Inverse scattering problem in presence of a conducting cylinder

Jianhua Shen; Xudong Chen; Yu Zhong; Lixin Ran

doi:10.1364/OE.19.010698

1. Introduction

Inverse scattering approach has been of considerable interest to obtain high resolution image of samples [1–10]. To improve image resolution, several imaging modalities introduce auxiliary scatterers to increase the interaction between sample and electromagnetic waves. In total internal reflection tomography, a denser substrate is placed beneath the sample, assisting in providing larger transverse wave vector when the incidence angle is larger than critical angle [7,8,10]. In [3], a grating substrate is placed beneath the sample, where higher Block mode exhibits a significant larger transverse component of wave vector. In [4, 5], a controlled scattering experiment is proposed to directly observe the chiral nature of charge carriers in grapheme layers or the electrons through a grapheme device, where the charge located on a scanning probe microscope (SPM) tip acts as a scattering center with controllable position on the graphene sheet. In [11–13], the knowledge of external information included in the data to process is used to improve the final results. In [6], a perfect conducting cylinder is introduced as an auxiliary scatterer, which helps the linear sampling method (LSM) to determine the boundary of sample. Other methods such as the Born Iterative Method [14] and the other self-consistent methods based on optimizing Frechet derivatives [15] are also widely used in solving the inverse scattering problems.

In this paper, a known perfectly electric conducting (PEC) cylinder is chosen as an auxiliary scatterer and it is placed near the sample. Due to multiple scattering effect, the radius of the conducting cylinder and its distance to sample play an important role in inverse scattering problem. The paper investigates the role of the perfectly electric conductor under different arrangement of transmitting/receiving antennas. In this paper, we use the subspace-based optimization method (SOM) to solve the inverse scattering problem [16–18]. We compare the reconstruction results with and without the conducting cylinder. We also investigate the reconstruction results for different distribution of antennas. It is found that the conducting cylinder located in the vicinity of the scatterers acts as a secondary source, and it may significantly perturb the distribution of the incident field. It is worth utilizing this effect to improve the quality of the reconstructions. Through theoretical analysis and numerical simulations, several observations are found as follows:

The roles of the radius of the PEC cylinder and its distance to samples are numerically investigated. The reconstruction results (both the objective function and relative error of permittivity) are not a monotonic function of either the radius of cylinder or the separation distance;
In some scenarios, by using only partial aperture of antenna array, we are able to design a combination of the cylinder radius and the distance of it to the samples, so that we still obtain a reconstruction result comparable to the situation with full aperture array;
It is worth mentioning that the auxiliary cylinder is not always helpful in the inverse scattering problems. We observe that when antennas are close to the line connecting the PEC cylinder and samples, there is detrimental effect to the reconstruction.

We expect that this paper could shed some light on the design of the inverse scattering experimental system, in which better imaging results can be obtained by using fewer antennas.

2. Forward and inverse scattering problems in presence of a conducting cylinder

In this section, we consider a two-dimensional (2D) inverse scattering problem under the transverse magnetic (TM) time harmonic incidence. In free space background, a PEC circular cylinder is centered at the origin O, and its radius is R. Nonmagnetic scatterers are located in the domain of interest D ⊂ R ². A total number of N _i line sources, which are located at ${\bar{r}}_{p}^{i}$ ,p = 1,2, ...,N _i, illuminate on this domain and the scattered electric field is measured by N _r receiving antennas, which located at ${\bar{r}}_{q}^{'}$ , q = 1, 2, ..., N _r. The problem is to determine the permittivity distribution of the domain ε(r̄), r̄ ∈ D, with a set of N _i N _r scattering data ${\bar{E}}_{p}^{sca} ({\bar{r}}_{q}^{'})$ . In practice, the domain D is usually discretized into many small subunits (compared to the wave length), which are centered at r̄_m, m = 1, 2, …, M.

