sparse reconstruction from gpr data with applications to rebar detection

1070 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 60, NO. 3, MARCH 2011

Sparse Reconstruction From GPR Data WithApplications to Rebar Detection

Francesco Soldovieri, Member, IEEE, Raffaele Solimene, Lorenzo Lo Monte, Member, IEEE,Massimo Bavusi, and Antonio Loperte

Abstract—The problem of detecting and localizing 2-D thinscatterers (i.e., elongated scatterers whose cross sections are smallin terms of the probing wavelength) from scattered field mea-surements is considered. To this end, a linear model that neglectsmutual scattering and is based on a distributional representationof the unknown is established. An improved imaging techniquebased on a minimization algorithm, which takes advantage of theinherent sparseness of the considered ground-penetrating radarscenario, is presented and compared to a classical migration al-gorithm. The comparison is achieved for both synthetically gener-ated and experimental data collected in realistic conditions undera multimonostatic/multifrequency configuration.

Index Terms—Experiments in realistic conditions, ground-penetrating radar (GPR), intrawall imaging, inverse scattering.

I. INTRODUCTION

THE suitability of ground-penetrating radar (GPR) as adiagnostic tool in civil engineering, cultural heritage mon-

itoring, and other research areas related to nondestructive diag-nostics is now well assessed as shown by a larger number ofworks present in the scientific literature [1].

In this framework, we focus the attention on rebar detectionand localization in concrete structures. This is an application ofsignificant interest in diagnostic operations not only to assessthe structural stability of concrete structures [2] but also toinspect them after disaster events (e.g., earthquakes) [3].

As well known, this task entails dealing with a nonlinearinverse scattering problem [4]. However, to avoid the draw-backs that affect the nonlinear inversion schemes, we adoptan approximate linear model to describe the scattering phe-nomenon. The corresponding reconstruction algorithms offerrelevant advantages in terms of the computational effectiveness

Manuscript received January 31, 2010; revised May 30, 2010; acceptedJuly 19, 2010. Date of publication October 28, 2010; date of current versionFebruary 9, 2011. The research leading to these results has received fundingfrom the European Community’s Seventh Framework Programme (FP7/2007-2013) under Grant Agreement n◦ 225663 Joint Call FP7-ICT-SEC-2007-1.The Associate Editor coordinating the review process for this paper wasDr. Sergey Kharkovsky.

F. Soldovieri is with the Institute for Electromagnetic Sensing of the En-vironment (IREA), Italian National Research Council (CNR), 80124 Naples,Italy.

R. Solimene is with the Department of Information Engineering, SecondUniversity of Naples, 81031 Aversa, Italy.

L. Lo Monte is with the General Dynamics Information Technology, 5100Springfield Pike, Dayton, OH 45431 USA.

M. Bavusi and A. Loperte are with the Institute of Methodologies forEnvironmental Analysis (IMAA), Italian National Research Council (CNR),85050 Tito Scalo, Italy.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2010.2078310

and reliability and allow us to detect the scatterers even whennonlinearity terms are not negligible [5].

Accordingly, the linear model is established as follows. First,mutual scattering among the different scatterers is neglected.Then, assuming that the scatterers are thin, i.e., infinitely longand with smaller cross sections as compared to the probingwavelength, we set up a model that explicitly accounts for thisfeature. Indeed, a distributional unknown is introduced, whichis only supported over the scatterers cross section centers.Hence, the problem is cast as the inversion of a linear integraloperator acting over a distributional unknown [5]. Many linearinversion algorithms are available in the literature, which rangefrom migration [6] to inverse filtering (e.g., Tikhonov regular-ization, truncated-singular value decomposition) [7], [8].

Here, we propose a data processing algorithm that fallswithin the framework of the sparse minimization; this class ofapproaches is gaining an increasing interest in various applica-tive fields, including inverse scattering and radar imaging [9]–[12]. It is particularly suited for our problem since it accountsfor the sparse nature of the scatterers in the solution domain. Infact, the typical scenario in a rebar localization problem consistsof localized and isolated targets.

The new reconstruction scheme is compared with a classicaldiffraction summation migration algorithm [8] by using bothsynthetic and experimental data.

The experimental data are collected thanks to high-frequencyGPR for diagnostics of a fractured joist of a public buildingwithin the operations for the monitoring and assessment of thepublic infrastructures damaged during the L’Aquila earthquakein April 2009 [3].

II. MATHEMATICAL MODEL

In this section, we establish the linear model for the scatter-ing phenomenon upon which the reconstruction procedures arebased.

To this end, we consider a two-layered background mediumwhere the upper layer is free space, with dielectric permittivityand magnetic permeability denoted as ε0 and μ0, respectively,The lower one has dielectric permittivity and electric conduc-tivity given by εb and σb and magnetic permeability equals theone of the free space (see Fig. 1).

