.003

.

As it an be observed from Fig. 13,where the proles when

SNR = 20 dB

[Figs. 13(a)(b)℄and

SNR = 10 dB

[Figs. 13( )(d)℄ re onstru ted by means of the IMSA-LS [Figs. 13(a)( )℄ and the Bare-LS [Figs. 13(b)(d)℄ are shown, the integrated strategy usually over omes the standard one both in lo ating the obje t and in estimating the shape. In parti ular, when

SNR = 20 dB

, the distribution in Fig. 13(a) is a faithful estimate of the s atterer under test (

δ⌋

IM SA−LS

= 1.25

and

∆⌋

IM SA−LS

= 3.13

). On the ontrary, the re onstru tion with the Bare-LS is very poor (

δ⌋

Bare−LS

= 65.2

and

∆⌋

_Bare−LS

= 34.39

). Certainly, a smaller

SNR

value impairs the inversion as shown

in Fig. 13( ) [ ompared to Fig. 13(a)℄. However, in this ase, the IMSA-LS is able to properly lo ate the obje t (

δ⌋

IM SA−LS

= 1.7

vs.

δ⌋

Bare−LS

= 65.9

) giving rough but useful indi ations about its shape (

∆⌋

IM SA−LS

= 7.6

vs.

∆⌋

Bare−LS

= 34.55

).

3.4. Syntheti Data - Multiple S atterers

The lastsyntheti test ase is aimedatillustratingthe behavior of the IMSA-LS when dealingwith

P = 3

s atterers(

ǫ

C

= 2.0

)distan edfromone another. Thetestgeometry is hara terizedby anellipti o- entered ylinder,a ir ularo- entereds atterer, and a square o- entered obje t lo ated in a square investigation domain hara terized by

L

D

= 3λ

. By adopting the same arrangement of emitters and re eivers as in Se tion

3.3, the investigationdomainisdis retized with

N

IM SA

= 23 × 23

and

N

Bare

= 31 × 31

square ells for the IMSA-LS and the Bare-LS, respe tively. Moreover,

∆t

1

is set to

0.03

.

Figures 14 and 15 show the results from the omparative study arried out in orresponden e with dierent values of signal-to-noise ratio. As shown by the re onstru tions (Fig. 14) andas expe ted,the multi-resolutionapproa hprovidesmore a urate results and appears to be more reliablethan the Bare-LS espe ially with low

(Fig. 15), for whi h the IMSA-LS a hievesa lowerlo alizationerror as well asa lower area error than those of Bare-LS, espe ially for

SNR = 5 dB

. On the other hand, both algorithmsprovide goodestimates of the s atterer undertest when inverting data ae ted by lownoise [

SNR = 20 dB

- Fig. 14(a)vs. Fig. 14(b);Fig. 15(a)and (b)℄.

3.5. Laboratory-Controlled Data

In order to further assess the ee tiveness of the IMSA-LS also in dealing with experimental data, the multiple-frequen y angular-diversity bi-stati ben hmark provided by Institut Fresnel in Marseille (Fran e) has been onsidered. Withreferen e to the experimental setup des ribed in [34℄, the dataset dielTM_de 8f.exp has been pro essed. The eld samples [

M = 49

,

V = 36

℄ are related to an o- entered homogeneous ir ular ylinder

ρ = 15

mm in diameter, hara terized by a nominal value of the obje t fun tion equal to

τ (r) = 2.0 ± 0.3

, and lo ated at

x

c

= 0.0

,

y

c

= −30

mm within an investigation domain assumed in the following of square

geometry and extension

20 × 20

m

2

.

By setting

ǫ

C

= 3.0

, the re onstru tions a hieved are shown in Fig. 16 (left olumn) ompared to those from the standard LS (right olumn) at

F = 4

dierent operation frequen ies. Whatever the frequen y, the unknown s atterer isa urately lo alizedand both algorithmsyield,at onvergen e, stru turesthat o upy alarge subsetof the true obje t. Su hasimilarityofperforman es,usuallyveriedinsyntheti experimentswhen the value of

SNR

is greaterthan

20 dB

,seems to onrmthe hypothesisof alow-noise environment as already eviden ed in[35℄.

Finally, also in dealing with experimental datasets, the IMSA-LS proves its e ien y sin e the overall amount of omplex oating point operations still remains two orders inmagnitude lowerthan the one of the Bare-LS (Tab. 5 -Fig. 17).

4. Con lusions

In this paper, a multi-resolution approa h for qualitative imaging purposes based on shape optimization has been presented. The proposed approa h integrates the multi-s ale strategy and the level set representation of the problem unknowns in order to protablyexploittheamountofinformation olle tablefromthes atteringexperiments as wellas the available a-priori information onthe s atterer under test.

