A finite element-based technique for microwave

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A Finite Element-Based Technique for Microwave Imaging 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 estimating the spatial distributions of the permittivity and the conductivity of a scatterer, by post-processing electromagnetic scattered field data, is presented. For the description of the direct scattering problem, the differential formulation is applied. This allows the use of the finite element method. During the inversion, the computation of the derivative of the finite element solution with respect to the parameters, which describe the scatterer, is required. This task is performed by a finite element-based sensitivity analysis scheme, which is enhanced by applying the adjoint state vector methodology. The merits of the proposed technique are examined by applying it to both transverse magnetic and transverse electric polarization cases. Finally, the technique is adopted by a frequencyhopping approach to cope with multifrequency inverse scattering problems. Index Terms—Finite element method, gradient methods, image reconstruction, inverse scattering, microwave imaging, optimization.

I. INTRODUCTION

T

HE development of inverse scattering techniques for estimating the material properties of inaccessible regions is of great importance. In particular, microwave imaging is related to the reconstruction of the spatial distributions of the constitutive parameters of an unknown scatterer by post-processing scattered field data. The latter are obtained by illuminating the scatterer domain with incident electromagnetic waves. The starting point for all the inverse scattering methodologies is the description of the direct problem. Most of the proposed approaches [1]–[3] are based on the integral formulation of the problem (Lippmann-Schwinger equation), which is treated numerically by applying the method of moments (MoM). This approach results in the solution of dense systems of equations that require a lot of computation time and memory storage capacity. In this paper, an alternative approach is investigated. The direct scattering problem is described by the differential formulation. 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 results in a reduction of the storage-capacity and computation-time demands. The inversion is based on the minimization of an error function, which describes the discrepancy between the measured and the 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 Thessaloniki, 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 algorithm [5]. During each iteration of the minimization process, the derivative of the error function with respect to the parameters that describe the scatterer domain is computed by performing a sensitivity analysis scheme based on the FEM [6]. In this work, the proposed methodology is applied to the case of two-dimensional (2-D) scatterers, while the scattered field data are synthetic and corrupted by additive white noise. In particular, we examine the following combinations of incidences and field data: 1) transverse magnetic (TM) incidences—electric field measurements; and 2) transverse electric (TE) incidences—magnetic field measurements. Furthermore, the proposed technique is adopted by a frequency-hopping approach [7] in order to cope with multifrequency microwave imaging problems. II. FORMULATION OF THE DIRECT PROBLEM Let us consider an infinitely long scatterer (along the z -axis of a Cartesian coordinate system) of bounded cross section D; which is invariant along z (Fig. 1). The scatterer is nonmagnetic and isotropic, while its dielectric permittivity, "s ; and conductivity, s ; vary only with respect to the transverse coordinates (x and y): Furthermore, we assume that the scatterer is embedded in a homogeneous surrounding medium having dielectric permittivity, "b; and conductivity, b: Hence, the electromagnetic properties of the scatterer can be described by the spatial distribution of the relative complex permittivity (RCP) given by

"(x; y) =

"s (x; y) 0 js (x; y)=! "b 0 jb =!

(1)

where ω is the excitation frequency. If the scatterer domain is illuminated by a TM incident wave ~ inc = (having the electric field polarized in the z direction, E

S0018-9456/00$10.00 © 2000 IEEE

REKANOS AND TSIBOUKIS: FINITE ELEMENT-BASED TECHNIQUE

z

235

E inc 0 ) and the time dependency has the form exp(j!t); then ~ = E 0 ; is given by the solution the scattered electric field, E of the Helmholtz equation

z

r

0 1)E b where kb = (!2 0 "b 0 j!0 b)1=2 and Imfkbg  0: 2E + k2 "E = b

0

k2("

inc

distribution of the RCP inside the scatterer domain. Actually, the vector " is composed of the values of the RCP, which are assumed constant inside each element of the mesh. If we are interested in calculating the scattered field at positions outside the FEM-mesh, then we can apply the HelmholtzKirchhoff theorem, given by the integral

p) =

E(

C0

p

p )@G(p; p )=@n 0G(p; p )@E (p )=@n ] dl :

[E (

0

0

0

0

0

0

(4)