Let us consider as scatterers N perfectly conducting infi-nitely long cylinders buried in the lower half-space. They havecircular cross sections of radii an, with n ∈ (1, 2, . . . , N), theiraxes are parallel to the y-axis and their centers are located atthe points rn. The buried scenario is probed by a source located

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SOLDOVIERI et al.: SPARSE RECONSTRUCTION FROM GPR DATA WITH APPLICATIONS TO REBAR DETECTION 1071

Fig. 1. Geometry of the problem.

at the separation interface. In particular, a 2-D filamentarycurrent directed along the y-axis, which radiates within thefrequency band ω ∈ [ωmin, ωmax], is considered. The scatteredfield is collected at the same position as the source whilethe latter moves. Therefore, a multimonostatic/multifrequencyconfiguration within a 2-D and scalar geometry is considered.

Let us denote as rO = (xO, 0) the generic source/measurement position and as Σ = [−xM , xM ] the synthesizedmeasurement line (of course rO ∈ Σ). Accordingly, the fieldimpinging in lower half-space can be conveniently expressed interms of the plane wave spectrum representation as [4]

Einc(x, z, xO, ω) = −ωμ0I

2π

kb∫−kb

exp(−jwbz)w0 + wb

× exp [−ju(x − xO)] du (1)

where r = (x, z) is the field point, I is the current amplitude,u is the spectral variable conjugated of the spatial coordinatex, wi =

√k2

i − u2, with i ∈ (0, b), and ki denotes the wave-number relative to the layer that it refers to. Note that in (1), theevanescent part of the spectrum has been discarded as it givesnegligible contribution for scatterers located at more than one-half wavelength from the source.

It is well known that the inversion of scattering data requiressolving a nonlinear inverse problem whose nonlinearity mainlyarises from mutual scattering between different scatterers [4].Therefore, to avoid drawbacks related to nonlinear inversiona linear model can be established. The first step toward sucha task is to neglect mutual scattering between the cylinders.Accordingly, the scattered field can be expressed by simplysuperimposing the field scattered by each of them as they werescattering alone, i.e.,

ES(xO, ω) =N∑

n=1

ESn(xO, ω) (2)

with ESn(·) being the field scattered by the nth scatterer.To find ESn(·), it is helpful to rewrite (1) as

Einc(x, z, xO, ω) = −ωμ0I

2π

kb∫−kb

exp(−jwbzn)w0 + wb

× exp [−ju(xn − xO)] exp [−j(ux′ + wbz′)] du (3)

where x′ = x − xn and z′ = z − zn are the field point coordi-nates in terms of the nth scatterer local reference frame centeredat the nth cylinder center rn = (xn, zn). Now, ESn(·) can befound by forcing the boundary condition on the nth scatterer’scontour, i.e.,

Einc(x′, z′, xO, ω) = −ESn(x′, z′, xO, ω) (4)

over the circle of radius equal to an.Now, by expanding the exponential term exp[−j(ux′ +

wbz′)] of (3) in cylindrical harmonics, (4) can be further

arranged as

ωμ0I

2π

∞∑m=−∞

j−m

kb∫−kb


exp [−ju(xn − xO)]

× Jm(kban) exp [jm(ϕ′ − ψ)] du

=∞∑−∞

cnmH(2)m (kban) × exp(jnϕ′) (5)

where ψ = ψ(u) is the impinging direction of the plane waveexp[−j(ux′ + wbz

′)], and ϕ′ denotes the angular position overthe scatterer’s contour. Moreover, Jm(·) and H

(2)m (·) are Bessel

functions and Hankel functions of the second kind, respectively.The cnm are the scattering coefficients (the first index refers tothe nth scatterer, whereas the second one refers to the harmonicorder).

Note that the scattered field has been given in terms of outgo-ing cylindrical waves only, which corresponds to neglecting theinteraction between the scatterers and the half-space interface.

In view of the type of scatterers that we are interested in,we can consider an � λb ∀n, where λb is the wavelength inthe lower half-space. In other words, the cylinders are assumedthin. This implies that the zero-order term in (5) dominates, andhence, the relevant expansion coefficient is given as

cn = cn0 =ωμ0IJ0(kbam)

2πH(2)0 (kbam)

×kb∫

−kb


exp [−ju(xn − xO)] du

=J0(kbam)

H(2)0 (kbam)

Einc(xn, zn, xO, ω). (6)

Accordingly, the field scattered by the nth scatterer and propa-gated above the interface is given by

ESn(xO, ω) = G(xO, xn, zn, ω)Einc(xn, zn, xO, ω)bn (7)

where G(·) is the two-layered medium Green’s function (inparticular, for the case at hand, the field and the source pointsare at the opposite sides of the interface) and is related to theincident field (using reciprocity arguments) as

G(xO, xn, zn, ω) =Einc(xn, zn, xO, ω)

−jωμ0I(8)


and bn = ωμ0IJ0(kbam)/2πH(2)0 (kbam) is the amplitude

factor that depends on the size of the scatterers.The field scattered from the total ensemble of thin scatterers

is then

ES(xO, ω) =N∑

n=1

G(xO, xn, zn, ω)Einc(xn, zn, xO, ω)bn.

