622 J. Opt. Soc. Am. A/Vol. 21, No. 4 /April 2004 Crocco et al.

Inverse scattering from phaseless measurements ofthe total field on a closed curve

Lorenzo Crocco

Istituto per il Rilevamento Elettromagnetico dell’Ambiente, Consiglio Nazionale delle Ricerche, Via Diocleziano328, 80124 Napoli, Italy

Michele D’Urso and Tommaso Isernia

Dipartimento in Ingegneria Elettronica e delle Telecomunicazioni, Universita degli Studi di Napoli, Federico II,Viale Claudio 21, 80125 Napoli, Italy

Received May 22, 2003; revised manuscript received October 23, 2003; accepted December 2, 2003

A new approach for quantitative electromagnetic imaging of scatterers located in free space from phaselessdata is proposed and discussed. The procedure splits the problem into two steps. In the first one, we solve aphase-retrieval problem for the total field, thus estimating the amplitude and phase of the scattered field.Careful analysis of properties and possible representations of both scattered and incident fields allow us tointroduce a criterion for an optimal choice of the measurement setup and a successful retrieval. Then thecomplex permittivity profile is reconstructed in the second step by use of the estimated scattered field. Nu-merical examples are provided to check the whole chain in the presence of noise-corrupted data. © 2004 Op-tical Society of America

OCIS codes: 290.3200, 100.5070, 100.6950, 100.3190.

1. INTRODUCTIONInverse-scattering problems are of great interest becauseof their potential applications in biomedical imaging, non-destructive testing and diagnostics, remote-sensing appli-cations, applied geophysics, and noninvasive subsurfacemonitoring.1–6

In all these cases, the availability of more efficient com-puting facilities and evolved radar systems, as well as ad-vancements in theoretical understanding, encourages theexploitation of inverse-scattering techniques, which havegained a great deal of attention owing to their capabilityof achieving a quantitative description of the electricaland geometrical characteristics of the investigated region.

Roughly speaking, besides dealing with new applica-tions, inverse-scattering research is nowadays focused onthree main directions, i.e., development of

1. New reliable and accurate solution approaches,2. New computationally effective solution approaches

(such as solutions of actual three-dimensional problems),3. Low-cost and effective hardware setups.

With reference to the third point, note that in severalareas of applied science it can be very difficult or very ex-pensive to measure the phase of a field. Therefore itsamplitude is all we can work with. As a matter of fact,for frequencies above tens of gigahertz, it becomes diffi-cult to measure the phase of (total or scattered) fields di-rectly. Therefore holographic7 and interferometric8 tech-niques are generally used to obtain the phase informationin optical applications.

As phaseless measurements may be much more simpleand less invasive, in this paper we want to discuss a newapproach to inverse scattering that uses only amplitudemeasurements of the total field. In particular, we want

1084-7529/2004/040622-10$15.00 ©

to explore the possibility of obtaining the required infor-mation about the region under test (i.e., its dielectricproperties) via phaseless measurements of the total fieldunder different illumination conditions. The basic ideais to reconstruct the complex scattered field from only am-plitude measurements of the total field first and thensolve a standard inverse-scattering problem.

The proposed procedure is therefore organized in twosteps. In the first step the complex scattered field is es-timated by minimizing a functional that is the norm ofthe discrepancy between the measured pattern amplitudedistribution and the calculated one. In such a step, deci-sive effectiveness is gained by exploitation of accurateand nonredundant finite-dimensional representation ofthe scattered field.9 The second step, dealing with astandard inverse-scattering problem, allows us to obtainan estimate of the dielectric properties of the region undertest.

Several contributions can be found in the literature toinverse-scattering problems from use of only amplitudedata.10–12 Some of these contributions are based on ap-proximations of the scattering equations and deal withthe weak-scattering case. Maleki and coworkers, for in-stance, have proposed two alternative methods10,11 for ob-jects within the first-order Born or Rytov approximations.The first method10 is an intensity-only reconstruction al-gorithm for determining the (complex-valued) index-of-refraction distribution from observations of the far-fieldintensities of the optical scattered field. The reconstruc-tion of the object’s index-of-refraction distribution is per-formed directly from the far-field intensity of the scat-tered wave and does not require direct measurement orretrieval of the phase of the scattered field. The secondmethod11 uses an iterative phase-retrieval algorithm to