•

innovative multi-level representation of the problem unknowns in the

shape-deformation-based re onstru tion te hnique;

•

ee tive exploitation of the s attering data through the iterative multi-step

strategy;

•

limitationof theriskof beingtrappedinfalsesolutionsthankstotheredu ed ratio

between data and unknowns;

•

useful exploitationofthe a-priori information(i.e.,obje thomogeneity) about the

s enariounder test;

•

enhan ed spatial resolution limitedto the region of interest.

From the validation on erned with dierent s enarios and both syntheti and experimental data, the following on lusions an be drawn:

•

the IMSA-LS usually proved more ee tivethan thesingle-resolution

implementa-tion, espe ially whendealing with orrupted data s attered fromsimple as wellas omplex geometries hara terized by one or several obje ts;

•

the integratedstrategy appearedless omputationally-expensivethanthe standard

approa h in rea hing a re onstru tion with the same level of spatial resolution within the support of the obje t.

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a

( )

b

( )

c

( )

Actual Scatterer

Υ

1

(e

x

c

s=1

, e

y

s=1

c

)

r

_n

_s=1

r

n

s=1

R

s=2

φ

_k

opt

s=1

φ

k

s=1

s = 1

,

k = 1

s = 1

,

k = k

opt

s = 2

,

k = 1

D

I

DI

D

I

e

L

s=1

r

_n

_s=2

φ

k

s=2

Figure 1. Graphi alrepresentationofthe IMSA-LS zoomingpro edure. (a)Firststep (

k = 1

): the investigationdomainisdis retizedin

N

sub-domainsand a oarse solution is looked for. (b) First step (

k = k

opt

): the region of interest that ontains the rst estimateof the obje tis dened. ( )Se ond step (

k = 1

): anenhan ed resolution level is used onlyinside the region of interest.

Field Distribution Updating

Determine

Cost Function Evaluation

Level Set Update

Problem Unknown Representation

FALSE

TRUE

TRUE

Stop

Initialization

FALSE

IMSA

Stopping Criterion

Stopping Criteria

LS

Compute e

E

v

k

i

r

n

i

r

n

i

∈ D

I

e

x

c

s

,e

y

c

s

,e

L

s

Determine e

ξ

v

k

s

r

m

v

and compute

Θ {φ

k

s

}

r

m

v

∈ D

O

Compute

φ

k

s

from

φ

k

s

−1

e

τ

k

s

(r) =

P

s

i=1

P

N

I M SA

n

i

=1

τ

k

i

B r

n

i

r

n

i

∈ D

I

s-th resolution level

k

s

= k

s

+ 1

Υ

s

opt

Υ

s

= Υ

1

k

s

= 0

s = s + 1

γ

x

_e

c

,γ

e

y

c

,γ

e

L

γ

τ

, γ

Θ

, γ

th

, K

max

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

R

(2)

1

τ (x, y)

0

1

τ (x, y)

0

(a) (b)

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1

τ (x, y)

0

1

τ (x, y)

0

( ) (d)

Figure 3. Numeri al Data. Cir ular ylinder (

ǫ

C

= 1.8

,

L

D

= λ

, Noiseless Case). Re onstru tions with IMSA-LS at (a)

s = 1

and (b)

s = s

opt

= 2

, ( ) Bare-LS. Optimalinversion (d).

10

-4

10

-3

10

-2

10

-1

10

0

0

50

100

150

200

Θ

Iteration, k

s=1

s=2

IMSA (

Θ

_opt

)

IMSA (

Θ

)

BARE

Figure 4. Numeri al Data. Cir ular ylinder (

ǫ

C

= 1.8

,

L

D

= λ

, Noiseless Case). Behavior of the ost fun tion.

R

(2)

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1

τ (x, y)

0

1

τ (x, y)

0

(a) (b)

(3)

R

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

(2)

R

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1

τ (x, y)

0

1

τ (x, y)

0

( ) (d)

(3)

R

R

₍₂₎

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1

τ (x, y)

0

1

τ (x, y)

0

(e) (f)

Figure 5. Cir ular ylinder (

ǫ

C

= 1.8

,

L

D

= λ

, Noisy Case). Re onstru tions with IMSA-LS (left olumn) and Bare-LS (right olumn) for dierent values of

SNR

[

SNR = 20 dB

(top),

SNR = 10 dB

(middle),

SNR = 5 dB

(bottom)℄.

10

-3

10

-2

10

-1

10

0

10

1

0

20

40

60

80

100

Θ

opt

Iteration, k

SNR = 20 dB

SNR = 10 dB

SNR = 5 dB

s = 1

s = 2

s = 1

s = 2

s = 2

s = 1

Figure 6. Numeri alData. Cir ular ylinder (

ǫ

C

= 1.8

,

L

D

= λ

). Behaviorof the ost fun tion versus the noise level.

