A New Integral Equation Method to Solve Highly ... - IEEE Xplore

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL 64, NO. 5, MAY 2016

A New Integral Equation Method to Solve Highly Nonlinear Inverse Scattering Problems Yu Zhong, Marc Lambert, Dominique Lesselier, Senior Member, IEEE, and Xudong Chen, Senior Member, IEEE Abstract—A family of new integral equations (NIE) is proposed in this paper, which are transformed from the original Lippmann– Schwinger integral equation. It can be shown that the NIE can effectively reduce the nonlinearity of inverse scattering problems by reducing the global nonlinear effects, introduced by the multiple scattering behaviors, in estimating the contrast. Equipping the previously proposed twofold subspace-based optimization method with such NIE, the new inversion method is able to solve inverse scattering problems with strong scatterers, like with high contrast and/or large dimensions (in terms of wavelength) ones. Furthermore, such a family of NIE could provide a convenient tool to appraise reconstructed results. Several representative numerical tests are carried out, using both synthetic and experimental data, to verify the efficacy of the new inversion method. Index Terms—Inverse scattering problems, new integral equations (NIE), strong nonlinearity.

I. I NTRODUCTION

I

NVERSE medium scattering problems have been of interest for decades in applied mathematics, engineering, and physics, due to vast potential applications in many fields, such as geophysical explorations, medical imaging, nondestructive testing, etc. To solve inverse scattering problems, an objective function is usually constructed to appraise mismatches between the calculated physical model and the measured one, the minimization of which results in an optimal distribution of scatterers that fits the physical model and the measured data. As analyzed in the literature, difficulties in practically solving inverse medium scattering problems lie in handling two intrinsic properties: ill-posedness and nonlinearity [1], [2]. For ill-posedness, regularization techniques [3] have been proposed and developed to build a sufficiently stable solver for problems that are not severely nonlinear, like the well-known Tikhonov regularization methods [4], Newton-type methods [5], total Manuscript received September 26, 2014; revised November 05, 2015; accepted February 15, 2016. Date of publication February 29, 2016; date of current version May 03, 2016. Y. Zhong was with National University of Singapore, 119077 Singapore. He is now with the Institute of High Performance Computing, A*STAR, 138632 Singapore (e-mail: [email protected]). M. Lambert is with Group of Electrical Engineering - Paris, UMR 8507, CentraleSupélec, Univ. Paris-Sud, Université Paris Saclay, Sorbonne Université, UPMC Univ. Paris 06, 91192 Gif-sur-Yvette CEDEX, France (e-mail: [email protected]). D. Lesselier is with Laboratoire des Signaux et Systèmes, UMR8506, CNRS-CentraleSupélec-Univ. Paris-Sud, Université Paris Saclay, 91192 Gif-sur-Yvette, France (e-mail: [email protected]). X. Chen is with the Department of Electrical and Computer Engineering, National University of Singapore, 117583 Singapore (e-mail: elechenx@nus. edu.sg). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2016.2535492

variation methods (the multiplicative regularization method) [6], and a recently proposed twofold subspace-based optimization method (TSOM) [7] and its Fourier bases version (FFT-TSOM) [8]. All these regularization techniques modify objective functions so as to regularize them in the sense that outputs of the optimization, scatterer distributions, are stabilized in terms of input measured scattered fields. On the other hand, to solve aforementioned minimization problems, two types of optimization methods are usually employed, stochastic, and deterministic ones. When dealing with nonlinear problems, stochastic optimization methods have better chance of finding the wanted global solution. Several types of stochastic methods in solving inverse scattering problems have been tried out. However, since in such type of problems the number of unknowns is not small, especially with three-dimensional (3-D) problems, this requires quite a large computational cost, including CPU time and memory [9]. Outputs of the deterministic type of optimization, though efficient to find a local minimum, are quite sensitive to initial guesses. By far, many inversion methods using deterministic optimization schemes have been proposed, like distorted Born iterative method (DBIM) [4], modified gradient method [10], contrast source inversion (CSI) method [11], [12], Newtontype method [5], [13], inexact Newton-based method [14], and recently proposed subspace-based optimization method (SOM) [15], [16] and its twofold versions [7], [8]. All such methods strongly depend on initial guesses. So, they usually can handle problems without strong nonlinearity [17]. With the deterministic type of optimizations, inversion methods can be further grouped into two types, i.e., the Newton-type methods and the CSI-type methods. By Newton-type methods, we mean conventional inversion methods with contrasts as the only unknowns in the objective function, which are usually minimized using the Newton-type nonlinear optimization methods. On the other hand, developed from the modified gradient method [10], the CSI-type methods consider both contrasts and physical quantities, either electric fields or induced currents (or contrast sources), as unknowns and update them alternatively during the optimization. Compared to the Newton-type methods, the CSI-type methods avoid to solve forward problems in each iteration of updating unknowns, though they usually need a larger number of iterations. By considering physical quantities as unknowns, the CSI-type methods provide additional margin in modeling wave behaviors to tackle major difficulties in inversions, such as in [8] the FFT-TSOM low-dimensional subspace constraints onto induced currents for better stability of inversions. And as shown in [18], such an effective regularization method

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ZHONG et al.: NIE METHOD TO SOLVE HIGHLY NONLINEAR INVERSE SCATTERING PROBLEMS

provides different regularization effects compared to those based on the mathematical constraints, like total variation ones and Tikhonov ones, meaning that they can work together to provide more robust inversions. Such new properties allow the CSI-type inversion methods to tackle more difficult inverse problems (say, more nonlinear and/or with noisier measured data) compared to the Newton-type methods. With the Lippmann–Schwinger integral equation (LS-IE) being used in aforementioned methods that are based on IE modeling, as mentioned in [17], the nonlinearity of inverse scattering problems comes from the structure of LS-IE. To be more specific, when using the CSI-type methods, the nonlinearity mainly comes from the unknown multiple scattering effects (the global wave behaviors within the domain of interest). This means that with large permittivity and/or large dimensions, the nonlinearity of inverse scattering problems could increase significantly due to the large unknown portion of the multiple scattering effects. To reduce the nonlinearity, in [19]–[21] and references therein, another type of IE is proposed, the contrast sourceextended Born (CS-EB) type. Numerical tests show that the CS-EB type equation succeeds to some extent. Also, in [22], the LS-IE is rewritten by extracting the singularity of the Green’s function, and, for the DBIM, the nonlinearity of inverse scattering problems can be shown reduced. Besides, other pioneering works in rewriting the LS-IE exist with different purposes, such as modified Born series for solving forward problems with large contrast [23], for tackling remote sensing problems [24], etc. Motivated by the contraction integral equation (CIE) [25]– [27] in tackling forward problems with strong conductivities, in this paper we propose a family of new integral equations (NIE), which are transformed from and equivalent to LS-IE. The aforementioned CS-EB type equation can be shown as a specific case of the family. In NIE, by introducing a new local term, global multiple scattering contributions in estimating the contrast in CSI-type methods can be effectively alleviated via a properly chosen parameter yet without compromising the accuracy of the physical model. To properly regularize the inversion solver, we use the NIE with our previously proposed regularization scheme, the FFT-TSOM [8], and, as expected, the resultant new CSItype inversion method is able to handle problems that are severely nonlinear for CSI-type inversion methods using the ordinary LS-IE and Newton-type conventional methods. In brief, new properties of the proposed inversion method include the following. 1) No special initial guess is needed, and only the background medium is used as initial guess. 2) Strong and/or large scatterers (the latter in terms of wavelength) can be effectively handled. 3) Moderate scatterers can be efficiently handled. 4) A convenient way to appraise the reconstruction results by using different NIE is provided. The paper is organized as follows. In Section II, the NIE is introduced. In Section III, inversion methods using the LS-IE and the NIE are thoroughly discussed together with the nonlinearity of the concerned problems, and reasons to choose the

