a finite element-based technique for microwave imaging of two-dimensional objects

234 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 49, NO. 2, APRIL 2000

A Finite Element-Based Technique for MicrowaveImaging of Two-Dimensional Objects

Ioannis T. Rekanos, Student Member, IEEE,and Theodoros D. Tsiboukis, Senior Member, IEEE

Abstract—In this paper, a microwave imaging technique for esti-mating the spatial distributions of the permittivity and the conduc-tivity of a scatterer, by post-processing electromagnetic scatteredfield data, is presented. For the description of the direct scatteringproblem, the differential formulation is applied. This allows the useof the finite element method. During the inversion, the computa-tion of the derivative of the finite element solution with respect tothe parameters, which describe the scatterer, is required. This taskis performed by a finite element-based sensitivity analysis scheme,which is enhanced by applying the adjoint state vector method-ology. The merits of the proposed technique are examined by ap-plying it to both transverse magnetic and transverse electric po-larization cases. Finally, the technique is adopted by a frequency-hopping approach to cope with multifrequency inverse scatteringproblems.

Index Terms—Finite element method, gradient methods, imagereconstruction, inverse scattering, microwave imaging, optimiza-tion.

I. INTRODUCTION

T HE development of inverse scattering techniques for es-timating the material properties of inaccessible regions is

of great importance. In particular, microwave imaging is relatedto the reconstruction of the spatial distributions of the consti-tutive parameters of an unknown scatterer by post-processingscattered field data. The latter are obtained by illuminating thescatterer domain with incident electromagnetic waves.

The starting point for all the inverse scattering methodologiesis the description of the direct problem. Most of the proposedapproaches [1]–[3] are based on the integral formulation of theproblem (Lippmann-Schwinger equation), which is treated nu-merically by applying the method of moments (MoM). This ap-proach results in the solution of dense systems of equations thatrequire a lot of computation time and memory storage capacity.

In this paper, an alternative approach is investigated. The di-rect scattering problem is described by the differential formu-lation. This allows the application of the finite element method(FEM) for the numerical treatment of the problem [4]. Thus,we obtain sparse systems of equations; a property that resultsin a reduction of the storage-capacity and computation-time de-mands.

The inversion is based on the minimization of an error func-tion, which describes the discrepancy between the measured andthe estimated values of the scattered field data, by applying the

Manuscript received May 26, 1999; revised November 5, 1999.The authors are with the Division of Telecommunications, Department

of Electrical and Computer Engineering, Aristotle University of Thessa-loniki, Thessaloniki 54006, Greece (e-mail: [email protected];[email protected]).

Publisher Item Identifier S 0018-9456(00)02424-4.

Fig. 1. Geometric configuration of the problem.

Polak-Ribière nonlinear conjugate gradient optimization algo-rithm [5]. During each iteration of the minimization process, thederivative of the error function with respect to the parametersthat describe the scatterer domain is computed by performing asensitivity analysis scheme based on the FEM [6].

In this work, the proposed methodology is applied to the caseof two-dimensional (2-D) scatterers, while the scattered fielddata are synthetic and corrupted by additive white noise. In par-ticular, we examine the following combinations of incidencesand field data: 1) transverse magnetic (TM) incidences—elec-tric field measurements; and 2) transverse electric (TE) inci-dences—magnetic field measurements. Furthermore, the pro-posed technique is adopted by a frequency-hopping approach[7] in order to cope with multifrequency microwave imagingproblems.

II. FORMULATION OF THE DIRECT PROBLEM

Let us consider an infinitely long scatterer (along thez-axisof a Cartesian coordinate system) of bounded cross sectionD;which is invariant alongz (Fig. 1). The scatterer is nonmagneticand isotropic, while its dielectric permittivity,"s; and conduc-tivity, �s; vary only with respect to the transverse coordinates(xandy): Furthermore, we assume that the scatterer is embeddedin a homogeneous surrounding medium having dielectric per-mittivity, "b; and conductivity,�b: Hence, the electromagneticproperties of the scatterer can be described by the spatial distri-bution of the relative complex permittivity (RCP) given by

"(x; y) ="s(x; y)� j�s(x; y)=!

