Evaluation study of the effectiveness of the integrated genetic-algorithm-based strategy for the...

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Benedetti et al. Vol. 23, No. 6 /June 2006/J. Opt. Soc. Am. A 1311

Evaluation study of the effectiveness of theintegrated genetic-algorithm-based strategy for the

tomographic subsurface detection of defects

Manuel Benedetti, Massimo Donelli, Gabriele Franceschini, and Andrea Massa

Department of Information and Communication Technologies, University of Trento, Via Sommarive 14,38050 Trento, Italy

Matteo Pastorino

Department of Biophysical and Electronic Engineering, University of Genoa, Via Opera Pia 11/A,16145 Genoa, Italy

Received July 27, 2005; revised November 14, 2005; accepted November 18, 2005; posted December 9, 2005 (Doc. ID 63678)

An assessment is presented of the integrated genetic-algorithm strategy based on a numerically computedGreen’s function for subsurface inverse scattering problems arising in nondestructive evaluation/testing indus-trial applications. To show the effectiveness and current limitations of such a microwave technique in dealingwith various scenarios characterized by lossless and lossy host media as well as in noisy environments, severalnumerical experiments are considered. The results obtained confirm the effectiveness of the approach in fullyexploiting the available a priori information through a suitable scattering model, which allows a nonnegligibleenhancement of the reconstruction accuracy as well as a reduction of the overall computational burden withrespect to standard imaging approaches. © 2006 Optical Society of America

OCIS codes: 100.3010, 110.0110, 110.6960, 290.0290, 290.3200.

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. INTRODUCTIONubsurface sensing involves the detection, localization,iscrimination, and identification of unknown (or par-ially known) objects beneath and on a surface. Variouspplications of related technologies in many scientific andngineering branches have been reported. We considerhe framework of nondestructive testing (NDT) for civilonstructions, railroads, maritime structures, demining,aterial inspection, surface metrology, geology, archeol-

gy, industrial process monitoring, etc.An area that is strongly dependent on the development

nd application of advanced subsurface and surface sens-ng technologies is the detection and reconstruction (al-hough limited to a set of characteristic parameters) ofamage in industrial manufactures. This topic has re-eived great interest as both a major safety and an eco-omic concern. Considerable advances have beenchieved in monitoring and detecting flaws and changesn structures, thanks to the availability of wireless tech-ologies, mems/micro/nano sensors, and the evolution ofata postprocessing techniques (e.g., data fusion, dataanagement, etc.) allowing distributed sensing capabili-

ies and quick structure recovery. As far as real-timeonitoring is concerned, the development of advanced

ensors1 and the assessment of imaging techniques forhe solution of the inverse scattering problem are key is-ues to be addressed further.

In the framework of microwave imaging methods, sev-ral iterative inversion procedures have been proposed.rom a computational point of view, these techniques cane grouped into two main categories according to the solv-

1084-7529/06/061311-15/$15.00 © 2

ng algorithm adopted. The first includes approachesased on local inversion algorithms.2–9 The other consid-rs global inversion procedures.10–13

In general, whatever the strategy, an imaging proce-ure is aimed at determining the object function in theverall investigation domain. However, such a require-ent is unnecessary in industrial processes or materialonitoring since a large portion of the investigation do-ain is known. Thus it is profitable to develop customized

nverse scattering approaches able to exploit fully thevailable a priori information.In Ref. 14, de Oliveira et al. developed an effectiveethod for the eddy-current evaluation of a defect in a

onductive half-space by modeling the test zone as a dis-ribution of either void or metallic cells and successivelyolving the subsequent optimization problem by means of

simulated annealing algorithm. The problem of low-requency eddy-current nondestructive evaluation (NDE)f damaged metal structures has also been addressed ineveral papers through deterministic or stochastic tech-iques (see for example Refs. 15–18).Within the same framework, but considering micro-

ave frequencies, Caorsi et al.19 proposed a genetic-lgorithm (GA)-based approach called free-space Green’sunction approach (FGA) in which the defect was param-terized by a set of significant parameters to be deter-ined during an iterative minimization process. In such aethod, the reference background was assumed to be co-

ncident with the free space. Several numerical test casesonfirmed the effectiveness of the approach in dealingith a free-space scenario or a weak host medium (in