2004 Optical Society of America

Crocco et al. Vol. 21, No. 4 /April 2004 /J. Opt. Soc. Am. A 623

retrieve the phase of the total scattered field from its in-tensity distribution and takes advantage of a prioriknowledge of the support of the object. The reconstruc-tion of the complex index-of-refraction distribution of thescattering object is then performed from the recoveredfield via the filtered backpropagation algorithm of diffrac-tion tomography.13

Takenaka and co-workers have proposed a completelydifferent solution procedure12 in which, opposite to theprevious cases, the problem is tackled in a more generalfashion, as the approach aims to reconstruct the refrac-tive index of a cylindrical object from the intensity mea-surements of the total field even when the scatterer is be-yond the first-order Born or Rytov approximations. Theapproach is based on the minimization of a functionalthat is the norm of the discrepancy between the measuredpattern function and the calculated one for an estimatedobject function. In this approach the problem is solved ina single step, without the estimation step of the scatteredfield. Moreover, the outer boundary of the object is usedas a priori information in the solution procedure.

The approach proposed herein has interesting andcomplementary characteristics with respect to the abovetechniques. First, it can be used in absolutely generalcases. Second, it allows us to exploit all the huge amountof available knowledge in both phase-retrieval andinverse-scattering problems. As a consequence, the splitin the two different steps allows better control of the over-all nonlinearity of the problem with respect to a single-step procedure. In fact, a convenient exploitation of the-oretical results on the inversion of quadratic operators14

and on field properties9 allows one to successfully solvethe first step. Moreover, all previous experience in solv-ing inverse-scattering problems (in their full nonlinear-ity) can be exploited in the second step. Note that thefirst step, while similar in nature, is conceptually differ-ent from the standard phase-retrieval problem, as boththe phase and the amplitude of the unknown functionhave to be determined herein.

On the basis of the above theoretical analysis, as wellas on the basis of cost analysis and consideration of mea-surement errors, the paper provides guiding rules tochoose the measurement setup in an optimal fashion. Asa matter of fact, the given rules suggest the minimumnumber of probes and incident fields that will guarantee asuccessful estimation of the scattered field while provid-ing an intensity of this field as large as possible. Thislatter circumstance is essential in order to reduce asmuch as possible the effects of the unavoidable measure-ment errors.

In this paper all the analysis is limited to the casewherein primary sources and measurement probes are lo-cated on a curve enclosing the scatterer under test. Gen-eralization to the case of transmitters and receivers lo-cated on (different) open lines is addressed, in its generalguidelines, in Ref. 15.

The paper is organized as follows. In Section 2 geom-etry of the problem is described, formulation of the prob-lem is given, and properties of fields are recalled. In ad-dition, the adopted solution approach is introduced anddiscussed, with emphasis on its difference from previousapproaches. Then in Section 3 the first step of the pro-

posed approach, amounting to an estimate of the scat-tered field, is dealt with. In particular, similarities anddifferences with the so-called phase-retrieval problems14

are discussed and exploited to devise proper solution ap-proaches. Section 4 is concerned with the second step ofthe overall procedure, i.e., retrieval of the unknown con-trast function from the estimated scattered fields. Inparticular, the adopted solution approach for such a stepis briefly outlined. To show the feasibility of the overallidea while containing the length of the paper, we haveadopted a simple solution procedure16,17 (which does notexploit any a priori information); our more recent ap-proaches to standard inverse-scattering problems havebeen described elsewhere.18–20 Finally, Section 5 is con-cerned with a numerical analysis confirming the feasibil-ity of the whole chain and effectiveness of the proposedtwo-step approach. Conclusions are presented in Section6.

2. FORMULATION OF THE PROBLEM ANDFIELD PROPERTIESFor the sake of simplicity, although much of the discus-sion also applies to the generic three-dimensional inverse-scattering problems, we refer in the following to the two-dimensional scalar case.

As in a standard inverse-scattering problem, let us con-sider a region V enclosing one or more homogeneous ob-jects characterized by a complex dielectric permittivityequal to er(r)eb , where eb is the permittivity of the back-ground. Moreover, let us assume that the magnetic per-meability is everywhere equal to m0 .

The aim of the problem is to reconstruct the dielectricprofile of the region under test from the knowledge of theincident fields and from the measurements of the square-amplitude distributions of the corresponding total fields.

In this kind of problem the so-called contrast function,defined as

x~ • ! 5 @er~ • ! 2 1#, (1)

is usually assumed as the unknown to be determined.