"b � j�b=!(1)

whereω is the excitation frequency.If the scatterer domain is illuminated by a TM incident wave

(having the electric field polarized in thez direction, ~Einc =

S0018-9456/00$10.00 © 2000 IEEE

REKANOS AND TSIBOUKIS: FINITE ELEMENT-BASED TECHNIQUE 235

Eincz0) and the time dependency has the formexp(j!t); thenthe scattered electric field,~E = Ez0; is given by the solutionof the Helmholtz equation

r2E + k2b"E = �k2b(" � 1)Einc (2)

wherekb = (!2�0"b � j!�0�b)1=2 and Imfkbg � 0:

The numerical solution of (2) is dealt with by the FEM,while the mesh is truncated by an absorbing boundary con-dition (ABC). In particular, the second-order ABC proposedby Bayliss and Turkel has been applied [8]. By selecting thislocal-type ABC we can preserve the sparsity of the systems ofequations obtained by the FEM.

After, the application of the Galerkin formulation to the dif-ferential equation (2), the scattered electric field is given by thesolution of the sparse system

S(""")E = b(""";Einc) (3)

where the vectorsE andEinc represent the scattered and theincident field, respectively, at the nodes of the mesh. The matrixS and the vectorb depend on the vector"""; which describes thedistribution of the RCP inside the scatterer domain. Actually,the vector""" is composed of the values of the RCP, which areassumed constant inside each element of the mesh.

If we are interested in calculating the scattered field at posi-tions outside the FEM-mesh, then we can apply the Helmholtz-Kirchhoff theorem, given by the integral

E(p) =

IC0

[E(p0)@G(p;p0)=@n0

�G(p;p0)@E(p0)=@n0] dl0: (4)

In (4), the positionp is placed outside the mesh, and the closedline integral is evaluated along the curveC0; which lies entirelywithin the finite element region. Furthermore,G(p;p0) is thesurrounding-space Green’s function, andn0 is the unit vectornormal to the curveC0:

Thus, the calculation of the scattered electric field,Ef atMpositions outside the FEM-mesh, can be written in the matrixform

Ef = [Ef1

Ef2

� � � EfM ]T = QE (5)

whereQ is obtained by approximating the line integral (4), andis sparse.

In the case, where the incident field is TE polarized( ~Hinc =Hincz0); we describe the scatterer by means of the inverse rel-ative complex permittivity (IRCP)

(x; y) ="b � j�b=!

"s(x; y) � j�s(x; y)=!: (6)

Hence, the differential formulation of the direct scatteringproblem is given by

r� ( r� ~H)� k2b ~H = r� [(1� )r� ~Hinc] (7)

where ~H = Hz0 is the scattered magnetic field. Conse-quently, if we apply (as in the TM case) the FEM and the

Helmoltz–Kirchhoff theorem we obtain analogous equationsfor the FEM system

S( )H = b( ;Hinc): (8)

and for the computation of the scattered magnetic field outsidethe FEM-mesh

Hf = [Hf1

Hf2

� � � HfM ]T = QH: (9)

In (8), the vectorγ represents the spatial distribution of the IRCPwithin the scatterer domain.

III. M ICROWAVE IMAGING

The proposed microwave imaging technique will be pre-sented for the case where the incident field is TM polarizedand we measure the scattered electric field component. Asimilar analysis is valid for the case of TE polarization andmeasurements of the scattered magnetic field. Assuming thatthe scatterer is illuminated from a set ofI distinct directionsaround its domain, and that, for each incidence,i; a set ofMmeasurements of the scattered field,Emi ; are obtained, then thevector""" is reconstructed by minimizing the error function

F (""") = I�1IXi=1

kEmi k�2kEmi �QEi(""")k

2+rkD"""k2: (10)

In (10), the second term is added to regularize the ill-posed in-verse problem [9]. In particular, the regularization term is re-lated to the spatial gradient of the RCP, over the scatterer do-mainD; and its influence is tuned via the positive regularizationfactor,r: The required gradient of the RCP is approximated bydifferences, which are implemented by the matrixD. If DX andDY are the matrices that implement the differences along thexand they direction, respectively, then the last term of (10) canbe written as follows:

kD"""k2 = """HDTD""" = """HDT

XDX""" + """HDT

YDY""": (11)

