988 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 64, NO. 3, MARCH 2016

Higher Order Sparse Microwave Imagingof PEC Scatterers

Marija Nikolic Stevanovic, Lorenzo Crocco, Senior Member, IEEE, Antonije R. Djordjevic,and Arye Nehorai, Fellow, IEEE

Abstract—In this paper, we present an innovative algorithmfor sparse imaging of perfect electric conducting (PEC) targets.The algorithm is based on higher order sources or multipoles,rather than zero-order ones as in the standard sparse imaging. Wedemonstrate that, due to the directivity of higher order sources,the imaging capabilities of the algorithm are enhanced. Particularimprovements are achieved when dealing with complex-shapedtargets. As a matter of fact, through several illustrative exam-ples, we show that the zero-order sources are more suitable for thereconstruction of the convex parts of target boundaries, whereasthe higher order sources are more appropriate for the concaveparts of the boundaries. Taking advantage of some analyticalconsiderations in a canonical case, we devise a strategy to selectthe optimal orders to consider in the imaging procedure withoutneeding any a priori information on the target.

Index Terms—Microwave imaging, multipole expansion, sparseprocessing.

I. INTRODUCTION

I N many localization problems, targets occupy only a frac-tion of the observed domain. Consequently, compressed

sensing (CS) techniques, which take advantage of sparsenessor compressibility of the unknown function in the search space,have gained significant attention [1]–[6]. Recently, benefits ofsparse processing have been recognized in microwave imag-ing [7]–[19], wherein the reconstruction algorithms use eitherunknown contrast or unknown currents induced in the targets,assuming that they are sparse. As far as induced currents areconcerned, they are called sparse if they are bound to a smallportion of the region under test, which is the case that weconsider in this paper. Notably, if such an assumption is not

Manuscript received March 16, 2015; revised November 21, 2015; acceptedJanuary 10, 2016. Date of publication January 26, 2016; date of current versionMarch 01, 2016. This work was supported in part by NSF under Grant CCF-0963742 and Grant CCF-1014098, in part by AFOSR under Grant FA9550-11-1-0210, and in part by Serbian Ministry of Science and Education under GrantTR32005. This work was partially developed in the framework of COST ActionTD1301 MiMed.

M. N. Stevanovic is with the School of Electrical Engineering, University ofBelgrade, Belgrade 11120, Serbia (e-mail: mnikolic@etf.bg.ac.rs).

L. Crocco is with CNR-IREA, National Research Council of Italy, Institutefor Electromagnetic Sensing of the Environment, Naples 80124, Italy (e-mail:crocco.l@irea.cnr.it).

A. R. Djordjevic is with the School of Electrical Engineering, University ofBelgrade, Belgrade 11120, Serbia, and also with Serbian Academy of Sciencesand Arts, Belgrade 11120, Serbia (e-mail: edjordja@etf.bg.ac.rs).

A. Nehorai is with Preston M. Green Electrical and Systems EngineeringDepartment, Washington University in St. Louis, St. Louis, MO 63130 USA(e-mail: nehorai@ese.wustl.edu).

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

Digital Object Identifier 10.1109/TAP.2016.2521879

fulfilled, the induced currents may become sparse after apply-ing suitable transformations [12], [17], [18]. Utilizing this priorknowledge, in the form of an l1 constraint, yields clean andfocused images.

In the simplest form, the induced currents may be approxi-mated by pulse basis functions [10]. In two-dimensional (2-D)space, a pulse basis function is equivalent to an infinitely longline current. If the problem is sparse, it is sufficient to use a fewbasis functions to represent the electric field scattered from theobject. Due to its straightforwardness, such a model is amenableto signal processing. However, in most real-life scenarios, thescattered field has significant spatial variations. Hence, it is hardto obtain an accurate approximation with such a simple model.As a consequence of this information loss, targets with differentshapes may have identical images.

In order to improve the imaging accuracy, with respect tothe CS-based imaging approach developed in [10], we explorethe possibility of incorporating higher order basis functions intothe sparse algorithm. Therefore, instead of line currents, wealso consider current multipoles [20], i.e., dipoles, quadrupoles,etc. In contrast to the line sources, multipoles produce direc-tive electric fields. Hence, by combining multipoles of differentorders and orientations, we can obtain arbitrary scattered fieldpatterns. The idea of using higher order sources has been alsoconsidered in [21], as a tool for enhancing the linear samplingmethod (LSM). The authors demonstrated significant improve-ments in the shape estimation of complex targets, with respectto the standard formulation of LSM, by exploiting a suitablecombination of the results achieved with the sources of differentorders.

