Two-dimensional inverse profiling problem using phaseless data

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phaseless data

Amelie Litman, Kamal Belkebir

To cite this version:

Amelie Litman, Kamal Belkebir. Two-dimensional inverse profiling problem using phaseless data.

Journal of the Optical Society of America. A Optics, Image Science, and Vision, Optical Society of

America, 2006, 23 (11), pp.2737 - 2746. �10.1364/JOSAA.23.002737�. �hal-00082943�

Amélie Litman and Kamal Belkebir

Institut Fresnel, UMR-CNRS 6133,

Campus de Saint Jérme, ase 162

Université de Provene, 13397 Marseille Cedex, Frane

This paperdeals with the haraterization of two-dimensionaltar-

gets from their dirated intensity. The target haraterization is

performed by minimizing an adequate ost funtional, ombined

with a level-set representation if the target is homogeneous. One

keyissueinthisminimizationisthehoieofanupdatingdiretion,

whih involves the gradient of the ost funtional. This gradient

anbeevaluatedusingatitiouseld,solutionofanadjointprob-

lem where reeivers at as soures with a spei amplitude. We

explore the Bornapproximation for the adjoint eld and ompare

various approahes for awide varietyof objets.

OCIS odes: 290.3200, 110.6960

1. Introdution

In some pratial appliations, the phase measurement of the sattered elds are too or-

rupted by noise to be useful or even there is no phase measurement at all, e.g., optial

measurement setup. Even if there is some eort nowadays to provide experimental setups

whihmeasure allomponents ofthe satteredelds 1 ,2

,our purpose hereinis toinvestigate

a method that image samples from the modulus of the sattered eld only. Indeed, it has

been shown that the sattered intensity ould provide useful informationon the obstales.

3

Insteadof extrating somephase informationfrommeasurements 4

,and then solving the

inversesattering problemfromthemeasuredintensityandthe preliminaryretrievedphase,

we diretly retrieve the targets under test from the sattered intensity. Following the ideas

ofRefs.5,6, theapproah,suggested herein,buildsupthe parameterofinterest,namelythe

ontrastof permittivity,iteratively. Itisgraduallyadjustedbyminimizingaost funtional

properlydened.

Thisminimizationunderonstraintsis reformulatedintermsof aLagrangianfuntional,

whosesaddlepointleadstothedenitionofanadjointproblem.

7

Byvirtueofthereiproity

priniple, thisadjointproblemisequivalenttoaforward satteringproblemwherereeivers

at as soures with aorretly dened amplitude. It willbeshown that the only dierene

between a standard minimization proess using modulus-phase data and this algorithm

is expressed in these weighting oeients. This implies that passing from full data

to amplitude data requires only one line hange in a software program if an adjoint eld

formalism isused.

and a step prole. The rst ase is solved thanks to a onjugate-gradient type algorithm.

For the seond ase, a level-set representation is introdued, whih fully takes intoaount

prior information stating that the obstale is homogeneous.

8

Results using modulus only

measurements will then be analyzed in a free spae onguration for those two ases of

permittivityprole. In partiular, we highlightwith various numerialexamples the eet

on the gradient omputation and on the onvergene, of physial approximations suh as

Born approximation for both the forward and adjoint elds. We also introdue a new

initial guess based on an appropriate use of topologial derivative, whih is no more than

the variationof the ost funtional due tothe inlusions of smalldieletri balls.

9

The following paper is organized as follows. In the rst setion, a desription of the ge-

ometry isprovided. The seondsetionisdevoted tothe denition of theinverse sattering

problem, with the introdution of the ost funtional and the assoiated Lagrangian for-

mulation. The gradientexpression is then provided and several hoies of omputation are

disussed. The third setionfouses on the appliation of this gradient omputationto the

ase of heterogeneous obstales by means of onjugate-gradientalgorithm orto the ase of

homogeneous obstales by meansof level-sets. The way the initialguess is obtained is also

explainedinthesamesetion. Finally,the lastsetionprovidesnumerialexamplesforboth

homogeneous and heterogeneous obstales, with and without noise, showing the eets of

a orretgradient omputationaswell asthe appropriate use of ana prioriinformationon

the nature of the satterers.

