compressive sensing in electromagnetics a review.pdf

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

Compressive Sensing inElectromagneticsVA Review

Andrea Massa, Paolo Rocca, and Giacomo Oliveri

ELEDIA Research Center, Department of Information Engineering and Computer Science,University of Trento, 38123 Trento, Italy,

E-mails: [email protected]; [email protected]; [email protected]

Abstract

Several problems arising in electromagnetics can be directly formulated or suitably recast for an effective solutionwithin the compressive sensing (CS) framework. This has motivated a great interest in developing and applying CSmethodologies to several conventional and innovative electromagnetic scenarios. This work is aimed at presenting,to the best of the authors knowledge, a review of the state-of-the-art and most recent advances of CS formulationsand related methods as applied to electromagnetics. Toward this end, a wide set of applicative scenarios comprisingthe diagnosis and synthesis of antenna arrays, the estimation of directions of arrival, and the solution of inversescattering and radar imaging problems are reviewed. Current challenges and trends in the application of CS to thesolution of traditional and new electromagnetic problems are also discussed.

Keywords: Antenna and array diagnosis; antenna arrays; compressive sensing (CS); direction-of-arrival (DoA) estimation;inverse problems; radar imaging; sparse problems

1. Introduction and Scenario

The Compressive Sensing (CS) paradigm has enabled thedevelopment of completely new approaches in severalresearch areas of signal processing, information theory, com-puter science, and electrical engineering [1, 2]. The promise toovercome the common wisdom in data acquisition based onShannons celebrated theorem [1] and to allow one to recover(certain) signals/phenomena from far fewer measurements thantraditional techniques has attracted considerable interest (seeFigure 1). Therefore, its study/application to those systemsusually severely constrained by the Nyquists sampling rate [1, 2](e.g., imaging [3, 4], audio/video capture [5], and communica-tions [6]) has been strongly motivated.

CS-based techniques build upon the assumption that manyphysical quantities are intrinsically or extrinsically sparse andthey can be represented by few nonzero expansion coefficients,with respect to suitable expansion bases [1]. Indeed, CS ap-proaches essentially look for an approximate solution x of alinear system y Ax, while requiring that x has the minimumnumber of nonzero entries [see (8)]. If the acquisition process(i.e., the input/output transformation matrix A) fits suitable work-ing conditions, a high-dimensional signal can be exactly retrievedfrom a small set of measurements through efficient deterministic/Bayesian search strategies [1, 712].

In addition to their fundamental advantages over Shannonstheorem-based methodologies, the success of CS formulations isalso related to their intrinsic flexibility and generality [1, 11, 12],the effectiveness and the numerical efficiency of the corre-sponding retrieval techniques [10], and the wide availability ofsoftware (SW) libraries implementing state-of-the-art CS algo-rithms [1315] for effectively dealing with complex engineeringproblems [10].

In this framework, Electromagnetics represents a relativelynew field of application. While CS methods have been earlyadopted in some specific electromagnetic-related applicative do-mains such as radar imaging [16], their exploitation has beeninitially limited to those recovery problems that naturally fit thestandard CS requirements (i.e., linearity and unknowns sparse-ness [1, 16]). However, starting from the consideration thatseveral conventional electromagnetic problems can be properlyreformulated (e.g., using suitable approximations or if some apriori knowledge is available [1720]) to still lie within the set ofthe CS-tractable ones, CS has been recently extended to applica-tions beyond intrinsically CS-compliant problems, and interestingresults have been obtained, ranging from antenna array diagnosisand synthesis [2123] or direction-of-arrival (DoA) estimation[18, 20] until inverse scattering and microwave imaging [2426].

The aim of this work is then that of reviewing, to the bestof the authors knowledge, the state-of-the-art and the more

IEEE Antennas and Propagation Magazine, Vol. 57, No. 1, February 2015 1045-9243/15/$26.00 2015 IEEE 1


recent advances of CS techniques, as applied to electromag-netics, focusing on three main fields of research, i.e., antenna ar-rays, inverse scattering, and radar imaging. The most widelyadopted solution strategies, current challenges, and limitationswill be discussed by also envisaging future trends of the CS re-search topic.

Toward this end, the paper is organized as follows: Aftera short resume of CS problem statements and fundamentaltheorems (see Section 2), the CS reconstruction strategies usuallyadopted in electromagnetics are surveyed in Section 3. Section 4presents a review of the applications of CS in the electromag-netics framework starting from the diagnosis and the synthesisof antenna arrays (see Section 4.1) and the DoA estimation (seeSection 4.2) to the solution of inverse scattering (see Section 4.3)and radar imaging (see Section 4.4) problems. Some conclu-sions are eventually drawn (see Section 5).

2. CSVProblem Statement

2.1 Definition of the Signal Model

Let us consider a real-valued signal f r [without lossof generality, space-dependent signals will be assumed here-inafter. However, the same formulation applies to other sce-narios, by replacing r with the suitable variable (e.g., time,frequency, wavenumber, etc.)] represented in a suitable ex-pansion basis nr by means of an N -dimensional vectorf ffn 2 R; n 1; . . . ;Ng complying with

f r XNn1

fnnr (1)

whose image yr is related to f r through

yr L f rf g (2)

with Lfg being the linear observation operator. Let us as-sume that the signal yr is acquired through the followingmeasurement operation:

ym yr; mrh i m 1; . . . ;M (3)

where h; i is the inner product, and fmr;m 1; . . . ;Mgare the M sensing waveforms (e.g., Dirac delta functions) being

M N : (4)

By virtue of the linearity of the operators at hand, one can substi-tute (2) and (1) in (3) to yield the following relationships:

ym L f rf g; mrh i

LXNn1

fnnr( )

; mr* +

m 1; . . . ;M

XNn1

fnL nrf g; mr* +

XNn1

fn L nrf g; mrh i XNn1

mnfn

whose matrix form looks as follows:

y f (5)

with y fym 2 R;m 1; . . . ;Mg being the vector of observa-tions, and

mn L nrf g; mrh i;fm 1; . . . ;M ; n 1; . . . ;N

o(6)

is the sensing matrix.

Moreover, let us suppose that f is S-sparse [1], with respectto the signal basis fnk 2 R; k 1; . . . ;K; n 1; . . . ;Ng,so that it can be expressed as

f x (7)

where fn PN

k1 nkxk , and x fxn 2 R; n 1; . . . ;Ng hasonly S nonzero entries (i.e., kxk0

PNn1 jxnj0 S, with k k0

being the 0-norm). Under such hypotheses, (5) and (7) can becombined to give the following observation equation [7]:

y x Ax (8)

where A fAmn 2 R;m 1; . . . ;M ; n 1; . . . ;Ng is the ob-servation matrix whose (m, n)th entry is equal to

Amn XNk1L krf g; mrh ikn : (9)

Equation (8) points out two fundamental requirements of thestandard CS paradigm: (a) the relation between unknowns

Figure 1. Number of CS-related papers published each year(based on IEEE Xplore databases).

IEEE Antennas and Propagation Magazine, Vol. 57, No. 1, February 20152


(x) and data (y) is linear (i.e., it can be represented bymeans of a matrix multiplication) and (b) the unknown vec-tor (x) is sparse (i.e., a suitable signal matrix is known sothat x has only few nonzero entries).

2.2 Sampling and Recovery Problems

Equation (8) can be used to formulate two different CSproblems: (i) the CS sampling problem and (ii) the CS recov-ery problem.

The CS sampling problem is mainly concerned with thedesign of the signal acquisition system, and it requires specifyingthe type [i.e., mr] and the minimum number of measurementsM that allows one the exact recovery of the unknown S-sparsevector x for a given combination of Lfg, fnr; n 1; . . . ;Ng,and . Mathematically, the sampling problem can be statedas follows.

CS Sampling Problem

Given Lfg, fnr; n 1; . . . ;Ng, and , find M andfmr, m 1; . . . ;Mg, such that (7) is a well-posed problemwhen x is S-sparse.Such a problem can be exactly solved (i.e.,a unique and exact solution can be found) due to the followingtheorem.

Theorem 1 [7]

A necessary and sufficient condition for the well-posednessof (8), when x is S-sparse, is that A complies with the RestrictedIsometry Property (RIP) of order 2 S.

The observation matrix A satisfies the RIP of order S withconstant 0 G G 1 if, for all S-sparse v 2 RN , the followingcondition holds true [1, 7]:

1 kAvk2kvk2 1 (10)

with k k2 being the 2-norm. Intuitively, such a conditionguarantees that A preserves the length (i.e., the correspond-ing k k2 value) of every S-sparse signal v after its projectionin the lower (M N ) M -dimensional space of the observations.This does not verify if x is not S-sparse, and the equation y Ax turns out to be still ill posed since the kernel of a rectangular(M G N ) matrix A is not empty.

On the other hand, it is worthwhile to notice that check-ing (10) is, in practice, numerically unfeasible even for smallA matrices [7], while observation matrices that a priori fit theRIP (e.g., random Gaussian matrices [27]) are usually consid-ered in CS signal processing/compression problems [27]. Un-fortunately, electromagnetic applications do not usually allowthe direct user-definition of the observation matrix [21] since

it is mainly dictated by the physics of the problem at hand (i.e.,the form of the operator Lfg) and it can be only slightly con-trolled in an indirect way (see Equation (4)). Since no generalrules exist or have been yet proposed in such a latter direc-tion, the CS sampling problem (i) will not be further discussedin the following.

As concerns the retrieval of an S-sparse vector from aset of measurements (CS recovery problem), it is usually for-mulated in its canonical form as follows.

CS Recovery Problem

Given y 2 RM , find x 2 RN complying with (8) andsuch that x is S-sparse, whose unique solution is guaranteedunder the hypotheses of Theorem 1. Nevertheless, real-worldproblems cannot be exactly formulated in the form (8) be-cause of the unavoidable measurement noise. Therefore, thenoise formulation considers the form [10]

ey Ax z (11)with z 2 RM being the stochastic or deterministic unknownnoise term. Accordingly, the recovery problem turns out tobe generally formulated as follows.

