shape reconstruction of a perfectly conducting scatterer using differential evolution and particle...

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 7, JULY 2008 1967

Shape Reconstruction of a Perfectly ConductingScatterer Using Differential Evolution and

Particle Swarm OptimizationIoannis T. Rekanos, Member, IEEE

Abstract—The shape reconstruction of a perfectly conducting2-D scatterer by inverting transverse magnetic scattered fieldmeasurements is investigated. The reconstruction is based onevolutionary algorithms that minimize the discrepancy betweenmeasured and estimated scattered field data. A closed cubicB-spline expansion is adopted to represent the scatterer contour.Two algorithms have been examined the differential-evolution(DE) algorithm and the particle swarm optimization (PSO). Nu-merical results indicate that the DE algorithm outperforms thePSO in terms of reconstruction accuracy and convergence speed.Both techniques have been tested in the case of simulated measure-ments contaminated by additive white Gaussian noise.

Index Terms—Differential evolution (DE), evolutionary algo-rithms, inverse scattering, particle swarm optimization (PSO),shape reconstruction.

I. INTRODUCTION

THE RECONSTRUCTION of the electromagnetic prop-erties of unknown scatterers by inverting scattered field

measurements is of great interest because it is associated withnumerous applications in biomedical imaging, nondestructivetesting, geophysical exploration, etc. In general, inverse scat-tering is a nonlinear and ill-posed problem [1], and it is solvedby means of iterative optimization algorithms combined withregularization schemes [2].

The scattered field measurements depend on the scattererproperties as well as the total field inside the scatterer domain.Moreover, the total field in the scatterer domain also depends onthe scatterer properties. Hence, estimating the scatterer profilefrom measurements becomes a nonlinear problem that is usu-ally solved by minimizing a cost function, which is related tothe discrepancy between measured and estimated scattered fielddata. The latter ones are obtained by assuming trial solutionsof the scatterer properties that are updated iteratively. This op-timization problem has been treated by means of deterministic[3]–[11] or stochastic methods (see [12] and references within).The major drawback of deterministic methods is that the finalestimate of the scatterer profile is highly dependent on the initialone. This problem is attributed to the fact that deterministicoptimization methods are trapped in local minima of the costfunction. On the other hand, stochastic methods, such as geneticalgorithms (GAs), conduct global-search optimization, while

Manuscript received May 9, 2007; revised December 19, 2007.The author is with the Physics Division, Department of Mathematics, Physics

and Computational Sciences, School of Engineering, Aristotle University ofThessaloniki, 54124 Thessaloniki, Greece (e-mail: [email protected]).

Digital Object Identifier 10.1109/TGRS.2008.916635

getting trapped in local minima is avoided. However, comparedto deterministic methods, the stochastic ones require the solu-tion of numerous direct scattering problems in each iteration.

Because the initial development of global optimization tech-niques mimics evolutionary mechanisms, these techniques havebeen applied successfully in the solution of design and opti-mization problems in electromagnetics as well as in inversescattering [12]–[23]. Among these techniques, the differential-evolution (DE) algorithm [24] and the particle swarm opti-mization (PSO) [25], [26] are the most recent ones. Comparedwith GA, the DE and PSO are much easier to implementand converge faster. Concerning the shape reconstruction ofconducting scatterers, the DE has been investigated [15], [16],whereas the PSO has been utilized in the reconstruction ofdielectric scatterers [19]–[22], [27]. In both cases, the reportedresults indicate that the DE and PSO are reliable tools forinverse-scattering applications. Moreover, it has been shownthat both DE and PSO outperform real-coded GA in termsof convergence speed [14], [15], [20], [21]. However, to ourknowledge, a comparative study about the performances ofthese two techniques when applied to inverse-scattering prob-lems has not yet been reported.