2. Statement of the problem

The geometry of the problem studied in this paper is shown in Fig. (1) where a two-

dimensionalobjetofarbitraryross-setion

Ω

D

beddingmedium

Ω b

ε b = ε ₀ ε br

and of permeability

µ = µ ₀

ε ₀

µ ₀

mittivity distribution

ε( r ) = ε ₀ ε r ( r )

µ = µ ₀

right-handed Cartesian oordinate frame (

O, u

x , u

y , u

z

O

either inside or outside the satterer and the

z

satterer. Thepositionvetor

OM

OM = r + z u

z .

generatetheeletromagnetiexitationareassumedtobelinesparalleltothe

z

at

( r _l ) 1≤l≤L

exp(−iωt)

ase, the time-harmoniinidenteletri eld reated by the

l ^th

E ⁱ _l ( r ) = E _l ⁱ ( r ) u _z = P ωµ ₀

4 H ₀ ⁽¹⁾ (k b | r − r _l |) u _z ,

where

P

ω

H ₀ ⁽¹⁾

funtion of zero order and of the rst kind and

k b

For the inverse sattering problem, we assume that the unknown objet is suessively

illuminated by

L

Γ

M

oupledontrast-soureintegralrelations: theobservationequation(Eq.2)andtheoupling

equation (Eq. 3)

E _l ^s ( r ∈ Γ) = k ₀ ² Z

D

χ( r ^′ ) E l ( r ^′ ) G( r , r ^′ ) d r ^′ ,

E l ( r ∈ D) = E _l ⁱ + k ₀ ²

Z

D

χ( r ^′ ) E l ( r ^′ ) G( r , r ^′ ) d r ^′ ,

where

χ( r ) = ε r ( r ) − ε br

D ⊃ Ω

G( r , r ^′ )

k 0

wavenumber. Forthe sake of simpliity,the equations (Eq. 2)and (Eq. 3) are rewritten as

E _l ^s = K χ E l ,

E l = E _l ⁱ + G χ E l .

3. Inverse sattering problem

The inverse sattering problemis stated as ndingthe permittivity distributionin the box

D

and the observation equation mathes the data. Is proposed herein an iterative approah

to solve this ill-posed and non-linear problem. The rst step onsists in the denition of

a disrepany riteria between the measured elds and the simulated ones. This riteria

depends on the amount of available data, e.g., modulus and phase or modulus only. The

derivativeof this ost funtionalmust then be expliitlyobtained andit willbeshown that

it introdues an adjoint state equation where reeivers at as soures with an amplitude

mainlydependingon the expression ofthe ost funtional.

A. Cost funtional denition

The parameter of interest, namely the ontrast

χ

ost funtional

J (χ) = P L

l=1 F (E _l ^s (χ))

both amplitudes and phase must be mathed, the ost funtionalreads as

J (χ) = 1 2

L

X

l=1

w l kE _l ^obs − E _l ^s (χ)k ² _Γ

where

E ^obs

w l

w ⁻¹ _l = kE _l ^obs k ² _D

funtionalreads as

J (χ) = 1 2

L

X

l=1

w l kI _l ^obs − |E _l ^s (χ)| ² k ² _Γ

where

I ^obs

w _l ⁻¹ = kI _l ^obs k ² _Γ

B. Gradient expression

This minimizationproblemunder onstraints anbereformulatedusingaLagrangian fun-

tional

L

L(χ, E ^s , E, U ^s , U ) =

L

X

l=1

{F (E _l ^s )

+ < U _l ^s | E _l ^s − K χE l > _Γ + < U l | E l − E _l ⁱ − G χE l > D

where

χ

F

E ^s

E

to the simulated sattered and total elds,

U ^s

U

< | > Γ

the salarprodut on

Γ < u | v > Γ = R

Γ u ^∗ ( r )v( r )d r

and

< | > D

D

< u | v > D = R

D u ^∗ ( r )v( r )d r

. This Lagrangian is used to express rst-order and seond-

order onditions for a loal minimizer,whih are linked to the existene of a saddle-point.

Thissaddle-pointprovidesaneientwaytoompute thegradientoftheostfuntionalby

introduinganadjointeld. Theadjointeld, due tothe reiproitypriniple, isequivalent

to the diret eld where reeivers at are soures with an amplitude linked to the ost

funtionalexpression

P l = P _l ⁱ + G χ P l

P _l ⁱ = − K ^t ∇F (E _l ^s )

Ifboth amplitudes and phase must bemathed, the inident adjointeld is given by

P _l ⁱ = w l K ^t (E _l ^obs − E _l ^s )

Ifsattered intensity must be mathed, the souresfor the adjoint problemread as

P _l ⁱ = 2w l K ^t E _l ^s (I _l ^obs − |E _l ^s | ² )

Therefore the adjointmethodisaveryonvenient way foromputingderivativesfor several

typesof ost funtional.