(9)

Although mutual scattering has been neglected in (9), retrievingthe scatterer locations still represents a nonlinear problem.This drawback can be avoided by adopting a distributionalrepresentation of the unknown [5]. Following this approach, theunknown cylinders’ positions rn are represented as the supportof δ(·) distributions, and (9) can be rewritten as

ES(xO, ω)=∫∫

D

G(xO, x, z, ω)Einc(x, z, xO, ω)γ(x, z)dxdz

(10)

where

γ(x, z) =N∑

n=1

bnδ(x − xn, z − zn) (11)

and D = [−a, a] × [zmin, zmax] is a spatial region in the lowerhalf-space, which is denoted hereafter as the investigationdomain, where the scatterers are assumed to reside.

Note that the unknown coefficients bn are, indeed, frequencydependent. However, this dependence can be considered weaksince an � λb; accordingly, it has been discarded in (11).

Eventually, the problem of imaging thin scatterers is for-mulated as the inversion of the linear operator in (10) for thedistributional unknown γ(·).

Some comments are now necessary. As described above,linear scattering model in (10) has been obtained by neglectingmutual scattering. This assumption is reasonable when the scat-terers are not too close to each other. Of course, this statementshould be quantified. Having done it, however, one did notgenerally know a priori if this is the case. Therefore, it wouldbe better to study the effect of the model error arising fromneglecting mutual scattering on the linear inversion schemes.This has been already done in [5], where analytical as well asnumerical arguments were used to show that mutual scatteringcan give rise to spurious oscillations appearing in the linear re-constructions or/and introduce masking toward the more deeplylocated scatterers. However, unless the number of scatterers ishigh, this generally does not impair the detection capability.

III. IMAGING APPROACHES

As shown in Section II, the inverse scattering problem athand has been reduced to the inversion of the linear operatorin (10). Of course, before attempting such an inversion, theproblem must be discretized by suitably choosing a finitedimensional representation for the data ES(·) and the unknownγ(·). To this end, we denote with xOn and ωm the spatial points(belonging to Σ) and the frequencies (within the work range[ωmin, ωmax]) upon which measurements are taken. Moreover,

a pixel representation is assumed for the unknown. Accord-ingly, the discretized counterpart of (10) is given by

ES = Aγ (12)

where ES and γ are staked vectors that represent the collectedscattered field data and the discretized version of the unknown,respectively, and A is the matrix that represents the discretizedlinear operator in (10).

Now, as the operator in (10) is compact [5], [7], its inversionis an ill-posed problem. This also entails that the inversion ofits discretized version [see (12)] is ill-conditioned. Therefore, tocounteract the instability arising from ill-conditioning a regular-ized inversion procedure must be applied [7]. This can also beachieved by employing another class of inversion algorithms,largely widespread in different scientific contexts and used “asa standard tool” in applications, i.e., the one referred in theliterature as migration (see [8] for a comprehensive treatise ofvarious migration algorithms and a comparison with the inversefiltering). They are so popular because their understandingis quite simple and founded on wave-front back-propagationinterpretation. Moreover, they do not suffer from the prob-lem of the selection of the regularizing parameter affectinginverse filtering. In addition, they are easy to implement andare often very fast (particularly when the background mediumis lossless). For this reason, we adopt a migration algorithmas a benchmark to compare the performance achievable bythe sparse optimization inversion scheme. In particular, weimplement the version addressed in the literature as diffractionsummation [8]. Under this imaging algorithm, the unknown isretrieved as

γ̃(xl, zk) =∑

n

∑m

G∗(xOn, xl, zk, ωn)

× E∗inc(xl, zk, xOn, ωm)ES(xOn, ωm) (13)

where ∗ denotes the complex conjugation, xl and zk are thespatial coordinates of lk-pixel within the investigation domainD, and γ̃(·) is the unknown reconstructed version.

As seen in (13), the migration is substantially equivalentto the inversion of (12) when the inverse operator is approx-imated by the adjoint of the forward scattering matrix A [7].In particular, by exploiting the matrix notation as in (12), (13)can be rewritten as

γ̃ = AHES (14)

where AH is the Hermitian of A [7], [8].Obviously, the singularity of γ(·) cannot be recovered.

Indeed, as a compromise between stability and accuracy, theretrieved function γ̃(·) will be a filtered/smoothed version ofγ(·), which will exhibit, besides lobes just centered over thescatterers’ positions, spurious sidelobes that could be erro-neously interpreted as actual scatterers. The height of sidelobes depends on the regularization and the noise. Therefore,to mitigate the presence of such artifacts, a threshold will beapplied on the image after reconstructing γ̃(·). This meansthat all those parts of the reconstruction whose level is belowthe chosen threshold will be set to zero. In particular, the


threshold is chosen by following the guidelines that are pointedout in [13].

