Microwave imaging of piecewise constant objects in a 2D-TE configuration O. F´eron, B. Duchˆene and A. Mohammad-Djafari Laboratoire des signaux et syst`emes (CNRS-Sup´elec-UPS), Plateau de Moulon, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France. Abstract. In this paper we propose a stochastic algorithm applied to an electromagnetic inverse scattering problem. The objective is to characterize an unknown object from measurements of the scattered fields at different frequencies and for several illuminations. This inverse problem is known to be nonlinear and ill-posed. It then needs to be regularized by introducing prior information. The particular prior information we account for is that the object is composed of a finite known number of different materials distributed in compact regions. The algorithm is applied to the inversion of experimental data collected at the Institut Fresnel (Marseille) and has already provided satisfactory results in a 2D-TM configuration. Herein, the goal is to test the same kind of method in a 2D-TE configuration.
Key Words: Microwave imaging, Bayesian estimation, Markov Random Fields.
I. Direct model We deal herein with an electromagnetic inverse obstacle scattering problem where the goal is to characterize unknown targets from measurements of the scattered fields that result from their interactions with a known interrogating (or incident) wave in the microwave frequency range. The configuration considered herein is that of the Institut Fresnel experiment: an incident time-harmonic wave at angular frequency ω (the time-dependence exp(-iωt) is implied) ~ axis and of arbitrary cross-section shape Ω in illuminates a cylindrical target (considered as infinite along the Oz the (x, y) plane) at Nq frequencies in the range 1 - 18 GHz and the resulting electric field is collected around the target on a circular measurement domain S (of radius Rmes = 1.67 m) at Nr = 241 receiver positions. The propagation direction φ of the incident wave lies in the (x, y) plane and can be varied (φ ∈ [0, 2π]), Nv views being carried out at varying incidence for each frequency q. The different media are supposed to be linear isotropic and nonmagnetic and are characterized by their propagation constant k(~r) such that k 2 (~r) = k02 = ω 2 ε0 µ0 , ~r = (x, y) 6∈ Ω, where ε0 and µ0 represent the dielectric permittivity and the magnetic permeability of vacuum, respectively, or 2 k 2 (~r) = kΩ (~r) = ω 2 ε(~r)µ0 , ~r ∈ Ω, where ε is the complex permittivity (ε(~r) = ε0 εr (~r) + iσ(~r)/ω) and εr and σ represent the relative permittivity and the conductivity of the target, respectively. ~ is The situation is such that a 2D configuration is considered: the TM polarization case, where the electric field E ~ parallel to the cylinder axis Oz, has been the subject of a previous paper [1] published in a special section dedicated to the test of inversion algorithms against experimental data [2]; we consider, herein, the TE polarisation case, where the electric field lies in the (x, y) plane. Let us notice that, in that case, the data consist of the electric field component tangential to the measurement circle. Modelling is based upon a domain integral representation obtained by applying Green′ s theorem to the Helmholtz wave equations satisfied by the fields and accounting for continuity and radiation conditions. This leads to two coupled contrast-source integral equations, denoted as the state and observation equations, that express the total ~ inside the domain Ω occupied by the object and the scattered field E ~ dif observed on the measurement electric field E domain S, respectively: Z Z ~ r ′ ) d~r ′ + 1 ∇ ~ r) = E ~ inc (~r) + ~ r ′ ) d~r ′ ~ ∇. ~ G(~r, ~r ′ )J(~ E(~ G(~r, ~r ′ )J(~ ~r ∈ Ω (1) 2 k Ω Ω 0 Z Z 1 ~~ ~ dif (~r) = ~ r ′ ) d~r ′ E G(~r, ~r ′ )J~(~r ′ ) d~r ′ + 2 ∇ ∇. G(~r, ~r ′ )J(~ ~r ∈ S, (2) k0 Ω Ω