Here, we investigate the application of the multipoles in thesparse-processing framework. This paper represents an exten-sion of the work in [22], where we presented the basic idea ofhigher order sparse imaging. Nevertheless, we propose severalmajor novelties with respect to [22]. First, we use another formof higher order basis functions and, consequently, develop adifferent sparse model. Second, by relying on analytical consid-erations and a physical explanation of the multipole modeling,we introduce a multistep algorithm to tackle the higher ordersparse processing in an almost automated fashion. In particu-lar, we estimate the convex hull of the object from the standard(zeroth-order) image and use this information to estimate themaximum order of the radiating multipoles associated with theunknown target. For both the highest order and the immedi-ately lower one, we compute the corresponding images. Then,we obtain the final image by superimposing the higher orderresults and the zeroth-order one.

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STEVANOVIC et al.: HIGHER ORDER SPARSE MICROWAVE IMAGING OF PEC SCATTERERS 989

Fig. 1. Target illuminated by incident electromagnetic field and the associatedcoordinate system.

Fig. 2. (a) Current dipole and (b) its radiation pattern.

Fig. 3. Grid of line currents (zero-order multipoles).

Fig. 4. Grid of orthogonal dipoles.

The above procedure has been tested on several numericalexamples, in the presence of noise, which confirmed that theexploitation of higher order processing improves the imagingresults. In particular, we show that jointly exploiting the stan-dard and higher order processing is appropriate to achieve thereconstruction of both convex and concave boundaries of tar-gets. Interestingly, this is consistent with the findings of [21],which have been developed in the completely different contextsof LSM.

This paper is organized as follows. In Section II, webriefly review the multipole expansion of the electric field. InSection III, we define the higher order sparse processing. InSection IV, we provide some analytical considerations in thecanonical case of a circular cylinder that are helpful in the phys-ical interpretation of the method. The multistep algorithm isdetailed in Section V, while Section VI presents several charac-teristic examples computed using different noise levels. Finally,we discuss the obtained results in conclusion.

Fig. 5. Circular array of current dipoles (multipoles of the order n = 1).

Fig. 6. Cylindrical body surrounded by an array of sensors.

II. MULTIPOLE EXPANSION

We consider an arbitrary object illuminated by an incidentelectromagnetic field, as depicted in Fig. 1. For the sake of sim-plicity, we assume 2-D geometry and transverse-magnetic (TM)polarization. In terms of the multipole expansion [20], the elec-tric field produced by the induced currents in the object is givenby

E (r) =∞∑

n=0

H(2)n (βr) (an cos (nφ) + bn sin (nφ)) (1)

where H(2)n is the nth order Hankel function of the second kind,

β is the wave number, an and bn are the expansion coefficients,and r = (r, φ) is the observation point in the polar coordinatesystem. In the far field, (1) becomes approximately

E (r) ≈∞∑

n=0

exp (−jβr)√r

(an cos (nφ) + bn sin (nφ)) . (2)

In (1) and (2), the term associated with n = 0 corresponds tothe field radiated by a line source, whereas the terms associatedwith n = 1 correspond to the fields radiated by two orthog-onal line dipoles. Assuming that the radius of the minimumcircle covering the object is a, only the components with indicessmaller or equal to n = βa are significant. The components ofthe order of n > n have a low radiation efficiency, as they giverise to fields that vanish exponentially with the order [21].

Fig. 2 shows a current dipole and its radiation pattern.The radiation patterns of higher order sources become moredirective as their order increases.

990 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 64, NO. 3, MARCH 2016

Fig. 7. Reconstructions of the portion of the cylindrical body computed using different multipole orders (n) for γ = 1.

Fig. 8. Reconstructions of the whole cylindrical body computed using different multipole orders (n) for γ = 1.

Fig. 9. Cross section of the cross-shaped target.

Fig. 10. Computation of the regularization parameter for the standard sparseprocessing.