As a second approach for inverting (12), we consider aconstrained optimization method that exploits the a prioriinformation about the sparse nature of the solution (i.e., asolution that contains the least number of nonzero elements)as a regularizing tool. More in detail, the general regularizationproblem can be cast as

γ̃ = arg min ‖Aγ − ES‖p + α‖γ‖q (15)

with α being the regularization parameter. When p = 2, q =0, the solution is guaranteed to have the smallest number ofnonzero pixels, but the minimization problem becomes NP-hard. Recent advances in compressive sensing have demon-strated that when the number of nonzero pixels is assumedsparse, the reconstructed image using p = 2, q = 1 and p = 2,q = 0 are similar in quality [14]. Moreover, optimization in (15)for p = 2, q = 1 is inherently convex and can be easily imple-mented by fast algorithms. Therefore, we chose this strategy toperform the inversion which is also equivalent to solve

minimize: ‖Aγ − ES‖22

subject to: ‖γ‖1 < τ. (16)

This method is generally called least absolute shrinkage andselection operator (LASSO) [15] in applied mathematics andcan be solved using a vast number of convex optimizationtechniques.

In our case, we solved the LASSO problem using the spectralprojected gradient (SPG) method as described in [16], [17]because of its efficiency and simplicity of implementation.The SPG method relies upon the computation of a projectedvector onto a unit ball in l1-norm at each iteration. Since wehandle complex vectors (because both A and γ are complexvalued), we prefer to utilize a fast complex one-norm projectionalgorithm for SPG that is clearly described in [17].

Note that the sparseness of the reconstructed image is strictlycontrolled by the parameter τ : the lower τ is, the sparser thereconstruction becomes, and vice versa. Therefore, the choiceof τ is a crucial question that strongly affects reconstructionquality.

To cope with this question, we exploit reconstructions re-turned by migration for τ selection, in the following way.

Let us denote by γm and γS the reconstructions returnedby the migration and the sparse minimization, respectively.Moreover, let us assume that γm roughly coincides with thegeneralized solution of (12). Then, it is obvious that

‖γS‖1 ≤ ‖γm‖1. (17)

Moreover, as the generalized solution is the minimum l2-normsolution, it also results to

‖γm‖2 ≤ ‖γS‖2 ≤ ‖γS‖1. (18)

In other words, (17) and (18) dictate an upper and a lower boundfor τ , i.e.,

‖γm‖2 ≤ τ ≤ ‖γm‖1. (19)

Accordingly, we solve the optimization reported in (16) fordifferent values of τ ranging within the interval dictated by (19).Then, among the corresponding reconstructions, we choose theone which returning the largest improvement on the achievableresolution (when compared to the migration reconstruction).

IV. NUMERICAL ANALYSIS

This section is devoted to presenting a comparison betweenthe performances of the two approaches for the case of datagenerated synthetically.

The data have been simulated in time domain with a finite-difference time domain code [18] accounting explicitly for thehalf-space geometry of the background scenario. The simula-tion refers to a 2-D geometry where the source is a filamentarycurrent radiating a Ricker wavelet at a central frequency of1 GHz. In particular, since the inversion algorithms require datain the frequency domain, we have the necessity to pass fromthe total field in time domain to the scattered field in frequencydomain. The strategy to accomplish this task consists in firstgating the time domain raw data, so to erase the reflection formthe interface, then the gated signal is Fourier transformed.

The following test cases refer to a scenario made up of a half-space geometry where the upper medium is free space and thelower medium is a soil with dielectric permittivity and conduc-tivity equal to εb = 10ε0 and σb = 0.005 S/m, respectively.

As to the measurement configuration, 81 source/observationpoints, taken uniformly at a spatial step of 0.01 m, are exploited.Moreover, according to the choice of the excitation in FDTD,the frequency band 500–1500 MHz is selected for the recon-struction procedures and sampled at a step of 50 MHz.

Finally, perfect electric conducting cylinders with circularcross sections of radius 0.01 m are considered buried withinthe investigation domain D = [−0.4, 0.4] × [0.01, 0.5] m2.

