Reciprocity gap method for an interior inverse scattering problem

1 Introduction

Inverse scattering problems have wide applications such as radar, medical imaging, geophysical explorations, etc. Most of them are exterior inverse problems due to the fact that both incident fields and measurements are outside the target. In this paper we consider the interior inverse scattering problem of determining the shape of a cavity. In contrast to the exterior case, the scattering object is illuminated by incident waves which are generated by dipole sources from interior of the cavity. The measurements are also distributed inside the cavity. The study of such problems is necessary for non-destructive testing in industrial applications such as monitoring the structural integrity of the fusion reactor [8].

The interior inverse scattering problem has similar mathematical background as exterior problem. However, it has its own characteristics since now the scattered fields are “trapped” inside the cavity. This new topic has received considerable interests recently. In [8], Jakubik and Potthast used the solutions of the Cauchy problem by potential methods and the range test to study the integrity of the boundary of some cavity by acoustics. In [16, 17], Qin and Colton applied the linear sampling method to reconstruct the shape of cavities with sound-soft boundary condition and the surface impedance boundary condition, respectively. Recently, the linear sampling method was used to deal with the interior inverse scattering problem for partially coated cavity and penetrable cavity by Hu, Cakoni and Liu [7], Qin and Liu [18] and Cakoni, Colton and Meng [1]. In addition, the factorization method [10, 11] and near-field imaging method [9] were applied to determine the shape of cavities. For the case of one point source and several measurements, nonlinear integral equations [15] and a decomposition method [20] have been used. The electromagnetic case for a PEC cavity was considered in [19] using the linear sampling method.

In this paper, we study the determination of the shape of a cavity using the reciprocity gap method due to Colton and Haddar [2], which previously applied to a wide range of applications for exterior inverse scattering problems (cf. [6, 13, 12]). This method is well suited to the case when there is a lack of information of the physical properties of the scattering object, and we will show in this paper that it works well for the case when the innermost property of the cavity is unknown. Furthermore, there is no need to consider the background Green’s functions which may penalize the efficiency or are not even known.

The rest of our paper is organized as the following. In Section 2, we consider the interior scattering problem of an inhomogeneous cavity with Dirichlet boundary condition. In Section 3, a reciprocity gap method based on a linear integral equation is introduced. We provide some preliminary numerical examples to show the viability of the method in Section 4. Finally in Section 5, we make conclusions and discuss some future works.

2 The scattering problem for an inhomogeneous cavity

We consider the inverse scattering problem of the cavity D with sound-soft boundary condition. In precise, let D be a simply connected bounded Lipschitz domain in ℝ2 and B be a region inside D which is a piecewise inhomogeneous medium, i.e., the index of refractive n⁢(x) is piecewise continuous. Outside B, the index of refraction is constant, which can be complex for absorbing media. Let k be the wavenumber. If ui is the Green’s function with point source x0 on a smooth curve C contained in D\B¯, we can formulate the direct scattering problem of finding a solution u∈H1⁢(D\{x0}) of

(2.1)

For consistency and simplicity, we assume n⁢(x)=1 in ℝ2\B¯ for the rest of the paper. Let ν be the unit normal to the indicated curve directed outward to the region bounded by the same curve. Furthermore, we assume that B has finitely many components, and the curves across which n⁢(x) is discontinuous are piecewise smooth. Note that us is the scattered field due to the scattering of the incident field ui by D (we could also have written the scattering problem (2.1) with ui replaced by the free space Green’s function). We have chosen the formulation (2.1) in order to simplify the analysis which follows. We note that ui can be written in the form

for x≠x0, where

Here H0(1) is a Hankel function of the first kind of order zero. We remark that 𝔾 is the solution to

where r=|x|. And ui and us satisfy the reciprocity relations

for x and x0 in D\B¯. If k2 is not a generalized Dirichlet eigenvalue (see Definition 1 below) of negative Laplacian in D, the well-posedness of (2.1) is known [3, 4].

Figure 1

Explicative picture. The cavity is denoted by D. The domains Dc and Ω are contained in D and Dc⊂Ω. Point sources and measurements are distributed on the boundaries C and ∂⁡Ω, respectively.

The corresponding exterior boundary value problem is to find u∈Hloc1⁢(ℝ2\D¯) satisfying

(2.2)

where f∈H1/2⁢(∂⁡D). It was shown in [4] that (2.2) has a unique solution and the solution depends continuously on f.

Now let Ω be a bounded Lipschitz domain in D such that Dc⊂Ω⊂D (see Figure 1), where Dc is the interior of C. The inverse scattering problem we are interested in is to determine the shape of the scattering object from the knowledge of Cauchy data of the total field u on ∂⁡Ω without a prior knowledge of the innermost physical properties of the object.

Now we show the uniqueness theorem for the inverse scattering problem. Since the boundary ∂⁡D of the object is assumed to be Lipschitz class and therefore, following the ideas of Qin and Liu [18], we use weak solutions instead of classical solutions and arrive at a contradiction using the limit in H1/2 norm.

Note that the requirement that k2 is not a generalized Dirichlet eigenvalue in Dc or Ω is non-essential since we have the freedom to choose C and ∂⁡Ω.