In (4), the position is placed outside the mesh, and the closed line integral is evaluated along the curve C 0; which lies entirely within the finite element region. Furthermore, G( ; 0) is the surrounding-space Green’s function, and 0 is the unit vector normal to the curve C 0: Thus, the calculation of the scattered electric field, E f at M positions outside the FEM-mesh, can be written in the matrix form

pp

n

Q

Ef = [E1f

E2f

111

f T EM ]

=

QE

(5)

where is obtained by approximating the line integral (4), and is sparse. ~ inc = In the case, where the incident field is TE polarized (H inc H 0 ); we describe the scatterer by means of the inverse relative complex permittivity (IRCP)

z

(x; y) =

"b 0 jb =! : "s (x; y) 0 js (x; y)=!

(6)

Hence, the differential formulation of the direct scattering problem is given by

r 2 ( r 2 H~ ) 0 kb2H~

z

~ inc] = r 2 [(1 0 )r 2 H

(8)

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

S(")E = b("; Einc) (3) where the vectors E and Einc represent the scattered and the incident field, respectively, at the nodes of the mesh. The matrix S and the vector b depend on the vector "; which describes the

0

S( )H = b( ; Hinc):

(2)

The numerical solution of (2) is dealt with by the FEM, while the mesh is truncated by an absorbing boundary condition (ABC). In particular, the second-order ABC proposed by Bayliss and Turkel has been applied [8]. By selecting this local-type ABC we can preserve the sparsity of the systems of equations obtained by the FEM. After, the application of the Galerkin formulation to the differential equation (2), the scattered electric field is given by the solution of the sparse system

I

Helmoltz–Kirchhoff theorem we obtain analogous equations for the FEM system

(7)

~ = H 0 is the scattered magnetic field. Consewhere H quently, if we apply (as in the TM case) the FEM and the

Hf = [H1f

111

H2f

f T HM ]

=

QH:

(9)

In (8), the vector γ represents the spatial distribution of the IRCP within the scatterer domain. III. MICROWAVE IMAGING The proposed microwave imaging technique will be presented for the case where the incident field is TM polarized and we measure the scattered electric field component. A similar analysis is valid for the case of TE polarization and measurements of the scattered magnetic field. Assuming that the scatterer is illuminated from a set of I distinct directions around its domain, and that, for each incidence, i; a set of M measurements of the scattered field, im ; are obtained, then the vector " is reconstructed by minimizing the error function

E

F (") = I 01

I X i=1

kEmi k02 kEmi 0 QEi (")k2 + rkD"k2:

(10)

In (10), the second term is added to regularize the ill-posed inverse problem [9]. In particular, the regularization term is related to the spatial gradient of the RCP, over the scatterer domain D; and its influence is tuned via the positive regularization factor, r: The required gradient of the RCP is approximated by differences, which are implemented by the matrix D. If X and Y are the matrices that implement the differences along the x and the y direction, respectively, then the last term of (10) can be written as follows:

D

D

kD"k2 = " HDT D" = "H DTXDX" + " HDTY DY" :

(11)

Since (10) is nonlinear with respect to " ; the error function is minimized by using an iterative optimization technique. In this study, the Polak-Ribière nonlinear conjugate gradient algorithm has 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. For convenience, both gradients can be represented by introducing the operator

=" F (") =



@ @""R

@ +j @""I



F ("):

(12)

If the discrepancy between the measurements and the estimated values of the field, for the ith incidence is

E

Fi(" ) = k m i 0

QEi(")k2

(13)

then we can prove [6] that

=

3

( " Fi ) =

ZTi (=E F )3 i

i

(14)

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

where

Z

i

=

@ Ei @""

=

 @E

i

@"1

@ Ei @"2

111

@ Ei @"N

where

 (15)

is the Jacobian matrix describing the sensitivity of the FEM solution with respect to the N -dimensional vector representing the RCP distribution. The columns of (15) can be achieved by differentiating [5] the FEM system of (3) with respect to each component of "

S(@Ei =@"n) = @bi=@"n 0 (@S=@"n)Ei;

1

 n  N: (16)

