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2566 J. Opt. Soc. Am. A/Vol. 23, No. 10 /October 2006 Bucci et al.

Inverse scattering from phaseless measurements ofthe total field on open lines

Ovidio Mario Bucci

Istituto per il Rilevamento Elettromagnetico dell’ Ambiente, Consiglio Nazionale delle Ricerche (IREA-CNR),Via Diocleziano 328, 80124, Napoli, Italy, and DIET, Università degli Studi di Napoli, Federico II

Viale Claudio 21, 80125, Napoli, Italy

Lorenzo Crocco

Istituto per il Rilevamento Elettromagnetico dell’ Ambiente, Consiglio Nazionale delle Ricerche (IREA-CNR),Via Diocleziano 328, 80124, Napoli, Italy

Michele D’Urso

DIET, Università degli Studi di Napoli, Federico II, Viale Claudio 21, 80125, Napoli, Italy

Tommaso Isernia

DIMET, Università Mediterranea di Reggio Calabria, Via Graziella, località Feo di Vito, Reggio Calabria, Italy

Received February 27, 2006; accepted April 13, 2006; posted May 11, 2006 (Doc. ID 68278)

A new solution approach to inverse scattering from aspect-limited phaseless measurements of the total field isintroduced and discussed. In analogy with the case of measurements on closed curves [J. Opt. Soc. Am. A 21,622 (2004)], the procedure splits the problem into two different steps. In the first step, amplitude and phase ofthe scattered field are estimated from only amplitude information of the total field. By properly extending theconcept of reduced radiated field to the case of scattered fields (as a function of both illumination and mea-surement variables) and taking advantage of the properties of the square amplitude distribution of the totalfield, criteria are given for an optimal choice of the measurement setup and a successful retrieval. Then thecomplex permittivity profile is reconstructed in the second step, starting from the scattered fields estimated inthe previous step. Numerical examples are provided to assess the effectiveness of the whole chain in the pres-ence of noise-corrupted data and the relevance of the representation introduced for the scattered fields.© 2006 Optical Society of America

OCIS codes: 290.3200, 100.5070, 100.6950, 100.3190.

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. INTRODUCTIONraditionally, electromagnetic inverse scattering prob-

ems consist in retrieving a quantitative description of thelectrical and geometrical characteristics of an investi-ated region from the knowledge of a set of incident fieldsnd measures of the corresponding total or scatteredelds on a generic surface lying outside the region underest.

The development of accurate and reliable techniquesor solving this kind of problem is an important challengeecause of their potential applications in biomedical im-ging, applied geophysics, noninvasive subsurface moni-oring, and nondestructive testing and diagnostics.1–6 Inact, in all the above-mentioned applications, inverse scat-ering approaches can open the way to new advanced im-ging techniques. On the other hand, as is well known,olving an inverse scattering problem generally meansacing an ill-posed nonlinear inverse problem.7,8

Many solution procedures have been introduced andxtensively tested in the past for both the weak scatteringase, wherein the inverse scattering problem can be lin-arized, and for the case of scatterers that do not fulfill

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he weak scattering (or Born) approximation. In these lat-er cases, most solution approaches recast the inversecattering problem as an optimization problem where aroperly defined cost functional, whose global minimumefines the solution, has to be minimized.2–6

By leaving aside peculiar characteristics of the differ-nt solution approaches, one of the main drawbacks of thesual procedures resides in the need to measure both themplitude and the phase of the scattered fields. As a mat-er of fact, accurate knowledge of the field phase involvesophisticated measurement equipment, which is morend more expensive as the working frequency increases,o that phaseless measurements are indeed mandatory atptical frequencies. In addition, the existence of mini-ally invasive (only amplitude) probes strongly suggests

he adoption of phaseless techniques also at microwaverequencies. In fact, these probes considerably simplifyhe electromagnetic case with respect to classical (ampli-ude and phase) probes since they avoid multiple interac-ions and do not require probe compensation.

Several solution approaches have been proposed in theiterature concerning inverse scattering from phaseless

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Bucci et al. Vol. 23, No. 10 /October 2006 /J. Opt. Soc. Am. A 2567

easurement of either the scattered field9,10 or of the to-al field.11,12 In the two-dimensional (2D) case, a new in-erse scattering technique based on amplitude-only mea-urements of the total field on a closed curve surroundinghe scatterer has been recently proposed by some of theresent authors.1 Such an approach, unlike some previ-us contributions,9,10 can be applied to any kind of dielec-ric scatterers and it splits the problem into two steps. Inhe first step, the scattered field is estimated from theeasurement of the square amplitude of the total field,hile the second step is aimed at estimating the dielectricroperties of the region under test starting from the esti-ated scattered field. Note that separation of the problem

nto two different steps allows for better control of theverall nonlinearity of the problem with respect to aingle-step procedure.11,12 In fact, a convenient exploita-ion of theoretical results on the inversion of quadraticperators13 and on field properties (andepresentation)14–16 allows one to successfully solve therst step, while all the available knowledge about inversecattering problems (in their full nonlinearity) can be ex-loited in the second one.In this paper, we deal with the case wherein transmit-

ers and receivers are placed on two different open lineshat enclose the scatterer (see Fig. 1). Note that thisransmission mode measurement arrangement can be re-lized in a relatively easy fashion with respect to theather canonical geometry analyzed in our previousontribution,1 so that it has an even greater interest. Onhe other hand, the aspect-limited nature of measure-ents entails a reduction of the available independent

ata for the overall imaging problem, thus making itore difficult to solve.As a matter of fact, each step of the adopted solution

rocedure is much harder to solve with respect to the casef a circular observation domain.1 Such a circumstance isell known as far as the second step is concerned,17–19 as

n this paper an inverse scattering problem, with (ampli-ude and phase) limited aperture data, has to be solved.his kind of problem is itself challenging, since without

ull aperture measurements, the ill-posedness of the in-erse problem becomes more severe. As a consequence, toeach the desired reconstruction, one generally has to ex-loit

Fig. 1. Geometry of the problem.

ultifrequency data or some a priori information on theature of the target (such as, for example, lossless naturer positivity constraints on the unknown functions).

