Multicrack Detection in Two-Dimensional Structures by Means of Ga-Based Strategies

UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38123 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

MULTICRACK DETECTION IN TWO-DIMENSIONAL

STRUCTURES BY MEANS OF GA-BASED STRATEGIES

M. Benedetti, M. Donelli, and A. Massa

January 2007

means of GA-based Strategies

M. Benedetti, M.Donelli, Member, IEEE,and A.Massa, Member, IEEE,

Department of Information and Communi ation Te hnology University of Trento,Via Sommarive 14,38050 Trento - Italy Tel. +39 0461 882057,Fax +39 0461 882093

E-mail: andrea.massaing.unitn.it,

{manuel.benedetti,massimo.donelli}dit.unitn.it Web-site: http://www.eledia.ing.unitn.it

means of GA-based Strategies

M. Benedetti, M.Donelli, and A. Massa

Abstra t

Thispaperproposesamethodologi alapproa hforthedete tionofmultipledefe ts insidediele tri or ondu tivemedia. Twoinnovativealgorithmsaredeveloped start-ing fromthe inverse s attering equations solved by means of dierent optimization strategies. In the rst implementation, a hierar hi al strategy based on parallel-subpro esses is onsidered, whereas the se ond algorithm employs a single-pro ess ar hite ture. Whatever the implementation, the arising ost fun tion isminimized through a suitable hybrid- oded geneti algorithm, whose individuals en ode the problem unknowns. In order to a hieve a omputational saving, the formulation based on the inhomogeneous Green's fun tion is adopted and ea h ra k-region is parametrizedbymeans ofasele tedsetofdes riptiveparameters. The approa has well as its dierent implementations are assessed through a sele ted set of numer-i al experiments and in omparison with previously developed single- ra k inverse s attering methods.

Key-words: Non-destru tive Testing and Evaluation, Mi rowave Imaging, Multi ra k Dete tion

Nondestru tiveEvaluationandTesting (NDE/NDT)is aninterdis iplinaryresear harea devoted to the development of advan ed sensors, measurement te hnologies, and imag-ing te hniques for the hara terization of materials and stru tures in a non-destru tive fashion. Non-destru tive evaluation (NDE) and testing (NDT) are mandatory in many industrialpro essesthatrequireana urateanalysisofdiele tri or ondu tivestru tures (e.g., industrialprodu ts and artefa ts).

As far as the state-of-the-art is on erned, ultrasoni s [1℄,

χ

and

γ

-rays [2℄[3℄, infrared [4℄ and eddy urrents [5℄, are the methodologiesmainlyused indealing with NDE/NDT problems. Re ently, some emerging te hnologies su h as mi rowaves are appearing in Subsurfa e Sensing methods for the nondestru tive evaluation (see [6℄[7℄[8℄[9℄[10℄ and the referen es therein forageneral overview) and now, insome appli ations,the employ-mentofinterrogatingmi rowavesisre ognizedasasuitablediagnosti tool[11℄-[13℄. The main reasonsof thegrowing interest andrapid developmentof mi rowave-based method-ologies an be summarized by the following key-points: (a) ele tromagneti elds in the mi rowavesrangepenetrateallmaterials(unlessideal ondu tors)andthes atteredelds arerepresentativeoftheoverallvolumeoftheobje tundertestandnotonlyofitssurfa e; (b)mi rowaveimagingmodalitiesare verysensitivetothewater ontentofthe spe imen; ( ) mi rowave sensors an be used without me hani al onta ts with the spe imen, as well. Moreover, mi rowave te hnologies an be onsidered omplementary approa hes to onventionalinspe tionte hniques guaranteeingnon-invasivemeasurementsand avoiding ollateralee ts onthe spe imen under test (being safe non-ionizingradiations).

Intheframeworkofmi rowavemethodologies,afurtheradvan eisrepresentedbyimaging te hniques based on inverse s attering approa hes [14℄-[18℄, where a omplete image of the stru ture under test is looked for. Unfortunately, these te hniques are hara terized by several drawba ks su h as ill-position and non-linearity as well as the presen e of lo al minima that partially prevent their use in industrial appli ations (unlike passive te hniques) [11℄. Therefore, in order to allow an ee tive te hnologi al transfer in the frameworkofindustrialpro esses,otherdevelopmentsaremandatory. Letus onsiderthe area of post-pro essing te hniques for the diagnosis of the spe imen under test starting

bythelow-speedof there onstru tion methods. Moreover, the wavelengthof theprobing ele tromagneti sour estronglylimitsthe a hievable spatialresolutionoritrequireshigh omputational osts for obtaininga detailedre onstru tion.

However, in the framework of inverse s attering te hniques, dealing with the dete tion of defe ts (also indi ated as  ra ks in the following) in known host stru tures seems to be loser to a realisti appli ation be ause of the large amount of available a-priori information on the problem in hand. Su h a topi has been ee tively addressed in [19℄, [20℄, and [21℄. However, the proposed approa hes demonstrated their feasibility and ee tiveness insimpliedgeometri ongurations hara terizedby the presen eof a single defe t.

In orderto onsider more omplexand realisti s enarios (e.g.,multiple defe ts,irregular shapes of the defe ts, et ...), this work presents two innovative NDE/NDT strategies aimed at dete ting the presen e of more than one defe tive region inside a diele tri or ondu tive host-medium. Sin e there is the a-priori knowledge of the unperturbed geometry, the ra ks are dened as in lusions in a known stru ture and approximated with a limited set of essential parameters. Su h a parameterization and the use of a suitableGreen's fun tionallowa redu tionof the numberof unknowns and onsequently a non-negligible omputational saving during the re onstru tion pro ess arried out in terms of the optimization of asuitable ost fun tion.

As far as the proposed implementations are on erned, the main dieren e lies in the ar hite ture of the solution pro edure and, onsequently, in ustomized and innovative optimizations based on a Geneti Algorithm (GA). The former strategy is based on a hierar hi al implementation, whi h onsiders a set of parallel sub-pro esses, ea h one inspe ting on a solutionwith a dierent xed number of ra ks. The latter deals with a single optimizationpro ess aimed atlookingfor the best re onstru tion among dierent ra k-length solutions.