Itan be shown (Se. A)that the gradient of the ost funtionalis given by

< ∇J (χ) | δχ > D = −ℜe <

L

X

l=1

E l P l | δχ > D

In the ase of intensity measurements, this gradient shows the ambiguity of the ost fun-

tional. On one hand, the ost funtional an be redued if the omputed eld is lose to

the measured eld. On the other hand, the ost funtionalan be redued if the size of the

satterer is very smalland we an negletits ontribution. In that ase, the adjointeld is

null aswell asthe gradient.

C. Gradient approximation

The gradient evaluation requires the omputation of two forward problems. The rst one

omputes the diret eld

E l

presribed amplitude, providesthe adjointeld

P l

Three ases an be onsidered: (i) no approximation is done for the diret and adjoint

eld omputation(notedas the Full-Fullase inthe following),(ii)Bornapproximation

is only made for the adjoint eld omputation (Full-Born) and (iii) nally the Born

approximation is applied for both elds (Born-Born). In the last ase, the gradient is

identialtotheone whihwouldbeobtained by assumingfromthebeginningthattheBorn

approximationwasvalid. Asexpeted, the way the gradientisomputedwillhaveaneet

ontheminimizationproessasitwillbehighlightedinSe.5withsomenumerialexamples.

4. Minimization sheme

Onethe disrepany riteriahas been dened and itsderivativeomputed, aminimization

algorithmanthen be applied,whih an bespeied aordingtothe a prioriinformation

available. Forexample, ifthe permittivity proleof the unknown obstale isassumed tobe

ontinuous, a standard onjugate-gradient type algorithman be used. If, onthe ontrary,

one isinterested inlookingathomogeneousby part obstales, this a prioriinformationan

beintroduedviaalevel-set formulationwherethe ostfuntionalderivativeisstillneeded.

Inallases, theinitialguessseletionisakeypointfor theonvergene ofthe minimization

proess.

A. Initial guess seletion

The initialguess omputationis based on topologial asymptotiexpansion results 9

. The

topologialderivativeaimsatintroduingsomesmalldieletriballsofonstantpermittivity

ε r

ξ( r )

eletromagneti elds whih are expressed via a topologial asymptotiexpansion formula.

Letusdenoteby

B ρ

ρ|B |

r

|B|

of a referene ball

B

r ∈ B ρ ⊂ B ρ

0 < ρ < ρ ^′ < 1

asymptotiexpansion of our ost funtion an then be expressed by 10

J

χ = (ε r − ε br ) 1 B

(ξ − ε br ) 1 D\B

(12)

− J {χ = (ξ − ε br )1 D }

= −ρ ² ℜe(ε r − ε br )k ₀ ² |B|(

L

X

l=1

E l P l ) + o(ρ ³ )

where

1

E l

P l

with

χ( r ) = ξ( r ) − ε br

∀ r ∈ D

topologyofthe satterers. Infat,if weassume that

ξ = ε br

the rst step of the inversion proess, as expressed in (Eq. 11), assuming that there is no

initialguess.

Using this topologial derivative, as we do not know the value of

ε r

initialguess with

χ ₀ ( r ) = η ℜe

L

X

l=1

E l ( r )P l ( r )

where

η

J (χ 0 )

E l

P l

diret and adjoint elds omputed for

χ( r ) = ξ − ε br

∀ r ∈ D

ξ

ε br

It would have been more natural to use

χ = 0

D

have orrespond to

Ω = ∅

measurements, the adjointeld would have been nullas wellas the topologialderivative.

If a priori information on the nature of the satterer is given, suh as the obstale is

homogeneous, a trunation atmidvalue isperformedto obtaina binary image.

B. Retrieval of an inhomogeneousprole

Ifnoa prioriinformationisavailableonthe natureof thesatterer,asequene

{χ n }

up iteratively aording tothe following relation

χ n = χ n−1 + α n d n ,

where

d n

α n

by minimizingthe ost funtional

J (χ n )

α n

E

d n

authors take aPolak Ribière onjugatediretion

d n = g n + γ n d n−1 , γ n = < g n | g n − g n−1 > D

kg _n−1 k ² _D ,

where

g n

J (χ)

χ

an be exatlyomputed or approximated.

