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In many applications of microwave imaging, there is the need of confining the device in order to shield it from environmental noise as well as to host the targets and the medium used for impedance matching purposes. For instance, in MWI for biomedical diagnostics a coupling medium is typically adopted to improve the penetration of the probing wave into the tissues. From the point of view of quantitative imaging procedures, that is aimed at retrieving the values of the complex permittivity in the domain under test, the presence of a confining structure entails an increase of complexity of the underlying modelling. This entails a further difficulty in achieving real-time imaging results, which are obviously of interest in practice. To address this challenge, we propose the application of a recently proposed inversion method that, making use of a suitable preprocessing of the data and a scenario-oriented field approximation, allows obtaining quantitative imaging results by means of quasi-real-time linear inversion, in a range of cases which is much broader than usual linearized approximations. The assessment of the method is carried out in the scalar 2D configuration and taking into account enclosures of different shapes and, to show the method’s flexibility different shapes, embedding nonweak targets.

Microwave imaging (MWI) is an emerging and appealing technique to retrieve morphological and quantitative maps of the electromagnetic properties of notaccessible domains. The physical phenomenon which MWI is based on is the electromagnetic scattering, which is due to differences in the electromagnetic properties of the targets with respect to those of the background. Based on this principle, there is a number of applications in which MWI is finding place, ranging from civil engineering to cultural heritage and archaeology, from security to medical diagnostics [

Owing to such a remarkably broad range of applications, several efforts have been carried out and are ongoing in the design of accurate imaging devices and in the development of reliable strategies for the processing of the data. In particular, these latter represent a challenging task, as they have to cope with the nonlinearity and ill-posedness of the inverse problem underlying MWI.

Among the various applications, one that has gained a growing interest is that of biomedical diagnostics, wherein the interest in MWI is mainly motivated by the nonionizing nature of microwave radiations and their potential of providing quantitative images through relatively low cost devices.

In such a framework, microwave imaging systems are typically confined, that is, delimited by a metallic or dielectric casing. The role of such a casing is twofold. On the one hand, it is meant to isolate the apparatus from external interference. On the other, it has to contain the coupling medium (usually liquid) in which the target of the diagnostic survey is embedded and which is adopted to facilitate the penetration of the probing wave inside the inspected tissues and maximize their interaction. Some examples of such systems are the system for breast cancer imaging developed at Dartmouth College [

On the other hand, the presence of a confining structure implies a considerable increase in the complexity of the interaction between the field and the target. This entails that, in the development of any inversion strategy of the data, an adequate electromagnetic modeling of the actual system is necessary.

While canonical cases (such as a cylindrical circular cross section embedding) can be accounted for by means of analytic tools [

While the abovementioned approaches are certainly effective ways to tackle the inverse scattering problem in a noncanonical configuration, such as the one in which the system is enclosed into a shielding, they have a drawback related to their nonnegligible computational cost deriving from their iterative nature. This circumstance prevents them from providing real-time or quasi-real-time imaging results. In addition, as they tackle the nonlinear problem through a local optimization scheme, they are prone to the occurrence of false solutions, which is an obviously detrimental outcome in any imaging application, but it is even worse in medical applications.

On the downside, approaches based on the linearization arising from the Born approximation (BA) [

With respect to such a scenario, this paper presents the application of a recently proposed inversion method [

As a consequence of the above, the adopted method allows tackling the inversion task in a computationally inexpensive fashion (in quasi-real-time) and free from false solutions, thus overcoming the aforementioned drawbacks. On the other side, the proposed method still relies on an approximation, so that its validity is in any case limited as compared to the full-wave iterative inversion schemes recalled above [

The paper is structured as follows. Section

Let us consider the canonical 2D scalar inverse scattering problem in a PEC enclosure.

Let

The scatterers

By assuming the TM polarization for the electric field, the equations governing the scattering phenomenon in the considered geometry can be expressed in an integral form as

In free space, that is, in absence of any enclosure, such Green’s function is

For noncanonical casing, instead, Green’s function is not known in closed form, so its numerical evaluation is needed.