The paper is stru tured as follows. Se tion 2 provides the mathemati al formulation for the inverse s attering approa h to the re onstru tion of multiple-defe ts in a two-dimensional s enario. In Se tion3,the two implementationsof the methodare presented

twostrategies indealing withNDE/NDT problems are analyzedinSe tion 4. Finally,in Se tion 5,a dis ussion follows and possible future developments are sket hed.

2 Mathemati al Formulation

Let us onsider a ylindri geometrywhere anarea under test

H

is o upied by a known hostmedium hara terizedbyanobje tfun tion

τ

H

(x, y) = ε

H

−1−j

σ

H

2πf ε

0

,

ε

H

and

σ

H

be-ingitsdiele tri permittivityand ondu tivity,respe tively(

f

is theworking frequen y). As shown in Fig. 1, su h a region lies in a homogeneous ba kground whose ele tromag-neti properties, without loss of generality, are that of the va uum

(ε

0

, σ

0

)

. A set of

C

defe ts

D

i

,

i = 1, ..., C

,belongsto

H

. The geometri and ele tromagneti hara teristi s of the regions

D

i

are unknown as well as their number. The two-dimensional s enario is probed by

V

ele tromagneti TM plane waves with ele tri in ident elds linearly-polarizedalong the axisof symmetry of the stru ture under test,

E

v

inc

(r) = E

inc

v

(x, y) ˆ

z

,

v = 1, ..., V

. A ording to the inverse s attering equations [22℄, the ele tri eld indu ed

in the investigation domain

H

is equivalent to that radiated in the free-spa e by an equivalent urrent density

J

v

_{(x, y)}

E

tot

v

(x, y) = E

v

inc

(x, y) +

Z Z

H

J

v

(x

′

_{, y}

′

_)G

0

(x, y/x

′

, y

′

) dx

′

dy

′

(1)

where

G

0

is the free-spa e Green's fun tion [19℄ and

J

v

_{(x, y) = τ (x, y) E}

v

tot

(x, y)

being

τ (x, y) = ε (x, y) − 1 − j

σ(x,y)

2πf ε

0

. Equation (1) an be reformulated in terms of a dier-ential equivalent urrent density

J

v

D

i

(x, y)

dened in

D

i

,

i = 1, ..., C

and radiatingin an inhomogeneous medium [23℄. A ordingly, the ele tri eld an be expressed as

E

v

tot

(x, y) = E

inc

v

(x, y) +

R R

H

τ

H

(x

′

, y

′

) E

v

tot

(cf )

(x

′

, y

′

) G

0

(x, y/x

′

, y

′

) dx

′

dy

′

+

+

P

C

_i=1

R R

_D

i

τ

D

i

(x

′

_{, y}

′

_{) E}

v

tot

(c),i

(x

′

, y

′

) G

1

(x, y/x

′

, y

′

) dx

′

dy

′

(2)

where the se ondterm onthe right side of(2)is theele tri elds attered fromthe host

E

v

_{(x, y)}

turbed domain

H

. The lastterm of (2) is on erned with the eld distribution radiated by

J

v

D

i

(x, y) = τ

D

i

(x, y) E

v

tot

(c),i

(x, y)

where

τ

D

i

(x, y) = τ (x, y) − τ

H

(x, y)

,

(x, y) ∈ D

i

,

i = 1, ..., C

[20℄arethe dierentialobje tfun tions,

G

1

being theinhomogeneous Green's

fun tion. Byassumingtheknowledgeofthehostmedium(generallyavailable),itisuseful to rewrite(2)as follows:

E

v

tot

(x, y) = E

inc(cf )

v

(x, y) +

C

X

i=1

Z Z

D

i

τ

D

i

(x

′

_{, y}

′

_{) E}

v

tot

(c),i

(x

′

, y

′

) G

1

(x, y/x

′

, y

′

) dx

′

dy

′

(3)

where

E

v

inc

(cf )

(x, y)

istheele tri eldinthes enarioundertestwithoutthedefe t,whi h an be omputed o-lineon e.

Inordertonumeri allydeal withthes atteringequations,letusdis retize

H

in

N

square sub-domains [24℄. Therefore,

D

i

turns out to be lled by

P

i

square pixels a ording to the ra k size and the dis retized operator

G

1

an be easily omputed [20℄ and stored in a

N × N

matrix,

[G

1

]

. Thus, (3) be omes

[E

v

tot

] =



E

v

inc

(cf )



+

C

X

i=1

[G

1,i

] [τ

D

i

]



E

v

tot,i



(4) where:

•

[E

v

tot

]

is a

N × 1

array whose

n

-thelement is

E

v

tot

(x

n

, y

n

)

,

(x

n

, y

n

) ∈ H

;

•

h

E

v

inc(cf )

i

, [E

v

inc

] + [G

o

] [τ

H

]

h

E

v

tot

(cf )

i

isa

N × 1

arraywhose

n

-thentryisthe ele -tri eldwithoutthe defe tatthe

n

-thsubdomainof

H

given by

E

v

inc

(cf )

(x

n

, y

n

) =

E

v

inc

(x

n

, y

n

) +

R R

H

τ

H

(x

′

, y

′

) E

v

tot

(cf )

(x

′

, y

′

) G

0

(x

n

, y

n

/x

′

, y

′

) dx

′

dy

′

;

•



E

v

tot,i



is a

P

i

× 1

ve tor whose

p

i

-th entry identies the value of the ele tri eld in the

p

i

-thpixelof

D

i

,

i = 1, ..., C

;

•

[τ

D

i

]

is a

P

i

× P

i

diagonal matrix, whose non-nullelementsare the values of obje t

fun tion

τ

D

i

in the

P

i

pixels of

D

i

;

•

[G

1,i

]

is the

i

-th inhomogeneous spa e Green's matrix of

N × P

i

elements derived

from

[G

1

]

by sele ting the rows related to the positions

p

i

(

p

i

= 1, ..., P

i

) of the pixels of

D

i

in

H

.