C. Retrieval of a binary prole

Asthenonlinearinverseproblemstatedaboveishighly ill-posed,allavailableinformationis

useful toimprovethe qualityof the reonstrutions. Insome ases,it ispossible toassume

that the dieletri properties of the obstale are known and furthermore that this obstale

ishomogeneous. Theontrast ofpermittivitywillthenbeabinaryfuntionof thefollowing

form:

χ( r ) = ε r − ε br r ∈ Ω, χ( r ) = 0 r ∈ / Ω

where

ε r

Ω

the same ost funtional as previously under the onstraints of (Eq. 2) and (Eq. 3). An

sequene of shapes

{Ω n }

F (Ω n )

doing so,several elementsare neessary : (i) theshape representation, (ii)the omputation

of the derivative of the ost funtionalaording to shape,(iii) and the onstrution of the

iterative sequene. To represent the shape, let us introdue an auxiliary funtion alled a

level-set funtion

φ

Ω = { r ∈ D

φ( r ) < 0} .

This representation handles naturally all topologial hanges suh as fusion or separation

and does not require to know in advane the number of satterers as well as their enters

positions. The derivation of the ost funtional

J

φ

done aording tothis level-set representation, toobtain

< ∇J (φ) | δφ > D = −ℜe(ε _r − ε br )

< δ(φ)|∇φ|

L

X

l=1

E l P l | δφ > D

where

δ(φ)

φ = 0

∂Ω

omputedorapproximated. Anartiialtimevariable

t

is done by nding the steady state solutionof

φ t = −∇J (φ)

assuming that the

δ(φ)

D

is solved using the Osher-Sethian numerialsheme desribed inRef. 11.

5. Numerial experiments

In this setion we report examples of reonstrutions of dieletri samples to illustrate the

eieny of the inversion algorithms presented in the previous setions. In all ases, syn-

theti data are generated thanks to a fast forward solver desribed in details in Ref. 12.

This forward solver is based on a seond-order aurate spae-disretisation whih is a-

pable of handling homogeneous as well as inhomogeneous proles. The onvolution-type

struture of the integral equation is exploited and solved via a Conjugate Gradient-Fast

Fourier Transform (CG-FFT) method. Moreover, a speial extrapolation proedure is

used,by"marhing-on-in"soure position,togenerateaurate initialestimatesforthe CG

method to redue the omputation time. On the ontrary, the inversion solveris based on

a standard method of moment without any use of the CG-FFT method 12

. This solver is

needed for omputing both the internal and adjoint elds. The dieletri permittivity as

wellas theeletromagneti eld areinterpolated by pieewise onstantbasis funtionswith

olloationpointtest funtions.

The reeivers aswell asthe soures are assumed to be innitelines loated ona irle

Γ

of radius

1.5λ

λ

64

and reeivers evenly distributedonthe measurement irle

Γ

solver togenerate data is

λ/64

D

2λ

subdivided for numerial purposes into

30

λ/15

forinversionshemes. Consequently,the meshsizeusedintheinversionisdierentfromthe

one usedtogeneratedata, preventing any inverse rime. Inallthe followingexamples,the

initialguess ishosenasdesribed inSe. 4A with aninitialontrast of

χ = ξ − ε br = 1.01

Forsuhontrastvalue,the Bornapproximationisappliable. Finally,alliterativeshemes

have been onduted up to the

512

one, is ahieved. In all ases, the evolution of the ost funtion is presented. By letting

the inversion algorithmruns, we then have agoodindiationof the onvergene speed, the

disrepany auray and the trends of the methods. In partiular, we an hek if we

reahed aplateau orif the algorithmisunstable.

A. Reonstrution of spatially homogeneousproles

1. TheHomoCyl16objet

As a rst example, we onsider two irular homogeneous ylinders of radii

a ₁ = 0.15λ

and

a ₂ = 0.3λ

ε r = 1.6

(−0.2 λ, 0.2 λ)

(−0.3 λ, − 0.3 λ)

objet under test is referred asHomoCyl16 objet.

Toemphasize the inuene of thephase information,two initialestimates obtained with

thesametopologialexpansionmethodareplottedinFig.2fortheHomoCyl16objet. In

Fig.2(a),onlymodulusinformationisusedasinFig.2(b),modulusandphasearetaken into

aount. It is lear that the phase ontains important topologial information. Therefore,