The analysis in this paper is based on the method of moment (MOM) [19]. Note that MOM is valid for both weak scattering and multiple scattering problems. For the pth incidence, the total incident field upon the mth subunit ${\bar{E}}_{p}^{inc} ({\bar{r}}_{m})$ consists of the direct incident field from the sources and the scattered field from the conducting cylinder, ${\bar{E}}_{p}^{inc} ({\bar{r}}_{m}) = {\bar{E}}_{d, p}^{inc} ({\bar{r}}_{m}, {\bar{r}}_{p}^{i}) + {\bar{E}}_{s, p}^{inc} ({\bar{r}}_{m}, {\bar{r}}_{p}^{i})$ , p = 1,2, ...,N _i, where

{\bar{E}}_{d, p}^{inc} ({\bar{r}}_{m}, {\bar{r}}_{p}^{i}) = - \frac{k η_{0}}{4} H_{0}^{(1)} (k | {\bar{r}}_{m} - {\bar{r}}_{p}^{i} |),

and

{\bar{E}}_{s, p}^{inc} ({\bar{r}}_{m}, {\bar{r}}_{p}^{i}) = - \frac{k η_{0}}{4} \sum_{l = - \infty}^{+ \infty} S_{l} H_{l}^{(1)} (k ρ_{p}^{i}) e^{- i l θ_{p}^{i}} H_{l}^{(1)} (k ρ_{m}) e^{i l θ_{m}},

where k is the free space wave number, η ₀ is the impedance of the free space,

S_{l} = - J_{l} (k R) / H_{l}^{(1)} (k R)

,

(ρ_{p}^{i}, θ_{p}^{i})

represents cylindrical coordinate for

{\bar{r}}_{p}^{i}

, and (ρ_m, θ_m) for r̄_m.

In the process of the discretization of the domain, the dimension of every subunit is much smaller than the wave length and the polarization current (also known as contrast current) on the mth subunit is approximated as a constant within the subunit. With the total electric field ${\bar{E}}_{p}^{tot} ({\bar{r}}_{m})$ in the mth subunit, the polarization current is induced as

{\bar{I}}_{p}^{d} ({\bar{r}}_{m}) = - i ω ε_{0} [ε_{r} ({\bar{r}}_{m}) - 1] {\bar{E}}_{p}^{tot} ({\bar{r}}_{m}),

where ω is the angle frequency of the incident field, ε ₀ is the permittivity of the free space and ε _r (r̄_m) denotes the relative permittivity of the mth subunit. The total electric field upon the mth subunit

{\bar{E}}_{p}^{tot} ({\bar{r}}_{m})

is composed of the total incident field upon this subunit from the sources

{\bar{E}}_{p}^{inc} ({\bar{r}}_{m})

and the scattered field upon this subunit due to the radiation of the induced current in the domain

{\bar{E}}_{p}^{sca} ({\bar{r}}_{m})

, the latter being composed of the directly scattered wave

{\bar{E}}_{d, p}^{sca} ({\bar{r}}_{m})

and indirectly scattered wave via PEC cylinder

{\bar{E}}_{s, p}^{sca} ({\bar{r}}_{m})

. Thus the total field is written as

{\bar{E}}_{p}^{tot} ({\bar{r}}_{m}) = {\bar{E}}_{p}^{inc} ({\bar{r}}_{m}) + {\bar{E}}_{d, p}^{sca} ({\bar{r}}_{m}) + {\bar{E}}_{s, p}^{sca} ({\bar{r}}_{m}),

where

{\bar{E}}_{d, p}^{sca} ({\bar{r}}_{m}) = - \frac{k η_{0}}{4} \sum_{n = 1}^{M} \int_{cell} \int_{n} H_{0}^{(1)} (k | {\bar{r}}_{n} - {\bar{r}}_{m} |) {\bar{I}}_{p}^{d} ({\bar{r}}_{n}) d σ,

{\bar{E}}_{s, p}^{sca} ({\bar{r}}_{m}) = - \frac{k η_{0}}{4} \sum_{n = 1}^{M} \sum_{l = - \infty}^{+ \infty} S_{l} \int_{cell} \int_{n} H_{l}^{(1)} (k ρ_{n}) e^{- i l θ_{n}} {\bar{I}}_{p}^{d} ({\bar{r}}_{n}) d σ H_{l}^{(1)} (k ρ_{m}) e^{i l θ_{m}} .