III. MEASUREMENT MODEL

A. Standard Sparse Imaging

From the spatial distribution of the equivalent sources orline sources with significant currents, we learn about the targetshape. The locations of the equivalent sources, as well as theircurrents, are unknown. Therefore, as illustrated in Fig. 3, wedefine as a search space an even grid of line currents. Initially,we start with an internodal distance equal to λ/4, where λ isthe wavelength. Then, we refine the estimation by considering

a denser grid. We estimate the currents on the grid by exploit-ing the fact that only a few of them have nonzero values. Thefar-field measurement model reads

es = W0c0 (3)

with

es =[Es

1 . . . EsM

] T(4)

c0 =[C0

1 . . . C0L

] T(5)

W0 = Z

√jβ8π

⎡⎢⎣

e−jβd11/√d11 . . . e−jβd1L/

√d1L

.... . .

...e−jβdM1/

√dM1 . . . e−jβdML/

√dML

⎤⎥⎦

dmn = ‖rm − tl‖ , m = 1, 2, . . . , M, l = 1, 2, . . . , L(6)

where c0 is the unknown vector of the zero-order multipolecoefficients, es is the known vector of the scattered electric fieldat the sensor locations, rm is the location of the mth sensor, andtl is the location of the lth grid node. Further, M is the totalnumber of the sensors, L is the total number of the grid nodes,and Z is the wave impedance of a vacuum. In the following, we

omit the multiplicative factor Z√

jβ8π .

We look for the solution of (3) under the sparse processingframework, i.e.,

c = minc

{‖es −W0c0‖22 + γ‖c0‖1

}(7)

where c is the estimated coefficient vector and γ is the reg-ularization parameter. The regularization parameter balancesbetween the data fidelity and the solution sparsity.

STEVANOVIC et al.: HIGHER ORDER SPARSE MICROWAVE IMAGING OF PEC SCATTERERS 991

Fig. 11. Reconstruction of the cross-shaped cylindrical body obtained using the standard sparse processing.

Fig. 12. Computation of the threshold and the corresponding convex hull.

B. Higher Order Sparse Imaging

In order to get more insight about the target, instead ofline currents, we consider a grid of multipoles of the order n.According to (1), we define two types of dictionaries associatedwith the cosine and sine expansion terms, namely

Wcn =⎡

⎢⎣e−jβd11 cos (nΔφ) /

√d11 . . . e−jβd1L cos (nΔφ)/

√d1L

.... . .

...e−jβdM1 cos (nΔφ)/

√dM1 . . . e−jβdML cos (nΔφ)/

√dML

⎤⎥⎦

(8)

Wsn =⎡

⎢⎣e−jβd11sin(nΔφ)/

√d11 . . . e−jβd1L sin (nΔφ) /

√d1L

.... . .

...e−jβdM1 sin (nΔφ)/

√dM1 . . . e−jβdML sin(nΔφ)/

√dML

⎤⎥⎦

Δφ = φ− φ′ (9)

where φ is the observation angle and φ′ is the angle of inci-dence. The remaining variables are the same as in (6). We alsodefine the vectors containing the multipole coefficients

cn =[Cn

1 . . . CnL

]T, sn =

[Sn1 . . . Sn

L

]T(10)

related to the cosine (Wcn) and sine (Ws

n) dictionaries, respec-tively. Fig. 4 gives the physical interpretation for the casen = 1. Two dictionaries correspond to a grid in which eachnode consists of two orthogonal dipoles. In the estimationprocess, we use the sine and cosine bases jointly

W =[Wc

n Wsn

](11)

c =[cTn sT

n

]T(12)

where W is the new dictionary matrix and c is the vector con-taining all unknown multipole coefficients. The cost function tobe minimized reads

c = minc

{‖es −Wc‖22 + γ‖kn‖1

}(13)

where

kn =[kn1 . . . knL

] T, knl =

√(Cn

l )2+ (Sn

l )2. (14)

To solve (13), we use the CVX package [23], [24]. We per-form the minimization (13) for each incident angle or for agroup of close incident angles. The corresponding image isobtained as

I(l) =

M∑i=1

∥∥∥c(i)l

∥∥∥22

(15)

where I(l) is the lth pixel, and the superscript (i) denotes theindex of the transmitting sensor.

IV. INSIGHT INTO HIGHER ORDER SPARSE IMAGING

A. Behavior of a Circular Array of Multipoles

To set the ground for higher order sparse imaging, we con-sider a canonical case of a circular array of multipoles of theorder n. In Fig. 5, we show an example of such an array forn = 1. Without loss of generality, we consider only the cosinepart of the multipole expansion (1). Up to the scaling constant,the electric field produced by the cosine multipole located at(a, φ′) is given by

cos (nφ)exp (−jβR)√

R

≈ cos (nφ)exp (−jβr) exp (jβa cos (φ− φ′))√

r(16)

assuming r >> a. In the limiting case, in which the multipolesare continuously distributed along the circle, the total electricfield is proportional to

992 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 64, NO. 3, MARCH 2016

Fig. 13. Reconstructions of the cross-shaped cylindrical body using different multipole orders.