006 Optical Society of America

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1312 J. Opt. Soc. Am. A/Vol. 23, No. 6 /June 2006 Benedetti et al.

erms of dielectric characteristics). However they alsoighlighted the computational limitations of the FGA dueo the large dimensions of the unknown arrays. To reducehe computational burden of the FGA while maintainingts key features, an improved microwave imaging proce-ure called inhomogeneous-space Green’s function ap-roach (IGA) has been recently presented.20 A prelimi-ary assessment has been performed by focusing on theelationship between the reconstruction accuracy and therack characteristics (dimensions and orientation).

In this paper, to complete the numerical validation,reat care is devoted to determining the dependence ofhe retrieval capabilities of the IGA on the dielectric pa-ameters and geometric features of the host medium inhich the defect is buried.The manuscript is organized as follows. In Section 2,

he subsurface inverse scattering problem arising inDE/NDT industrial applications is presented and math-

matically formulated in terms of an optimization proce-ure that considers a suitable scattering model based on aumerically computed inhomogeneous Green’s function.hen the iterative solution process based on a customizedybrid GA procedure is briefly summarized and assessedhrough a large set of numerical experiments aimed atointing out the range of applicability of the approachSection 3). Finally, some remarks and conclusions are of-ered (Section 4).

. MATHEMATICAL FORMULATIONn the following, the mathematical formulation of the in-egrated optimization approach previously presented inef. 20 will be briefly summarized. Let us consider the cy-

indrical geometry shown in Fig. 1. DD is a fixed area on alane orthogonal to the cylinder axis (the z axis) occupiedy a host object and characterized by a host medium of

Fig. 1. Geometry of the problem.

Fig. 2. Hybrid-coded, variab

nown dielectric properties ��D�0 ,�D ,�D�0� embedded inhomogeneous background or external medium

�B�0 ,�B ,�B�0�. DD includes, at an unknown position, theross section of a cylindrical homogeneous crack whosehape and material characteristics ��C�0 ,�C ,�C�0� arenknown. A set of V incident electromagnetic fields,

vinc�r� ,v=1, . . . ,V, is used to successively illuminate the

catterer. The waves are TM, i.e., Evinc�r�=Ev

inc�x ,y�z, with=xx+yy. The frequency and the corresponding free-pace wavelength are denoted by f and �0, respectively.

With these notations, the z component of the electriceld Etot�r� corresponding to the incident field Einc�r� cane expressed as21

Evtot�r� = Ev

inc�r� +� �DD

��r��Evtot�r��G0�r/r��dr�, �1�

here G0 is the two-dimensional Green’s function and � isescribed by

�C�r� =1

j2�f�0��C�r� − �B� + j2�f�0��C�r� − �B�r�� 2�

or r belonging to the crack area and

�D�r� =1

j2�f�0��D�r� − �B� + j2�f�0��D�r� − �B�r�� 3�

therwise.In order to detect the presence of a homogeneous crack

n the original scatterer, the crack is approximated by anbject of rectangular shape and parameterized by length, width w, orientation �, and center coordinates �xC ,yC� toe determined during the reconstruction process. Underhis hypothesis, the object function describing the defecturns out to be

�D˜�x,y� = ��C, X � −

l

2,

l

2 and Y � �− w/2,w/2�

�D, otherwise

,

here X= �x−xC�cos �+ �y−yC�sin �, Y= �x−xC�sin �+ �yyC�cos �, and �C is a constant value.The internal electric field for the flaw configuration is

nknown. Consequently, the reconstruction process isimed at searching for the unknown array

� = �xC,yC,w, � ,�,�C,�nv,n = 1, . . . ,Nt; v = 1, . . . ,V��,

�4�

here nv is the value of the total electric field inside the

th subdomain of DD (DD is partitioned into Nt subdo-ains). Furthermore, according to the adopted discretiza-

ion, it is convenient to assume that � ,w ,� are discreteariables �= j ; j=1, . . . ,L ; w= i , i=1, . . . ,W ; �= t� ,

th chromosome for the IGA.
le-leng

tsvc

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f

wg

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m

w=bd

tg

scia

m

wi=q


=1, . . . ,T; where =�An , An being the area of the nthubdomain and � the angular step used for the multi-iew process. Thus, the unknown array turns out to be a
ombination of integer values, binary coded, and real-
t

wa

Ffa�=

alued quantities (i.e., the values of the total electric field;ee Fig. 2).

