Reconstructing the contour of metallic planar objects from only intensity scattered field data over...

Reconstructing the contour of metallicplanar objects from only intensity

scattered field data over a single plane

F. Soldovieri†, R. Solimene‡, R. Pierri‡

† Istituto per il Rilevamento Elettromagnetico dell’Ambiente (IREA-CNR) Via Diocleziano328, 80124 Napoli, Italy

‡ Dipartimento di Ingegneria dell’Informazione, Seconda Universit di Napoli Via Roma 29,81031, Aversa, Italy

[email protected]

Abstract: The inverse scattering problem of recovering the contour ofplanar metallic scattering objects from only the amplitude of the scatteredfield is considered. A two step reconstruction procedure is proposed: first thephase of the scattered field is retrieved by solving a phase retrieval problem;then the objects’ supports are reconstructed from the retrieved scatteredfield. Differently form previous approaches, (see [11] for example), herethe amplitude of the scattered field is assumed known over a single plane innear zone but at two different frequencies. In this way, while the frequencydiversity increases the number of independent data, relevant for ensuringthe reliability of the phase retrieval stage, to perform measurements on asingle plane allows some practical advantages. Numerical results show theperformances achievable by the proposed reconstruction scheme with re-spect to the local minima problem and the stability against the noise on data.

© 2008 Optical Society of America

OCIS codes: (100.3200) Image reconstruction-restoration; (100.5070) Phase retrieval;(100.3190) Inverse problems.

References and links1. R. Pierri, A. Liseno, R. Solimene, and F. Soldovieri, “Beyond physical optics SVD shape reconstruction of

metallic cylinders,” IEEE Trans. Antennas Propag. 54, 655–665 (2006).2. R. E. Kleinman and P. M. van den Berg, “Two-dimensional location and shape reconstruction, Radio Sci. 29,

1157–1169 (1994).3. E. A. Marengo, F. K. Gruber, and F. Simonetti, “Time-Reversal MUSIC Imaging of Extended Targets, IEEE

Trans. Imag. Process. 16, 1967–1984 (2007).4. A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett.

62, 2385–2388 (1989).5. A. J. Devaney, “Diffraction tomographic reconstruction from intensity data,” IEEE Trans. Imag. Process. 1, 221–

228 (1991).6. M. H. Maleki, A. J. Devaney, and A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,”

J. Opt. Soc. Am. A 9, 1356–1363 (1992).7. T. Takenaka, D. J. N. Wall, H. Harada, and M. Tanaka, “Reconstruction algorithm of the refracrive index of

a cylindrical object from the intensity measurement of the total field,” Microw. Opt. Techn. Lett. 14, 182–188(1997).

8. L. Crocco, M. D’Urso, and T. Isernia, “Inverse scattering from phaseless measurements of the total field on aclosed curve,” J. Opt. Soc. Am. A 21, 622–631 (2004).

9. T. Isernia, G. Leone, and R. Pierri, “Phase retreival of radiated fields,” Inv. Probl. 11, 183–203 (1995).

#93867 - $15.00 USD Received 17 Mar 2008; revised 27 Apr 2008; accepted 14 May 2008; published 12 Jun 2008

(C) 2008 OSA 23 June 2008 / Vol. 16, No. 13 / OPTICS EXPRESS 9468

10. E. A. Marengo, R.D. Hernandez, and H. Lev-Ari, “Intensity-only signal-subspace-based imaging,” J. Opt. Soc.Am. A 24, 3619–3635 (2007).

11. F. Soldovieri and R. Pierri, “Shape reconstruction of metallic objects from intensity scattered field data only,”Opt. Lett. 33, 246–248 (2008).

12. G. Hislop, G. J. James, and A. Hellicar, “Phase retrieval of scattered fields,” IEEE Trans. Antenn. Propag. 55,2332–2341 (2007).

13. P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retreival using multiple illumination wavelengths,” Opt. Lett.33, 309–311 (2008).

14. G. Leone, R. Pierri, and F. Soldovieri, “Reconstruction of complex signals from intensities of Fourier-transformpairs,” J. Opt. Soc. Am. A 13, 1546–1556 (1996).

15. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley and Sons, Hoboken, NJ, 1989).16. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Field (Wiley-IEEE Press, Hobo-

ken, NJ, 1996).17. I. Sabba Sefanescu, “On the phase retrieval problem in two dimensions,” J. Math. Phys. 26, 2141–2160 (1985).18. T. Isernia, G. Leone, and R. Pierri, “Phaseless near field techniques: uniqueness condition and attainment of the

solution,” J. Elettromagn. Waves Appl. 8, 889–908 (1994).19. O. M. Bucci, C. Gennarelli, and C. Savarese, “Representation of electromagnetic fields over arbitrary surfaces

by finite and nonredundant number of samples,” IEEE Trans. Ant. Prop. 46, 351–359 (1998).20. A. Requicha, “The zeros of entire functions: theory and engineering applications,” Proc. of IEEE 68, 308–328

(1980).21. R. Pierri and F. Soldovieri, “On the information content of the radiated fields in the near zone over buonded

domains,” Inv. Prob. 14, 321–337 (1998).22. D. Luenberger, Linear and Nonlinear Programming. Reading, (Addison-Wesley, 1987).23. R. Pierri, G. D’Elia, and F. Soldovieri, “A two probes scanning phaseless near-field far-field transformation

technique,” IEEE Trans. Ant. Prop. 47, 792–802 (1999).

1. Introduction

Inverse scattering problems amount to recovering scattering objects (in terms of their positions,shape and possibly physical parameters) by scattered field measurements. Such problems arerelevant in all those applicative contexts where it is convenient or one is forced to inspect with-out physically penetrating inside the object under test. A few of examples are remote sensing,non-destructive diagnostic and microwave tomography.

In this framework many inversion methods have been developed ranging from linearized ap-proaches [1], to non-linear inversion schemes which perform the reconstruction by minimizingcost functionals accounting for the mismatch between the data and the model [2], and morerecently, to the so-called qualitative methods, which still in the framework of non-approximatemodels for the scattering, aim at reconstructing the scatterers’ support by means of certainindicator functions [3] 1.

As is well known, inverse scattering problems suffer from ill-posedness and non-linearity.Moreover, in some situations, one also has to tackle the further difficulty that only phaselessdata are available. This occurs when frequency grows up and the field phase can be over-whelmed by noise or could require too expensive measurement setup.

Several contributions can be found in the literature which address inverse scattering prob-lems from phaseless data. Some of the pioneering contributions in the case of weak scatteringobjects are reported, for example, in [4] and [5]. In particular, in [4], under the Born model, thescattering amplitude is approximated in terms of the total field amplitude and the reconstructionis obtained by backpropagating such an approximate scattering amplitude. In this case, it wasshown that the reconstruction so obtained are affected by an error term which becomes negligi-ble, within the object spatial domain, if the scattering amplitude is collected in far zone. A sim-ilar procedure is also presented in [5] where the Rytov approximation is employed. Moreover,

1Note that references [1], [2] and [3] are just few examples taken from the very large inverse scattering literature.To give a more complete account of the literature and of the different methods is not within the schedule of the presentpaper.



in [6] a hybrid approach is presented to deal with the breakdown of the Rytov approximation infar-field zone.

A more general approach, in the sense that it can deal with scatterers beyond the linear scat-tering regimes, is presented in [7] where the solution is searched for by a single step inversionprocedure consisting of the minimization of a suitable cost functional accounting for the mis-match between the intensity-only data and the the ones predicted by the model acting on anestimated scattering object.

Alternatively, in [8], a two step inversion procedure is proposed. First, the scattered field is re-trieved (both in amplitude and in phase) from the square amplitude of the total field (scatteredplus incident fields). Then, starting from the estimated scattered field a further optimizationstage leads to the reconstruction. This approach has the advantage that splitting the reconstruc-tion procedure in two steps allows one to better control the overall non-linearity of the problemas compared to the single step procedure. In particular, the first step amounts to a phase retrievalproblem which takes advantage from the presence of an interfering term (between the scatteredand the incident fields) which increases the data dimensionality and thus increase the reliabilityof solution against local minima [9]. Finally, it is worth mentioning the very recent approachpresented in [10] where a signal sub-space based method is presented.

When the available data concerns the amplitude of the scattered field, the phase retrievalproblem becomes more challenging with respect to the case of total field amplitude data [11,12]. In fact, in this case, measurements under different scattering conditions are required toincrease the ratio between the number of data and the number of unknowns in order to obtainreliable reconstructions. In particular, in [11] and [12] a two step reconstruction procedure, asdiscussed in connection to the work in [8], is adopted where the phase retrieval stage is tackledby collecting the scattering amplitude over two different planar bounded measurement domainin the near zone of the scatterers.