Moreover, considerable problems also arise as far ashe first step of the procedure is concerned. As extensivelyiscussed in our previous contribution,1 reliability andccuracy in retrieving the scattered field depend onhe ratio between the essential dimension of the datapace and the dimension of the space wherein theolution is looked for (which may vary when the repre-entation of the unknown is changed). In particular, whenuch a ratio is larger than or equal to 3, occurrence ofalse solutions is avoided, whereas false solutions can oc-ur (they do in fact occur) when such a ratio is lower than.13 Accordingly, an important point arises in exploitingr, possibly, developing efficient (i.e., with the minimalossible redundancy) representations of the unknowncattered field, while increasing as much as possible thessential dimension (i.e., the degrees of freedom8) of theeasured data.With respect to the representation of the fields, in this

aper, as a first contribution, we exploit and generalizehe concept of reduced field and effective bandwidth intro-uced by Bucci and co-workers14–16 to deal with the inci-ent and scattered fields arising from multiview experi-ents, as well as with the corresponding squared

mplitude of the total field. This allows a number of rel-vant points to be answered. First, a minimally redun-ant representation for the unknowns of the first stepthe scattered fields) can be developed. Second, as theumber of terms of this representation coincides with theegrees of freedom of the scattered fields (as a function ofoth the incidence and the observation variables), theaximum amount of information on the scatterer that

an be extracted from scattering measurements (i.e., theaximum number of unknowns of the second step) is also

etermined.Last, but not least, since the amount of independent

ata depends on both the scattered and the incident fieldshrough the corresponding squared amplitude of the totaleld, properties and possible representation of the latterust also be investigated.By exploiting the properties of both the scattered and

he incident fields, as well as the essential dimension ofhe data, we can provide rules to choose the measurementetup in an optimal fashion. In particular, the givenuidelines allow avoiding false solutions in the first stephile guaranteeing at the same time a scattered field as

ntense as possible (which is of interest to alleviate the ef-ects of measurements errors) and the minimum numberf sources and probes (which is obviously of interest forhe overall costs).

The paper is organized as follows. In Section 2, the ge-metry of the problem is described, the formulation of theroblem is given, and the relevant properties of fields areecalled. Then, in Section 3, the problem of determiningamplitude and phase of) the scattered fields from thequare amplitude distribution of the total field is dis-ussed. In Section 4, the problem of reconstructing theontrast function starting from the estimated scatteredelds in the first step is briefly dealt with. Finally, Sectionis concerned with a numerical analysis confirming the

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easibility of the whole chain and the relevance of the rep-esentation adopted for the scattered fields. Conclusionsollow.

. FORMULATION OF THE PROBLEM ANDIELD PROPERTIESet us consider a region under test � enclosing one orore nonmagnetic objects characterized by a complex

ermittivity ��r�=��r�− j��r� /�, where ��r� and ��r� de-ote the permittivity and electrical conductivity, respec-ively, and r= �x ,y�. If we denote with �b the permittivityf the background medium, it is possible to introduce theontrast function as

��r� = ��r�/�b − 1�, r � �. �1�

With reference to the 2D geometry depicted in Fig. 1, inhe following we assume that

The domain under test � is a circle with radius a.The incident fields are TM-polarized unitary monochro-atic cylindrical waves, radiated by elementary probes

ocated in rs= �−xs ,ys� on the line �s of extent Ls, withs� �−Ls /2 ,Ls /2� and fixed xs=−d /2. Accordingly, Einc�r�−jk2 /4Ho

�2��k�r−rs��, where k=��0�b is the backgroundavenumber and Ho

�2� is the zero-order Hankel function ofhe second kind.

Elementary measurement probes are located on theine �o of extent Lo, in the generic coordinate ro= �xo ,yo�,ith yo� �−Lo /2 ,Lo /2� and fixed xo=d /2.

Under these hypotheses and omitting the exp�j�t� timeependence, the scattering equations describing the totaleld for each illumination condition can be expressed as20

J�rs,r� = ��r�Einc,i�rs,r� + ��r� � ��

g�r − r��J�rs,r��dr�

= ��r�Einc,i�rs,r� + ��r�Ai�J��rs,r�, r � �, �2a�

tot�rs,ro� = Einc,e�rs,ro� + Es�rs,ro�

= Einc,e�rs,ro� +� ��

g�ro − r��J�rs,r��dr�

= Einc,e�rs,ro� + Ae�J��rs,ro�, rs � �s, ro � �o,

�2b�

here Einc,i and Einc,e are the incident fields as evaluatedn � and on the observation domain, respectively. J�·��·�E�·� is the contrast source function, defined as theroduct between the unknown contrast and the total elec-ric field. Etot is the total field on the observation domain,hile Es is the scattered field. Ai :L2��→L2�� ande :L2��→L2��o� are the internal and external radi-tion operators, which relate the contrast source to thecattered field in � and on �o, respectively. g�r−r��−j�k2 /4�H0

�2��k�r−r�� is Green’s function of the homoge-eous background.The overall aim of the problem is to determine the 2D

ontrast function ��r� in �, starting from the knowledge

f the incident field and from an incomplete (because onlyfinite number of measurements can be performed) and

naccurate (because the measurements are error-affected)nowledge of �Etot�rs ,ro��2 on �o.To develop and discuss the two steps of the proposed

rocedure, it proves fruitful to briefly recall propertiesnd possible representations of both scattered and inci-ent fields. The aim is to use a number of scattering ex-eriments (i.e., a number of incident fields) and a numberf measurements to collect as much information as pos-ible about the scatterer while being nonredundant.

