Dealing with Multi-Frequency Scattering Data Through the IMSA

UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38123 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

DEALING WITH MULTI-FREQUENCY SCATTERING DATA

THROUGH THE IMSA

D. Franceschini, M. Donelli, R. Azaro, and A. Massa

January 2011

the IMSA

Davide Fran es hini, Massimo Donelli,Renzo Azaro, and Andrea Massa

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,

{davide.fran es hini,massimo.donelli,renzo.azaro}dit.unitn.it

the IMSA

Davide Fran es hini, Massimo Donelli,Renzo Azaro, and Andrea Massa

Abstra t

This ontribution presents a set of representative results obtained withtwo

multi-resolution strategies developed for dealing with multi-frequen y inverse s attering

experiments starting from the iterative multi-s aling approa h previously studied

for mono hromati illuminations. Therst approa h is on erned withthe

integra-tionoftheiterativemulti-s alingalgorithm intoafrequen y-hoppingre onstru tion

s heme, while the se ond one allows a multi-resolution simultaneous pro essing of

multi-frequen ydata. Thenumeri alandtheexperimentalanalysisareaimedat

as-sessingboththere onstru tionee tivenessandtherequired omputational ostsof

the two- and three-dimensional implementations of theproposedinversion s hemes

in omparison with the mono hromati multi-step pro ess and to the single-step

optimization strategy,aswell.

Key words:

Mi rowave imaging te hniques are aimed at pro essing the s attered ele tromagneti

ra-diation olle ted in a non-invasive fashion on an external measurement region for

deter-mining the ele tromagneti propertiesof an ina essible area of investigation [1℄-[3℄.

A key problem related to su h te hniques is the la k of olle table information on the

s enario undertest. As amatterof fa t,the information ontentavailablefromthe

s at-tering experiments annotbe arbitrarilyin reased by oversamplingthe s attered eld [4℄

and, sin e single-illuminationmeasurement setups do not usually allow adetailed

re on-stru tion of unknown s atterers, multi-view and multi-illumination a quisition systems

have been proposed [5℄for in reasing the amount ofinformativedata. In su h a ontext,

severalworkshaveshown thatfrequen y-hoppingstrategies(e.g.,[6℄) orthesimultaneous

pro essing of multi-frequen y data [7℄-[9℄ an improve the re onstru tion apabilities of

inverse s attering algorithms.

However, multi-frequen y strategies have some intrinsi drawba ks to be arefully

ad-dressed for realizing ee tive and reliable inversion s hemes. From the point of view

of software/hardware resour es, the in rease ofinversion dataunavoidablyneeds of

addi-tional omputationalandstorageresour es withrespe ttomono hromati re onstru tion

methodologies. Moreover, the obje t fun tion modeling the diele tri and ondu tivity

parameters of the medium varies in relation to the working frequen y and a ording to

the dispersionmodel. Therefore, suitable ountermeasures are ommonlyadopted, whi h

onsider a narrow range of frequen ies or assume a Maxwellian model for des ribing the

relationship between diele tri parameters and operatingfrequen y (e.g., [7℄and [8℄).

On the other hand, even though possible advantages of using multi-frequen y measures

havebeen deeply investigated inthe frameworkofsingle-step inversionalgorithms,a

lim-ited literature deals with the exploitation of multi-frequen y information through

multi-resolution strategies [10℄. Therefore, this paper is aimed at developing su h an issue by

investigating how a multi-frequen y approa h an be protably integrated into a

multi-step inversion algorithm(

IMSA

)[11℄[12℄previously studiedfa ingmono hromati data.

This interest is mainlymotivated from the non-negligible advantages that are expe ted.

[11℄-[14℄) taking into a ount the enlarged set of information omingfrom the s attering

experimentsat dierent frequen ies.

The paper is organized asfollows. The mathemati alformulationof the multi-frequen y

approa hes based on the

IMSA

is briey summarized in Se t. 2 as far as the

two-dimensional ase is on erned, while a set of representative syntheti as well as

experi-mentalresults(regardingtwo-andthree-dimensional ongurationsofdiele tri andlossy

bodies) are shown and dis ussed in Se t. 3. Finally, in Se t. 4, some on lusions and

further possible developments willbeoutlined.

2 Mathemati al Formulation

Forsimpli ity,letusrefertoatwo-dimensionals enario(Fig. 1) hara terizedbyan

inho-mogeneous rossse tion

D

I

lyinginahomogeneoushostmedium(

ε

0

,

µ

0

)andilluminated

by a set of

P

mono hromati (

f

p

being the working frequen y of the

p

-th illumination,

p = 1, ..., P

) in ident ele tri elds

T M

-polarized impinging from

V

dierent dire tions

(

E

inc

v,p

(x, y) = E

v,p

inc

(x, y)ˆz

,

v = 1, ..., V

,

p = 1, ..., P

). The multi-frequen y information

available fromthe s attered radiation

E

tot

v,p

(x

m

v,p

, y

m

v,p

) = E

tot

v,p

(x

m

v,p

, y

m

v,p

)ˆz

, olle ted at

m

v,p

= 1, ..., M

v,p

points belonging to the observation domain

D

O

, has to be e iently

exploited for re onstru ting the frequen y-dependent ontrast fun tion

τ

p

(x, y) = ε

r

(x, y) − 1 − j

σ (x, y)

2πf

p

ε

o

p = 1, ..., P

(1)

ε

r

(x, y)

and

σ (x, y)

beingtherelativepermittivityandthe ondu tivity,respe tively. The

hara teristi softhes enarioundertest

τ

p

(x, y)

