Phaseless imaging with experimental data : facts and challenges

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challenges

Michele d’Urso, Kamal Belkebir, Lorenzo Crocco, Tommaso Isernia, Amelie Litman

To cite this version:

Michele d’Urso, Kamal Belkebir, Lorenzo Crocco, Tommaso Isernia, Amelie Litman. Phaseless imaging with experimental data : facts and challenges. Journal of the Optical Society of Amer- ica. A Optics, Image Science, and Vision, Optical Society of America, 2008, 25 (1), pp.271 - 281.

�10.1364/JOSAA.25.000271�. �hal-00438268�

Mihele D'Urso

DIET - Dipartimento di Ingegneria Elettronia e delle Teleomuniazioni,

Università degli Studi di Napoli Federio II,

Via Claudio 21, I-80125 Naples, Italy

Kamal Belkebir

Institut Fresnel, UMR-CNRS 6133,

Campus de Saint Jérme, ase 162,

Université de Provene, F-13397 Marseille Cedex, Frane

Lorenzo Croo

Istituto per il Rilevamento Elettromagnetio dell'Ambiente (IREA) CNR,

via Dioleziano 328, I-80124 Naples, Italy

Tommaso Isernia

DIMET - Università Mediterranea of Reggio Calabria,

Lo. Feo di Vito, I-89060 Reggio Calabria, Italy

Amélie Litman

Institut Fresnel, UMR-CNRS 6133,

Campus de Saint Jérme, ase 162,

Université de Provene, F-13397 Marseille Cedex, Frane

The aim of this paper is to disuss the haraterization of two-

dimensionaltargets using inverse prolingapproahes with phase-

less data. Data orrespond to the total elds intensity, whih are

the atual objets of the measurements devies in many applia-

tions. Two dierent inversion shemes are presented, disussed,

ompared and validated by using experimental data. In the rst

one,theintensity-onlydataareexploitedinaminimizationsheme,

thanks to a proper denition of the ost funtional and the evalu-

proess takes denite advantage of a suitable normalization and

of an useful starting guess whih allows us to irumvent the use

of a global optimization shemes, whih are time onsuming. In

the seond sheme, suggested in [1℄, one exploits the properties of

the sattered elds and the theoretial results on the inversion of

quadratioperatorstoderiveatwo-stepssolutionstrategy,wherein

the (omplex) sattered elds embedded in the available data are

retrieved rst and then a traditionalinverse sattering problem is

solved. Inbothases, theanalytialpropertiesandrepresentations

ofthe involved elds allowtoproperlyx the measurement set-up

andtoidentify themore onvenient solutionstrategytoadopt. In-

diationsonthe numberand typeofprimary souresandreeivers

tobeusedarealsogiven. Resultsfromexperimentaldatashowthe

eieny of the above approahes and of the introdued tools.

OCIS odes: 290.3200, 110.6960

1. Introdution

In inverse sattering problems, one looks for a quantitatively aurate desription of the

eletrial and geometrial properties of an investigated region given a set of inident elds

and measures (both in amplitude and phase) of the orresponding sattered elds on a

generi surfae lyingoutside the region under test [2℄. Dueto their wide range of potential

appliations, the development of aurate and reliable tehniques for solving this kind of

problems is nowadays astill important hallenge[35℄.

By leaving aside peuliar harateristis of the dierent approahes proposed in the

literature, one of the main ommon drawbaks resides in the need of measuring both

amplitude and phase of the sattered elds. As a matterof fat, inseveral areas of applied

siene, the phase distribution of the sattered elds is often too orrupted by noise to be

useful, oreven thereisnophase measurementatall,e.g., optialmeasurementsetup. Even

if thereissomeeort nowadays toprovideexperimentalsetupsapableof measuringallthe

that image samples from only amplitude data, as these latter would open the way tomore

simple and ost eetive experimental set-ups. In addition, it is also important to remark

that, in most appliations, the atual measured quantity is the total eld. In fat, unless

the inident eld is provided by a diretive antenna, the measured eld in presene of the

target ontains both the inident and the sattered eld, so that the total eld has to be

proessed insteadof the sattered eld asusually done.