A particular kind of inverse-scattering problem is the recon-struction of the shape of perfectly conducting scatterers [15],[16], [28], [29]. In this paper, the shape reconstruction is basedon the application of both DE and PSO. The scatterer is con-sidered 2-D, and it is illuminated by transverse magnetic (TM)plane waves from different directions of incidence, whereasthe generated scattered field measurements are utilized for theinversion. The measurements are simulated by means of themethod of moments (MoM) [30]. The scatterer shape is pa-rameterized by closed cubic B-spline expansion [31], [32],which involves a set of weighting parameters. The objectiveof both DE and PSO is to minimize the discrepancy betweenmeasured and estimated scattered field data with respect to theparameters of the spline expansion. Numerical results show thatthe DE outperforms the PSO in terms of shape reconstructionaccuracy, giving a lower reconstruction error for the samenumber of iterations. Moreover, both techniques have beenapplied in cases where the measurements are noisy. The resultsindicate that both techniques are efficient and give adequate re-construction, even when the signal-to-noise ratio (SNR) is low.

II. DESCRIPTION OF THE PROBLEM

We assume a 2-D perfectly conducting scatterer, which isparallel to the z-axis, whose cross section on the xy-plane

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1968 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 7, JULY 2008

Fig. 1. Geometric configuration of the 2-D scattering problem.

is invariant. The scatterer is illuminated by a monochromaticTM incident plane wave with the electric field aligned alongthe z-axis (see Fig. 1). Under this assumption, the problemformulation with respect to the electric field becomes scalar.Concerning the direct scattering problem, if C is the contour ofthe scatterer, then the scattered field E at any position r on thescatterer surface or outside the scatterer domain is given by theclosed line integral

E(r) = −ωµ0

4

∮C

J(r′)H(2)0 (k0‖r − r′‖) dl′ (1)

where r′ ∈ C, J(r′) is the equivalent surface-current density, k0

is the wavenumber, and H(2)0 is the Hankel function of second

kind and zeroth order. Note that the temporal dependenceexp(jωt) has been adopted. Because the total field on thesurface vanishes (E(r) + Ei(r) = 0), J(r′) is derived by thesolution of the integral equation

Ei(r) =ωµ0

4

∮C

J(r′)H(2)0 (k0‖r − r′‖) dl′ (2)

where r, r′ ∈ C, and Ei(r) are the TM incident fields onC. To compute the scattered field, (2) is solved with re-spect to J(r′), which is then substituted into (1). The equiva-lent surface-current density can be computed by applying theMoM [30] to (2).

A. Cubic B-Spline Representation of the Scatterer Shape

The contour of the scatterer is described by a parametricmodel, utilizing closed cubic B-splines with N control parame-ters {p0, p1, . . . , pN−1} [31], [32]. According to this approach,the contour is assumed smooth because its second derivative iscontinuous, whereas its third derivative is piecewise continu-ous. In particular, the contour is written as

C(φ) = R

(N

2πφ

), φ ∈ [0, 2π] (3)

where the closed curve R(τ) (τ ∈ [0, N ]) is composed of Ncurve segments rn(t), i.e.,

R(τ) =N−1∑n=0

rn(τ − n). (4)

By definition, a single curve segment has a compact support do-main. In particular, the segment rn(τ − n) is nonzero only fort ∈ [0, 1], where t = τ − n. Hence, R(τ) = rn(τ − n) whenn ≤ τ ≤ n + 1. Moreover, according to (3), the curve seg-ment rn(τ − n) is identical to the contour representation C(φ)when the angle variable φ belongs in the domain [n(2π/N),(n + 1)(2π/N)].

Each curve segment rn(t) is a linear combination of fourcubic polynomials, which are defined in the normalized domaint ∈ [0, 1], i.e.,

rn(t) = pn−1Q0(t) + pnQ1(t) + pn+1Q2(t) + pn+2Q3(t)(5)

where 0 ≤ n ≤ N − 1, p−1 = pN−1, pN = p0, and pN+1 =p1. The cubic polynomials in (5) are given by

Q0(t) =16(1 − t)3 (6)

Q1(t) =12t3 − t2 +

23

(7)

Q2(t) = − 12t3 +

12t2 +

12t +

16

(8)

Q3(t) =16t3. (9)

Therefore, the contour C is described by the N -dimensionalvector p = [p0, p1, . . . , pN−1] of the control parameters, i.e.,C = C(p). From (5), we conclude that the control parameterpn contributes to the formation of the curve segments rn−2,rn−1, rn, and rn+1. In other words, pn affects the contour rep-resentation C(φ) when φ ∈ [(n − 2)(2π/N), (n + 2)(2π/N)].