In this paper, we are concerned with the inverse scattering problem of reconstructing the con-tour of metallic planar objects from the scattered field amplitude measurements. In particular,as previously discussed, we adopt a two step procedure where first the phase of the scatteredfield is retrieved. However, at variance of all the above mentioned contributions where a single-frequency measurement configuration is employed, here, instead, we adopt frequency diversity.More in detail, in order to accommodate the need of independent data for avoiding local min-ima problem in the phase retrieval stage we collect the scattered field amplitude over a singleplanar bounded measurement domain at two different working frequencies.

As is shown afterwards, the problem can be cast as done in [11] so it is expected the recon-struction algorithm shares the same features in terms of local minima and achievable perfor-mances of the method reported in [11]. However, as now the measurements are taken over asingle domain the proposed technique has the advantage of requiring a more practical and lessexpensive measurement set up.

It is worth mentioning that phase retrieval using multiple illumination wavelengths has re-cently presented in [13]. In such a paper, a successive alternating projections technique is em-ployed and the wavelength is changed after each couple of projections by converting the phaseof the field obtained at a given wavelength at the object plane for the next employed wavelength.The algorithm was tested for 20 different wavelengths.

The method we propose herein, instead, is developed, and is shown to work, for only twodifferent frequencies (even though it can be easily generalized for more frequencies) which areexploited simultaneously for phase retrieval. More in detail, the phase retrieval problem is castas the inversion of a quadratic non-linear operator by minimizing a cost functional accountingfor the square amplitude of the scattered field as data of the problem. This formulation hasallowed a thorough analysis of the effect of the parameters of the problem on the local minima



problem [9, 14]. In particular, it has been pointed out how the increase in the ratio betweenthe number of independent data and the number of unknowns allows us to achieve a favorableeffect on the local minima problem.

The paper is organized as follows. Section 2 is devoted to present the formulation of theinverse scattering problem and to the description of the inversion algorithm. In particular, as weare interested in metallic planar scatterers, the Physical Optics approximation will be adoptedin setting the mathematical relationship linking the scatterers to their scattered field. In Section3, details concerning the implementation of the inversion procedure emphasizing the requireddiscretization related on the representation for the unknown and the data sets are given. Finally,a numerical analysis which outlines the feasibility and reliability of the proposed algorithm isprovided in Section 4. Conclusions follow.

2. Formulation of the problem

We are concerned with the problem of determining the contour of planar metallic scatterersfrom the knowledge of only the amplitude of their scattered field collected over a boundedplanar domain in near-field zone.

More in detail, we consider the following scattering configuration. The planar metallic ob-jects reside within the so-called object-aperture O = [−XO,XO]× [−YO,YO] which is a subsetof the x−y plane located at z = 0. Whereas the amplitude of the scattered field is collected overa single planar measurement-domain D = [−XM,XM]× [−YM,YM] in the x− y plane at z = z1.As incident field we consider a normally impinging plane wave (i.e., along −z direction) at twodifferent frequencies, say f1 and f2, with f2 > f1. The background medium is assumed to bethe free-space whose dielectric permittivity and magnetic permeability are denoted by ε 0 andμ0, respectively.

Accordingly, say Γ ⊆ O the scatterers’ supports (not necessarily connected in the case ofmore than one scatterers) and UΓ the corresponding characteristic function, that is

UΓ(r′) ={

1 f or r′ ∈ Γ0 otherwise

. (1)

then the problem can be rigorously stated as follows: retrieve UΓ from the amplitude of the scat-tered field M1 and M2 collected over the measurement aperture D at two different frequenciesf1 and f2.

In setting up the approach, first we need the mathematical relationship between U Γ and Mi,with i ∈ (1,2). Hence, we start by considering the expression of the scattered field which arisesfrom the interaction between the incident plane wave and the metallic planar scatterers.

As aforementioned, the incident field is a plane wave of the type E inc = Einc exp( jkiz)ix, withki = 2π fi

√ε0μ0 being the wavenumber of the free-space at the frequency f i. Note that the timedependence exp( j2π f it) has been assumed and suppressed.