Similarly, for each incidence, the scattered field measured at each receiving antenna ${\bar{E}}_{q}^{sca} ({\bar{r}}_{q}^{'})$ is written as

{\bar{E}}_{p}^{sca} ({\bar{r}}_{q}^{'}) = {\bar{E}}_{d, p}^{sca} ({\bar{r}}_{q}^{'}) + {\bar{E}}_{s, p}^{sca} ({\bar{r}}_{q}^{'}) .

It is convenient to write Eqs. (3) and (6) in compact form

{\bar{I}}_{p}^{d} = \bar{\bar{ξ}} \cdot ({\bar{E}}_{p}^{inc} + {\bar{\bar{G}}}_{D} \cdot {\bar{I}}_{p}^{d}) = \bar{\bar{ξ}} \cdot [{\bar{E}}_{p}^{inc} + ({\bar{\bar{g}}}_{D} + {\bar{\bar{h}}}_{D}) \cdot {\bar{I}}_{p}^{d}],

where

{\bar{I}}_{p}^{d} = {[{\bar{I}}_{p}^{d} ({\bar{r}}_{1}), {\bar{I}}_{p}^{d} ({\bar{r}}_{2}), \dots, {\bar{I}}_{p}^{d} ({\bar{r}}_{M})]}^{T}

,

{\bar{E}}_{p}^{inc} = {[{\bar{E}}_{p}^{inc} ({\bar{r}}_{1}), {\bar{E}}_{p}^{inc} ({\bar{r}}_{2}), \dots, {\bar{E}}_{p}^{inc} ({\bar{r}}_{M})]}^{T}

and the diagonal matrix ξ̿ consists of ξ_m (which is –iωε ₀ [ε _r (r̄_m) – 1]). The Green function G̿ _D is composed of g̿ _D which denotes the scattering directly into the considered subunit and h̿ _D the scattering via the cylinder. The superscript T denotes the transpose operator. Similarly, Eq. (7) is written as

{\bar{E}}_{p}^{sca} = {\bar{\bar{G}}}_{S} \cdot {\bar{I}}_{p}^{d} = ({\bar{\bar{g}}}_{S} + {\bar{\bar{h}}}_{S}) \cdot {\bar{I}}_{p}^{d}

where

{\bar{E}}_{p}^{sca} = {[{\bar{E}}_{p}^{sca} ({\bar{r}}_{1}), {\bar{E}}_{p}^{sca} ({\bar{r}}_{2}), \dots, {\bar{E}}_{p}^{sca} ({\bar{r}}_{M})]}^{T}

. Similarly, the Green function G̿ _S here is also composed of two parts: g̿ _S and h̿ _S.

To solve inverse scattering problem, we adopt the recently proposed SOM algorithm. The SOM method has good reconstruction quality and is robust against noise. It has been widely used for solving 2D inverse scattering problems in both TM scenario [16] and TE scenario [20], and it is able to deal with many kinds of scatterers, such as isotropic scatterers, anisotropic scatterers [21], and even PEC scatterers [22]. Recently, a 3D SOM was proposed, with reduced computational cost, for solving 3D inverse scattering problems [23]. Moreover, based on the SOM, a new twofold SOM expressed better stability in the inversion and high robustness against noise [18]. The SOM method is of great potential in solving inverse scattering problems.

Following the SOM algorithm proposed in [17], the induced current in the domain D is decomposed into two parts, the deterministic current ${\bar{I}}_{p}^{s}$ and the ambiguous current ${\bar{I}}_{p}^{n}$ . From the singular value decomposition (SVD) of ${\bar{\bar{G}}}_{S} ({\bar{\bar{G}}}_{S} = \sum_{m} {\bar{u}}_{m} σ_{m} {\bar{v}}_{m}^{*})$ , the deterministic part is determined by ${\bar{I}}_{p}^{s} = \sum_{j = 1}^{L} \frac{{\bar{u}}_{j}^{*} \cdot {\bar{E}}_{p}^{sca}}{σ_{j}} {\bar{v}}_{j}$ , and the ambiguous part is constructed as ${\bar{I}}_{p}^{n} = {\bar{\bar{V}}}^{n} \cdot {\bar{α}}_{p}^{n}$ , where L(L < N _r) is determined by the noise level of the scattered field, V̿ ⁿ is composed of the last M − L right singular vectors (v̿_m) and ${\bar{α}}_{p}^{n}$ is an M − L dimensional vector. We follow the criteria described in [17] to choose the value of L. Thus the ε _r (r̄_m) of the scatterer can be determined by minimizing the objective function