A significant advantage of the cubic B-spline representationof the scatterer contour is that the cubic polynomials (6)–(9)are nonnegative. Thus, the reasonable constraint C(p) ≥ 0 (theradius is positive) can be simply imposed by requiring allcontrol parameters pn to be positive. This approach of imposingthe constraint C(p) ≥ 0 is based on the property

min0≤n≤N−1

{pn} ≤ C(p) ≤ max0≤n≤N−1

{pn}. (10)

B. Inverse Scattering

Let us assume that the scatterer is illuminated by a number ofincident fields and, that for each incidence, the scattered field ismeasured at a set of measurement positions around the scattererdomain. The objective is to reconstruct the scatterer shape fromthe total set of measurements. To cope with this inverse prob-lem, a cost function representing the discrepancy between themeasured and estimated scattered fields is introduced. In this

REKANOS: SHAPE RECONSTRUCTION OF SCATTERER USING DIFFERENTIAL EVOLUTION AND OPTIMIZATION 1969

Fig. 2. Schematic presentation of the mutant vector construction uGk , which

corresponds to the primary parent pGk . The dots represent points in the solution

space.

paper, the shape of the scatterer is reconstructed by minimizingthe cost function

F (p) =∑I

i=1

∑Mm=1

[Ed

im − Eeim(p)

]2∑Ii=1

∑Mm=1

(Ed

im

)2 (11)

where I is the total number of incidences, M is the numberof measurements per incidence, Ed denotes the measured scat-tered field, and Ee is the estimated one. In particular, becausethe contour is a function of p = [p0, p1, . . . , pN−1], (11) isminimized with respect to p.

III. DE ALGORITHM

The DE algorithm is a global optimization technique, belong-ing to the general class of evolutionary algorithms. Accordingto the DE, an initial set of K candidate solutions {pk : k =1, 2, . . . ,K}, composing a generation, is formed randomly.Then, new generations are produced iteratively by applyingmutation, crossover, and selection operators to the current setof candidate solutions. During mutation, each vector pG

k , whereG denotes the generation, is selected as a primary parent. Foreach primary parent, a mutant vector uG

k is generated given by

uGk = pG

r1+ q

(pG

r2− pG

r3

)(12)

where r1, r2, and r3 ∈ {1, 2, . . . ,K} are selected randomlyand are mutually different, as well as different from k. Theparameter q is positive and tunes the impact of the differencepG

r2− pG

r3on the construction of uG

k . A schematic presentationof the mutant vector construction mechanism is shown inFig. 2. The derived mutant vector uG

k is considered as thesecondary parent. Then, the crossover operator is applied tothe primary and secondary parents, resulting in the offspringvector vG

k = [vGk0, v

Gk1, . . . , v

Gk(N−1)]. The components of vG

k

are inherited either from the primary or secondary parent,according to a probabilistic scheme. To ensure that theoffspring is different from the primary parent, the offspringinherits at least one component from the secondary parent. Theindex l of this particular component is selected randomly fromthe set {0, 1, . . . , N − 1}. Thus, vG

kl = uGkl, whereas the rest

components of the offspring are given by the following scheme:

vGkn =

{uG

kn, if hn ≤ HpG

kn, if hn > Hn �= l (13)

where hn is a random number that is uniformly distributedwithin [0, 1], and H ∈ (0, 1) is a predefined crossoverprobability. Finally, the offspring vector vG

k competes with itscorresponding primary parent pG

k for a position in the nextgeneration. If vG

k is fitter than pGk (gives a lower value of

the cost function), then it replaces pGk in the next generation

(G + 1). Otherwise, pGk survives and remains in the next

generation, i.e.,

pG+1n =

{vG

k , if F(vG

k

)≤ F

(pG

k

)pG

k , if F(vG

k

)> F

(pG

k

).

(14)

From (14), we conclude that the best candidate solution(the one with the lowest cost function) of the new generationperforms at least as good as the best candidate of the pre-vious generation. Hence, the cost function is monotonicallydecreasing with respect to the number of generations. Theaforementioned steps of the DE are repeated iteratively until thecost function of an individual vector is lower than a predefinedthreshold or until a predefined total number of generations havebeen generated.