As the scatterers we are considering are metallic and planar their scattered field can be wellexpressed by adopting the Physical Optics (PO) approximation [15]. By doing so, then, thescattered field can be written as

ES(r, fi) = − j2π fiμ0

∫O

G(r− r′, fi)JPO(r′, fi)dr′, (2)

where r is the field point within the observation aperture D , G(r− r ′, fi) is the dyadic Green’sfunction and JPO(r′, fi) is the current induced over the scattering objects under the PO approx-imation. In particular,

JPO(r′, fi) = 2in ×Hinc(r′, fi)UΓ(r′) =

2ζ

iz ×EincUΓ(r′)iy =2ζ

EincUΓ(r′)ix, (3)



where ζ =√

μ0/ε0 is the free-space characteristic impedance, r ′ ranges within O and in is theunitary vector normal to the scatterers’ surface, which for the case at hand coincides with iz.

Hereinafter, we choose to exploit only the x-component of the scattered field. Hence, theproblem turns to be scalar. For such a case, by exploiting Eq. (3), Eq. (2) simplifies as

ES(r, fi) = − j4π fiμ0

ζ

∫O

Gxx(r− r′, fi)EincUΓ(r′)dr′, (4)

where for simplicity of notation we have denoted the x-component of the scattered field still asES but discarding the vectorial notation.

By resorting to the spectral representation of the Green’s function G xx(r − r′, fi) [16] thescattered field over the measurement aperture D at z = z1 can be rewritten as

ES(x,y,z1, fi) = (LiJ)(x,y,z1, fi) = Fi

∫ ∞

−∞

∫ ∞

−∞

v2 + w2i

wi×

exp(− jux− jvy− jwiz1)UΓ(u,v)dudv, (5)

where u, v and wi are the spectral variables conjugated of those spatial x, y and z, respectively,

with wi =√

k2i −u2 − v2, ki = 2π/λi and λi are the free-space wavenumber and the correspond-

ing wavelength, respectively, at the frequency f i, Fi = 14π2ki

and UΓ is the Fourier transform ofthe induced current support

U(u,v) = FUΓ =∫ ∞

−∞

∫ ∞

−∞EincUΓ(x′,y′)exp( jux′ + jvy′)dx′dy′, (6)

with F being the two dimensional Fourier transform operator.Now, starting from the amplitude of the scattered field over D at the two frequencies f 1 and

f2 we aim at reconstructing the function UΓ. To this end, a reconstruction procedure in twosteps is adopted. In the first step we retrieve UΓ by the data M1 and M2. Then, UΓ is retrievedfrom UΓ.

It is clear that the first step of the reconstruction procedure consists in solving a phase re-trieval problem. Such a problem has been extensively studied and many approaches are presentin the literature. Here, we adopt an approach similar to the one already exploited for the phaseretrieval problem in antenna diagnostics and, very recently, in inverse scattering problem [11].

Accordingly, in the first step the problem is cast as the inversion for UΓ of the followingnon-linear operator

A : UΓ → (M21 ,M2

2) = (|L1UΓ|2, |L2UΓ|2), (7)

achieved by searching for the global minimum of the cost functional

Φ(UΓ) = ‖(M21 ,M2

2 )− (|L1UΓ|2, |L2UΓ|2)‖2 = ‖|L1UΓ|2 −M21‖2 +‖|L2UΓ|2 −M2

2‖2. (8)

where ‖ · ‖ is the usual quadratic norm in the data space.Note that in eqs. (7) and (8) the constant Einc has been assumed equal to one and the square

amplitude of the scattered field M21 and M2

2 have been considered as data instead of M1 and M2.Such a choice offers the advantage to deal with the inversion of a non-linear quadratic opera-

tor for which some conclusions have been derived to overcome the problems of false solutionscorresponding to the local minima of the non-quadratic functional (8) within which the min-imizing procedure can be trapped. More in detail, based on the geometrical features of the



mapping (7), it was shown in a number of papers (see [9] fore example) that the problem oflocal minima can be avoided if an adequate ratio between independent data and unknowns isavailable. This circumstance can be meet by assuming the knowledge of the scattered field am-plitude in more than one measurement condition. In the paper by Soldovieri and Pierri [11] thiswas achieved by collecting the scattered field amplitude data at a single frequency over twodifferent measurement apertures still located in the near-field zone of the scatterers.