Noisy CS Recovery Problem

Given ey 2 RM , find x 2 RN complying with (11) andsuch that x is S-sparse.

3. CS Recovery Algorithms

This section is aimed at reviewing the basic ideas behindCS techniques usually adopted in electromagnetics rather thanproviding an exhaustive discussion of the existing methodolo-gies to address the Noisy CS Recovery Problem (the interestedreader is referred to [10] and the references therein for an in-depth introduction on computational methods for solving CSproblems). Toward this end, CS recovery methods belongingto both deterministic (see Section 3.1) and Bayesian (seeSection 3.2) classes [10] will be briefly recalled hereinafter.

3.1 Deterministic CS Strategies

From a deterministic point of view, the solution of theCS recovery problem turns out to be [10]

bx arg minxkxk0h in o

subject to kAx eyk2 : (12)Since the minimization of the 0-norm functional in (12) is anNP-hard problem [7, 11, 28], it cannot be easily/profitably(computationally) directly addressed. Therefore, greedy pursuit

IEEE Antennas and Propagation Magazine, Vol. 57, No. 1, February 2015 3


methods (e.g., orthogonal matching pursuit (OMP) [29], stage-wise OMP (StOMP) [30], and compressive sampling matchingpursuit (CoSaMP) [31]) aimed at finding the sparsest x throughan iterative search have been proposed. They are usually basedon the iterative refinement of the estimated solution bx (i.e.,bxi; i 0; . . . ; I) as schematized in the following [10, 29, 32].1. Initialization: Initialize the guess solution (bx0 0),

the residual (0 ey), the set of nonzero coeffi-cients (0 ;), and the iteration index (i 1).

2. Identification: Find the nth column of A,an fAmn 2 R;m 1; . . . ;Mg, such that n 6i1 and an is most strongly correlated withi1 (i.e., n argfmaxni1 ang). Updatei (i i1 [ n).

3. Estimate: Find the ith guess solution bxi, such thatbxi arg min

xey Aix

2

h in o(13)

with Ai fAimn 2 R;m 1; . . . ;M ; n 1; . . . ;Ngbeing the ith observation matrix whose generic en-try is given by

AimnAmn; if n 2 i0; elsewhere

; m 1; . . . ;M ; n1; . . . ;N :

(14)

4. Update: Set i ey Aibxi, update the iterationindex (i i 1), and repeat steps 24 until a suit-able stopping criterion (e.g., based on the maximumnumber of iterations) is satisfied.

Alternatively, convex relaxation (CR) approaches basedon the Basis Pursuit (BP) [33] have been widely used bysubstituting the 0-norm functional in (12) with an 1-normfunctional (relaxation) to yield [10, 11]

bx arg minxkxk1 n o

subject to kAx eyk2 : (15)The diffusion and success of such a latter formulation ismainly motivated by the fact that the unique solution of thelinear problem associated to (15) exactly coincides with thatof (12) in the noiseless case and if the RIP holds true. More-over, due to the convexity of (15), the arising optimizationproblem enables the use of local search algorithm [10], andefficient-solution SW packages have been developed (e.g.,the well-known L1-Magic tool [13]).

Still within the CR framework, alternative formulationsto (15) have been then employed/proposed. Let us considerthe minimum 1-error technique [28]

bx arg minxkAx eyk1 n o (16)

the 1-regularized least-square method [34]

bx arg minxkAx eyk2 kxk1 n o (17)

with being a regularization parameter; the least absoluteshrinkage and selection operator (LASSO) [35]

bx arg minxkAx eyk2 n o subject to kxk1G

(18)

and the Total Variation (TV) [36, 37]

bx arg minxkxk1 rAxk k1h in o

subject to kAx eyk2 : (19)It is worth pointing out that a shared key-feature of these ap-proaches is the exploitation of numerically efficient local searchtechniques [10, 38], whose implementation is often available[1315] because of their nature of linear- or quadratic-programmingproblems. Toward this end, let us recall interior-point methods [13]and gradient techniques [10, 39]. As an example and with refer-ence to (17), this latter computes the next estimate bxi1 ac-cording to the following rule [10]:

xi arg minc c bxi 0A0 Abxi ey hn

12i c bxi 22kck1iobxi1 bxi i xi bxi 8>>>>>:

where i and i are scalar user-defined parameters [10]. Insuch a case, the convergence to an optimal or quasi-optimal so-lution holds true when A satisfies the RIP [10], and the conver-gence rate can be significantly improved for a warm starting,i.e., when a good initial estimate bx0 is available [10]. Unfortu-nately, both previous assumptions are rarely satisfied in CS-basedelectromagnetics.

3.2 BCS Techniques

Whether deterministic CS recovery algorithms provide re-liable and computationally efficient solutions to CS under RIPconditions [11], these latter techniques cannot be generally sat-isfied or a priori efficiently verified in several electromag-netic problems [21, 40, 41]. Moreover, since deterministicapproaches do not usually provide any estimation on the confi-dence level of the estimated solution bx, their exploitation is notadvisable whenever some sort of (a priori) reliability assess-ment of the CS result is mandatory [42]. Therefore, alternativeCS recovery algorithms have been studied in CS electromag-netics literature that, on the one hand, do not rely on the RIP ofA to yield accurate and stable results, and, on the other hand,naturally provide the degree of confidence of bx [8]. Let us refer



to a Bayesian perspective resulting in the following formulationof the CS retrieval [8, 4346]:

bx arg maxxP xjey n o subject to x Px (20)

where stands for distributed as, and Px is a suitableprior used to enforce the sparsity of x [8]. In such a way, theRIP of A is not required, and the arising Bayesian CS (BCS)approach implicitly calculates the full posterior density func-tion for bx, i.e., Pxjey, rather than a single-point estimationbx, giving the confidence level for each nth component of bx, i.e.,bxn, as a function of the covariance matrix C of Pxjey [8]. In-deed, it is possible to prove that if Px complies with thefollowing zero-mean Gaussian density function [8]:

Px QN

n1ffiffiffiffiffihnp

exp hnx2n2

2N2(21)

where hn is a Gamma-distributed hyperparameter, thenPxjey turns out to be a multivariate Gaussian distributionwith mean

bx b1CA0ey (22)with 0 being the transpose operator, and covariance

C b1A0A bh 1 (23)where the auxiliary parameters bh; b are determined throughdedicated algorithms [8] solving (24), shown at the bottom ofthe page.

bh; b arg maxh;

N log 2 log I A diagh 1A0 ey0 I A diagh 1A0h i1ey

2

26643775

8>>>:9>>=>>;:

(24)

In addition to the information on the reliability of bx, the knowl-edge of C can be also used to extend the capabilities of the re-trieval algorithm, with respect to deterministic CS strategies. Asa matter of fact, since C yields a direct evaluation on the levelof uncertainty of the estimation, it indirectly acts as an indica-tor about the enhancement or usefulness of a set of measure-ments depending on the higher/lower level of uncertainty of theresulting estimation [8]. Following this line of reasoning, anadaptive BCS algorithm can also be envisaged in terms of aniterative application of the BCS, where, at each step, the choiceof additional measurements is based on the information contentof the added information (i.e., uncertainty reduction) throughthe computation of (23).

Within the BCS formulation (20), several variations havebeen proposed to include a priori information in the inversionprocess through suitable prior definitions [9, 47, 48]. For in-stance, hierarchical-Laplace priors on x have been includedin [8, 9] by replacing (21) with

Px 2exp

2kxk1

(25)

where is a hyperprior usually distributed according to theGamma density function [8]. Such a formulation has the ad-vantage of being mathematically equivalent to (17), thus en-abling a direct link between Bayesian and deterministic CSretrieval strategies [9]. However, the BCS inference cannot becarried out in closed form [8] since (25) is not conjugate tothe Gaussian likelihood function usually considered to modelthe noise distribution.

Otherwise, the inversion of K correlated CS tasks (e.g.,repeated MRI images of the same scene [47]) has been alsoconsidered by statistically linking the Pxk, k 1; . . . ;K asfollows:

PxkZPxkhPhdh (26)

with h being the shared hyper-prior, to derive the so-calledmultitask BCS (MT-BCS) strategy [47]. Furthermore, tempo-ral correlation between successive CS inversion processes hasbeen also considered, by introducing a block-sparse Bayesianframework [48]. It is also worth noticing that greedy-like al-gorithms to solve the CS problems formulated in a Bayesianframework have been developed [49].

Due to their effectiveness, which often outperforms deter-ministic CS strategies in terms of reconstruction accuracy androbustness [8, 9, 47, 48, 51], as well as the availability of stan-dard implementations of BCS and MT-BCS techniques [50],BCS with Laplace prior algorithms [51], and fast Bayesianmatching pursuit [52] techniques, Bayesian approaches havebeen widely adopted in electromagnetics (see Section 4).

4. Application of CS to Electromagnetics

This section is aimed at providing a review, to the best ofthe authors knowledge, on the use of CS strategies to electro-magnetic problems, by focusing on four main applicative do-mains: diagnosis and synthesis of antenna arrays (see Table 1in Section 4.1), DoA estimation (see Table 2 in Section 4.2),electromagnetic inverse scattering (see Table 3 in Section 4.3),and radar imaging (see Table 4 in Section 4.4).

4.1 CS in Antenna Array Analysisand Synthesis

In antenna arrays, the excitation coefficients are linearlyrelated to the radiated field y through the M N observationmatrix A3, whose (m, n)th entry is given by

Amn exp j2dn cosm (27)



where dn is the nth element position (in wavelengths), and mis the mth observation angle (in radians) [for notation simplic-ity, far-field measurements in a 1-D arrangement are assumed.However, both near-field data and more complex array layoutscan be dealt with, and they have been already addressed in theCS literature]. Starting from the linearity of such a relationship,two standard array problems can be formulated within the CSframework. The former is related to the diagnosis of isolated el-ement failures (see Section 4.1.1), while the other is concernedwith the synthesis of sparse arrangements (see Section 4.1.2).