Sin e the inhomogeneous operator

[G

1,i

]

determines the ee ts of the i-th dierential equivalent urrent density lo atedin the unknown region

D

i

, the s attering problem an be reformulated as the re onstru tion of the dierential obje t fun tion

τ

D

i

in the set of pixels

p

i

= 1, ..., P

i

of

H

where the defe t is lo ated. Moreover, in order to further de rease the number of unknowns by adding some a-priori assumptions, ea h region

D

i

is approximated by a re tangular homogeneous domain properly parametrized. In

parti ular, letus des ribethe

i

-thdefe t with the oordinates of the enter of

D

i

,

(x

i

, y

i

)

, the length

L

i

, its side

W

i

, and the orientation

θ

i

. Then, the

P

i

diagonal entries of

[τ

D

i

]

turn out tobe

τ

D

i

⌋

n,n

=











τ (x

n

, y

n

) − τ

H

(x

n

, y

n

)

if

ξ ∈



−

L

i

2

,

L

i

2



and

ζ ∈



−

W

i

2

,

W

i

2



0

otherwise (5)

where

ξ = (x

n

− x

i

)

os

θ

i

+ (y

n

− y

i

)

sin

θ

i

,

ζ = (x

n

− x

i

)

sin

θ

i

+ (y

n

− y

i

)

os

θ

i

, and

n =

1, . . . , P

i

.

Moreover, sin e the ele tri eld indu ed in

D

i

is unknown aswell,the set of parameters to be retrieved duringthe re onstru tion is

χ =



C; Υ

i

, i = 1, ..., C;



E

tot,i

v



, i = 1, ..., C

(6) where

Υ

i

= [(x

i

, y

i

) ; L

i

; W

i

; θ

i

]

and



E

v

tot,i



= {E

v

tot

(x

p

i

, y

p

i

) , p

i

= 1, . . . , P

i

}

,

i = 1, . . . , C

. In order to determine the optimal solution

χ

opt

of the re onstru tion problem from the knowledge of the eld measured in anexternal observation domain

O

[i.e.,

E

v

tot

(x

m

, y

m

)

,

m = 1, ..., M

,

(x

m

, y

m

) ∈ O /

∈ H

℄,oftheeldatthesamelo ationsbutwithoutthedefe t,

E

v

tot

(cf )

(x

m

, y

m

)

, and of the eld without the defe t in the investigation domain

H

[i.e.,

E

v

inc

(x

n

, y

n

)

,

n = 1, ..., N

,

(x

n

, y

n

) ∈ H

℄,the probleminhand isre ast asanoptimization one through the denition of a suitable ost fun tion

Ω(χ) = α



k

[E

v

tot

]−

[

E

tot

v

(cf )

]

−

P

C

i=1

[G

1,i

]

[

τ

_Di

][

E

v

tot,i

]k

2

O

k

[E

v

tot

]−

[

E

inc

v

]k

2

O



+β



k[

E

v

tot

(cf )

]

+[E

v

tot

]−

P

C

i=1

[G

1,i

]

[

τ

Di

][

E

tot,i

v

]k

2

H

k[

E

v

inc

]k

2

H



(7)

the in ident (

β

)orthe s attered (

α

)data depending onthe un ertainties or thenoise levelasso iated with both of them. Moreover, equation(7) isthe sum of two normalized least-square terms quantifying the errors when mat hingthe s attering data.

In order to look for

χ

opt

[ orresponding to the global minimum of the nonlinear ost fun tion (7)℄, a suitable global minimizationstrategy has to be used. Towards this end, twodierentGA-basedapproa heshavebeen developedand theywillbedes ribedinthe followingsub-se tions.

2.1 Hierar hi al Strategy (HS)

Let us assume that the number of defe ts lying in

H

is lower than a xed integer

C

max

(

C

max

≥ C

). Under su h a hypothesis, the hierar hi al approa h onsiders

C

max

parallel re onstru tion sub-pro esses ea h one aimed at investigating the presen e and the har-a teristi s of adierent number of defe ts from

1

up to

C

max

. As far asthe

j

-thpro ess is on erned (

j = 1, . . . , C

max

), a populationof

Q

j

trial solutions oding a xed number of ra ks,

j

, is onsidered

χ

_j

_{= {χ}

j,q

; q = 1, . . . , Q

j

} =

n

j; Υ

i

, i = 1, ..., j;



E

tot,i

v



, i = 1, ..., j

_q

; q = 1, . . . , Q

j

o

.

(8)

Starting from a set of randomly generated solutions

χ

0

j

, the

j

-th population iteratively evolves(

χ

k

j

j

=⇒ χ

k

j

+1

j

,

k

j

beingtheiterationindexofthe

j

-thpro ess)untilastopping ri-terion[

k

j

= K

max

or

Ω {χ

j,opt

} ≤ Ω

th

,

χ

j,min

=

arg

n

min

q=1,..,Q

j

h

min

k

j

=1,...,K

max



Ω

n

χ

k

j

j,q

oio

℄ is rea hed. At the

k

j

-thstep the

(k

j

+ 1)

-thpopulationis generatedby meansof aset of suitablegeneti operatorsdenoted by

ℑ {·}

(

χ

k

j

+1

j

= ℑ

n

χ

k

j

j

o

)and thebesttrialsolution dened so far is stored inthe array

χ

j,min

. When allthe

C

max

pro esses are terminated, the onvergen e solutionisdeterminedastheminimumamongtheset ofsolutions

χ

j,min

,

j = 1, . . . , C

. Thewholeminimizationpro edure anberesumed bythefollowing

pseudo- ode:

for

j = 1, ..., C

max

do

k

j

= 0

χ

0

j

= ℜ

n

χ

_j

o

χ

j,min

=

arg



min

q=1,..,Q

j



Ω



χ

0

j,q



while

h

(k

j

< K

max

)

and



Ω

n

χ

k

j

j,min

o

> Ω

th

i

do

k

j

= k

j

+ 1

χ

k

j

j

= ℑ

n

χ

k

j

−1

j

o

χ

k

j

j,min

=

arg

n

min

q=1,..,Q

j

h

Ω

n

χ

k

j

j,q

oio

if

h

Ω

n

χ

k

j

j,min

o

> Ω

n

χ

k

j

−1

j,min

oi

then

χ

j,min

= χ

k

j

−1

j,min

else

χ

j,min

= χ

k

j

j,min

endif enddo

χ

opt

=

arg

{

min

j=1,..,C

max

[Ω {χ

j,min

}]}

enddo

where

ℜ

is the random fun tion. As far as the implementation of the geneti operators is on erned, the hierar hi al approa h makes use of a multi ra k variable-length hybrid oding (Figure 2). Ea h parameter of the ra k in (6) is binary en oded by onsidering

x

i

,

y

i

,

L

i

,

W

i

,and

θ

i

asdis retevariables:

x

i

= l∆

B

,

l = 1, . . . , B

;

y

i

= l∆

B

,

l = 1, . . . , B

;

L

i

= l∆

D

,

l = 1, . . . , D

;

W

i

= l∆

D

,

l = 1, . . . , D

, and

θ

i

= l∆

Θ

,

l = 1, . . . , Θ

. Moreover,

a real representation is used for oding the eld unknowns,

E

v

tot

(x

p

i

, y

p

i

)

,

p

i

= 1, . . . , P

i

,

i = 1, . . . , j

.

