UNIVERSITY
OF TRENTO
DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE
38123 Povo – Trento (Italy), Via Sommarive 14
http://www.disi.unitn.it
EFFECTIVE EXPLOITATION OF THE INFORMATION CONTENT
OF NOISY DATA IN INVERSE SCATTERING PROBLEMS: A
FUZZY-LOGIC-BASED STRATEGY
A. Casagranda, M. Donelli, D. Franceschini, and A. Massa
January 2011
CONTENT OF NOISY DATA IN INVERSE SCATTERING
PROBLEMS: A FUZZY-LOGIC-BASED STRATEGY
Aronne Casagranda*, Massimo Donelli*, Davide Fran es hini*, and Andrea Massa*
*ELEDIAResear h Group DIT
UniversityofTrento-ViaSommarive,14-38050Trento-ITALY
E-mail: {aronne. asagranda, massimo.donelli, davide.fran es hini, andrea.massa}dit.unitn.it Web-site: http://www.eledia.ing.unitn.it
Abstra t. One of the main drawba ks when dealing with inverse s attering problems is their in-trinsi ill- onditioned nature, emphasized by the presen e of the noise in the measured data. In order to avoid/limit this drawba k, this paper presents an innovative fuzzy-logi -based approa h. The proposed fuzzy system allows totake into a ount the orrupted nature of the data in a simple and ee tive way. The ee tiveness of the proposed inversion strategy is shown by omparing the a hieved results withthose obtained withstate-of-the-art inverse s attering te hniques.
1. Introdu tion
In the framework of the mi rowave imaging, some of the main di ulties in the re onstru tion pro ess are ertainlydue to theill-posedness oftheinverse problemand to thenoisynature ofthe available measureddata.
Inpra ti alsituations,experimentalandenvironmentalnoisesaddtothes atteredsignalsduetothe me hani al positioning of the ele tromagneti eld sensors or to the ele tromagneti interferen es in the test- hamber. These fa tors signi antly ae t the result of the ele tromagneti imaging pro ess ( hara terized byan highintrinsi instability), leading to ina urate re onstru tions ofthe s enariounder test.
In the framework of spatial-domain pro edures, this drawba k is generally over ome by reformu-latingtheoriginal inverse s attering probleminthe minimizationof asuitable ost fun tion. Su h a fun tion provides ameasure of the tting between there onstru ted and theavailable data or-ruptedbythenoise. Itis omposedoftwoterms: thedatatermandthestatetermwhi hdependon thes attered eldinthe observation domain andon thein ident eld intheinvestigation domain, respe tively. Suitable multipli ative regularization parameters allows one to weight more the one or theotherterm, dependingon theun ertainties asso iatedwith both of them.
In order to properly exploit the information- ontent available in thenoisy data, theproposed ap-proa h avoids a dire t estimation of the reliability of thes attering data (be ause of the ost and the omplexity of su h an estimate) and it takesinto a ount the presen e of thenoise through a strategy based ona fuzzy logi [1 ℄. The fuzzy systemdeterminesthe values of theweighting regu-larization parameters estimating, in an unsupervised way, the degree of reliability of theavailable data.
The paper is organizedasfollows. In Se tion 2, themathemati al formulation of theinverse s at-tering problem is brie y resumedand ades ription of the fuzzy-logi -based approa h is presented (Se t. 3). Se tion4shows asetofsele tednumeri al resultsinordertopointout theimprovement brought by the fuzzysystem. A naldis ussionis reportedin Se tion 5.