f ({\bar{α}}_{1}^{n}, {\bar{α}}_{2}^{n}, \dots, {\bar{α}}_{N_{i}}^{n}, \bar{\bar{ξ}}) = \sum_{p = 1}^{N_{i}} (\frac{{‖ {\bar{\bar{G}}}_{S} \cdot {\bar{\bar{V}}}^{n} \cdot {\bar{α}}_{p}^{n} + {\bar{\bar{G}}}_{S} \cdot {\bar{I}}_{p}^{s} - {\bar{E}}_{p}^{sca} ‖}^{2}}{{‖ {\bar{E}}_{p}^{sca} ‖}^{2}} + \frac{{‖ \bar{\bar{A}} \cdot {\bar{α}}_{p}^{n} - {\bar{B}}_{p} ‖}^{2}}{{‖ {\bar{I}}_{p}^{s} ‖}^{2}})

where A̿ = V̿ ⁿ – ξ̿ · (G̿ _D · V̿ ⁿ) and

{\bar{B}}_{p} = \bar{\bar{ξ}} \cdot ({\bar{E}}_{p}^{inc} + {\bar{\bar{G}}}_{D} \cdot {\bar{I}}_{p}^{s}) - {\bar{I}}_{p}^{s}

. The objective function is minimized by alternatively updating the ᾱⁿ and the ξ̿ using the conjugate-gradient (CG) method [17]. The flowchart of the optimization can be arranged as the following steps:

Step 1: Calculate G̿ _S, G̿ _D and the SVD of G̿ _S. Obtain the parameter L and the deterministic current ${\bar{I}}_{p}^{s}$ .
Step 2: Initial step, n = 0. The initial values ${\bar{α}}_{p, 0}^{n}$ and ξ̿ ₀ are set to zero.
Step 3: n=n+1. Calculate the gradient ${\bar{g}}_{p, n} = \nabla_{{\bar{α}}_{p}^{n}} f$ and then the coefficients ${\bar{α}}_{p, n}^{n}$ can be obtained as done in [17]. Then update new induced current ${\bar{I}}_{p . n}^{d} = {\bar{I}}_{p}^{s} + {\bar{\bar{V}}}^{n} \cdot {\bar{α}}_{p, n}^{n}$ and total field in Eq. (4). The scattering strength ξ̿_n can be obtained by the least square method.
Step 4: If the termination condition is satisfied, stop the iteration. Otherwise, go on to Step 3.

3. Simulation setup

The configuration for the numerical simulations, including the cylinder, the test domain D and the antennas, is illustrated in Fig. 1. Throughout the paper, the unit of the length is chosen as wavelength λ. The cylinder with a radius of R is centered at the origin O. The test domain D under study is a square of 2λ × 2λ and it is centered at (r, 0). The background medium is air. A number of 20 line sources and 40 line receivers are evenly distributed along a circle with a radius 5λ centered at (r, 0). In the following calculations, a 100 × 100 grid mesh is adopted in the forward problem and a 64×64 grid mesh is adopted in the inverse problem in order to avoid inverse crime. An additive white Gaussian noise $\bar{κ}$ _p is added to the scattered field ${\bar{E}}_{p}^{sca}$ , and the sigal-to-noise ratio quantified by $20 log \frac{{‖ {\bar{E}}_{p}^{sca} ‖}_{F}}{{‖ {\bar{κ}}_{p} ‖}_{F}}$ is 20dB [24], where ||·||_F denotes the Frobenius norm of a matrix. The value of L is determined from the spectrum of G̿ _S according to the noise level in the scattered field [17].