(2)

R

(3)

R

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

1

τ (x, y)

0

(a)

x

y

λ

λ

−0.5

0.5

0.5

−0.5

1.0

−1.0

−1.0

1.0

1

τ (x, y)

0

(b)

Figure 7. Numeri al Data. Cir ular ylinder (

ǫ

C

= 1.8

,

L

D

= 2λ

,

SNR = 5 dB

). Re onstru tions with (a) IMSA-LS and (b)Bare-LS.

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1

τ (x, y)

0

1

τ (x, y)

0

(a) (b)

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1

τ (x, y)

0

1

τ (x, y)

0

( ) (d)

Figure 8. Numeri alData. Re tangular ylinder(

ǫ

C

= 1.8

,

L

D

= 3λ

,NoiselessCase). Re onstru tions with IMSA-LS for the dierent settings of Tab. 3[(a)

Γ

1

, (b)

Γ

2

, ( )

10

-4

10

-3

10

-2

10

-1

10

0

10

1

0

200

400

600

800

1000

1200

1400

Θ

opt

Iteration, k

Γ

₃

Γ

₂

Γ

₄

Γ

₁

Γ

1

Γ

₂

,

Γ

₃

Γ

₄

Figure 9. Numeri alData. Re tangular ylinder(

ǫ

C

= 1.8

,

L

D

= 3λ

,NoiselessCase). Behavior of the ost fun tion of IMSA-LS forthe dierent settingsof Tab. 3.

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1

τ (x, y)

0

1

τ (x, y)

0

(a) (b)

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1

τ (x, y)

0

1

τ (x, y)

0

( ) (d)

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1

τ (x, y)

0

1

τ (x, y)

0

(e) (f)

Figure 10. Numeri al Data. Re tangular ylinder (

ǫ

C

= 1.8

,

L

D

= 3λ

, Noisy Case). Re onstru tions with IMSA-LS (left olumn)and Bare-LS (right olumn)for dierent values of

SNR

[

SNR = 20 dB

(top),

SNR = 10 dB

(middle),

SNR = 5 dB

(bottom)℄.

10

0

10

1

10

2

10

3

5

10

15

20

δ

SNR [dB]

IMSA

BARE

(a)

2.5

2.0

1.5

1.0

0.5

0.0

5

10

15

20

∆

SNR [dB]

IMSA

BARE

(b)

Figure 11. Numeri al Data. Re tangular ylinder (

ǫ

C

= 1.8

,

L

D

= 3λ

, Noisy Case). Values of the error guresversus

SNR

.

10

-3

10

-2

10

-1

10

0

10

1

0

200

400

600

800

1000

1200

1400

Θ

opt

Iteration, k

IMSA

BARE

(a)

10

-2

10

-1

10

0

10

1

0

100

200

300

400

500

600

Θ

opt

Iteration, k

IMSA

BARE

(b)

10

-1

10

0

10

1

0

100

200

300

400

500

600

Θ

opt

Iteration, k

IMSA

BARE

( )

Figure 12. Numeri al Data. Re tangular ylinder (

ǫ

C

= 1.8

,

L

D

= 3λ

, Noisy Case). Behavior of the ost fun tion versus the iteration index when (a)

SNR = 20 dB

, (b)

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

2

τ (x, y)

0

2

τ (x, y)

0

(a) (b)

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

2

τ (x, y)

0

2

τ (x, y)

0

( ) (d)

Figure 13. Numeri al Data. Hollow ylinder (

ǫ

C

= 2.5

,

L

D

= 3λ

, Noisy Case). Re onstru tions with IMSA-LS (left olumn)and Bare-LS (right olumn)for dierent values of

SNR

[

SNR = 20 dB

(top),

SNR = 10 dB

(bottom)℄.

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R

(3)

R

(3)

R

(3)

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1

τ (x, y)

0

1

τ (x, y)

0

(a) (b)

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R

₍₃₎

R

₍₃₎

R

₍₃₎

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1

τ (x, y)

0

1

τ (x, y)

0

( ) (d)

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

R

₍₃₎

R

₍₃₎

R

₍₃₎

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

1

τ (x, y)

0

1

τ (x, y)

0

(e) (f)

Figure 14. Numeri al Data. Multiple s atterers (

ǫ

C

= 2.0

,

L

D

= 3λ

, Noisy Case). Re onstru tions with IMSA-LS (left olumn)and Bare-LS (right olumn)for dierent values of

SNR

[

SNR = 20 dB

(top),

SNR = 10 dB

(middle),

SNR = 5 dB

(bottom)℄.

2.0

2.5

3.0

3.5

4.0

5

10

15

20

δ

SNR [dB]

IMSA

BARE

(a)

1.0

2.0

3.0

4.0

5.0

6.0

5

10

15

20

∆

SNR [dB]

IMSA

BARE

(b)

Figure 15. Multiple s atterers (

ǫ

C

= 2.0

,

L

D

= 3λ

, Noisy Case). Values of the error gures versus

SNR

.