The optimum value of � is obtained by minimizing aunctional defined as

�� =�

MV�v=1

V �m=1

M �Evscatt�rm� − �n=1

Nt�D˜�rn�n

vAnGmn0 �2

�Evscatt�rm��

�+

NtV�v=1

V �n=1

Nt �nC

v − Evinc�rn� − �p=1

Nt�D˜�rp�p

vAnGnp0 �2

�Evinc�rn��2

� , �5�

here rn is the center of the nth subdomain and Gnp0 is

iven by

Gnp0 =�

Dn

G0�k�p�dr�, n,p = 1, . . . ,Nt,

n being the domain of the nth subdomain and �p the dis-ance between points rp and r�. This approach is the FGAnd requires the entire discretization of DD.On the other hand, it is possible rewrite Eq. (1) in theore convenient form

Evtot�r� = Ev

inc�r� +� �DD

�D�r��Evtot�cf��r��G0�r/r��dr�

+� �DC

�C�r��Evtot�r��GI�r/r��dr�, �6�

here �C is a “differential object” function such that �C

�C−�D, and Evtot�cf� denotes the electric field that would

e present in the unperturbed (crack-free) configurationue to Ev

inc.Finally, GI�r /r�� is the Green’s function for the unper-

urbed configuration, which satisfies the following inte-ral equation

GI�r/r�� = G0�r/r�� +� �DD

�D�r��GI�r�/r��G0�r/r��dr�.

�7�

Since the sum of the first two terms on the right-handide of Eq. (6) can be seen as the “incident” field on therack, once GI has been computed, the scattering problems limited to the “differential object” occupying the defectrea DC.Equation (7) can be numerically solved by using Rich-ond’s theory,22 and it assumes the algebraic form

GhhI = Gnh

0 + �m=1�m�k�

Nt

�mDGmh

I � �Am

G0�r/r��dr�, �8�

ith n=1, . . . ,Nt ; h=k+�hn ; k=1, . . . ,Nt−1; with �ij=0f i� j or �ij=1 if i� j. In Eq. (8), Gnh

I =GI�rn /rh� , Gnh0

G0�rn /rh�, and �mD =�D�rm�. The computation of Gnt

I re-uires the solution of N systems of equations given by

�Gn�gn = g0n, n = 1, . . . ,Nt, �9�

here gn and g0n are �Nt−1��1 arrays whose elementsre given by �gn�h=Gnh

I and �g0n�h=Gnh

0 , respectively.

ig. 3. Numerically computed inhomogeneous Green’s functionor a source located at x=y=−0.025�0. The host medium is char-cterized by �a� �D=0.0, �D=1.0 (free-space Green’s function); �b�D=0.0, �D=2.0; �c� �D=0.0, �D=5.0; �d� �D=0.1, �D=2.0; �e� �D0.5, � =2.0; �f� � =1.0, � =2.0.
D D D

Mtwiomsad

t=2(SttFm=c(b

et

qsanqGdtp�

Tslltnotnnln

ttvj=s

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oreover, the �Nt−1�� Nt−1� elements of the square ma-rix �Gn� are given by ��Gn��mk=�mk−�h

D��AhG0�ru /r��dr�,

here u=m+�km ; m=1, . . . ,Nt−1. Fortunately, as shownn Ref. 23, if �G� is an Nt�Nt matrix from which �Gn� isbtained by deleting the nth row and the nth column, theatrix �Gn�−1 can be easily determined by neglecting the

ame row and column of the matrix �F��G�−1, where �F� israther empty matrix whose nonvanishing elements are

irectly obtained from the elements of �G� as follows:

��F��mk =�0 if m � k and k � j

1 if m = k and k � j

−��G��mk

��G��kk

if m � k and k = j

1

��G��kk

if m = k and k = j

,

m, k, j = 1, . . . ,Nt. �10�

As an example, Fig. 3 shows some pictorial representa-ions of the Green’s function for a source located at x=y−0.025�0 and computed in a square domain of area.4�0�2.4�0 containing a host medium DD of side 0.8�0�0 being the wavelength in vacuum). As mentioned inection 1, this different approach has been called IGA. Inhis case, to detect the presence of a homogeneous crack,he same array � [Eq. (4)] is looked for. However, unlikeGA, the unknown values n

v are limited to the subdo-ains belonging to the crack area �n=1, . . . ,N ;v1, . . . ,V ;N�Nt� with a large computational saving. Ac-ordingly, the functional to be minimized is given by Eq.5) in which again, Nt is replaced by N and Gij

0 is replacedy Gij

I .The functional � is iteratively minimized, and the it-

ration step is indicated by k. Consequently, all the quan-ities that change at each iteration are indexed by k.

Because of the presence of local minima in the subse-uent cost function (see Fig. 4, where a representativeample of the behavior of the cost function is reported)nd because the unknown array turns out to be a combi-ation of integer values, binary coded, and real-valueduantities, the minimization is carried out by applying aA.24,25 The algorithm processes a population of candi-ate solutions ��k�= ��q

�k� ; q=1, . . . ,Q�, Q being the popula-ion dimension. The quality of the solutions of the kthopulation is evaluated by computing the Q scalar values,

q�k�=��q

�k�� , q=1, . . . ,Q, according to relationship (5).

Table 1. Summary of the Param

est Case Figure xC /�0 yC /�0 � AC /�02

5, 6 0.15 0.1 0 0.047 0.15 0.1 0 0.02258 0.26 0 � /4 0.049 0.26 0 � /4 0.0410 0.26 0 � /4 0.0411 0.26 0 � /4 0.0412 0.26 0 � /4 0.0413 0.26 0 � /4 0.04

a� and � indicate the relative permittivity of the host medium and of the crack
D C
he individuals that achieve higher fitness values (corre-ponding to lower values of the cost function) are moreikely to be selected as parents and they generate new so-utions (called offspring) by means of crossover and muta-ion. Generally, crossover promotes the exchange of ge-etic information among elements of the population. Theffspring are subject to mutations that randomly modifyhe genetic structures of trial solutions in order to createew variants. The current population is replaced by theewly generated group of offspring ��k�Ü ��k+1�. The evo-

utionary procedure terminates either when a maximumumber of generations elapses �k=K� or a fixed value of

he cost function is reached (�opt�k*�=minq��q

�k*�� ,� beinghe convergence threshold and k=k* the iteration of con-ergence). In Fig. 4, a representative example of the “tra-ectory” of the optimal trial solution (i.e., �opt

�k�

arg�minq��q�k��) in a plane of the solution space is

hown.In the IGA approach, it is necessary to customize the

A by defining a suitable encoding procedure and geneticperators. In particular, the chromosomes are variable-ength strings. This is due to the variable dimensions ofhe domain where the unknown field is computed. TheGA approach, on the other hand, considers hybrid coded,xed-length chromosomes.

s Employed in the Test Casesa

�02 �D �C �D �S/m� �C �S/m� SNR �dB�

2.25 2 1 0 0 2.5÷30.04 1÷5 1÷5 0 0 154 1÷5 1 0 0 2.5÷30.04 2 1÷5 0 0÷1 154 2 1÷5 0÷1 0 154 1÷5 1 0 0÷1 154 1÷5 1 0÷1 0 154 2 1 0÷1 0÷1 15

tively.

ig. 4. Example of the behavior of the cost function in the planexC /�0 ,yC /�0� of the solution space.

eter

AD /

0.25÷0.60.60.60.60.60.60.6

, respec

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With respect to the genetic operators, mutation is de-ned according to Ref. 19, whereas the crossover is modi-ed in order to allow variable-length chromosomes. Aingle-point crossover is used. When the cross positionies in the binary part, there are three possibilities. Sincehe number of subdomains of the defect depends on therack dimensions at the current iteration, the two off-

ig. 5. Reconstruction accuracy versus the area of the host merror in the crack area estimate �A. �a� , �c� FGA; �b� , �d� IGA; �e�

pring of the new population may correspond to larger,qual-sized, or smaller cracks.

