Shape Reconstruction of Thin Electromagnetic Inclusions via Boundary Measurements : Level-Set Method Combined with the Topological Derivative

An inverse problem for reconstructing arbitrary-shaped thin penetrable electromagnetic inclusions concealed in a homogeneous material is considered in this paper. For this purpose, the level-set evolution method is adopted.The topological derivative concept is incorporated in order to evaluate the evolution speed of the level-set functions. The results of the corresponding numerical simulations with and without noise are presented in this paper.


Introduction
In general, one of the main purposes of the inverse scattering problem is to determine the characteristics of an unknown target, for example, the shape, location, and internal constitution, based on the measured scattered field data.Therefore, there exist a considerable number of significant inverse scattering problems for identifying the information of arbitraryshaped thin penetrable electromagnetic inhomogeneities or perfectly conducting cracks concealed in structures such as concrete walls of buildings and bridges (refer to [1] and the references therein).
In order to solve related inverse problems, various algorithms have been developed.Among them, an iterative algorithm based on level-set evolution has been established in [2] and many generalizations have been investigated (refer to [3] and the references therein).However, most research studies have focused on the shape reconstruction of volumetric targets.Therefore, in order to reconstruct crack-like targets via level-set evolution, a new concept is required.
In recent work [4], arbitrary-shaped thin penetrable conductivity inclusions have been successfully reconstructed via the level-set method by adopting two different levelset functions.Shape reconstruction of thin electromagnetic inclusions is considered in [5].However, in [5], only the case of the purely permittivity contrast or permeability contrast with respect to the background material is considered.So, extension to both contrast cases is still remaining as a research topic.
Motivated by the mentioned above, the main objective of this paper is to reconstruct the shape of thin electromagnetic inclusions by adopting the level-set method in the case of both the dielectric permittivity and the magnetic permeability contrasts with respect to the background material.Note that, in general research papers, the evolution speed is evaluated via a complex calculation of the Fréchet derivatives using an adjoint technique.Instead of such evaluations, we incorporate a topological derivative to derive the evolution speed of two level-set functions introduced in [6][7][8][9][10].
This paper is organized as follows.In Section 2, we introduce two-dimensional direct scattering from an arbitraryshaped thin penetrable electromagnetic inclusion and the level-set functions for describing such inclusions.In Section 3, the evolution speed of the level-set functions is derived via the topological derivative.We present some numerical simulation results in Section 4 to show the effectiveness of the level-set method.Finally, we provide short conclusions in Section 5.

Mathematical Problems in Engineering
Figure 1: Illustration of the thin inclusion Γ using two level-set functions.

Direct Scattering Problem and Two
Level-Set Functions: A Brief Survey where n(r) is a unit normal to  at r. Throughout this paper, we assume that Γ does not touch Ω; that is, we assume that Γ ∩ Ω = 0.In this paper, the electric permittivities and magnetic permeabilities of Γ and Ω at a given nonzero frequency are assumed to be known; they are finite valued and they differ (either one or both of them) from the ones of the homogeneous embedding medium.For the sake of simplicity, let  0 and  0 denote the permittivity and permeability of Ω, and let  and  denote those of Γ, respectively.Then, we can define the following piecewise constants: Let  () Γ (r) denote a time-harmonic total field that satisfies the following Helmholtz equation: with boundary condition where  0 =  √  0  0 , ](r) denotes the unit outward normal to r ∈ Ω, and d  denotes a vector on the unit circle S 1 .In the same manner, we denote  () (r) = exp( 0 d  ⋅ r) as a background solution of (3) with boundary condition (4).

Description of Thin Inclusion
Using Two Level-Set Functions.Now, we describe the thin inclusion Γ by two different level-set functions (see [4,5] for a detailed description).From now on, we assume that the level-set function  is a C 1 function.With this assumption, we introduce the zero levelset function L 0 and its boundary L 0 as respectively.Then, a thin region L  of thickness 2ℎ and inner boundary L 1 can be written as follows: respectively.
In order to describe a thin region of finite length (which is connected or disconnected) in Ω, we assume that another level-set function  is also a continuously differentiable function.Let us define a band structure L  ⊂ R 2 by Then, we can describe Γ as and the boundary L 2 of Γ for evolving at inclusion tips as Figure 1 shows the illustration of the thin inclusion Γ using two level-set functions.