,

p = 1, ..., P

,arerelatedtothes attering

data (namely, the s attered eld in the observation domain,

E

scatt

v,p

(x, y)

,

(x, y) ∈ D

O

,

and the in ident eld in the investigation domain,

E

inc

v,p

(x, y)

,

(x, y) ∈ D

I

) through the

Lippmann-S hwingers attering equations [11℄

E

scatt

v,p



x

m

v,p

, y

m

v,p



= S

ext

v,p

n

τ

p

(x, y)E

v,p

tot

(x, y)

o

(x, y) ∈ D

I



x

m

v,p

, y

m

v,p



∈ D

O

(2)

E

inc

v,p

(x, y) = E

v,p

tot

(x, y) − S

v,p

int

n

τ

p

(x, y)E

v,p

tot

(x, y)

o

(x, y) ∈ D

I

(3) where

S

ext

v,p

and

S

int

v,p

denotetheexternalandinternals atteringoperator,respe tively,and

E

scatt

v,p

is dened as

E

scatt

v,p

(x, y) = E

v,p

tot

(x, y) − E

v,p

inc

(x, y)

.

In order to ee tively deal with multi-frequen y data by solving (2) and (3) in terms of

the unknown ontrast fun tion

τ

p

(x, y)

,

p = 1, ..., P

, two multi-resolution methodologies

based onthe

IMSA

willbe proposed in the following.

2.1 Frequen y-HoppingIterativeMulti-S aling Approa h

(IMSA-FH)

The rst inversion algorithm is on erned with an integration of the

IMSA

into a

fre-quen y hopping s heme. In su h an integration the multi-frequen y data (

p = 1, .., P

)

are used in a as ade fashion starting from the lowest available frequen y (

f = f

1

). At

ea h stage of the hopping strategy (

p = 1, ..., P

), a nested multi-s alingpro edure of

S

p

optimization steps pro esses the mono hromati data-set related to the

p

-th frequen y.

More indetail, the pro edure an besummarized as follows

(a) Frequen y-Hopping Re onstru tion Loop (

p = 1, ..., P

)

(a.1) Initialization (

s

p

= 0

)

The investigationdomain

D

I

is partitioned into

N

p

sub-domains a ording to the

infor-mation ontent of the s attered eld [8℄ at

f = f

p

and, onsequently, a suitable set of

re tangular basis fun tions [

B

n

(x, y)

,

n = 1, ..., N

p

℄ is dened. Moreover, the problem

unknownsare initializedtothe free-spa e onguration, bysetting

E

tot

v,p,s

p

(x, y)

k

p=1,s

p

=0

=

E

inc

v,p

(x, y)

k

p=1

and

τ

p,s

p

(x, y)

k

p=1,s

p

=0

= τ

0

(x, y)

, when

p = 1

, otherwise the prole

re- onstru ted atthe onvergen e step of the

(p − 1)

-thfrequen y stageis suitably mapped

intothe investigationdomain [i.e.,

E

tot

v,p,s

p

(x, y)

k

s

p

=0

= E

tot

v,p−1

(x, y)

and

τ

p,s

p

(x, y)

k

s

p

=0

=

τ

p−1

(x, y)

℄;

(a.2) Low-Order Re onstru tion (

s

p

= 1

)

τ

p,s

p

(x, y)

k

s

p

=1

=

P

N

1

n=1

τ

p,s

p

(x

n

, y

n

)

k

s

p

=1

B

n

(x, y) ,

(x, y) ∈ D

I

(4)

E

tot

v,p,s

p

(x, y)

k

s

p

=1

=

P

N

1

n=1

E

v,p,s

tot

p

(x

n

, y

n

)

k

s

p

=1

B

n

(x, y) ,

(x, y) ∈ D

I

(5)

is yielded by determining the set of unknown oe ients

f

p,s

p



s

p

=1

=



τ

p,s

p

(x

n

, y

n

)

k

s

p

=1

,

E

tot

v,p,s

p

(x

n

, y

n

)

k

s

p

=1

;

n = 1, ..., N

p

,

v = 1, ..., V }

,throughtheminimizationofthelow-order

ost fun tion

Ψ

n

f

_p,1

o

=

V

X

v=1

P

Mv,p

mv,p=1

|

E

scatt

v,p

(

x

mv,p

,y

mv,p

)

−S

v,p

ext

{

τ

p,1

(x

n

, y

n

)E

tot

_v,p,1

(x

n

, y

n

)

}|

2

V

X

v=1

P

Mv,p

mv,p=1

|

E

scatt

v,p

(

x

mv,p

,y

mv,p

)|

2

+

V

X

v=1

P

Np

n=1

|

E

inc

v,p

(x

n

, y

n

)−E

_v,p,1

tot

(x

n

, y

n

)+S

int

v,p

{

τ

p,1

(x

n

, y

n

)E

_v,p,1

tot

(x

n

, y

n

)

}|

2

V

X

v=1

P

Np

n=1

|

E

v,p

inc

(x

n

, y

n

)

|

2

(6)

(a.3) Multi-S aling Pro ess (

s

p

= 2, ..., S

p

)

(a.3.I)Multi-S aling ProleRe onstru tion

Starting from the solution

f

p,s

p

−1

omputed at the previous step (

s

p

− 1

), a set of

Q

s

p

regions-of-interest (

RoI

s),

D

(q)

O(s

p

−1)

, is dened a ording to the lustering pro edure

de-s ribed in [12℄ where the resolution level is in reased [

R

s

p

← R

s

p

+ 1

℄. Therefore a

multi-resolutionrepresentation of the unknowns

τ

p,s

p

(x, y) =

P

R

sp

r

sp

=1

P

N

(

r

sp

)

n

(

r

sp

)