IV. PSO ALGORITHM

According to the PSO algorithm, the candidate solutions{pk : k = 1, 2, . . . ,K} compose a swarm, which is generatedrandomly. Each particle pk denotes a position in the solutionspace, and it has its own velocity vk, which is also selectedrandomly. The velocity allows the particle to travel in the so-lution space, searching for better solution positions in terms ofits performance with respect to the cost function. Each particleremembers its individual best position ever visited fk, whereasthe best position g among all particles, namely, the global best,is communicated to all particles. After the initialization stepof the PSO, the algorithm enters an iterative process, whereall particles update their velocities and positions. In particular,during each iteration of the PSO, the velocities and the positionsare updated according to the scheme

vG+1kn =A · vG

kn + B1 · r1 ·(fkn − pG

kn

)+ B2 · r2 ·

(gG

n − pGkn

)(15)

pG+1kn = pG

kn + vG+1kn (16)

where 1 ≤ k ≤ K, 0 ≤ n ≤ N − 1, and G denotes the itera-tion. In (15), A is the inertia, B1 is the cognitive, and B2 isthe social parameter, whereas r1 and r2 are random numbersuniformly distributed between zero and one. If the new positionof a particle is better than its best position ever visited, then itscurrent position is considered as the individual best one, i.e.,if F (pG+1

k ) < F (fk), then fk = pG+1k . Also, the global best

position is updated, i.e.,

gG+1 = pG+1b (17)

where

b = arg mink

[F

(pG+1

k

)]. (18)

It has to be mentioned that as the particles travel around the so-lution space, they may visit positions that are worse comparedwith their previous ones. Thus, according to (17), it is possible


Fig. 3. Schematic presentation of the updating procedure of the PSO. The dotsrepresent points in the solution space. The vector di stands for the motion dueto inertia, whereas dc and ds are related to the attraction toward the individualbest position fk and the global best position gG, respectively. The updatedvelocity vector vG+1

kis equal to di + dc + ds.

that the new global best position gG+1 is worse than gG. As aresult, and in contrast to the DE algorithm, the cost functionis not monotonically decreasing with respect to the numberof iterations. The adopted scheme for updating the global bestposition, which is presented in (17) and (18), is different fromthe one used in the original PSO version. In the original PSO,the global best position is updated only if the best positionwithin the new generation outperforms the current global bestposition, i.e.,

gG+1 ={

pG+1b , if F

(pG+1

b

)≤ F (gG)

gG, if F(pG+1

b

)> F (gG).

(19)

It is obvious that following the scheme in (19), the cost functionis monotonically decreasing. In our simulations, we found thatthe adopted updating scheme compared with the original oneenhances the convergence of the PSO to lower values of the costfunction. This could be attributed to the fact that the adoptedscheme allows the global best position to avoid local minima. Itis also mentioned that in cases where gG+1 is worse than gG,the latter characteristic position is not lost, because it survivesas the best individual position of a particle.

The inertia parameter A describes the tendency of the particleto travel along the same direction it has been traveling [26].A large inertia value allows wide-range search in the solutionspace, whereas a small one facilitates local exploration. Itis generally accepted that a large inertia is suitable for thebeginning of the PSO algorithm, while its value should slowlydecrease as the iterations of the algorithm proceed. The cog-nitive parameter B1 controls the attraction of a particle towardthe best position it has ever visited, whereas the social parame-ter B2 tunes the attraction of the particles toward the globalbest position [26]. A schematic presentation of the updatingprocedure of the PSO is shown in Fig. 3. According to theupdating scheme described in (15) and (16), it is possible forall particles in the swarm to profit from their individual, as wellas the swarm community, discoveries about the solution space.

It has to be mentioned that the position-updating scheme ofthe PSO (16) as well as the mutation operator of the DE (12)may give candidate solutions that are not feasible. For example,(16) and (12) may result in negative values for some controlparameters of the cubic B-spline representation. In such case,

the candidate solution is corrected by placing it back into theadmissible solution space. This can be done by applying the“reflecting wall” boundary condition [17] Thus, for both DEand PSO, if pG+1

kn = w < 0, then pG+1kn is set equal to −w.