Inorder tooverometheabovelimitations,several approahes forsolving inverse sattering

problems from intensity-only data have been proposed in the literature [1, 813℄. Among

them, an approah based on only amplitude measurements of the total elds has been

reently proposed, rst with referene to the ase of measures taken on a losed urve

surrounding the domain under test [1℄ and then to that of transmitters and reeivers

plaed over two trunated lines somehow enlosing the investigatingdomain [13℄. In both

ases, the proposed proedures splits the imaging problem into two dierent steps. In

the rst one, the sattered eld is estimated from the measures of the square amplitude

distributions of the total eld, while the seond step is aimed at estimating the unknown

dieletri properties fromthe estimated sattered elds (modulus and phase). In summary,

the rst step allows us toestimatethe input data for the seondone, whihis atraditional

inverse sattering problem. Notably, as realled throughout this paper and in previous

ontributions [1, 13℄, the separation of the problem into two dierent steps allows a better

ontrol of the overall non-linearity of the inverse problem with respet to single step

proedures. In fat, the exploitation of theoretial results on the inversion of quadrati

operators [14℄ and eld properties representations [15, 16℄, leading todesign onstraints on

the measurement set-up, allows to suessfully solve the rst step, while all the available

knowledge about traditional inverse sattering problems is exploited in the seond one.

More reently, suh an imaging tehnique has been extended to a three-steps proedure,

where a Phase Retrieval (PR) problem is preliminary solved to estimate the phase of the

inidenteld fromitsmeasured amplitude[17℄. Bydoing so,theresultingimagingstrategy

ompletelyrelies on onlyamplitude data.

However, the above mentioned inversion approahes [1, 13, 17℄ an be atually applied

provided that some onditions on the measurement set-up are satised. As a matter of

fat, when these onditions do not hold true, the estimation of the sattered eld from the

develop new, aurate and eetive inverse proling approahes based on only amplitude

information of the total eld, one known or estimated the inident eld in the sattering

domainand onthemeasurementurve. In suhapproahes, the aimis tosolvethe imaging

problem in a single step, without previously estimating the sattered eld embedded in

the measures. This would require toreformulate the inverse sattering problemin order to

take into aount that the available data are intensity only. On the other hand, at least

in priniple, partiular onstraints onthe measurement set-up are not required. Therefore,

these approahes are expeted tobe useful inallthose aseswherein the two-stepsstrategy

[1,13℄ orits generalization[17℄ annotbeused.

The aim of this paper is therefore to introdue a novel one-step imaging strategy based on

only-amplitude total eld data and ompare and disuss, by using experimental data, its

performanes with that of the two-steps strategy.

It isworth notingthatthe idea of diretlyinorporatingthe square amplitude distributions

of the total eld inthe inversion sheme is not new in the literature [1012℄. With respet

totheseontributions,theapproahes proposed anddisussed inthis paperhaveinteresting

and omplementaryharateristiswith respet to the aboveones. First, unlike[10℄, we do

not makeuse of a priori informationinthe inversionproess, but werather takeadvantage

of a suitable starting guess ahieved by means of a simple modiation of the widely used

bakpropagation solution [18℄. Moreover, unlike [12℄, the adopted minimization sheme

exploits a loal optimization proedure based on an eient CG-FFT sheme and thus

avoids the use of time onsuming global optimization algorithms. In this respet, it is

also worth notiing that the use of a proper weighting of the ost funtional to minimize,

derived from the propertiesof the intensity only data pattern, aswell asfrom the available

knowledge in Phase Retrieval proedures [14℄, allows us to improve the data tting and of

ourse the nal reonstrutions in terms of permittivity and ondutivity of the unknowns

targets. As a last but not least point, let us remark that our approahes are based on

the Contrast Soure-Extended Born (CS-EB) inversion sheme, whih allows to redue

the degree of non-linearity [19℄ of the inverse sattering problem and ahieves improved

permittivityand ondutivity maps reonstrutions inmany ases [20℄.

The paper is organized as follows. In Setion 2, the adopted geometry onguration is

presented and the mathematial model is given. The sampling properties and representa-

inversion sheme is thoroughly desribed, together with the weighting strategy and the

adopted modied bakpropagation as initial solution. The features and limitations of the

two-steps approah are briey skethed in Setion 4. Setion 5 is devoted to assess and

ompare theperformanesof thetwo approahesbymeansof experimentaldataonerning

metallianddieletri inhomogeneoustargets,olletedattheInstituteFresnelofMarseille.

Conlusions follow.