Here, the role of the two set of data collected over two different planar domains is played bythe two set of data acquired on a single measurement domain but at two different frequencies.

This is made it possible as the the unknown of the problem is the support function U Γ whichdoes not depend on the frequency whereas the way the induced surface current depends onfrequency is a priori known due to the adopted PO approximation. In particular, for the config-uration considered in this paper, the induced surface current does not depend on the frequency(as is evident from Eq. (3)). Accordingly, observations (of the scattered field intensity) at dif-ferent frequencies can be exploited for conveying an increased amount of information on theunknown, and simultaneously adopted in the reconstruction procedure. From a mathematicalpoint of view, this leads to the the inversion of an operator similar to the one in [11] but now thediversity between the two set of intensity data is achieved by exploiting the frequency diversity.

In other words, for the case considered herein, diversity between the two sets of intensitydata stems from the differences in the kernels of L1 and L2, that is from the exponential termsexp [− jz1(w2 −w1)], whereas in the approach presented in [11] this same role was played bythe exponential term exp [− jw(z2 − z1)], with w defined as above but at the single employedfrequency, and z2 and z1 were the quotas of the two planar measurement domains.

Therefore, since exp [− jz1(w2 −w1)] has unit modulus (in the visible domain), the possibilityto achieve diversity between the two sets of near field data is related to the variability of its

phase. By rewriting such a term as exp [− j2πz1

(1

λ2− 1

λ1

)(w2 −w1)], with λ1 and λ2 being the

wavelengths at f1 and f2 respectively, it is evident the positive role of the increase in the spacingbetween f1 and f2. Indeed, as f1 and f2 are more far apart, the phase term is increasingly varyingin the spectral plane and more different the two sets of the intensity data are.

It is worth recalling that, the possibility of exploiting intensity data collected over two dif-ferent apertures, and equivalently for two different frequencies in this case, is also helpful inresolving the intrinsic ambiguities of the phase retrieval problem [17], aside from very particu-lar (for instance symmetric) situations. This, indeed, results from the further constraint providedby the knowledge of a second set of intensity data (over a different aperture or at a differentfrequency) to which the solution has to obey [18].

Therefore, the proposed two frequencies approach shares the main advantages of the methodadopted in [11] as to the local minima problem and phase ambiguities. In addition it also allowsone to adopt a more practical and convenient configuration as data over a single aperture, eventhough at two different frequencies, are required.

Once UΓ has been retrieved, the current support is obtained by passing UΓ through the adjointof the Fourier operator F , that is

UΓ(x,y) = F †UΓ. (9)

UΓ being the reconstructed version of UΓ.The implementation of the reconstruction algorithm and the details related to the discretiza-

tion of Eq. (8) are discussed in next section.



3. The solution algorithm

In this section we aim at describing the discretizing procedure upon which the the reconstruc-tion scheme rely on.

Accordingly, the phase retrieval stage is achieved by minimizing a suitably discretized ver-sion of the functional (8), that is

Φ(UΓ) = ‖(M21,M

22)− (|L

1UΓ|2, |L 2

UΓ|2)‖2 = ‖|L1UΓ|2 − M2

1‖2 +‖|L2UΓ|2 − M2

2‖2.(10)

with UΓ, M21 and M2

2 being finite dimensional (discretized) version of the corresponding en-tities and L

1and L

2are the propagator operators at the two different frequencies as in (5)

discretized according to the representation adopted for the data and the unknown. Note, thatM

21 and M

22 are corrupted (by measurement errors and noise) version of M 2

1 and M22 .

As is known, while numerically implementing an algorithm for minimizing Eq. (10), data andunknown representations play a crucial role in ensuring the reliability of the solution. Indeed,the ratio between the number of data and the number of unknowns is strongly related to theoccurrence of false solutions. Moreover, due to the ill-posedness of the problem a judiciouschoice of such representations helps in restoring the stability of the solution.

To this end, mathematical features of the involved entities have to adopted. However, optimalnonredundant representation schemes [19], that exploit precisely the mathematical features andthe available a priori information, not necessarily are simple to be implemented. Therefore, herewe choose a driscretizing scheme which exploit the a priori information as guide line makingsafe the possibility to implement the reconstruction algorithm easily.