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NIE are given. Several benchmark tests, against both synthetic and experimental data, in Section IV verify the interests. A brief conclusion follows in Section V. In this paper, we discuss two-dimensional (2-D) scalar problems only, and time-harmonic fields are assumed with exp(−iωt) suppressed. II. A N EW T YPE OF I NTEGRAL E QUATION Let us assume a 2-D domain D with a background medium with permittivity 0 and permeability μ0 . Some unknown scatterers with different permittivity r (r)0 (r ∈ D) from the background medium might appear in the domain. As mentioned in the Introduction, the LS-IE that describes the scattering behavior is used to build up the objective function in solving inverse medium scattering problems, such as for total electric fields E(r) in the domain of calculation inc G(r, r )χ(r )E(r )dr (1) E(r) = E (r) + D

or its variant for the induced current I(r) inc G(r, r )I(r )dr I(r) = χ(r)E (r) + χ(r)

(2)

D

where the contrast function χ(r) = r (r) − 1, incident field E inc (r), and Green’s function of the background medium G(r, r ) are present. E sct (r) = D G(r, r )I(r )dr is the 1 is actually I(r) = scattered field, so the induced current inc sct χ(r)E(r) = χ(r) E (r) + E (r) . Either one of the two equations is directly used in most of inversion methods, such as (1) in DBIM and (2) in the CSI and the SOM where contrast source (the induced current) is considered as unknown together with the contrast. In [25]–[27], another type of IE, the CIE, is proposed to solve the forward scattering problems in the low-frequency band with strong conductivity in the background medium. We adopt this methodology, and by changing the coefficient we propose a new type IE as follows. By multiplying both sides of (2) by a function −1 β(r) [β(r)χ(r) + 1] , and after straightforward algebraic manipulations β(r)I(r) = R(r)β(r)I(r) inc G(r, r )I(r )dr + R(r) E (r) +

(3)

D

−1

where R(r) = β(r)χ(r) [β(r)χ(r) + 1] is the modified contrast function. Here, we mention them as a family of NIE, since different functions β leads to different IE. Please note that the only condition for β is that βχ + 1 = 0 and β is not the wavenumber as in most engineering literature. Though they describe the same wave behavior (no approximation in (3)), the differences between (2) and (3) are mainly in two aspects. The first is on the new term R(r)β(r)I(r), 1 This is the so-called contrast source, and the difference between the physical induced current and the contrast source is a constant coefficient −iω0 absorbed by the Green’s function here.

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which actually is a local effect of the induced current instead of a global one. The second difference is the modified contrast function R(r), the norm of which is always about or less than 1 provided that β(r)χ(r) has a positive real part. Since we aim at the high-contrast scatterers whereas χ(r) has a positive real part and a nonnegative imaginary part (lossless or lossy medium in the assumed time convention), such a condition is easy to fulfill with a β having a positive real part and a nonpositive imaginary part. By properly choosing β, the local wave effect could dominate the global one in (3), which will be analyzed in detail in next section. Compared with the CS-EB type equation proposed in [19]– [21], one sees that the CS-EB type equation is a specific case of (3) by letting β(r) = −fΩ (r) with fΩ defined therein. III. I NVERSION M ETHOD To reconstruct the unknown scatterer distributions in the domain D when incidences, background medium, and measured scattered fields are known, differences between the two inversions with LS-IE and NIE are discussed in this section. A. State of the Problem All equations in Section II need to be discretized first before any numerical calculation, like using the method of moments (MoM) as in [8]. Similarly, we choose a rectangular domain in this paper, and the domain of interest D is discretized into M small cells centered at rm,n , within each one of which the induced current is assumed to be constant, with M1 and M2 being the number of cells along the x and y direction (so M = M1 × M2 and 1 ≤ m ≤ M1 and 1 ≤ n ≤ M2 ). Assume that there are Ni incidences onto the domain. From (2), the discretized form of LS-IE can be written as follows, for the lth incidence ¯ inc + A¯ ¯ ¯m,n E I¯l;m,n = χ (4) m,n Il l;m,n where the double bar ¯· denotes a 2-D tensor, subscript m, n denotes the indexes of the element in the tensor, 1 ≤ m ≤ M1 and 1 ≤ n ≤ M2 , and A¯m,n I¯l = M1 M2 ¯ m =1 n =1 GD (rm,n , rm ,n )Il;m ,n is the Green’s function operator mapping the induced current to the scattered fields in the domain of calculation. Similarly, the NIE in (3) can be discretized as ¯ ¯ ¯ inc + A¯ ¯ ¯ β¯m,n I¯l;m,n = R m,n βm,n Il;m,n + E m,n Il l;m,n

(5) −1 ¯ ¯ ¯ ¯m,n βm,n χ ¯m,n + 1 . where Rm,n = βm,n χ It is well-known that the induced current can be divided into two parts as a radiating part and a non-radiating part, as analyzed in [28] and [29]. Such a partition of the induced current is mainly due to the existence of a null subspace of the scattering operator, i.e., the subspace with null singular values in which the current cannot generate any scattered fields that can be observed in the far-field measurement domain. In [7] and [15], this partition is further stretched to cope with the white

noise and measurement setup, by separating the current subspace with large singular values of the scattering operator from the subspace with small and null singular values. By doing so, the current portion in the former current subspace is not severely contaminated by the noise provided that the noise is a white one, and, therefore, is well perceptible in the far-field measurement domain. Such a current portion is defined as the deterministic part of the induced current (DPIC). Consequently, the remaining portion of the induced current is within the subspace with small singular values and null singular values, and it is imperceptible in the far-field measurement domain due to the noise effect and the compactness of the scattering operator. Such a current portion is defined as an ambiguous part of the induced current (APIC) in [7] and [15]. Though terminologies used for the two current portions may not fully reflect their natures, we still follow the convention in our previous papers and use them here. Assuming that there are Nr measurements for each incidence and starting from the so-called data equation Elsct (r) = G(r, r )Il (r )dr with r in the far-field measurement D ¯ · I¯ , ¯ sct = G domain, with proper discretization, we have E S l l ¯ is the scattering operator that where the Nr × M matrix G S maps the induced current to measured scattered fields,

the and M × 1 vector I¯l = vec I¯l is the induced current vector with vec {} the vectorization operator. Given the singular ¯ = value (SVD) of the scattering operator, G S decomposition ∗ ¯j v¯j with σj the singular values, u ¯j and v¯j the left j σj u and right singular vectors, and the superscript ∗ the complex conjugate transpose, we have the DPIC as L ∗ ¯ sct u ¯ j · El d ¯ Il = v¯j (6) σj j=1 where the singular value series σj is arranged as a nonincreasing one, and parameter L is chosen according to the noise level ¯ and the measured [15]. So, once the scattering operator G S scattered fields are given, the DPIC is known. According to the FFT-TSOM, the APIC can be represented using the remaining right singular vectors as