V. NUMERICAL RESULTS

As a first application example of microwave imaging bymeans of DE and PSO, the reconstruction of the contour ofa single 2-D perfectly conducting scatterer is considered. Thescatterer is illuminated by a number of I = 7 TM plane-waveincidences, which are uniformly distributed around the scattererdomain. The maximum radius of the scatterer is approximately1.5λ, where λ is the wavelength of the monochromatic in-cidences. For each incidence, the scattered field is measuredat M = 32 positions, which are uniformly placed around thescatterer at a distance from the scatterer center equal to 10λ.These measurements were simulated by means of the MoM.For the reconstruction of the scatterer shape, the contour hasbeen represented by N = 10 control parameters of the splineexpansion.

The initial values of the components of the solutions thatcorrespond to the shape control parameters were chosen ran-domly (uniformly distributed in the range [0, 2]). The sameinitial population was used for both DE and PSO, and the totalnumber of iterations was 200. To quantify the accuracy of theshape reconstruction, the reconstruction error is defined as

E(C) =

(∑Ll=1

[Cd(φl) − Ce(φl)

]2L

)1/2

(20)

where φl = 2πl/L, and Cd(φl) and Ce(φl) stand for theoriginal and estimated contours of the scatterer, respectively.In (20), L is the total number of distinct points where theradius is evaluated, and in the application examples, it wasequal to 60.

Concerning the DE implementation, the parameter q was setequal to 0.5. This value belongs in the range [0.4, 1.0], as sug-gested in [16]. Other choices of the parameter q from the samerange have been tested and resulted in similar convergencespeed. The crossover probability H was 0.8, which means thatthe offspring statistically inherits more genes from the mutantthan from the primary parent.

In the PSO, the initial values of the components ofthe particle velocities were chosen randomly (uniformlydistributed in the range [−1, 1]). Parametric studies [33], [34]have shown that no optimum values for B1 and B2 really existand are actually problem dependent (usually both are set equalto 2.0 [25] or 1.49 [17]). As it has been mentioned in [26],setting B1 = B2 = 0.5 might give better convergence resultscompared with the aforementioned choices. In this paper, threescenarios, i.e., B1 = B2 = 2.0, B1 = B2 = 1.49, and B1 =B2 = 0.5, have been tested, and the highest convergence speedwas achieved when B1 = B2 = 0.5. Thus, in the followingreconstruction applications, the cognitive and social parameterswere kept constant (B1 = B2 = 0.5), whereas the inertia pa-rameter was initially A = 1.0 and dropped linearly to 0.7 after200 iterations.

Two different original scatterers have been examined,which are a bean-shaped scatterer and a butterfly-shaped one.


TABLE ICOST FUNCTION F (p) AND RECONSTRUCTION ERROR E(C) FOR THE

BEAN-SHAPED SCATTERER AFTER 200 ITERATIONS OF DEAND PSO. THE RESULTS OBTAINED FOR THE CASE OF

N = 10 CUBIC B-SPLINE PARAMETERS AND

DIFFERENT POPULATION SIZES K

TABLE IICOST FUNCTION F (p) AND RECONSTRUCTION ERROR E(C) FOR THE

BUTTERFLY-SHAPED SCATTERER AFTER 200 ITERATIONS OF DEAND PSO. THE RESULTS OBTAINED FOR THE CASE OF

N = 10 CUBIC B-SPLINE PARAMETERS AND

DIFFERENT POPULATION SIZES K

Fig. 4. Original contour of the (©) bean-shaped scatterer, (+) best ini-tial estimates, and (×) finally reconstructed shapes, which are derived after200 iterations of (a) the DE and (b) the PSO.

Fig. 5. (a) Cost function and (b) the reconstruction error versus the number ofiterations of the (dashed line) DE and the (solid line) PSO when applied to thereconstruction of the bean-shaped scatterer.

Three different population sizes have been tested, namely, K =10, 30, and 60. The cost function and the reconstruction errorafter 200 iterations of DE and PSO for the bean- and butterfly-shaped scatterers are presented in Tables I and II, respectively.It is evident that for both scatterers and all population sizestested, the DE results in lower reconstruction error comparedwith the PSO. For the smallest population size (K = 10), thereconstruction is poor, particularly when the PSO is utilized.However, when K = 30, the final reconstruction error is, inall cases, lower than 5.5 × 10−2. Finally, when K = 60, thereconstruction error is not significantly reduced compared withthe K = 30 case, although the computation time is doubled.Thus, a population size of 30 candidate solutions is consideredadequate for the dimension of the adopted solution space(N = 10).