~ r ′ ) = χ(~r ′ )E(~ ~ r ′ ) are the induced currents, χ(~r ′ ) = k 2 (~r ′ ) − k 2 is a contrast function null outside Ω, where J(~ 0 G(~r, ~r ′ ) = 4i H01 (k0 R) is the free space Green′ s function, and R = |~r − ~r ′ |. The contrast function χ characterizes OIPE 2006 ”The 9th Workshop on Optimization and Inverse Problems in Electromagnetics” - September 13th – 15th, Sorrento (Italy)
the unknown object and the inverse problem then consists in retrieving this function. In a TM configuration ([1]), these coupled equations become simple and scalar, whereas, in the TE configuration considered herein, the electric ~∇ ~ field is a two-component vector and the problem is then more involved. It can be noticed that, as the operator ∇ acts upon a convolution product, several strategies [4] can be used to solve equations (1) and (2): i) both of the differential operators can be applied to the Green’s function ([5]) or ii) one can be applied to the latter and the other to the induced currents ([6]). The latter strategy is adopted herein. Equation (1) then reads: Z Z 1 ′ ~ ′ ′ inc ~ ~ ~ ′ G(~r, ~r ′ )∇ ~ ′ . J(~ ~ r ′ ) d~r ′ G(~r, ~r )J (~r ) d~r − 2 E(~r) = E (~r) + ∇ ~r ∈ Ω (3) k0 Ω Ω
~ ′ means a derivation with respect to the variable ~r ′ , and the observation equation is of the same form. where ∇ With in mind the fact that we want to solve the above equations by means of the method of moments, we partition ~ ′ . J(~ ~ r ′) the test domain into elementary square pixels Ωm of constant complex permittivity. Then, by expressing ∇ ~ = εE, ~ we found that the former vanishes everywhere except at the frontier C as a function of the electric flux D ~ across the between pixels of different contrasts. By accounting for the continuity of the normal component of D latter, this reads: −
1 ~′ ~ ′ ~ r ′ ).∇ ~ ′ 1 = ε(~r ′ ) E(~ ~ r ′ ). ~nC ζ(~r ′ ) δC (~r ′ ), ∇ . J(~r ) = ε(~r ′ ) E(~ 2 k0 ε(~r ′ )
(4)
where ~nC is the normal to the contour C, ζ(~r ′ ) is the jump of the function 1/ε(~r ′ ) across the latter and δC is the Dirac delta function centered on C. This means that the second integral in (3) is reduced to a line integral along the contours of the above-mentionned pixels. By applying the method of moments with point matching and pulse basis functions Hm defined as Hm (~r ) = 1 if ~r ∈ Ωm , 1/2 if ~r ∈ Cm and 0 elsewhere, where Ωm and Cm represent the pixel m and its contour, respectively, we obtain discrete versions of the observation and state equations : ~ r) E(~ ~ dif (~r) E
~ inc (~r) + GΩ E(~ ~ r) = E ~ r) = GS E(~
~r ∈ Ω
(5)
~r ∈ S,
(6)
These two equations are of the same kind as in the TM configuration, but obviously the expressions of the operators GΩ and GS , that act from Ω onto itself and from Ω onto S, respectively, are a little bit more intricate and 4 times bigger. Equation (5) (as well as equation (6)) can be decomposed into two scalar equations of the form :
Eu (~r) = Euinc (~r) +
NΩ X r , ~rm )Eu (~rm ) + GΩ r , ~rm )Ex (~rm ) + GΩ r , ~rm )Ey (~rm ) , u = x or y χ(~rm )GΩ uy (~ ux (~ 0 (~
(7)
m=1
where NΩ is the number of pixels that partition Ω, ~rm is the center of the mth pixel, GΩ 0 results from the first Ω integral in (3) and is obtained analytically by replacing the square pixel by a disc of same area [7] and GΩ vx and Gvy result from the second integral in (3) and correspond to line integrals along the parts of the pixel contour that are perpendicular to x and y, respectively. It can be noticed that the the last two terms are obtained by means of fast Fourier transforms, integration being performed analytically along the exact contour of the pixels after a spectral decomposition of the Green’s function as in [4]; they are of the form: GΩ r , ~rm ) = ε(~rm ) [ζ(~rm + a~v )Quv (~r, ~rm + a~v ) − ζ(~rm − a~v )Quv (~r, ~rm − a~v )] , u = x or y, v = x or y uv (~