Fig. 14. Filtered images computed using the multipoles of the orders n = 0, 3, 4 for (a) SNR = 20 dB; (b) SNR = 10 dB; and (c) SNR = 5 dB.

Etot (r)

≈ cos (nφ)exp (−jβr)√

r

∫φ′

g (φ′) exp (jβa cos (φ− φ′)) dφ′

(17)

where g (φ′) is a function proportional to the multipole coef-ficients. Then, by exploiting the Taylor series to expand theexponential function, (17) becomes

Etot (r) ≈ cos (nφ)exp (−jβr)√

r

·∫φ′

g (φ′)

(1 + jβa cos (φ− φ′)

+(jβa)2

2cos2 (φ− φ′) + · · ·

)dφ′.

(18)

From (18), we can draw the following observations. First, ifthe radius of the array is electrically small, i.e., βa → 0, thefield is completely described by the first term of the expansion.Hence, (18) becomes

Etot (r) ≈ cos (nφ)exp(−jβr)√

r

2π∫0

g (φ′) dφ′

= A cos (nφ)exp(−jβr)√

r(19)

where A is the multiplicative factor independent of φ. In thiscase, the electric field is proportional to the field radiated by thenth order multipole positioned in the origin. Hence, as expected,

Fig. 15. (a) Cross section of the U-shaped cylinder and (b) binary convex hull.

an electrically small array of the nth order multipoles producesthe electric field that has only the nth harmonic.

Second, when the radius of the array becomes larger, moreterms have to be retained in the series. For instance, consideringthe expansion up to the second order, after some straightforwardmanipulations, we get

Etot (r) ≈ exp (−jβr)√r

(A cos (nφ)+ B cos ((n+ 1)φ)

+C cos ((n− 1)φ) +D sin ((n+1)φ) + E sin ((n− 1)φ))(20)

where A, B, C, D, and E are certain complex numbers, inde-pendent of φ. If a further increases, it is necessary to keep thethird term in the expansion (18). Hence, the radiated field willalso contain the harmonics n− 2 and n+ 2, and so on. Forsufficiently large values of the radius (w.r.t. the wavelength), anarray of multipoles of the order n can radiate a field having allharmonic components lower than n. This suggests that, whendealing with extended targets, we can use the sparse process-ing model of the highest order that is allowed for the unknowntarget, say n.

STEVANOVIC et al.: HIGHER ORDER SPARSE MICROWAVE IMAGING OF PEC SCATTERERS 993

Fig. 16. (a) Reconstruction of the U-shaped cylinder computed using the standard sparse processing. The final results were obtained using the multipoles of theorders n = 0, 8, 9. The adopted SNR was (b) 20; (c) 10; and (d) 5 dB.

Fig. 17. (a) Cross section of the S-shaped cylinder and (b) binary convex hull.

A general current distribution, confined within a circle of aradius a, can efficiently radiate fields having harmonic com-ponents up to n = βa [21]. This also holds for the array ofthe multipoles. Hence, in order to capture the whole spectrumof the field scattered from the target, we have 2a ≥ Dmax,where Dmaxis the maximal distance between any two pointsbelonging to the target.

The above outlined strategy relies only on the largest multi-pole order, based on the consideration that an array of higherorder multipoles is in principle capable of radiating all har-monic components of lower order. However, we cannot foreseea priori if all orders will be radiated with the same efficiency.Accordingly, to take full advantage of both the highest-ordermodel and the lower-order ones, we also consider the resultfrom the standard sparse imaging (zeroth-order) and the resultfrom the sparse imaging of the order n− 1. We obtain the finalresult by juxtaposing the three partial results.

B. Standard and Higher Order Imaging of a Circular Cylinder

To provide further insight into the higher order sparse pro-cessing, based on the observations in the previous section,we investigated the reconstruction of a circular cylinder ofthe radius a = 7.5 cm. The target was probed by an array ofhalf-wavelength dipoles, as shown in Fig. 6. The radius ofthe antenna array was 0.75 m, and the operating frequencywas f = 2 GHz. To calculate the array response, we used thecommercial software WIPL-D Pro [25]. The simulated datawere corrupted by the additive Gaussian noise, where thesignal-to-noise ratio (SNR) was 20 dB.