Fig. 1. Geometry of the problem.

624 J. Opt. Soc. Am. A/Vol. 21, No. 4 /April 2004 Crocco et al.

With reference to the geometry shown in Fig. 1, in the fol-lowing we assume that

• The domain under test is a circle V with radius a.• The incident fields are TM-polarized unitary plane

waves, E inc(r, u i) 5 exp( jkIrI), where rI 5 (x, y), kI5 (k cos ui , k sin ui), and u i is the generic incidence angle,with u i P @0, 2p#.

• Receivers are located on a circle G, with radius b, con-centric with V. The observation angle u0 P @0, 2p#scans the measurement domain.

Under these hypotheses, scattering equations21 de-scribing the total field for each illumination condition are

E~rI , u i! 5 E inc,i~rI , u i! 1 k2

3 EV

G~rI 2 rI 8!x~rI 8!E~rI 8, u i!drI 8

5 E inc,i~rI , u i! 1 Ai@xE~u i!#

rI P V, (2a)

Etot~u0 , u i! 5 E inc,e~u0 , u i! 1 Ed~u0 , u i!

5 E inc,e~u0 , u i! 1 k2

3 EV

G~rI 2 rI 8!x~rI 8!E~rI 8, u i!drI 8

5 E inc,e~u0 , u i! 1 Ae@xE~u i!#

rI P G, rI 5 ~b cos u0 , b sin u0!, (2b)

where, for each u i , E inc,i and E are the incident and thetotal fields in V, respectively, and E inc,e and Etot are thesame quantities as evaluated on the observation domain.Moreover, Ed is the scattered field from the unknown ob-ject. Ae and Ai are integral operators relating the con-trast source xE to the scattered fields outside and insidethe domain under test, respectively. Both of them are de-fined on X 3 T, with X , L`@V# as the subspace of thepossible contrast functions and T , L2@V# as a propersubspace for the total electric field inside the domain un-der test. Both of them have values in Se , L2@G# andSi , L2@V#, which are two proper subspaces for the scat-tered fields outside and inside the domain under test, re-spectively. k 5 vAm0eb is the wave number depending onthe working frequency, and G(rI 2 rI 8) 5 2( j/4)H0

(2)(kurI2 rI 8u) is the Green’s function in the external homoge-neous medium at the prescribed frequency, with H0

(2) asthe Hankel function of zero order and second kind.

Note that the incident internal field and the total inter-nal field are functions depending on (x, y) variables ofthe domain under test and on the u i variable, whereas theexternal incident field and the scattered one are two-dimensional functions depending on u i and u0 . There-fore the total field outside the domain under test is a two-dimensional function as well. This circumstance plays akey role in the first step of the proposed procedure. Infact, unlike one-dimensional problems, two-dimensionalphase-retrieval problems (to which the first step is re-lated, see Section 3) have a unique solution, but for azero-measure set of cases and trivial ambiguities.14,22,23

In standard inverse-scattering problems, one startsfrom the knowledge of the total fields in both amplitudeand phase. As incident fields are known, this is equiva-lent to knowledge of the scattered fields for each illumi-nation condition. The problem we want to solve hereinamounts to retrieving the dielectric characteristics of agiven region under test, starting from measurements ofthe square-amplitude distribution of the total fields corre-sponding to given and known incident fields. Formally,our aim is hence to determine the two-dimensional con-trast function x(x, y) in V, starting from knowledge ofuEtot(ui , u0)u2 on G. To do that, under actual field condi-tions, we first have to come to a convenient finite-dimensional approximation of the problem. This ap-proximation should lead to a finite number ofexperiments and measurements while possibly not leav-ing aside any useful information. To this end, it provesfruitful to briefly recall properties and possible represen-tations of both scattered and incident fields, which alsowill allow us to develop and discuss in an accurate fashionthe two steps mentioned in Section 1. As we are going toshow, the analysis allows the use of a number of scatter-ing experiments (i.e., a number of incident fields) and anumber of receivers for the measurements such as toguarantee an accurate solution of the first step (and col-lect all the available information about the scatterer)while being nonredundant.