A ording to this representation, suitable sto hasti operators (denoted by

ℑ {·}

) are needed. The ospring are generated from their parents by means of a ustomized multi- ra k rossover,whereas standard elitism, sele tion and mutation are adopted [19℄. Be ause ofthe hybrid oding,a single-point rossover (binary rossover)

Φ

b

isperformed with probability

π

b

between the binary sequen es of two parents, namely

χ

k

j

q

a

and

χ

k

j

q

b

. By imposing that the ross-position 

cp

 falls only ona boundary between two adja ent genes [Figure3(a)℄, the binary rossover returnsthe following hildren

χ

k

j

+1

q

a

= {j; (Υ

1

)

k

j

q

a

,

h

(x

i

, y

i

)

k

q

a

j

; (L

i

)

k

j

q

a

;







_cp

(W

i

)

k

q

b

j

; (θ

i

)

k

j

q

b

i

, . . . ,

(Υ

j

)

k

_q

_b

j

;



E

v

_tot,1



k

j

q

a

,

h

E

v

tot,i

i

k

j

+1

q

a

, . . . ,

h

E

v

tot,j

i

k

j

q

b

}

χ

k

j

+1

q

b

= {j; (Υ

1

)

k

j

q

b

,

h

(x

i

, y

i

)

_q

k

_b

j

; (L

i

)

k

_q

_a

j

;







_cp

(W

i

)

k

_q

_a

j

; (θ

i

)

k

_q

_a

j

i

, . . . ,

(Υ

j

)

k

_q

_a

j

;



E

v

_tot,1



k

_q

j

b

,

h

E

_tot,i

v

i

k

j

+1

q

b

, . . . ,

h

E

_tot,j

v

i

k

j

q

a

}

(9) where



E

v

tot,i



k

j

+1

q

a

and



E

v

tot,i



k

j

+1

q

b

are omputed extending to the multi- ra k ase the relationship reported in [20℄ [Eq. (11)℄ and pi torially des ribed in Fig. 3(a) (

r

being a random number uniformlydistributed in

[0; 1]

).

If the binary rossover

Φ

b

has not been applied,the double-point rossover is performed with probability

π

d

on that part of the hromosomes on erned with the eld unknowns and a ording to Eq. (13) of [20℄.

2.2 Integrated Strategy (IS)

Unlike the hierar hi al approa h, the integrated strategy onsiders a single optimization pro ess. Towards this purpose, the populationof

Q

trial solutions is omposed by  het-erogeneous hromosomesea h of them oding adierentnumberof ra ks

χ = {χ

q

; q = 1, . . . , Q} =



C

q

; Υ

i

, i = 1, ..., C

q

;



E

v

tot,i



, i = 1, ..., C

q

; q = 1, . . . , Q

(10)

C

q

being an integer value randomly hosen inthe range between

1

and

C

max

. Moreover,

startingfromarandomlygeneratedset ofsolutions

χ

0

,theiterative(

k

beingtheiteration index) optimization pro ess evolvesfor a hieving the  onvergen e ondition (

k = K

max

or

Ω



χ

k

opt

≤ Ω

th

) a ording to the following instru tions

while



(k < K

max

)

and

Ω



χ

k

opt

> Ω

th



do

k = k + 1

χ

k

b

= Φ

b



ℵ



χ

k−1

opt



χ

k

o

= ℘



χ

k−1

opt

χ

k

p

=

n

χ

k

b

, χ

k

o

, χ

k−1

o

χ

k

= ℑ

n

χ

k

p

o

χ

k

opt

=

arg



min

q=1,..,Q



Ω



χ

k

q



where

χ

k

o

and

χ

k

b

are two populations of

Q

2

trial solutions generated from the optimal trial solution rea hed at the

(k − 1)

-th step,

χ

k−1

opt

, by means of the operators

℘ {·}

and

Φ

b

[ℵ {·}]

, respe tively.

More indetail,

χ

k

b

isapopulationwhoseindividuals odesolutionswiththesame number of ra ks as

χ

k−1

opt

and it is generated by randomly modifying through the operator

ℜ (·)

all genes of

χ

k−1

opt

ex ept that oding

C

k−1

opt

χ

k−1

b,q

= ℵ



χ

k−1

opt

,

n

C

k−1

opt

; ℜ Υ

k−1

opt,i

, i = 1, ..., C

k−1

opt

; ℜ



E

v

tot,i



k−1

opt



, i = 1, ..., C

k−1

opt

o

q = 1, ...,

Q

₂

.

(11)

and su essively, applying the binary rossover

χ

k

b,q

= Φ

b



χ

k−1

b,q

. As far as the sub-population

χ

k

o

is on erned, it onsists of

(C

max

− 1)

equally parti-tioned sub-sets, ea h of them with individuals having the same number of ra ks

C

l

,

l = 1, ..., (C

max

− 1)

, and dierent from

C

k−1

opt

. These trialsolutionsare generated a ord-ing tothe followingrules:

•

If anindividualbelongsto the

l

-thsubset hara terized by

C

l

< C

k−1

opt

,then

χ

k

o,q

=

n

C

l

; Υ

k

i

= Υ

k−1

opt,r

, i = 1, ..., C

l

;



E

tot,i

v



k

=



E

tot,s

v



k−1

opt

, i = 1, ..., C

l

o

(12)

r

and

s

being integer randomnumbers between

1

and

C

k−1

opt

[Fig. 3(b)℄;

•

Otherwise (i.e.,

C

l

> C

k−1

opt

),the trialsolution is obtained by adding suitable genes to the hromosome oding

χ

k−1

opt

[Fig. 3( )℄. Randomlyin that part on ernedwith the ra k parametersand fromthe eld distributionof the unperturbed s enarioin the remainingpart:

χ

k

o,q

=











C

l

;

Υ

k

i

= Υ

k−1

opt,i

,

i = 1, ..., C

opt

k−1

Υ

k

i

= ℜ,

i = C

opt

k−1

+ 1, ..., C

l

;



E

v

tot,i



k

=



E

v

tot,i



k−1

opt

,

i = 1, ..., C

k−1

opt



E

v

tot,i



k

=

h

E

v

tot

(cf ),i

i

k−1

opt

, i = C

k−1

opt

+ 1, ..., C

l











(13)

Finally, ea h iterative loopis terminated by pro essing the heterogeneous population

χ

k

p

throughstandardsele tion,elitism,andmutation(no rossoveroperationsareperformed) as des ribed in[19℄[20℄(

χ

k

p

= ℑ

n

χ

k−1

p

o

). 3 Numeri al Validation

As far as the validation of the proposed strategies is on erned, a numeri al assessment has been arried out by onsidering dierent ongurations of the defe ts and various hara teristi s of the host medium in order to verify the possibility and feasibility of dealing with more general (and probably more realisti ) multiple-defe ts s enarios. On the other hand, the robustness against blurred s attering data has been evaluated by addingarandomGaussiannoisewithaxedsignal-to-noiseratio(

SNR

)tothemeasured eld samples [20℄.

For quantifying the performan e of the proposed implementations and be ause of the multiple- ra k geometries, suitable error indexes have been dened extending those re-portedin [19℄:

•

Multi- ra k Lo alization Error,

δ

:

δ =

P

C

c=1

C







q

(b

x

c

− x

c

)

2

+ (b

y

c

− y

c

)

2

d

max

× 100







(14)

d

max

beingthe maximumlinear dimensionof

H

and

(b

x

c

, b

y

c

)

the estimated enter of

c

-th ra k;

•

Multi- ra k Area Error,

∆

:

∆ =

P

C

c=1

C











 b

A

c

− A

c







A

c

× 100







(15)

R =

Ψ

opt

Ψ

× 100

(16)

Ψ

opt

and

Ψ

being the number of su essful dete tions and the total number of repeated

simulations with the same geometryand onditions,respe tively.

For the numeri al validation, if it is not spe ied, the following referen e s enario has been onsidered. A square homogeneoushost mediumof side

L

H

= 0.8λ

(

d

max

=

√

2L

H

)

hara terized by a diele tri permittivity equal to

ε

H

= 2.4

and homogeneous defe ts (

ε

D

i

= 1.0

and

σ

D

i

= 0.0

,

i = 1, ..., C

). Su h a s enario has been illuminated by

V = 4

dire tions with sour esradiating ele tri in ident elds

E

v

inc

(x, y) = e

−jk

0

(

x

os

γ

v

_+y

sin

γ

v

₎

,

γ

v

= (v − 1)

2π

V

,

v = 1, ..., V

,

k

0

being the free-spa e wavenumber. Furthermore, the

samples of the s attered ele tri eld have been olle ted at

M = 50

equally-spa ed positionslo atedona ir le

ρ = 0.64λ

inradius.

A ording to the guidelines suggested in [25℄-[28℄, the following parameters for the GA-based multi- ra k optimization has been assumed:

Q = 80

,

π

b

= π

d

= 0.7

,

π

m

= 0.4

(mutation probability),

K

max

= 600

, and

Ω

th

= 10

−5

.

3.1 Test Case #1 - Re onstru tion of a Single-Cra k

Congura-tion

As a rst test ase, letus onsider a omparative study onthe ee tiveness of the multi- ra k strategiesversus ustomizedsingle- ra kte hniques previouslydeveloped and are-fullyassessed (i.e.,the

F GA

[19℄andthe

IGA

[20℄approa hes). Towardsthispurpose,an unknown defe t (

C = 1

) of area

A

c

λ

2



c=1

= 2.25 × 10

−2

has been lo ated at

x

c

λ



c=1

= 0.22

and

x

c

λ



c=1

= 0.15

in a lossy (

σ

H

= 0.1 [S/m]

) host medium. Moreover, the s attering data have been blurred with an in reasing level of additive noise (from

SNR = 30 dB

up to

SNR = 5 dB

). Con erning multi- ra k implementations,

C

max

has been xed to

C

max

= 3

, thusthe numberof ra ks lyinginthe investigationdomain

H

isanunknown,

as well.

[Fig. 4(b)℄. Due tothe intrinsi natureboth of the

GA

-based strategies and of the noise, these results are average values of the exe ution of ea h algorithm for ten independent realizations of the random pro ess blurringthe s attering data.

As expe ted and already demonstrated in [20℄,

IGA

-based approa hes (i.e., the single- ra k

IGA

and the multi- ra k strategies) allow a non-negligible improvement in the lo alizationa ura y. Asamatteroffa t,theaveragevalueofthelo alizationerror

hδi

F GA

turns out to be greaterthan

25 %

, whereas

IGA

-based algorithmsprovide alo alization a ura y with an error index lower than

20 %

whatever the noise level. Moreover, the ee tiveness of

IGA

-basedte hniques improves (

δ ≈ 5 %

) foran in reasing of the signal-to-noise-ratio (

SNR > 15 dB

). Furthermore, multi ra k implementations prove a better robustness against higher noisy ondition (

δ

IGA

> δ

IGA−HS

≥ δ

IGA−IS

when

SNR ≤

12 dB

) despite the enlargement of the unknowns spa e (sin e

C

max

6= 1)

with respe t to

the

IGA

single- ra k strategy.

As farasthe estimationof the dimensionof the defe tis on erned, Fig. 4(b) shows that the behaviors of the error guresof the single- ra k

IGA

approa hand of the integrated strategy (

IG

) are quite similar in the range of noise variations, whereas the hierar hi al approa h (

HS

)generally doesnot rea h the a ura y of single- ra kalgorithms.