2. Mathematial model and eld properties

The geometry of the problem studied in this paper is shown in Fig. 1 where one or more

two-dimensional objets of arbitrary ross-setion

Ω

D

The embedding medium

Ω

ε

= ε

ε

µ = µ

ε

µ

apermittivitydistribution

ε( r ) = ε

ε

( r )

µ = µ

A right-handed Cartesian oordinate frame (

O, u

x

, u

y

, u

z^{) is dened. The origin}

O

either inside or outside the satterer and the

z

satterer. Thepositionvetor

OM

OM = r + z u

z

.

that generate the eletromagneti exitation (denoted as

T

probes olleting the data (

R

( r

)

Γ

R

exp(iωt)

time-harmoniinident eletrield reated by the

l

E

( r ) = E

( r ) u

z

= A ωµ

4 H

(k

| r − r

| ) u

z

,

where

A

ω

H

funtionof zero-order and seondkindand

k

Underthesehypothesesand omittingthe

exp(iωt)

tion ondition, the sattering equations desribing the total eld an be formulated as two

oupledontrast-soureintegralrelations [18℄: the observationordataequationEq.(2)and

the ouplingor state equation Eq.(3), whih are

E

( r ∈ Γ) = E

( r ∈ Γ) + E

( r ∈ Γ) = E

( r ∈ Γ) + ZZ

D

G( r , r

) J

( r

), d r

,

J

( r ∈ D) = χ( r ∈ D)E

( r ∈ D) + χ( r ∈ D)

ZZ

D

G( r , r

) J

( r

) d r

,

where

χ( r ) = ε

( r ) − ε

D

G( r , r

)

J( r ) = χ( r )E( r )

tothe ontrast soure.

The overall aimof the problemis to determine the two dimensional ontrast funtion

χ( r )

in

D

E

( r ∈ Γ)

Γ

and from aninomplete (beause only a nite number of measurements an be performed)

and inaurate (beause the measurements are error-aeted)knowledgeof the intensity of

the total elds

| E

( r ∈ Γ) |

, l ∈ (1, . . . , L)

Tothisend,as

| E |

= | E

|

+ | E

|

+ 2 ℜ e(E

E

)

andpossiblerepresentationsofboth sattered andinidentelds, andthen of

| E

|

| E

|

well as of the interferene term

ℜ e(E

E

)

will allow to quantify the amount of independent data at our disposal for solving the

imaging problem at hand, to sample the intensity data in an aurate and non-redundant

fashion and to determine the maximum amount of information about the targets one an

extrat from the available data. Moreover, asin [1,13,17℄, exploitationof these properties

providesthe guidelines todesign aneetive measurement set-up.

With referene to the geometry depited in Fig. 1, it is known that the sattered eld

orresponding to a given soure an be aurately represented with a nite number of

Fourier harmonis given by

2k

R

R

the targets [16℄. As a Fourier series an be turned into a Dirihlet sampling series,

2k

R

samples uniformlyspaed in angle aurately represent eah sattered eld as well. From

reiproity [16℄, the number of non-superdiretive independent inident elds impinging

on the domain under test is

2k

R

2k

R

inident elds. Therefore, as a funtion of the inident angle

ϑ

ϑ

(2k

R

) × (2k

R

) = (2k

R

)

As far as the inident elds measured on

Γ

fat,by parallelingthe above reasoning tothe representation ofthe inidenteld in

D

an prove that eah inident eld on

Γ

2k

R

samples, and that

2k

R

sattered eld, the inident eld on

Γ

ϑ

ϑ

aurately represented by a number of samples given by

(2k

R

) × (2k

R

) = (2k

R

)

Notethat, also inthis ase, only one half of these samplesis atuallyindependent [1℄.

When onsideringthe square amplitudepatterns of the aboveelds, the numberof samples

required for a faithful representation beomes four times larger (with respet to amplitude

and phase measurements) as the sampling step has to be halved along eah of the two

oordinates. Therefore,

| E

( r ∈ Γ) |

(4k

R

) × (4k

R

) = (4k

R

)

| E

( r ∈ Γ) |

(4k

R

) × (4k

R

) = (4k

R

)

In order to aurately represent

| E |

Γ

| E |

= | E

|

+ | E

|

+ 2 ℜ e(E

E

)

needs a number of samplesequal tothe maximumbetween

(4k

R

)

(2k

(R

+ R

))

the latter being the number of samples required to represent the term

2 ℜ e(E

E

)

Γ

[13℄. Of ourse,only a halfof these samples isindependent [1℄.

