# Topological Derivative for Imaging of Thin Electromagnetic Inhomogeneity: Least Condition of Incident Directions

Authors: Won-Kwang Park
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

It is well-known that using topological derivative is an effective noniterative technique for imaging of crack-like electromagnetic inhomogeneity with small thickness when small number of incident directions are applied. However, there is no theoretical investigation about the configuration of the range of incident directions. In this paper, we carefully explore the mathematical structure of topological derivative imaging functional by establishing a relationship with an infinite series of Bessel functions of integer order of the first kind. Based on this, we identify the condition of the range of incident directions and it is highly depending on the shape of unknown defect. Results of numerical simulations with noisy data support our identification.


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## Body

## 1. Introduction

In this paper, we consider topological derivative [1] based imaging technique for thin, curve-like penetrable electromagnetic inhomogeneity with small thickness. Originally, this has been considered for shape optimization problems [2–7] and was then successfully combined with the level-set method (see [8–13]) for various inverse scattering problems. Surprisingly, throughout some researches [14, 15], it has been confirmed that topological derivative is also one of noniterative imaging techniques and very effective algorithm. However, as we can see [15–21], the most work has considered full-aperture problems.Throughout results of numerical simulations, it has been confirmed that topological derivatives can be applied in limited-aperture problems, and an analysis in limited-aperture problems has been performed in [22]. In this interesting research, a relationship between topological derivative imaging function and an infinite series of Bessel functions of first kind has been established and, correspondingly, a sufficient condition of the range of incident directions for application has been identified theoretically. However, a least condition of application still remains unknown. Motivated by this fact, we identify a least condition of the range of incident directions for a successful application in limited-aperture inverse scattering problem and confirm this condition is highly depending on the unknown shape of thin inhomogeneity.The remainder of paper is organized as follows. In Section2, we survey two-dimensional direct scattering problem and topological derivative based imaging technique. In Section 3, we investigate a least condition of the range of incident directions and discuss its properties. In Section 4, several results of numerical simulations with noisy data are presented in order to support our investigation. A brief conclusion is given in Section 5.


## 2. Introduction to Direct Scattering Problem and Topological Derivative

LetΩ⊂R2 be a homogeneous domain with smooth boundary ∂Ω that contains a homogeneous thin inhomogeneity Γ with a small thickness 2h. That is,(1)Γ=x+ηnx:x∈σ,-h≤η≤h,where n(x) is the unit normal to σ at x and σ denotes the a simple, smooth curve in R2 which describes the supporting curve of Γ. In this contribution, we assume that the applied angular frequency is of the form ω=2πf. We assume that all materials are characterized by their dielectric permittivity and magnetic permeability at frequency of operation ω; we define the piecewise constant permittivity ε(x) and permeability μ(x) as(2)εx≔ε0forx∈Ω\Γ¯ε⋆forx∈Γ,μx≔μ0forx∈Ω\Γ¯μ⋆forx∈Γ,respectively. For this sake, we set ε0=μ0≡1 and denote k=ωε0μ0=2π/λ as the wavenumber, where λ is a given wavelength and satisfies h≪λ.Letunx;ω be the time-harmonic total field that satisfies Helmholtz equation(3)∇·1μx∇unx;ω+ω2εxunx;ω=0,x∈Ωwith boundary condition(4)1μ0∂unx;ω∂νx=1μ0∂eiωθn·x∂νx,x∈∂Ωand with transmission conditions on the boundary of Γ. Here, θn=cos⁡θn,sin⁡θn denotes a two-dimensional vector on the connected, proper subset of unit circle S1 such that(5)θn=θ1+n-1N-1θN-θ1,0<θN-θ1<2π.Similarly, let uBnx;ω=eiωθn·x be the background solution of (3) with boundary condition (4).Now, we introduce the basic concept of topological derivative operated at a fixed single frequency. The problem considered herein is the minimization of the tracking type functional depending on the solutionunx;ω:(6)EΩ;ω≔12∑n=1Nunx;ω-uBnx;ωL2∂Ω2=12∑n=1N∫∂Ωunx;ω-uBnx;ω2dSx.Assume that an electromagnetic inclusionΣ of small diameter r is created at a certain position z∈Ω\∂Ω, and let Ω∣Σ denote this domain. Since the topology of the entire domain has changed, we can consider the corresponding topological derivative dTEz based on EΩ with respect to point z as(7)dTEz;ω=limr→0+⁡EΩ∣Σ;ω-EΩ;ωφr;ω,where φr;ω→0 as r→0+. From (7), we can obtain an asymptotic expansion:(8)EΩ∣Σ;ω=EΩ;ω+φr;ωdTEz;ω+oφr;ω.In [21], the following normalized topological derivative imaging function ETD(z;ω) has been introduced:(9)ETDz;ω=12dTEεz;ωmax⁡dTEεz;ω+dTEμz;ωmax⁡dTEμz;ω.Here, dTEεz;ω and dTEμ(z;ω) satisfying (8) for purely dielectric permittivity contrast (ε≠ε0 and μ=μ0) and magnetic permeability contrast (ε=ε0 and μ≠μ0) cases, respectively, are explicitly expressed as (see [21])(10)dTEεz;ω=Re⁡∑n=1NuAnz;ωuBnz;ω¯,dTEμz;ω=Re⁡∑n=1N∇uAnz;ω·∇uBnz;ω¯,where uAnx;ω satisfies the adjoint problem(11)ΔuAnx;ω+ω2uAnx;ω=0inΩ∂uAnx;ω∂νx=unx;ω-uBnx;ωon∂Ω.


