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An analytical approach to location and shape reconstruction of dielectric scatterers, that was recently proposed, is tested against experimental data. Since the cross-sections of the scatterers do not depend on the

In the last decades, inverse electromagnetic scattering and near field imaging have been widely studied research topics [

One of the aim of an inverse electromagnetic-scattering imaging system is to find the actual position of a dielectric object inside a bounded space region, as well as its shape. The techniques applied are based on the use of a known incident field illuminating the space region containing the object. By suitably measuring the scattered field, a solution to the problem can be derived.

The main difficulties with solving an inverse electromagnetic-scattering problem result from its nonlinearity and instability. To overcome these negative features, in the scientific literature, several regularization techniques have been proposed. Furthermore, the a priori knowledge usually available could be included in the formulation of the resolving algorithm. The problem could be recast as a global nonlinear optimization problem, and stochastic, as well as deterministic approaches can be used to achieve a solution. The interested reader is referred, for example, to [

Another common technique to overcome nonlinearity is based on the transformation of the original problem into an inverse source one, by replacing the scatterer with an equivalent (induced) source. Although the equivalent source density linearizes the inverse scattering equation, the transformation of the inverse scattering problem into an inverse source one involves the solution of an integral equation of the first kind. Hence, the resulting operator is ill-posed, and its solution is usually unstable and nonunique [

In the present work, an approach to the inverse source problem that allows to clearly reformulate it in terms of both the radiating and nonradiating parts of the induced sources is used. Through a singular value decomposition (SVD) of the scattering operator, a closed-form solution to the radiating sources is reached. By using such a solution, an expression for the reconstructed scattered field inside the dielectric object can also be obtained in closed form.

While the theory that is at the base of this work was already presented and tested by numerical simulations [

Although the present work is at an early stage and only some features of the theoretical model have been checked against the experimental data, the method seems to perform well, and its capabilities to reconstruct both the location and the shape of the induced sources are demonstrated. Furthermore, thanks to the closed-form formulation of the solution, results are available in a very short time that can allow for a quasi real-time imaging system.

In this section, the closed-form solution to the inverse scattering problem is briefly resumed. For more details the reader is referred to [

Let us consider an inhomogeneous object whose dielectric parameters are unknown, and which is irradiated by an electromagnetic field produced by a known source. The presence of the object leads to a total electromagnetic field that is different from the one that the same source would radiate in a free-space environment. In particular, the scattered field due to the object can be modeled by means of the volume equivalence theorem [

The problem is simplified into a two-dimensional and scalar one, when

the scatterers are infinitely extended along

the scattering potential does not depend on

the incident electric field does not depend on

As it was already mentioned, the inverse problem given by (

The problem can be linearized by using the equivalent source

The radiating part

For a lossless propagation medium, the singular system

the functions

the functions

the numbers

A solution to the radiating source contribution

Since

As for the nonradiating sources

By using the expression given by (

The method was tested against three sets of experimental data involving dielectric scatterers. In particular, the dielectric targets of the first Fresnel dataset [

It should be pointed out that the present theory assumes that data are available over the full circumference. Instead, in any dataset, scattering field measures were taken over an arc of fixed radius with an angular span of only

The tests were focused on the reconstruction of the radiating equivalent source, and, in particular, three parameters involved in the process were considered: the number of singular values used in computing

Results are shown by means of normalized maps of the reconstructed domain. On the same maps also the shape and the location of the scattering object are shown, in order to give an immediate feeling about the fairness of the result.

Although both [^{2} around the scatter is shown. Anyway, it should be stressed that the value of the reconstructed equivalent source outside this area is negligible in any case.