Fig. 1 A view to the configuration with the test domain, the conducting cylinder and the distribution of the sources and the receivers.

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In numerical simulations, we use ’Austria’ profile, which consists of two discs and one ring, as illustrated in Fig. 2. The discs with a radius of 0.2λ are centered at (−0.3λ + r, 0.6λ) and (0.3λ + r, 0.6λ) respectively. The ring with an exterior radius of 0.6λ and an inner radius of 0.3λ, is centered at (0λ +r, 0.2λ). The relative permittivities of the scatterers are ε _r = 2. To be noticed, in the following figures the coordinates are all normalized to the center of the domain (r, 0).

Fig. 2 The ’Austria’ profile adopted for the numerical simulations.

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Since different incident angle introduces different field pattern inside the test domain, we are interested in investigating reconstruction results when antennas are distributed in some partial apertures. To be specific, as illustrated in Fig. 1, the measure domain is divided into four regions: region I, region II, region III and region IV. The size of each region is determined by the angle θ marked in Fig. 1. We investigate the influence of the PEC cylinder on image reconstruction when antennas are distributed only in some of the regions.

4. Numerical results

This section presents some numerical results to evaluate the influence of a PEC circular cylinder that is placed around the test domain D. In view of the configuration of the simulation setup, there are mainly three controllable parameters relating to the quality of numerical results: the radius of the cylinder, the distance between the cylinder and the obstacles, and the distribution of the antennas. To investigate the performance of different combinations of the aforementioned parameters, we consider three cases: Case I, based on the description in Fig. 1, there are 20 transmitters and 40 receivers, as introduced in Section 3. The radius of the cylinder R varies from 0 to 1λ with increment 0.1λ and the distance r varies from 2.1λ to 3.9λ with increment 0.05λ, which yields 11 × 37 sets of scattering data; Case II, based on the description in Case I, we choose θ to be 90°, and those antennas in region III are removed; Case III, based on the description in Case I, we choose θ to be 135°, and those antennas in region I and III are removed.

In order to evaluate the quality of the reconstructed profiles, we introduce an error function E ^2D that indicates the difference between the reconstructed permittivity distribution ${\bar{\bar{ε}}}_{m, n}^{sim}$ and the actual permittivity distribution ${\bar{\bar{ε}}}_{m, n}^{act}$ :

E^{2 D} = \sqrt{\frac{1}{n u m} \sum_{m = 1}^{64} \sum_{n = 1}^{64} {| \frac{{\bar{\bar{ε}}}_{m, n}^{act} - {\bar{\bar{ε}}}_{m, n}^{sim}}{{\bar{ε}}_{m, n}^{act}} |}^{2}},

where num is the total number of meshes of the test domain that is equal to 64 × 64 here.

For case I, when the antennas are evenly distributed on a circle, the error of the reconstructed results is illustrated in Fig. 3. Three important points can be concluded from the figure: (1) For all the combinations of the aforementioned parameters, none yields a better result than the non-cylinder case (the last row of Fig. 3 where the radius of the cylinder R = 0); (2) Most of combinations are only slightly inferior to the case without the conducting cylinder. For example, for a cylinder of a radius 0.5λ placed at a distance of 3.15λ, the result is very similar with the non-cylinder case as illustrated in Fig. 4. The dash line in the figure is the real shape of the scatterer for a clear comparison. However, when the cylinder is nearer and larger, the reconstruction tends to get intolerantly worse; (3) The reconstruction result is not a monotonic function of either the radius of cylinder or the separation distance.

Fig. 3 The simulated errors of the reconstruction results for the combinations described in case I.

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Fig. 4 The simulation results when the antennas are distributed as described in case I. (a) is for the combination of R = 0.5λ,r = 3.15λ and (b) is for the case with no cylinder.

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For case II, the antennas that are behind the cylinder, mainly distributed in region III, are moved away. It is predicable that the reconstruction results become inferior compared with Case I due to the partial illumination and the smaller aperture of the measurement. The error of the reconstruction results for case II is illustrated in Fig. 5. It is observed that for a PEC cylinder of small radius, the reconstruction results are only slightly different from the case without the conducting cylinder. Under the effect of the auxiliary cylinder with radius of 0.5λ placed at a distance of 2.1λ, the reconstruction results are slightly better than the non-cylinder case as shown in Fig. 6.