As far as scattered fields are concerned, for a circulardomain under test and a circular observation domain, itcan be proved that the scattered field corresponding to agiven incident field can be accurately represented with anumber of (azimuthal) Fourier harmonics given by 2ba,where a is the radius of the minimum circle enclosing thescatterer.9,24 As a Fourier series can be easily turnedinto a Dirichlet sampling series,25 2ba uniformly spacedsamples accurately represent each scattered field as well.By using reciprocity,25 one also proves that the number ofnonsuperdirective independent incident fields impingingon the domain under test is 2ba as well. Therefore, byexcluding superdirective sources, 2ba uniformly spacedin angle plane waves form a complete family of indepen-dent incident fields. Hence, as a function of both the in-cident angle u i and the observation angle u0 , the scat-tered field can be accurately represented by a number ofsamples given by (2ba)(2ba) 5 (2ba)2. Note that, byvirtue of reciprocity, only half of these samples are actu-ally independent.25

As far as the incident fields as measured on the obser-vation curve are concerned, a different result holds true.In fact, by paralleling results25 leading to representationof E inc in V, one can prove that each incident field on Gcan be accurately represented by 2bb Dirichlet samples(b is the radius of the circle G) on the observation curveand that 2bb (nonsuperdirective) independent incidentfields (constituted by plane waves uniformly spaced inangle) exist therein. Therefore, as discussed for the scat-tered field, as a function of both u i and u0 , the incidentfield on G can be accurately represented by a number ofsamples given by (2bb)(2bb) 5 (2bb)2. Note that alsoin this case only half of these samples are actually inde-pendent by virtue of reciprocity.25

As shown in the following, these properties play a key

Crocco et al. Vol. 21, No. 4 /April 2004 /J. Opt. Soc. Am. A 625

role in achieving a reliable and accurate reconstruction ofthe scattered field from only amplitude data.

3. STEP ONE: ESTIMATING THESCATTERED FIELD FROM PHASELESSMEASUREMENTS OF THE TOTAL FIELDThe problem of reconstructing a signal when only ampli-tude information is available has received great interestin many fields. First results come from the opticsliterature,26–29 in which a number of solution techniqueshave been introduced. In all of them, however, somea priori information about the unknown is required forsuccessful reconstructions. The problem has been con-sidered also in crystallography,30,31 in which the phase re-covery from phaseless data plays a key role in the inter-pretation of crystal diffraction data. The problem isrelevant at microwave frequencies as well. In the an-tenna community, a number of solution techniques havebeen developed,32–38 even in recent times,37,38 to simplifymeasurement setups in both near-field techniques and di-agnostics of radiotelescopes.

In the considered problem, the first step of the proposedapproach amounts to retrieving Ed(u0 , u i) from theknowledge of E inc(u0 , u i) and uEtot(u0 , ui)u2 on G. As ex-traction of Ed(u0 , u i) from Etot(u0 , ui) would be relativelysimple, the problem could be seen as a phase-retrievalone, amounting to estimating Etot(u0 , ui) from error-affected knowledge of uEtot(u0 , ui)u2. As such, it wouldshare all the difficulties of standard phase-retrievalproblems14 including nonlinearity and ill-posedness.

On the other hand, owing to the presence of E inc(u0 , u i)(and of its properties) in Etot(u0 , ui), it is simple to showthat the problem at hand exhibits some interesting char-acteristics that make its solution remarkably simpler. Tobetter appreciate similarities and differences with respectto the standard problem,14 let us cast the problem as theinversion of the operator:

B@Ed~u0 , u i!#,uEd~u0 , u i!u2

1 2 Re@Ed~u0 , u i!E inc~u0 , u i!* #. (3)

In particular, one wants to solve

B@Ed~u0 , u i!# 5 M2~u0 , u i! 2 uE inc~u0 , u i!u2, (4)

where M2(u0 , u i) are the error-affected data of the prob-lem.

As a first, more obvious, difference, note that the prob-lem at hand differs from the usual one14 as the operator tobe inverted has both a nonlinear term due to the squareamplitude of the scattered field (i.e., the unknown of theproblem) and a linear one, given by the interference be-tween the known incident field and the unknown scat-tered field. As a consequence, whenever the intensity ofthe scattered field does not overcome that of the incidentone, the linear term is dominant in the operator to be in-verted, so that, by use of this information, the solution ofEq. (4) is relatively simple.