Thanks to the closed-form formulation of the solution, all the results were obtained in a very short computation time. In particular, on a single-core class PC, a single view reconstruction takes about

Since reconstructions are evaluated from the data collected by a single experiment, only a partial information is available at each view. Furthermore, inspections at different frequencies also provide different information. To enhance the overall result, three different possibilities were considered: summing the partial results over the different views at a fixed frequency, summing over the frequencies at a fixed view, and summing over both frequencies and views. These approaches are summarized by the expressions in (

The first results are about the inversion of the scattering data produced by a singular dielectric cylinder with a circular cross section of radius

A single-view, single-frequency reconstruction is shown in Figure

Reconstruction of the radiating source for a single dielectric cylinder. Single view

Reconstruction of a single dielectric cylinder. Multiview; map of the

The reconstruction using the multifrequency-multiview approach provided the result shown in Figure

Reconstruction of a single dielectric cylinder. Multiview-multifrequency. Map of the

This configuration was made by a couple of twin dielectric cylinders whose centers are spaced by

Single view reconstructions, at each frequency, clearly show the presence of two distinct objects. As an example, in Figure

Reconstruction of the radiating source for a couple of twin dielectric cylinders. Single view

Reconstruction of a couple of twin dielectric cylinders. Multifrequency. Map of the

Multiview reconstructions can provide a much more regular result, with a limited background noise. The case of

Reconstruction of a couple of twin dielectric cylinders. Multiview; map of the

Finally, in Figure

Reconstruction of a couple of twin dielectric cylinders. Multiview-multifrequency. Map of the

The third considered scatterer was the “FoamDielExt” object, included in the second Fresnel dataset [

For this case, an accurate investigation about the right number of singular values was needed to achieve a good reconstruction. As an example, the limiting values of

Reconstructions of the radiating source for the FoamDielExt object. Single view

Another example of reconstruction is given in Figure

Reconstruction of a couple of the FoamDielExt object. Multiview; map of the

A further enhancement, with a pretty nice contrast between the two scattering cylinders, is achieved when results are combined also in frequency. The related map is shown in Figure

Reconstruction of a couple of twin dielectric cylinders. Multiview-multifrequency. Map of the

The metallic object was an U-shaped cylinder, having a cross-section with dimensions ^{2}, and the thickness of the metal was

Information provided by multifrequency can give a sharp reconstruction of a partial object, as it is shown in Figures

Reconstruction of a metallic U-shaped cylinder. Multifrequency; map of the

Reconstruction of a metallic U-shaped cylinder. Multifrequency; map of the

As already mentioned, instead, multiview gives a much more better overall result. In particular, in Figure

Reconstruction of a metallic U-shaped cylinder. Multiview; map of the

As it was in the cases of dielectric objects, the multiview-multifrequency approach can smooth the noise in the background. However, in this case, the results is not as good as it was for dielectric objects and the true object is smoothed, too. This result is shown in Figure

Reconstruction of the shape of a metallic U-shaped cylinder. Multiview-multifrequency. Map of the

In this work, an analytical method for inverse scattering problems was used to successfully reconstruct both homogeneous and inhomogeneous targets from multifrequency multistatic experimental data. The proposed method uses the singular values decomposition of the scattering operator to achieve a closed solution to the integral equation of the electromagnetic scattering. The derivation of the radiating source density inside the domain of investigation is the first step of the work, as well as the preliminary result useful to locate a scattering object inside such a region.

The dependence of the reconstructions on various parameters was investigated. In particular, results have shown that a correct choice of the significant number of singular values in the solution can lead to an optimal reconstruction, however; the method has proved robust enough to provide good reconstructions for a wide range of parameters.

Dielectric objects as well as objects made of conductive material were successfully tested.

As a general comment it should be stressed that the algorithm is very robust, and that, even for the very large domains used and with a suboptimal number of singular values, the object location was always clearly recovered.

Furthermore, thanks to the closed-form formulation of the solution, all the results were obtained in a very short computation time. It must also be pointed out, in particular, that, once the operating frequency and the dimensions of the investigation domain have been chosen, the singular system of the problem can be computed “offline.” For the same reason also some other operations, for example, the inspection of the singular values, can be performed before the reconstruction process. As for the “online” part of the algorithm, also in this case things go very fast. As an extra bonus, since the closed-form virtually provides the solution at any point inside the investigation domain, high resolution maps of the reconstructed area can be obtained, without the constraints on grid dimensions and on the number of pixels usually imposed by algorithms based on numerical discretization of the problem.

Future work will be focused on the reconstruction of the scattering potential of the objects under test and on the possible use of a set of nonradiating sources to refine the results. It is expected that these steps can provide a notable enhancement in the results, as it was already proved by numerical simulations.

A further goal will be also testing the method on other experimental datasets, for example, on the series provided by the University of Manitoba [