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

Authors: Won-Kwang Park
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

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.


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## Body

## 1. 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 arbitrary-shaped 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 level-set 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 [6–10].This paper is organized as follows. In Section2, we introduce two-dimensional direct scattering from an arbitrary-shaped 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.


## 2. Direct Scattering Problem and Two Level-Set Functions: A Brief Survey

### 2.1. Two-Dimensional Direct Scattering Problem

LetΩ be a two-dimensional homogeneous domain with a smooth boundary ∂Ω. Throughout this paper, we assume that Ω conceals an electromagnetic inclusion Γ with thickness 2h, which is characterized in the neighborhood of the smooth curve σ:
(1)Γ:={r+ηn(r):r∈σ,η∈(-h,h)},
where n(r) is a unit normal to σ at r. Throughout this paper, we assume that Γ does not touch ∂Ω; that is, we assume that Γ∩∂Ω=∅.In this paper, the electric permittivities and magnetic permeabilities ofΓ 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 ε0 and μ0 denote the permittivity and permeability of Ω, and let ε and μ denote those of Γ, respectively. Then, we can define the following piecewise constants:
(2)ε(r)={ε0forr∈Ω∖Γ-,εforr∈Γ,μ(r)={μ0forr∈Ω∖Γ-,μforr∈Γ.LetuΓ(n)(r) denote a time-harmonic total field that satisfies the following Helmholtz equation:
(3)∇·(1μ(r)∇uΓ(n)(r))+ω2ε(r)uΓ(n)(r)=0inΩ
with boundary condition
(4)1μ0∂uΓ(n)(r)∂ν(r)=1μ0∂exp(ik0dn·r)∂ν(r)=g(r)on∂Ω,
where k0=ωε0μ0, ν(r) denotes the unit outward normal to r∈∂Ω, and dn denotes a vector on the unit circle 𝕊1. In the same manner, we denote u(n)(r)=exp(ik0dn·r) as a background solution of (3) with boundary condition (4).


### 2.2. Description of Thin Inclusion Using Two Level-Set Functions

Now, we describe the thin inclusionΓ by two different level-set functions (see [4, 5] for a detailed description). From now on, we assume that the level-set function φ is a 𝒞1 function. With this assumption, we introduce the zero level-set function ℒ0 and its boundary ∂ℒ0 as
(5)ℒ0:={r∈ℝ2:φ(r)≤0},∂ℒ0:={r∈ℝ2:φ(r)=0},
respectively. Then, a thin region ℒT of thickness 2h and inner boundary ∂ℒ1 can be written as follows:
(6)ℒT:=ℒ0∩{r-ηn(r):r∈∂ℒ0,0≤η≤2h},∂ℒ1:=∂ℒT∖∂ℒ0,
respectively.In order to describe a thin region of finite length (which is connected or disconnected) inΩ, we assume that another level-set function ψ is also a continuously differentiable function. Let us define a band structure ℒB⊂ℝ2 by
(7)ℒB:={r=(r1,r2)∈ℝ2:ψ(r1)<0}.
Then, we can describe Γ as
(8)Γ=(ℒT∩ℒB)∩Ω
and the boundary ∂ℒ2 of Γ for evolving at inclusion tips as
(9)∂ℒ2:={r=(r1,r2)∈∂Γ:ψ(r1)=0}.Figure1 shows the illustration of the thin inclusion Γ using two level-set functions.Illustration of the thin inclusionΓ using two level-set functions.
(a)
(b)


## 2.1. Two-Dimensional Direct Scattering Problem

LetΩ be a two-dimensional homogeneous domain with a smooth boundary ∂Ω. Throughout this paper, we assume that Ω conceals an electromagnetic inclusion Γ with thickness 2h, which is characterized in the neighborhood of the smooth curve σ:
(1)Γ:={r+ηn(r):r∈σ,η∈(-h,h)},
where n(r) is a unit normal to σ at r. Throughout this paper, we assume that Γ does not touch ∂Ω; that is, we assume that Γ∩∂Ω=∅.In this paper, the electric permittivities and magnetic permeabilities ofΓ 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 ε0 and μ0 denote the permittivity and permeability of Ω, and let ε and μ denote those of Γ, respectively. Then, we can define the following piecewise constants:
(2)ε(r)={ε0forr∈Ω∖Γ-,εforr∈Γ,μ(r)={μ0forr∈Ω∖Γ-,μforr∈Γ.LetuΓ(n)(r) denote a time-harmonic total field that satisfies the following Helmholtz equation:
(3)∇·(1μ(r)∇uΓ(n)(r))+ω2ε(r)uΓ(n)(r)=0inΩ
with boundary condition
(4)1μ0∂uΓ(n)(r)∂ν(r)=1μ0∂exp(ik0dn·r)∂ν(r)=g(r)on∂Ω,
where k0=ωε0μ0, ν(r) denotes the unit outward normal to r∈∂Ω, and dn denotes a vector on the unit circle 𝕊1. In the same manner, we denote u(n)(r)=exp(ik0dn·r) as a background solution of (3) with boundary condition (4).