=1

τ

p,s

p



x

_n

₍

_r

sp

), y

n

(

r

sp

)



B

_n

₍

_r

sp

) (x, y)

(x, y) ∈ D

I

E

tot

v,p,s

p

(x, y) =

P

R

sp

r

sp

=1

P

N

(

r

sp

)

n

(

r

_sp

)

=1

E

tot

v,p,s

p



x

_n

₍

_r

sp

), y

n

(

r

sp

)



B

_n

₍

_r

sp

) (x, y)

(7)

r

s

p

being the resolution index at the

s

p

-th step ranging from the largest hara teristi

length s ale (

r

s

p

= 1

) up to the smallest basis-fun tion support (

r

s

p

= R

s

p

= s

p

), is

Ψ



f

_p,s

p



=

V

X

v=1

P

Mv,p

mv,p=1









E

scatt

v,p

(

x

mv,p

,y

mv,p

)

−S

ext

v,p



τ

p,sp



x

n

(

_rsp

)

, y

n

(

_rsp

)



E

tot

v,p,sp



x

n

(

_rsp

)

, y

n

(

_rsp

)









2

V

X

v=1

P

Mv,p

mv,p=1

|

E

scatt

v,p

(

x

mv,p

,y

mv,p

)|

2

+

1

V

X

v=1

P

Rsp

rsp =1

P

N

(

rsp

)

n

(

_rsp

)

=1









E

inc

v,p



x

n

(

_rsp

)

, y

n

(

_rsp

)









2

V

X

v=1

P

R

sp

r

sp

=1

P

N

(

r

sp

)

n

(

r

sp

)

=1



w



x

_n

₍

_r

sp

), y

n

(

r

sp

)











E

inc

v,p



x

_n

₍

_r

sp

), y

n

(

r

sp

)



− E

tot

v,p,s

p



x

_n

₍

_r

sp

), y

n

(

r

sp

)



+

S

int

v,p



τ

p,s

p



x

_n

₍

_r

sp

), y

n

(

r

sp

)



E

tot

v,p,s

p



x

_n

₍

_r

sp

), y

n

(

r

sp

)









2

)

(8) where

w



x

_n

₍

_r

sp

), y

n

(

r

sp

)



=















0 if



x

_n

₍

_r

sp

), y

n

(

r

sp

)



/

∈ D

(q)

_O(s

_p

₋₁₎

1 if



x

_n

₍

_r

sp

), y

n

(

r

sp

)



∈ D

(q)

_O(s

_p

−1)

q = 1, ..., Q

s

p

(a.3.II)Multi-S aling Termination

The multi-s ale pro ess is stopped (

s

p

= S

opt

p

< S

p

) when a set of stability riteria

on the re onstru tion [11℄ hold true and the rea hed solution,

f

p,S

p

opt

is assumed as the

estimatedsolutionatthe

p

-thfrequen ystage[i.e.,

τ

p

(x, y) = τ

p,S

opt

p

(x, y)

and

E

tot

v,p

(x, y) =

E

tot

v,p,S

p

opt

(x, y)

℄. Moreover, if

p = P

then the loop over the frequen ies (

f = f

1

, ..., f

P

) is

terminated and the nal diele tri prole returned,

τ

opt

_{(x, y) = τ}

p

(x, y)

, as well as the

eld distributions

E

opt

v,p

(x, y) = E

v,p

tot

(x, y)

,

p = 1, ..., P

.

2.2 Multi-Frequen y Iterative Multi-S aling Approa h

(IMSA-MF)

Unlike the

IMSA − F H

, the

IMSA − MF

represents a generalization of the

IMSA

pro essingproposedin[11℄forthe re onstru tionof singles atterersand extended in[12℄

for multiple-obje ts from single-frequen y data. A ordingly, some assumptions on the

s attering fun tion (1) and a dierent data pro essing are needed. In the following, the

distin tiveissues of su h anmethodwillbebriey resumed.

dealing with multi-frequen y data, the maxwellian modelfor the dispersion relationship is assumed

τ

p

(x, y) = Re [τ

ref

(x, y)] + j

f

ref

f

p

Im [τ

ref

(x, y)]

(9)

where the value of the ontrast at the

p

-th frequen y,

τ

p

(x, y)

, is related to that of the

referen efrequen y

f

ref

,

τ

ref

(x, y)

. Therefore, theinversionpro ess aimedatdetermining

the unknown referen e ontrast fun tion,

τ

ref

(x, y)

, insteadof

P

distributions, thus

prof-itably exploitingthe multi-frequen y data spa e and limiting the growing of the number

of unknowns (with respe t to the mono-frequen y ase) to the internal elds,

E

tot

v,p

(x, y)

,

p = 1, ..., P

, that have to be estimated for ea h working frequen y (

p = 1, .., P

) of the

illuminationsetup;

(b) Multi-Frequen y Re onstru tion Loop (

s = 1, ..., S

)

Likewise the standard

IMSA

, a multi-s aling pro edure of

S

(

s = 1, ..., S

) steps is

per-formed in order todene the multi-resolutionexpansion of the unknown parameters

τ

ref

(x, y) =

P

R

r

s

s

=1

P

N (r

s

)

n(r

s

)=1

τ

ref



x

n(r

s

)

, y

n(r

s

)



B

n(r

s

)

(x, y)

(x, y) ∈ D

I

E

tot

v,p

(x, y) =

P

R

s

r

s

=1

P

N (r

s

)

n(r

s

)=1

E

tot

v,p



x

n(r

s

)