In what follows, the presented reconstruction results havebeen derived by setting the population size equal to 30. Theoriginal contour of the bean-shaped scatterer, the best estimateswithin the initial population of DE and PSO, and the finallyreconstructed shapes (after 200 iterations) derived by the DEand the PSO are shown in Fig. 4. The cost function and thereconstruction error versus the number of iterations for bothDE and PSO are shown in Fig. 5. As in Fig. 4, Fig. 6 showsthe original contour of the butterfly-shaped scatterer, the bestestimates within the initial set of candidate solutions, and the


Fig. 6. Original contour of the (©) butterfly-shaped scatterer, (+) bestinitial estimates, and (×) finally reconstructed shapes, which are derived after200 iterations of (a) the DE and (b) the PSO.

finally reconstructed shapes derived by DE and PSO. Thecorresponding cost function and the reconstruction error versusthe number of iterations are shown in Fig. 7.

In Figs. 4 and 6, it is evident that both DE and PSO succeededin reconstructing the original shape of both bean- and butterfly-shaped scatterers. However, as it can be seen in Figs. 5 and 7,the DE outperforms the PSO because both the cost functionand the reconstruction error decrease faster. Also, the finalvalues of the cost function and the reconstruction error after200 iterations are lower for the DE case compared with thePSO one. Apart from the two typical scatterer shapes presentedearlier, the reconstruction of 30 different realizations of theoriginal scatterer shape has been examined. In all realizations,the DE resulted in lower values of the cost function andreconstruction error compared with the PSO.

The case of reconstructing an off-centered bean-shaped scat-terer, whose center is positioned at (x0, y0) = (1, 1), has alsobeen investigated. Now, apart from the control parameters p ofthe spline expansion, two additional parameters are required forthe description of the scatterer contour, namely, the coordinatesof the scatterer center. Thus, the dimension N of the solutionspace is equal to 12. In the initial population of DE andPSO, the values of both x0 and y0 coordinates of the scatterer

Fig. 7. (a) Cost function and (b) the reconstruction error versus the number ofiterations of the (dashed line) DE and the (solid line) PSO when applied to thereconstruction of the butterfly-shaped scatterer.

center are selected randomly (uniformly distributed within therange [−2, 2]). All parameters related to the implementation ofDE and PSO are the same to the ones adopted in the previousexamples, whereas the population size K is equal to 30. Theoriginal contour of the off-centered scatterer, the best estimateswithin the initial population of DE and PSO, and the finallyreconstructed shapes (after 200 iterations) are shown in Fig. 8.For the DE, the final values of the cost function and the recon-struction error were equal to F (p) = 7.2 × 10−5 and E(C) =9.7 × 10−3, respectively, whereas for the PSO, the correspond-ing values were F (p) = 8.2 × 10−2 and E(C) = 5.9 × 10−1.

The performance of the DE and PSO in case of noisyscattered field data has been examined. A circular perfectlyconducting scatterer, with radius equal to λ, has been consid-ered and reconstructed, following the same measurement setupas in the previous application examples. The measurementshave been corrupted by additive white Gaussian noise withdifferent SNRs. The SNR level in decibels is given by

SNR = 10 log10

∑Ii=1

∑Mm=1

∣∣∣Ed(noiseless)im

∣∣∣22IMσ2

(dB) (21)

where σ2 is the variance of the Gaussian noise. The parametersrelated to the DE and PSO implementation have also been the


Fig. 8. Original contour of the (©) off-centered bean-shaped scatterer,(+) best initial estimates, and (×) finally reconstructed shapes, which arederived after 200 iterations of (a) the DE and (b) the PSO.