(8)
where a is the pixel half-side, ~v is a unitary vector oriented along the x or y axis and Quv is given by: +∞
sin (αa) iβ |(~r−~r ′ ).~v| iα((~r−~rm ).~u) e e dα, v = x or y, u 6= v, Fuv 2 −∞ q ~r ′ = ~rm ± a~v , β = k02 − α2 , ℑm(β) ≥ 0, Fuv = 1/β, Fvv = sign((~r − ~r ′ ).~v ))/α
Quv (~r, ~r ′ ) =
1 2π
Z
(9)
It can be noticed (see equation (8)) that, through ζ(~rm ± a~v ) = 1/ε(~rm ± 2a~v ) − 1/ε(~rm ), this model introduces an intrinsic correlation between neighbouring pixels, and also a nonlinearity with respect to the contrast function χ, which brings a lot of difficulties into the implementation; that is why, in a first approach of inversion, we propose to do a bilinearized approximation of the direct model, neglecting these correlation terms. The bilinearized model we use can be considered with respect to the contrast and the induced currents in a vectorial form: Eudif Ju
= GS0 Ju
(10)
= χEuinc + χGΩ 0 Ju
(11)
OIPE 2006 ”The 9th Workshop on Optimization and Inverse Problems in Electromagnetics” - September 13th – 15th, Sorrento (Italy)
and, in the following, we renote GS = GS0 and GΩ = GΩ 0 . In order to test the validity of the above considerations, we have compared the results obtained by means of the direct model described by equations (5) and (6) to the Institut ~ r = ~rm ), m = 1, ..., NΩ , and then E ~ dif (~r), ~r ∈ S, Fresnel experimental results. Hence, we first solve (7) for E(~ is obtained directly from the discrete counterpart of (6). Solving (7) requires the knowledge of the incident field ~ inc (~r) inside Ω. The latter is supposed to be of the form: E ~ inc (~r) = f (θ) H01 (k0 R) ~uθ E (12)
180
0.1
phase (◦ )
modulus (V/m)
~ = ~r − ~re = (R, Φ), ~re = (re = Rmes , φe ) is the position of the emitting antenna, θ = Φ − φe + π, ~uθ is a where R ~ and f (θ) is a directivity factor obtained from the incident field measured on S. unitary vector such that ~uθ ⊥ R Figure 1 displays the results obtained for the configuration studied in ([3], fig. 9), the latter being chosen for comparison purposes: the target under consideration corresponds to the data set F oamDielExtT E in the Institut Fresnel data-base. The target is illuminated by an emitting antenna located at φe = 270◦ and operating at a 8 GHz frequency. The domain Ω is a rectangular area partitioned into 34 × 25 pixels with side 2a = 3.5 mm, which corresponds to a discretization step of λ/10.7, where λ is the wavelength.
0.05
0
−180 120
60
40
0
θr (◦ )
120
60
40
60
120 180
0.14
40
0
θr (◦ )
120
60
40
3
0.12 2
0.1
phase (◦ )
modulus (V/m)
0
0.1
0.08
0.06
0.05 0.04
1
0
0
−1
−2
0.02
0 120 0
−3
−100
−50
60
0
40
0
50
◦
θr ( )
40
100
60
120
−180 120
−100
−50
60
0
40
0
θr (◦ )
50
40
100
60
120
Figure 1: Modulus (left) and phase (right) of the computed (- - -) by the exact direct model (top) and the bilinearized ) scattered fields versus the position of the receiver relative to the direction of model (bottom) and measured ( illumination (θr = φr − φe + π). It can be concluded that the exact model we propose yields good results, as compared to the experimental ones, which are similar to those obtained in [3] by means of a different model based upon contour integrals. The bilinearized model we consider is quite worse but seems to be satisfactory enough for the inversion task.