We observe that the computation of the Jacobian matrix requires the solution of N systems of equations. Since S is the same for all of these systems and only the second part of (16) changes, we can apply the adjoint state vector methodology in order to reduce the computational burden. According to this methodology, during each iteration, we solve for each incidence, i; the system

Svi = 02QT(Emi 0 QEi)3 (17) and compute the adjoint vectors vi : Finally, the gradients of the error function are given by

X =" F (") = I 01 i=1 I

kEm k02UHv3 + 2rDTD" i

i

i

(18)

Ui =

 @ b

0 @S E @"1 @"1 i i

  @ b @ S  1 1 1 @" i 0 @" Ei : N N

(19)

We should also mention that S does not depend on the incidence. Thus, the computation time can be further reduced if the solution of the forward problem and the computation of sensitivities is based on a factorization of S. In this work, we have applied the Cholesky factorization. IV. MULTIFREQUENCY MICROWAVE IMAGING In microwave imaging applications, by using a high excitation frequency we can obtain more information about the scatterer domain since the resolution of the reconstruction becomes finer. However, by increasing the excitation frequency the inversion procedures might diverge or converge to a local minima far from the real solution, due to the nonlinearity of the problem. A powerful methodology that allows the use of high frequency data measurements without the aforementioned difficulties is the frequency-hopping technique. According to this technique the reconstruction of the scatterer profile is based on measurements that are obtained by applying a set of distinct excitation frequencies f!k : !k < !k+1 g; k = 1; 2; 1 1 1 ; K: Then, we define a set of cost functions I X 0 1 Fk (" ) = I i=1

kEmki k02kEmki 0 Qk Eki (")k2 + rkD"k2 (20)

REKANOS AND TSIBOUKIS: FINITE ELEMENT-BASED TECHNIQUE

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where each one of these functions is related to the measurements, 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 the one corresponding to the lowest frequency and hopping from lower to higher frequencies. Hopping takes place each time a cost function in not reduced any further, or after a prespecified number of iterations. In this work, each cost function is minimized by applying the proposed methodology.

V. NUMERICAL RESULTS In this study, the proposed technique has been applied to various inverse scattering problems covering the cases of: different polarizations of the incident field, measurements of different components of the scattered field, and multifrequency microwave imaging based on the frequency-hopping technique. A. TM Incidence—Electric Field Measurements In the following, we present the reconstruction of a lossy scatterer embedded in free space. It is assumed that the scatterer lies entirely within a 2λ × 2λ square domain (λ is the wavelength of the incident field in free space). Inside a λ × λ region centered in the scatterer domain the RCP of the scatterer is equal to 1.6-j 0.2, whereas outside it equals to 1.3-j 0.4 (Fig. 2). The scatterer is illuminated by TM plane waves, while the scattered electric field is measured. A number of 30 directions of incidence and 30 positions of measurements are considered. The angles of incidence are uniformly distributed around the scatterer domain, while the measurement positions are equispaced on a circle of radius 8λ. Finally, a 30 × 30 grid of square subdivisions is used to discretize the scatterer domain, resulting in a set of 900 complex unknowns. In this example, the measurements are assumed uncorrupted by noise, and no regularization is applied. Starting from an initial guess of the scatterer profile equivalent to its absence, the reconstruction process is stopped after 48 iterations, when no further reduction of the error function is achieved. The reconstructed profiles (Fig. 2) of the RCP prove the efficiency of the proposed technique. B. TE Incidence—Magnetic Field Measurements In the second example we investigate the case of TE polarization. In particular, we assume two distinct lossy scatterers of square cross section having a side equal to 2λ/3. The RCP’s of the two scatterers are equal to 1.8-j 0.4 and 1.4-j 0.2. The scatterers lie inside a 2.5λ × 2.5λ square domain, which is divided by a 30 × 30 grid. This region is illuminated from 35 directions by TE plane waves, while 35 measurements of the scattered magnetic field are obtained for each incidence. The synthetic measurements have been corrupted by additive gaussian noise resulting in a 10dB signal-to-noise ratio, while two different values of the regularization factor have been examined (0.0 and 0.1). The reconstructed profiles of the two scatterers after 32 iterations are illustrated in Fig. 3. We observe that the regularization improved the reconstruction.