Since (10) is nonlinear with respect to"""; the error function isminimized by using an iterative optimization technique. In thisstudy, the Polak-Ribière nonlinear conjugate gradient algorithmhas been implemented. During the application of this algorithm,the gradient of the FEM solution with respect to both the real,"""R; and the imaginary part,"""I ; of """ has to be computed. Forconvenience, both gradients can be represented by introducingthe operator

="""F (""") =

�@

@"""R+ j

@

@"""I

�F ("""): (12)

If the discrepancy between the measurements and the estimatedvalues of the field, for theith incidence is

Fi(""") = kEmi �QEi(""")k2 (13)

then we can prove [6] that

(="""Fi)� = ZTi (=Ei

Fi)� (14)


Fig. 2. (a) Original RCP of the lossy scatterer. (b) Reconstructed profile after 48 iterations.

where

Zi =@Ei

@"""=

�@Ei

@"1

@Ei

@"2� � �

@Ei

@"N

�(15)

is the Jacobian matrix describing the sensitivity of the FEM so-lution with respect to theN -dimensional vector representing theRCP distribution. The columns of (15) can be achieved by dif-ferentiating [5] the FEM system of (3) with respect to each com-ponent of"""

S(@Ei=@"n) = @bi=@"n � (@S=@"n)Ei; 1 � n � N:

(16)

We observe that the computation of the Jacobian matrix re-quires the solution ofN systems of equations. SinceS is thesame for all of these systems and only the second part of (16)changes, we can apply the adjoint state vector methodology inorder to reduce the computational burden. According to thismethodology, during each iteration, we solve for each incidence,i; the system

Svi = �2QT(Emi �QEi)� (17)

and compute the adjoint vectorsvi: Finally, the gradients of theerror function are given by

="""F (""") = I�1IXi=1

kEmi k�2UH

i v�

i + 2rDTD""" (18)

where

Ui =

��@bi@"1

�@S

@"1Ei

��

�@bi@"N

�@S

@"NEi

��: (19)

We should also mention thatS does not depend on the inci-dence. Thus, the computation time can be further reduced if thesolution of the forward problem and the computation of sensi-tivities is based on a factorization ofS. In this work, we haveapplied the Cholesky factorization.

IV. M ULTIFREQUENCYMICROWAVE IMAGING

In microwave imaging applications, by using a high excita-tion frequency we can obtain more information about the scat-terer domain since the resolution of the reconstruction becomesfiner. However, by increasing the excitation frequency the inver-sion procedures might diverge or converge to a local minima farfrom the real solution, due to the nonlinearity of the problem.

A powerful methodology that allows the use of highfrequency data measurements without the aforementioneddifficulties is the frequency-hopping technique. According tothis technique the reconstruction of the scatterer profile is basedon measurements that are obtained by applying a set of distinctexcitation frequenciesf!k: !k < !k+1g; k = 1; 2; � � � ;K:Then, we define a set of cost functions

Fk(""") = I�1IXi=1

kEmkik�2kEmki �QkEki(""")k

2 + rkD"""k2

(20)


where each one of these functions is related to the measure-ments,Em

ki; which are associated with only one of the distinct

frequencies.The concept of the frequency-hopping technique is to

minimize successively the cost functions,Fk; starting form theone corresponding to the lowest frequency and hopping fromlower to higher frequencies. Hopping takes place each time acost function in not reduced any further, or after a prespeci-fied number of iterations. In this work, each cost function isminimized by applying the proposed methodology.

V. NUMERICAL RESULTS

In this study, the proposed technique has been applied tovarious inverse scattering problems covering the cases of: dif-ferent polarizations of the incident field, measurements of dif-ferent components of the scattered field, and multifrequency mi-crowave imaging based on the frequency-hopping technique.