In order to numerically implement the solution algorithm the first choice concerns the rep-resentation of the unknown function UΓ. In this case the a priori knowledge we have says usthat UΓ, being the Fourier transform of a function compactly supported over O , is an entireband-limited function [20]. Accordingly, UΓ can be expanded in a sampling series as

UΓ(u,v) =N

∑−N

M

∑−M

UΓnmsinc(uXO −nπ)sinc(vYO−mπ), (11)

where UΓnm are the samples of UΓ taken at the Nyquist rate, the truncation indexes N and M arechosen as explained below and sinc(x), as usual, is defined as

sinc(x) =sinx

x. (12)

In particular, as the measurement aperture D is chosen some wavelengths apart from theaperture O where the scatterers reside, it can be argued that the integration appearing withinthe propagator operators L1 and L2 can be limited to only the visible part of the plane wavespectrum as evanescent waves give rise to an exponentially decaying (with z) contribution.Therefore, the truncation indexes are chosen so that

(nπ/XO)2 +(mπ/YO)2 ≤ k22 ∀(n,m) : n ∈ (−N,N),m ∈ (−M,M). (13)

Choice in Eq. (15) entails that the spectrum samples searched for as actual unknowns arethe ones belonging to the visible domain attained at the maximum adopted frequency. This, onone side allows us to implicitly regularize the problem. On the other side this choice fixes theachievable resolution while, from UΓ, UΓ is reconstructed by means of (9).

It can be also argued that, fixed N and M, the best way (in terms of the norm of the represen-tation error) to represent a band-limited function projected over a compact support (in our case



the circle of radius k2 in the u− v plane) is by means of a finite series of the singular functionsof the operator PVCFPO , where F is defined as above and PO and PVC are projector operatorsover the investigation domain O and the circular visible domain, respectively. However, therepresentation adopted in Eq. (11) is useful to achieve a very efficient implementation of thesolution algorithm as such a choice enables most of the operations to be performed thanks tothe FFT technique.

As we have assumed that evanescent waves are negligible then the scattered field, as functionof x and y, can be considered an almost (related to the actual negligibility of the evanescentpart) bandlimited function whose band is enclosed in the square [−k i,ki]× [−ki,ki] (for eachadopted frequency). Accordingly, the scattered field square amplitude has a bandwidth whichis enclosed in the square [−2ki,2ki]× [−2ki,2ki] and this suggest to acquire data at the Nyquistrate π/(2ki) = λi/4 along x and y.

This representation may results redundant. It has been shown that exploiting the a prioriinformation concerning the scatterer dimension allows to reduce the number of measurementsbut generally require a nonuniform sampling step in x and y [19, 21]. Therefore, once again,our choice simplifies the implementation operation as it allows for an heavy use of the FFTalgorithm.

As far as the minimization of the functional in (10) is concerned, an iterative procedure basedon the Polak-Ribiere method [22] is applied which requires the computation of the gradient ofthe cost function (10) with respect to the real and imaginary parts of the Fourier coefficientsUΓnm to evaluate the updating direction in the minimizing procedure.

As mentioned, the increase of the ratio between the number of independent data and thedimension of the unknown space has a favorable effect on the local minima problem. Thus, forthe test cases presented below, we adopt a minimization strategy based on L minimization stepswhere the accuracy of the solution is improved by gradually increasing the number of unknowns[23]. In particular, at the l-th step we consider N = 2l +1 and M = 2l +1. The obtained solutionis then used as starting point in the next step by considering N = 2(l + 1)+ 1 and M = 2(l +1)+ 1. This procedure continues until all the samples falling within the visible domain at thehigher frequency, as dictated by Eq. (15), are considered. Accordingly, a resolution of λ 2/2,along both x and y, is expected while reconstructing UΓ.

Finally, the result can be further improved by adopting the weighted formulation [14]

Ψ(UΓ) = ‖(|L1UΓ|2 − M

21)./(M2

1 + η)‖2 +‖(|L2UΓ|2 − M

22)./(M2

2 + η)‖2, (14)

where c = a./b means a vector whose components are given as c n = an/bn and η is factoravoiding division by zero.

After that the phase retrieval problem has been solved and an estimate of the characteristicfunction is obtained, the shape of the metallic plate is determined as the region where thereconstruction is significantly different from zero. To this end, we introduce a threshold underwhich the reconstruction is discarded. In particular, the threshold is chosen at −13dB below themaximum of the reconstruction which corresponds to the sidelobe level of the expected pointspread function.