, y

n(r

s

)



B

n(r

s

)

(x, y)

(10)

being

N(R

s

) = max

p

{N

p

}

and

R

s

= s

, by minimizingat ea h step of the re onstru tion

loopthe Multiple-Frequen y Multi-Resolution Cost Fun tion dened as follows

Ψ

n

f

_s

o

=

P

X

p=1

V

X

v=1

P

Mv,p

mv,p=1

|

E

scatt

v,p

(

x

mv,p

,y

mv,p

)

−S

v,p

ext

{

τ

ref,s

(

x

n(rs)

, y

n(rs)

)

E

v,p,s

tot

(

x

n(rs)

, y

n(rs)

)}|

2

P

X

p=1

V

X

v=1

P

Mv,p

mv,p=1

|

E

scatt

v,p

(

x

mv,p

,y

mv,p

)|

2

+

1

P

X

p=1

V

X

v=1

P

Rs

rs=1

P

N(rs)

n(rs)=1

|

E

v,p

inc

(

x

n

(rs)

, y

n

(rs)

)|

2

V

X

v=1

P

R

s

r

s

=1

P

N (r

s

)

n(r

s

)=1

n

w



x

n(r

s

)

, y

n(r

s

)









E

inc

v,p



x

n(r

s

)

, y

n(r

s

)



− E

tot

v,p,s



x

n(r

s

)

, y

n(r

s

)



+

S

int

v,p

n

τ

ref,s



x

n(r

s

)

, y

n(r

s

)



E

tot

v,p,s



x

n(r

s

)

, y

n(r

s

)

o





2



(11)

where the unknown array turns out to be

f

s

=

n

τ

ref



x

n(r

s

)

, y

n(r

s

)



,

E

tot

v,p,s



x

n(r

s

)

, y

n(r

s

)



;

n (r

s

) = 1, ..., N (r

s

)

,

r

s

= 1, ..., R

s

,

p = 1, ..., P

,

v = 1, ..., V }

.

3 Numeri al Analysis

In this Se tion, a sele ted set of representative inversion results are presented and

dis- ussed. The aim is to assess the re onstru tion apabilities of the proposed

multi-resolution strategies (

IMSA − MF

and

IMSA − F H

)in omparisonwith the standard

IMSA

[referred in the following as Single-Frequen y

IMSA

(

IMSA − SF

)℄ and with

respe t to traditionalsingle-step methods. As far as two-dimensional ongurations are

on erned, both syntheti and experimental data are presented and the limitations in

re onstru ting lossy stru tures are dis ussed. Moreover, as arepresentative example,the

re onstru tion of athree-dimensional layered stru tures isreported, aswell.

3.1 Two-Dimensional Congurations

3.1.1 Numeri al Testing - Layered Diele tri Prole

The geometryoftherst test ase isshowninFig. 2(a). Alayered ylinderliesat

x

ref

c

=

−y

ref

c

= −0.6 λ

(

λ

beingthe wavelength at

f = 6 GHz

)in asquare investigationdomain

L

DI

= 3.0 λ

-sided. The obje t fun tion of the inner square layer (

L

in

= 0.6 λ

inside) is

τ

in

= 0.5

, while that of the outer layer (

L

out

= 0.9 λ

-sided) is equal to

τ

out

= 2.0

. The

investigationdomainhasbeenilluminatedbyasetofin identplanewavesimpingingfrom

V = 8

equally-spa ed dire tions. For ea h illumination, multi-frequen y data

(1)

(

P = 3

,

f

1

= 5 GHz

,

f

2

= 6 GHz

,

f

3

= 7 GHz

) have been simulated in a set of measurement

points lo ated on a ir le

r

D

O

= 3 λ

in radius. Be ause of the dierent information

ontent available at ea h frequen y [4℄[8℄,

M

v,1

= 31

,

M

v,2

= 37

and

M

v,3

= 44

eld

sampleshavebeen taken intoa ount at

f

1

,

f

2

and

f

3

,respe tively. Therefore,

N

1

= 121

,

N

2

= 144

,and

N

3

= 169

basisfun tionshavebeenusedforthe lower-orderre onstru tion

(

IMSA − F H

and

IMSA − SF

), while

N(R

s

) = N

3

,

s = 1, ..., S

, has been hosen for

the

IMSA − MF

inversion.

Intheframeworkofthe omparativestudy,theresultsoftheinversionpro essbothforthe

(1)

IMSA−SF

simulations[Figs. 2(b)-(d)℄andforthe

IMSA−MF

approa h[Fig. 2(e)℄are

showninFig. 2. Asexpe ted,thegray-levelimagesoftheobje tfun tionpi toriallypoint

outthat there onstru tionbenetsofnon-negligibleimprovementswhenmulti-frequen y

data are exploited both in rening the outer and the inner layer. Su h a qualitative

im-pression is onrmed by the orresponding values of the error gures reported in Tab. I

and dened as in [11℄. Whatever the frequen y used for the

IMSA − SF

inversion, the

IMSA − MF

turns out tobebetter guaranteeingenhan ed performan es. In parti ular

min

p=1,2,3

n

χ

IM SA−SF

p

tot

o

≥ 2.21χ

IM SA−MF

tot

,

min

p=1,2,3

n

χ

IM SA−SF

p

int

o

≥ 1.20χ

IM SA−MF

int

and

min

p=1,2,3

n

χ

IM SA−SF

p

ext

o

≥ 1.58χ

IM SA−MF

ext

. Su essively, inorder togeneralize these

indi- ations, the whole set of simulations in the frequen y range between

3 GHz

and

7 GHz

has been performed. The results are resumed in Fig. 3 in terms of the total

re onstru -tion error versus frequen y [or the referen e frequen y

f

ref

for the

IMSA − MF

when

the

P = 3

adja ent frequen ies (

f

p−1

= f

ref

− 1 GHz

,

f

p

= f

ref

,

f

p+1

= f

ref

+ 1 GHz

)

are simultaneouslypro essed℄. As it an beobserved, thea ura yofthe multi-frequen y

strategy is on average of about

△χ

IM SA−SF/IMSA−MF

tot

= 40

% (being

△χ

i/j

tot

=

χ

i

tot

−χ

j

tot

χ

i

tot

)

betterthan that of single-frequen y multi-resolutionmethods.