II. Prior information and Bayesian approach Let q and v be respectively the indices of frequency and views. We note A = {Au,q,v }u=x,y, q=1,...,Nq , v=1,...,Nv any vector regrouping all the directions, frequencies and views. We consider an additive noise on the observation equation in order to model the noise measurements and also the incertainty due to the fact that the model is not perfect: dif Eu,q,v = GSu,q,v Ju,q,v + bu,q,v . (13) Assuming the noise to be Gaussian, white and centered bu,q,v ∼ N (0, ρb Id ) and independant along the frequencies and the different views, we can express the likelihood function: ) ( 1 X dif S 2 dif kEu,q,v − Gq,v Ju,q,v k (14) p(E |J, ρb ) ∝ exp − 2ρb q,v ~ within The inverse problem in microwave imaging is known to be very ill-posed, and also nonlinear, the total field E Ω being unknown. We then need to introduce additional information in order to regularize the problem. There exists a lot of work that introduce prior information such as the positivity of the real and imaginary parts of the contrast function or the smoothness of the latter, or impose an edge-preserving constraint. Herein, we introduce a particular additional information: although the materials that compose the object are not known, their finite number Nl is supposed to be known. We introduce this prior information by modelling the contrast image by a hidden Markov random field. This consists in considering a hidden variable z(~r) which represents a classification of the contrast image. This hidden variable takes discrete values between 1 and Nl , each value being associated to a given material. OIPE 2006 ”The 9th Workshop on Optimization and Inverse Problems in Electromagnetics” - September 13th – 15th, Sorrento (Italy)
Each material will thus be identified by a label n in the image z and characterized by a mean complex value mn (associated to its estimated complex permittivity) and a variance ρn (which allows some fluctuations around the mean value in the contrast image χ). In this work we consider that all the labels are characterized by the same variance, ρn = ρ, ∀n = 1, . . . , Nl . Therefore we define a prior conditional probability distribution of the contrast as a Gaussian law: for each pixel ~r of the contrast image, p(χ(~r)|z(~r) = n, mn , ρ) = N (mn , ρ). We can thus define the distribution of the entire vector χ given the entire vector z, m = {mn }n=1,...,Nl and ρ: p(χ|z, m, ρ) = N (mz , ρId ) 1 ∝ exp − kχ − mz k2 , 2ρ
(15)
with mz (~r) = mn if z(~r) = n. We focus herein on the joint estimation of the contrast and the induced currents related to the contrast by the state equation (11). The prior distribution on the unknowns we propose is: 1 1 X 2 inc 2 , (16) − χq GΩ J k − kJu,q,v − χq Euq,v kχ − m k p(J, χ|z, m, ρ) ∝ exp − z q u,q,v 2ρη u,q,v 2ρ where the first term in the exponential accounts for the information issued from the state equation and the second term expresses the Hidden Markov model on the contrast. Obviously we have also to estimate the classification z and all the parameters mn and ρ, as the materials themselves are not known. Prior probability distributions have then to be assigned to these variables. As the materials are supposed to be distributed in compact regions, a local spatial correlation on the pixels of the classification is introduced by modelling z with a Potts Markov random field: X X δ(z(~r) − z(~r ′ )) , (17) p(z) ∝ exp ′ ~ r ∈S ~ r ∈V(~ r)
where δ is the kronecker function and V(~r) is the set made of the four neerest neighbors of the pixel ~r. The prior probability distributions of the parameters mn and ρ are chosen to be in the so-called ”conjugate priors” family in order to render their estimation easier. All the previous definitions allow us to define a joint prior distribution p(J, χ, z, m, ρ) of all the unknowns we have to estimate that takes into account the state equation (11) of the modelling and all the information introduced earlier. By and using the Bayes formula, this allows us to express the a posteriori distribution of all the unknowns, given the scattered field data: p(J, χ, z, m, ρ|E dif ) ∝ p(E dif |J, ρb ) p(J, χ|z, m, ρ) p(z) p(m, ρ)
(18)
This posterior distribution represents all the information we have on the unknowns which comes from our a priori knowledge and from the data. The Bayesian approach consists then in using this posterior distribution to define an estimator for the unknowns. We propose to estimate the posterior mean by means of a Gibbs sampling algorithm. This consists in split the entire set of variables into subsets and alternately sample these subsets from their conditional probability distribution. In this work we decompose the set of variables into thre subsets: J, (χ, z) and ˆ (0) and zˆ(0) , (m, ρ). The Gibbs sampling algorithm we propose then reads: given an initialization Jˆ(0) , χ repeat 1. 2. 3.