The crack dimension is defined by the binary part ofhe chromosome. The binary parts of the two offspring areomputed by the usual interchange of bits as in the classicinary GA. On the other hand, the real elements of thehromosome corresponding to the field values at the cen-

or different values of SNR. �a� , �b� Localization error �C; �c� , �d�f� �A.

dium f� ; �

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at

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cp

FS


ers of the subdomains of the crack are obtained by sum-ing each gene of a chromosome, multiplied by a random

alue r� �0,1�, with the corresponding gene of the otherarent chromosome multiplied by �1−r�.If the numbers of subdomains of the offspring are less

han or equal to those of the parent chromosomes, onlyhe first values of the field elements of the parents aresed. Otherwise, all the field values of the parents aresed. The remaining values, corresponding to subdo-ains in excess (not present in the cracks defined by the

arent chromosomes), are filled with the field values ofhe unperturbed configuration that correspond to theame subdomains now “occupied” by the new crack.

Finally, to ensure that the solution can only improveuring the genetic evolution, a standard elitism strategys implemented over the generations.

. NUMERICAL RESULTSn this section, the results of several numerical simula-ions are reported. In the following, AD and AC denote thereas of the host medium �DD� and of the crack �DC�, re-pectively. Moreover, symbols AD , AC , xC, and yC aredopted to indicate estimated quantities.To evaluate the effectiveness of the IGA method and its

etter performance with respect to the FGA technique, aeference geometry is considered in which a square hostedium includes a square defect.

ig. 6. Reconstruction accuracy versus the area of the host mediNR=13.75 dB. Behaviors of �c� � , �d� � versus SNR when A
C A D 0
In all experiments, the following values of the GA pa-ameters are assumed. The population size is set to Q80, the maximum number of generations is K=200, and

he crossover and mutation rates are Pc=0.7 and Pm0.05, respectively. These values arose from a large set ofxperiments and they are in good agreement with refer-nce parameters encountered in the literature on theubject.26,27

The numerical results and the achieved reconstructionccuracy are evaluated in terms of the following quanti-ative error figures: the defect-localization error

�C =��xC − xC�2 + �yC − yC�2

dmax� 100 �11�

nd the defect-dimension error

�A = �AC − AC

AC� � 100. �12�

oreover, in order to give a clear picture of the compara-ive study, the differences between the error figures,

�C = �CIGA − �C

FGA,

�A = �AIGA − �A

FGA, �13�

oncerned with the IGA and FGA approaches are also em-loyed.

different values of SNR. Behaviors of �a� �C, �b� �A versus AD for25.

um for/�2=1.

wrmti

a�

Ft


Since simulated measurements using a forward solverith the actual parameters are generated instead of car-

ying out real experiments, a Gaussian noise with zeroean value and variance �2 is successively added in order

o simulate experimentally acquired data. The noise levels fixed by the signal-to-noise ratio (SNR) defined as

ig. 7. Effects of the dielectric permittivity on the reconstructiohe crack area estimate �A. �a� , �c� FGA; �b� , �d� IGA; �e� �C; �f�

SNR = 10 log10

�v=1

V �m=1

M�Ev

scatt�xm,ym��2

2MV�2 , �14�

nd the reference scattering scenario (�B=1.0,�B=0, and=� =� =1.0) is illuminated by microwave sources at

racy �SNR=15 dB�. �a� , �b� Localization error �C; �c� , �d� error in
n accu�A.
B D C

fi

mva=

u2sstst

Fp


=6 GHz. The values of the other parameters character-zing each test case are given in Table 1.