For ompleteness,the

IMSA−MF

hasbeen omparedtotheother

multi-resolution/multi-frequen y approa h (

IMSA − F H

). Moreover, as a referen e, the inversion has been

performed with single-step onjugate gradient-based strategies (also referred as bare

approa hes sin e a homogeneous dis retization, a ording to the Ri hmond's riterion

[15℄,ofthe

D

I

hasbeen adopted)bothintheirfrequen y hopping(

CG − F H

)and

multi-frequen y (

CG − MF

)implementations. In Fig. 4(a)the diele tri prole retrieved with

the

IMSA−F H

strategyisshown. Asit anbenoti ed(and onrmedbytheerrorvalues

giveninTab. I),the

IMSA

slightlytakesadvantageofthefrequen yhoppings heme

[ ar-ried out by onsidering the

P = 3

dierent frequen iesused forthe mono hromati

simu-lationswhoseresultsarepresentedinFigs. 2(b)-(d)℄. Asamatteroffa t,itturnsoutthat

min

p

n

△χ

IM SA−SF

p

/IM SA−HF

tot

o

= 8.7

%and

max

p

n

△χ

IM SA−SF

p

/IM SA−HF

tot

o

= 34

%versus

min

p

n

△χ

IM SA−SF

p

/IM SA−MF

tot

o

= 54

% and

max

p

n

△χ

IM SA−SF

p

/IM SA−MF

tot

o

= 67

%.

On the other hand, it should be noti ed that multi-resolution strategies (

IMSA − MF

in omparison with the orresponding bare approa hes [

CG − MF

- Fig. 4(b) and

CG − HF

-Fig. 4( )℄sin e

△χ

CG−F H/IMSA−F H

tot

= 36

%and

△χ

CG−MF/IMSA−MF

tot

= 55

%.

As far as the omputational issues are on erned, a general overview is given in Tab.

II where a set of representative parameters are reported. In parti ular, the number

of unknowns (

U

), the total number of onjugate-gradient iterations needed for

rea h-ing the onvergen e solution (

K

tot

), the mean time per iteration (

t

k

), and the global

CP U

time (

T

tot

) are given. Unavoidably, be ause of the larger dimension of the set of

s attering data, the simultaneous multi-frequen y pro essingrequires greaterstorage

re-sour es (

U

IM SA−MF

_{= 8500}

and

U

CG−MF

_{= 18052}

vs.

U

IM SA−SF

_{= U}

IM SA−F H

_{= 3042}

and

U

CG−F H

_{= 6515}

) and an in reasing of the omputational burden for ea h

itera-tion (

t

IM SA−MF

k

= 4.5 sec

and

t

CG−MF

k

= 20.2 sec

vs.

t

IM SA−SF

k

= t

IM SA−F H

k

= 0.5 sec

and

t

CG−F H

k

= 1.32 sec

). On the other hand, as expe ted and also demonstrated in the

mono hromati ase, the multi-resolution te hnique limit the omputational osts both

in terms of unknowns (

U

IM SA−MF

_{≃ 2.12 U}

CG−MF

and

U

IM SA−F H

_{≃ 2.14 U}

CG−F H

) and

CP U

time(

t

CG−MF

k

≃ 4.5 t

IM SA−MF

k

and

t

CG−F H

k

≃ 2.64 t

IM SA−F H

k

).

3.1.2 Numeri al Testing - Lossy Prole

The se ond test ase deals with a lossy prole. A square ylindri al stru ture, whose

dimensionsarethoseoftheinnerlayerusedofthepreviousexperiments,and hara terized

by an obje t fun tion

τ = 0.5 − j

σ

2πf

p

ε

o

has been onsidered. For a sensitivity study,

σ

has been varied in the range

0.0

S

m

≤ σ ≤ 1.0

S

m

.

The experiments arried out through the

IMSA − MF

indi ate that the obje t under

test is quite faithfully retrieved, even though there is an in reasing of the values of the

re onstru tion errors [

χ

re

tot

- Fig. 5(a);

χ

im

tot

- Fig. 5(b)℄ when the ondu tivity grows.

Moreover, the omparisonswith other

IMSA

-basedapproa hes shows that the

IMSA −

MF

s hemeusuallyimprovestheee tiveness ofthe retrievalpro edureandthea ura y

in re onstru ting the imaginary part is lower [Fig. 5(b)℄ than that in dealing with the

real part [Fig. 5(a)℄, espe ially when the

F H

s heme is exploited. Therefore, the latter

does not seem the most ee tive methodology for obtaining a good estimation of the

[

χ

im

int

- Fig. 5(d)℄ and total [

χ

im

tot

- Fig. 5(b)℄ errors present apeak below

σ = 0.1

S

m

. This

is duetothe di ulty of ana uratere onstru tion of the imaginarypart whenitsvalue

is mu h lower than that of the real part.