Fig. 3. (a) Original RCP of the two lossy scatterers. Reconstructed profiles 10 dB) with (b), and (c) without based on noisy measurements (SNR regularization.

=

C. Frequency Hopping Finally, the proposed methodology has been applied to a multifrequency inverse scattering problem. We assume the scatterer examined in the first example [Fig. 2(a)]. Synthetic scattered field measurements obtained at three distinct frequencies are used. The lowest excitation frequency, !1 , corresponds to a wavelength (in free space) equal to the side of the square domain of investigation (1 = d): The other two frequencies are !2 = 2!1 and !3 = 3!1: A number of 30 directions of incidence and 30 positions of measurements for each excitation frequency are considered. First, the inversion has been performed without regularization by using noiseless single-frequency data. The reconstruction results after 48 iterations using the lowest and the highest frequency are illustrated in Fig. 4. We observe that the use of the lowest frequency results in rather qualitative reconstruction results. On the other hand, when high frequency data are used the inversion diverges. To overcome these limitations, the frequency-hopping approach is applied (Fig. 5). The first 16 iterations of the inversion are based on data obtained at the frequency !1: This gives a rather poor reconstruction. Then, another 16 iterations are carried out by using data obtained at !2: This improves further

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

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. Finer resolution is achieved after the last hopping, from !2 to !3 :

VI. CONCLUSION An inverse scattering method which is based on the differential formulation of the direct scattering problem has been pre-

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sented. The method combines the FEM and the Polak-Ribière optimization algorithm and is characterized by the solution of sparse systems. Thus, the computational burden is lower compared to the rather traditional approach, which is based on the integral formulation of the problem and the MoM. Moreover, the introduction of the adjoint state vector methodology resulted in a reduction of the total number of systems of equations that have to be solved. The method has been applied successfully to the reconstruction of 2-D lossy scatterers where both the TM and the TE polarization of the incident field have been investigated. Finally, the proposed method has been adopted by a frequency-hopping technique and its application to multifrequency inverse scattering has been proven very promising.

Ioannis T. Rekanos (S’92) was born in Thessaloniki, Greece, in 1970. He received the Diploma degree (with honors) in electrical engineering, in 1993, and the Ph.D. degree in electrical and computer engineering, in 1998, both from the Aristotle University of Thessaloniki, Greece. From 1993 to 1998, he was a Research and Teaching Assistant in the Department of Electrical and Computer Engineering of the same university. From 1995 to 1998, he was a scholar of the Bodosaki’s Foundation. His research interests include inverse electromagnetic problems, computational electromagnetics, optimization techniques, neural network applications in eddy current non destructive testing, and signal processing. Dr. Rekanos is a member of the American Geophysical Union and the Technical Chamber of Greece. In 1995, he received the URSI, Commission B, Young Scientist Award.

REFERENCES

Theodoros D. Tsiboukis (M’81–M’91–SM’99) was born in Larissa, Greece, on February 25, 1948. He received the Diploma degree in electrical and mechanical engineering from the National Technical University of Athens, Greece, in 1971, and the Dr. Eng. degree from the Aristotle University of Thessaloniki, Greece, in 1981. During the academic year 1981–1982, he was a Visiting Research Fellow at the Electrical Engineering Department of the University of Southampton, U.K. Since 1982, he has been working at the Department of Electrical and Computer Engineering of the Aristotle University of Thessaloniki, where he is now a Professor. His research interests include electromagnetic field analysis by energy methods, computational electromagnetics (FEM, BEM, Vector Finite Elements, MoM, FDTD, ABC’s), inverse problems, and adaptive meshing in FEM analysis. He is the author of six books. He has authored or coauthored more than 60 refereed journal articles, and more than 60 conference papers. Dr. Tsiboukis was the Guest Editor of a special issue of the International Journal of Theoretical Electrotechnics (1996) and the Chairman of the local organizing committee of the Eighth International Symposium on Theoretical Electrical Engineering (1995). He has also organized and chaired conference sessions and was awarded a number of distinctions. From 1993 to 1997, he was the Director of the Division of Telecommunications at the Department of Electrical and Computer Engineering, of the Aristotle University of Thessaloniki. From 1997 he is the Chairman of the above department. He is a member of various societies, associations, chambers, and institutions.

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