In test case 1, the effect of the dimension of the hostedium was evaluated. The first experiment deals with a

oid square defect located at xC=0.15�0 and yC=0.10�0 ofrea AC=0.04�0

2 and lying in a lossless host medium �D0.0 ��D=2.0�. Figure 5 gives the values of the error fig-

ig. 8. Reconstruction accuracy versus the values of the dielectrermittivity of the host medium �D: �a� , �b� localization error �C;� ; �f� � .
C A
res obtained in the range 0.25�AD /�02�2.25 and when

.5 dB�SNR�30 dB. For comparison purposes, theame quantities concerning the FGA are also reported inome figures. As expected, when the value of AD is small,he localization of the crack is quite good; it deteriorateslightly for the smallest values of SNR. However, whilehe localization accuracy of the FGA is significantly af-

e host medium for different SNR values. Effects of the dielectricerror in the crack area estimate �A. �a� , �c� FGA; �b� , �d� IGA; �e�

ic of th�c� , �d�

ff[

tdf

qMuttg

o

F� in the


ected by the dimension of the host medium, the IGA per-orms very efficiently, at least for the highest values SNRFigs. 5(a)–5(f)].

Except for the highest noise levels, taking into accounthe Green’s function of the host medium results in betterefect retrievals as confirmed by the values of �C and�A shown in Figs. 5(e) and 5(f), respectively. In order to

urther assess such behavior, Figs. 6(a) and 6(b) show the

ig. 9. Dependence of the reconstruction accuracy on crack dieleC and conductivity �C: �a� , �b� localization error �C; �c� , �d� error

uantitative behavior of �C versus AD for SNR=13.75 dB.oreover, for AD /�0

2=1.25, Figs. 6(c) and 6(d) give the val-es of �A for different SNRs. Clearly, for SNR�2.5 dB,he noise level is such that the positive effect of the use ofhe inhomogeneous Green’s function is completely ne-ated.

It should be observed that the three-dimensional plotsf Fig. 5 (as well as those of the following figures) are not

aracteristics �SNR=15 dB�. Effects of the dielectric permittivitycrack area estimate �A. �a� , �c� FGA; �b� , �d� IGA; �e� �C; �f� �A.

ctric ch

stor

pet�t

FdI


mooth because of the rather limited number of repeatedests for each simulation (because of the stochastic naturef the approach ten simulations are run for each configu-ation).

In the second experiment, the effects of the ratio of the

C A

ermittivity of the host medium to that of the defect werevaluated. For these simulations, the following configura-ion was assumed: AD /AC=28.45, xC=0.15�0, yC=0.10�0,C=0.0 S/m, and �D=0.0 S/m, and SNR=15 dB (Table 1,est case 2).

ig. 10. Dependence of the reconstruction accuracy on crack and host medium dielectric characteristics �SNR=15 dB�. Effects of theielectric permittivity �C and conductivity �D: �a� , �b� localization error �C; �c� , �d� error in the crack area estimate �A. �a� , �c� FGA; �b� , �d�GA; �e� � ; �f� � .

tssms7

w�I→i

FdI


Figure 7 reports the results obtained. The behavior ofhe localization error [Fig. 7(b)] turns out to be irregularince the defect constitutes alternately a weak or a strongcatterer with a dielectric permittivity relative to the hostedium ��C /�D� sometimes �1 and sometimes �1. The

ame irregular behavior is present in the plots of �A [Fig.

ig. 11. Dependence of the reconstruction accuracy on crack anielectric permittivity �D and conductivity �C: �a� , �b� localizationGA; �e� �C; �f� �A.

(d)] as well. However, in such a case, larger errors occur s

hen the FGA is applied for the highest values of �D andC as seen in the plot of �A [Fig. 7(f)]. As expected, theGA and FGA tend to provide the same results when �D

1, since the two involved Green’s functions tend to co-ncide.