Su essively,alayered ongurationforboththepermittivityand ondu tivityproleshas

beentakenintoa ount. Inparti ular,therealpartofthe ontrastisdistributedasinthe

test aseofFig. 2(a),whiletheimaginarypartisshowninFig. 6(a),wheretheinnerlayer

is hara terizedby

σ

in

= 0.5

S

m

andtheouteroneby

σ

out

= 0.1

S

m

. Su hareferen eprole

has been re onstru ted with the

IMSA − MF

and the results reportedin Figs. 6(b)( ).

As it an be noti ed, the layered distribution has been satisfa torily imaged. Moreover,

inordertogiveawideroverview onpotentialitiesand urrentlimitationsofthe proposed

strategies, the behaviorsof the error gures when varying

σ

in

and

σ

out

from

0.0

S

m

up to

0.5

_m

S

are showninFig. 7. As expe ted,they onrmthat the a ura yinestimatingthe

ondu tivitydistributionislowerthanthat inretrievingthanthe permittivityprole. As

a matterof fa t,

χ

re

tot

< 3.8

% while

χ

im

tot

< 11

%.

3.1.3 Experimental Testing - Multiple S atterers

Inordertofurtherassesstheindi ationsdrawnfromthesyntheti testing,anexperimental

validation has been then performed. The s attering data are kindly available from the

Institute Fresnel in Marseille (Fran e) and the onguration of the measurement

set-up has been arefully des ribed in [16℄. The measurements have been olle ted in the

frequen y rangefrom

2 GHz

and

10 GHz

at

M

v,p

= 241

positionsfor ea hof the

V = 18

illuminations. As a representative ben hmark, the

T M

-polarized data on erned with

the so- alled FoamDielExtTM geometry [Fig. 8(a)℄ have been pro essed. In su h a

test ase, the s atterer under test is omposed by two ylinders hara terized by obje t

fun tions:

τ

1

= 0.45

and

τ

2

= 2.0

. Moreover, the two-dimensional assumption an be

made sin e the s atterers are long with respe t tothe wavelengthin the

z

-dire tion.

As far as the omparative study is on erned, let us onsider as a referen e result that

obtained pro essing mono hromati data (

f = 5 GHz

) with the

IMSA − SF

[Fig. 8(b)℄

and by onsideringaninitialdis retizationgridof

N = 256

sub-domains(more resultsat

[Fig. 8( )℄ has been arriedout by using

P = 4

frequen ies in the range between

f

min

=

2 GHz

and

f

max

= 5 GHz

with afrequen y step of

△f = 1 GHz

. Moreover, the analysis

has been ompleted with the re onstru tions by means of the

IMSA − F H

[Fig. 9(a)℄,

the

CG − MF

[Fig. 9(b)℄, and the

CG − F H

[Fig. 9( )℄. It is worth noting that the

investigation domain

D

I

for the bare approa hes has been partitioned into

N = 400

sub-domains inorder to rea h a satisfa tory spatialresolution in the estimated prole.

From a qualitative point of view, the gray-level plots onrm that usually the use of

multi-frequen y data improves the a ura y of the re onstru tions. More in detail, the

re onstru ted proles better reprodu e the a tual distribution espe ially on erning the

homogeneity of the s atterers ex ept for the

IMSA − F H

. In su h a ase, the hopping

strategy turns out to be ee tive in dening the stru ture and the ontrast value of the

stronger s atterer, while some problems o ur in shaping the weaker obje t. On the

other hand, it should be noti ed that single step algorithms [Fig. 9(b) e Fig. 9( )℄

underestimate the value of obje t fun tion inside the support of the smaller s atterer

(

τ

CG−MF

2

≃ τ

2

CG−F H

≃ 1.5

)and they requiremore omputationalresour es than

IMSA

-based pro edures aspointed out inthe previous sub-se tion.

3.2 Three-Dimensional Conguration

Thisse tionpresentssomerepresentativeresultsand onsiderationsonthere onstru tion

of three-dimensionalmultilayerstru tures frommulti-frequen ydata. In su h a ase, the

investigation domain is a ube

L

ind

= 1.2 λ

-sided and it has been illuminated by

V = 4

planewavesimpingingfrom

θ

inc

= π/2

,

φ

inc

=

(v−1)π

2

lo kwisewithrespe t tothe

z

axis.

A set of points-like re eivers (

M

v,p

= 21

) has been lo ated in

G = 3

rings

(ρ

m,v,p

= 2.93λ

in radius) at the positions

φ

scatt

m,v,p

= φ

inc

v,p

+ 2πG

(m

v,p

−1)

M

v,p

and

z

v,m

=

h

1 − G

(m

v,p

−1)

M

v,p

i

z

0

,

z

0

= 0.06λ

. Moreover, the olle ted data have been blurred by adding a Gaussian noise

with

SNR = 30 dB.

Insu hanarrangement,amultilayerproleofvolume

0.6×0.6×0.6λ

3

hasbeenpositioned

at

(x

b

= y

b

= z

b

= 0.0)

. The obje t fun tion of the three layers is

τ

1

= 0.5

,

τ

2

= 1.0

and

τ

3

= 1.5

, respe tively, and the volume of ea h layer is

0.2 × 0.6 × 0.6λ

3

. The referen e

been initially uniformly dis retized into

N = 5 × 5 × 5

ubi ells and a 3D frequen y

hopping s heme has been adopted (

P = 2

-

f

1

= 1 GHz

,

f

2

= 2 GHz

). The retrieval

obtained at the se ond hop (

s

2

= 2

) of the

IMSA − F H

(Fig. 11) learly points out

the layered stru ture of the obje t under test. The use of the hopping s heme together

withthe multi-s alingapproa himprovesthea ura yofthe re onstru tionpro edureas

indi ated by the behaviors of the error guresin Tab. III.

