Fast shape reconstruction of perfectly conducting cracks by using a multi-frequency topological derivative strategy

Fast shape reconstruction of perfectly conducting cracks by using a multi-frequency topological derivative strategy


This paper concerns a fast, one-step iterative technique of imaging extended perfectly conducting cracks with Dirichlet boundary condition. In order to reconstruct the shape of cracks from scattered field data measured at the boundary, we introduce a topological derivative-based electromagnetic imaging function operated at several nonzero frequencies. The properties of the imaging function are carefully analyzed for the configurations of both symmetric and non-symmetric incident field directions. This analysis explains why the application of incident fields with symmetric direction operated at multiple frequencies guarantees a successful reconstruction. Various numerical simulations with noise-corrupted data are conducted to assess the performance, effectiveness, robustness, and limitations of the proposed technique.

Perfectly conducting cracks, Topological derivative, Multiple frequencies, Numerical experiments


1 Introduction

One goal of the inverse scattering problem is to find the shape of bulk or flaw in a medium using the scattered field data measured at the boundary. This is an old but it has several of applications such as detection of cracks in material engineering, ultrasound imaging in medical sciences, and scanning anti-personnel mines hidden in the ground in military services. Related researches may be found in (3); (7); (16); (17); (18); (37) and the references therein. In many studies, various shape reconstruction algorithms have been developed, most of which are focused on the minimization of the least-square functional by using a Newton-type iteration strategy, e.g., level-set method (2); (18); (35). The merits of the iteration strategy is that it does not require a very large number of boundary measurements for complete shape reconstruction; instead, in order to guarantee successful reconstruction, it demands very high computational expenditures, optimal regularization terms related to the problems at hand, calculation of a complex Fréchet derivative at each iteration, and a priori information of the unknown target to be reconstructed. Nevertheless, even if these conditions are satisfied, the iteration strategy must begin with an initial guess that is close to the true one in order to avoid undesirable situations such as non-convergence or falling into the local minima. Therefore, development of a mathematical theory and an algorithm for generating a good initial guess is an important research topic.

Conversely, certain non-iterative shape reconstruction algorithms have been proposed; for example, MUltiple SIgnal Classification (MUSIC) algorithm (6); (10); (32); (34), linear sampling method (15); (21), Kirchhoff migration (6); (28); (29); (33), and Fourier inversion based one (8); (9); (11). In contrast with the iterative strategy, these algorithms require a large amount of incident field data and a significant number of boundary measurements. However, if these conditions are satisfied, non-iterative shape reconstruction algorithms can prove to be effective, fast, and easy to extend to multiple targets.

Topological derivative concept is a non-iterative strategy. This concept was originally developed for the shape optimization problem, but its application to rapid shape reconstruction has only recently been proven. Related works can be found in (5); (10); (12); (13); (14); (24); (25); (27); (30); (31); (38) and references therein. One of the advantages of topological derivative concept is that it does not require a large amount of many incident field data; however, a reduction in the amount of this data has been reported to result in poor resolution of the reconstructed shape.

The purpose of this literature is to establish an effective and fast reconstruction algorithm for detecting both the location and shape of perfectly conducting crack(s) from boundary-measured scattered field data by using only a small amount of incident field data. Based on the structure of topological derivative imaging functional operated at a fixed-frequency (24) and multi-frequency imaging schemes (6); (26), we introduce a multi-frequency topological derivative concept. Note that recent work ((4), Formula (1.33)) has briefly introduced a multi-frequency topological derivative but through a more detailed analysis, we investigate the structure of the proposed multi-frequency topological derivative and its benefits over the traditional topological derivative.

The remainder of this paper is organized as follows. Section 2 briefly introduces the traditional topological derivative by creating a linear, narrow crack. Section 3 describes the design of a multi-frequency topological derivative imaging function and analyzes it to investigate its properties. Section 4 presents various numerical experiments to assess the performance of the proposed imaging function under various situations. Section 5 contains a short conclusion and some remarks on future work.

2 Topological derivative: an inspection

Assume that a perfectly conducting crack is completely hidden in a homogeneous domain with a smooth boundary . We denote as a two-dimensional vector that lies on crack .

Throughout this paper, we consider the so-called Transverse Magnetic (TM) polarization case, letting be the (single-component) electric field that satisfies the following boundary value problem:

where represents the unit outward normal to and , , denotes the two-dimensional vector on the unit circle . Throughout this paper, we let denote a strictly positive wavenumber and presume to not be an eigenvalue of (2). Similarly, let be the solution of equation (2) without . Then, the problem we consider here is the computation of the topological derivative of the residual function depending on the solution :


where .