3.2 Test Case #2 - Dependen e of the Re onstru tion A ura y

on the Number of Defe ts

C

These ondtest aseisaimedatevaluatingthefeasibilityoftheproposedapproa hin deal-ing with multiple-defe t ongurations by omparing the hierar hi al and the integrated implementations. Under the assumption that

C

max

= 3

, three geometries hara terized by the presen e of a number of ra ks from

C = 1

up to

C = 3

(Fig. 5) have been onsidered. Thepositionand sizeof ea h defe t

D

i

,

i = 1, . . . , C

,aresummarizedinTab. I.

In ordertoassess theee tiveness indete tingthe numberofdefe ts, thepre ision-re all index

R

has been evaluatedforea hexperiment(

C = 1, 2, 3

)andin orresponden e with dierent signal-to-noise ratios (

SNR = 10, 20, 30 dB

). The results of su h an analysis are reported in Fig. 6. As far as the

HS

is on erned [Fig. 6(a)℄,

R⌋

when

SNR ≤ 10 dB

whatever the number of ra ks. Otherwise (

SNR ≥ 20 dB

), the e a y of the algorithm improves espe ially for a smaller value of

C

. Unlike the

HS

, the integrated implementation generally provides better performan es very lose to the optimal value (

R = 100 %

) ex ept for worst ongurations hara terized by a higher noise (

SNR < 10 dB

) and smallerdefe ts.

The dependen e of the re onstru tion a ura yonthe numberof defe ts and the level of noise anbeestimatedfromthe plotsshown inFig. 7. Theresultsare presented interms of

δ

- left olumn[Figs. 7(a), 7( ),and 7(e)℄ -and of

∆

- right olumn [Figs. 7(b),7(d), and 7(f)℄ for

C = 1

- rst row [Figs. 7(a) and 7(b)℄,

C = 2

- se ond row [Figs. 7( )and 7(d)℄,and

C = C

max

= 3

-thirdrow[Figs. 7(e)and7(f)℄. Bothimplementationsprovide a satisfa tory lo alization(

δ⌋

IS

< 18 %

and

δ⌋

HS

< 29 %

) and the enters of the ra ks are a urately retrieved when

SNR > 15.0 dB

(

δ < 7 %

).

On the otherhand,the dimensioningof thedefe ts turnsout tobemore di ultandthe performan es of the multi- ra k strategies get worse. However, for omparison purposes, it should be noti ed that the

IS

onsiderably over omes the

HS

and the arising error

∆⌋

IS

is always lower than

60 %

. Inparti ular,

∆⌋

IS

< 20 %

when

SNR > 15.0 dB

. From a omputational point of view, on e again, the

IS

turns out to be more ee tive than the

HS

both in terms of onvergen e rate and time per iteration. As one example, Figure 8 shows the plot of the required CPU-time for ea h iteration of a representative simulation (

C = 3

and

SNR = 5.0 dB

).

3.3 Test Case #3 - Dependen e of the Re onstru tion A ura y

on the Host Medium Properties (

σ

)

The test ase #3 is devoted at evaluatingthe multi- ra k strategies for dierent ong-urations of the host medium. Towards this end, the ele tri ondu tivity

σ

H

has been variedfrom

0.1 [S/m]

up to

1.0 [S/m]

and thearrangementof the ra ks wasthe same as shown in gure 5(Tab. I).

The olor-levelrepresentationsofthere onstru tionerrors

δ

and

∆

arereportedinFigure 9. As expe ted, sin e the multi- ra k strategies are based on the omputation of the inhomogeneous Green's fun tion as well as the single- ra k

IGA

, both the

IS

and the

HS

give a urate lo alizations of the defe ts, whi h result in low average values of the

orresponding indexes:

hδi

HS

= 9.49 %

and

hδi

IS

= 8.15 %

. Con erning the dependen e on the ondu tivity of the host medium and on the

SNR

, the quality of the estimation of the defe ts oordinates

(x

i

, y

i

)

,

i = 1, .., C

,

C = 3

, enhan es as

SNR

in reases and

σ

de reases. A similar behavior holds true also for the ra ks dimensioning as shown in

Figs. 9( )-9(d), but the errors signi antly grow also on average (

h∆i

IS

= 52.19 %

vs.

h∆i

HS

= 74.73 %

).

3.4 Test Case #4 - Dependen e of the Re onstru tion A ura y

on the Defe ts Conguration

Thelasttest aseisaimedattestingtheresolution apabilitiesoftheproposedapproa hes. Asamatteroffa t,areliablere onstru tionpro essshouldbeabletodistinguishadja ent defe ts avoiding the in orre t dete tion of a single ra k instead of a multiplegeometry. In order to verify su h a feature, the same s enario asfor test ase #2 (

C = 3

) has been onsidered, but the defe ts havebeen pla ed loser the ones tothe others asindi ated in Tab. II and shown in Fig. 10.

As expe ted, the obtained results worsen with respe t to those of Se t. 3.2. Figure 11(a) gives the values of the pre ision-re all index for dierent signal-to-noise ratios. The integrated approa h turns out to be more e ient than the hierar hi al strategy. It a hievesavalueof

R = 90 %

in orresponden ewiththehighest

SNR

valueand

R > 70 %

whateverthe noise level.

In order to point out the reliability of the

IS

versus the

HS

, it is interesting to better detail the a hieved resultsfor a xed

SNR

. Letus onsider the ase of

SNR = 10 %

. In su h a ase, the integrated approa h always dete ts a multiple-defe t onguration and the fault per entage [i.e.,

F ⌋

(C=2)

IS

= 30 %

℄ is related to a geometry with two ra ks. On the ontrary, the probability of estimating one or two ra ks is equalfor the hierar hi al strategy to

F ⌋

(C=1)

HS

= 13.7 %

and

F ⌋

(C=2)

HS

= 46.3 %

[

1 − R =

P

C

max

C=1

F

(C)

℄,respe tively. Con erning the re onstru tion errors, by omparingthe resultsshown in Figs. 11(b)and 11( ) to those in Figs. 7(e) and 7(f), it is evident the worsening in the lo alizationand in the estimation of the ra ks dimensions. Moreover, the apabilities in lo alizing and

dimensioning the defe ts of the

IS

are almost independent fromthe noise level.

4 Con lusions

In this work, the problem of dete ting and re onstru ting multiple defe ts in a known host medium has been analyzed. Starting from the inverse s attering equations and by onsideringanee tiveintegralformulationbasedonthedenitionoftheinhomogeneous Green's fun tion, the problem in hand has been addressed with a GA-based te hnique implemented through two innovativestrategies.