The inverse scattering problem at hand is then cast as retrieving the unknown contrast

The LSM belongs to the class of qualitative reconstruction methods, as it provides an estimate of the targets support, but not its electric properties. The LSM requires solving an auxiliary linear inverse problem rather than the nonlinear one formulated through (

Due to compactness of

The estimated support is achieved by evaluating the energy (i.e., the

The explicit expression of the LSM indicator reads

In order to better understand how the quantitative LSM (QLSM) works, let us note that

by construction,

given the linearity of the relationship between incident fields and scattered ones,

in those points

These simple observations have an interesting implication. As a matter of fact, as long as the LSM equation (

This concept can be exploited to cast a new approximation of the relationship between the contrast and the scattered fields. In particular, one can assume that when using

By relying on the above concepts, the data-to-unknown relationship can be recast as

With respect to the nature of the introduced approximation, it is worth remarking that the total field given in (

In this subsection, we recall the inversion method which relies on the LSM based approximation introduced in the previous subsection. To this end, let us first note that

to achieve the new data equation (

by considering different sampling points for which the LSM equation is properly solved, it is possible to devise several “virtual” experiments, so as to rearrange the available multiview data into

Let us now assume that the original multiview multistatic configuration consists of

The first step of the proposed method consists in the application of the LSM to the

Then, we select a subset of

In (

By denoting with

The truncation index

In this section we present some numerical examples in order to assess the performance of the QLSM, recalled in previous section, in noncanonical embedding systems. The case is supposed to be a perfectly electric conductor (PEC), filled by a matching medium whose electromagnetic features are

The transmitter/receiver antennas are infinite-length filamentary sources located inside the cavity. When an antenna is transmitting, all the others are in receiving mode.

Figure

Considered measurement configurations: ((a), (b), and (c)) measurement configurations with probes on a circle located around the imaging domain (configuration A); ((d), (e)) measurement configurations with probes on lines parallel to the edges of the PEC at a distance equal to

It is important to underline that the number of probes has been chosen by relying on the degrees of freedom theory [

In each example, two different discretization grids for the generation of data and for the inversion procedure have been adopted, so as to avoid the so-called inverse crime. In particular, the data and Green’s functions were generated by using a nodal grid, produced by means of the mesh generator GiD, which is available online [

All numerical examples have been performed on a standard laptop with a 2.20 GHz Intel Core Duo CPU, 4 GB RAM, and equipped with a 64-bit Microsoft Windows 7 OS, taking computational time lower than one minute per each scenario.

For each configuration and enclosure, two different examples have been considered, as detailed in the following.

In the first example, two circular scatterers, each one of radius equal to

In the second example, the reference scenario consists of two objects: one is L-shaped object with

Example 1: reference profile. (a) Real and (b) imaginary parts of reference contrast function.

Figure

Example 2: reference profile. (a) Real and (b) imaginary parts of reference contrast function.

All the data employed for the inversion procedure have been corrupted by additive white Gaussian noise, with a SNR value equal to 20 dB.

The performance of the inversion method is quantitatively assessed by means of the normalized mean square error on the contrast function defined as

As mentioned in Section

Example 1: LSM reconstructions by using measurement configuration A and corresponding pivot points. (a) Circular scanner, (b) square scanner, and (c) triangular scanner.

Example 1: LSM reconstructions by using measurement configuration B and corresponding pivot points. (a) Square scanner and (b) triangular scanner.

Example 2: LSM reconstructions by using measurement configuration A and corresponding pivot points. (a) Circular scanner, (b) square scanner, and (c) triangular scanner.

Example 2: LSM reconstructions by using measurement configuration B and corresponding pivot points. (a) Square scanner and (b) triangular scanner.

As detailed in Section

In order to investigate the fidelity of field approximation we can get by means of virtual experiments, among all the selected pivot points and for each example and measurement configuration, we have considered the two virtual experiments pertaining to the sampling points in which the residual of the far field equation is minimum and maximum, respectively. In this way, we analyze the accuracy of the field approximation in the best and in the worst case.

As an example, in Figure

Performance assessment of the field approximation: comparison for the triangular cavity, example 2. ((a), (c)) Real and imaginary parts of total field in the minimum residual pivot point; ((b), (d)) real and imaginary parts of virtual field in the minimum residual pivot point; ((e), (g)) real and imaginary parts of total field in the maximum residual pivot point; ((f), (h)) real and imaginary parts of virtual field in the maximum residual pivot point.

As expected and predicted in Section

Similar results have been obtained in all the examples and are herein omitted for the sake of brevity. However, the mean square error of the field approximation is below 10% for all the analyzed cases.

Before presenting quantitative reconstructions obtained by means of the QLSM, let us observe that the efficacy of the method depends on the accuracy with which we are able to approximate the field inside the investigated domain. Ideally, if we assume the exact knowledge of the field’s distribution, no approximation is needed and the linear inversion of (

Ideal reconstruction error.