(7) I¯la (γ¯l ) = I¯M − V¯S+ V¯S+∗ · vec {IDFT {γ¯l }} where V¯S+ = [¯ v1 , . . . , v¯L ], I¯M is the M -dimensional identity matrix, and IDFT {} is the inverse discrete Fourier transform operator. The coefficient tensor γ¯l is only with particular nonnull elements for low-frequency Fourier bases. Constructing the APIC in this way, we actually constrain the APIC within a proper low-dimensional current subspace that has strong influence on scattered fields within the domain of interest but has minor effects on scattered fields in the measurement domain. Such a constraint provides quite effective regularization to the inversion, since only the most important portion of wave behaviors is used [8]. ¯ While only the thin-type SVD of the scattering operator G S is sufficient to obtain the DPIC and the APIC (since we only need the first L singular vectors), the calculation of (7) can be efficiently carried out as in [8] [the (6) therein].


B. Nonlinearity In [17], [19], and [21], it is proposed that the degree of the nonlinearity of inverse scattering problems can be measured by the norm of the multiplication of the contrast operator and the scattering operator that maps induced currents to scattered fields inside the domain of interest. Here, we wish to enlarge the view a bit more on this aspect. As mentioned in the Introduction section, it seems to us that the nonlinearity of the problem comes from the unknown portion of the multiple scattering effects, and this can be shown as follows. From (4), after separating the current into two portions as mentioned above, the LS-IE becomes

d a ¯ inc + A¯ ¯d + A¯ ¯m,n E I I¯la I¯l;m,n + I¯l;m,n =χ m,n m,n l;m,n l (8) with I¯la = ten I¯la with ten {} the inverse operation of vec {}, and I¯ld = ten I¯ld . Since I¯ld , l = 1, . . . , Ni are known already, ¯ and the the unknowns to be reconstructed are the contrast χ APIC I¯la , l = 1, . . . , Ni . In (8), we see that the only term that includes the two types

¯m,n A¯m,n I¯la , of unknowns, the contrast and the APIC, is χ where they are multiplied with each other through the global ¯ When simultaneously reconstructing both types operator A. of unknowns, as in the CSI-type inversion methods, this term introduces the nonlinearity of the optimization. On the other hand, if we separate the induced current into DPIC and APIC in (5), we arrive at d a ¯ ¯ ¯d β¯m,n I¯l;m,n + β¯m,n I¯l;m,n =R m,n βm,n Il;m,n

a ¯ inc + A¯ ¯d + A¯ ¯a . (9) +β¯m,n I¯l;m,n +E m,n Il m,n Il l;m,n Compared to (8), two more terms appear on the right hand side, ¯ ¯ ¯ ¯ ¯d ¯a R m,n βm,n Il;m,n and Rm,n βm,n Il;m,n . Though the latter term includes both types of unknowns, it introduces a local nonlinearity, instead of a global one as mentioned previously that ¯ affects through the Green’s operator A. Besides, we actually can further see that the known information from the measured scattered fields, i.e., the DPIC, and the incident fields plays an important role here in either type of equations. Generally speaking, the more information known, the easier to reconstruct the unknowns. In this section, compared to the analysis given in [17], we see that instead of looking into the norm of the contrast operator and the scattering operator, we identify the sources of the nonlinearities of reconstructions, including the global one and the local one, and we also differentiate the known information from the unknowns. We will see how the LS-IE and NIE handle such two types of nonlinearities in the following sections.

¯ = f (γ¯1 , . . . , γ¯Ni , χ)

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Ni l=1

a sct 2 ¯ inc 2 cur ¯a ¯ ¯ ¯ ¯ ¯ E I I ( γ ) / + Δ ( γ ), χ / E Δﬁe l l l l l l l l (10) a 2 ¯ sct ¯ ¯a ¯ where Δﬁe I¯l (γ¯l ) = E − GS · Il (γl ) + I¯ld as the l l data-equation mismatch,2 and ¯ Δcur I¯la (γ¯l ), χ l 2

¯ inc ¯ ⊗ A¯ I¯la (γ¯l ) + I¯ld − χ ¯⊗E = I¯la (γ¯l ) + I¯ld − χ l (11) as the current-equation mismatch, where ⊗ is the element wise multiplication as shown in (4). The inversion is to minimize the objective function (10) by alternatively updating the current ¯ as shown in [8] and [15]. Since coefficients γ¯l and the contrast χ we do not have any prior information about the unknowns, the initial guesses of both are zeros. Here, we might think of using some techniques to have a possibly better initial guess, such as the back-propagation method. According to our experience, the back-propagation method may work when handling only weak scatterers, and it appears to produce useless or even misleading initial guesses when handling strong scatterers. The above inversion method works pretty well when the unknown scatterers are not so strong. For instance, at 400-MHz frequency, given the initial guesses as the background air and null APIC, the standard Austria profile (the detail of which will be introduced in Section IV) with relative permittivity r = 2 can be effectively reconstructed as shown in [8]. However, upon increasing the relative permittivity to r = 2.2 (not to mention a higher one), the profile cannot be reconstructed at 400 MHz using the above objective function given such initial guesses. This means that, even using the FFT-TSOM technique as of (7) to constrain the APIC, we are not able to tackle problems with strong scatterers. This is due to the fact that the nonlinearity of the problem strongly interferes with the optimization and the initial guess of the profile (and the null APIC) is not in the proximity of the global minimum but of a certain local minimum where the optimization is trapped. This actually can be seen from (8). In this equation, the known quantities are the incident fields and the DPIC. In the CSI-type methods, like CSI [11], and SOM [15], when updating the contrast at the pth iteration, the least squares solution of (8) is used, i.e.

∗ ¯ inc a(p) (p) I¯l;m,n El;m,n + A¯m,n I¯ld + A¯m,n I¯l l (p) ¯m,n = χ

2 ¯ inc ¯ ¯d + A¯ ¯a(p) E A I I + m,n m,n l l;m,n l l (12)

C. Traditional Inversion With LS-IE

(p) a(p) a(p) the APIC at the pth iteration. with I¯l = I¯ld + I¯l , and I¯l

Given the two portions of the induced current, the inversion can be carried out as in [8] with an objective function constructed from the LS-IE

2 Note also that the inclusion of the ambiguous part of induced currents I a into the data-equation mismatch is to further regulate such portion of currents. Otherwise, if one knows that the noise level is very high in the measured data, such inclusion might introduce instability and, therefore, could be removed.