The first test case refers to three targets located at the samedepth of 0.1 m and equally spaced by 0.2 m. The correspondingreconstructions are reported in Fig. 2. Fig. 2(a) reports the resultgiven by the migration algorithm, while Fig. 2(b) reports thereconstruction obtained by means of the sparse optimizationscheme. Reconstruction obtained by both methods are givenin terms of the normalized (to their maximum) amplitude andclearly allow us to detect and localize the scatterers. However,the sparse optimization procedure permits to obtain a finer reso-lution of the reconstructed spots. This is particularly remarkablesince, whereas migration reconstruction has been thresholded ata level of 0.21 (to cutoff spurious artifacts) [13], no thresholdhas been applied for sparse minimization. Moreover, accordingto results presented in [5], due to mutual scattering, somespurious artifacts are expected to appear in the reconstructions.Nonetheless, threshold adopted after migration algorithm re-moved such unwanted oscillations. Interestingly, sparse mini-mization reconstruction also did not exhibit spurious artifactsalthough a threshold was not used. The improvement on theachievable resolution can be better appreciated by looking atFig. 3, where the cut views along x (at z = 0.1 m) and alongz (at x = 0 m) of the two reconstructions are reported. Seealso Table I for a measure of the resolution improvement. Asmentioned above, for the sparse reconstruction method, the


Fig. 2. Reconstructions of three aligned rebars. (a) Migration reconstruction (right). (b) Sparse reconstruction (left).

Fig. 3. Cut views of the reconstructions of three aligned rebars. (a) Cut along x at z = 0.1 m. (b) Cut along z at x = 0 m. Migration reconstruction (with nothreshold applied) is denoted by red dotted lines instead sparse minimization reconstruction by solid blue lines.

TABLE ICOMPARISON BETWEEN SPARSE MINIMIZATION AND MIGRATION IN TERMS OF THE ACHIEVABLE RESOLUTION

parameter τ plays the role of the regularization parameter,and its choice severely affects the reconstruction quality. InFig. 2(b), we adopted the strategy sketched at the end of theprevious section based on the choice of different values of τwithin the interval reported in (19). Then, having performedthe minimization for each of these values, we selected the re-construction that allowed us to obtain more resolved scatterers(as compared to what migration did). To show how sparseminimization works for different values of τ , reconstructionsobtained with τ larger than the optimal one (in the sense ex-plained above) are reported in Fig. 4. In particular, in Fig. 4(a),we adopted τ = ‖γm‖1. For such a case, the l1-norm constraint,of course, did not work at all, and the reconstruction is iden-tical to the one returned by migration without the use of anythreshold on the image. By lowering the value of the parameterτ , the reconstruction starts improving since the targets appearmore resolved, and above all, image is “clearer” as artifactspractically disappear [see Fig. 4(b)]. For this reconstruction, the

l1-norm of the solution is about one-half that of the solutionobtained by the migration.

In the second test case, we increase to five the number oftargets so that the two added targets are close to the edgesof the investigation domain. The corresponding reconstructionsare reported in Fig. 5 (see also Table I for performance compar-ison). Clearly, the same conclusions as in the previous case canbe substantially traced. However, we also note that the sparseminimization procedure succeeds in retrieving the scatterers atalmost the same level [see Fig. 5(b)]. Conversely, the migrationalgorithm returns a reconstruction where the added scatterersare reconstructed at a lower level [see Fig. 5(a)]. This is per-fectly consistent with the analysis shown in [19], where it wasshown how the migration procedure entails a spatially varyingfiltering on the unknown.

As a further test case, we consider a scatterer ensemblearranged over two layers. The upper layer is the same as thefirst case (the one addressed in Fig. 2), while the more deeply


Fig. 4. Role of the parameters τ . (a) Sparse minimization with a large value of the parameter τ (left). (b) Sparse minimization with an intermediate value of theparameter τ (right).

Fig. 5. Reconstructions of five aligned rebars. (a) Migration reconstruction (right). (b) Sparse reconstruction (left).

Fig. 6. Reconstructions of five rebars located over two lines. (a) Migration reconstruction (right). (b) Sparse reconstruction (left).

located one is made up of two targets at a depth of 0.2 m. Fig. 6reports the reconstructions for such a case.

In this case, both the procedures permit to detect and lo-calize all the five targets and previous statements still hold. Inaddition, in this case, while achieving the sparse reconstructionalgorithm, particular care must be taken on the choice of theparameter τ . In fact, when the value of τ is not sufficiently high,the approach could retrieve only the more strongly scatteringtargets, i.e., the upper rebars (not shown here). On the contrary,by increasing τ the lower layer can also be detected, but atthe cost of deteriorated resolution, particularly for the scatterersarranged over the upper layer. However, the proposed procedurefor τ selection allowed us to detect all the scatterers and toimprove resolution as compared to migration (see Table I).

Finally, we consider the case of four targets where the twocentral are more closely located so as to be spaced by 0.1 m.This example has been devised to increase the mutual scat-tering effect and hence to check how the reconstructionmethods behave. The corresponding reconstructions are shownin Fig. 7.

By comparing Fig. 7(a) and (b), it is seen that while for themigration reconstruction a spurious artifact appears betweenthe central scatterers (that the threshold was not able to re-move), the sparse reconstruction is free from this ghost scatterer[see Fig. 7(b)]. This outcome can be basically explained in viewof the previous discussion regarding the choice of the τ . Indeed,for the case at hand, the spurious artifact has a lower magnitudeas compared to the true targets; therefore, an effect similar to


Fig. 7. Reconstructions of more closely located scatterers. (a) Migration reconstruction (right). (b) Sparse reconstruction (left).

the one highlighted for the case of two-layered ensemble ofscatterers arises when the sparse constraint is used.