Unfortunately, this hypothesis is not generally satis-fied, so that the problem to be solved is nonlinear. There-fore, besides requiring some form of regularization totackle ill-posedness, the inversion of B(Ed) (which is a

nonlinear operator) is subject to possible false solutions,which correspond to local minima of the cost functionalwhose global minimum defines the solution. Luckily, asecond (less obvious) difference with respect to the stan-dard problem comes into play. In particular, it is worthnoting that the interference term between the incidentand the scattered fields, by virtue of properties discussedin Section 2, makes the essential dimension of the spaceof the data larger as compared with the case wherein onlythe square amplitude of the unknown (scattered) field isknown. This circumstance turns out to be of the outmostimportance, as it has been proved that in solving qua-dratic inverse problems of this kind false solutions can beavoided, provided that a sufficiently large ratio betweenthe number of (linearly) independent data and the num-ber of unknowns is available.14

To better understand such a point, it is worthwhile toexploit properties and possible representations of bothscattered and incident fields recalled in Section 2. Thisanalysis allows us to get three important results. First,for any given value of a and b, these properties and theirsampling requirements allow us to fix locations of pri-mary sources and measurement probes such as to collectall the available information while being nonredundant.Second, it allows us to prove that the present field recon-struction problem is much easier to solve than the stan-dard one.14 Third, together with results of Ref. 14, theyallow us, for any given a, to choose the value of b in anoptimal fashion.

To demonstrate the first statement, let us look to theproperties of the different contributions to uEtotu2. Ac-cording to Section 2, uEdu2 requires (4ba)(4ba) samples,as uEdu2 has a double spatial bandwidth with respect toEd . By reasoning on spatial bandwidths, one also seesthat uE incu2 requires (4bb)(4bb) samples, whereas2 Re(EincEd* ) needs @2b(a 1 b)#2 samples. Therefore therange of B [see expression (3)], is accurately representedin a space of @2b(a 1 b)#2 samples. Note that, with thelack of errors in data, the bandwidth of M2 exceeding2b(a 1 b) is exactly canceled out by 2uE incu2. Thereforethe essential dimension of data is bounded by @2b(a1 b)#2, that is, the number of Nyquist samples for therange of B. However, by virtue of reciprocity, only half ofthese samples are actually independent.25 For any givenvalue of a and b, these properties already allow us tochoose the optimal number and kind of incident fields andmeasurement probes. Both of them should be equal to2b(a 1 b), and they have to be uniformly spaced inangle.

By use of the same properties, an interesting result canbe derived about the false-solution problem, which provesthe second statement above.

In the standard problem,14 one wants to recover a(band-limited) complex function from its square-amplitude distribution. Assuming, for the sake of sim-plicity of explanation, one wants to recover Ed from uEdu2,the essential dimension of data would be (4ba)2, as theoperator to be inverted represents the square amplitudeof the unknown function, and therefore it has a doubleband with respect to the unknown function. The un-knowns are the (2ba)2 complex harmonics of the un-known function. Therefore, in a standard phase-

626 J. Opt. Soc. Am. A/Vol. 21, No. 4 /April 2004 Crocco et al.

retrieval problem, the ratio among independent data andnumber of real unknowns is given by

Rs 5~4ba !2

2~2ba !25

16~ba !2

8~ba !25 2. (5)

As discussed in Ref. 14, this ratio is not enough to guar-antee the lack of local minima. In fact, it has been shownthat one needs a ratio as large as 3 to avoid false solu-tions, so that some additional information is needed.

If we turn back to the actual problem at hand and cal-culate the ratio among independent data and number ofreal unknowns, we find

Ra 5@2b~a 1 b !#@2b~a 1 b !#

2

2

2~2ba !2

5@2b~a 1 b !#@2b~a 1 b !#

2~2ba !25

a2 1 b2 1 2ab

2a2.

(6)

If b 5 a 1 d, we have

Ra 51

2

~2a 1 d !2

a25 2 1 2

d

a1

1

2

d2

a2. (7)

Therefore, because of the presence of E inc in the data, R isalways greater than 2. Moreover, by acting on b, we canenlarge this ratio at our will. On the other hand, en-largement of b has two drawbacks. First, one needs amuch larger number of incident fields and measurementprobes. Second, the scattered field becomes weaker andweaker, and it becomes more and more difficult to get anaccurate estimate of it in the presence of the (unavoid-able) measurement error on uEtotu2.