On theotherhand, itshouldbepointedout that ertainlythesimultaneous pro essingof

multi-frequen ydata would allowa further improvement of the re onstru tion a ura y,

but this would unavoidably require additional omputationalresour es. Inorder to limit

the impa t of su h an issue in multi-frequen y approa hes, fast and e ient numeri al

methods are needed to be fully exploited through a multi-resolution allo ation of the

unknowns. Apreliminaryindi ationofsu hapossibilityandofthearisingadvantages,let

us onsiderthere onstru tionofthepreviouss atterings enariowhenboththeimaginary

and the real part of the ontrast fun tion are unknown. By onsidering a bare single

frequen y approa h (

CG − SF

),

U

GC−SF

_{= 3250}

are the unknowns parameters and the

mean time per iteration is of about

t

CG−SF

k

≃ 5.80 s

. Dealing with a multi-frequen y

implementation, the dimension of the unknowns spa e be omes

U

GC−MF

_{= 25600}

and

t

CG−MF

_k

≃ 45.64 s

(2)

. On the ontrary, an exploitation of the proposed

IMSA − MF

strategywouldrequireameantimeperiterationof

t

IM SA−MF

k

≃ 11.14 s

forre onstru ting

U

IM SA−MF

_{= 6250}

parameters (su h a onguration provides a resolution level of the

same orderofthe

CG − MF

)withoutleadingtoanin reasingofthe omputational osts

(with respe t tothe standard single-step approa h), thanks tothe adaptive allo ationat

ea h step of the unknowns.

4 Con lusions

In this paper, starting from a set of representative test ases, a omparative assessment

of two multi-resolutionapproa hes exploiting multi-frequen y data has been arriedout.

The obje tive was that of giving some indi ations on the most suitable strategy able to

of omputational resour es ompared to that of single-step methodologies. As a matter

of fa t, unlike the

IMSA − F H

,the

IMSA − MF

provided numeri alproofs of the

en-han ed ee tiveness in pro essing multi-frequen y information. Moreover, the numeri al

and experimental analysis pointed out urrent limitations of the proposed approa h in

re onstru ting lossyprolesand the omputationalneeds ofthe three-dimensional

multi-frequen y multi-s aling pro edure that, even though demanding, is onsiderably more

ee tive than the single step pro edure. On the other hand, although the

IMSA − MF

demonstrated ana eptable robustness to the hoi e of illuminationfrequen ies, the

nu-meri al and experimental analysis pointed out that great are should be exer ised in

determining the informative amount inmulti-frequen ys attering data and how (and in

whi hquantity)to pro ess dierentdata-sets. As a matterof fa t, su hissues, urrently

under study, strongly ae t the performan e of ea h multi-resolution multi-frequen y

pro edure sin e they play a key-role both in allowing a good trade-o between suitable

resolution level inthe re onstru tion, o urren e of lo alminima, and ill- onditioning of

the wholeinversion.

Future developments of su h a resear h work will be aimed at theoreti ally (without

heuristi rules or expensive test-and-trial numeri al investigations) quantifying the

di-mension of the s attering data spa e as well as the independent information ontent of

ea hfrequen ydata-setforamoreee tiveexploitationofmulti-frequen ymeasurements

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Multi-resolution iterativeinversion of real inhomogeneous targets, Inverse Problems, vol.

•

Figure 1. Sket h of the two-dimensional imaging onguration.

•

Figure 2. Numeri al Testing - Layered prole. (a) Referen e distribution of the

obje tfun tion. Re onstru tedproles(

SNR = 20dB

)bymeansofthe

IMSA−SF

at the working frequen y of (b)

f = 5 GHz

, ( )

f = 6 GHz

and (d)

f = 7 GHz

.

(e) Comparison with the results obtained by means of the

IMSA − MF

(

f =

5, 6, 7 GHz

).

•

Figure 3. Numeri al Testing - Layered prole. Behavior of the quantitative error

guresversus theworkingfrequen y (

IMSA−SF

)and thereferen efrequen y

f

ref

(

IMSA − MF

).

•

Figure 4. Numeri al Testing - Layered prole. Re onstru tions obtained with (a)

IMSA − F H

, (b)

CG − MF

,and ( )

CG − F H

,respe tively.

•

Figure 5. Homogeneous lossy prole. Behavior of the error gures versus the

ondu tivity. (a)

χ

re

tot

, (b)

χ

im

tot

,( )

χ

re

int

and (d)

χ

im

int

.

•

Figure 6. Layered lossy prole. (a) Imaginary part of the referen e distribution

of the obje t fun tion. Re onstru ted prole (

SNR = 20dB

) by means of the

IMSA − MF

(

f = 5, 6, 7 GHz

) [(b) real and ( ) imaginary part℄ when

σ

int

=

0.5 S/m

and

σ

ext

= 0.1 S/m

.

•

Figure 7. Layered lossy prole. Behavior of (a)

χ

re

tot

and (b)

χ

im

tot

versus the

on-du tivity.

•

Figure 8. Experimental Testing - 'FoamDielExtTM' Conguration. (a) Sket h

of the obje t under test. Re onstru tion results obtained with (b)

IMSA − SF

(

f = 5 GHz

)and ( )

IMSA − MF

(

f = 2, 3, 4, 5 GHz

).

•

Figure9. ExperimentalTesting-'FoamDielExtTM'Conguration. Re onstru tion

resultsobtainedwith(a)

IMSA−F H

,(b)

CG−MF

,and( )

CG−F H

pro essing

•

Figure 10. Multi-layer diele tri ube - A tual prole: (a)

z − plane

and (b)

x − plane

.