## 3. Least Condition of Incident Directions

In this section, we identify the least condition of incident directions for applying topological derivative. For this, we introduce the structure ofdTEεz;ω and dTEμz;ω as follows.Lemma 1 (see [20, 21]).
Suppose thatN and ω are sufficiently large; then(12)dTEεz;ω≈Re⁡∑n=1N∫σε⋆-ε0eiωθn·z-xdσxdTEμz;ω≈Re⁡∑n=1N∫σΦtx,nx,θneiωθn·z-xdσx,where t(x) and n(x) are unit vectors that are, respectively, tangent and normal to the supporting curve σ at x, and(13)Φtx,nx,θn=21μ⋆-1μ0θl·tx+21μ0-μ⋆μ02θl·nx.Based on Lemma1, the structure of (9) in limited-aperture problem can be represented as follows. This result plays an important role in identifying least condition of incident directions. For a detailed proof, we refer to [22].Theorem 2.
Letθ=cos⁡θ,sin⁡θT and x-z=rcos⁡ϕ,sin⁡ϕT. If ω and N are sufficiently large, the structure of (9) becomes(14)ETDz;ω=12W1z,ωmax⁡W1z,ω+W2z,ωmax⁡W2z,ω,where(15)W1z,ω≈∫σε⋆-ε0J0ωx-z+Dωx-z,θ1,θNdσxwith(16)Dωx-z,θ1,θN=2θN-θ1∑m=1∞-1mmsin⁡mθN-θ1×cos⁡mθN+θ1-2ϕJ2mωx-zand the term W2(z,ω) does not contribute to the imaging performance.In recent work [20], it has been confirmed that the application of multifrequencies guarantees better a imaging performance than the application of a single frequency. Therefore, we consider the following normalized multifrequency topological derivative: for several frequencies ωf:f=1,2,…,F, define(17)Ez;F≔1F∑f=1FdTETDz;ωfmax⁡dTETDz;ωf.Based on the structure ofETDz;ω in Theorem 2, we can observe that the terms J0ωfx-z and Dωfx-z,θ1,θN contribute to and disturb the imaging performance, respectively, for f=1,2,…,F. Hence, eliminating the term Dωfx-z,θ1,θN will guarantee a good result. This means that the least condition comes from(18)∑m=1∞sin⁡mθN-θ1cos⁡mθN+θ1-2ϕJ2mωfx-z≡0for all z∈Ω and f=1,2,…,F. Note that since Jm satisfies the asymptotic property(19)Jmx≈2xcos⁡x-mπ2-π4for x≫m2-0.25, good results can appear in the map of Ez;F when ωf≈+∞. Unfortunately, this is an ideal condition. Furthermore, since z is arbitrary, we cannot control the value of the term J2mωfx-z. So, we must find a condition of θ1 and θN:(20)sin⁡mθN-θ1cos⁡mθN+θ1-2ϕ≡0for all m∈N. A simple way is to select θ1 and θN such that(21)θN-θ1=π2,θN+θ1-2ϕ=π2,that is, for any α∈R, a selection θ1=ϕ+α and θN=ϕ-α+π/2 will guarantee good results via topological derivative. Based on this, we can obtain the following theoretical result of the range of incident directions, which has been examined heuristically.Theorem 3 (least condition of range).
The least range of incident directions for successful application of topological derivative is highly depending on the shape of unknown thin inhomogeneity and the range of directions must be wider thanπ/2.