A comparison between the two methods is summarized inTable I. In particular, for each addressed case, we report theratios ΔxS/Δxm and ΔzS/Δzm averaged over the number ofscatterers. Here, ΔxS and Δxm are the half-width of the mainbeams along the x-axis obtained by sparse minimization andmigration, respectively. Analogously, ΔzS and Δzm refer to thez-axis. Accordingly, such ratios measure the improvement ontransverse and along depth resolutions.

V. REAL DATA PROCESSING

In this section, we check the imaging algorithms using exper-imental data. To this end, we process measurements collectedwith the aim of acquiring information about the inner status ofa reinforced concrete element of a public building in the city ofL’Aquila, Italy.

This GPR survey falls within the framework of the non-invasive diagnostics investigations of public infrastructures thatwere performed with the aim of providing an early damage as-sessment following the earthquake occurred in Abruzzo Region(Italy) on April 6, 2009 [3].

The structural element considered here was a cracked joisthaving a section of 30 cm × 100 cm, and the aim of theinvestigation was to gain information about the presence andlocation of the rebar layers and evaluate the result of injectionsof epoxy resin in the cracks.

A Geophysical Survey Systems Inc. SIR-300 system witha 1500-MHz antenna was used to gain information about thecorner formed by the joist and a pillar. Here, we focus on tworadargrams collected over the opposite faces of the pillar and atthe same height (see Fig. 8).

The parameters of the reconstruction procedure (necessaryto fill the matrix) are the same adopted for the synthetic data.In particular, the dielectric permittivity is chosen according tothe values reported in literature on concrete walls [20], [21].

Fig. 9 depicts the raw data for the first profile. Three hyper-bolas are present in the shallower part while the lower part ofthe radargram accounts for the reflection from the other face ofthe pillar.

Fig. 10(a) depicts the migration reconstruction that permitsto point out three targets at the depth of 0.1 m, while the lower

Fig. 8. Picture of the experimental site.

Fig. 9. Raw data for the first profile.

part of the reconstruction is concerned with the lower side ofthe pillar. When sparse reconstruction approach is performed,the reconstruction is made noticeably clearer. Moreover, it ispossible also to detect three spots that, as will be shown byprocessing the other data set, are concerned with three rebarslocated close to the opposite side of pillar.

Fig. 11 depicts the raw data for the second profile collectedat the same height and on the opposite face of the pillar.Similarly to the profile above, three hyperbolas are presentin the shallower part, whereas in the lower part, a reflectionappears from the other face of the pillar.


Fig. 10. Reconstructions by employing the first set of measurements.(a) Migration reconstruction. (b) Sparse reconstruction.

Fig. 11. Raw data for the second profile.

Fig. 12(a) depicts the migrated reconstruction and permits topoint out three targets at the depth of 0.1 m and the three rebarsin the deeper part of the reconstruction scene. When sparsereconstruction approach is performed, the reconstruction [seeFig. 12(b)] is slightly clearer. Finally, for these experimentalreconstructions, the average resolution improvement is 1.7 forΔxS/Δxm and 1.6 ΔzS/Δzm.

VI. CONCLUSION

This paper has dealt with the problem of rebar detectionand localization starting for GPR measurements collected at theair/structure interface.

Fig. 12. Reconstructions by employing the second set of measurements.(a) Migration reconstruction. (b) Sparse reconstruction.

First, a model tailored to account for the properties of thesought targets, which have small cross sections in terms ofprobing wavelength, has been presented. This has led to a linearscattering model linking the scattered field data to a distribu-tional unknown just supported over the scatterers’ centers.

Accordingly, the detection/localization problem has beencast as a linear inverse problem. To solve it, the following twodifferent techniques have been applied: 1) a usual migrationalgorithm (used as benchmark) and 2) a very recent techniquebased on an optimization that is able to perform a sparsereconstruction where the “sparse-discrete” nature of the targetsis accounted for by a constraint in the l1-norm.

The comparison between the two techniques has been con-ducted for both synthetic and experimental data collected fora realistic scene. It turned out that the sparse minimizationperforms better in localizing the targets allowing, indeed, fora better resolution (see Table I). However, such an inversionscheme has the drawback of the regularization parameter choicewhich strongly affects the reconstruction quality. To cope withthis question, by exploiting the migration reconstruction and byintroducing simple reasoning, we identified an interval withinto search for τ . Unfortunately, the procedure still requiresperforming different sparse reconstructions [in correspondenceto different values of τ taken within the interval reported in(19)] to select the best one (according to some figure). Here, weadopted resolution as a criterion to achieve such a task. There-fore, the choice of the regularization parameter still deservesfurther, possibly theoretical, study.