Let us start by considering the scattered field.The problem of determining nonredundant representa-

ions (i.e., to involve a minimum number of parameters)f radiated or scattered fields in a homogeneous losslesspace over arbitrary curves or surfaces has been thor-ughly investigated in the past. In a number ofontributions14–16 by Bucci and co-workers it has beenhown that they can be accurately represented by a finiteumber of samples on any observation domain, includingruncated and unbounded ones. When the observation do-ain totally encircles the source, this number is essen-

ially coincident with the number of degrees of freedom ofhe field and depends only on source (or scatterer) sizend shape.14

To achieve a nonredundant representation of the scat-ered field (in the 2D case at hand), let us first considerhe case of a fixed position of the primary source. Then, ashe scattered fields are the fields radiated from the in-uced currents, one has to consider16 along the observa-ion curve C, described by a proper analytic parameter-zation r=r��, the generalized reduced field F��E�r��exp�j�� obtained from the actual field by ex-

racting a suitable phase function ��.16 As a matter ofact, provided that the parameterization � and the phaseunction �� do not introduce spurious singularities, F��s an analytic function on the observation domain,16

hich can be closely approximated by a band-limitedunction Fw��, with an error that becomes negligible ashe bandwidth exceeds a critical value W�. The latter,hose general expression and properties are recalled inppendix A, is equal to the maximum of the local band-idth (which accounts for the local spectral behavior of

he reduced field). The approximation error can be effec-ively controlled by choosing a bandwidth W�, where 1 is the enlargement bandwidth factor,16 and decreases

xponentially with �−1�3/2W�,16 which means that in the

ase of electrically large objects it becomes negligible foralues of that are only slightly larger than a unit.16 Ac-ordingly, W� can be identified as the effective bandwidthf the field corresponding to the chosen parameterizationnd phase function, and the generalized reduced fieldlong C can be represented by a cardinal sampling seriesith Nyquist step ��= /W�. A proper choice of thehase function and the curve parameterization allow uso minimize the number of required samples, thus obtain-ng a nonredundant sampling representation of theeld.16 In particular,

Fw�� = �n=−N

N

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here �n=n / �W��, sinc�x�=sin�x� /x,21 and �2N+1� ishe number of samples lying on the observation intervalf interest (plus, possibly, a few extra samples to keep theruncation error low).

It must be stressed that the overall number of samplesn Eq. (3) is finite even when the observation curve C isnbounded.The general expressions of the optimal phase function

nd parameterizations are recalled in Appendix A. In par-icular, it is worth noting that the resulting optimal aux-liary variables and phase functions depend only on theize and shape of the source and on the observationurve.16 These geometrical parameters also determine theinimum number and optimal positions of the samples

equired to represent the field scattered over C. The opti-al functions have been explicitly determined for a large

umber of relevant geometries.16

By exploiting reciprocity arguments,8 similar reasoningan be made for the source variable, which allows us toxtend the above theory and results to the case of scat-ered fields collected under multiview arrangements,eading to the reduced scattered field, defined as

Fs��s,�o� = Es��s,�o�exp�jo��o��exp�js��s��. �4�

According to the above, the reduced scattered field (ando the field) can be closely approximated by means of aand-limited function, both in �s and in �o.The above theory is of course absolutely general and

an be applied to arbitrary observation and source geom-tries. In the following, we will particularize it to theeasurement configuration of Fig. 1, assuming, by the

ake of simplicity, that Ls=Lo=L.Let us consider first the case where the lines �o and �s

re unbounded. In this case, by assuming that the (un-nown) scatterers are enclosed within a circle (centered

n the origin) of radius a, the optimal auxiliary variable(s)nd the phase function(s) on both �o and �s (and the cor-esponding effective bandwidth) are given by16

�s = �o = �, s�r� = o�r� = k�r2 − a2 − ka cos−1�a/r�,

WEs= ka, �5�

here r denotes the distance of the generic observationor source point) from the origin, � is the usual polar co-rdinate, and cos−1�·� is the inverse function of cos�·�.uch a choice allows one to approximate the reduced scat-ered field with a spatially band-limited function with aandwidth WEs

for each adopted illumination.16 Alsoote that, on the basis of the last part of Eqs. (5), ka islso the total (i.e., 2N+1� number of samples to be consid-red in the expansion given in Eq. (3).

When the source and measurement lines are truncated,he number of relevant samples decreases accordingly,eading to a reduction of the number of degrees of freedomf the field. In particular, exploiting Ref. 16, we get Nint�ka�AP / �, wherein �AP=tan−1�L /d� is the semi-

ngle subtended by the observation line (see Fig. 1) and s the usual enlargement bandwidth factor.

Note that in both the above cases, the samples are uni-ormly angularly spaced, so that they are not uniformlypaced in terms of the original y variable.
o
Let us now turn to the incident fields. To this end, notehat the incident field over an unbounded observationpen line due to a collection of transmitters located on adifferent) source line, of finite extension, can be seen ashe field radiated by a linear source in a linear scanningeometry. The set of transmitters is the linear source withength Ls placed at d /2 from the y axis (see Fig. 1), d be-ng the distance between the source line �s and the obser-ation one �o. Starting from these considerations, and bytill using the concept of reduced radiated fields,16 it fol-ows that by introducing an auxiliary variable �o (whichakes the place of yo) and extracting a proper phase func-ion o��o�, one can define a reduced incident field

inc,e��o ,ys�, which can be closely approximated by meansf a band-limited function in term of �o, and therefore cane represented by a finite number, say �2M+1�, ofamples (for each ys).

Applying the general equations in Appendix A to thebove outlined source and scanning geometries, it followshat the auxiliary optimal variable(s) and the phase func-ion (and the corresponding effective bandwidth) areiven by16

�o�yo� = �R1 − R2�

2Ls, o�yo� =

k

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kLs

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R1 = �d2 + �yo + Ls/2�2, R2 = �d2 + �yo − Ls/2�2, �6�

here R1 and R2 are the maximum and minimum dis-ances between the generic observation point on �o andhe linear source of extension Ls, respectively (see Fig. 1).

Again, when truncation of the measurement line isaken into account, as for the scattered fields, a reductionf the number of degrees of freedom of the incident field isbserved. In particular, we get M=int�k�R1−R2�� /2 �,herein R1 and R2 are evaluated for yo=L /2 and the

2M+1� samples are uniformly spaced in the coordinateystem defined by the functions � and , but, again, non-niformly spaced samples in terms of the original yo vari-ble.Then, as for the scattered field and still exploiting reci-

rocity arguments,8 by introducing an additional auxil-ary variable �s (which takes the place of ys) and theroper phase function s��s�, we can define the reduced in-ident field from �o to �s as

Finc,e��s, �o� = Einc,e��s, �o�exp�jo��o��exp�js��s��, �7�

here the auxiliary variables(s) and the phase function(s)re given by Eqs. (6). According to the above, the incidenteld can be conveniently represented taking advantage ofhis band-limited function.