Evolution Speed of the Level-Set Functions and the Topological Derivative
The evolution of the level-set function  satisfies the Hamilton-Jacobi type equation: where  is  or  introduced in Section 2.2 and F(r) is given by (see, e.g., [4,5]) the following.
(i) For level-set function (r) and r ∈ L 0 , (ii) For level-set function (r) and r = ( 1 ,  2 ) ∈ L 2 , Here, S(r) denotes the speed function of the level sets that we have to evaluate.In a recent work [5], S(r) has been derived for the purely permittivity contrast ( ̸ =  0 and  =  0 ) or permeability contrast ( =  0 and  ̸ =  0 ) cases by evaluating the Fréchet derivative.In order to evaluate S(r) for both the permittivity and permeability contrast cases, we introduce the following discrepancy functional: Then, we adopt the following relationship between the evolution speed of level sets and the topological derivative.
Hence, we can evaluate the evolution speed S(r) by computing the topological derivative   E(r).The result is as follows.
Theorem 2. In both the permittivity and permeability contrast cases,   E(r) can be written as where V satisfies the following adjoint problem: Proof.See the appendix.
Note that, if either  =  0 or  =  0 , we can obtain the Fréchet derivatives based on [5, Theorem 3.1] or [5,Theorem 3.2], respectively.This result is identical to that of [11,Theorem 3.2], but the proof is considerably easier.

Numerical Simulations
In this section, we choose the background material Ω as a unit circle centered at origin with  0 =  0 = 1.For thin inclusions Γ  , the thickness is set to ℎ = 0.02 and the parameters of are chosen as   =   = 5, where   and   are the permittivity and permeability contrasts of Γ  , respectively.The supporting curve   is selected as For a given inclusion Γ  , let  true (r) be the solution to (3) with boundary condition (4).In the same manner, for an inclusion Γ ()   at the iteration step  (if  = 0, it is an initial guess), let  ()  comp denote the solution to (3) with boundary condition (4).For the least-square functional at a given frequency, we introduce a normalized error function: where In this section, we adopt  = 16 and the locations r  ∈ Ω are selected as A finite-difference time discretization form of (10) is For the evolving level sets  and , the time step is chosen as Δ() = 3 × 10 − (see [5,12] for a detailed discussion).With this configuration, the forward and adjoint problems are solved via the finite element method (FEM) to avoid an inverse crime, and a 20 dB white Gaussian random noise is added to the boundary measurements data in order to present the robustness.
First, we consider the shape reconstruction of Γ 1 .In this example, we assume that the location of the end points of Γ 1 has been identified based on the location search algorithm in [18][19][20].Thus, by connecting them by a straight line segment, a good initial guess can be obtained.
Figure 2 shows the shape reconstruction of Γ 1 at the operated frequency  = 2/0.6.Based on this result, we can confirm that the shape of Γ 1 is successfully reconstructed and the residual considerably decreases accordingly.
Next, we consider the reconstruction of Γ 2 .In contrast to the previous example, we apply the topological derivative strategy introduced in [10,30] to obtain a good initial guess.In Figure 3, the obtained shape of Γ 2 via the topological derivative with  = 16 and  = 2/0.4 is illustrated.In this result, we can construct a supporting curve via well-identified points as for  ∈ [−0.2451, 0.6646] (see Figure 9 in [10]).Here,   () denotes the Chebyshev polynomials of the first kind.Considering this curve as the supporting curve of the initial guess, the reconstruction of the complete shape of Γ 2 is illustrated in Figure 3.Note that, because of the considerably close initial guess, only 31 iterations are sufficient to obtain a good result.

Conclusion Remarks
In this paper, a level-set method to perform the shape reconstruction of thin electromagnetic inclusion using both the dielectric permittivity and magnetic permeability contrasts with respect to the homogeneous domain was considered.
The relationship between the topological derivative and the evolution speed of level-set functions allowed us to reconstruct the complete shape of the thin inclusion with a good initial guess.It is discovered from the numerical experiments that the proposed technique is stable with respect to random noise.Further, incorporating the topological derivative guarantees a successful reconstruction procedure.
In this paper, we performed shape reconstruction with a priori information of unknown targets, for example, thickness, permittivity, and permeability values.Performing simultaneous reconstruction without a priori information will be a challenge (refer to [3,Section 10.3]).

Appendix
Proof of Theorem 2. In this appendix, we prove Theorem 2. In order to derive   E(r) of ( 13), we construct a small electromagnetic inclusion Σ = r + D in Ω with the same permittivity and permeability of Γ and denote this domain by Ω ∨ Σ.Then, by virtue in [10], the following asymptotic expansion holds: where () → 0 as  → 0+.Let  () Σ (r) be a solution to (3) with boundary condition (4) in the existence of Σ.Then, the following asymptotic expansion formula holds (see [13]): where N(r, r  ) denotes the Neumann function that is the solution to Then, applying (A.2) to ( 13), we can compute where Then, applying boundary condition ( 16) and asymptotic formula (A.2) yields First, applying integration by parts and (A.3), we can compute E  (r) as  This completes the proof.