In order to compute the topological derivative, let us create a small linear crack of length centered at such that

and denote as that domain. Then, because of the change in the topology of , we can consider the topological derivative based on the residual function with respect to point as follows:


where as . From relationship (2), we have the following asymptotic expansion:

Then, the topological derivative with different incident waves at a given wavenumber is as follows (see (24) for its derivation).

Lemma 2.1 (Topological derivative)

Let denote the real part of . Then, the topological derivative corresponding to (1) is given by


where satisfies the following adjoint problem:

3 Multi-frequency topological derivative: introduction and analysis

In this section, we introduce a multi-frequency topological derivative imaging function and explore its properties. Based on recent research (5); (10); (30); (31), a sufficiently large number of incident directions are required to obtain a good result. This means that if is small, an image with poor resolution will be reconstructed. Moreover, when boundary measurement data are affected by a large amount of random noise, (3) will fail to reconstruct a good image. Inspired by multi-frequency imaging algorithms (6); (28); (29); (33), we consider the following multi-frequency topological derivative for several wavenumbers :


where is given by (3) for , .

In this section, we investigate the structure of (4). For this purpose, we recall a useful result (see (24)) as follows.

Lemma 3.2

Let imply that there exists a constant such that . Then, for and , (3) satisfies


Here denotes the two-dimensional time-harmonic fundamental solution (or Green’s function) of Helmholtz equation

where , , and denote the Hankel function of the first kind, Bessel function, and Neumann function of order zero, respectively.

Consequently, (4) is analyzed for the following three cases of interest.

Theorem 3.3 (Case 1: Symmetric incident directions)

Under the configuration of a small number of symmetric incident directions, the structure of (4) is

for a sufficiently large and .


[Proof.] Assume that the symmetric incident directions are of the form:

where is an even number, say, . Based on this form, satisfies for .

In contrast with Lemma 3.2, we assume that is not sufficiently large. Let denote the imaginary part of then since

(5) becomes


Let us recall that for (see ((1), formula 11.4.37 (p. 487)))

Let , , and . Then since


by letting , (4) becomes

Theorem 3.4 (Case 2: Non-symmetric incident directions)

Under the configuration of a small number of non-symmetric incident directions, the structure of (4) becomes

for a sufficiently large and .


[Proof.] In constrast with Theorem 3.3, we assume that the incident directions are non-symmetric. Then, (5) can be written as

and therefore

Let us recall an integral formula in ((1), formula 11.4.40 (p. 487)),

Then, if , we can evaluate


Hence, by combining (6), (7), and (8), we can obtain

Theorem 3.5 (Case 3: Sufficiently large number of incident directions)

Under the configuration of a sufficiently large number of incident directions, the structure of (4) becomes

for an adequately large . Here, is defined as


and denotes the Dirac delta function.


[Proof.]Suppose that is large enough, i.e., , then by ((19), Lemma 4.1), (5) can be approximated as

Assume that is large enough and that . Then applying an integral formula of the Bessel function ((36), page 35)

and a change of variable yields


where is given in (10). Note that the term

can be negligible since and , refer to (20).

Finally, by applying in (11), we can immediately obtain the following result:

Remark 3.6

Based on structures (7), (9), and (11), we can summarize some properties of (4) as follows:

  1. Formulas (7) and (9) reach their maximum value at that satisfies

    Therefore, at , the map of (4) will yield that the location of can be accurately determined. However, at point such that

    the map of (4) will show a large magnitude such that unexpected replicas will appear (see Figure 1).

  2. If satisfies

    then both the numerator and denominator of (9) become . Hence, at this point, (9) will yield a large value at not only but also other locations in . Therefore, the configuration of symmetric incident directions will yield a better imaging result (see Figure 5).

  3. If the number of incident directions is increased to as many as possible, i.e., , it is clear that a more accurate location of can be obtained. In this case, an odd number of will not affect the imaging performance significantly.

  4. Application of multiple frequencies enhances the imaging performance.

Figure 1: Illustration of Remark 3.6. At the location such that (black colored vector) and (green colored vector), true shape of and ghost replicas will appear, respectively.

4 Numerical experiments

In this section, we exhibit some numerical examples to demonstrate the effectiveness of the proposed algorithm. Throughout this section, the homogeneous domain is chosen as the interior region of the two-dimensional unit disk centered at the origin, i.e.,

The adopted wavenumber has the form for ; here, is the given wavelength that is equi-distributed between and . Based on Theorem 3.3, the incident direction is selected as

for an even number .