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

x

y

λ

λ

−0.25

0.25

0.25

−0.25

0.50

−0.50

−0.50

0.50

3

τ (x, y)

0

3

τ (x, y)

0

(a) (b)

x

y

λ

λ

−0.5

0.5

0.5

−0.5

1.0

−1.0

−1.0

1.0

x

y

λ

λ

−0.5

0.5

0.5

−0.5

1.0

−1.0

−1.0

1.0

3

τ (x, y)

0

3

τ (x, y)

0

( ) (d)

Figure 16(I). Experimental Data (Dataset Marseille [34℄). Cir ular ylinder (dielTM_de 8f.exp). Re onstru tionswithIMSA-LS(left olumn)andBare-LS (right olumn)at dierent frequen ies

f

[

f = 1 GHz

(a)(b);

f = 2 GHz

( )(d)℄.

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

x

y

λ

λ

−0.75

0.75

0.75

−0.75

1.50

−1.50

−1.50

1.50

3

τ (x, y)

0

3

τ (x, y)

0

(e) (f)

x

y

λ

λ

−1.0

1.0

1.0

−1.0

2.0

−2.0

−2.0

2.0

x

y

λ

λ

−1.0

1.0

1.0

−1.0

2.0

−2.0

−2.0

2.0

3

τ (x, y)

0

3

τ (x, y)

0

(g) (h)

Figure 16(II). Experimental Data (Dataset Marseille [34℄). Cir ular ylinder (dielTM_de 8f.exp). Re onstru tionswithIMSA-LS(left olumn)andBare-LS (right olumn)at dierent frequen ies

f

[

f = 3 GHz

(e)(f);

f = 4 GHz

(g)(h)℄.

M. Benedetti et al.

10

-2

10

-1

10

0

10

1

10

2

0

200

400

600

800

1000

Θ

opt

Iteration, k

IMSA

BARE

10

-3

10

-2

10

-1

10

0

10

1

10

2

0

200

400

600

800

1000

Θ

opt

Iteration, k

IMSA

BARE

(a) (b)

10

-3

10

-2

10

-1

10

0

10

1

10

2

0

200

400

600

800

1000

Θ

opt

Iteration, k

IMSA

BARE

10

-3

10

-2

10

-1

10

0

10

1

10

2

0

200

400

600

800

1000

Θ

opt

Iteration, k

IMSA

BARE

( ) (d) 17. Exp erimen tal Data (Dataset Marseille [34 ℄). Cir ular ylinder 8f.exp ). Beha vior of the ost fun tion v ersus the n um b er of iterations ( a )

f

=

1

G

H

z

, ( b )

f

=

2

G

H

z

, ( )

f

=

3

G

H

Z

, and ( d )

f

=

4

G

H

z

.

IMSA − LS

Bare − LS

s = 1

s = 2

δ

6.58 × 10

−6

2.19 × 10

−6

5.21 × 10

−1

∆

2.36

0.48

0.64

M.

Benedetti

et

al.

SNR = 20 dB

SNR = 10 dB

SNR = 5 dB

IMSA − LS

Bare − LS

IMSA − LS

Bare − LS

IMSA − LS

Bare − LS

δ

5.91 × 10

−1

_2.72

_2.28

_2.45

_{6.78 × 10}

−1

_1.63

∆

0.98

1.28

1.07

1.80

1.50

2.07

2. Numeri al Data. Cir ular ylinder (

ǫ

C

=

1.

8

, Noisy Case ). V alues of the indexes for dieren t v alues of

S

N

R

.

Set of Parameters

γ

Θ

γ

τ

γ

x

_e

c

, γ

e

y

c

γ

e

L

Γ

₁

0.5

0.05

0.01

0.05

Γ

2

0.2

0.02

0.01

0.05

Γ

3

0.2

0.02

0.1

0.5

Γ

₄

0.02 0.002

0.01

0.05

Table 3. Numeri al Data. Re tangular ylinder (

ǫ

C

= 1.8

,

L

D

= 3λ

, Noiseless Case). Dierent settings forthe parametersof the stopping riteria.

M.

Benedetti

et

al.

SNR = 20 dB

SNR = 10 dB

SNR = 5 dB

IMSA − LS

Bare − LS

IMSA − LS

Bare − LS

IMSA − LS

Bare − LS

k

tot

1089

41

393

53

410

28

N

361

1089

361

1089

361

1089

f

pos

1.02 × 10

11

1.02 × 10

11

3.70 × 10

10

1.37 × 10

11

3.86 × 10

10

7.23 × 10

10

4. Numeri al Data. Re tangular ylinder (

ǫ

C

=

1.

8

,

L

D

=

3λ

, Noisy Case ). indexes for dieren t v alues of

S

N

R

.