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

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.

Throughout results of numerical simulations, it has been confirmed that topological derivatives can be applied in limited-aperture problems, and an analysis in limitedaperture 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 Section 2, 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.

Introduction to Direct Scattering Problem and Topological Derivative
Let Ω ⊂ R 2 be a homogeneous domain with smooth boundary Ω that contains a homogeneous thin inhomogeneity Γ with a small thickness 2ℎ.That is, where n(x) is the unit normal to  at  and  denotes the a simple, smooth curve in R 2 which describes the supporting curve of Γ.In this contribution, we assume that the applied angular frequency is of the form  = 2.We assume that 2 Advances in Mathematical Physics all materials are characterized by their dielectric permittivity and magnetic permeability at frequency of operation ; we define the piecewise constant permittivity (x) and permeability (x) as respectively.For this sake, we set  0 =  0 ≡ 1 and denote  =  √  0  0 = 2/ as the wavenumber, where  is a given wavelength and satisfies ℎ ≪ .
Let  () (x; ) be the time-harmonic total field that satisfies Helmholtz equation with boundary condition and with transmission conditions on the boundary of Γ.Here,   = [cos (  ), sin (  )] denotes a two-dimensional vector on the connected, proper subset of unit circle S 1 such that Similarly, let  () B (x; ) =    ⋅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 solution  () (x; ): Assume that an electromagnetic inclusion Σ of small diameter  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   E(z) based on E(Ω) with respect to point z as where (; ) → 0 as  → 0+.From (7), we can obtain an asymptotic expansion: In [21], the following normalized topological derivative imaging function E TD (z; ) has been introduced: Here,   E  (z; ) and   E  (z; ) satisfying ( 8) for purely dielectric permittivity contrast ( ̸ =  0 and  =  0 ) and magnetic permeability contrast ( =  0 and  ̸ =  0 ) cases, respectively, are explicitly expressed as (see [21]) where  () A (x; ) satisfies the adjoint problem

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 of   E  (z; ) and   E  (z; ) as follows.
Lemma 1 (see [20,21]).Suppose that  and  are sufficiently large; then where t(x) and n(x) are unit vectors that are, respectively, tangent and normal to the supporting curve  at x, and Based on Lemma 1, the structure of ( 9) in limitedaperture 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. Let  = [cos , sin ]  and x − z = [cos , sin ]  .If  and  are sufficiently large, the structure of ( 9) becomes where with and the term W 2 (z, ) 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 {  :  = 1, 2, . . ., }, define Based on the structure of E TD (z; ) in Theorem 2, we can observe that the terms  0 (  |x − z|) and D(  |x − z|,  1 ,   ) contribute to and disturb the imaging performance, respectively, for  = 1, 2, . . ., .Hence, eliminating the term D(  |x − z|,  1 ,   ) will guarantee a good result.This means that the least condition comes from for all z ∈ Ω and  = 1, 2, . . ., .Note that since   satisfies the asymptotic property for  ≫ | 2 − 0.25|, good results can appear in the map of E(z; ) when   ≈ +∞.Unfortunately, this is an ideal condition.Furthermore, since z is arbitrary, we cannot control the value of the term  2 (  |x − z|).So, we must find a condition of  1 and   : sin for all  ∈ N. A simple way is to select  1 and   such that that is, for any  ∈ R, a selection  1 =  +  and   =  −  + (/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 than /2.

Simulations Results and Discussions
In this section, some results of numerical simulations are exhibited to support identified condition mentioned in previous section.The homogeneous domain Ω is chosen as a unit circle centered at the origin, and two supporting curves   of the Γ  are selected as The thickness ℎ of all Γ  is set to 0.02, and parameters  0 ,  0 are chosen as 1.Let   and   for  = 1, 2, 3 denote the permittivity and permeability of Γ  , respectively.The applied frequency is selected as   = 2/  ,  = 1, 2, . . ., (= 10) with  1 = 0.7,  10 = 0.4, and  = 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 inhomogeneity Γ 1 .In this case, since  ≡ 0, when x, z ∈ Γ 1 ,  1 =  and   =  + (/2) will be a good choice for any .Corresponding results for  = 0,  = /6,  = /4, and  = 3/4 are exhibited in Figure 1.Based on these results, we can observe that, for any value of , the shape of Γ 1 was retrieved satisfactorily via E(z; ) with the /2 range of incident directions.However, although one can recognize the existence of inhomogeneity, the shape of Γ 1 cannot be reconstructed satisfactorily when  = 3/4.Based on the results in Figure 1, we can conclude that one can identify the shape of existence of Γ 1 for any choice of .But if the values of  1 and   are satisfying the result via the map of E(z; ) is very poor.This is the worst choice of the selection.In contrast, if the values of  1 and   are satisfying where an acceptable result should be obtained via the map of E(z; ); refer to Figure 2.With this observation, we can conclude that if  1 and   are satisfying a good result of Γ 1 can be obtained; refer to Figure 3. Now, 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 inclusion Γ 2 and compare maps of E(z; ) for various range of incident directions.Figure 4 shows maps of E(z; ) for where  = /12, /6, /4, and /2.Based on these result, we can observe that if the range of directions is narrow, we  cannot recognize the existence of Γ 2 ; refer to Figures 4(a) and 4(b).Note that when the range of directions satisfies the sufficient condition in [22], the shape of Γ 2 can be identified; refer to Figure 4(d).However, if one selects the optimal range, adopted for imaging of Γ 1 , the result is still poor (see Figure 4(c)).
In order to find the least condition, let us reconsider the imaging of Γ 1 .In this case, the selection of (24) was a good choice.Following this observation, one of the possible choices of  is that, for n(x) = [cos( x ), sin( x )]  , x ∈  2 , select Then, it is expected that identified shape of inhomogeneity will be close to the shape of Γ 2 .Notice that, throughout the numerical computation, Thus, selection of ( 27) is the least condition of the range of incident directions.Figure 6 exhibits maps of E(z; ) for  1 and   in (27) and for  1 = 0 and   = 2 (see Figure 5 for instance).Since these conditions satisfy least condition, the shape of Γ 2 seems retrieved well, and this result supports Theorem 3.
For the final example, let us consider the imaging of two, nonoverlapped thin inhomogeneities Γ 2 and Γ 3 , where the supporting curve  3 of Γ 3 is and  2 =  3 =  2 =  3 = 5.Based on the results in Figure 7, we can conclude that it is hard to recognize the shape of Γ 2 ∪Γ 3 when the range   −  1 < , but we can identify when    −  1 ≈ .With this, we end up this section with the following remark.
Remark 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 to , which is the sufficient condition of range of application.Related results of numerical simulations can be found in [22] also.

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 1 = /2 and  2 =

Figure 1 :
Figure 1: Maps of E(z; ) for Γ 1 .Violet-colored solid line describes the range of incident directions.

Figure 2 : 6 −
Figure 2: Description of best (blue-colored range) and worst (red-colored range) choice of the range of incident directions.

Figure 3 :
Figure 3: Same as Figure 1 except the range of the incident directions.

Figure 4 :Figure 5 :
Figure 4: Maps of E(z; ) for Γ 1 .Violet-colored solid line describes the range of incident directions.