. ESTIMATING THE SCATTERED FIELDSn this section we deal with the problem of retrieving themplitude and phase of E �r ,r � from the knowledge of
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inc,e�rs ,ro� and from an error-affected knowledge ofEtot�rs ,ro��2. As shown in our previous paper,1 the prob-em at hand can be conveniently formulated as an inver-ion of the operator:

B�Es�rs,ro�� = �Es�rs,ro��2 + 2 Re�Es�rs,ro�Einc,e�rs,ro�*�,

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here Re�·� denotes the real part of the argument.In particular, one wants to solve

B�Es�rs,ro�� = M2�rs,ro� − �Einc,e�rs,ro��2, �9�

here M2 are the error-affected data of the problem.Since B�Es� is a nonlinear operator involving both a lin-

ar term and a quadratic one, its inversion, in addition toequiring some form of regularization to tackle ill-osedness, is subject to possible false solutions. The latterorrespond to local minima of the cost functional whoselobal minimum defines the regularized solution.

Now, it has been proved that false solutions in solvinguadratic inverse problems can be avoided provided thatsufficiently large ratio between the number of indepen-

ent data and the number of the unknowns is available.13

herefore, the problem arises of enlarging as much asossible this ratio to solve Eq. (9) in a robust way.To this end, it immediately appears that the nonredun-

ant representation of the unknown scattered field, giveny the reduced scattered field, is certainly convenient tohis aim, since it allows us to keep the number of un-nowns as low as possible.To characterize the range of the operator B�Es� and to

nderstand in which way one can take advantage of theresence of the incident fields to make the ratio betweenhe independent data and unknowns as large as possible,roperties (and possibly sampling representations) of theecond member of Eq. (8) have to be inspected. A suitablend convenient way to do this in an analytic fashion is toxtend the concepts of local and effective bandwidth tohe two terms on the right-hand side of Eq. (8), i.e., to theroduct of the fields.According to the general strategy of Ref. 16, whoseain steps are recalled in Appendix A, to find a minimally

edundant representation, one should extract a phase fac-or and find a suitable parameterization such that theverall bandwidth is reduced as much as possible. As aonsequence, the spectrum arising from the optimal rep-esentation will have a symmetric extension (so that theandwidth cannot be ameliorated by a linear phase) andhe optimal parameterization will also be such that theorresponding local bandwidth becomes constant (so thathe maximum of the local bandwidth cannot be further re-uced). Now, as both terms at the second member of Eq.8) are real, both their local22 and global spectrum haveermitian symmetry (i.e., are centered on the origin) so

hat it makes no sense to extract a phase factor.Moreover, as we are interested in determining the over-

ll number of samples required to represent these twoerms, and the latter depends on the local bandwidth inhe arc-length parameterization [see Eqs. (A3) and (A4)],e will consider such a parameterization in the following.

Let us consider �Es�2 first. By virtue of the fact that theandwidth of a square amplitude function doubles, the lo-al bandwidth of �Es�2=Es ·Es

* is given by

w�Es�2�s� = 2wEs�s�. �10�

or the interference term, let us start by noting that

Re�EsEinc,e* � = 1/2�EsEinc,e

* + Es*Einc,e�. �11�

oreover, one also has

Es�s�Einc,e�s�* = Fs�s�Finc,e�s�* expj��s� − �s��, �12�

here and are the phase function of the reduced scat-ered and incident fields, respectively, as given by Eqs. (5)nd (6), respectively.Then, from Eqs. (11) and (12), and the well-known

roperties of bandwidths, it also follows that

wInt�s� = wEs�s� + wEinc,e

�s� + � �� − �

�s� . �13�

Equation (13) has a simple interpretation. As a matterf fact, wEs

�s�+wEinc,e�s� is the local bandwidth of the prod-

ct of the reduced fields Fs and Finc,e. The presence of theactors exp±j��s�− �s�� in the interference terms

sEinc,e* and Es

*Einc,e shifts their spectrum to the points��− � /�s�, leading to the overall bandwidth given byq. (13).Therefore, for the local bandwidth of the right-handember of Eq. (8), wtot�s�, we get

wTot�s� = max�wInt�s�,w�Es�2�s��. �14�

y using the fact that the local bandwidth in a point lyingn the observation curve is proportional to the angular ex-ension of the source as seen from the point itself (see Ref.2), it can be shown that w�Es�2�s� is not smaller thanInt�s� if and only if the source line, as seen from the cen-

ral point on the observation line, is completely shadowedy the scattering domain. In the geometry of our interest,his never happens provided that

L

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��d/2a�2 − 1. �15�

f inequality (15) holds, wtot�s�=wInt�s�, and we can evalu-te the optimal parameterization and the number of Ny-uist samples in the observation domain by exploitingqs. (A3) and (A4), respectively. In particular, as by virtuef Eqs. (5) and (6) ��− �� /�s does not change sign, weet

NTot = N + M + Int

�� − ��0

L/2� = N + M + �N. �16�

nd so, we conclude that, for each illumination, the rangef B�Es� is hosted in a real linear space of dimension2Ntot+1��2Ntot. Therefore, by again exploiting the reci-rocity arguments,8 it can be concluded that 4Ntot

2 (real)erms are needed to accurately represent the interferenceerm, i.e., the right-hand side of Eq. (8), by means of a lin-ar expansion [see discussion after Eq. (14)], thus sug-

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esting the minimum number of measures of the squaremplitude distribution of the total field able to take intoccount all the available independent information.Explicit evaluation of Ntot shows that, because of the

relatively similar) behavior of and , �N / �N+M��1, sohat N+M provides a simple satisfactory lower bound for

tot.Assuming the ��2N+1�� 2N+1��4N2 samples of the

educed scattered field as the actual (complex) unknownsf our problem, we can now evaluate the ratio Ra amongndependent data and number of (real) unknowns as