Motivated by (22), we choose two single cracks and as

and inspired by (34), we choose multiple cracks as

Note that in order to show the robustness of the proposed algorithm, a white Gaussian noise with dB signal-to-noise ratio (SNR) is added to the unperturbed data.

First, let us consider the imaging result of . The left-hand side of Figure 2 shows the map of for different directions. In this result, because of its small magnitude at , appears divided. Moreover, owing to the large magnitude at certain points (for example, ), we cannot recognize its shape at this stage. The right-hand side of Figure 2 is the map of . The figure shows that there is undivided or whole ; this means that we obtained a more accurate result than the previous one. However, because certain points have a large magnitude, we cannot determine the true shape of .

Figure 2: Map of with (left) and (right) for .

Figure 3 shows the map of for crack . In contrast with the result of , it is difficult to identify the shape of the crack because of the great number of points with large magnitudes. Fortunately, similar to the results in Figure 2, map of yields a better image than that obtained by the map of .

Figure 3: Map of with (left) and (right) for .

The application of multiple cracks is shown in Figure 4. This figure shows the maps of and . Throughout these results, we can observe that the map of yields a poor result even if the number of incident directions is large; conversely, the multi-frequency imaging function yields an accurate result even if is small.

Figure 4: Maps of with (top, left) and (top, right), and maps of with (bottom, left) and (bottom, right) for .

In Figure 5, maps of with number of incident directions are displayed for cracks and in order to verify the (P2) of Remark 3.6. By comparing Figures 2 and 4, it is difficult to discern the shape of true crack(s) because of the appearance of unforseen replicas with large magnitudes. Note that if is an odd number but is sufficiently large enough, the map of yield a good result, refer to Figure 6.

Figure 5: Map of with for (left) and (right).
Figure 6: Map of with for (left) and (right).

At this moment, we increase the number of applied frequencies and incident directions . Figures 7, 8, and 9 depict the map of when and or . As we expected in (P3) of Remark 3.6, the results yielded by improve with increasing number of and/or .

Figure 7: Map of with for (left) and (right).
Figure 8: Map of with for (left) and (right).
Figure 9: Map of with (left) and (right) for .
Remark 4.7 (Limitation of the multi-frequency topological derivative)

Although multi-frequency topological derivative imaging yields very good imaging results for not only single but also multiple cracks, it still has the following limitations.

  1. Let us consider the imaging of a crack of a large curvature near one of the crack tips. To illustrate this, we choose a crack from (23):

    From the results in Figure 10, the imaging results yielded by are satisfactory. However, at the point of the large curvature, the map of reconstructs a slightly different shape of the crack.

  2. We apply the proposed algorithm for imaging an oscillation crack represented as follows:

    Figure 11 shows the map of . The obtained result by the proposed algorithm does not improve even when the number of and are increased. The proposed algorithm requires further optiization.

Figure 10: Map of for with (top, left), (top, right), (bottom, left) and true shape (bottom, right).
Figure 11: Same as Figure 10 except the crack is .

5 Concluding remarks

We improved the traditional topological derivative concept to determine location and reconstruct the shape of arbitrary-shaped curve-like perfectly conducting cracks. For this purpose, we designed a reconstruction algorithm based on the multi-frequency topological derivative, analyzed its structure, and investigated its properties. We confirmed that the multi-frequency topological derivative contains singularity on the cracks, and therefore it reaches its maximum value at the location of the cracks. Moreover, we demonstrated that why ghost replicas appeared in the neighborhood of cracks and why the configuration of symmetric incident directions yields better results than the non-symmetric configuration. Numerical simulations under various situations depicted both the benefits and limitations of the proposed algorithm. Because the algorithm requires only a one-step iteration procedure, it is very fast in terms of shape identification but does not yield the complete shape of cracks. However, by adopting the obtained result as a starting point of a Newton-type based algorithm (22), it is expected that complete shape reconstruction can be successfully accomplished.

This paper deals with the reconstruction of perfectly conducting cracks with Dirichlet boundary condition. Accordingly, the extension of this research to the crack with Neumann boundary condition will be an interesting work. We believe that applying higher-order terms in the asymptotic expansion formula (9) give a higher order topological derivative (13). The calculation and analysis of higher-order topological derivative will be a valuable research topic. According to the Statistical Hypothesis Testing (6), multi-frequencies are expected to enhance the imaging performance. The corresponding work will include a careful stability and resolution analysis of the multi-frequency topological derivative.


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