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1) Effects of Nonlinearity: Due to the CSI-type optimization scheme used here, the contrast is updated using the least squares solution whereas the induced current is updated using the Polak–Ribière conjugate gradient method, meaning that the contrast is much more thoroughly optimized compared to the induced current in each iteration. Also, when updating the contrast, the information from different incidences is directly glued together. Therefore, the update of the contrast might tell the key cause of the nonlinearity, which can be seen from the balance of the three terms in the brackets in (12). We see that for weak or not-too-strong scatterers, i.e., when a(p) the contribution from the APIC, A¯m,n (I¯l ), is small compared to the sum of the other two terms in the square brackets, ¯ inc + A¯ ¯d E m,n Il , the estimation of the contrast will be l;m,n accurate enough to lead the inversion to the global solution. Such a situation includes the Born-approximation case, where the total induced current is mainly excited by the incident fields. However, when scatterers become stronger, the APIC becomes not anymore a small portion compared to the DPIC, and it could influence the whole domain via the global Green’s ¯ At the beginning of the inversion, since we do operator A. not have any information about the APIC, the estimation of contrasts becomes less accurate and farther from the global minimum. When the scatterers are strong enough,3 the obtained contrasts can find themselves associated to a local minimum from which the deterministic optimization scheme does not enable to escape. The aforementioned example, the Austria profile with r = 2 at 400-MHz frequency, is a very typical scatterer profile, which represents the limit of the reconstruction ability of most inversion methods with deterministic optimization schemes, such as the DBIM and the FFT-TSOM using the LS-IE. Since such a scatterer profile is neither too strong nor too weak, we refer to it as a moderate scatterer. Those with larger dimensions (in terms of wavelength) and/or higher relative permittivity than this moderate scatterer are referred to as strong scatterers, and those with smaller dimensions and/or lower relative permittivity than this as weak scatterers. Note that such Austria profile is not only with these properties (dimensions and relative permittivity) but also with a complicated topological distribution, made of three parts: two disks and an annulus. D. Inversion With NIE In this section, we first show how the NIE (5) can reduce the nonlinearity of inverse scattering problems, and then we propose the new inversion method using the NIE. Using the NIE, the objective function shares the same form as (10) but with a different current-equation mismatch as ¯ = I¯la (γ¯l ), R Δcur β¯ ⊗ I¯la (γ¯l ) + I¯ld l

¯ ⊗ β¯ ⊗ I¯a (γ¯ ) + I¯d −R l

l

l

2 ¯ inc +A¯ I¯la (γ¯l ) + I¯ld + E . l

(13)

3 Here by strong scatterers, we mean those generating strong APIC at least in part of the domain, which could affect the inversion in the whole domain via ¯ the global operator A.

The update of the induced current using the Polak-Ribière conjugate gradient method [12] needs to be revised accordingly also, and the detail is omitted here for conciseness. The ¯ is updated, at the pth iteration, as modified contrast function R ¯ (p) = R m,n

l

a(p) d Γ∗l;m,n β¯m,n I¯l;m,n + β¯m,n I¯l;m,n 2 l |Γl;m,n |

(14)

¯ ¯ inc + A¯ ¯d ¯a(p) + β¯ I¯d with Γl;m,n = E m,n Il + Am,n Il m,n l;m,n l;m,n a(p) ¯ ¯ +β I , which, similarly as in the traditional inversion, m,n l;m,n

provides the dominant effect in inversions. 1) Handling of the Nonlinearity: From (9), the contributions of the induced current (either the DPIC or the APIC) to the “fields” (Γl;m,n ) consists of two parts: the global part through the Green’s function operator, and the new local part amplified by the factor β¯m,n . This also presents two types of nonlinearities, as mentioned in the previous section, the global type and the local type. Consequently, if β¯m,n is chosen such that the local effect becomes

dominant compared to the global one, i.e., a(p) A¯m,n I¯l is a small term and does not change much of the value of Γl;m,n , it is possible that the reconstructed modified ¯ contrast function R m,n at the beginning of the inversion could be a good estimation, which could lead the optimization to the global solution, even when the scatterers are strong. This can be seen from the analyses next. Since the global nonlinearity caused by the APIC is suppressed, the missing factor that counts here is the APIC’s local a(p) nonlinearity, i.e., β¯m,n I¯l;m,n in Γl;m,n . For strong scatterers, we know that the DPIC is a small portion compared to the APIC (in the sense of the energy of each portion in the whole domain of interest). However, this happens in the cells in the scatterers. The local APIC in cells outside scatterers (cells with no scatterer) are just the additive inverse of the DPIC (due to the null total induced current there), its effect on the estimation is limited. From this point, we can tell that these cells that are outside scatterers can be reconstructed in a satisfactory way at the beginning of the inversion. This effectively facilitates the inversion by focusing the inversion solver onto the subdomain where scatterers are actually distributed. For cells on the scatterers, since cells outside scatterers are well identified such that the affect from them could be minimized, the difficulties of estimating the contrast on the scatterers are mainly the local nonlinearity of the missing APIC and strong near-field effects from the nearby cells. From this point of view, when using the NIE, even when scatterers are strong, we still have better chance to identify these cells as scatterers though the estimation might not be accurate enough. And more importantly, from aforementioned analyses, such inaccurate estimation of the contrast is a localized effect. This might mean that the NIE could facilitate the deterministic optimization by properly utilizing the known information comparing to the case with the LS-IE. In brief, global multiple scattering contributions in estimating the contrast in CSI-type methods can be effectively


Fig. 1. Real part and the imaginary part of β¯ in CS-EB equation with optimal radius chosen as indicated in [21]. The x and y-axes are in meters.

Fig. 2. Real part, the imaginary part, and the absolute value of for a rectangular domain (2.67λ × 2.67λ).

D

G(r, r )dr

alleviated by using the NIE and therefore the corresponding global nonlinearity is suppressed. 2) Choice of β : Choosing a proper β¯m,n allows the local effect of the current to overcome the global one, meaning the value of β¯m,n cannot be too small. On the other hand, it cannot be too large either, since a very large β¯m,n could significantly suppress the global effect and imperil the reverse mapping from ¯ ¯m,n . modified contrast function R m,n to physical contrast χ Considering that all small cells in the domain of interest are equivalent in terms of the possibility of being a scatterer or not, we choose the values of β¯ in all such small cells to be the same. For comparison, we plot the values of β(r) = −fΩ (r) in (1) the CS-EB method with fΩ (r) = i πk20 R H1 (k0 R)J0 (k0 r) − 1 with optimal R chosen, as shown in Fig. 1, which exhibits a strong inhomogeneity in the whole domain (2.67 λ × 2.67 λ with λ = 0.75 m the wavelength in the background medium). ¯ as analyzed above, the global effects are supUsing such a β, pressed with different “amount” on different cells, and the cells with less suppression are less likely to be identified. Such an effect could further complicate the problem by introducing additional nonlinearity. Taking β¯ on all the small cells at the same value might not be optimal but is a reasonable option. With a constant β = β0 , one needs to provide a proper value of β0 . As mentioned above, the β is used to reduce global non¯ I) ¯ (or G(r, r )I(r )dr ), so we plot the linear effects by A( D values of this operator with a unit input, as in Fig. 2, for the same rectangular domain used above. From such results, one finds that the largest values are around 3. One also has carried out tests with different domains, larger and smaller, and one observes the same values. Therefore, while choosing the optimal β¯ remains an open problem, in the inversion, we use a unique value for all β¯m,n , and as mentioned above, in order to compensate the global nonlinear effects, we initially choose β0 > 3, and from the analysis given above, β0 cannot be too large as well. From comprehensive numerical tests, a value less than 10 appears to be reasonable. 3) Regularization: Due to the amplification of the local effect and the missing APIC, via (14) the contrast could be unstably estimated, i.e., the estimated contrast values could vary within a large interval. To avoid such an instability, an