4. Numerical analysis

This section is devoted to present the feasibility of the solution procedure by considering twotest cases concerned with different shapes of the metallic planar scatterers.

For both the test cases, we consider two frequencies such that f 2 = 2 f1, a measurement aper-ture D = [−16λ2,16λ2]× [−16λ2,16λ2] located at the quota z1 = 5λ2 and an object-apertureO = [−15λ2,15λ2]× [−15λ2,15λ2]. Accordingly, the scattered field data at the lower frequency



x/λ2

y/λ 2

a

−10 0 10

−10

0

10

−10 0 10

−10

0

10

d

x/λ2

y/λ 2

−10 0 10

−10

0

10

c

x/λ2

y/λ 2

−10 0 10

−10

0

10

b

x/λ2

y/λ 2

−10

−5

0

−10

−5

0

−10

−5

0

−10

−5

0

Fig. 1. a: actual scatterer. b: reconstruction obtained by considering up to (2N1 +1)(2M1 +1) unknown coefficients for theUΓ representation in the minimization of the cost functional(10). c: reconstruction obtained by enlarging the number of unknown coefficients up to(2N +1)(2M+1). d: reconstruction obtained by simultaneously searching for all the (2N +1)(2M + 1) unknowns by first minimizing the cost functional (10) and then adopting theresults as starting point for minimizing the weighted functional (14). Results are shown indB scale.

f1 consist of 64× 64 measurements taken λ2/2 apart each other both along x and y, while forthe frequency f2 we have 128×128 measurements spaced by λ2/4.

In all the cases the starting point of the minimization procedure is assumed in a random way.The first test case refers to a square plate scatterer whose support is given by Γ =

[−5λ2,5λ2]× [−5λ2,5λ2] (see panel a of Fig. 1). The corresponding reconstructions are re-ported in panels b, c and d of the same figure whereas the behavior of the cost functional Φ (asgiven by Eq. (10)) normalized to the square norm data, that is Φ(UΓ)/‖(M2

1,M22)‖2, is shown as

a blue line in Fig. 2 as the minimization procedure evolves. Markers ∗ denote the transition be-tween two successive minimization steps, namely when the minimization employing N = 2l+1and M = 2l +1 arrests and the one with an enlarged number of unknowns N = 2(l +1)+1 andM = 2(l +1)+1 starts. In particular, panel b of Fig. 1 refers to the case in which the minimiza-tion procedure considers up to the unknown coefficients UΓnm laying within the visible domainrelated to the frequency f1, that is

(nπ/XO)2 +(mπ/YO)2 ≤ k21 ∀(n,m) : n ∈ (−N1,N1),m ∈ (−M1,M1). (15)

Instead, in panel c of the same figure the reconstruction have been obtained by enlarging thenumber of unknowns so as to consider all the visible part dictated by f 2. As can be seen, even



0 200 400 600 800 1000 1200 1400 1600 1800−80

−60

−40

−20

0

20

40

60

80

100

Iteration number

dB

Fig. 2. Behavior of the error in the data space defined as Φ(UΓ)/‖(M21,M

22)‖2. Blue line

refers to the case reported in Fig. 1 whereas red line refers to the case reported in Fig. 3.Markers ∗ denotes the transition by two different minimization steps accordingly to theprogressive increasing of the searched for number of unknown coefficients.

though the reconstruction error quantified as ‖UΓ − UΓ‖2/‖UΓ‖2 is not particularly low (forexample in the case of panel c it is equal to 0.12) the reconstruction results allow us to correctlydetermine the location and the geometry of the metallic plate.

The advantage offered by the adopted minimization procedure against false solutions is madeit evident by comparing the reconstructions reported in panels c and d of Fig. 1. In fact, thereconstruction shown in panel d have been obtained by a single minimization of (10) by si-multaneously searching for all the 2N × 2M unknowns. In particular, in order to improve thereconstruction, the procedure have been followed by a weighted minimization according to thefunctional in Eq. (14) (instead for the case of panel c such further step did not give a relevantimprovement and hence is not reported here). In this case, the error in the data space evaluatedas for previous case, when the procedure arrests returns −34.7 dB which is much higher thanthe one obtained for the case reported in panel c (see blue line of Fig. 2). Indeed, due to thereduced ration between the number of data and unknowns the procedure was not able to achievethe true solution.