## 2.2. Description of Thin Inclusion Using Two Level-Set Functions

Now, we describe the thin inclusionΓ by two different level-set functions (see [4, 5] for a detailed description). From now on, we assume that the level-set function φ is a 𝒞1 function. With this assumption, we introduce the zero level-set function ℒ0 and its boundary ∂ℒ0 as
(5)ℒ0:={r∈ℝ2:φ(r)≤0},∂ℒ0:={r∈ℝ2:φ(r)=0},
respectively. Then, a thin region ℒT of thickness 2h and inner boundary ∂ℒ1 can be written as follows:
(6)ℒT:=ℒ0∩{r-ηn(r):r∈∂ℒ0,0≤η≤2h},∂ℒ1:=∂ℒT∖∂ℒ0,
respectively.In order to describe a thin region of finite length (which is connected or disconnected) inΩ, we assume that another level-set function ψ is also a continuously differentiable function. Let us define a band structure ℒB⊂ℝ2 by
(7)ℒB:={r=(r1,r2)∈ℝ2:ψ(r1)<0}.
Then, we can describe Γ as
(8)Γ=(ℒT∩ℒB)∩Ω
and the boundary ∂ℒ2 of Γ for evolving at inclusion tips as
(9)∂ℒ2:={r=(r1,r2)∈∂Γ:ψ(r1)=0}.Figure1 shows the illustration of the thin inclusion Γ using two level-set functions.Illustration of the thin inclusionΓ using two level-set functions.
(a)
(b)


## 3. Evolution Speed of the Level-Set Functions and the Topological Derivative

The evolution of the level-set functionϕ satisfies the Hamilton-Jacobi type equation:
(10)∂ϕ(r)∂t=𝔽(r)|∇ϕ(r)|=0,
where ϕ is φ or ψ introduced in Section 2.2 and 𝔽(r) is given by (see, e.g., [4, 5]) the following.(i)
For level-set functionφ(r) and r∈∂ℒ0,
(11)𝔽(r)=Re(𝕊(r)[δ(r;∂ℒ0)-δ(r+2hn(r);∂ℒ0)]).(ii)
For level-set functionψ(r) and r=(r1,r2)∈∂ℒ2,
(12)𝔽(r)=Re(∫∂ℒ2‍𝕊(r)δ(r;∂ℒ2)χ(r;ℒ)dr2).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 (ε≠ε0 and μ=μ0) or permeability contrast (ε=ε0 and μ≠μ0) 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:
(13)𝔼(Ω):=12∑n=1N‍∥uΓ(n)(r′)-u(n)(r′)∥L2(∂Ω)2=12∑n=1N‍∫∂Ω‍(|uΓ(n)(r′)-u(n)(r′)|2)dS(r′).
Then, we adopt the following relationship between the evolution speed of level sets and the topological derivative.Theorem 1.
LetDT𝔼(r) denote the topological derivative of (13); then DT𝔼(r) satisfies
(14)DT𝔼(r)={-𝕊(r)forr∈Γ𝕊(r)forr∈Ω∖Γ¯.Proof.
A rigorous derivation can be found in [6, 8, 9].Hence, we can evaluate the evolution speed𝕊(r) by computing the topological derivative DT𝔼(r). The result is as follows.Theorem 2.
In both the permittivity and permeability contrast cases,DT𝔼(r) can be written as
(15)DT𝔼(r)=ℜ𝔢∑n=1N‍(μ-μ0μ+μ0ω2(ε-ε0)|𝒟|v(n)(r)u(n)(r)¯-2|𝒟|μ-μ0μ+μ0∇v(n)(r)·∇u(n)(r)¯),
where v satisfies the following adjoint problem:
(16)1μ0Δv(n)(r)+ω2ε0v(n)(r)=0inΩ,1μ0∂v(n)∂ν(r)=uΓ(n)(r)-u(n)(r)on∂Ω.Proof.
See the appendix.Note that, if eitherμ=μ0 or ε=ε0, 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.