As a trade-off between the need of enlarging Ra (toguarantee reliability) and the necessity of keeping itsvalue as small as possible (to guarantee accuracy andavoid redundancies), a threshold value can be defined forb, say, bcr , such that R 5 3, which implies bcr 5 (211 A6)a. This value, suggested by the analysis in Ref.14, allows us to avoid false solutions while reducing asmuch as possible the number of required experiments(and measurements) and the effect of measurement noiseas well. It is therefore an optimal choice for b. Note (seeSection 5) that numerical analysis fully confirms the the-oretical arguments. As a matter of fact, reconstructionprocedures are always successful for b . bcr , whereasthey may get stuck into false solutions for b , bcr .

4. STEP TWO: INVERSE SCATTERINGFROM THE ESTIMATED SCATTERED FIELDThe second step of the proposed procedure amounts to re-constructing the dielectric profile of the investigated re-gion, starting from the estimated scattered field. As inthe first step, we have to solve a nonlinear and ill-posedinverse problem.

For each illumination v, the integral equations for theinverse-scattering problem can be expressed in a compactfashion16 as

Edv 5 Ae~xEv!,

0 5 Eiv 1 Ai~xEv! 2 Ev, (8)

wherein Eiv is the incident field, Ev and Ed

v represent, forthe vth view, the total field inside the region V under testand the scattered field along the measurement circle ofradius b (as estimated in the previous step), respectively.

To achieve a quantitative reconstruction of the domainunder test, in terms of shape, location, and electricalproperties of the inclusions, we then have to solve the sys-tem of integral equations (8) (v 5 1 ... V) in terms of thecontrast function. In this step, one needs a set of inci-dent fields such that collects all the available informationabout the scatterer while being nonredundant. There-fore, by virtue of the properties in Section 2, only V5 2ba incident fields will be considered from now on.In particular, we will consider plane waves coming fromV 5 2ba different angles equispaced in [0, 2p]. Notethat the corresponding scattered fields can be extractedfrom Ed(u0 , u i) estimated in the previous step.

The approach we adopt is the one introduced in Ref. 16,which belongs to the class of modified gradient ones anddoes not introduce any approximation (but for the dis-cretization) in the solution procedure. In this approach,both the contrast x and the total field inside the scattererEv are assumed as unknowns, and the ill-posedness of theproblem is dealt with by looking for finite-dimensionalrepresentations for both the unknowns. We thus define

x~x, y ! 5 (l51

L

clc l~x, y !, (9)

Ev~x, y ! 5 (q51

Q

eqvwq~x, y !, v 5 1 ... V, (10)

wherein $ c l% l51L and $ wq%q51

Q are two orthonormal basesand the sequences c 5 $c1 , ..., cL% and e5 $e11 , ..., e1Q , ..., eVQ% contain the coefficients repre-senting the contrast and the internal fields (for all theviews), respectively. For each illumination, the numberQ of terms in Eq. (10) (i.e., the field unknowns) is given bythe proper discretization of Eq. (2a) performed accordingto the approach proposed in Ref. 39, and the number ofcontrast unknowns L has to be less than the essential di-mension of data.16,17,25

Owing to the lack of a priori information on the un-known scatterers, spatial Fourier harmonics have beenadopted as basis functions in both Eqs. (9) and (10).

Since projection of the unknowns over a finite-dimensional basis is a necessary but not sufficient condi-tion to prevent ill-conditioning of the solution procedure,a generalized solution defined as the global minimum of aproper cost functional has to be introduced. According tothe approach proposed in Ref. 16, the problem can be castas the global minimization of the cost functional:

f~ c, e ! 5 (v51

V iEiv 1 Ai~ xEv! 2 Evi2

iEivi2

1iAe~ xEv! 2 Es

vi2

iEsvi2

, (11)

Crocco et al. Vol. 21, No. 4 /April 2004 /J. Opt. Soc. Am. A 627

where i • i2 denotes the usual quadratic norm. Thelarge number of unknowns prevents us from using globalminimization approaches, and we adopt a conjugate-gradient fast-Fourier-transform approach whose detailsand features are reported in Ref. 16.

Since a local optimization technique has to be adopted,the minimization procedure may get stuck into a localminimum of Eq. (11), giving rise to a false solution. Ow-ing to the lack of a suitable starting guess and/or properpreprocessing,18–20 a general rule to reduce false-solutionoccurrences consists of keeping the ratio among (indepen-dent) data and the number of unknowns as large aspossible.17 This can be accomplished by use of a numberof contrast unknowns as low as possible. In this respect,an effective strategy to solve the problem amounts to pro-gressively increasing the number of contrast unknowns(i.e., L) and using as a starting-point guess at each stepthe estimated contrast achieved in the previous one.16

Extensions of this solution approach to the case of ho-mogeneous scatterers and multifrequency data are dis-cussed in Refs. 18 and 40, respectively.