The main features of the proposed approa h are the following:

•

apability to dete tmultiple as wellas single defe ts;

•

apabilitytore onstru tmultipledefe tsdierentinshapeaswellasindimensions;

•

exploitationof the a-priori informationand omputationalsaving;

•

apability to operatein the presen e of diele tri as well as ondu tive host media;

•

robustness to blurreddata.

Con erning the methodologi alnovelties of this work, some key-issues should be pointed out:

•

spe i formulationofthe multi- ra kre onstru tion problemwithintheframework

of inverse s attering te hniques;

•

original implementations of the data pro essing through innovative ar hite tures

based onGAs;

•

denition of ustomized GA-operatorsfordealing withheterogeneous and

variable-length hromosomes.

From thenumeri alexperiments arriedoutondierent ongurationsand s enarios,the following on lusions an bedrawn:

•

the proposed multi- ra k approa h proved ee tive providing, in its integrated

im-plementation,bothhighdete tiona ura y,goodre onstru tions,andasatisfa tory robustness;

•

the

HS

showed good results,but ingeneral, they were onaverage inferior tothose

obtained withthe integratedalgorithm(IS).Su h abehavior,althoughthe IS hro-mosomes ould be onsidered as a subset of the whole set of trial solutions oded by the HS,seems tobe relatedtoamore ee tivesamplingofthe solutionspa e in the limited amount of iterations;

•

the

IS

revealed very good in terms of omputational osts oering an a eptable

trade-o between a ura yand onvergen e rate of the optimizationpro ess;

•

the multi- ra k strategies exhibited good a ura ies in dealing with single-defe t

geometries andtheirperforman es turnedout omparablewiththoseof ustomized single-defe t te hniques;

However, although the proposed multi- ra k dete tion te hnique seems to be a very promising tool for unsupervised and automati appli ations, several improvements for its industrialimplementationare mandatory.

Towards this end, future developments of this work will be oriented in the following dire tions:

•

developing apro edurethat parameterizes the ra k by means ofmore generaland

omplex des riptors;

•

extending the pro edureto athree-dimensional s enario;

•

des ribinginana urateanddetailedfashiontheoverallmeasurementsetupinorder

to take intoa ount other sour es of noise and ina ura iesin the data olle tion.

A knowledgements

A. Massa wishes to thank E.Vi o for her support. Moreover, the authorsare grateful to Ing. D. Pregnolato for kindlyprovidingsome numeri alresults of omputer simulations.

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[23℄ S. Caorsi, G. L. Gragnani, M. Pastorino, and M. Rebagliati, A model-driven ap-proa htomi rowavediagnosti sinbiomedi alappli ations, IEEETrans.Mi rowave Theory Te h., vol. 44,pp. 1910-1920, 1996.

[24℄ J.H.Ri hmond,S atteringbyadiele tri ylinderof arbitrary ross-se tionshape, IEEE Trans.Antennas Propagat., vol.13, pp. 334-341,1965.

[25℄ R.L.HauptandS.E.Haupt.Pra ti alGeneti Algorithms,John Wiley&SonsIn ., New York, 1998.

[26℄ D. S. Weile and E. Mi hielssen, Geneti algorithmoptimization applied to ele tro-magneti s: a review, IEEE Trans.Antennas Propagat.,vol.45, pp. 343-353,1997.

[27℄ J.M.JohnsonandY.Rahmat-Samii,Geneti algorithmsinengineering ele tromag-neti s, IEEE Trans. Antennas Propagat. Magaz., vol. 39,pp. 7-25, 1997.

[28℄ Y. Rahmat-Samii and E. Mi hielssen. Ele tromagneti Optimization by Geneti Al-gorithms, John Wiley &Sons In ., New York,1999.

•

Figure 1. Multi- ra k problemgeometry(

C = 2

).

•

Figure 2. Example of a multi ra k hybrid oded variable-length hromosome.

•

Figure 3. Example of (a) the binary rossover. Generation of trialsolutions of the

sub-population

χ

k

o

when (b)

C

l

< C

k−1

opt

and when ( )

C

l

> C

k−1

opt

.

•

Figure 4. Test Case #1. Behavior of (a)

δ

and (b)

∆

versus

SNR

when

C = 1

and for

F GA

,

IGA

,

HS

, and

IS

.

•

Figure 5. Test Case #2. Referen e geometry.

•

Figure 6. Test Case #2. Behaviorofthepre ision-re allindex

R

: (a)Hierar hi al

Strategy and (b) and Integrated Strategy.

•

Figure7. TestCase#2. Behaviorofre onstru tionerrorsversus

SNR

fordierent

number of defe ts.

C = 1

: (a)

δ

and (b)

∆

;

C = 2

: ( )

δ

and (d)

∆

;

C = 3

: (e)

δ

and (f)

∆

.

•

Figure 8. CPU times. Comparison among the

IS

and the

HS

when

C = 3

and

SNR = 5 dB

.

•

Figure 9. Test Case #3. Behavior of the re onstru tion errors versus

SNR

and

the ondu tivity of the host medium,

σ

H

. Hierar hi al Strategy: (a) lo alization error

δ

and (b) area error

∆

. Integrated Strategy: ( ) lo alizationerror

δ

and (d) area error

∆

.

•

Figure 10. Test Case #4. Referen e geometry.

•

Figure 11. Test Case #4. Behavior of the error indexes versus

SNR

for the

HS

and the

IS

: (a)pre ision-re allindex

R

,(b)lo alizationerror

δ

,and( )area error

∆

.

•

Table I. Test Cases #1 and #2 - Positions and sizes of the defe ts.

L

y

y

W

x

H

θ

y

x

θ

O

Y

X

Y

X

τ

L

x

W

S

1

1

1

1

1

2

2

2

2

2

D

1

D

2

H

. . .

. . .