## 4. Numerical Simulations

In this section, we choose the background materialΩ as a unit circle centered at origin with ε0=μ0=1. For thin inclusions Γj, the thickness is set to h=0.02 and the parameters of are chosen as εj=μj=5, where εj and μj are the permittivity and permeability contrasts of Γj, respectively. The supporting curve σj is selected as
(17)σ1={(x,0.5x2+0.1):x∈[-0.5,0.5]},σ2={(x+0.2,x3+x2-0.6):x∈[-0.5,0.5]}.For a given inclusionΓj, let utrue(r) be the solution to (3) with boundary condition (4). In the same manner, for an inclusion Γj(s) at the iteration step s (if s=0, it is an initial guess), let ucomp(s) denote the solution to (3) with boundary condition (4). For the least-square functional at a given frequency, we introduce a normalized error function:
(18)ℝ(s)=∥utrue(rn)-ucomp(s)(rn)∥ℓ22∥ucomp(s)(rn)∥ℓ22forrn∈∂Ω,
where
(19)∥f(rn)∥ℓ22:=∑n=1N‍|f(rn)|2.
In this section, we adopt N=16 and the locations rn∈∂Ω are selected as
(20)rn=(cos2nπN,sin2nπN),n=1,2,…,N.A finite-difference time discretization form of (10) is
(21)ϕ(s+1)(r)=ϕ(s)(r)+Δt(s)𝔽(s)(r)|∇ϕ(s)(r)|.
For the evolving level sets φ and ψ, the time step is chosen as Δt(s)=3×10-s (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.Remark 3 (generating a good initial guess).
It is worth mentioning that, for a successful performance of the iterative based algorithm, one must start the iteration procedure with a good initial guess that is close to the unknown target. Without this guess, one will encounter unexpected scenarios such as nonconvergence (refer to Figures  7 and  14 in [5]). There are various noniterative shape reconstruction algorithms such as the multiple signal classification (MUSIC)-type algorithm [13–17], the end-point identification algorithm [18–20], the multifrequency based subspace migration imaging technique [21–27], the topological derivative strategy [10, 28–32], and the gradient-for-the-initiation (GFI) method [33].First, we consider the shape reconstruction ofΓ1. In this example, we assume that the location of the end points of Γ1 has been identified based on the location search algorithm in [18–20]. Thus, by connecting them by a straight line segment, a good initial guess can be obtained.Figure2 shows the shape reconstruction of Γ1 at the operated frequency ω=2π/0.6. Based on this result, we can confirm that the shape of Γ1 is successfully reconstructed and the residual considerably decreases accordingly.Reconstruction ofΓ1 with noisy data.
(a)
Initial guess
(b)
Shape ofΓ1(20)
(c)
Shape ofΓ1(60)
(d)
Shape ofΓ1(100)
(e)
Shape ofΓ1(140)
(f)
Shape ofΓ1(160)
(g)
Shape ofΓ1(200)
(h)
Shape ofΓ1
(i)
Value ofℝ(s)Next, we consider the reconstruction ofΓ2. 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 Γ2 via the topological derivative with N=16 and ω=2π/0.4 is illustrated. In this result, we can construct a supporting curve via well-identified points as
(22)σ2(0)=-0.4308T0(x)+0.4160T1(x)+0.1482T2(x)+0.1895T3(x)=0.7580x3+0.2964x2-0.1525x-0.5790
for x∈[-0.2451,0.6646] (see Figure  9 in [10]). Here, Tj(x) 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 Γ2 is illustrated in Figure 3. Note that, because of the considerably close initial guess, only 31 iterations are sufficient to obtain a good result.Reconstruction ofΓ2 with noisy data.
(a)
Identified points
(b)
Initial guess
(c)
31 iterations
(d)
Value ofℝ(s)
(e)
Shape ofΓ2(0) and Γ2
(f)
Shape ofΓ2(31) and Γ2


## 5. 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 witha 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]).



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