Finally, we assumed to precisely known the backgroundmedium. However, in practical scenario, this is not the case.


This occurred, for example, for the experimental data usedherein, for which not only the concrete electromagnetic param-eters were unknown (we set them according to literature) butalso the joist’s thickness (finite in the experiment) has beenignored in the model (as we considered a half-space). Yet,reconstructions allowed us to localize rebars within. However,medium uncertainties can degrade performance. This has beenstudied for linear inverse filtering algorithms [22], [23]. A sim-ilar study should also be conducted for the sparse minimizationalgorithm.

REFERENCES

[1] D. J. Daniels, Ground Penetrating Radar, 2nd ed. London, U.K.: IEEPress, 2004.

[2] J. Hugenschmidt and R. Mastrangelo, “The inspection of large retainingwalls using GPR,” in Proc. 4th Int. Workshop Adv. Ground PenetratingRadar, Jun. 27–29, 2007, pp. 267–271.

[3] V. Lapenna, M. Bavusi, A. Loperte, and C. Moroni, “Ground penetratingradar and microwave tomography for high resolution post-earthquakedamage assessment of a public building in L’Aquila City (AbruzzoRegion, Italy),” in Proc. AGU Fall Meeting, San Francisco, USA,Dec. 2009.

[4] W. C. Chew, Waves and Fields in Inhomogeneous Media. Piscataway,N.J.: IEEE Press, 1995.

[5] R. Pierri, R. Solimene, A. Liseno, and J. Romanano, “Linear distributionimaging of thin metallic cylinders under mutual scattering,” IEEE Trans.Antennas Propag., vol. 53, pp. 3019–3029, 2005.

[6] N. Bleinstein and S. H. Gray, “From the Hagedoorn imaging technique toKirchhoff migration and inversion,” Geophys. Prospect., vol. 49, pp. 629–643, 2001.

[7] M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging.Bristol, U.K.: IOP, 1998.

[8] F. Soldovieri and R. Solimene, “Ground penetrating radar subsurfaceimaging of buried objects,” in Radar Technology. Vedran Kordic:In-Tech, 2010, pp. 105–126.

[9] E. A. Marengo, “Inverse scattering by compressive sensing and signalsubspace methods,” in Proc. 2nd IEEE Int. Workshop CAMPSAP,Dec. 12–14, 2007, pp. 109–112, DOI: 10.1109/CAMSAP.2007.4497977.

[10] M. A. Herman and T. Strohmer, “High-resolution radar via compressedsensing,” IEEE Trans. Signal Process., vol. 57, no. 6, pp. 2275–2284,Jun. 2009.

[11] W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman,“Terahertz imaging with compressed sensing and phase retrieval,” Opt.Lett., vol. 33, no. 9, May 1, 2008.

[12] L. Carin, D. Liu, and Y. Xue, “In situ compressive sensing,” InverseProbl., vol. 24, no. 1, p. 015 023, Feb. 2008.

[13] A. Liseno, F. Soldovieri, and R. Pierri, “Improving a shape reconstructionalgorithm with thresholds and multi-view data,” AEU—Int. J. Electron.Commun., vol. 58, no. 2, pp. 118–124, 2004.

[14] R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag., vol. 24,no. 4, pp. 118–121, Jul. 2007.

[15] R. Tibshirani, “Regression shrinkage and selection via the Lasso,” J. R.Stat. Soc. B, vol. 58, no. 1, pp. 267–288, 1996.

[16] E. G. Birgin, J. M. Martinez, and M. Raydan, “Nonmonotone spectralprojected gradient methods on convex sets,” SIAM J. Optim., vol. 10,no. 4, pp. 1196–1211, 2000.

[17] E. van den Berg and M. P. Friedlander, “Probing the Pareto frontier forbasis pursuit solutions,” SIAM J. Sci. Comput., vol. 31, pp. 890–912,Nov. 2008.

[18] A. Giannopoulos, GprMax2D/3D, User’s Guide, 2002. [Online].Available: www.gprmax.org

[19] R. Solimene and R. Pierri, “Imaging thin PEC cylinders via a linearinversion scheme and a spatially varying threshold,” Proc. Eur. Microw.Assoc., vol. 4, pp. 11–20, Mar. 2008.

[20] C. A. Grosvenor, R. T. Johnk, J. Baker-Jarvis, M. D. Janezic, andB. Riddle, “Time-domain free-field measurements of the relative permit-tivity of building materials,” IEEE Trans. Instrum. Meas., vol. 58, no. 7,pp. 2275–2282, Jul. 2009.

[21] J. Davis, S. G. Millard, Y. Huang, and J. H. Bungey, “Determination ofdielectric properties of in situ concrete at radar frequencies,” in Proc. Int.Symp. Non-Destructive Test. Civil Eng., 2003, p. 8p.