4.1.1 Array Diagnosis

Let us consider the problem of detecting the failed ele-ments in an N -sized linear array [indicated as array under

test (AUT)] with excitation coefficients fwtn; n 1; . . . ;Ngstarting from a set of M far-field measurements [40, 53]

FtmXNn1

wtn exp j2dn cosm zm m1; . . . ;M :

(28)

More in detail, the problem at hand consists in finding the arrayelements for which wtn 6 wrn, with wrn being the nth excitationcoefficient of the undamaged array (denoted as reference ar-ray). Despite the linearity of (28) as pointed out in Section 4.1, the problem at hand does not fit CS-applicability hypothesesbecause of the dense nature of the unknown vector x fwtn; n 1; . . . ;Ng due to the generally small number of failuresin realistic structures. Nevertheless, the array diagnosis can becasted as a CS one by exploiting a differential approach [40, 53].Instead of determining the AUT coefficients fwtn; n 1; . . . ;Ngfrom the measured data fym Ftm; m 1; . . . ;Mg, the setof differential excitations x fxnwrn wtn; n 1; . . . ;Ng thatradiate the differential field

eymFrm ym Frm Ftm (29)is looked for, with Frm being the mth far-field pattern mea-surement of the reference array [40]. Due to this formulation,the unknown vector x turns out to be now sparse since the num-ber of failed elements S is much more smaller than N [40],while the relationship between data y and unknowns x is stilllinear (27). Thus, CS retrieval techniques can be applied, pro-vided that (27) is transformed in a real-valued form to enable theexploitation of state-of-the-art algorithms (see Section 3). Thiscan be easily done by introducing the fictitious matrix

A RfAg IfAgIfAg RfAg

(30)

Table 2. List of scientific publications on CS as applied toDoA estimation.

Table 3. List of scientific publications on CS as applied toelectromagnetic inverse scattering.

Table 1. List of scientific publications on CS as applied toarray diagnosis/synthesis.



and rearranging the differential excitation and differential fieldvectors in double-length vectors comprising their real and imagi-nary components [53], i.e., x fxn 2 R; n 1; . . . ; 2Ng, where

xnR wrn wtn

n 1; . . . ;NI wrnN wtnN

n N 1; . . . ; 2N ;

and ey feym 2 R; m 1; . . . ; 2Mg, whereeym R Frm Ftm m1; . . . ;MI FrmM FtmM mM 1; . . . ; 2M .

Accordingly, deterministic CS strategies have been first appliedfor detecting the array failures. In [40], the diagnosis of planararrays from a small number (M N ) of near-field mea-surements has been carried out by means of a reweighted 1-minimization algorithm [54] based on the solution of a sequenceof convex problems formulated as (15) through an interior

point method called log-barrier algorithm [13]. The proposedapproach proved to outperform standard fast Fourier transform(FFT) or matrix inversion methods, when highly undersampleddata (i.e., M=N 0:1) affected by noise (e.g., SNR 35 dB)are at hand and a very small number of failures is presentwithin the array layout (i.e., S=N0:01) [40].

The main limitation of such an approach is that it cannota priori guarantee its effectiveness since a proof of the RIP forthe observation matrix is generally not available. To overcomesuch a drawback, BCS methodologies have been recently ap-plied [53]. More specifically, a BCS strategy [8] has been con-sidered in [53] to diagnose linear arrays from near-/far-fieldmeasurements with nonnegligible improvements in terms of de-tection accuracy over 1-based minimization strategies (e.g., thediagnosis error reduced of above 50%) and truncated singularvalue decomposition (SVD) techniques and an enhanced ro-bustness to the noise with good performance also whenSNRG30 dB. Moreover, suitable guidelines for the nonuniformselection of the data measurement (i.e., the angular samplepoints m, m 1; . . . ;M ) have been derived to yield a goodaccuracy (e.g., failure retrieval error below 1%) with a moderatenumber of failures (S=N 0:1) even when limited measure-ments (M=N 0:3) are at disposal. Furthermore, the capabilityof CS strategies to detect partial failures (i.e., amplitude errorsrather than element shutdown) has been assessed [53].

4.1.2 Sparse Array Synthesis

The design of unequally spaced arrays through CS strate-gies has attracted a great attention within the electromagneticscommunity, as confirmed by the recent list of publications (e.g.,[2123, 5558]).

Let us consider the Pattern Matching problem [21],where the positions and the excitations of S array elementsare computed, so that (a) S is minimized, (b) the element loca-tions belong to a (user-defined) candidate set of positionsfdn; n 1; . . . ;Ng, and (c) the corresponding array patternFt matches a reference one Fr in a set of user-defineddirections f m;m 1; . . . ;Mg [21]. Mathematically, sucha synthesis problem can be directly formulated as a CS oneby setting (a) the nth unknown entry xn of the unknown vectorx to the excitation coefficient of the nth candidate antenna ele-ment located at dn, (b) the observation matrix as (27), and (c)the observation/measurement vector y to the reference patternsamples, i.e., fym Frm; m 1; . . . ;Mg [21]. As a matterof fact, x turns out linearly linked to y and, if a suitable sam-pling of the desired antenna aperture is used (i.e., N is chosento be sufficiently large), also sparse [21]. Accordingly, the re-trieved bx provides both the actual element positions (i.e., thecandidate locations dn for which bxn 6 0; n 1; . . . ;N ) aswell as the excitations of the array elements (bxn 6 0).

Following this line of reasoning, the synthesis of maxi-mally sparse arrays has been first solved as a fully real variableproblem in [21, 55]. Symmetric linear [21, 55] and planar [55]

Table 4. List of scientific publications on CS as applied toradar imaging.



arrangements with real excitations have been synthesizedby considering:

Amn cos 2dn cosm 1-D Case (31)

and symmetric real-valued patterns as references [i.e.,Frm Frm, Frm 2 R]. When applied to such aframework, BCS implementations [8] proved to be a conve-nient solution, as compared to state-of-the-art strategies, interms of accuracy, array sparseness, and computational efficiency[21, 55]. More in detail, a satisfactory matching (i.e., normalizedintegral errors below 104) has been reached with a 35%40%element reduction with respect to the corresponding fully popu-lated arrangements affording the same patterns [21, 55]. Similarresults have been also yielded by combining the BCS method withother techniques such as the matrix pencil method (MPM) [58].

Although efficient, these strategies [21, 55] cannot be ef-fectively employed when complex excitations (e.g., those as-sociated to asymmetrically shaped beams) are of interest [56]since they rely on the real-valued nature of the synthesis prob-lem. To overcome such a constraint, the pattern matching prob-lem has been recently formulated for asymmetric linear [56]and planar [57] arrangements within the MT-BCS framework[47]. Since complex arrangements comprise weights that fre-quently exhibit nonnegligible real and imaginary components atthe same spatial locations [56], the correlation between the twocomponents (i.e., the real part and the imaginary one) of the ar-ray excitations has been directly included into the prior defini-tion, Px. Due to this approach, performance close to those ofthe fully-real synthesis has been reached despite the highercomplexity of the problem at hand [56, 57].

A further extension of CS formulations has been more re-cently proposed [22, 23] to address the Mask-Constrainedsyntheses, as well. Toward this end, the following mathematicalformulation has been considered [22, 23]:

bx arg minxkxk1 n o

subject toFtm 1 m 1Ftm m m 2; . . . ;M

(32)

where 1 is the steering angle, m is an angular direction belong-ing to the sidelobe region S (m 2 S, m 2; . . . ;M ), and is the user-defined target mask. By suitably casting (32) into asecond-order cone problem, a CS-based approach has been im-plemented by applying a reweighted 1-based algorithm [54] forthe minimization [22, 23]. Effective performance in terms ofcomputation time (e.g., several orders in magnitude smaller thanprocedures involving global optimization methods), flexibility(i.e., arbitrary user-defined geometries/constraints), and easy calibra-tion (i.e., very few parameters to be tuned) has been yielded [22].

4.2 DoA Estimation

Unlike CS applications to antenna array synthesis, DoAestimation through CS has been widely investigated in the

literature [1720]. As a matter of fact, one of the earliest ap-plications of CS theory in electromagnetics [17, 19] wasconcerned with the relationships between the achievable per-formance of DoA retrieval techniques when applied to ran-dom sensor arrays and CS. Nevertheless, it is worth noticingthat the exploitation of CS strategies to DoA problems is notas direct as for the diagnosis and the synthesis of antenna ar-rays (see Section 4.1.2).

With reference to a linear array composed of M isotro-pic sensors located at dm, m 1; . . . ;M and measuring theincident field due to S monochromatic plane waves comingfrom (unknown) directions s, s 1; . . . ; S, the mth receivedvoltage at the kth temporal snapshot is given by [20]

eykm XSs1

Eks exp j2dm coss zkmm 1; . . . ;M ; k 1; . . . ;K (33)

where Eks is the (unknown) amplitude of the sth incident waveat the kth snapshot, and zkm is the additive noise term at the mtharray element and kth instant. Since the measurement vector atthe kth instant, i.e., yk fykm;m 1; . . . ;Mg, is not linearlydependent on the incident directions fs; s 1; . . . ; Sg of theincoming signals (33), a suitable reformulation of the problemis mandatory to enable the use of CS-based methods [20]. To-ward this end and similarly to Capon or multiple signal classifica-tion (MUSIC) algorithms [59], the angular range is oversampledwith N S samples to rewrite (33) as follows [20]:

eykm XNn1

xkn exp j2dm cosn zkm;

m 1; . . . ;M ; k 1; . . . ;K (34)

where

xkn Eks ; if n s

0; otherwise

k 1; . . . ;K (35)

is the (sparse) vector whose nth entry is the (unknown) ampli-tude of the signal impinging from the direction n. Accordingly,the DoA problem is that of recovering the sparse signal vectorxk fxkn; n 1; . . . ;Ng linearly related to the measurementvector yk through the observation matrix A of entries [20]

Amnexp j2dm cosn

; m1; . . . ;M ; n1; . . . ;N :(36)

Chronologically, the first application of CS to DoA signal de-tection [18, 60] was concerned with the exploitation of an1-regularized least square and CR [39, 61]. In order to en-hance the robustness and the angular resolution accuracy, aniterative CS-based approach has been adopted in [18]. At eachstep, the DoA problem is formulated as the minimization ofan 1-SVD functional corresponding to an even finer grid follow-ing a multiresolution scheme similar to that in [62] for inversescattering. Still dealing with an 1-regularized formulation, a



truncated Newton technique belonging to the gradient-basedstrategies [39] has been adopted in [60]. Both methods provedto be a competitive alternative to classical DoA techniques, in-cluding Estimation of Signal Parameters via Rotational VarianceTechniques (ESPRIT) and MUSIC, in terms of angular resolu-tion, robustness to noise, and tradeoff between acquisition time(i.e., number of snapshots) and estimation reliability [18].