In order to focus on the effects of the host medium,

t medium dielectric characteristics �SNR=15 dB�. Effects of theC; �c� , �d� error in the crack area estimate �A. �a� , �c� FGA; �b� , �d�

d hoserror �

ome of the previous simulations have been repeated for

dA(

itc

bf

�r

m

Fp


ifferent noisy environments. In such experiments,D /AC=16.0, xC=0.26�0, yC=0.0�0, �C=1.0, and �C=0.0

Table 1, test case 3).In general, the IGA performs better than the FGA, and

ts behavior is less sensitive to the noise superimposed onhe input data [Figs. 8(a)–8(d)]. Specific quantitative con-lusions regarding the advantages of the IGA method can

ig. 12. Dependence of the reconstruction accuracy on host meermittivity �D and conductivity �D: �a� , �b� localization error �C;� ; �f� � .
C A
e drawn from Figs. 8(e) and 8(f) where the absolute dif-erences between the error figures are shown.

Lossy materials (0.0 S/m��C�1.0 S/m and 0.0 S/m�D�1.0 S/m) are now considered and different configu-

ations are analyzed (Table 1, test cases 4–8).Figures 9(a)–9(d) report the localization and area esti-ate errors versus the variation of �C and �C (Table 1,

dielectric characteristics �SNR=15 dB�. Effects of the dielectricerror in the crack area estimate �A. �a� , �c� FGA; �b� , �d� IGA; �e�

dium�c� , �d�

tccptlM

tmrt�

v

Fl �a� , �c�


est case 4). The FGA algorithm achieves a maximum lo-alization error �C=38% for �C=0 and �C�2.4, but as therack becomes conductive, its localization capabilities im-rove (the error is lower than 20%). On the other hand,he IGA algorithm results in errors exceeding 30% for theowest values of the crack permittivity when �C increases.

oreover, in such a range of dielectric parameters, �

ig. 13. Dependence of the reconstruction accuracy on dielectriocalization error �C; �c� , �d� error in the crack area estimate �A.

C

urns out to be greater than 25% [Fig. 9(e)]. In the esti-ation of the area [see Fig. 9(f)], the IGA provides better

esults than the FGA, for which �A seems to depend onhe crack conductivity [Fig. 9(c)]. In particular, ��A�FGA120% and ��A�IGA�50%.In Fig. 10, the reconstruction capabilities for different

alues of the dielectric permittivity of the crack � and

cteristics ratio �SNR=15 dB�. Effects of the ratio �C /�D: �a� , �b�FGA; �b� , �d� IGA; �e� �C; �f� �A.

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he electric conductivity of the host medium �D are as-essed (Table 1, test case 5). The FGA technique results inlocalization error quite constant and close to the aver-

ge value ��C�FGA=17.64% [Fig. 10(a)]. In particular,ax��C�FGA=36.83% for �C=4. The IGA algorithm, how-

ver, is characterized by �C�10% for �D�0.3 and �C2.5. The average error is ��C�IGA=17.69% andax��C�IGA=39.75%. To give a detailed picture of the com-

arison between the two methodologies, Fig. 10(e) showshe behavior of �C. As for the previous experiment, thencrease of the conductivity affects the localization accu-acy of the IGA more significantly than that of the FGA.

For completeness, Fig. 10(c) shows the dependence ofA for the FGA for different values of the host mediumonductivity. For �D�0.2,�A grows up to max��A��480%.he IGA algorithm achieves max��A��200%, and ��A�IGA60%.Figure 11 represents the error figures for 0.0 S/m�C�1.0 S/m and 1.0��D�5.0 (Table 1, test case 6). It

hows that in both cases �C and �A are mainly related toD. In particular, �C grows up to 30% for the FGA for �D4. For �D�1,�C increases for both IGA and FGA. For

omparison purposes, Fig. 11(e) gives the plot of �C. Asxpected, in general, the localization accuracy of the IGA

ig. 14. Evolution of the crack detection and reconstructionSNR=15 dB� during the iteration process (k. iteration number).a��b� k=1, �c��d� k=50, �e��f� k=100, �g��h� k=k*. �i� Actual con-guration. �a��c��e��g� FGA, �b��d��f��h� IGA.

urns out to be better than that of the FGA. For the esti-ate of the crack dimension, the performances of the two

pproaches are quite similar as indicated by the differ-nce error �A shown in Fig. 11(f).