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appropriate solution is to constrain the induced current in a stable current subspace, such that the resultant estimated contrast will be a stable one. This brings the need of a good regularization on the APIC, such as the FFT-TSOM [8]. In this paper, we only adopt the FFT-TSOM type regularization technique, but this is not a limitation. As shown in [18], if the multiplicative total variation technique is used together with FFT-TSOM, we believe that even better stability could be achieved. 4) Inversion Algorithm With NIE: With background medium and the zero APIC as the initial guess, we start the inversion with a large β0 and small MF (a parameter in FFT-TSOM to control the number of Fourier bases being used to represent the APIC [8]), meaning that the optimization is carried out in a small current subspace for the APIC and with a limited global nonlinearity. The optimization is terminated when the change of the APIC is smaller than a predefined threshold δ 2D [8], letting Ni 2 1 ¯ ¯ γ − γ l;p l;p−1 δ 2D = (15) 2 Ni γ¯l;p−1 l=1 for the pth iteration. When having some rough yet good estimation of the scatterers (and the APIC) as the initial guess of the optimization in the next round, we decrease the β0 (it can be smaller than 3) and increases the MF , such that the inversion could have a larger current subspace and more global interactions for the induced current. Doing so, finer structures of the scatterers are to be reconstructed. If two rounds of optimization are inadequate, we could carry out the third or even the fourth rounds of optimization. In all, the most important step would be the first round of the optimization, which gives a rough estimation of the true scatterers. 5) Limit of the Method: Before closing this section, we discuss the limit of the proposed inversion method. As seen above, the stronger the scatterers, the higher the parameter β0 needed, and therefore the stronger regularization on the APIC. However, on the other hand, the stronger the scatterers, the more complicated the induced current. To approximate such a complicated induced current, a low-dimension subspace may not be sufficient. This means that there is a conflict between the demands on the regularization and the approximation of a complicated induced current, which indicates the limit of the proposed inversion method. Such a limitation might be overcome by using additional regularization techniques, such as the scheme proposed in [18]. Note that, if the scatterer is not strong all over the whole domain but only a part, i.e., the approximation of APIC is inaccurate in part of the domain, the missing information (in higher-dimension or higher-frequency subspace) represents the fine features of the scatterer in that part. Consequently, the proposed inversion method might still find a solution that is close to the true one, without the fine features in that part, instead of jeopardizing the reconstruction in the whole domain. As analyzed previously, this is due to the localized nonlinearity effects (not propagating the error to the whole domain). We will propose such an example in the numerical section.

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IV. N UMERICAL T ESTS In this section, we first employ synthetic data to compare performances of two inversion methods, using the family of NIE, (3), and using the original LS-IE, (2), on the same problems, either with the strong scatterers or with the moderate scatterers. Then, the new inversion method is verified against experimental laboratory-controlled data. The reconstruction qualities are quantified by the mean square errors given below 2 1 ¯est ¯tr r;m,n − r;m,n (16) Ertot = ¯tr M m

n

r;m,n

where the summation of indexes m and n are over the whole domain of interest. To further evaluate the reconstruction qualities on the unknown scatterers, we define another type of error, Ersct , where the summation of indexes m and n are over only the domain where scatterers distribute (and divided by the number of cells in the scatterers). In all tests, the termination conditions of both methods are the same, i.e., δ 2D = 0.01 for the first round of optimization and δ 2D = 0.001 for the subsequent rounds4 In the four tests with synthetic data, the domain of interest is a 2 m × 2 m square centered at the origin. The scatterers are illuminated with 20 plane waves at 400 MHz for the first two tests and at 750 MHz for the third test, incident at different angles evenly distributed in [0, 2π). The scattered waves are collected by an antenna array with 40 antennas uniformly placed on a circle of radius 3.75 m (5λ with λ = 0.75 m, the wavelength at 400 MHz in the background medium air) and centered at the origin. A 64 × 64 grid mesh of the domain of interest is used for the reconstructions. The synthetic scattered fields are calculated by the MoM using a 150 × 150 grid mesh of the domain of interest. As in [8], we choose L = 15, and 10% additive white Gaussian noise is added to the synthetic data unless otherwise stated. In the first comparison, we use the “Austria” profile consisting of an annulus, two disks and the background air. The two disks are of same radius 0.2 m and their centers locate at (−0.3 m, 0.6 m) and (0.3 m, 0.6 m). The annulus is centered at (0 m, −0.2 m) with inner radius of 0.3 m and outer radius of 0.6 m. All three scatterers are of same relative permittivity r = 3.5, as shown in Fig. 3(a). In the first-round optimization, the initial guess of the profile is the background air and the APIC are zeros, and the results of the profile and the APIC obtained after the first-round optimization are used as the initial guess of the second round, and so on. The values of β0 for the three rounds of optimization are 6, 2, and 0.5, coupled with MF being 5, 7, and 10, respectively. 4 The reason of choosing such a setting is to accelerate the optimization and to increase the stability of the first round of optimization, since with lowdimensional current subspace too much optimization might involve unwanted noise effects. This is a compromise due to the FFT-TSOM technique, where the used current subspace is an approximate subspace so as that it can be efficiently obtained, and this leads to some small leakages of current subspace with less ¯ [30]. When the dimension of the current subdominant singular values of A space is not so small in the second round or the third round of optimization, such leakages will not be an important effect.

Fig. 3. “Austria” profile with r = 3.5 [(a) exact profile] and reconstructed results at 400 MHz using (b) FFT-TSOM with the NIE and (c) with the LS-IE. The 1st, 2nd, and 3rd rows correspond to β0 being 6, 2, and 0.5 [for (b) only], coupled with MF being 5, 7, and 10 [for (b) and (c)], respectively.