3 The reciprocity gap functional

Denote by ℍ⁢(ℝ2\Ω¯) the set

and by U the set of solutions to (2.1) for all x0∈C. For v∈ℍ⁢(ℝ2\Ω¯) and u∈U we define the reciprocity gap functional by

where ν is the unit outward normal to ∂⁡Ω. The functional ℛ⁢(u,v) can be viewed as an operator

given by

for all point sources x0∈C since u depends on x0. In the following, we show that the operator R is injective and has a dense range if k2 satisfies a prior assumption.

In the following, we set v to be a single layer potential defined by (3.4) and we have that vg∈ℍ⁢(ℝ\Ω¯). Our aim is to find an (approximate) solution g∈L2⁢(C) to

where Φz:=Φ⁢(⋅,z) for z in the exterior of Ω. In particular, we will show how such a function g can be used to characterize ∂⁡D.

We note that in the special case of a homogeneous cavity, i.e., n⁢(x)≡1 in D and U is the set of solutions to (2.1) with ui=𝔾⁢(x,x0) replaced by the free space Green’s function ui=Φ⁢(x,x0) for x0∈C. Equation (3.5) leads to the ill-posed equation for the linear sampling method. In particular, the set of solutions can be parameterized by x0 and we have for u⁢(⋅,x0)∈U,

and

Therefore, it follows from (3.5) that

which is the near field equation of the linear sampling method [16].

In general, the integral equation (3.5) has no solution. However it is possible to prove the existence of an approximate solution if k2 is not a Dirichlet eigenvalue in Dc. We now proceed to the main result of this paper.

4 Numerical examples

In this section, we provide some numerical examples. We choose the wave number k=2 for all examples. The forward problems are computed using a linear finite element method. The sizes of triangulations of forward problems are fine enough such that the numerical approximation error can be ignored.

Equation (3.5) can be written in the form

where 𝒜:L2⁢(C)→L2⁢(C) is the integral operator with kernel A:C×C→ℂ defined by

and

From Theorem 3 we expect that, as in the case of the linear sampling method [16], for g⁢(⋅,z) being the solution to (4.1) where 𝒜 is replaced by a regularized operator, ∥g⁢(⋅,z)∥L2⁢(C) should be large (depending on the size of the regularization parameter) for z∈D\Ω¯ and bounded for z∈ℝ2\D¯. In particular, we use the Tikhonov regularization method for (4.1). The regularization parameter is chosen by Morozov’s discrepancy principle.

We consider four cavities. The first cavity is a square given by (-3,3)×(-3,3). The second is a triangle whose vertices are

The third one is an ellipse given by

The fourth is a kite given by

We use 40 point sources uniformly distributed on C, a circle with radius 0.7. Inside C, we choose a disk with radius 0.5 as B, an inhomogeneous medium. The measurements are 40 points uniformly distributed on ∂⁡Ω, a circle with radius 1.0. We add 3% noises to the data and employ the reciprocity gap method to reconstruct the cavities. Note that the reconstruction does not improve if more measurement points are used. In addition, it is not necessary to have the same number of measurement and source points.

It is clear that the boundary of the cavity is outside the measurement curve. Hence we choose a sampling region to be a domain outside the circle with radius 1.1. In all examples, we choose the sampling region to be

with 1681 (41×41) uniformly distributed sampling points.

We first consider the case that there is no inhomogeneous medium inside C. In Figure 2, we show the reconstructions of the four cavities. To get a better visualization, we take the indicator function as

at z in the sampling region. The solid lines are the exact boundaries.

Figure 2

The contour plots of the indicator functions and the exact boundaries (solid line).

Next we consider the case when there is an inhomogeneous medium inside C. This is the case the linear sampling method cannot be employed directly and background Green’s functions are necessary. In fact, this is the case the reciprocity gap method is advantageous over the linear sampling method. The medium B is a disk with radius 0.5 with index of refraction n=4 . In Figure 3, we show the reconstructions using contour plots of the indicator function.

Figure 3

The contour plots of the indicator functions and the exact boundaries (solid line). The index of refraction of medium in B is n=4.

We repeat the numerical experiments when the medium is absorbing. The index of refraction is set to be n=4+i and all other parameters keep unchanged. The reconstructions are shown in Figure 4.

Considering the wavelength of the incident wave, λ=2⁢πk≈3.14, and the sizes of the cavities, all the above examples give reasonable reconstructions of the position and the shape of the target.

As the last example, we consider a case when the cavity is not at the center of the sampling region. We use the same triangle as above but shift its center to (3,2). The sampling region is now

The reconstruction is shown in Figure 5. It can be seen that, when cavity is not centered, we can still obtain the correct location and rough shape.

5 Conclusions and future work

In this paper, an interior inverse scattering problem for impenetrable cavity with inhomogeneous medium is considered. We prove an uniqueness theorem of the inverse problem. Then we employ the reciprocity gap method to reconstruct the shape of the cavity. Numerical examples are provided to show the viability of the method.

Figure 4

The contour plots of the indicator functions and the exact boundaries (solid line). The medium in B is absorbing (n=4+i).

Figure 5

The contour plot of the indicator function and the exact boundary of the triangle. The index of refraction of medium in B is n=4.

In general, qualitative methods such as the reciprocity gap method performs well over an interval of wave numbers. However, our numerical experience indicates that this admissible interval for the interior inverse scattering problems is significantly smaller than the exterior inverse scattering problems. This is an indication that the “trapped” scattered fields inside the cavity indeed bring extra difficulty for inverse problems. We plan to extend the reciprocity gap method to penetrable cavities with inhomogeneous medium which is currently under our consideration.