## 4. Simulations Results and Discussions

In this section, some results of numerical simulations are exhibited to support identified condition mentioned in previous section. The homogeneous domainΩ is chosen as a unit circle centered at the origin, and two supporting curves σj of the Γj are selected as(22)σ1=s,0.2T:-0.5≤s≤0.5σ2=s+0.2,s3+s2-0.6T:-0.5≤s≤0.5.The thicknessh of all Γj is set to 0.02, and parameters ε0, μ0 are chosen as 1. Let εj and μj for j=1,2,3 denote the permittivity and permeability of Γj, respectively. The applied frequency is selected as ωf=2π/λf, f=1,2,…,F(=10) with λ1=0.7, λ10=0.4, and N=16 different incident directions chosen. In order to show the robustness, a white Gaussian noise with 20 dB signal-to-noise ratio (SNR) was added to the unperturbed boundary data.First, let us consider the imaging of straight line shaped thin inhomogeneityΓ1. In this case, since ϕ≡0, when x,z∈Γ1, θ1=α and θN=α+π/2 will be a good choice for any α. Corresponding results for α=0, α=π/6, α=π/4, and α=3π/4 are exhibited in Figure 1. Based on these results, we can observe that, for any value of α, the shape of Γ1 was retrieved satisfactorily via Ez;F with the π/2 range of incident directions. However, although one can recognize the existence of inhomogeneity, the shape of Γ1 cannot be reconstructed satisfactorily when α=3π/4.Figure 1
Maps ofE(z;F) for Γ1. Violet-colored solid line describes the range of incident directions.
(a)
θ 1 = 0 and θ2=π/2 (b)
θ 1 = π / 6 and θ2=2π/3 (c)
θ 1 = π / 4 and θ2=3π/4 (d)
θ 1 = π / 2 and θ2=πBased on the results in Figure1, we can conclude that one can identify the shape of existence of Γ1 for any choice of α. But if the values of θ1 and θN are satisfying(23)θ1=ξt-π4,θN=ξt+π4,wheretx=cos⁡ξt,sin⁡ξtT,the result via the map of E(z;F) is very poor. This is the worst choice of the selection. In contrast, if the values of θ1 and θN are satisfying(24)θ1=ξn-π4,θN=α+π4,wherenx=cos⁡ξn,sin⁡ξnT,an acceptable result should be obtained via the map of Ez;F; refer to Figure 2. With this observation, we can conclude that if θ1 and θN are satisfying(25)0≤θ1≤ξn-π4,ξn+π4≤θN≤2π,a good result of Γ1 can be obtained; refer to Figure 3.Figure 2
Description of best (blue-colored range) and worst (red-colored range) choice of the range of incident directions.Figure 3
Same as Figure1 except the range of the incident directions.
(a)
θ 1 = π / 6 and θ2=5π/6 (b)
θ 1 = 0 and θ2=πNow, let us consider the influence of range of incident directions when the shape of thin inhomogeneity is no more straight line. For this purpose, we choose thin inclusionΓ2 and compare maps of Ez;F for various range of incident directions. Figure 4 shows maps of Ez;F for(26)θ1=π4-α,θN=π4+α,where α=π/12, π/6, π/4, and π/2. Based on these result, we can observe that if the range of directions is narrow, we cannot recognize the existence of Γ2; refer to Figures 4(a) and 4(b). Note that when the range of directions satisfies the sufficient condition in [22], the shape of Γ2 can be identified; refer to Figure 4(d). However, if one selects the optimal range, adopted for imaging of Γ1, the result is still poor (see Figure 4(c)).Figure 4