5. NUMERICAL RESULTSIn this section we show some reconstructions obtained byusing simulated data to check the effectiveness of the pro-posed procedure.

In all the following examples, the unknown scatterersare positioned inside a square domain, of side l. We illu-minate the domain under test with plane waves at a fixedfrequency, impinging from a set of known angles.

In all the examples, we first show the scattered field es-timated in the first step. Then, by using this estimationas input of the inverse-scattering algorithm, we presentthe achieved reconstruction. The scattered field is recon-structed on a circle having the same center of the squareinvestigated region and a radius b greater than the se-midiagonal of the square, according to the criteria dis-cussed in Section 2 (2a 5 A2l).

The synthetic data (i.e., the samples of the square am-plitude of the total field) have been generated by use of anexact forward solver in which the domain under test isdiscretized according to the rules discussed in Ref. 39. Inall the examples, synthetic data have been corrupted withan additive noise.

Fig. 2. Real part of the reference profile.

As far as the first step of the proposed procedure is con-cerned, the accuracy of the reconstruction of the scatteredfield is evaluated by means of the normalized error de-fined as

eEd 5uEd,rec 2 Ed,acu2

uEd,acu2, (12)

where Ed,rec is the reconstructed scattered field and Ed,acis the actual one.

As far as the inverse-scattering step is concerned, theaccuracy of the reconstruction is evaluated by means ofthe normalized error defined as

Fig. 3. Comparison between (a) amplitude and (b) phase of thereconstructed scattered field (dashed curve) and those of the ac-tual one (solid curve); u i 5 p and u0 P @0, 2p#.

Fig. 4. Real part of the reconstructed profile.

628 J. Opt. Soc. Am. A/Vol. 21, No. 4 /April 2004 Crocco et al.

e 5uxrec 2 xacu2

uxacu2, (13)

where xrec is the reconstructed contrast and xac is the ac-tual one. In all the examples, Mi 3 Ni denotes the num-ber of sampling points for the internal equation for eachillumination condition, L is the number of unknown con-trast parameters, and Q is the number of total internal-field unknown parameters for each view. Moreover, Vdenotes the number of views, and M denotes the numberof measures of the scattered field for each view.

As a first example, consider the reference profile shownin Fig. 2, where a square void is embedded in a square re-

Fig. 5. Real part of the reference profile.

Fig. 6. Comparison between (a) amplitude and (b) phase of thereconstructed field (dashed curve) and those of the actual one(solid curve); u i 5 3p/5 and u0 P @0, 2p#.

gion of side l 5 1l, with l as the background wavelength,filled up with a lossless dielectric material of relative per-mittivity equal to 2.8. The side of the crack is l/2 long,and Mi 3 Ni 5 16 3 16, b 5 1.5l, V 3 M 5 10 3 10,and Q 5 15 3 15.

In Fig. 3 we show the amplitude and phase of the re-constructed scattered field for u i 5 p after solving thefirst step of our diagnostic technique. As can be seen, asatisfactory reconstruction is achieved in both amplitudeand phase. Similar results are obtained for all other in-cidence angles. Note that initial square-amplitude datawere affected by noise (20%) and that the final normalizederror, as defined in Eq. (12), is equal to 3.92 3 1022.

By using the estimated values of the samples of thescattered fields, we next considered the profile-estimationproblem. To avoid false solutions, we initially looked fora low-resolution approximation of the contrast. To thisend, a Fourier expansion with just 3 3 3 spatial harmon-ics has been used in the sought approximation of the con-trast. Then we considered a larger number of harmonics(5 3 5 first and then 7 3 7) by using the previously ob-tained profile as the initial guess. The final reconstruc-tion is shown in Fig. 4. As no edge preserving41 or binaryregularization18 has been used, the result is quite satis-factory. For this second step, the final normalized errordefined in Eq. (13) is equal to 8.85 3 1022.

As a second example, let us now consider the profile ofFig. 5, consisting of two square cylinders of sides 0.3l and

Fig. 7. Comparison between (a) amplitude and (b) phase of thereconstructed field (dashed curve) and those of the actual one(solid curve); u i 5 7p/5 and u0 P @0, 2p#.