Ra =4NTot

2

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2�NTot

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N�2

. �17�

s expected, the presence of the incident fields in theunctional to be inverted gives the opportunity of actingn the effective dimension of the space of the availableata. As a matter of fact, provided that the geometric pa-ameters and the measurement setup are chosen in suchway to guarantee the relation M�N, we can enlarge Ra

t will by acting on M.On the other hand, the enlargement of M (i.e., the en-

argement of �o and �s to which M is related) has tworawbacks. First, one needs a much larger number of in-ident fields and measurement probes. Second, since thecattered field becomes weaker at the extremes of the ob-ervation lines and sources, it becomes more difficult toetrieve it in the presence of the (unavoidable) measure-ent error on �Etot�rs ,ro��2.Therefore, to take into account the trade-off that arises

etween the need to enlarge Ra to guarantee reliableesults13 and the necessity of keeping its value as small asossible to guarantee accuracy and avoid redundancies, ahreshold value can be defined for M /N, say �M /N�cr, suchhat Ra=3, which gives �M /N�cr=−1+�6.

Properties of quadratic operators13 ensure that thisalue allows false solutions to be avoided, while the aboveonsiderations show that at the same time it reduces asuch as possible the number of required experiments

and measures) and the effect of measurement noise asell.It is therefore an optimal choice for M /N and so, due to

he relation between N and M with the geometric param-ters [see Eqs. (5) and (6) for more details], for the mea-urement setup.

A large numerical analysis (see Section 5) has fully con-rmed this theoretical statement. As a matter of fact, theeconstruction procedure of the first step is always suc-essful for �M /N�� M /N�cr, whereas it may get stuck intoalse solutions if M /N is lower than the critical value.

. ESTIMATING THE CONTRASTUNCTIONfter performing the estimation of the scattered fields ac-ording to the criteria discussed above, to achieve a quan-itative reconstruction of the domain under test, in termsf shape, location, and electrical properties of the inclu-ions, we have to solve the system of integral Eqs. (2) (forach considered incident field) in terms of the contrast

unction embedding the unknown permittivity profile. Asn the first step, we have to solve a nonlinear and ill-posednverse problem.

First, let us start by noting that in the inverse scatter-ng step one needs a set of incident fields to collect all thevailable information about the targets while being non-edundant. Therefore, taking advantage of properties re-alled in Section 2, only 2N int�2ka�AP / � incidentelds are necessary. In particular, we will consider uni-ary cylindrical waves coming from different nonuni-ormly spaced positions on the line Ls, as suggested by thenalysis in the first part of Section 2.Second, we have to choose a convenient solution ap-

roach. Among the several approaches available in theiterature,6,23 in the following we have adopted a new in-ersion method, called the contrast source-extended BornCS-EB) inversion method, recently introduced by some ofhe authors.2 The latter has proved to be an effective toolor the solution of both forward-scattering as well as of in-erse scattering problems.2 In particular, the rewriting ofhe traditional Eqs. (2) brings us to the introduction of anuxiliary function that is related in a nonlinear way tohe usually adopted contrast.2 In many cases of practicalnterest, the adoption of this latter function as the un-nown of the inverse problem implies a reduction of theverall degree of nonlinearity24 of the relationship amonghe unknown and the data, thus allowing, at least in prin-iple, increased robustness against false solutions in thenversion process.24 It is worth noting that, although theerivation of the CS-EB model relies on some mathemati-al and physical considerations that are well suited whenosses are present in the hosting medium and/or in thenknown scatterers, a recent experimental validation hashown that very accurate results can also be achieved forhe case of purely dielectric targets in free space.25 Theolution of the new system of equations is then recast asn optimization problem wherein a proper functionalwhose global minimum defines the solution of the secondtep of the proposed procedure) has to be minimized. Be-ause of the large amount of unknowns to be determined,conjugate gradient fast-Fourier-transform-based inver-

ion scheme has been adopted to this end (the reader iseferred to Ref. 26 for more details about the adoptedinimization scheme).

. EXAMPLES OF PHASELESS MICROWAVEOMOGRAPHY

n this section we present some numerical examples con-rming the feasibility of performing faithful phaseless to-ography starting from the knowledge of the incidentelds (both in amplitude and phase) and by measuringhe corresponding square amplitude distributions of theotal field on the observation curve.

The analysis in the following is subdivided in two parts.n the first part, numerical examples confirming the ca-ability of the proposed approach to estimate both ampli-ude and phase of the scattered field (for each view) fromspect-limited phaseless data are shown and discussed.n this first step, the accuracy of the reconstruction of thecattered field is evaluated by means of the normalized

ei

frssfiwa

ttpsiObbAoarsa

+�

wapowstsfi=iashtmrept

t

F(

Fa


rror defined as errEs= �Es,rec−Es,ac�2 / �Es,ac�2, where Es,rec

s the estimated scattered field and Es,ac is the actual one.In the second part, numerical examples confirming the

easibility of the overall phaseless imaging approach areeported. In particular, the aim is to use in the inversecattering step the scattered fields estimated in the firsttep. To evaluate the accuracy in the second step, we de-ne a normalized error given by err�= ��rec−�ac�2 / ��ac�2,here �rec is the estimated contrast function and �ac is thectual one.As a first example, let us consider the problem of re-

rieving both the amplitude and the phase of the scat-ered field from a circular cylinder of radius 1.7� and com-lex permittivity �=2.25− j0.225. The measurement andource lines are 20� long, located at 3.3� apart from thenvestigated region, which is a square domain of side 4�.n the basis of the results discussed in Sections 2 and 3,y considering the same enlargement bandwidth factor inoth cases �=1.15�, we have N=7, M=13 while �N=4.ccordingly, 2NTot=40 measures of the square amplitudef the total field allow us to accurately consider all thevailable information. Therefore, 64 transmitters andeceivers27 nonuniformly spaced along the source and ob-ervation lines have been used. In particular, the samplesre positioned at yn=d /2tg��n�, with �m=�m=m� / �2�N

ig. 2. Comparison between the real part of the (a) actual andb) estimated scattered field for the case of a single object.