same as in the previous application. The cost function and thereconstruction error, which are derived by the application ofthe DE and the PSO after 200 iterations, are presented inTable III. It is interesting to notice that when the SNR isreduced, the values of the cost function increase significantly,whereas the corresponding reconstruction errors are kept lowerthan 0.01 and 0.014 for the DE and PSO, respectively. Thiscan be attributed to the fact that the number of parametersthat describe the scatterer shape is low (N = 10); thus,the reconstructed shape is regularized. In fact, because theoriginal scatterer shape is circular, the smallest numberof parameters required for an accurate cubic B-splinerepresentation of the original scatterer is N = 3. However,the shape of the original scatterer is unknown a priori.To investigate the regularization achieved by limiting thenumber of free optimization parameters, different values ofN (N = 3, 10, 20, 30, 40) have been examined in the case ofhighly contaminated measurements (SNR = 5 dB). We haveconsidered highly contaminated measurements, because in thiscase, the regularization effect becomes more evident. The costfunction and the reconstruction error after 200 iterations of DEand PSO with constant population size K = 30 and differentvalues of N are presented in Table IV.

The results in Table IV underline the regularization effectderived when the dimension of the solution space N is reduced.

TABLE IIICOST FUNCTION F (p) AND RECONSTRUCTION ERROR E(C)

AFTER 200 ITERATIONS OF DE AND PSO FOR VARIOUS

MEASUREMENTS OF SNR LEVELS

TABLE IVCOST FUNCTION F (p) AND RECONSTRUCTION ERROR E(C) AFTER

200 ITERATIONS OF DE AND PSO IN THE CASE OF 5-dB SNR AND FOR

DIFFERENT NUMBERS OF CUBIC B-SPLINE PARAMETERS N

In particular, when highly contaminated measurements areconsidered and if the dimension of the solution space increases,then the cost function is reduced. On the contrary, when Nincreases, the reconstruction error also increases. In all casespresented in Table IV, the DE outperforms PSO in terms ofboth final cost function and reconstruction error.

Finally, the computational burden, which is related to thegeneration of new candidate solutions from the previous ones,is almost identical for DE and PSO. For the reconstructionproblem studied, the computation time is governed by theMoM solution of the direct-scattering problems. Thus, thecomputation time is proportional to the population size K, andit is the same for both DE and PSO.

VI. CONCLUSION

The problem of shape reconstruction of perfectly conduct-ing scatterers has been investigated by applying the DE andPSO techniques. Numerical results show that the DE results inbetter reconstruction compared with the PSO when the samenumber of total iterations is applied. However, both techniquesare reliable and give accurate shape reconstruction even whenthe scattered field measurements are corrupted by additivewhite Gaussian noise. This fact is actually attributed to therepresentation of the scatterer contour by means of a limitednumber of parameters. Finally, the computational burden ofboth algorithms when applied to the scatterer reconstructionproblem is almost identical, because the computation time isgoverned by the direct-scattering problem solution.

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Ioannis T. Rekanos (S’92–A’00–M’02) was bornin Thessaloniki, Greece, in 1970. He received theDiploma degree (with honors) in electrical engineer-ing and the Ph.D. degree in electrical and com-puter engineering from the Aristotle University ofThessaloniki (AUTH), Thessaloniki, in 1993 and1998, respectively. He has been a scholar of theBodossaki Foundation, Athens, Greece.

From 1993 to 1998, he was a Research and Teach-ing Assistant with the Department of Electrical andComputer Engineering, AUTH. From 2000 to 2002,

he was a Senior Researcher with the Radio Laboratory, Helsinki University ofTechnology, Espoo, Finland, holding a Marie Curie Postdoctoral Fellowship.From 2002 to 2006, he was an Assistant Professor with the Departmentof Informatics and Communications, Technological Educational Institute ofSerres, Serres, Greece. Since 2006, he has been an Assistant Professor withthe Physics Division, Department of Mathematics, Physics, and Computa-tional Sciences, School of Engineering, AUTH. His current research interestsinclude electromagnetic and acoustic wave propagation, inverse scattering,computational electromagnetics and acoustics, and digital signal processing inbiomedicine.

Dr. Rekanos is a member of the American Geophysical Union, theMarie Curie Fellowship Association, and the Technical Chamber of Greece. In1995, he received the Union Radio–Scientifique Internationale Commission BYoung Scientist Award.

shape reconstruction of a perfectly conducting scatterer using differential evolution and particle...

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