In Fig. 12, the changes of the dielectric characteristicsf the host medium are considered (Table 1, test case 7).s can be observed, the IGA gives a better estimate of

xC ,yC� when the host medium conductivity �D is lesshan 0.2. However, when �D increases, the localization er-or becomes �5% also for �D�0.5 [Fig. 12(b)]. On thether hand, the accuracy of the FGAs localization is notffected by the variations in the host medium [Fig. 12(a)].s a matter of fact, �C is quite constant and near to theverage value ��C�FGA=16.26%. Figures 12(c) and 12(d)how the error �A for these simulations, while Fig. 12(f)eports the relative error �A. Finally, the average errorsre 280% and ��A�FGA�280% and ��A�IGA�50%.Figure 13 shows the reconstruction results for

.0 S/m��C�1.0 S/m, 0.0 S/m��D�1.0 S/m, AD /AC16.0, xC=0.15�0, yC=0.10�0, �C=1.0, �D=2.0, and SNR15.0 dB (Table 1, test case 8). As expected, the accuracy

n the computation of the inhomogeneous Green’s func-ion is far less than in the lossless cases, since the solu-ion of a forward-scattering problem is required that in-olves strong attenuations along paths connecting sourcend field points. Such difficulties are confirmed by the be-aviors of the error figures [Figs. 13(b)–13(d) versus Figs.3(a)–13(c)]. Furthermore, as far as the localization accu-acy is concerned, Fig. 13(e) indicates that there is nouarantee that the IGA outperforms the FGA at all if thelectric conductivity is not small enough.

Finally, as an example of the typical behavior of the re-onstruction process, Fig. 14 compares the retrieved im-ges at different iterations when the FGA [Figs. 14(a),4(c), 14(e), and 14(g)] and the IGA [Figs. 14(b), 14(d),4(f), and 14(h)] are applied (AD /AC=28.44, xC=0.15�0,C=0.10�0, �C=3.5, �C=0.0, �D=1.5, �D=0.1 S/m, andNR=15.0 dB). As a reference, the actual profile is re-orted in Fig. 14(i).

. CONCLUSIONSn this paper, an inverse-scattering-based procedure forhe detection of buried defects has been studied. The hostedium, including the defect, is inspected by means of in-

errogating microwaves. The solution of the problem is re-ast as an optimization problem that is solved through austomized genetic algorithm. The key point of the ap-roach is the use of an integrated genetic-algorithm strat-gy based on a numerically computed Green’s function forhe inhomogeneous configuration constituted by the hostedium in the external homogeneous background. Such aethodological approach represents a direct way to in-

lude a priori information in the scattering model, and itllows a nonnegligible enhancement of the reconstructionccuracy as well as a reduction of the overall computa-ional burden.

Several numerical results have been reported concern-ng noiseless as well as noisy environments with losslessnd lossy materials to evaluate the effectiveness of theethod dependent on the characteristics of the host me-

ium. As a matter of fact, such an analysis completes the

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ssessment of the proposed technique presented in Ref.0 and focused on the relationship between the perfor-ance and the properties and dimension of the defect.oreover, for comparison purposes, the method has been

ompared with a more classic approach based on the free-pace Green’s function, which requires taking into ac-ount the complete investigation domain and therefore re-uires large computational resources. As indicatedreliminarily in Ref. 20 (although there the main empha-is was on the dependence of the performance on therack characteristics), the results arising from such aomparative assessment also generally confirm a greaterfficacy of the approach based on the inhomogeneousreen’s function. However, this study pointed out that be-

ause of some difficulties in the computation of the inho-ogeneous Green’s function, the IGA turns out to be more

ensitive than the FGA to the variations in the conductiv-ty of the scheme.

Corresponding author Andrea Massa may be reachedt the address on the title page, by phone at 39-0461-82057, fax at 39-0461-882093, or e-mail [email protected].

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