Results obtained by the proposed method are shown in Fig. 3(b), where results in the same column share the same color bar (this applies to all reconstructed results in this section, unless otherwise stated). From these results we observe that the reconstructions are rather satisfactory. However, if we use the same MF values for the FFT-TSOM method with original LSIE, we obtain results shown in Fig. 3(c), from which we see that the lower part of the profile, the ring, is not properly retrieved. We have also tried other options for the MF here for the traditional inversion but none of them produced a satisfactory reconstruction. Note that in this example, after looking into the values of the induced currents (the contrast sources), we find that I a is much larger than I d in the scatterers, and the absolute values of the former are larger than 4 while the latter are about 0.3. This tells that scatterers concerned here are very strong. To further confirm our reconstructed results, we change the values of β0 to 5, 3, and 1 for the three rounds of optimizations (MF still being 5, 7, and 10), and results are shown in Fig. 4(a), in which we see quite similar reconstruction as given in Fig. 3(b). This means that such a reconstruction can be achieved by using different equations, and it tends to be the true scatterer. Also, with the same FFT-TSOM setup (on L and MF values), we test the CS-EB equation in this case, and the unsatisfactory reconstructed results obtained are given as well in Fig. 4(b). The objective function values through the optimizations for these tests are shown in Fig. 5(a), illustrating


Fig. 4. Reconstructed results at 400 MHz using (a) FFT-TSOM with NIE and (b) CS-EB equation for Austria type scatterers with r = 3.5. The 1st, 2nd, and 3rd rows correspond to MF being 5, 7, and 10 for both tests, and with β0 being 5, 3, and 1 for the results with NIE.

Fig. 5. Objective function values and errors of reconstructions for Austria type scatterers with r = 3.5. (a) Objective function values of different inversions for Austria type scatterers with r = 3.5. (b) Errors for the whole domain. (c) Errors for scatterers’ domain.

that the proposed method provides a better convergence. The quantitative evaluations of above reconstructions can be found in Fig. 5(b) and (c), where we further confirm that the inversions with NIE are much better than the one with LS-IE and the one with CS-EB. To illustrate the effect of the subspace constraint on the APIC when using the FFT-TSOM, we use the SOM, in which the APIC is optimized without any constraint, with the NIE for this example, and the reconstructed results are shown in Fig. 6. In this figure, we plot the result reconstructed after three rounds of optimization, using β0 as 6, 2, and 0.5, respectively. In the first row, the linear plot of the results is given. In the second row, the log 10 scale plot of the absolute value of the reconstructed profile is given, and in the third row, we restrict the color bars

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Fig. 6. Reconstructed results using the SOM coupled with the NIE for Austria type scatterers with r = 3.5.

for the real and the imaginary parts of the reconstructed results between −10 and 10 to depict the linear plot of the results. From these plots, we see that without the proper constraint on the APIC from the FFT-TSOM, the reconstructed profile becomes very unstable and presents a large value in some parts of the scatterer (mainly at the boundary). In the second test, we increase the relative permittivity of the Austria profile from 3.5 to 4. The values of β0 for the three rounds of optimization are 8, 2, and 0.5, coupled with the MF being 5, 8, and 10, respectively. The results obtained by the two different methods are shown in Fig. 7(a) and (b), and the values of the objective functions are in Fig. 7(c). From these results we see that there is a nonexisting scatterer appearing inside the annulus. We have tried other values of β0 and MF , and obtained similar or even worse results. Such results show that we probably have reached the limit of the proposed method. That is, the scatterer is too strong, and part of the induced current cannot be well approximated by such a low-dimension subspace (MF = 5 and 8) since they are in the subspace spanned by high-frequency Fourier bases that are contaminated by noises (since the nonexisting scatterer is within the annulus). But, on the other hand, since we have used a high β0 here (lower ones produce even worse results), a low MF is needed to regularize the solver. Nevertheless, due to the localized global nonlinarities, the optimization converges to a local minimum quite close to the wanted solution (other parts of the scatterers are well reconstructed). The values of objective functions and the errors during the inversions are depicted in Fig. 7(c) and (d), where we see that although the Ertot by the inversion with NIE is quite high, due to the dummy scatterer within the annulus, the Ersct by the inversion with NIE is quite low (0.263). To further verify our aforementioned guess that the current that cannot be properly retrieved is in the current subspace spanned by high-frequency Fourier bases, we carry out a test

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Fig. 7. Reconstructed results at 400 MHz using (a) FFT-TSOM with NIE and (b) with LS-IE for Austria type scatterers with r = 4. The 1st, 2nd, and 3rd row correspond to MF being 5, 8, and 10, and β0 being 8, 2, and 0.5, respectively. In (c) and (d) the corresponding objective function values and reconstruction errors are plotted, respectively.

Fig. 8. Reconstructed results for Austria type scatterers with r = 4 at 400 MHz: (a) are results with noise-free data; (b) shows the comparisons with further optimization with δ 2D = 10−4 for the last round optimizations in 10% noise, 3% noise, and noise-free cases (first, second, and third row, respectively).

on the noise-free synthetic data, and reconstruction results are shown in Fig. 8(a), where we see that they are quite similar to those obtained previously with data with 10% noise.

However, if we further optimize the results (to decrease the δ 2D to 10−4 ), we obtain results shown in Fig. 8(b). From these results, including an inversion with data with 3% noise, we see that the less noise the better the reconstruction (the nonexisting scatterers within the annulus appear weaker, especially the imaginary parts). This also reflects on the reconstruction errors: Ertot = 1.14, 0.82, 0.42 and Ersct = 0.29, 0.26, 0.18 for the mentioned three cases, respectively. This example indicates that when the scatterers are quite strong, the proposed inversion method can still reconstruct the profile that is close to the exact one by localizing the reconstruction errors within a small domain (such as the dummy scatterer within the annulus). As analyzed in the previous section, the inversion method matches the physical model and the measured data in a low-dimension subspace, where the main features of the scatterers can be found, yet the fine features of the scatterers cannot be fully retrieved. This might tell that we encounter the “plateau” mentioned in [1], where several possible solutions coexist in the solution space, the difference of which is not (fully) reflected in the observed scattered data due to either the noise effect or the physics itself. In the third example, we use the standard Austria profile, i.e., the one with r = 2, but increase the frequency to 750 MHz. This is actually another type of strong scatterers, the one with large dimensions. The physical setting, except the working frequency, is the same as in the first two examples. This means that, at 750 MHz frequency, the domain is 5λ × 5λ. The values of β0 are chosen as 8, 2, and 0.5, coupled with MF as 6, 8, and 10. The results obtained by the two methods are shown in Fig. 9, and values of the objective functions and errors in Fig. 9(c) and (d). Again, the FFT-TSOM method using the original LS-IE cannot fully reconstruct the profile but only the two disks on the top. In the fourth example, we use a moderate scatterer, the Austria profile with r = 2, to compare the efficiency of the two methods using different equations. For the NIE, β0 = 6 is used, and for both inversions MF = 5 is chosen. We choose the frequency f = 400 MHz, which is the limit case that the FFTTSOM with LS-IE can handle. Results shown in Fig. 10(a) tell that the convergence of the FFT-TSOM with new IE is very good, i.e., after 2 iterations the algorithm obtains a promising result, and after 10 iterations a very satisfactory one. Fig. 10(b) shows the test results obtained by the FFT-TSOM with the LSIE, in which we see after two iterations that the reconstructed result is not close to the true profile, and a satisfactory result can only be obtained after 50 iterations in this case. The comparison shows that the NIE can significantly reduce the nonlinearity of the objective function. We have also tested another profile, which is given in [8], with two disks and a lossy coated rectangle. We observe the same phenomenon as above, i.e., the FFT-TSOM with the LS-IE needs about 200 iterations to find a meaningful reconstruction, whereas the FFT-TSOM with the NIE only needs about 20 iterations. For conciseness of the present paper, we omit the results. In the last test, we use experimental data from Institut Fresnel [31]. We choose the complicated profile for medium scatterers, the FoamTwinDiel in TM case with three scatterers inside, where two of them are with r = 3 and the coated layer foam


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Fig. 9. Reconstructed results at 750 MHz using (a) FFT-TSOM with NIE and (b) with LS-IE for Austria type scatterers with r = 2. The first, second, and third rows correspond to MF being 6, 8, and 10, and β0 being 8, 2, and 0.5. In (c) and (d), the corresponding objective function values and reconstruction errors are plotted, respectively.