•

Figure 11. Multi-layer diele tri ube - Re onstru tion obtained at

s

2

= S

opt

= 2

:

•

TableI.Numeri alTesting-Layeredprole. Valuesofthequantitativeerrorgures

at the onvergen e.

•

TableII.Numeri alTesting -Layeredprole. Valuesofthe omputationalindexes.

•

Table III. Multi-layer diele tri ube - Error gures vs steps atthe dierent hops

Scatterer

D

I

D

O

Receivers

Electromagnetic Source

p

p=1,...,P

f ,

x

y

y

x

2.3

Re {τ (x, y)}

0.0

(a)

y

x

y

x

2.3

Re {τ (x, y)}

0.0

2.3

Re {τ (x, y)}

0.0

(b) ( )

y

x

y

x

2.3

Re {τ (x, y)}

0.0

2.3

Re {τ (x, y)}

0.0

(d) (e)

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

3

3.5

4

4.5

5

5.5

6

6.5

7

χ

tot

f [GHz]

IMSA-SF

IMSA-MF

y

x

2.3

Re {τ (x, y)}

0.0

(a)

y

x

y

x

2.3

Re {τ (x, y)}

0.0

2.3

Re {τ (x, y)}

0.0

(b) ( )

0.2

0.4

0.6

0.8

1

1.2

0

0.2

0.4

0.6

0.8

1

χ

tot

re

σ

IMSA-MF

IMSA-SF 5GHz

IMSA-SF 6GHz

IMSA-SF 7GHz

IMSA-FH

0

0.5

1

1.5

2

2.5

3

0

0.2

0.4

0.6

0.8

1

χ

tot

im

σ

IMSA-MF

IMSA-SF 5GHz

IMSA-SF 6GHz

IMSA-SF 7GHz

IMSA-FH

(a) (b)

10

12

14

16

18

20

22

24

26

28

0

0.2

0.4

0.6

0.8

1

χ

int

re

σ

IMSA-MF

IMSA-SF 5GHz

IMSA-SF 6GHz

IMSA-SF 7GHz

IMSA-FH

0

5

10

15

20

25

30

35

40

45

50

0

0.2

0.4

0.6

0.8

1

χ

int

im

σ

IMSA-MF

IMSA-SF 5GHz

IMSA-SF 6GHz

IMSA-SF 7GHz

IMSA-FH

( ) (d)

y

x

0.0

Im {τ (x, y)}

−1.8

(a)

x

y

x

y

2.3

Re {τ (x, y)}

0.0

0.0

Im {τ (x, y)}

−1.8

(b) ( )

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

χ

tot

re

σ

ext

σ

int

=0.0

σ

int

=0.01

σint

=0.1

σ

int

=0.5

(a)

0

2

4

6

8

10

12

14

16

18

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

χ

tot

im

σ

ext

σ

int

=0.0

σ

int

=0.01

σint

=0.1

σ

int

=0.5

(b)

30mm

80mm

=0.45

τ

τ

=2

(a)

y

x

y

x

2.6

Re {τ (x, y)}

0.0

2.6

Re {τ (x, y)}

0.0

(b) ( )

y

x

2.6

Re {τ (x, y)}

0.0

(a)

y

x

y

x

2.6

Re {τ (x, y)}

0.0

2.6

Re {τ (x, y)}

0.0

(b) ( )

y/

λ

x/

λ

−0.6

0.6

0.6

−0.6

1.5

Re{τ (x, y, z = 0.0)}

0.0

z/

λ

y/

λ

−0.6

0.6

0.6

−0.6

1.5

Re{τ (x = 0.0, y, z)}

0.0

y/

λ

x/

λ

−0.6

0.6

0.6

−0.6

1.5

Re{τ (x, y, z = 0.0)}

0.0

y/

λ

x/

λ

−0.6

0.6

0.6

−0.6

1.5

Re{τ (x = 0.0, y, z)}

0.0

χ

_tot

χ

_int

χ

_ext

IM SA − SF (f = 5 GHz)

6.40

41.51

3.03

IM SA − SF (f = 6 GHz)

5.32

37.26

2.69

IM SA − SF (f = 7 GHz)

4.59

27.35

2.34

IM SA − M F (f = 5, 6, 7 GHz)

2.07

23.10

1.48

IM SA − F H (f = 5, 6, 7 GHz)

4.19

28.69

1.77

CG − M F (f = 5, 6, 7 GHz)

4.62

25.12

2.60

CG − F H (f = 5, 6, 7 GHz)

6.62

25.32

4.77

able I -D. F ran es hini et al ., De aling with Multi-F r e quen y S attering Data...  32

U

K

tot

t

k

[s]

T

tot

[s]

IM SA − SF (f = 7 GHz)

3042

1820

0.5

0.8 × 10

3

IM SA − M F (f = 5, 6, 7 GHz)

8500

4553

4.5

20.5 × 10

3

IM SA − F H (f = 5, 6, 7 GHz)

3042

7287

0.5

3.4 × 10

3

CG − M F (f = 5, 6, 7 GHz)

18052

1356

20.2

27.4 × 10

3

CG − F H (f = 5, 6, 7 GHz)

6515

2180

1.32

2.9 × 10

3

able I I -D. F ran es hini et al ., De aling with Multi-F r e quen y S attering Data...  33

p

s

p

χ

tot

χ

int

χ

ext

1

1

19.31

53.01

38.87

1

2

8.45

15.68

24.06

2

1

7.21

13.11

21.17