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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
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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).
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(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
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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
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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)
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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.
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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.
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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
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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
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(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.
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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].
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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.
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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.
IEEE Antennas and Propagation Magazine, Vol. 57, No. 1, February
2015 15