Fig. 11. (a) Exact profile “FoamTwinDiel” and the reconstructed results using (b) the FFT-SOM with NIE and (c) with LS-IE on experimental data at 8 GHz. The first, second, and third rows correspond to MF being 6, 10, and 16, and with β0 = 6, 2, and 0.5 for the proposed method. In (d) and (e), the corresponding objective function values and reconstruction errors are plotted, respectively.

Fig. 10. Reconstructed results for Austria type scatterers with r = 2 at 400 MHz using (a) the FFT-TSOM with the NIE (the first and the second row being the reconstructed results after 2 and 10 iterations, respectively) and (b) with the LS-IE (the first and the second row being the reconstructed results after 2 and 50 iterations, respectively).

is with r = 1.45. The experimental data provides the measurements from 2 to 10 GHz. In the reconstruction, we use a 20 cm × 20 cm domain of interest with 60 × 60 small cells inside, and we set L = 5. Choosing a small value of L is mainly due ¯ for each incidence to the change of the scattering operator G S in this experiment. With background air and null APIC as initial guesses, we tried the FFT-TSOM with the NIE on each

frequency data (note again that frequency hopping technique is not used here), and we find that the highest frequency which the proposed method can handle is 8 GHz, meaning that the domain of interest is about 5.3 λ × 5.3 λ. When using the LSIE, the inversion only succeeds on data up to 4 GHz. Note again that all reconstructions start with background medium and null APIC as initial guess. The reconstructed results at 8 GHz shown in Fig. 11(b) are obtained by the FFT-TSOM with the NIE using MF = 6, 10, and 16 and β0 = 6, 2, and 0.5. Those in Fig. 11(c) are obtained by the FFT-TSOM using the LS-IE, which are far from satisfactory results. Objective function values and reconstruction errors are displayed in Fig. 11(d) and (e). V. C ONCLUSION In this paper, we propose a family of NIE, which could be used to solve inverse medium scattering problems with strong scatterers. As shown in the analyses in Section III, the

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NIE could reduce the global nonlinear effects, introduced by the multiple scattering behaviors, in estimating the contrast in the CSI-type inversion methods via introducing new local terms. This effectively reduces the nonlinearity of the inverse scattering problem with respect to using the original LS-IE. Equipping our previously proposed regularization scheme FFTTSOM with such NIE, the new inversion method is able to reconstruct strong scatterers. When handling moderate scatterers, this new inversion method enjoys a much faster convergence compared to the traditional one. Also, since by changing β there is a series of IE, it is convenient to appraise the reconstructed results by using different IE, i.e., if similar reconstructions are obtained by using different IE, such results might be surmised to be the true scatterer profile. Numerical tests using synthetic data and the experimental data confirm the interest of the proposed method, and these results can be also used as benchmarks for other inversion methods. With such NIE, together with the proposed inversion method, we believe that the efficacy of solving the inverse scattering problem can be significantly improved. Though only a 2-D problem is discussed, extension of the proposed inversion method to the three-dimensional (3-D) case could be a direct one, though there are some aspects which need to be carefully addressed, such as the polarization effect. This is a subject of future works. ACKNOWLEDGMENT The authors would like to thank anonymous reviewers for their constructive comments. R EFERENCES [1] R. Snieder, “The role of nonlinearity in inverse problems,” Inverse Probl., vol. 14, pp. 387–404, 1998. [2] P. C. Sabatier, “Past and future of inverse problems,” J. Math. Phys., vol. 41, pp. 4082–4124, 2000. [3] P. Mojabi and J. Lo Vetri, “Overview and classification of some regularization techniques for the Gauss–Newton inversion method applied to inverse scattering problems,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2658–2265, Sep. 2009. [4] W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imag., vol. 9, no. 2, pp. 218–225, Jun. 1990. [5] A. Lakhal, “KAIRUAIN-algorithm applied on electromagnetic imaging,” Inverse Problems, vol. 29, p. 095001, 2013. [6] P. M. van den Berg, A. Abubakar, and J. T. Fokkema, “Multiplicative regularization for contrast profile inversion,” Radio Sci., vol. 38, no. 2, pp. 23-1–23-10, 2003, doi: 10.1029/2001RS002555. [7] Y. Zhong and X. Chen, “Twofold subspace-based optimization method for solving inverse scattering problems,” Inverse Probl., vol. 25, p. 085003, 2009. [8] Y. Zhong and X. Chen, “An FFT twofold subspace-based optimization method for solving electromagnetic inverse scattering problems,” IEEE Trans. Antennas Propag., vol. 59, no. 3, pp. 914–927, Mar. 2011. [9] P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” Inverse Probl., vol. 25, p. 123003, 2009. [10] R. E. Kleinman and P. M. van den Berg, “A modified gradient method for two-dimensional problems in tomography,” J. Comput. Appl. Math., vol. 42, pp. 17–35, 1992. [11] P. M. van den Berg and R. E. Kleinman, “A contrast source inversion method,” Inverse Probl., vol. 13, pp. 1607–1620, 1997. [12] P. M. van den Berg, A. L. Broekhoven, and A. Abubakar, “Extended contrast source inversion,” Inverse Probl., vol. 15, pp. 1325–1344, 1999.