As for multisnapshot DoA architectures, customized ap-proaches have been recently developed [20, 48, 59], as well.In [59], the multisnapshot DoA estimation has been for-mulated as the minimization of the following mixed 2;0-norm functional:

bX arg minXXNn1

exp PK

k1 xkn

22

" #keYAXk2F

" #( )(37)

where Xfxk ; k 1; . . . ;Kg, eYfeyk ; k 1; . . . ;Kg, k k2F isthe Frobenius norm, and are control parameters, and K isthe total number of snapshots [59]. Toward this end, a joint0-approximation (JLZA) algorithm [64] has been adopted byalso comprising the grid refinement strategy proposed in [18].As a result, the JLZA-DoA performances turned out to besignificantly better than those of existing strategies for theminimum/optimal number of snapshots, K, the accuracy inestimating correlated sources, and the suppression of aliasingeffects due to widely spaced (i.e., up to three wavelengths)sensors [59].

The use of Bayesian formulations (20) has been pro-posed in [48] to deal with the same multisnapshot processing,but including the temporal correlation of the sources withinthe estimation strategy through a block-sparse prior over X[48]. The arising evidence maximization problem has thenbeen solved through an ad-hoc BCS strategy called temporalmultiresponse sparse Bayesian learning (T-MSBL) and de-rived from the empirical Bayesian technique in [65]. The nu-merical results have shown that temporally correlated sourcescan be more effectively resolved than throughout state-of-the-artDoA techniques [48].

Still dealing with multiple snapshots, the DoA problemhas been also recently addressed by exploiting hierarchicalsparseness hyperpriors [20]. The multitask version of the BCSformulation [47] has been adopted by setting the prior as in(26) and using an ad-hoc relevance vector machine (RVM)solver for optimizing the arising MT-BCS functional [20]. Nu-merical comparisons with the single-task BCS (ST-BCS) [8]implementation of the same approach and state-of-the-art DoAalgorithms proved the effectiveness and robustness of the ap-proach with a reduction of the root-mean-square error of morethan one order in magnitude (K 20) [20].

When the observation matrix A is affected by noise,modeling for instance the nonidealities of the receivers or theangular grid mismatches, etc., suitable CS-based strategies havebeen recently discussed also [6669]. In [66], the sparse total

least square (S-TLS) method has been proposed to solve thefollowing single-snapshot problem:

bxk arg minxkkxkk1

eyk eAxk 22

1 xkk k22

264375

8>:9>=>; (38)

where eA A Z is the perturbed observation matrix, andZfzmn; m 1; . . . ;M ; n 1; . . . ;Ng is the noise term. De-spite the single-snapshot data at hand [66], the proposed ap-proach was shown to overcome LASSO strategies [35]. Asimilar perturbed CS problem has been also dealt with in[67] through the following CR formulation:

bxk arg minxkkxkk1

subject to eAxk eyk 2 (39)

successively solved with the Alternating Algorithm for Per-turbed Basis Pursuit Denoising (AA-P-BPDN) [67]. Beyondthe effectiveness of the DoA estimation, it is worth noticingthat theoretical conditions for the a priori estimate of the recov-ery error have been deduced [67]. In addition, BCS approaches[9] have been investigated for solving the perturbed CS prob-lem [68]. For example, the off-grid DoA detection has beenformulated with Laplace priors in [9] with enhanced perfor-mances with respect to [18], but at the expense of a slowerspeed, particularly when a dense angular grid (i.e., a large N ) isat hand [68]. Within the same line of reasoning, the use of anexpected likelihood approach [70] in a two-step BCS proce-dure has been considered to mitigate the bias on A [69]. Onceagain, the resulting approach outperformed the spatiallysmoothed MUSIC for the accuracy, but with a higher computa-tional complexity [69].

The use of CS for DoA has not been limited to simpletest cases, but different operative conditions have been takeninto account, ranging from narrow-/wideband signals, linear/planar arrays, up to fixed/dynamic sensors, assessing the flexi-bility of the CS tool besides its effectiveness. For instance, theuse of CS to deal with dynamic sensor arrays (i.e., maneuver-ing receiver) has been addressed in [63], by using a CR strat-egy [10]. The arising spatial CS (SCS) technique turned outto overcome several state-of-the-art methods [e.g., MUSIC, fo-cal undetermined system solver (FOCUSS), Wagstaff (WS),and spatial steered covariance matrix (SSTCM)] in terms of an-gular resolution and ambiguity resolution [63].

4.3 Inverse Scattering

Electromagnetic inverse scattering problems have beenwidely investigated in the CS literature [2426], [41, 42],[7173]. Although the standard framework of an imaging prob-lem cannot be directly tackled with CS strategies because of itsintrinsic nonlinearity [74], several alternative formulations havebeen proposed either within the fully-nonlinear framework



(see Section 4.3.1) or assuming some approximations (seeSection 4.3.2) in modeling the relationships between the data(usually, the scattered field) and the unknowns.

4.3.1 Fully Nonlinear Formulations

Dealing with the reconstruction of dielectric profilesof penetrable objects through CS strategies and within thefully nonlinear framework (i.e., without approximations), theso-called contrast source formulation [74] has been usuallyused [25, 75]. With reference to a 2-D scenario and undertransverse-magnetic (TM) illumination, the scattered electricfield Ekx; y in correspondence with the kth probing sourceillumination (k 1; . . . ;K) complies with the followingData equation:

eEk rkm ZD

Jkr0G rkm=r0

drkm z rkm

rkm 2 O (40)

where frkm;m 1; . . . ;Mg are the M measurement points ly-ing in the observation domain O external to the investigationdomain D, Gr=r0 is the 2-D free-space Greens function,zr is a zero-mean additive Gaussian noise, and J kr is the(unknown) Contrast Source. The inversion procedure is thenaimed at solving (40) by looking for the unknown contrastsource J kr starting from the knowledge of the incident electricfield and the samples of the scattered electric field at each illu-mination (the dielectric profile r is then retrieved by meansof the State equation [74]). Toward this end, the method-of-moments-based pulse-discretized version of (40) is considered[under such assumption, D is discretized in N subdomains Dn(D [Nn1Dn) centered at rn; n 1; . . . ;N , and the nth pulsefunction is defined as nrf1 if r 2 Dn; 0 otherwiseg [25]]and a set of K matrix equations (one for each kth illumina-tion) equivalent to (11) is derived being eykm eEkrkm, xkn J krn and where the (m, n)th entry of the observation matrixis given by

Akmn ZDn

G rkm=r0 drkm: (41)

Doing so, the problem can be still casted into the linear CS-based framework, provided that xk fxkn; n 1; . . . ;Ng issparse [25].

Following such a guideline, Bayesian strategies [8] havebeen applied to qualitatively image sparse (i.e., composed offew pixels) dielectric scatterers in 2-D scenarios [25]. By ex-ploiting a fast RVM technique [8] to solve (24) for retrievingthe unknown contrast source, the corresponding dielectric con-trast in D [25] resulted faithfully reconstructed with a reductionof one order in magnitude of the inversion error when com-pared to the state-of-the-art techniques. Moreover, the methodturned out to be more than three orders in magnitude fasterthan deterministic techniques when dealing with sparse profiles.

Furthermore, it proved to be more robust in high-noise condi-tions (i.e., SNR 5 dB) [25], as well.

Still using the contrast source BCS [75], transverse-electric(TE) data have been effectively processed. Toward this end, themultitask version of the Bayesian retrieval technique [8] hasbeen taken into account to mathematically model the relation-ships among the contrast currents induced by each kth illumi-nation. Despite the increased problem complexity due to thevectorial nature of the data [75], the MT-BCS-TE outperformedthe method in [25].

More recently, such a strategy has also been extended tolocalize sparse metallic scatterers [76] by combining the local-shape-function (LSF) formulation of the inverse scatteringproblem with the MT-BCS retrieval tool. The arising two-stepprocedure proved its effectiveness also when a low number ofilluminations/measurements were available [76].

4.3.2 Approximate Formulations

In order to recast the data and the state equations toprofitable forms for applying CS retrieval tools, several ap-proximate formulations have been considered for linearizationpurposes. Early developments included the application of CSstrategies to imaging problems linearized through the Born ap-proximation (BA) and comprising Laplace priors [9], as envis-aged in [7779]. In such a framework, a 1-D inverse problemhas been solved in [80], by means of a greedy-pursuit strategy(i.e., the subspace pursuit technique [81]), while the use ofBCS strategies with hierarchical priors [8, 47] has been also in-vestigated in [42] and [72], where, besides the reconstruction ofthe contrast profile, an estimate of its confidence level [42]has been provided also.

Linearized formulation alternatives to BA have been re-cently considered [41, 71, 72]. A representative approach isthat discussed in [71], where the inversion strategy has beendeveloped within the Rytov approximation (RA) [82].