[22] R. Persico and F. Soldovieri, “Effects of uncertainty on background per-mittivity in one dimensional linear inverse scattering,” J. Opt. Soc. Amer.A, Opt. Image Sci., vol. 21, no. 12, pp. 2334–2343, Dec. 2004.

[23] F. Soldovieri, G. Prisco, and R. Persico, “Application of microwavetomography in hydrogeophysics: Some examples,” Vadose Zone J., vol. 7,no. 1, pp. 160–170, Feb. 2008.

Francesco Soldovieri (M’10) received the Laurea degree in electronic engi-neering from the University of Salerno, Salerno, Italy, in 1992 and the Ph.D.degree in electronic engineering from the University of Naples “Federico II,”Naples, Italy, in 1996.

In 2001, he became a Researcher with the Institute for ElectromagneticSensing of the Environment (IREA), Italian National Research Council (CNR),Naples, where he has been a Senior Researcher since 2006. He has been a GuestEditor for Special Issues of the Journal of Applied Geophysics, Near SurfaceGeophysics, and Advances in Geosciences and the Journal of Geophysicsand Engineering. His main scientific interests include electromagnetic diag-nostics, inverse scattering, ground penetrating-radar (GPR) and through wallimaging applications, antenna diagnostics and characterization, and securityapplications.

Dr. Soldovieri was the General Chair of the International Workshop onAdvanced Ground Penetrating Radar (IWAGPR) 2007 and General Cochair ofthe XIII International Conference on Ground Penetrating Radar, 2010. Since2002, he has been involved in the Technical Committees of the GPR Conferenceand IWAGPR. He was the recipient of the 1999 Honorable Mention for theH. A. Wheeler Applications Prize Paper Award of the IEEE Antennas andPropagation Society.

Raffaele Solimene was born in Italy in 1974. He received the Laurea (summacum laude) and Ph.D. degrees in electronic engineering from the SecondUniversity of Naples, Aversa, Italy, in 1999 and 2003, respectively.

From 2002 to 2006, he was an Assistant Professor with the UniversitàMediterranea di Reggio Calabria, Reggio Calabria, Italy. Since 2006, has beenwith the Department of Information Engineering, Second University of Naples.His main research interests include inverse scattering problems and resolutionpower in microwave imaging.

Lorenzo Lo Monte (S’05–M’09) received the B.Sc.and M.Sc. degrees (both summa cum laude) intelecommunications engineering from the Universityof Rome “Tor Vergata,” Rome, Italy, in 2003 and2005, respectively, and the Ph.D. degree in electricalengineering from the University of Illinois, Chicago,in 2009.

In 2005, he was with Rheinmetall AG, Rome,working on the synthesis of phased arrays for anti-aircraft systems. From 2007 to 2008, he was withPCTEL Inc. as an Antenna Engineer, developing a

new family of WiMax BTS sector panels. Since March 2008, he has been aMilitary Systems Engineer with General Dynamics Information Technology,Springfield Pike, Dayton, OH, working as a contractor for the Air Force Re-search Laboratory/RYRT branch at Wright Patterson Air Force Base, Dayton.In 2009, he joined the Rensselaer Polytechnic Institute, Troy, NY, and theWright State University, Dayton, OH, as a visiting scholar, performing researchin inverse problems for radar imaging. His main research interests includemultistatic radar imaging, ground penetrating radar, and inverse scattering.


Massimo Bavusi received the Laurea degree in geology and the Ph.D. degreein methods and technologies for environmental monitoring from the Universityof Basilicata, Potenza, Italy, in 2001 and 2006, respectively.

For eight years now, he has been with the Institute of Methodologies forEnvironmental Analysis (IMAA), Italian National Research Council (CNR),Tito Scalo, Italy, with several agreements, where he is currently under a non-tenure-track research fellowship. He is the author/coauthor of several scientificnational and international papers and proceedings submitted at several nationaland international workshops and conferences regarding the application ofelectromagnetic techniques for the characterization of polluted areas and engi-neered and archaeological structures. His research interests include the joint useof several electromagnetic techniques as a tool for the ground and groundwaterpollution study, archaeological research, and engineering structures, includingtransport infrastructures.

Antonio Loperte received the Laurea degree in geology from the University ofBasilicata, Potenza, Italy, in 2001.

Since 2002, he has been with the Institute of Methodologies for Environmen-tal Analysis (IMAA), Italian National Research Council (CNR), Tito Scalo,Italy, where he is currently under a non-tenure-track research fellowship. Heis the author/coauthor of several scientific national and international papersand proceedings submitted at several national and international workshops andconferences. His research interests include the joint use of several electromag-netic techniques as a tool for the ground and groundwater pollution study,archaeological prospecting, and engineered structures diagnostics.

Dr. Loperte is a member of the Geologists of Basilicata Region.

sparse reconstruction from gpr data with applications to rebar detection

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