[13] T. M. Habashy and A. Abubakar, “A general framework for constraint minimization for the inversion of electromagnetic measurements,” Prog. Electromagn. Res., vol. 46, pp. 265–312, 2004. [14] G. Bozza and M. Pastorino, “An inexact Newton-based approach to microwave imaging within the contrast source formulation,” IEEE Trans. Antenna Propag., vol. 57, no. 4, pp. 1122–1132, Apr. 2009. [15] X. Chen, “Subspace-based optimization method for solving inverse scattering problems,” IEEE Trans. Geosci. Remote Sens., vol. 48, no. 1, pp. 42–49, Jan. 2010. [16] G. Oliveri, Y. Zhong, X. Chen, and A. Massa, “Multiresolution subspacebased optimization method for inverse scattering problems,” J. Opt. Soc. Amer. A, vol. 28, pp. 2057–2069, 2011. [17] O. Bucci, N. Cardace, L. Crocco, and T. Isernia, “Degree of nonlinearity and a new solution procedure in scalar two-dimensional inverse scattering problems,” J. Opt. Soc. Amer. A, vol. 18, no. 8, pp. 1832–1843, Aug. 2001. [18] K. Xu, Y. Zhong, R. Song, X. Chen, and L. Ran, “Multiplicativeregularized FFT two-fold subspace-based optimization method for inverse scattering problems,” IEEE Trans. Geosci. Remote Sens., vol. 53, no. 2, pp. 841–850, Feb. 2015. [19] T. Isernia, L. Crocco, and M. D’Urso, “New tools and series for forward and inverse scattering problems in lossy media,” IEEE Geosci. Remote Sens. Lett., vol. 1, no. 4, pp. 327–331, Oct. 2004. [20] I. Catapano, L. Crocco, M. D’Urso, and T. Isernia, “On the effect of support estimation and of a new model in 2-D inverse scattering problems,” IEEE Trans. Antenna Propag., vol. 55, no. 6, pp. 1895–1899, Jun. 2007. [21] M. D’Urso, T. Isernia, and A. F. Morabito, “On the solution of 2D inverse scattering problems via source-type integral equation,” IEEE Trans. Geosci. Remote Sens., vol. 48, no. 3, pp. 1186–1198, Mar. 2010. [22] J. Ma, W. C. Chew, C.-C. Lu, and J. Song, “Image reconstruction from TE scattering data using equation of strong permittivity fluctuation,” IEEE Trans. Antenna Propag., vol. 48, no. 6, pp. 860–867, Jun. 2000. [23] R. E. Kleinman, G. F. Roach, and P. M. van den Berg, “Convergent Born series for large refractive indices,” J. Opt. Soc. Amer. A, vol. 7, no. 5, pp. 890–897, 1990. [24] L. Tsang and J. A. Kong, “Application of strong fluctuation random medium theory to scattering from vegetation-like half space,” IEEE Trans. Geosci. Remote Sens., vol. GE-19, no. 1, pp. 62–69, Jan. 1981. [25] O. V. Pankratov, D. B. Avdeyev, and A. V. Kuvshinov, “Electromagnetic field scattering in a heterogeneous earth: A solution to the forward problem,” Phys. Solid Earth, vol. 31, pp. 201–209, 1995. [26] G. Hursan and M. S. Zhdanov, “Contraction integral method in threedimensional electromagnetic modeling,” Radio Sci., vol. 37, p. 1089, 2002. [27] D. B. Avdeev, A. V. Kuvshinov, O. V. Pankratov, and G. A. Newman, “Three-dimensional induction logging problems—Part I: An integral equation solution and model comparisons,” Geophysics, vol. 67, pp. 413– 426, 2002. [28] W. C. Chew, Y. M. Wang, G. Otto, D. Lesselier, and J. C. Bolomey, “On the inverse source method of solving inverse scattering problems,” Inverse Probl. vol. 10, pp. 547–553, 1994. [29] O. M. Bucci and T. Isernia, “Electromagnetic inverse scattering: Retrievable information and measurement strategies,” Radio Sci., vol. 32, pp. 2123–2137, 1997. [30] Y. Zhong, “Subspace-based inversion methods for solving electromagnetic inverse scattering problems,” Ph.D. dissertation, Dept. of Electrical and Computer Engineering, Nat. Univ. Singapore, Singapore, 2009. [31] J.-M. Geffrin, P. Sabouroux, and C. Eyraud, “Free space experimental scattering database continuation: Experimental set-up and measurement precision,” Inverse Probl., vol. 21, pp. S117–S130, 2005. Yu Zhong was born in Guangdong, China. He received the B.S. and M.S. degrees in electronic engineering from Zhejiang University, Hangzhou, China, in 2003 and 2006, respectively, and the Ph.D. degree in electrical engineering from the National University of Singapore, Singapore, in 2010. He was a Research Engineer/Fellow with the National University of Singapore, from 2009 to 2013, during which time he was involved in a FrenchSingaporean MERLION Co-Operative Program. Since 2014, he has been a Scientist with the Institute of High Performance Computing, A*Star, Singapore. He has been invited to the Laboratoire des Signaux et Systèmes, Paris, France, as an Invited Senior Scientific Expert once per year since 2012. In August 2014, he was invited to present his works on inverse scattering problems at the 2014 Inverse Problems-From Theory to Application, the conference celebrating 30 years of the journal Inverse Problems. His research interests include numerical methods for inverse scattering problems, electromagnetic/acoustic/seismic modeling with complex materials, and nondestructive testing.


Marc Lambert received the Ph.D. degree in optics and photonics and HDR degrees from the University of Paris-Sud, Orsay, France, in 1994 and 2001, respectively. From 1995 to 2014, he carried out his research activities within the L2S as Charg de Recherche CNRS, Paris, France, and since 2015 he has been with the Group of Electrical Engineering, Paris, France. His research interests include solutions of direct and inverse scattering problems in electromagnetics and acoustics, and their applications to the characterization of objects buried in complex environments from limited datasets.

Dominique Lesselier (SM’00) was born in Lons-le-Saunier, France, in August 1953. He received the Engineering degree from Ecole Supérieure d’Electricité (Supélec), Paris, France, and the Doctorat d’Etat et Sciences Physiques degree from Université Pierre et Marie Curie, Paris, France, in 1975 and 1982, respectively. He is with the Centre National de la Recherche Scientifique (CNRS) as Researcher in October 1981 and Research Director in October 1988. He was a Visiting Scholar at the Department of Electrical Engineering, University of California at Los Angeles, Los Angeles, CA, USA, from 1982 to 1983. He is currently with the Laboratoire des Signaux et Systémes, jointly CNRS, CentraleSupélec, and Université Paris-Sud, Université Paris Saclay, Paris, France. As Director (2006–2009) of the Groupement de Recherche CNRS “GDR Ondes,” he managed a network of scientists involved in the science of waves. His research interests include development of solution methods of inverse problems, from mathematics to numerics to applications, and vice versa.

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Dr. Lesselier is a Fellow of the Institute of Physics and a member of the Electromagnetics Academy and the International Union of Radio Science, Commission B. He sits on the International Advisory Panel of Inverse Problems after serving on its Editorial Board (1997–2004), and since 2003, he has been an Associate Editor of Radio Science. Since 1998, he has been on the Standing Committee of the Electromagnetic Non-Destructive Evaluation Workshop Series and the International Steering Committee of the International Symposia on Applied Electromagnetics and Mechanics. He was the recipient of the R. W. P. King Award in 1982 from the IEEE Antennas and Propagation Society.

Xudong Chen (M’09–SM’14) received the Ph.D. degree in electrical engineering and computer science from the Massachusetts Institute of Technology, Cambridge, MA, USA, in 2005. Since then he joined the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, and he is currently an Associate Professor. He has authored about 130 peer-reviewed journal papers on inversescattering problems, material parameter retrieval, optical microscopy, and optical encryption. His research interests include electromagnetic inverse problems.