The size of the search space was 4λ× 4λ. According to thetheory of the degrees of freedom [26], the number of anten-nas that guarantees collecting all available information carriedby the scattered field is M = 2O/λ+ 1, where O denotes theperimeter of the search space. Here, the optimal number wasM = 33, but for convenience, we chose M = 36. In all theinversions, we set the regularization parameter in accordancewith the knee of the L-curve [27].

First, we assumed that the target was illuminated by a sin-gle dipole located in the negative direction of the x-axis. Giventhe size of the target, the expected maximum order of themultipoles was n = βa ≈ 3. Hence, we observed the resultsobtained by independently using the orders n = 0, 1, 2, 3 inthe inversion procedure. As expected, Fig. 7, the zero-ordersources exactly recover the portion of the target probed viaspecular reflection. Conversely, higher order sources give riseto reconstructions that surround the scattering center. The sizeof these surrounding contours increases with the consideredorder, which is in agreement with the theoretical predictionsof Section IV-A.

To complete the target image, we considered the whole arrayof probing dipoles and superimposed the images computedfor different incident angles (Fig. 8). Again, the image com-puted for n = 0 provided an accurate estimate, which couldnot be improved by means of the higher order sources. Theexplanation of this circumstance resides in the convexity of theunknown shape. As a matter of fact, as shown in [21], the zero-order sources are the most suitable for imaging convex bodies.In the following, we will show that higher order sources areinstead needed to retrieve complex, nonconvex shapes.

V. MULTISTEP INVERSION PROCEDURE

As mentioned in Section IV, one of the important parame-ters in the proposed inversion scheme is the maximum orderof the radiating multipoles, n. In practice, the electrical sizeof the target is unknown, but several methods exist to estimateit from the scattered-field data, such as the one in [28], basedon the spectral band limitedness of the radiated fields. Here,we calculate it using the standard sparse processing [10], as italso provides an estimate of the convex hull of the object underinvestigation.

994 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 64, NO. 3, MARCH 2016

Fig. 18. (a) Reconstruction of the S-shaped cylinder computed using the standard sparse processing. The final results were obtained using the multipoles of theorders n = 0, 11, 12. The adopted SNR was (b) 20 and (c) 5 dB.

Fig. 19. (a) Cross section of the star-like cylinder and (b) its convex hull.

The results of higher order sparse imaging may exhibit arti-facts lying outside of the convex hull. These “artifacts” are dueto the fact that higher order sources occupy large contours toaccurately represent the field scattered from the target. Thisphenomenon is visible in Fig. 8. Accordingly, the useful partof the higher order imaging result is its intersection with theestimated convex hull.

In such a sparse processing, we take a regularization param-eter value slightly larger than the knee of the L-curve, since,as is well known, this leads to smoother images,which facil-itates the estimation. Once we solve the sparse imaging, weassume that a pixel belongs to the convex hull if its value isabove a threshold. To set the threshold, we use the curve relat-ing the area of the convex hull to different threshold values.This curve has a reversed L shape, so that we can pick the knee-value for the threshold. Finally, we set to one the pixels insidethe convex hull and set to zero the pixels outside the convexhull. In our computations, we used the bwconvhull MATLABcommand [29].

From the estimated convex hull, we determine the multipoleorder as n ≈ 0.5βDmax, where Dmax is the maximum distancebetween any two pixels belonging to the estimated convex hull.The binarized convex hull is used as a digital filter to removethe pixels that are located outside of the convex hull.

The final result is the superposition of three images: the oneobtained using the multipoles of the highest order, as deter-mined from the estimated convex hull, n; the one obtained usingthe multipoles of the order n− 1; and the one resulting from thestandard sparse processing.

VI. NUMERICAL RESULTS

A. Cross-Shaped Cylinder

To illustrate complex target imaging, we considered a cross-shaped cylindrical body depicted in Fig. 9, where a = 0.2 m.The reconstruction of this shape is, in general, a difficult taskdue to the multiple scattering from the arms of the cross. In thecomputations, we used a 3λ× 3λ search space. The grid con-sisted of 40× 40 nodes, corresponding to the sampling rate of0.08λ. Given the size of the search space, the optimal numberof antennas foreseen by the theory was M = 8 · 3/λ+ 1 = 25.For convenience, we set M = 24.