The application of CS retrieval strategies to phaseless datahas been introduced [41], as well. In such a case, a two-step re-formulation of the problem has been adopted to retrieve point-like scatterers by solving of a linear system of equations theneffectively yielded through an 1-based CS minimization ap-proach (i.e., the Dantzig selector [83]) [41].

4.4 Radar Imaging

A wide set of the scientific literature on the application ofCS strategies to electromagnetics is concerned with radar imag-ing problems [16]. Since an exhaustive list of all the existing ap-proaches and implementations is almost impossible also due tothe limited space, a summary, to the best of the authors knowl-edge, of the CS techniques as applied to different radar imagingmodalities will be provided in the following.



4.4.1 Monostatic Radar Imaging

Conventional radar imaging problems [84] are usuallyformulated as the retrieval of a reflectivity map given a setof measurements of the scattered electric field [16]. Theycomprise a wide variety of scenarios differing in the mea-surement setup, the sensing objectives, the processing tech-niques, the propagation media, and the sensor configurations[16, 84].

Analogously to inverse scattering, the simplest model ofthe reflected radar signal for adapting the nonlinear formula-tion at hand to the CS framework is based on the BA appliedto the free-space propagation of narrowband plane waves im-pinging on slow targets located in the far field [16]. Withinsuch a framework, a regularized OMP strategy [29] has beenapplied in [85], showing improved discrimination capabilitieswith respect to matched filtering algorithms [85] when detect-ing close targets.

Otherwise, the retrieval of sparse scatterers located inthe Fresnel region has been carried out in [24], where theCR formulation (15) of the arising CS problem has been ad-dressed through the subspace pursuit approach in [81].

In order to effectively treat localized extended scatterers[26], these techniques have been more recently extendedto also deal with multishot single-inputsingle-output sam-pling schemes.

4.4.2 Spotlight SAR Imaging

The solution of Synthetic Aperture Radar (SAR) prob-lems through CS strategies has attracted great attention be-cause, on the one hand, most targets can be easily modeled assparse distributions in suitable representation bases (e.g., wave-let or complex wavelet [37]) and, on the other hand, it is possi-ble to linearly (approximated) describe the relation between thereceived demodulated signal and the unknown reflectivity field[37]. Typical applications are mainly concerned with spotlightSAR (where the traveling radar sensor continuously steers theantenna beam to illuminate the terrain patch being imaged)[37, 86, 87] where both TV (19) [37] and BCS (20) [86] for-mulations have been assessed in undersampling SAR data, in-creasing the robustness to noise, and reducing the sidelobes inretrieved images despite the use of simple, but efficient, re-construction algorithms [8], [36].

More recently, the presence of errors in the matrixA caused by misalignments between actual targets and pro-cessing grid (called gridding errors) has been addressed[87], as well. Similarly to the perturbed CS problem discussedin Section 4.2 for the DoA estimation, a CR formulation hasbeen presented in [87]. A greedy technique, the support-constrained OMP (SCOMP) [87], has then been introduced to

mitigate the gridding error instabilities, yielding better im-age reconstructions.

4.4.3 Tomo-SAR Imaging

In tomographic SAR (i.e., Tomo-SAR, which extends thesynthetic aperture principle by using acquisitions from slightlydifferent viewing angles in elevation to yield 3-D reconstruc-tions [88]), CS approaches have been considered to minimizethe number of signal measurements, while keeping the desiredaccuracy in image reconstruction [8890]. Toward this aim,CR formulations have been widely adopted [8890] to obtaina superior resolution, an improved robustness to noise, and ahigher computational efficiency with respect to state-of-the-art non-linear least square techniques [88] and truncated SVD (TSVD)methods [89]. Different greedy/local minimization solvers havebeen employed, ranging from BP [33, 90], BP DeQuantizer[88, 91], DouglasRachford splitting method [89, 92], up to theiterative shrinkage/thresholding method [89, 93].

Hybrid approaches based on CS retrieval tools havebeen also analyzed to further improve Tomo-SAR imaging[94. 95]. For instance, spectral estimation algorithms (e.g., the SL1MMER technique) have been proposed by combining 1-minimization CS steps with a model order reduction and amaximum-likelihood parameter selection [94]. Moreover, com-bining the NDOF-TSVD approaches [96] and CS techniques hasbeen discussed [95] by formulating the sparse problem at handas the second-order cone problem [97, 98].

4.4.4 ISAR Imaging

Dealing with CS applications, inverse SAR (ISAR) prob-lems (in which a fixed radar monitors a moving target to yieldhigh-resolution images [37]) have been widely discussed also[99, 100], because of their intrinsically sparse nature related tothe representation of the targets as few strong scatterers whosenumber is much more smaller than that of the pixels of the im-age under analysis [100]. Moreover, the ISAR signal model turnsout to be linear with respect to the (complex) amplitude of thescattering bins [100] in the rangeDoppler domain. Due to theseproperties, both Bayesian [99] and 1-regularized formulations[100104] have been adopted to yield a high range resolutionwith fewer data samples of stepped-frequency chirp signals [99]and to enhance the ISAR antijamming capabilities [102]. Morespecifically, BCS approaches [8] proved in [99] to be able togive an image detailed as those from state-of-the-art FFT recon-struction techniques, while more effectively suppressing theimage sidelobes.

An improvement of the ISAR image quality has also beenreached through greedy-pursuit strategies [33, 101] and localsearch techniques (e.g., half-quadratic regularization [105], con-jugate gradient [106], or convex programming [38] methods)when applied to 1-based functionals [101103].



Further enhancements in signal recovery and noise sup-pression have been yielded by solving the following weighted1-norm problem [100]:

bx arg minxkWxk1 n o

subject to kAx eyk2 (42)

by means of OMP techniques [29]. In (42), W fwnp; n 1; . . . ;N ; p 1; . . . ;Ng is the weighting matrix [100], wheresmall weights allow one to detect strong signal components,while large coefficients mitigate the noise components [100].

As for the phase errors due to unexpected target mo-tions in 1-norm regularized ISAR imaging [104], localsearch strategies based on quasi-Newton solvers have beenapplied, showing better compensation features with respectto state-of-the-art gapped-data amplitude and phase estima-tion (GAPES) techniques [104].

4.4.5 GPR Imaging

Data acquisition and imaging methods based on CS havebeen also applied to stepped-frequency continuous-wave groundpenetrating radars (SFCW-GPR) mainly to mitigate/overcometechnological issues related to the required high data acquisitionspeed [107110]. In such a framework, a popularly adoptedformulation is based on the Dantzig selector [83] dealt withconvex optimization tools [38] because of its higher stabilitywhen processing incomplete and noisy data as those of GPRapplications [107109]. Due to the CS-based imaging featuresand the intrinsic sparsity of GPR imaging problems, robust recon-structions with better resolutions than standard back-projectionalgorithms have been achieved also experimentally [107].

In order to minimize the acquisition time in SFCW-GPRprototypes [110], TV formulations solved with interior-pointmethods [98] have been investigated also.

4.4.6 TWRI

As far as through-the-wall radar imaging (TWRI) isconcerned, there has been an increasing interest within thescientific community [111113] to simplify high-resolutionultrawideband-TWRI imaging systems, in terms of acquisitiontime and hardware (HW) complexity, by leveraging on the fea-tures of CS [111]. As a matter of fact, the use of CS strategiesbased on Dantzig selector formulations [83] has been shownto outperform conventional delay-and-sum beamforming (DSBF)algorithms in both imaging accuracy and data reduction for theinversion. As for this latter issue, let us consider that the CS re-quires only 7.7% of the DSBF data [111]. Moreover, the use ofCR formulations solved through greedy techniques (e.g., OMP[29] and CoSaMP [31]) has been introduced, for mitigating the

wall backscattering [112] and effectively dealing with moving tar-gets when integrated with change-detection approaches [113].

5. Final Remarks and Future Trends

The CS paradigm has enabled a wide range of new appli-cative scenarios to be investigated and developed, due to its un-ique features. This paper was aimed at giving a short review ofthe applications of CS to electromagnetics starting from the CSsignal model definition and including the formulation of thesampling and recovery problems. While pointing out themandatory hypotheses for the CS exploitation, some indicationson the reliability of CS algorithms (mainly concerned with theproperties of the observation matrix) have been also recalled.Popular Bayesian and deterministic CS recovery algorithmshave shortly summarized, to point out the main features of eachsolution strategy as well as their advantages and limitations interms of efficiency and flexibility. Successively, the potentialitiesof CS strategies in solving sparse formulations naturally arisingin a broad class of electromagnetic problems (e.g., array synthe-sis and diagnosis) have been illustrated by reporting, to the bestof the authors knowledge, the most recent advances on thesetopics. Moreover, the possibility to apply CS techniques to non-linear problems concerned with DoA estimation, inverse scatter-ing, and radar imaging through suitable reformulations andapproximations has been discussed also mentioning the most dif-fused retrieval strategies usually adopted in these cases.

For the interested readers and potentially future practi-tioners, the key motivations and main advantages (e.g., numeri-cal efficiency, robustness to noise, flexibility, and accuracy) ofapplying CS in electromagnetics have been pointed out re-marked in the illustrated scenarios. Although, certainly, CSdoes not outperform all existing retrieval techniques, it has beenshown that it is able to yield enhanced performances with respectto several state-of-the-art strategies whenever a suitable sparsedescription of the problem is at hand. According to the resultsin the leading-edge researches, CS represents a reliable, effec-tive, and efficient paradigm/tool for properly addressing severalconventional electromagnetic problems as well as for envisagingnew applicative fields of research.

On the other hand, it cannot be neglected that both theo-retical issues, mathematical implementations, and numericalfeatures of CS applications to electromagnetics are still at thebeginning, and they have still to be carefully studied and ad-dressed in ongoing and future research activities. For instance,the solution of sampling problems arising in electromagnetics(e.g., the minimization of the number of required measure-ments in inverse scattering and array diagnosis) through astrategy that a priori guarantees the CS observation matrix tocomply with the RIP represents an interesting and challengingapproach to minimize HW and processing costs. Moreover,the extension of CS formulations to fully nonlinear problems,whether possible, could enable significant enhancements in awider set of applicative areas within electromagnetics.