s s s s AP

M�� for the transmitters and at yom=d /2 tan��o

m�, with

om=�o

m=m�AP / �2�N+M�� for the receivers.To show the effectiveness of the introduced approach,

e compare the behavior (in terms of the auxiliary vari-bles �o, �s) of the real (Fig. 2) and imaginary (Fig. 3)arts of the actual scattered field and of the estimatednes. As can be seen, although the data are corruptedith an additive noise (20%), a good reconstruction of the

cattered field is achieved for all the adopted illumina-ions. Figures 4 and 5 show some cuts of the achieved re-ults. In particular, Fig. 4 compares the actual scatteredeld and the estimated one for a source placed in ys7.3� as a function of the positions of the receivers, while

n Fig. 5 the same result for a source located in ys=3.5� islso given. As can be seen, a good reconstruction of thecattered field is found in both cases, and similar resultsold true for the other views, although worse reconstruc-ions are achieved for sources located at the end of theeasurement domain, as can be seen by comparing the

esults in Figs. 4 and 5. The final normalized error isqual to 3.22�10−2. Finally, note that the samplingoints are nonuniformly spaced in terms of the observa-ion variable (see Figs. 4 and 5).

Some important comments are now in order.First, note that the adopted parameters, in terms of ex-

ension of the measurement and source lines and their

ig. 3. Comparison between the imaginary part of the (a) actualnd (b) estimated scattered field for the case of a single object.

dbktwos

skaukp==

to

mpotcai

edt

3fhbk2fifcvs

wmattfiass

d

Fpfit

Ftct


istance from the domain under test, make the loweround for the ratio among independent data and un-nowns of the problem Ra larger than 3, thus allowing uso estimate the scattered field in an effective and accurateay. A large series of numerical examples has shown thatne is not able to achieve a faithful reconstruction of thecattered field when such a ratio is lower than 3.

Second, the representation introduced for the unknowncattered field (in terms of reduced fields) indeed plays aey role. As further proof of this latter statement, Figs. 6nd 7 show the result one would achieve (in terms of thesual coordinates yo and ys) when representing the un-nown field in terms of 64 uniformly spaced samples. Inarticular, in this case the sampling spacing is �s

n=ysn

nL / �2�N+M�� for the transmitters and �om=yo

m

mL / �2�N+M�� for the receivers.As can be seen, the field is not retrieved at all, both in

erms of the real part (see Fig. 6) and of the imaginaryne (see Fig. 7).

As a second example aimed at assessing the perfor-ance of the overall approach, let us now consider the

roblem of imaging two square cylinders of side 0.7� andf complex permittivity 2− j0.2. The two targets are posi-ioned in a square domain of side 3�. The real part of theontrast function is shown in Fig. 8. The measurementnd source lines are 12� long, located 2� apart from thenvestigated region.

ig. 4. Comparison between the (a) real parts and (b) imaginaryarts of the reconstructed field (dashed curves) and the actualeld (solid curves) for a source located in ys=7.3� as a function ofhe locations of the receivers.

Data collected at both 1 and 3 GHz have been consid-red to compensate in the inverse scattering step the re-uction of independent data due to the reduced observa-ion domain.

On the basis of the results discussed in Sections 2 andand by considering the same enlargement bandwidth

actor for the scattered and incident field �=1.15�, weave N=5, M=9 while �N=2, thus making the loweround for the ratio among independent data and un-nowns of the problem Ra equal to 3.9. Accordingly,NTot=28 measures of the square amplitude of the totaleld allow us to accurately consider all the available in-ormation. Therefore, 32 (Ref. 27) transmitters and re-eivers not uniformly spaced along the source and obser-ation lines are used here (see above for details about theampling spacing).

To show the effectiveness of the introduced approach,e first compare the real and imaginary parts of the esti-ated scattered field to the actual ones at the higher

dopted frequency (see Figs. 9 and 10). As can be seen, al-hough simulated data have been corrupted with an addi-ive noise (20%), a good reconstruction of the scatteredeld is achieved even though worse reconstructions arelso achieved for sources located at the end of the mea-urement domain in this case. The estimated field corre-ponds to an error equal to 3.84�10−2.

As far as the imaging step is concerned, multifrequencyata have been processed within a frequency-hopping

ig. 5. Behavior of the (a) real parts and (b) imaginary parts ofhe reconstructed field (dashed curves) and the actual one (solidurves) for a source located in ys=3.5� as a function of the loca-ions of the receivers.

sathNtb

ttttehlps

ftNtstsi

6Aahcssm

F(f

Fat

F


cheme.28 In particular, the estimated contrast functiont the first frequency has been used as a starting guess athe higher frequency, while the backpropagation solutionas been used as a starting guess at the first frequency.ote that use of multifrequency data has required the es-

imation of amplitude and phase of the scattered fields atoth the adopted frequencies.The above strategy allows us to accurately reconstruct

he unknown contrast as the final normalized reconstruc-ion error is as low as 6.03�10−2; the real part of the es-imated contrast is shown in Fig. 11. The smoothed na-ure of the reconstruction is not surprising, since nodge-preserving29 or binary regularization31 techniquesave been used. Of course, adoption of this kind of regu-

arization enforcing piecewise constant profiles is ex-ected to further improve the quality of the final recon-truction.

The satisfactory results achieved fully confirm the ef-ectiveness of the proposed approach and the relevance ofhe representation introduced for the scattered fields.ote that in all the examples the scatterers are beyond

he weak scattering regime. As a matter of fact, the actualcattered fields strongly differ from those correspondingo the Born approximation, thus not allowing us to applyolution procedures based on linearization of the scatter-ng equation.11,12

ig. 6. Comparison between the real part of the (a) actual andb) estimated scattered field by using a uniform representationor the scattered field in terms of the observation variable.

. CONCLUSIONS AND FUTURE WORKnew solution approach to inverse scattering from

spect-limited phaseless measurements of the total fieldas been introduced and discussed. In analogy with thease of measurements on closed curves,1 the procedureplits the problem into two different steps. In the firsttep amplitude and phase of the scattered field are esti-ated from only amplitude information of the total field.

ig. 7. Comparison between the imaginary part of the (a) actualnd (b) estimated scattered field by using a uniform representa-ion for the scattered field in terms of the observation variable.

ig. 8. Real part of the contrast for the case of a multiple object.