First, we applied the standard sparse processing. To calcu-late the regularization parameter, we used the L-curve method.The considered parameter values were in the range 2 ≤ γ ≤ 15,with a Δγ = 0.2 step. Fig. 10 shows the obtained curve, aver-aged with respect to different angles of incidence. The kneeof the L-curve corresponded to the value of the regulariza-tion parameter, γ0 ≈ 5. Fig. 11(a) and (b) shows the resultsof the standard sparse processing obtained for SNR = 20 dBand SNR = 5 dB, respectively. The results were stable withrespect to noise. Naturally, the images computed for γ0 yieldeda very sparse solution. For slightly smaller values of the regu-larization parameter than γ0, new pixels belonging to the targetcontour were revealed. On the opposite, when we increasedthe value of the regularization parameter beyond γ0, the sparsenature of the solution disappeared.

Irrespective of the value of the regularization parameter,in the considered range of SNR, the endings of the crosswere accurately estimated; however, the concave pits werenot always obvious. Therefore, we resorted to the higherorder sparse processing to improve the imaging accuracy. Asexplained in Section V, we first determined the convex hull ofthe object. For this purpose, we used the results of the sparseprocessing for which γ > γ0. The corresponding values of theregularization parameter belong to the vertical segment of theL-curve presented in Fig. 10. The associated images are givenin Fig. 11(c) and (d). We note that any value from this regionwould give the same result.

We then assume that pixels whose values are higher than athreshold belong to the convex envelope. To set the threshold,we compute the area of the convex hull as a function of all pos-sible threshold values. Typical appearance of the curve is given

STEVANOVIC et al.: HIGHER ORDER SPARSE MICROWAVE IMAGING OF PEC SCATTERERS 995

Fig. 20. (a) Reconstruction of the star-shaped cylinder computed using the standard sparse processing. The final results were obtained using the multipoles of theorders n = 0, 5, 6 for (b) SNR = 20 dB; (c) SNR = 10 dB; and (d) SNR = 5 dB.

TABLE ISPARSITY ORDER

in Fig. 12. To assure the tight convex hull, we use the knee pointas the threshold. Fig. 12 shows the related binary convex enve-lope. Using this result, we obtained n ≈ 4 which is indeed anaccurate estimate.

Fig. 13 presents the images associated with different multi-pole orders, n ≤ 4. The sources of the orders n = 3, 4 revealedthe true shape of the cross. Nevertheless, as expected, higherorder images contained “ghost” contours surrounding the tar-get. The size of the contours grew with the multipole order.To remove such artifacts, we applied the filter equivalent to thebinary convex hull. Finally, we superimposed the results of thestandard sparse processing, and the results obtained using themultipoles of the orders n and n− 1. Fig. 14 shows the finalresults of this multistage process for SNR = 20, 10, and 5 dB.The shape of the cross was clearly visible in the final images.For each order, we computed the regularization parameter usingthe L-curve.

We note that a further increase of the multipole order didnot improve the shape estimation. In this case, we foundthat all equivalent sources were outside of the body. This isin agreement with our expectations. It is worth noting that,the filtering step would in any case remove the contributions

from the multipoles of the order n > n, even if they aretaken into account (for instance if the maximum order is notaccurately set).

B. U-Shaped Cylinder

Fig. 15(a) shows the cross section of our next example, aU-shaped cylindrical body. The dimensions of the body werea = 0.26 m and b = 0.34 m. The search space was 6λ× 6λ.The grid consisted of 50× 50 nodes, corresponding to thesampling rate of 0.12λ. According to the theory, 49 antennaswere sufficient. Hence, we set M = 50. First, we computed theL-curve for 2 ≤ γ ≤ 8, with the step Δγ = 0.2. After averag-ing, with respect to different angles of incidence, we obtainedγ0 ≈ 6.4. Fig. 16(a) shows the result of the sparse process-ing computed for γ0 ≈ 6.4 and SNR = 20 dB. The zero-ordermultipoles estimated correctly the convex part of the body con-tour. As expected, the algorithm indicated strong scatteringcenters such as the wedges. However, irrespective of the valueof the regularization parameter, the zero-order sources were notable to reconstruct the concave part of the target. Moreover,due to the multiple scattering, a few false pixels appeared atthe target opening.