Maps ofE(z;F) for Γ1. Violet-colored solid line describes the range of incident directions.
(a)
θ 1 = 5 π / 12 and θ2=7π/12 (b)
θ 1 = π / 3 and θ2=2π/3 (c)
θ 1 = π / 4 and θ2=3π/4 (d)
θ 1 = π / 3 and θ2=5π/3In order to find the least condition, let us reconsider the imaging ofΓ1. In this case, the selection of (24) was a good choice. Following this observation, one of the possible choices of α is that, for n(x)=[cos(ξx),sin(ξx)]T, x∈σ2, select(27)θ1=min⁡ξx:x∈σ2-π4,θN=max⁡ξx:x∈σ2+π4.Then, it is expected that identified shape of inhomogeneity will be close to the shape of Γ2. Notice that, throughout the numerical computation,(28)θN-θ1=max⁡ξx:x∈σ2-min⁡ξx:x∈σ2+π2<π.Thus, selection of (27) is the least condition of the range of incident directions. Figure 6 exhibits maps of E(z;F) for θ1 and θN in (27) and for θ1=0 and θN=2π (see Figure 5 for instance). Since these conditions satisfy least condition, the shape of Γ2 seems retrieved well, and this result supports Theorem 3.Figure 5
Description of best choice of the range of incident directions.Figure 6
Map ofE(z;F) for when inhomogeneity is Γ2.
(a)
θ 1 and θN satisfy (27) (b)
θ 1 = 0 and θN=πFor the final example, let us consider the imaging of two, nonoverlapped thin inhomogeneitiesΓ2 and Γ3, where the supporting curve σ3 of Γ3 is(29)σ3=s-0.2,-0.5s2+0.5T:-0.5≤s≤0.5and ε2=ε3=μ2=μ3=5. Based on the results in Figure 7, we can conclude that it is hard to recognize the shape of Γ2∪Γ3 when the range θN-θ1<π, but we can identify when θN-θ1≈π. With this, we end up this section with the following remark.Figure 7
Maps ofE(z;F) for Γ2∪Γ3. Violet-colored solid line describes the range of incident directions.
(a)
θ 1 = π / 2 and θ2=π (b)
θ 1 = π / 4 and θ2=3π/4 (c)
θ 1 = π / 6 and θ2=5π/6 (d)
θ 1 = 0 and θ2=πRemark 4 (condition for imaging of multiple inhomogeneities).
Due to the shape dependency of the range of directions, when the shapes of thin inhomogeneities are not straight line, the range of directions must be close toπ, which is the sufficient condition of range of application. Related results of numerical simulations can be found in [22] also.


## 5. Conclusion

In this paper, we have considered the topological derivative in a limited-aperture inverse scattering problem for a noniterative imaging of thin inhomogeneity. Based on the relationship between topological derivative imaging function and infinite series of Bessel functions of integer order of the first kind, we discovered a least condition of the range of incident directions for successful application. We presented the results of some numerical simulations, which show that the discovered condition is valid for the imaging of a thin inclusion. Here, we have considered an imaging of thin penetrable inhomogeneity but the analysis could be carried out for a perfectly conducting crack. Furthermore, the extension to inverse elasticity problems will be an interesting subject. Finally, extension to the three-dimensional [23, 24] and real-world problem [25, 26] will be a remarkable research topic.



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