Ivtcsttm

ctticrtwqmc

ccpln

sf

piii

F(

Fto


n such a step the proposed procedure takes decisive ad-antage from properties and nonredundant representa-ions of both scattered and incident fields. Moreover, aonvenient exploitation of theoretical results on the inver-ion of quadratic operators13 and on field properties,14–16

ogether with considerations on the presence of noise onhe data, allow introduction and exploitation of an opti-ality criterion in the choice of the measurement setup.Then, the unknown complex permittivity profile is re-

onstructed in the second step, starting from the scat-ered fields estimated in the previous step and by usinghe recently introduced CS-EB inversion method.2 Sincen an aspect-limited measurement configuration mono-hromatic data cannot be sufficient to achieve an accurateeconstruction due to the lack of independent informa-ion, we have exploited the multifrequency processedithin a frequency-hopping scheme. Of course, this re-uires us to repeat the first step of the procedure to esti-ate the scattered field at each of the considered frequen-

ies.Numerical examples are provided to check the whole

hain in the presence of noise-corrupted data. Final re-onstructions achieve a low-pass version of the originalrofile, which is the best one can achieve because of theack of edge-preserving29 or binary30 regularization tech-iques. As a matter of fact, final reconstructions are es-

ig. 9. Comparison between the real part of the (a) actual andb) estimated scattered field for the case of a multiple object.

entially identical to the ones one would achieve startingrom amplitude and phase measurements.

The first possible developments of the proposed ap-roach concern its extension to the case of measurementsn a reflection mode and in particular to the problem ofmaging targets enclosed in a layered background, whichs relevant in through-wall-imaging applications.31

ig. 10. Comparison between the imaginary part of the (a) ac-ual and (b) estimated scattered field for the case of a multiplebject.

Fig. 11. Real part of the reconstructed contrast function.

(onthweaelostcat(ts3

ALtstdfdfpbpa

wptp

etgtwsta

w

�t

Tch

aaow

ATv

i

R

1


Extension of the whole approach to three-dimensional3D) geometry and to real-world applications using arraysf probes is the final challenge. In this respect, it is worthoting that progress of sciences on a nanometer scale inhe field of biology, new materials, and microelectronicsas stimulated a growing interest in imaging techniquesith a spatial resolution lower than 100 nm.32,33 Even if

lectron microscopy and atomic strength microscopy areble to produce images with such a resolution, they arexpensive, difficult to put into operation, and, last but noteast, do not allow one to retrieve internal profiles of 3Dbjects in a noninvasive way. Therefore, the use of inversecattering techniques based only on amplitude informa-ion of the measured total field appears as a stimulatinghallenge toward 3D imaging of complex structures withsub-100 nm resolution power. To this end, extension of

he overall approach to completely phaseless tomographyi.e., also removing the need of the phase distribution forhe incident fields) is mandatory. Preliminary results onuch a topic, confirming expectations, are reported in Ref.4.

PPENDIX Aet us consider an observation curve C and let us describe

his with a proper analytic parameterization r=r��. In-tead of considering the scattered field along the observa-ion curve, it is possible to introduce the generalized re-uced field F��=E�r��exp�j��, obtained by extractingrom the actual radiated field a phase function �� to beetermined.16 The generalized reduced field is an analyticunction on the observation curve that can be closely ap-roximated by a band-limited function with an error thatecomes negligible as the bandwidth governing the sam-ling representation exceeds the critical value W�, defineds16

W� = max�

�w�� = max� max

r��D�d��

d�− k

�R��,r��

�� ,

R = �r�� − r��, �A1�

here R is the distance between the generic observationoint P=P�� on C and a source point r�, and w�� is calledhe local bandwidth and is related to the local spectralroperties of the reduced field.22

The value W� in Eq. (A1) is naturally identified as theffective bandwidth of the reduced field corresponding tohe chosen parameterization and phase function. For anyiven observation curve, to get a nonredundant represen-ation, the phase factor must minimize the local band-idth w�� and the parameterization must be chosen in

uch a way as to make w�� constant with �. This leads tohe following expressions for the optimal phase functionnd parametrization (see Ref. 16 for more details):

�� =k

2�0

s�� maxr �D

�R

�s+ min

r �D

�R

�s �ds, �A2�
� �
� = ��s� =k

2W��

0

s maxr��D

�R

�s− min

r��D

�R

�s �ds, �A3�

here s is the arc length.For the case of a symmetrical observation interval

−s , s�, we get from Eq. (A3) that the number 2N of termso be considered in Eq. (3) can be computed according to

N�s� = Int ��s�W�

� = Int

k

2�0

s maxr��D

�R

�s

− minr��D

�R

�s �ds� = Int

�

0

s

w�s�ds� . �A4�

hese latter expressions fix the number of terms to beonsidered in a sampling representation of the field atand once the the optimal parametrization is adopted.It is worth noting that by virtue of the procedures

dopted to define the optimal parametrization, whichims at getting a constant local bandwidth, the last partf Eq. (A4) also holds true for the case of generic functionsith local bandwidth w�s�.

CKNOWLEDGMENTSommaso Isernia and Michele D’Urso thank Catello Sa-arese for interesting discussions on the field properties.

Corresponding author Lorenzo Crocco’s e-mail addresss [email protected].

EFERENCES AND NOTES1. L. Crocco, M. D’Urso, and T. Isernia, “Inverse scattering

from phaseless measurements of the total field on a closedcurve,” J. Opt. Soc. Am. A 21, 622–631 (2004).

2. T. Isernia, L. Crocco, and M. D’Urso, “New tools and seriesfor forward and inverse scattering problems in lossymedia,” IEEE Trans. Geosci. Remote Sens. Lett. 1, 331–337(2004).

3. R. F. Bloemenkamp, A. Abubakar, and P. van den Berg,“Inversion of experimental multi-frequency data using thecontrast source inversion method,” in special section“Testing Inversion Algorithm Against Experimental Data,”Inverse Probl. 17, 1611–1622 (2001).