To compute the convex hull of the object, we used thereconstruction results obtained for γ > γ0 [Fig. 15(b)]. Thecorresponding maximum order of the radiating field was n =0.5βDmax ≈ 9. Hence, we computed the images for n = 8, 9.Fig. 16(b)–(d) shows the final results of the multistage process,tested against different noise levels. Thanks to the use of thehigher order measurement models, the target image was sig-nificantly improved. In particular, the erroneous pixels closeto the opening of the U-shape were suppressed, thus allowingthe appreciation of the concavity of the target. The results weresatisfactory even for a noise level as low as SNR = 5 dB.

C. S-Shaped Cylinder

Next, we considered a complex shape with the cross sectiongiven in Fig. 17(a), where a = 0.26 m. In this case, we usedthe search space of the size 9λ× 6λ with the spacing of 0.15λ.The array consisted of M = 72 antennas, while the theory ofthe degrees of freedom predicted M = (9 + 6) 4 + 1 = 61.

996 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 64, NO. 3, MARCH 2016

From the estimated convex hull [Fig. 17(b)], we obtainedfor the highest order of the radiating multipoles, n ≈ 12. InFig. 18(a), we show the reconstruction calculated using thestandard sparse processing for SNR = 20 dB. Due to thestrongest scattering, the zero-order sources emphasized the cor-ners whereas the central segment was completely invisible. InFig. 18(b) and (c), we presented the superposition of the imagescomputed using the zero-order multipoles and the higher orderones. Again, the higher order multipoles were very helpful inthe reconstruction of the hidden, central part of the target. Bytesting the procedure against a value of the SNR as low asSNR = 5 dB, we demonstrated the robustness of the sparseprocessing with respect to the noise.

D. Star-Shaped Cylinder

As the concluding example, we considered a star-like cylin-der depicted in Fig. 19(a), where a = 0.12 m and b = 0.06 m.For the imaging, we used the search space of the size 6λ× 6λand the sampling rate of 0.1λ. The array consisted of M = 50antennas, whereas 49 was the optimal number according to thetheory. Fig. 19(b) presents the estimated convex hull. The asso-ciated order of the maximum field component was n ≈ 6. InFig. 20(a), we show the result of the standard sparse process-ing. In Fig. 20(b)–(d), we present the result of the superpositionof the images computed using the orders n = 0, 5, 6 for dif-ferent SNRs. In this case, the lowest value of SNR that allowedthe recovery of the star-like shape was SNR = 10 dB.

E. Sparsity Order

Finally, in Table I, we report the number of the significantcoefficients for each example and for each multipole order.Most of the nonzero coefficients belong to the outer contour;hence, the numbers are slightly larger than those obtained inthe standard sparse processing.

VII. CONCLUSION

This paper proposed a multipole-based sparse algorithm fortarget imaging and classification. In this context, the stan-dard sparse imaging with the pulse basis functions may beconsidered as the zero-order version of the algorithm. Byincluding higher order multipoles, the potential of the algo-rithm improved. In particular, by means of the multipoles of theorders close to n = 0.5βDmax, we were able to extract somepreviously inaccessible properties of the targets, namely theirconcave boundaries. On several illustrative examples, usingdifferent noise levels, we successfully tested the proposed algo-rithm. We also confirmed that the multipoles of the ordershigher than n = 0.5βDmax are inappropriate for the target esti-mation, being related to weakly radiating sources, whose fieldsquickly vanish outside of the target.

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Marija Nikolic Stevanovic received the B.Sc., M.Sc., and Ph.D. degrees fromthe University of Belgrade, Belgrade, Serbia, in 2000, 2003, and 2007, respec-tively, and the Ph.D. degree from Washington University in St. Louis, St. Louis,MO, USA, in 2011, all in electrical engineering.

In 2001, she joined the School of Electrical Engineering, University ofBelgrade, as a Teaching Assistant, and promoted to an Assistant Professor in2008. Her research interests include inverse scattering, array processing, andantennas analysis and design.

Lorenzo Crocco (SM’10) was born in Naples, Italy, in 1971. He received theLaurea degree (summa cum laude) in electronic engineering and the Ph.D.degree in applied electromagnetics from the University of Naples “FedericoII,” Naples, Italy, in 1995 and 2000, respectively.