6. References

[1] E. J. Candes and M. B. Wakin, An introduction to compressivesampling, IEEE Signal Process. Mag., vol. 25, no. 2, pp. 2130,Mar. 2008.

[2] R. Baraniuk, More is less: Signal processing and the data deluge,Science, vol. 331, no. 6018, pp. 717719, Feb. 2011.

[3] M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, Com-pressed sensing MRI, IEEE Signal Process. Mag., vol. 25, no. 2,pp. 7282, Mar. 2008.

[4] W. L. Chan, M. Moravec, R. Baraniuk, and D. Mittleman, Tera-hertz imaging with compressed sensing and phase retrieval, Opt.Lett., vol. 33, no. 9, pp. 974976, May 2008.

[5] M. Cossalter, G. Valenzise, M. Tagliasacchi, and S. Tubaro, Jointcompressive video coding and analysis, IEEE Trans. Multimedia, vol. 12,no. 3, pp. 168183, Apr. 2010.

[6] C. R. Berger, Z. Wang, J. Huang, and S. Zhou, Application ofcompressive sensing to sparse channel estimation, IEEE Commun.Mag., vol. 48, no. 11, pp. 164174, Nov. 2010.

[7] R. G. Baraniuk, Compressive sampling, IEEE Signal Process. Mag.,vol. 24, no. 4, pp. 118124, Jul. 2007.

[8] S. Ji, Y. Xue, and L. Carin, Bayesian compressive sensing, IEEETrans. Signal Process., vol. 56, no. 6, pp. 23462356, Jun. 2008.

[9] S. D. Babacan, R. Molina, and A. K. Katsaggelos, Bayesian com-pressive sensing using laplace priors, IEEE Trans. Image Process.,vol. 19, no. 1, pp. 5363, Jan. 2010.

[10] J. A. Tropp and S. J. Wright, Computational methods for sparse solu-tion of linear inverse problems, Proc. IEEE, vol. 98, no. 6, pp. 948958,Jun. 2010.

[11] R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hegde, Model-based compressive sensing, IEEE Trans. Inf. Theory, vol. 56, no. 4,pp. 19822001, Apr. 2010.

[12] M. F. Duarte and Y. C. Eldar, Structured compressed sensing: fromtheory to applications, IEEE Trans. Signal Process., vol. 59, no. 9,pp. 40534085, Sep. 2011.

[13] E. Candes and J. Romberg, L1-Magic Code. [Online]. Available:http://users.ece.gatech.edu/~justin/l1magic/.

[14] V. Stodden et al., SparseLab Code. [Online]. Available: http://sparselab.stanford.edu/.

[15] K. Koh, S.-J. Kim, and S. Boyd, ell-1LS Code [Online]. Available:http://www.stanford.edu/~boyd/l1_ls/.

[16] L. C. Potter, E. Ertin, J. T. Parker, and M. Cetin, Sparsity andcompressed sensing in radar imaging, Proc. IEEE, vol. 98, no. 6,pp. 10061020, Jun. 2010.

[17] L. Carin, On the relationship between compressive sensing and ran-dom sensor arrays, IEEE Antennas Propag. Mag., vol. 51, no. 5,pp. 7281, Oct. 2009.

[18] D. Malioutov, M. Cetin, and A. S. Willsky, A sparse signal recon-struction perspective for source localization with sensor arrays, IEEETrans. Signal Process., vol. 53, no. 8, pp. 30103022, Aug. 2005.

[19] L. Carin, L. Dehong, and G. Bin, Coherence, compressive sensing,and random sensor arrays, IEEE Antennas Propag. Mag., vol. 53, no. 4,pp. 2839, Aug. 2011.

[20] M. Carlin, P. Rocca, G. Oliveri, F. Viani, and A. Massa, Directions-of-arrival estimation through Bayesian compressive sensing strategies,IEEE Trans. Antennas Propag., vol. 61, no. 7, pp. 38283838, Jul. 2013.

[21] G. Oliveri and A. Massa, Bayesian compressive sampling for patternsynthesis with maximaly sparse non-uniform linear arrays, IEEETrans. Antennas Propag., vol. 59, no. 2, pp. 467481, Feb. 2011.

[22] B. Fuchs, Synthesis of sparse arrays with focused or shaped beampat-tern via sequential convex optimizations, IEEE Trans. Antennas Pro-pag., vol. 60, no. 7, pp. 34993503, Jul. 2012.

[23] G. Prisco and M. DUrso, Maximally sparse arrays via sequentialconvex optimizations, IEEE Antennas Wireless Propag. Lett., vol. 11,pp. 192195, Feb. 2012.

[24] A. Fannjiang, P. Yan, and T. Strohmer, Compressed remote sensingof sparse objects, arXiv, vol. 0904.3994v2, pp. 122, 2009.

[25] G. Oliveri, P. Rocca, and A. Massa, A Bayesian compressive sampling-based inversion for imaging sparse scatterers, IEEE Trans. Geosci. RemoteSens., vol. 49, no. 10, pp. 39934006, Oct. 2011.

[26] A. C. Fannjiang, Compressive inverse scattering II. SISO measure-ments with born scatterers, Inverse Probl., vol. 26, no. 3, pp. 117,Mar. 2010.

[27] R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, A simpleproof of the restricted isometry property for random matrices, Con-struct. Approx., vol. 28, no. 3, pp. 253263, Dec. 2008.

[28] E. J. Candes and T. Tao, Decoding by linear programming, IEEETrans. Inf. Theory, vol. 51, no. 12, pp. 42034215, Dec. 2005.

[29] J. Tropp and A. Gilbert, Signal recovery from random measurementsvia orthogonal matching pursuit, IEEE Trans. Inf. Theory, vol. 53,no. 12, pp. 46554666, Dec. 2007.

[30] D. L. Donoho, Y. Tsaig, I. Drori, and J. L. Starck, Sparse solutionof underdetermined systems of linear equations by stagewise or-thogonal matching pursuit, IEEE Trans. Inf. Theory, vol. 58, no. 2,pp. 10941121, Feb. 2012.

[31] D. Needell and J. A. Tropp, CoSaMP: Iterative signal recovery fromincomplete and inaccurate samples, Appl. Comput. Harmon. Anal.,vol. 26, no. 3, pp. 301321, May 2009.

[32] S. G. Mallat and Z. Zhang, Matching pursuits with time-frequency dic-tionaries, IEEE Trans. Signal Process., vol. 41, no. 12, pp. 33973415,Dec. 1993.

[33] S. Chen, Basis pursuit, Ph.D. dissertation, Dept. Stat., StanfordUniv., Stanford, CA, USA, 1995.

[34] S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, Aninterior-point method for large-scale 1-regularized least squares, IEEE J.Sel. Topics Signal Process., vol. 1, no. 4, pp. 606617, Dec. 2007.

[35] R. Tibshirani, Regression shrinkage and selection via the Lasso,J. Roy. Stat. Soc. B, vol. 58, no. 1, pp. 267288, 1996.

[36] J. Shi and S. Osher, A nonlinear inverse scale space method for aconvex multiplicative noise model, Comput. Appl. Math., Univ. Cali-fornia, Los Angeles, CA, USA, Tech. Rep. 0710, 2007.

[37] V. M. Patel, G. R. Easley, D. M. Healy, and R. Chellappa, Com-pressed synthetic aperture radar, IEEE J. Sel. Topics Signal Process.,vol. 4, no. 2, pp. 244254, Apr. 2010.

[38] M. Grant and S. Boyd, Graph implementations for nonsmooth con-vex programs, in Recent Advances in Learning and Control, Berlin,Germany: Springer-Verlag, 2008, pp. 95110.

[39] S. Nash, A survey of TruncatedNewton methods, Journal on Computa-tional and Applied Mathematics, vol. 24, no. 1/2, pp. 4559, Dec. 2000.

[40] M. D. Migliore, A compressed sensing approach for array diagnosisfrom a small set of near-field measurements, IEEE Trans. AntennasPropag., vol. 59, no. 6, pp. 21272133, Jun. 2011.

[41] L. Pan, X. Chen, and S. P. Yeo, A compressive-sensing-based pha-seless imaging method for point-like dielectric objects, IEEE Trans.Antennas Propag., vol. 60, no. 11, pp. 54725475, Nov. 2012.

[42] L. Poli, G. Oliveri, and A. Massa Microwave imaging within thefirst-order born approximation by means of the contrast-field Bayesiancompressive sensing, IEEE Trans. Antennas Propag., vol. 60, no. 6,pp. 28652879, Jun. 2012.

[43] M. E. Tipping, The relevance vector machine, in Advances in Neu-ral Information Processing Systems, vol. 12, S. A. Solla, T. K.Leen, and K.-R. Muller Eds., Cambridge, MA, USA: MIT Press,2000, pp. 652658.

[44] M. E. Tipping, Sparse Bayesian learning and the relevance vector ma-chine, J. Mach. Learn. Res., vol. 56, no. 6, pp. 211244, Jun. 2001.

[45] M. E. Tipping and A. C. Faul, Fast marginal likelihood maximiza-tion for sparse Bayesian models, in Proc. 9th Int. Workshop Artif. In-tell. Stat., C. M. Bishop and B. J. Frey, Eds., 2003, pp. 36.[Online]. Available: http://citeseer.ist.psu.edu/611465.html.

[46] A. C. Faul and M. E. Tipping, Analysis of sparse Bayesian learning,in Proc. Adv. NIPS, T. G. Dietterich, S. Becker, and Z. Ghahramani,Eds., 2002, pp. 383389. [Online]. Available: http://citeseer.ist.psu.edu/faul01analysis.html.