A. TM Incidence—Electric Field Measurements

In the following, we present the reconstruction of a lossy scat-terer embedded in free space. It is assumed that the scatterer liesentirely within a 2λ × 2λ square domain (λ is the wavelength ofthe incident field in free space). Inside aλ × λ region centered inthe scatterer domain the RCP of the scatterer is equal to 1.6-j0.2,whereas outside it equals to 1.3-j0.4 (Fig. 2). The scatterer is il-luminated by TM plane waves, while the scattered electric fieldis measured. A number of 30 directions of incidence and 30 po-sitions of measurements are considered. The angles of incidenceare uniformly distributed around the scatterer domain, while themeasurement positions are equispaced on a circle of radius 8λ.Finally, a 30 × 30 grid of square subdivisions is used to dis-cretize the scatterer domain, resulting in a set of 900 complexunknowns. In this example, the measurements are assumed un-corrupted by noise, and no regularization is applied.

Starting from an initial guess of the scatterer profile equiv-alent to its absence, the reconstruction process is stopped after48 iterations, when no further reduction of the error function isachieved. The reconstructed profiles (Fig. 2) of the RCP provethe efficiency of the proposed technique.

B. TE Incidence—Magnetic Field Measurements

In the second example we investigate the case of TE polar-ization. In particular, we assume two distinct lossy scatterers ofsquare cross section having a side equal to 2λ/3. The RCP’s ofthe two scatterers are equal to 1.8-j0.4 and 1.4-j0.2. The scat-terers lie inside a 2.5λ × 2.5λ square domain, which is divided bya 30 × 30 grid. This region is illuminated from 35 directions byTE plane waves, while 35 measurements of the scattered mag-netic field are obtained for each incidence.

The synthetic measurements have been corrupted by additivegaussian noise resulting in a 10dB signal-to-noise ratio, whiletwo different values of the regularization factor have been ex-amined (0.0 and 0.1). The reconstructed profiles of the two scat-terers after 32 iterations are illustrated in Fig. 3. We observe thatthe regularization improved the reconstruction.

Fig. 3. (a) Original RCP of the two lossy scatterers. Reconstructed profilesbased on noisy measurements (SNR= 10 dB) with (b), and (c) withoutregularization.

C. Frequency Hopping

Finally, the proposed methodology has been applied to a mul-tifrequency inverse scattering problem. We assume the scat-terer examined in the first example [Fig. 2(a)]. Synthetic scat-tered field measurements obtained at three distinct frequenciesare used. The lowest excitation frequency,!1, corresponds to awavelength (in free space) equal to the side of the square do-main of investigation(�1 = d): The other two frequencies are!2 = 2!1 and!3 = 3!1: A number of 30 directions of in-cidence and 30 positions of measurements for each excitationfrequency are considered.

First, the inversion has been performed without regulariza-tion by using noiseless single-frequency data. The reconstruc-tion results after 48 iterations using the lowest and the highestfrequency are illustrated in Fig. 4. We observe that the use ofthe lowest frequency results in rather qualitative reconstructionresults. On the other hand, when high frequency data are usedthe inversion diverges.

To overcome these limitations, the frequency-hopping ap-proach is applied (Fig. 5). The first 16 iterations of the inversionare based on data obtained at the frequency!1: This givesa rather poor reconstruction. Then, another 16 iterations arecarried out by using data obtained at!2: This improves further


Fig. 4. Single-frequency profile reconstruction after 48 iterations using the lowest and the highest frequency.

Fig. 5. Profile reconstruction based on frequency-hopping. 16 iterations are performed for each single-frequency data set.

the estimation of the RCP by reconstructing the edges. Finerresolution is achieved after the last hopping, from!2 to !3:

VI. CONCLUSION

An inverse scattering method which is based on the differen-tial formulation of the direct scattering problem has been pre-


sented. The method combines the FEM and the Polak-Ribièreoptimization algorithm and is characterized by the solution ofsparse systems. Thus, the computational burden is lower com-pared to the rather traditional approach, which is based on theintegral formulation of the problem and the MoM. Moreover,the introduction of the adjoint state vector methodology resultedin a reduction of the total number of systems of equations thathave to be solved. The method has been applied successfully tothe reconstruction of 2-D lossy scatterers where both the TMand the TE polarization of the incident field have been investi-gated. Finally, the proposed method has been adopted by a fre-quency-hopping technique and its application to multifrequencyinverse scattering has been proven very promising.

REFERENCES

[1] N. Joachimowicz, C. Pichot, and J.-P. Hugonin, “Inverse scattering: Aniterative numerical metod for electromagnetic imaging,”IEEE Trans.Antennas Propagat., vol. 39, pp. 1742–1752, Dec. 1991.