3. A single-step approah for intensity only inverse proling

Traditionally, in standard inverse sattering problems, one assumes the knowledge of the

total elds inboth amplitude and phase. Herein, the problemwe want tosolve onsists in

retrievingthe dieletriharateristiswithinaregionundertest frommeasurements ofthe

square amplitude distribution of the total eld, one known (or estimated as in [17℄) the

inidenteld. Theapproahdesribed inthissetionorrespondstoasingle-stepproedure,

basedontheminimizationofadisrepanyriterionbetweentheamplitudeofthesimulated

and measured total elds. This minimization problem is reast into a Contrast-Soure-

Extended-Born (CS-EB) formalismasin[20℄. Abriefreall ofderivationand mainfeatures

of the CS-EB sattering modelis reported in the Appendix A.

A. Unknowns representation

In the Contrast-Soure inversion method[18℄, both the ontrast

χ

J = χE

linearity [19℄ and thereforethe diultyof the inverse problemswith respetto parameters

embedding dieletri harateristis, the traditional sattering equation Eq.(3) is replaed

by a new oupling equation, the Contrast Soure - Extended Born (CS-EB) equation [20℄,

given by

J

( r ) − ξ( r )E

( r ) = ξ( r ) ZZ

D

G( r , r

)[J

( r

) − J

( r )]d r

= ξ( r ) G

(J

),

where

ξ( r ) = χ( r )

1 − χ( r )f

( r ) , f

( r ) = ZZ

D

G( r , r

)d r

. G

(J

) =

ZZ

D

G( r , r

)[J

( r

) − J

( r )]d r

= ZZ

D

G( r , r

)J

( r

)d r

− J

( r )f

( r ).

(5)

For the sake of simpliity, equations Eq.(2) and Eq.(4) an be rewritten using symboli

notations as

E

= K J

; J

= ξE

+ ξ G

(J

),

where

G

(J

)

tering domainto the sattered eld outside. Itis worth notiingthat, despitethe fatthat

the CS-EB model dened in Eq.(4) is just a simple rewriting of the traditional ontrast

soure model, ithas proved tobea moreeetive toolto formulate and solve both forward

and inverse sattering problems [20℄. Notably, while its derivation was inspired by some

mathematial and physial onsiderations relatedto presene of losses in the host medium

and/orin the targets[20℄, proessingof experimentaldata (both amplitudeand phase) has

shown that aurate and reliable results an be ahieved also for lossless inhomogeneous

targets infree spae [21℄.

The ill-posedness of the inverse sattering problem is also dealt with by looking for nite-

dimensional representations of both the unknowns [22℄. Wethus onsider

ξ( r ) =

P

X

p=1

a

ψ

( r )

J

( r ) =

Q

X

q=1

c

φ

( r ) ∀ l = 1, . . . , L

wherein

{ ψ

}

{ φ

}

ourse, aording to the above results onthe eld properties, the numberof the unknowns

oeients

{ a

}

{ c

}

partiular, note that, as far as the hoie of

P

elds realled in Setion 2 allow to state that, for any given

R

maximumamountofinformationwhihan beextratedinthe inverse satteringstep, thus

allowing to x the maximum number of unknowns oeients for the ontrast funtion in

Eq.( 7)whih an bereliably retrieved.

B. Disrepany riteria

The disrepany riteriabetween the measured elds and the simulated ones onsidered in

the followingis given by

J (ξ) =

L

X

l=1

α

k I

− | E

+ K J

(ξ) |

k

,

where

I

α

weighting oeient set in suh a way that the total eld intensities orresponding to the

dierentsattering experiments have anequal weight and

W

L

on

Γ

α

= k I

k

L

un-weighted one,isused beauseofthe fatthatthe adopted ostfuntionalEq.(9) embeds

the solution of a phase retrieval problem for the total eld. In these problems, the zeros

(or nearly zeros)of the data pattern(inour ase

I

hereinusefullyexploited. Aordingly,wehooseaweightingfuntionwhihemphasizesthe

ontributions to ost funtional orresponding to small amplitude data [14℄. In partiular,

the weighing funtion

w

( r ∈ Γ)

w

( r ∈ Γ) = 1

I

( r ∈ Γ) + ε

whereinthe positiveregularizationparameter

ε