2. Inverse Problem Formulation
Referringtoatwo-dimensional problem hara terized bya ylindri al geometry,asetof
T M
plane waves (E
inc
ele tromagneti parameters(ortheobje tfun tion
τ (x, y)
)havetobedetermined. Theba kground mediumis assumedto be homogeneous ( hara terized by adiele tri permittivityε
0
)and lossless. Therelation between diele tri propertiesof theinvestigation domainand s attered elds is math-emati ally des ribed through the well-known integral inverse s attering equations [2℄. Their dis- retized version, obtained subdividing the investigationdomain byN
re tangular basisfun tions, isreported in(1)and (3):ℑ
ST AT E
n
τ (x
n
, y
n
); E
v
tot
(x
n
, y
n
)
o
= E
v
tot
(x
n
, y
n
) − j
k
2
0
4
N
X
p=1
τ (x
p
, y
p
)E
v
tot
(x
p
, y
p
)G
v
2D
(x
n
, y
n
|x
p
, y
p
)
(1) whereG
v
2D
(x
m
, y
m
|x
n
, y
n
) =
(j/2)
h
πk
0
a
p
H
1
(2)
(k
0
a
p
) − 2j
i
if p = n
(jπk
0
a
n
/2)H
0
(2)
(k
0
ρ
v
mn
)J
1
(k
0
a
n
)
otherwise
(2) andℑ
DAT A
n
τ (x
n
, y
n
); E
v
tot
(x
n
, y
n
)
o
= j
k
2
0
4
N
X
n=1
τ (x
n
, y
n
)E
v
tot
(x
n
, y
n
)G
v
2D
(x
m
, y
m
|x
n
, y
n
)
(3) whereG
v
2D
(x
m
, y
m
|x
n
, y
n
) = (jπk
0
a
n
/2)H
0
(2)
(k
0
ρ
v
mn
)J
1
(k
0
a
n
)
(4)J
1
,H
(2)
0
,andH
(2)
1
beingtheBesselfun tionof rstkindandthe0-th and rst-orderHankelfun tion ofthese ondkind,respe tively;E
tot
v
(x
n
, y
n
)
istheunknownele tri eldintheinvestigationdomainand
ρ
v
mn
=
r
x
m
(v)
− x
n
2
+
y
m
(v)
− y
n
2
.Through a multi-view/multi-illumination measurement system the data of the inverse s attering problemarea quired. Thes attered ele tri eld
E
scatt
v
(x
m
(v)
, y
m
(v)
)
is olle ted atea h view,v =
1, ..., V
,inM
(v)
measurementpointsequally-spa edalonga ir ularobservationdomain. Moreover, thein ident eldE
inc
v
(x, y)
ismeasured, without thes atterer,intheinvestigation domain. Then, theobje tfun tionτ (x
n
, y
n
)
aswellasE
tot
v
(x
n
, y
n
)
areretrieved aftertheminimizationofa suitable ostfun tion de ned,a ording to thefuzzy-logi approa h, inSe tion 3.3. The Fuzzy-Logi -Based Approa h
Thefuzzy-logi systema tsbetweenthedataa quisitionandthede nitionofthe ostfun tiontobe minimized,bydeterminingthevaluesofasetofweighting oe ients(
α
m
v
andβ
n
v
,m
(v)
= 1, ..., M
(v)
,v = 1, ..., V
,n = 1, ..., N
) that expressthe reliability ofea h sampleofthemeasured s atteredand in ident eld. Consequently, the arising ost fun tion turnsout to beΦ
τ (x
n
, y
n
); E
v
tot
(x
n
, y
n
)
=
P
V
v
=1
P
M(v)
m
=1
α
m
v
|
E
scatt
v
(x
m
,y
m
)−ℑ
DAT A
{
τ(x
n
,y
n
);E
tot
v
(x
n
,y
n
)
}|
2
P
V
v=1
P
M(v)
m
=1
|E
scatt
v
(x
m
,y
m
)|
2
+
P
V
v
=1
P
N
n
=1
β
n
v
|
E
inc
v
(x
n
,y
n
)−ℑ
ST AT E
{
τ(x
n
,y
n
);E
tot
v
(x
n
,y
n
)
}|
2
P
V
v
=1
P
N
n
=1
|E
inc
v
(x
n
,y
n
)|
2
(5)thefollowing oe ients
η
v
m
=
E
scatt
v
(x
m
,y
m
)
E
tot
v
(x
m
,y
m
)
max {η
m
v
}
ξ
v
n
=
E
v
inc
(x
n
, y
n
)
max {ξ
n
v
}
v = 1, ..., V ; m
(v)
= 1, ..., M
(v)
(6)Then, the fuzzy ation step follows. Let us de ne a set of heuristi ally-de ned rules
ℜ
, om-posed by a set of ante edents (Λ