ˆ (n−1) , zˆ(n−1) , E dif ∼ p m, ρ|Jˆ(n−1) , χ ˆ (n) , zˆ(n) ) ∼ p χ, z|Jˆ(n−1) , m ˆ (n) , ρˆ(n) , E dif sample (χ ˆ (n) , zˆ(n) , m ˆ (n) , ρˆ(n) , E dif sample Jˆ(n) ∼ p J|χ ˆ (n) , ρˆ(n) sample m
Because we chose conjugate priors for the hyperparameters (m, ρ), the sampling step of p m, ρ|J, χ, z, E dif is easy an dcomes down to sample ρ from an Inverse Gamma law and the means mn from a Gaussian distribution. Applying the product rule we have: p(χ, z|J, m, ρ, E dif ) = p(χ|z, J, m, ρ, E dif ) p(z|J, m, ρ, E dif ) = p(χ|z, J, m, ρ) p(z|J, m, ρ)
(19)
ˆ (n) , zˆ(n) ) is obtained by first sample zˆ(n) from p(z|J, m, ρ) and after sample χ ˆ (n) from Thus, a joint sample (χ p(χ|z, J, m, ρ). Using the Bayes formula we have: p(z|J, m, ρ) ∝ p(J|z, m, ρ) p(z), OIPE 2006 ”The 9th Workshop on Optimization and Inverse Problems in Electromagnetics” - September 13th – 15th, Sorrento (Italy)
(20)
where p(z) is the Potts prior distribution of z, and the expression of p(J|z, m, ρ can be obtained by integrating p(J, χ|z, m, ρ) with respect to χ. The result of this integration gives that p(J|z, m, ρ) is a separable function of the pixels with respect to z. therefore the conditional distribution p(z|J, m, ρ) is a Markov random field with a neighboring system of four pixels, as its prios distribution. The sampling of this kind of distribution is easy [1]. Concerning the sampling of χ, the conditional distribution p(χ|z, J, m, ρ) is directly obtained from the joint prior distribution p(χ, J|z, m, ρ) and the product rule. Using the Bayes formula wa have: p(J|χ, z, m, ρ, E dif ) ∝ p(E dif |J) p(J|χ, z, m, ρ) X 1 1 (21) 2 Ω 2 dif S inc ∝ exp − kEu,q,v − G0 q,v Ju, q,v k + kJu,q,v − χq Euq,v − χq Gq Ju,q,v k 2ρb 2ρη u,q,v This conditional distribution is in fact Gaussian with a non diagonal covariance matrix. It is possible to obtain a sample of this kind of distribution by a maximisation technique.
III. Results The proposed method is applied to experimental data collected at the Institut Fresnel (Marseille, France). Four targets have been considered, named here DielExt, DielInt, TwinDiel and MetExt; they are composed of different dielectric and metallic cylinders. We refer the reader to [3] for a description of their basic features. The data have been measured both in TM and TE configurations, at several frequencies lying in between 2 and 17 GHz, and for various illuminations all around the object. The proposed method, which introduces the a priori knowledge of the number of materials, has already been applied to the TM configuration and has provided good results which have been the subject of paper [1]. DielExt
DielInt 2
2
1.5
χv
1
1
0.5
0
0
2
TwinDiel 2
2
1.5
1
2
2
1.5
1
1
1
0.5
0.5
0
0
2
2
2
1.5
1.5
0
2
0
2
1.5
ˆG χ 1
1
1
1
0.5
0.5
0.5
0
0
2
2
2
1.5
1.5
0
ˆ CSI χ
1
1
1
0
2
1
0
0
2
2
1.5
1
1
1
1
0.5
0.5
0.5
0
0
0
0
0
0
Figure 2: The true contrast (up), the estimated contraste by the proposed method (middle) and by the classical CSI method (down), for targets DielExt, DielInt and TwinDiel. The test domain is a 17.85 cm sided square divided into 51 × 51 pixels with sides 3.5 mm. Figures 2 and 3 display the results of reconstruction of the contrast by the proposed algorithm and the classical Contrast Source Inversion (CSI) method, for the four targets. We considered four frequency (3, 5, 7 and 9 GHz), 61 measuring points (4 degrees spaced) and 8 views arround the objects DielExt and DielInt and 9 views arround the objects TwinDiel and MetExt. As a general rule our method provides good results of shape reconstruction and localisation and the estimated contrast is composed of homogeneous regions. The contribution of the prior information allow us to obtain largely better results than the classical method of CSI. However, the results are not satisfactory enough for the target TwinDiel which presents a complexe geometry.