[47] S. Ji, D. Dunson, and L. Carin, Multi-task compressive sampling,IEEE Trans. Signal Process., vol. 57, no. 1, pp. 92106, Jan. 2009.

[48] Z. Zhang and B. D. Rao, Sparse signal recovery with temporally cor-related source vectors using sparse Bayesian learning, IEEE J. Sel.Topics Signal Process., vol. 5, no. 5, pp. 912926, Sep. 2011.

[49] P. Schniter, L. C. Potter, and J. Ziniel, Fast Bayesian matching pur-suit, in Proc. Inf. Theory Appl. Workshop, Jan. 27Feb. 1, 2008,pp. 326333.

[50] S. Ji, Y. Xue, and L. Carin, Bayesian Compressive Sensing Code.[Online]. Available: http://people.ee.duke.edu/~lcarin/BCS.html.

[51] R. R. McCormick, Bayesian Compressive Sensing Using Laplace PriorsCode. [Online]. Available: http://ivpl.eecs.northwestern.edu/node/1169.

[52] P. Schniter, L. C. Potter, and J. Ziniel, Fast Bayesian Matching Pur-suit Code. [Online]. Available: http://www2.ece.ohio-state.edu/~zinielj/fbmp/download.html.

[53] G. Oliveri, P. Rocca, and A. Massa, Reliable diagnosis of large lin-ear arrays-Bayesian compressive sensing approach, IEEE Trans. An-tennas Propag., vol. 60, no. 10, pp. 46274636, Oct. 2012.

[54] E. J. Candes, M. B. Wakin, and S. P. Boyd, Enhancing sparsity byreweighted 1 minimization, J. Fourier Anal. Appl., vol. 14, no. 5/6,pp. 877905, Oct. 2008.



[55] W. Zhang, L. Li, and F. Li, Reducing the number of elements inlinear and planar antenna arrays with sparseness constrained optimiza-tion, IEEE Trans. Antennas Propag., vol. 59, no. 8, pp. 31063111,Aug. 2011.

[56] G. Oliveri, M. Carlin, and A. Massa, Complex-weight sparse lineararray synthesis by bayesian compressive sampling, IEEE Trans. An-tennas Propag., vol. 60, no. 5, pp. 23092326, May 2012.

[57] F. Viani, G. Oliveri, and A. Massa, Compressive sensing patternmatching techniques for synthesizing planar sparse arrays, IEEETrans. Antennas Propag., vol. 61, no. 9, pp. 45774587, Sep. 2013.

[58] L. F. Yepes, D. H. Covarrubias, M. A. Alonso, and J. G. Arceo,Synthesis of two-dimensional antenna array using independent com-pression regions, IEEE Trans. Antennas Propag., vol. 61, no. 1,pp. 449453, Jan. 2013.

[59] M. M. Hyder and K. Mahata, Direction-of-arrival estimation using amixed 20 norm approximation, IEEE Trans. Signal Process., vol. 58,no. 9, pp. 46474655, Sep. 2010.

[60] D. Model and M. Zibulevsky, Signal reconstruction in sensor ar-rays using sparse representations, Signal Process., vol. 86, no. 3,pp. 624638, Mar. 2006.

[61] D. M. Malioutov, A sparse signal reconstruction perspective forsource localization with sensor arrays, M.S. thesis, Dept. Elect. Eng.Comput. Sci., MIT, Cambridge, MA, USA, Jul. 2003. [Online]. Avail-able: http://ssg.mit.edu/~dmm/publications/malioutov_MS_thesis.pdf.

[62] S. Caorsi, M. Donelli, D. Franceschini, and A. Massa, A new meth-odology based on an iterative multiscaling for microwave imaging, IEEETrans. Microw. Theory Tech., vol. 51, no. 4, pp. 11621173, Apr. 2003.

[63] I. Bilik, Spatial compressive sensing for direction-of-arrival estimationof multiple sources using dynamic sensor arrays, IEEE Trans. Aerosp.Electron. Syst., vol. 47, no. 3, pp. 17541769, Jul. 2011.

[64] M. M. Hyder and K. Mahata, An improved smoothed approximationalgorithm for sparse representation, IEEE Trans. Signal Process., vol. 58,no. 4, pp. 21942205, Apr. 2010.

[65] D. P. Wipf and B. D. Rao, An empirical Bayesian strategy for solv-ing the simultaneous sparse approximation problem, IEEE Trans. Sig-nal Process., vol. 55, no. 7, pp. 37043716, Jul. 2007.

[66] H. Zhu, G. Leus, and G. B. Giannakis, Sparsity-cognizant totalleast-squares for perturbed compressive sampling, IEEE Trans. SignalProcess., vol. 59, no. 5, pp. 22022016, May 2011.

[67] Z. Yang, C. Zhang, and L. Xie, Robustly stable signal recovery incompressed sensing with structured matrix perturbation, IEEE Trans.Signal Process., vol. 60, no. 9, pp. 46584671, Sep. 2012.

[68] Z. Yang, L. Xie, and C. Zhang, Off-grid direction of arrival estima-tion using sparse Bayesian inference, IEEE Trans. Signal Process.,vol. 61, no. 1, pp. 3843, Jan. 2013.

[69] E. T. Northardt, I. Bilik, and Y. I. Abramovich, Spatial compressivesensing for direction-of-arrival estimation with bias mitigation via expectedlikelihood, IEEE Trans. Signal Process., vol. 61, no. 5, pp. 1183106,Mar. 2013.

[70] Y. Abramovich and B. Johnson, GLRT-based detection-estimationfor under-sampled training conditions, IEEE Trans. Signal Process.,vol. 56, no. 8, pp. 36003612, Aug. 2008.

[71] G. Oliveri, L. Poli, P. Rocca, and A. Massa, Bayesian compressiveoptical imaging within the Rytov approximation, Opt. Lett., vol. 37,no. 10, pp. 17601762, May 2012.

[72] F. Viani, L. Poli, G. Oliveri, F. Robol, and A. Massa, Sparse scat-terers imaging through approximated multi-task compressive sensingstrategies, Microw. Opt. Technol. Lett., vol. 55, no. 7, pp. 15531558,Jul. 2013.

[73] A. C. Fannjiang, Compressive inverse scattering I. High frequencySIMO measurements, Inverse Probl., vol. 26, no. 3, pp. 130,Mar. 2010.

[74] P. M. van Den Berg and A. Abubakar, Contrast source inversion method:State of the art, Progr. Electromagn. Res., vol. 34, pp. 189218, 2001.

[75] L. Poli, G. Oliveri, P. Rocca, and A. Massa, Bayesian compressivesensing approaches for the reconstruction of two-dimensional sparsescatterers under TE illuminations, IEEE Trans. Geosci. Remote Sens.,vol. 51, no. 5, pp. 29202936, May 2013.

[76] L. Poli, G. Oliveri, and A. Massa, Imaging sparse metallic cylindersthrough a local shape function Bayesian compressive sensing approach,J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 30, no. 6, pp. 12611272,Jun. 2013.

[77] E. A. Marengo et al., Compressive sensing for inverse scattering,Proc. XXIX URSI Gener. Assem., Chicago, IL, USA, Aug. 716, 2008.

[78] E. A. Marengo, Compressive sensing and signal subspace methodsfor inverse scattering including multiple scattering, in Proc. IEEE Int.Workshop Comput. Adv. Multi-Sensor Adapt. Process., Boston, MA,USA, Jul. 711, 2008, pp. 109112.

[79] E. A. Marengo, Subspace and Bayesian compressive sensing methodsin imaging, in Proc. Progr. Electromagn. Res. Symp., Cambridge,MA, Jul. 26, 2008.

[80] A. Fannjiang, Compressed imaging of subwavelength structures,SIAM J. Imag. Sci., vol. 2, no. 4, pp. 12771291, 2009.

[81] W. Dai and O. Milenkovic, Subspace pursuit for compressive sens-ing signal reconstruction, IEEE Trans. Inf. Theory, vol. 55, no. 5,pp. 22302249, May 2009.

[82] A. J. Devaney, Inverse-scattering theory within the Rytov approxima-tion, Opt. Lett., vol. 6, no. 8, pp. 374376, Aug. 1981.

[83] E. Candes and T. Tao, The Dantzig selector: Statistical estimationwhen p is much larger than n, Ann. Stat., vol. 35, no. 6, pp. 23132351,Dec. 2007.

[84] G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing.Boca Raton, FL, USA: CRC Press, 1999.

[85] M. Tello Alonso, P. Lopez-Dekker, and J. J. Mallorqui, A novelstrategy for radar imaging based on compressive sensing, IEEE Trans.Geosci. Remote Sens., vol. 48, no. 12, pp. 42854295, Dec. 2010.

[86] J. Xu, Y. Pi, and Z. Cao, Bayesian compressive sensing in syntheticaperture radar imaging, IET Radar Sonar Navig., vol. 6, no. 1, pp. 28,Jan. 2012.

[87] A. Fannjiang and H.-C. Tseng, Compressive radar with off-gridtargets: A perturbation approach, Inverse Probl., vol. 29, no. 5,pp. 054008-1054008-23, May 2013.

[88] X. X. Zhu and R. Bamler, Tomographic SAR inversion by L1-normregularizationThe compressive sensing approach, IEEE Trans.Geosci. Remote Sens., vol. 48, no. 10, pp. 38393846, Oct. 2010.

[89] A. Budillon, A. Evangelista, and G. Schirinzi, Three-dimensionalSAR focusing from multipass signals using compressive sampling, IEEETrans. Geosci. Remote Sens., vol. 49, no. 1, pp. 488499, Jan. 2011.

[90] S. Xilong, Y. Anxi, D. Zhen, and L. Diannong, Three-dimensionalSAR focusing via compressive sensing: The case study of angel sta-dium, IEEE Geosci. Remote Sens. Lett., vol. 9, no. 4, pp. 759763,Jul. 2012.