) and relative onsequen es (C
). The system inputs (η
m
v
, ξ
v
n
) are transformedinto theirfuzzy ounterparts(ϕ (η
v
m
) , ϕ (ξ
n
v
)
)byasso iatingtoea hsampleagaussian membership fun tion [1℄. Su h a pro ess allows to a tivate a rule a ordingto thedegree of simi-larity betweenϕ(•)
and the ante edent of the rule inΛ
. An a tivated rule is hara terized byan a tivationvaluethatdeterminesthe degreeoftruthofevery onsequen eofC
. Finallya defuzzi er providesthereliabilityindex(α
v
m
, β
n
v
)by onsideringthe ompositionofthea tivated onsequen es ofC
.Su essively, the ost fun tion (whereea h measured data ontributes a ording to own reliability index)is minimizedbymeans ofa suitablegradient-based minimization algorithm [3℄.
4. Numeri al Results
To assess the ee tiveness of the proposed fuzzy-based methodology, the referen e square (
1.2 λ
0
-sided) ylindri alpro le(τ = 1.0 + j0.0
)hasbeen onsidered(Fig. 1(a)). Theinvestigationdomain (4 λ
0
in side),dis retizedinN = 100
square ells, hasbeen illuminatedthrough asetofV = 8
TM planewaves. Theinversiondata, hara terized byawhitegaussiannoise withSN R = 10 dB
,have been olle ted inM
(v)
= 35 (v = 1, .., V )
measurement points.For omparison purposes, theresults when the fuzzy system is onsidered have been ompared to those of a standard approa h. The quality improvement inthe nal re onstru tion an be learly observed in Fig. 1( ) (with respe t to Fig. 1(b) where the pro le retrieved with the standard pro edureisshown). For ompleteness,Figure2 showsa omparison between thede reasingofthe ost fun tionduring the iterative minimization pro esswithand without thefuzzysystem.
5. Con lusions
In this paper, an innovative approa h to inverse s attering problems has been presented. The approa h allowsasimple andee tiveintrodu tion oftheinformation onthereliabilityof olle ted noisy data. Su h an information is introdu ed into the ost fun tion to be minimized through suitableregularization oe ientsdetermined bymeansofafuzzy-logi system. Theadvantagesof thefuzzy-logi -based strategy overstandard inversionapproa hes have been preliminarilyassessed by onsidering aset ofnumeri al experiments.
x
y
x
y
x
y
1.0
Re {τ (x, y)}
0.0
1.0
Re {τ (x, y)}
0.0
1.0
Re {τ (x, y)}
0.0
(a) (b) ( )
Figure1. A tualpro le(a). Re onstru tedpro leobtainedwith(b)theStandardTe hniqueand( )with theFuzzyStrategy.
0.01
0.1
1
10
0
200
400
600
800
1000
1200
1400
Φ
iterations number
Standard
(Fuzzy-Logic)-based Approach
Figure 2. Behaviorofthe ostfun tion
Referen es
[1℄ L.A. Zadeh,Fuzzy Sets, Informationand Control, vol.8,pp.338-353, 1965.
[2℄ S. Caorsi, M. Donelli, D. Fran es hini, A. Massa, and M. Pastorino, Mi rowave imaging of two-dimensional s atterers using an iterative multi-s aling methodology, in Pro . Jina2002, pp.293-296, vol.2,Ni e,Fran e, 12-14November2002.
[3℄ H.Harada, D.J.Wall,T. T.Takenaka,andT. Tanaka,Conjugategradient methodapplied to inverse s attering problems, IEEE Trans. Antennas Propagat., vol. 43,pp.784-792, 1995.