4. Q. O. Liu, Z. Q. Zhang, T. T. Wang, J. A. Bryan, G. A.Ybarra, L. W. Nolte, and W. T. Joines, “Active microwaveimaging. I. 2-D forward and inverse scattering methods,”IEEE Trans. Antennas Propag. 50, 123–133 (2002).

5. S. Caorsi, M. Donelli, and A. Massa, “Detection, locationand imaging of multiple scatterers by means of theiterative multiscaling method,” IEEE Trans. MicrowaveTheory Tech. 52, 1217–1228 (2001).

6. W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distortedBorn iterative method,” IEEE Trans. Med. Imaging 9,218–225 (1999).

7. M. Bertero, “Linear inverse and ill-posed problems,” inAdvances in Electronics and Electron Physics (Academic,1989), Vol. 75, pp. 1–120.

8. O. M. Bucci and T. Isernia, “Electromagnetic inversescattering: retrievable information and measurementstrategies,” Radio Sci. 32, 2123–2138 (1997).

9. M. H. Maleki, A. J. Devaney, and A. Schatzberg,“Tomographic reconstruction from optical scatteredintensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).

0. M. H. Maleki and A. J. Devaney, “Phase-retrieval andintensity-only reconstruction algorithms from optical

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1

1

1

1

1

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2

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2

2

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2

2

3

3

3

3

3


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2. S. Caorsi, A. Massa, M. Pastorino, and A. Randazzo,“Electromagnetic detection of dielectric scatterers usingphaseless synthetic and real data and the memeticalgorithm,” IEEE Trans. Geosci. Remote Sens. 41,2745–2752 (2003).

3. T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, “Role ofthe support and zero locations in phase retrieval by aquadratic approach,” J. Opt. Soc. Am. A 16, 1845–1856(1999).

4. O. M. Bucci and G. Franceschetti, “On the spatialbandwidth of the scattered fields,” IEEE Trans. AntennasPropag. 35, 1445–1455 (1987).

5. O. M. Bucci and G. Franceschetti, “On the degrees offreedom of the scattered fields,” IEEE Trans. AntennasPropag. 37, 918–926 (1999).

6. O. M. Bucci, C. Gennarelli, and C. Savarese,“Representation of electromagnetic fields over arbitrarysurfaces by a finite and nonredundant number of samples,”IEEE Trans. Antennas Propag. 46, 351–359 (1998).

7. R. Snieder, “The role of nonlinearity in inverse problems,”Inverse Probl. 14, 387–404 (1998).

8. R. Potthast, “On a concept of uniqueness in inversescattering for a finite number of incident waves,” SIAM(Soc. Ind. Appl. Math.) J. Appl. Math. 58, 666–682 (1998).

9. I. Catapano, L. Crocco, and T. Isernia, “A simple two-dimensional inversion technique for imaging homogeneoustargets in stratified media,” Radio Sci. 39, RS1012, (2004).

0. D. Colton and R. Krees, Inverse Acoustic andElectromagnetic Scattering Theory (Springer-Verlag, 1992).

1. Strictly speaking, see Ref. 16; the Dirichlet polynomialDN�x�=sin��2N+1�x /2� / ��2N+1�sin�x /2�� should be used inEq. (3). However, this is really required only if theobservation domain totally encircles the source.

2. O. M. Bucci, A. Capozzoli, and G. D’Elia, “A novel effectiveapproach to scatterers localization problems,” IEEE Trans.Antennas Propag. 51, 2079–2090 (2003).

3. R. E. Kleinman and P. M. van den Berg, “An extendedmodified gradient technique for profile inversion,” RadioSci. 28, 877–884 (1993).

4. O. M. Bucci, N. Cardace, L. Crocco, and T. Isernia, “Degreeof nonlinearity and a new solution procedure in scalartwo-dimensional inverse scattering problems,” J. Opt. Soc.Am. A 18, 1832–1843 (2001).

5. L. Crocco, M. D’Urso, and T. Isernia, “Testing the contrastsource—extended Born inversion method against real data:the TM case,” in special section, “Testing InversionAlgorithm against Experimental Data: InhomogeneousTarget DATA 2005,” Inverse Probl. 21, S33–S50 (2005).

6. T. Isernia, V. Pascazio, and R. Pierri, “A non-linearestimation method in tomographic imaging,” IEEE Trans.Geosci. Remote Sens. 35, 910–923 (1997).

7. For an efficient exploitation of the fast-Fourier-transform(FFT) codes in the solution of Eq. (9), the first integer thatis a power of 2 not smaller than the needed number ofmeasures is considered in the numerical procedures. Notethat this convenient number is achieved by interpolatingthe measured independent samples via 2D FFT techniques.

8. W. C. Chew and J. L. Lin, “A frequency-hopping approachfor microwave imaging of large inhomogeneous bodies,”IEEE Microw. Guid. Wave Lett. 5, 439–441 (1995).

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0. L. Crocco and T. Isernia, “Inverse scattering with real data:detecting and imaging homogeneous dielectric objects,”Inverse Probl. 17, 1573–1583 (2001).

1. I. Catapano, L. Crocco, M. D’Urso, and T. Isernia,“Advances in microwave tomography: phaselessmeasurements and layered background,” in Proceedings ofthe 2nd International Workshop on Advanced GroundPenetrating Radar, A. Yarovoy, ed. (IEEE, 2003), pp.183–188.

2. N. Destouches, C. A. Guérin, M. Lequime, and H.Giovannini, “Determination of the phase of the diffractedfield in the optical domain. Application to thereconstruction of surface profiles,” Opt. Commun. 198,233–239 (2001).

3. P. C. Chaumet, K. Belkebir, and A. Sentenac,“Superresolution of three-dimensional optical imaging byuse of evanescent waves,” Opt. Lett. 29, 2740–2742 (2004).

4. I. Catapano, L. Crocco, M. D’Urso, and T. Isernia, “Faithfulphaseless microwave tomography,” in Progress inElectromagnetic Research (PIERS) 2006 CambridgeAbstracts (The Elecromagnetics Academy, 2006), p. 458.

inverse scattering from phaseless measurements of the total field on open lines

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