Let us turn to consider the second example. It refers to a metallic object consisting of a plateas in the previous case but, now, there is a hole at the center of side 4λ 2 (see Fig. 3, panel a).

The same analysis as in Fig. 1 has been performed and the reconstructions are shown in Fig.3, where the various panels are the analogues of the ones in Fig. 1. Also in this more criticalcase (note, by looking at the red line of Fig. 2, that when the minimization procedure arrests theresidual error in the data space is higher than the one of the previous case) the reconstructionprocedure works very well in retrieving the scatterer’s contour and we can appreciate the favor-able effect of the strategy based on the progressive enlargement of the unknowns to be searched



x/λ2

y/λ 2

a

−10 0 10

−10

0

10

−10 0 10

−10

0

10

d

x/λ2

y/λ 2

−10 0 10

−10

0

10

c

x/λ2

y/λ 2

−10 0 10

−10

0

10

b

x/λ2

y/λ 2

−10

−5

0

−10

−5

0

−10

−5

0

−10

−5

0

Fig. 3. a: actual scatterer. b: reconstruction obtained by considering up to (2N1 +1)(2M1 +1) unknown coefficients for theUΓ representation in the minimization of the cost functional(10). c: reconstruction obtained by enlarging the number of unknown coefficients up to(2N +1)(2M+1). d: reconstruction obtained by simultaneously searching for all the (2N +1)(2M + 1) unknowns by first minimizing the cost functional (10) and then adopting theresults as starting point for minimizing the weighted functional (14). Results are shown indB scale.

for on the reliability and accuracy of the retrieved solution.Finally, we verify the stability of the reconstruction procedure against noisy data. In Fig.

4 the reconstruction obtained by intensity data corrupted by an additive 10% uniformly dis-tributed noise is shown for the case reported in Fig. 1 (panel c). As expected in this case thereconstruction error increases (now ‖UΓ − UΓ‖2/‖UΓ‖2 returns 0.24 whereas in the previousexample it was 0.12). Nevertheless, the scatterer’s contour is retrieved and the reconstruction isvery similar to corresponding reconstruction reported in Fig. 1.

As concluding remarks we outline that the same performances in terms of reconstructionerror can be obtained by the single-frequency two plane configuration [11]. In fact, the recon-structions reported in this paper and those shown in [11] are very similar. However, we onceagain stress the practical advantages offered by the two-frequency single plane configurationaddressed herein.

5. Conclusion

The problem of reconstructing the support UΓ of metallic planar scatterers which are known toreside within the object aperture from the amplitude of their scattered field has been addressed.A two step inversion procedure has been presented. First, a phase retrieval problem has solved



x/λ2

y/λ 2

−15 −10 −5 0 5 10 15

−15

−10

−5

0

5

10

15

−12

−10

−8

−6

−4

−2

0

Fig. 4. Reconstruction for the same scatterer and for the same situation as in Fig. 1 panelc but data have been corrupted by an additive 10% uniformly distributed noise. Result isshown in dB scale.

to retrieve (both in amplitude and phase) the Fourier transform UΓ of the support function UΓ.Then, UΓ has been determined through F †UΓ. The phase retrieval stage has taken advantagefrom considering as data the square amplitude of the scattered field which has allowed to castthe problem as the inversion of quadratic operator whose features concerning the occurrenceof local minima has been well understood [9]. In particular, it is known that to avoid falsesolutions an adequate ratio of the number of data over the number of the unknowns is required.To this end, previous methods relying on a single frequency measurement set up [14, 12] needthe data to be collected over multiple measurement domains. Here, instead, we have consideredthe amplitude data acquired over a single measurement domain and two different frequencieshave been adopted to increase the number of independent data. By doing so we have obtainedthe advantage to deal with a more practical and less expensive measurement set up.

Numerical examples which have been reported testified the good performance achievable bythe proposed reconstruction method also against the case of data corrupted by noise.

Finally, it worth stressing that we have considered a canonical situation where measurementsare taken over a plane parallel to the target plane. This condition can be difficult to maintainexperimentally, specially in the optical regime. Therefore, as a future developments it could beinteresting to investigate tilting effect of the measurement plane on the reconstructed results.



Reconstructing the contour of metallic planar objects from only intensity scattered field data over...

Documents

Transcript of Reconstructing the contour of metallic planar objects from only intensity scattered field data over...