OIPE 2006 ”The 9th Workshop on Optimization and Inverse Problems in Electromagnetics” - September 13th – 15th, Sorrento (Italy)
real part
imaginary part 00 11 1.8 11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 1 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 0 0 1 0 1 1.8 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 1.8 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1.8
1.6
1.4
1.2
χv
1
0.8
0.6
0.4
0.2
0
1.8
1.6
1.4
1.2
ˆG χ
1
0.8
0.6
0.4
0.2
0
1.8
1.6
1.4
1.2
ˆ CSI χ
1
0.8
0.6
0.4
0.2
0
1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1
0.25
0.2
0.15
0.1
0.05
1 0 0 1 0 1 0.25 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
0.25
0.2
0.15
0.1
0.05
1 0 0 1 0 1 0.25 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1
0.25
0.2
0.15
0.1
0.05
Figure 3: The true contrast (up), the estimated contraste by the proposed method (middle) and by the classical CSI method (down), for target MetExt. The test domain is a 17.85 cm sided square divided into 51 × 51 pixels with sides 3.5 mm.
Conclusion In this paper we presented a new approach of inversion in a TE configuration. We proposed a solution to introduce a particular prior information: the object is composed of a finite known number of different materials distributed in compact regions. This information first allows us to determine a bilinearized direct model with respect to the induced currents and the contrast function. An appropriate Gibbs sampling algorithm has been implemented, taking into account the property of bilinearity and the prior information. The obtained results of this approach show the effectiveness of the proposed method. However, we can expect for better results if we consider the exact direct model and then the total fields and the contraste as unknowns.
References [1] O. F´eron, B. Duchˆene, A. Mohammad-Djafari, ”Microwave imaging of inhomogeneous objects made of a finite number of dielectric and conductive materials from experimental data”, Inverse Problems, vol. 21, n◦ 6, Dec. 2005, pp. S95–S115. [2] K. Belkebir and M. Saillard, ”Special section on testing inversion algorithms against experimental data: inhomogeneous targets. Guest Editors’ introduction”, Inverse Problems, vol. 21, n◦ 6, Dec. 2005, pp. S1–S4. [3] J.-M. Geffrin, P. Sabouroux and C. Eyraud, ”Free space experimental scattering database continuation: experimental set-up and measurement precision”, Inverse Problems, vol. 21, n◦ 6, Dec. 2005, pp. S117–S130. [4] N. Joachimowicz and C. Pichot, ”Comparison of three integral formulations for the 2-D TE scattering problem”, IEEE Trans. Microwave Theory Tech., vol. MTT-38, n◦ 2, Feb. 1990, pp. 178–185. [5] D. E. Livesay and K. M. Chen, ”Electromagnetic fields induced inside arbitrarily shaped biological bodies”, IEEE Trans. Microwave Theory Tech., vol. MTT-22, n◦ 12, Dec. 1974, pp. 1273–1280. [6] S. C. Hill, C. H. Durney and D. A. Christensen, ”Numerical solutions of low-frequency TE fields in arbitrarily shaped inhomogeneous lossy dielectric cylinders”, Radio Sci., vol. 18, n◦ 3, May - June 1983, pp. 328–336. [7] J. H. Richmond, ”TE-wave scattering by a dielectric cylinder of arbitrary cross-section shape”, IEEE Trans. Antennas Propagat., vol. AP-14, n◦ 4, July 1966, pp. 460–464.
OIPE 2006 ”The 9th Workshop on Optimization and Inverse Problems in Electromagnetics” - September 13th – 15th, Sorrento (Italy)