Fig. 5 The simulated errors of the reconstruction results for the combinations described in case II.

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Fig. 6 The simulation results when the antennas are distributed as described in case II. (a) is for the combination of R = 0.3λ,r = 2.1λ and (b) is for the case with no cylinder.

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For case III, we move away the antennas that are nearby the line connecting the PEC cylinder and the sample, i.e., in regions I and III. For comparison, the angle θ is broadened to 135°, with 16 transmitters and 32 receivers left that are of the same quantity and the same size of the aperture (270°) as those in case II. The error of the reconstruction results for case III is illustrated in Fig. 7. It is observed that most of combinations of the radius of PEC cylinder and the separation yield better reconstruction results than the case without the conducting cylinder. Moreover, compared to the results of case II, the errors of case III are significantly smaller. Although cases II and III use the same number of transmitters/receivers, the case III removes 45° aperture in both regions I and III. It is interesting to observe that, as illustrated in Fig. 8, when we chose a combination of the radius of 0.5λ and a distance of 2.1λ, the reconstruction result is much better than the case without the conducting cylinder and it is even comparable with the full-aperture non-cylinder case as shown in Fig. 4.

Fig. 7 The simulated errors of the reconstruction results for the combinations described in case III.

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Fig. 8 The simulation results when the antennas are distributed as described in case III. (a) is for the combination of R = 0.5λ,r = 2.1λ and (b) is for the case with no cylinder.

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The influence of the PEC cylinder to the scattering problem is also effective to other scatteres. Under the same establishment in case III, we change the scatters by two discs with a radius of 0.35λ centered at the position (r,0.4λ) and (r,–0.4λ) separately, which has a distance of 0.1λ to each other. The reconstruction result when the distance r is 2.1λ and the radius R is 0.3λ is illustrated in Fig. 9(a) and the reconstruction result for the case without the cylinder is illustrated in Fig. 9(b). In numerical value, the error of the result with the cylinder is 0.1390 and the error without the cylinder is 0.1520.

Fig. 9 The simulation results of new scatterers when the antennas are distributed as described in case III. (a) is for the combination of R = 0.3λ,r = 2.1λ and (b) is for the case with no cylinder.

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Considering the simulation results above, we see that a properly chosen PEC cylinder is helpful for the inverse scattering problems to improve the quality of the image and in the meanwhile to reduce the number of the antennas. This conclusion is also applicable to other auxiliary scatterers with different shape or materials. The multiple scattering between auxiliary scatterer and sample may be either constructive or destructive, and consequently the insertion of auxiliary scatterer is not guaranteed to improve reconstruction results. Form Fig. 3, Fig. 5, and Fig. 7, we observe that the best distance r is neither very far nor very near to the sample. We observe the tendency that the best r moves farer away when the size of the cylinder expands.

5. Conclusion

This paper investigates the influence of an auxiliary perfectly conducting cylinder in solving the inverse scattering problems with the subspace-based optimization method. Under different arrangements of transmitting/receiving antennas, we investigate the role of the radius of the PEC cylinder and its distance to sample. The multiple scattering between the PEC cylinder and the sample may be constructive or destructive, and insertion of auxiliary PEC cylinder is not guaranteed to improve reconstruction results. However, there are scenarios where a proper chosen combination of the radius of the PEC cylinder and its distance to sample yields better reconstruction results than the case without the conducting cylinder. In numerical simulations, we showed that by properly arranging antennas, a partial-aperture distribution of antennas may even produce comparable reconstruction results that obtained from full-aperture antennas, which is of great practical value.

Acknowledgments

This work was supported by the Singapore Temasek Defence Systems Institute under grant TDSI/09C001/1A and NSFC (No. 61071063).

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Inverse scattering problem in presence of a conducting cylinder

Abstract

1. Introduction

2. Forward and inverse scattering problems in presence of a conducting cylinder

3. Simulation setup

4. Numerical results

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Equations (11)

Optics Express