0

1

0

1

1

1

0

0

1

1

1

1

0

1

1

1

0

1

0

x

y

_L

_W

θ

x

y

_L

_W

θ

Crack 1

Crack C

Binary Part

0

Real Part

tot,1

ν

E

E

_tot,C

ν

1

1

1

1

1

C

C

C

C

C

[

]

[

]

2 -M. Benedetti et al. , Multi ra k Dete tion in T w o-Dimensional ... 25

Crossover Point

x

₁

a

y

₁

a

_L

a

₁

x

₁

b

y

₁

b

_L

b

₁

x

₂

a

y

₂

a

_L

a

₂

_W

₂

a

θ

a

₂

1

θ

1

a

W

a

x

₂

a

y

₂

a

_L

a

₂

_W

₂

a

θ

a

₂

1

θ

1

a

W

a

x

₂

b

y

₂

b

_L

b

₂

_W

₂

b

θ

b

₂

1

θ

1

b

W

b

x

₂

b

y

₂

b

_L

b

₂

_W

₂

b

θ

b

₂

1

θ

1

b

W

b

E

tot,1

v

[

]

a

E

tot,1

v

[

]

b

E

tot,2

v

[

]

a

r

2

E

tot,1

v

[

]

a

(1−r)

2

E

tot,2

v

[

]

a

E

tot,2

v

[

]

b

E

tot,1

v

[

]

b

(1−r)

2

r

2

E

tot,1

v

[

]

a

E

tot,1

v

[

]

b

E

tot,2

v

[

]

b

Parent q

Parent q

Child q

Child q

x

₁

a

y

₁

a

_L

a

₁

x

₁

b

y

₁

b

_L

b

₁

+

E

tot (cf)

v

[

a

b

a

b

+

]

(a) 3 ( I ) -M. Benedetti et al. , Multi ra k Dete tion in T w o-Dimensional ... 26

(b)

x

₂

y

₂

_L

₂

_W

₂

θ

₂

x

₂

y

₂

_L

₂

_W

₂

θ

₂

x

₁

y

₁

_L

₁

_W

₁

θ

₁

x

₃

y

₃

_L

₃

_W

₃

θ

₃

x

₁

y

₁

_L

₁

_W

₁

θ

₁

x

₃

y

₃

_L

₃

_W

₃

θ

₃

x

₁

y

₁

_L

₁

_W

₁

θ

₁

x

₂

y

₂

_L

₂

_W

₂

θ

₂

x

₃

y

₃

_L

₃

_W

₃

θ

₃

E

tot (cf)

v

[

]

C > C

_l

_opt

k−1

C < C

l

opt

k−1

Child

χ

k

_o

Parent

χ

_opt

k−1

x

₁

y

₁

_L

₁

_W

₁

θ

₁

Parent

χ

_opt

k−1

C < C

l

opt

k−1

C > C

_l

_opt

k−1

Random Selection

Random Selection

Random Generated

E

tot,1

v

[

]

E

tot,2

v

[

]

E

tot,3

v

[

]

E

tot,3

v

[

]

E

tot,1

v

[

]

E

tot,2

v

[

]

E

tot,1

v

[

]

E

tot,2

v

[

E

tot,1

v

[

]

Child

χ

k

_o

]

( ) 3 ( II ) -M. Benedetti et al. , Multi ra k Dete tion in T w o-Dimensional ... 27

0

5

10

15

20

25

30

35

40

5

10

15

20

25

30

δ

SNR [dB]

C = 1

Integrated Strategy

Hierarchical Strategy

IGA

FGA

(a)

0

5

10

15

20

25

30

35

40

5

10

15

20

25

30

∆

SNR [dB]

C = 1

Integrated Strategy

Hierarchical Strategy

IGA

FGA

(b)

y

x

1.0

ε (x, y)

2.4

0

10

20

30

40

50

60

70

80

90

100

C = 3

C = 2

C = 1

R [%]

SNR = 10.0 [dB]

SNR = 20.0 [dB]

SNR = 30.0 [dB]

(a)

0

10

20

30

40

50

60

70

80

90

100

C = 3

C = 2

C = 1

R [%]

SNR = 10.0 [dB]

SNR = 20.0 [dB]

SNR = 30.0 [dB]

(b)

0

5

10

15

20

5

10

15

20

25

30

δ

SNR [dB]

C = 1

Integrated Strategy

Hierarchical Strategy

0

5

10

15

20

25

30

5

10

15

20

25

30

∆

SNR [dB]

C = 1

Integrated Strategy

Hierarchical Strategy

(a) (b)

0

5

10

15

20

25

30

5

10

15

20

25

30

δ

SNR [dB]

C = 2

Integrated Strategy

Hierarchical Strategy

0

20

40

60

80

100

120

5

10

15

20

25

30

∆

SNR [dB]

C = 2

Integrated Strategy

Hierarchical Strategy

( ) (d)

0

5

10

15

20

5

10

15

20

25

30

δ

SNR [dB]

C = 3

Integrated Strategy

Hierarchical Strategy

0

20

40

60

80

100

5

10

15

20

25

30

∆

SNR [dB]

C = 3

Integrated Strategy

Hierarchical Strategy

(e) (f)

0

0.1

0.2

0.3

0.4

0.5

0

100

200

300

400

500

600

Arbitrary time unit [s]

Iteration, k

Integrated Strategy

Hierarchical Strategy

0.1

σ

H

1.0

0.1

σ

H

1.0

30

.0

S

N

R

5.

0

30

.0

S

N

R

5.

0

0

δ

42

0

∆

166

(a) (b)

0.1

σ

H

1.0

0.1

σ

H

1.0

30

.0

S

N

R

5.

0

30

.0

S

N

R

5.

0

0

δ

42

0

∆

166

( ) (d)

y

x

1.0

ε (x, y)

2.4

0

10

20

30

40

50

60

70

80

90

100

30

20

10

R [%]

SNR [dB]

Integrated Strategy

Hierarchical Strategy

(a)

0

5

10

15

20

5

10

15

20

25

30

δ

SNR [dB]

Integrated Strategy

Hierarchical Strategy

(b)

0

20

40

60

80

100

5

10

15

20

25

30

∆

SNR [dB]

Integrated Strategy

Hierarchical Strategy

( )

x

_c

λ

y

_c

λ

A

_λ

2

c

c = 1

0

.22

0

.15 0.0225

c = 2

0

_.0

_{−0.15 0.01}

c = 3 −0.26 0.15

0

.04

x

_c

λ

y

_c

λ

A

_λ

2

c

c = 1

0

_.102

0

_.102

0

_.04

c = 2

0

_{.046 −0.102 0.01}

c = 3 −0.102 0.046 0.0225