Crocco et al. Vol. 21, No. 4 /April 2004 /J. Opt. Soc. Am. A 629

0.5l, whose complex permittivities are e1 5 4 2 j0.4 ande2 5 2.5 2 j0.25, respectively. The two scatterers arepositioned in a square domain under test whose side is l5 1l. In this case Mi 3 Ni 5 20 3 20, b 5 1.8l, V3 M 5 10 3 10, and Q 5 19 3 19, and simulated datahave been corrupted with an additive noise (20%). InFig. 6 we show both amplitude and phase of the recon-structed scattered field achieved in the first step of ourprocedure for u i 5 3p/5, and, in Fig. 7, amplitude andphase of the reconstructed scattered field for u i 5 7p/5are shown. As can be seen, a satisfactory reconstructionof Ed is found in both cases, and similar results hold truefor the other views. The final normalized error for thefirst step is equal to 1.25 3 1022. In the following step

Fig. 8. Real part of the reconstructed profile.

Fig. 9. (a) Amplitude and (b) phase of the actual scattered fieldfor b 5 0.9l. For this simulation bcr 5 1.02l.

the same progressive enlargement strategy for the con-trast unknowns as in the first example has been adopted.Figure 8 shows the real part of the final achieved recon-struction when 5 3 5 spatial Fourier harmonics to repre-sent the unknown (similar results are obtained for theimaginary part) are considered. For the inverse-scattering step, the final normalized error is equal to 0.19.It can be noted that the overall approach correctly identi-fies (but for an unavoidable low-pass filtering) location,extension, and strength of the obstacles.

The satisfactory results achieved fully confirm effec-tiveness of the proposed approach.

Note that in both the reported examples the scatterersare beyond the weak-scattering regime. As a matter offact, the actual scattered fields differ from those corre-sponding to the Born approximation by more than 500%and 400%.

Finally, to show how the choice of the radius b of G af-fects the actual effectiveness in solving the first step, weconsider the reconstruction of the scattered field in a casewherein b , bcr . First, Fig. 9 shows the amplitude andphase of the actual scattered fields for the contrast de-picted in Fig. 5, and Fig. 10 shows the reconstructedfields. Although noise-free data have been considered,the first step gets stuck into a false solution when b, bcr .

Many other examples support the fact that bcr is a criti-cal threshold value.

Fig. 10. (a) Amplitude and (b) phase of the reconstructed scat-tered field for b 5 0.9l. Note that noise-free data have beenconsidered.

630 J. Opt. Soc. Am. A/Vol. 21, No. 4 /April 2004 Crocco et al.

6. CONCLUSIONSWe have proposed and discussed an inverse-scatteringtechnique based on only amplitude measurements of thetotal field. Our aim has been to overcome limitationscharacterizing the previous techniques, which either con-sider weakly scattering objects10,11 or are subject, at leastin principle, to possible false solutions.

To enlarge the class of complex profiles that can be re-liably reconstructed, we have split the problem into twosteps. In the first one, we solve a phase-retrieval prob-lem for the total field, thus estimating the scattered field.Careful analysis of the first step allows us to choose opti-mal locations (and kind) of primary sources and measure-ment probes, thus allowing an accurate reconstruction ofthe scattered fields. In particular, a careful analysis offield properties and consideration of measurement errorsallow us to introduce an optimality criterion in the choiceof the radius of the measurement circle. Then the com-plex permittivity profile is reconstructed in the secondstep by use of the estimated scattered field.

Final reconstructions achieve a low-pass version of theoriginal profile, which is the best one can achieve with thelack of edge-preserving41 or binary-regularizationtechniques.18 As a matter of fact, final reconstructionsare essentially identical to those that one would achievestarting from amplitude and phase measurements.

Possible developments concern the extension of the pro-posed approach to the case of measurements in a trans-mission mode and to the imaging of scatterers enclosed ina stratified background (such as, for instance, a concretewall). Preliminary results for the case of scatterers lo-cated in free space with probes and primary sources lo-cated on two different lines are addressed in Ref. 15.

Corresponding author Tommaso Isernia’s current ad-dress is Facolta di Ingegneria, Universita Mediterraneadi Reggio Calabria, Via Graziella, Loc. Feo di Vito, ReggioCalabria I-89100, Italy. E-mail, isernia@ing.unirc.it.The other authors may be reached by e-mail atcrocco.l@irea.cnr.it; micdurso@unina.it.

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