Example 1 | Example 2 | |||||
---|---|---|---|---|---|---|

CC | CQ | CT | CC | CQ | CT | |

Config. A | 0.27 | 0.29 | 0.25 | 0.22 | 0.35 | 0.29 |

Config. B | 0.24 | 0.30 | 0.30 | 0.38 |

Hence, starting from the the support estimation presented in the previous subsection, we selected 40 sampling points for the first example and about 24 for the second one. Then we have built and inverted the tomographic operator

Reconstruction error.

Example 1 | Example 2 | |||||
---|---|---|---|---|---|---|

CC | CQ | CT | CC | CQ | CT | |

Config. A | 0.28 | 0.32 | 0.26 | 0.29 | 0.46 | 0.48 |

Config. B | 0.28 | 0.32 | 0.43 | 0.44 |

Example 1: QLSM reconstructions by using measurement configuration A. ((a), (d)) Real and imaginary parts of retrieved profiles in a circular scanner; ((b), (e)) real and imaginary parts of retrieved profiles in a square scanner; ((c), (f)) real and imaginary parts of retrieved profiles in a triangular scanner.

Example 1: QLSM reconstructions by using measurement configuration B. ((a), (c)) Real and imaginary parts of retrieved profiles in a square scanner; ((b), (d)) real and imaginary parts of retrieved profiles in a triangular scanner.

Example 2: QLSM reconstructions by using measurement configuration A. ((a), (d)) Real and imaginary parts of retrieved profiles in a circular scanner; ((b), (e)) real and imaginary parts of retrieved profiles in a square scanner; ((c), (f)) real and imaginary parts of retrieved profiles in a triangular scanner.

Example 2: QLSM reconstructions by using measurement configuration B. ((a), (c)) Real and imaginary parts of retrieved profiles in a square scanner; ((b), (d)) real and imaginary parts of retrieved profiles in a triangular scanner.

As a first comment, let us again stress that the reconstructions obtained by using QLSM strongly depend on the LSM indicator: therefore, the better the LSM works, the better the QLSM recoveries will be (if appropriate number of sampling points is chosen).

As can be seen, satisfactory results have been achieved, as demonstrated also by the fact that the reconstruction errors in Table

To corroborate this ansatz, we have repeated example 2 “flipping” the object into the cavity. As shown in Figure

LSM indicator for flipped example 2 in a triangular cavity. (a) Measurement configuration A and (b) measurement configuration B.

Example 2 with flipped scatterers: QLSM reconstructions in a triangular enclosure. ((a), (c)) Real and imaginary parts of retrieved profiles in measurement configuration A; err = 0.35. ((b), (d)) Real and imaginary parts of retrieved profiles in measurement configuration B; err = 0.45.

Finally, a nonlinear inversion has been performed as well, by adopting the Distorted Born Iterative Method scheme [

Example 1: reconstructions in the circular cavity. ((a), (d)) Real and imaginary parts of retrieved contrast function assuming the “exact” scattering operator; ((b), (e)) real and imaginary parts of retrieved contrast function by measn of the DBIM; ((c), (f)) real and imaginary parts of retrieved contrast function via QLSM.

As can be seen, comparable results are obtained via DBIM and via QLSM and, as expected, in both cases such results are worse than the benchmark. However, the contrast function retrieved by means of the DBIM is slightly overestimated and this brings to a slightly larger reconstruction error

In this paper we have presented and assessed a simple and effective inversion strategy for quantitative microwave imaging in noncanonical PEC embedded systems. As a matter of fact, the presence of a confining structure is common to several applications of microwave imaging, as it allows shielding the imaging domain from environmental noise as well as containing a possible matching medium needed for coupling purposes. On the other hand, the embedded systems require a proper modeling of the scenario under test to make the processing of the acquired data reliable.

In this frame, we have generalized a recently proposed quantitative inversion strategy based on the LSM to the cases of PEC enclosures of arbitrary shapes. The main advantages of the QLSM stand in its simplicity, fastness, and effectiveness in a broad class of scenarios. The presence of the PEC enclosure has been modeled by exploiting a 2D TM full-wave forward based on the Finite Element Method (FEM), not only for the simulation of synthetic data, but also (and more important) to compute Green’s function of the actual system at hand, which is not known in analytic form but for the case of a circular cross section PEC embedding.

The performed numerical analysis confirmed the effectiveness of the QLSM as a way to reliable quantitative recovery strategy, also against complex scenarios and in presence of not canonical shaped enclosures. Future work will be concerned with the range of applicability of the method (with respect to the maximum contrast which can be reliably recovered), as already carried out in free space [

The authors declare that there is no conflict of interests regarding the publication of this paper.