[91] L. Jacques, D. K. Hammond, and J. M. Fadili, Dequantizing com-pressed sensing: When oversampling and non-Gaussian constraints com-bine, IEEE Trans. Inf. Theory, vol. 57, no. 1, pp. 559571, Jan. 2011.

[92] P. L. Combettes and J. C. Pesquet, A Douglas-Rachford splitting ap-proach to nonsmooth convex variational signal recovery, IEEE J. Sel.Topics Signal Process., vol. 1, no. 4, pp. 564574, Dec. 2007.

[93] M. Figueiredo and R. Nowak, An EM algorithm for wavelet-based im-age restoration, IEEE Trans. Image Process., vol. 12, no. 8, pp. 906916,Aug. 2003.

[94] X. X. Zhu and R. Bamler, Super-resolution power and robustness ofcompressive sensing for spectral estimation with application to space-borne tomographic SAR, IEEE Trans. Geosci. Remote Sens., vol. 50,no. 1, pp. 247258, Jan. 2012.

[95] R. Solimene, F. Ahmad, and F. Soldovieri, A novel CS-TSVD strat-egy to perform data reduction in linear inverse scattering problems,IEEE Geosci. Remote Sens. Lett., vol. 9, no. 5, pp. 881885, Sep. 2012.

[96] F. Soldovieri, R. Solimene, and F. Ahmad, Experimental validationof a microwave tomographic approach for through-the-wall radarimaging, in Proc. SPIE Defense Security Symp., vol. 7699, 2010,p. 76990C.

[97] Y.-S. Yoon and M. G. Amin, Through-the-wall radar imaging usingcompressive sensing along temporal frequency domain, in Proc.IEEE ICASSP, Mar. 2010, pp. 28062809.

[98] E. J. Candes, J. Romberg, and T. Tao, Robust uncertainty principles:Exact signal reconstruction from highly incomplete frequency informa-tion, IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489509, Feb. 2006.

[99] X.-C. Xie and Y.-H. Zhang, High-resolution imaging of movingtrain by ground-based radar with compressive sensing, Electron.Lett., vol. 46, no. 7, pp. 529531, Apr. 2010.

[100] L. Zhang et al., Resolution enhancement for inversed synthetic apertureradar imaging under Low SNR via improved compressive sensing, IEEETrans. Geosci. Remote Sens., vol. 48, no. 10, pp. 38243838, Oct. 2010.

[101] F. Ye, D. Liang, and J. Zhu, ISAR enhancement technology basedon compressed sensing, Electron. Lett., vol. 47, no. 10, pp. 620621,May 2011.

[102] L. Zhang, Z.-J. Qiao, M. Xing, Y. Li, and Z. Bao, High-resolutionISAR imaging with sparse stepped-frequency waveforms, IEEE Trans.Geosci. Remote Sens., vol. 49, no. 11, pp. 46304651, Nov. 2011.

[103] H. Wang, Y. Quan, M. Xing, and S. Zhang, ISAR imaging viasparse probing frequencies, IEEE Geosci. Remote Sens. Lett., vol. 8,no. 3, pp. 451455, May 2011.

[104] L. Zhang et al., High-resolution ISAR imaging by exploiting sparse aper-tures, IEEE Trans. Antennas Propag., vol. AP-60, no. 2, pp. 9971008,Feb. 2012.



[105] M. Cetin, W. C. Karl, and A. S. Willsky, Feature-preserving regulari-zation method for complex-valued inverse problems with application tocoherent imaging, Opt. Eng., vol. 45, no. 1, pp. 017003-1017003-11,Jan. 2006.

[106] O. Axelsson, Iterative Solution Methods. Cambridge, U.K.: CambridgeUniv. Press, 1994.

[107] A. C. Gurbuz, J. H. McClellan, and W. R. Scott, A compressivesensing data acquisition and imaging method for stepped frequencyGPRs, IEEE Trans. Signal Process., vol. 57, no. 7, pp. 26402650,Jul. 2009.

[108] M. A. C. Tuncer and A. C. Gurbuz, Ground reflection removal incompressive sensing ground penetrating radars, IEEE Geosci. RemoteSens. Lett., vol. 9, no. 1, pp. 2327, Jan. 2012.

[109] L. Qu and T. Yang, Investigation of air/ground reflection and an-tenna beamwidth for compressive sensing SFCW GPR migration im-aging, IEEE Trans. Geosci. Remote Sens., vol. 50, no. 8, pp. 31433149,Aug. 2012.

[110] A. B. Suksmono, E. Bharata, A. A. Lestari, A. G. Yarovoy, andL. P. Ligthart, Compressive stepped-frequency continuous-waveground-penetrating radar, IEEE Geosci. Remote Sens. Lett., vol. 7,no. 4, pp. 665669, Oct. 2010.

[111] Q. Huang, L. Qu, B. Wu, and F. Guangyou, UWB through-wallimaging based on compressive sensing, IEEE Trans. Geosci. RemoteSens., vol. 48, no. 3, pp. 14081415, Mar. 2010.

[112] E. Lagunas, M. G. Amin, F. Ahmad, and M. Najar, Joint wall miti-gation and compressive sensing for indoor image reconstruction, IEEETrans. Geosci. Remote Sens., vol. 51, no. 2, pp. 891906, Feb. 2013.

[113] F. Ahmad and M. G. Amin, Through-the-wall human motion indica-tion using sparsity-driven change detection, IEEE Trans. Geosci. Re-mote Sens., vol. 51, no. 2, pp. 881890, Feb. 2013.

Andrea Massa (M96) received the laureadegree in electronic engineering and the Ph.D.degree in electronics and computer sciencefrom the University of Genoa, Genoa, Italy,in 1992 and 1996, respectively.

From 1997 to 1999, he was an Assistant Pro-fessor in electromagnetic fields with theDepartment of Biophysical and ElectronicEngineering, University of Genoa, teaching theuniversity course of Electromagnetic Fields 1.From 2001 to 2004, he was an Associate Pro-fessor with the University of Trento, Trento,Italy. Since 2005, he has been a Full Professor

in electromagnetic fields with the University of Trento, where he currently tea-ches electromagnetic fields, inverse scattering techniques, antennas and wirelesscommunications, and optimization techniques. He is also the Director of theELEDIA Research Center at the University of Trento. Moreover, he is anAdjunct Professor at Pennsylvania State University, State College, PA, USA, andholder of a Senior DIGITEO Chair developed in co-operation between theLaboratoire des Signaux et Systmes in Gif-sur-Yvette and the DepartmentImagerie et Simulation for the Contrle of CEA LIST in Saclay (France) fromDecember 2014, and he has been a Visiting Professor at the Missouri Universityof Science and Technology, Rolla, MO, USA, at the Nagasaki University,Nagasaki Japan, at the University of Paris Sud, Orsay, France, and at theKumamoto University, Kumamoto, Japan. His research work since 1992 hasbeen principally on electromagnetic direct and inverse scattering, microwaveimaging, optimization techniques, wave propagation in presence of nonlinearmedia, wireless communications and applications of electromagnetic fieldsto telecommunications, medicine, and biology.

Prof. Massa is currently a member of the Progress in ElectromagneticsResearch Symposium (PIERS) Technical Committee and of the Inter-University Research Center for Interactions Between Electromagnetic Fieldsand Biological Systems (ICEmB), and he has served as Italian representa-tive in the general assembly of the European Microwave Association(EuMA). He serves as an Associate Editor of the IEEE TRANSACTIONS ONANTENNAS AND PROPAGATION.

Paolo Rocca (S08; M09; SM13) receivedthe M.S. degree (summa cum laude) in tele-communications engineering and the Ph.D.degree in information and communicationtechnologies from the University of Trento,Trento, Italy, in 2005 and 2008, respectively.

He is currently an Assistant Professor withthe Department of Information Engineering andComputer Science, University of Trento, wherehe is also a member of the ELEDIA ResearchCenter. He has been a Visiting Student at thePennsylvania State University, State College,PA, USA, and at the University Mediterranea of

Reggio Calabria, Reggio Calabria, Italy, and a Visiting Researcher at the colesuprieure dlectricit (SUPELEC), Paris, France. He is the author/coauthor ofover 200 peer-reviewed papers on international journals and conferences. Hismain interests are in the framework of antenna array synthesis and design, elec-tromagnetic inverse scattering, and optimization techniques forelectromagnetics.Dr. Rocca was a recipient of the best Ph.D. thesis award IEEE-GRS Central

Italy Chapter from the IEEE Geoscience and Remote Sensing Society and theItaly Section. He serves as an Associate Editor of the IEEE ANTENNAS ANDWIRELESS PROPAGATION LETTERS.

Giacomo Oliveri (S07; M09; SM13) re-ceived the B.S. and M.S. degrees in telecom-munications engineering and the Ph.D.degree in space sciences and engineering fromthe University of Genoa, Genoa, Italy, in 2003,2005, and 2009, respectively.

He is currently an Assistant Professor withthe Department of Information Engineeringand Computer Science, University of Trento,where he is also a member of the ELEDIAResearch Center. In 2012, 2013, and 2015, hewas a Visiting Researcher at the Laboratoiredes signaux et systmes (L2S), cole supr-

ieure dlectricit (SUPELEC), Paris, France. Moreover, in 2014, he was anInvited Associate Professor at the University of Paris Sud, Paris, France. He isan author/coauthor of over 200 peer-reviewed papers on international journalsand conferences. His research work is mainly focused on electromagnetic directand inverse problems, system-by-design and metamaterials, and antenna arraysynthesis. He is the Chair of the IEEE AP/ED/MTT North Italy Chapter.

Dr. Oliveri serves as an Associate Editor of the International Journalof Distributed Sensor Networks, of the International Journal of Antennasand Propagation and of the journal Microwave Processing.


compressive sensing in electromagnetics a review.pdf

Documents

application of cs

cs approaches

stateoftheart cs algori

radar imaging problems

based techniques

new electromagnetic

synthesis of antenna

small set of measurements