[2] S. Caorsi, G. L. Gragnani, and M. Pastorino, “Two-dimensionalmicrowave imaging by a numerical inverse scattering solution,”IEEETrans. Microwave Theory Tech., vol. 38, pp. 981–89, Aug. 1990.

[3] , “Reconstruction of dielectric permittivity distributions in arbitrary2-D inhomogeneous biological bodies by a multiview microwave nu-merical method,”IEEE Trans. Med. Imag., vol. 12, pp. 232–239, June1993.

[4] J. Jin, The Finite Element Method in Electromagnetics. New York:John Wiley & Sons, 1993.

[5] P. Neittaanmäki, M. Rudnicki, and A. Savini, “Inverse problems and op-timal design in electricity and magnetism,” inMonographs in Electricaland Electronic Engineering 35. Oxford, U.K.: Clarendon, 1996.

[6] I. T. Rekanos and T. D. Tsiboukis, “A combined finite element—Conju-gate gradient spatial method for the reconstruction of unknown scattererprofiles,” IEEE Trans. Magn., vol. 34, pp. 2829–2832, Sept. 1998.

[7] W. C. Chew and J.-H. Lin, “A frequency-hopping approach for mi-crowave imaging of large inhomogeneous bodies,”IEEE MicrowaveGuided Wave Lett., vol. 5, pp. 439–441, Dec. 1995.

[8] K. Bayliss and E. Turkel, “Radiation boundary conditions for wavelikeequations,”Commun. Pure Appl. Math., vol. 33, pp. 707–725, 1980.

[9] A. N. Tikhonov and V. Y. Arserin,Solutions of Ill-Posed Prob-lems. Washington, DC: Winston, 1977.

Ioannis T. Rekanos (S’92) was born in Thessa-loniki, Greece, in 1970. He received the Diplomadegree (with honors) in electrical engineering, in1993, and the Ph.D. degree in electrical and com-puter engineering, in 1998, both from the AristotleUniversity of Thessaloniki, Greece.

From 1993 to 1998, he was a Research andTeaching Assistant in the Department of Electricaland Computer Engineering of the same university.From 1995 to 1998, he was a scholar of theBodosaki’s Foundation. His research interests

include inverse electromagnetic problems, computational electromagnetics,optimization techniques, neural network applications in eddy current nondestructive testing, and signal processing.

Dr. Rekanos is a member of the American Geophysical Union and the Tech-nical Chamber of Greece. In 1995, he received the URSI, Commission B, YoungScientist Award.

Theodoros D. Tsiboukis(M’81–M’91–SM’99) wasborn in Larissa, Greece, on February 25, 1948. He re-ceived the Diploma degree in electrical and mechan-ical engineering from the National Technical Univer-sity of Athens, Greece, in 1971, and the Dr. Eng. de-gree from the Aristotle University of Thessaloniki,Greece, in 1981.

During the academic year 1981–1982, hewas a Visiting Research Fellow at the ElectricalEngineering Department of the University ofSouthampton, U.K. Since 1982, he has been working

at the Department of Electrical and Computer Engineering of the AristotleUniversity of Thessaloniki, where he is now a Professor. His research interestsinclude electromagnetic field analysis by energy methods, computationalelectromagnetics (FEM, BEM, Vector Finite Elements, MoM, FDTD, ABC’s),inverse problems, and adaptive meshing in FEM analysis. He is the authorof six books. He has authored or coauthored more than 60 refereed journalarticles, and more than 60 conference papers.

Dr. Tsiboukis was the Guest Editor of a special issue of theInternationalJournal of Theoretical Electrotechnics(1996) and the Chairman of the localorganizing committee of the Eighth International Symposium on TheoreticalElectrical Engineering (1995). He has also organized and chaired conferencesessions and was awarded a number of distinctions. From 1993 to 1997, he wasthe Director of the Division of Telecommunications at the Department of Elec-trical and Computer Engineering, of the Aristotle University of Thessaloniki.From 1997 he is the Chairman of the above department. He is a member of var-ious societies, associations, chambers, and institutions.

a finite element-based technique for microwave imaging of two-dimensional objects

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