Since 2001, he has been a Research Scientist with the Institute for theElectromagnetic Sensing of the Environment, National Research Council ofItaly (IREA-CNR), Naples, Italy. From 2009 to 2011, he was an AdjunctProfessor with Mediterranea University of Reggio Calabria, Calabria, Italy,where he is currently a Member of the Board of Ph.D. advisors. Since 2013, hehas been a Member of the Management Committee of COST Action TD1301on microwave medical imaging. In 2014, he was habilitated as a Professor ofElectromagnetic Fields, by the Italian Ministry of Research and University. Hisresearch interests include electromagnetic scattering problems, imaging meth-ods for noninvasive diagnostics, through the wall radar and ground-penetratingradar, as well as microwave biomedical imaging, and therapeutic uses ofelectromagnetic fields.

Dr. Crocco is a Fellow of The Electromagnetics Academy (TEA). He wasthe recipient of the Barzilai Award for Young Scientists from the ItalianElectromagnetic Society (2004), the Young Scientist Awardee at the XXVIIIURSI General Assembly (2005). In 2009, he was awarded as one of the TopYoung (under 40) Scientists of CNR (2009).

Antonije R. Djordjevic was born in Belgrade, Serbia, on April 28, 1952. Hereceived the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from theUniversity of Belgrade, Belgrade, Serbia, in 1975, 1977, and 1979, respectively.

In 1975, he joined the School of Electrical Engineering, University ofBelgrade, as a Teaching Assistant. He was promoted to an Assistant Professor,an Associate Professor, and a Professor, in 1982, 1988, and 1993, respec-tively. In 1983, he was a Visiting Associate Professor at Rochester Instituteof Technology, Rochester, NY, USA. Since 1992, he has been an AdjunctScholar with Syracuse University, Syracuse, NY, USA. In 1997, he becamea Corresponding Member of the Serbian Academy of Sciences and Arts, and aFull Member in 2006. He has authored or coauthored about 230 papers (61of which were published in leading international journals), and coauthored3 monographs, 2 chapters in monographs, 9 monographs with software (allmonographs published in the USA), and 13 textbooks at the university level(in Serbian). His research interests include numerical electromagnetics, in par-ticular applied to fast digital signal interconnects, wire and surface antennas,microwave passive circuits, and electromagnetic-compatibility problems.

Dr. Djordjevic has been a Reviewer for the IEEE TRANSACTIONS ON

MICROWAVE THEORY AND TECHNIQUES, the IEEE TRANSACTIONS ON

ANTENNAS AND PROPAGATION, and other journals.

Arye Nehorai (S’80–M’83–SM’90–F’94) received the B.Sc. and M.Sc.degrees from Technion, Haifa, Israel, and the Ph.D. from Stanford University,Stanford, CA, USA, in 1976, 1979, and 1983, respectively, all in electricalengineering.

He is the Eugene and Martha Lohman Professor and the Chair of PrestonM. Green Department of Electrical and Systems Engineering (ESE), and aProfessor with the Department of Biomedical Engineering (by courtesy) and theDivision of Biology and Biomedical Studies (DBBS), Washington Universityin St. Louis (WUSTL), St. Louis, MO, USA, where he serves as a Director ofthe Center for Sensor Signal and Information Processing. Under his leadershipas a Department Chair, the undergraduate enrollment has more than tripled inthe last four years. He was a Faculty Member of Yale University, New Haven,CT, USA, and the University of Illinois at Chicago, Chicago, IL, USA.

Dr. Nehorai has been a Fellow of the Royal Statistical Society since 1996,and a Fellow of the AAAS since 2012. He served as an Editor-in-Chief of theIEEE TRANSACTIONS ON SIGNAL PROCESSING from 2000 to 2002. From2003 to 2005, he was the Vice President (Publications) of the IEEE SignalProcessing Society (SPS), the Chair of the Publications Board, and a mem-ber of the Executive Committee of this Society. He was the Founding Editor ofthe special columns on Leadership Reflections in the IEEE Signal ProcessingMagazine from 2003 to 2006. He was elected as a Distinguished Lecturer ofthe IEEE SPS for a term lasting from 2004 to 2005. He was the PrincipalInvestigator of the Multidisciplinary University Research Initiative (MURI)project titled Adaptive Waveform Diversity for Full Spectral Dominance from2005 to 2010. In 2001, he was named University Scholar of the University ofIllinois. He was the recipient of the 2006 IEEE SPS Technical AchievementAward and the 2010 IEEE SPS Meritorious Service Award. He was also therecipient of several Best Paper Awards in the IEEE journals and conferences.