1. Introduction
Wireless sensor networks (WSNs) are nowadays widely adopted for various purposes in different environments and application fields. Their initial developments can be traced back to several years, as testified by the interesting review in [
1]. However, in recent times, there has been an increasing interest in the development of WSNs in more new areas [
2,
3,
4,
5], attesting to their ubiquitous role. The use of WSNs has had a further increase and acceleration in conjunction with the advent of the internet-of-things (IoT) [
6,
7].
In this framework, an interesting application is related to the possibility of retrieving information about the presence and properties of targets in the scenario in which the nodes are located [
8,
9,
10]. Such a possibility stems from the fact that the propagation of the electromagnetic waves (which are generated by the sensors) is significantly affected by the distribution of the dielectric properties in the scenario, and thus, by the presence and by the geometrical/physical properties of the targets. Consequently, by properly processing the received signals, it is in principle possible to extract some information about the objects that perturbate the field, e.g., their presence and possibly their dielectric properties. To this end, two main research directions have been followed. The first one concerns the development of algorithms able to provide just the position of the eventually present targets. Approaches belonging to this class are usually based on the analysis of the time-of-flight [
11,
12], on the estimation of the directions of arrival of the impinging waves [
13,
14,
15], or on the use of fingerprinting techniques relying on signal-strength measurements [
16,
17,
18]. Despite their robustness and accuracy, these methods do not usually provide information about the type of target, its extent, and so on. Moreover, depending on the adopted algorithm, it may be difficult to identify the presence of more than one target without a-priori information. The second class of approaches is based on the use of imaging techniques aimed at providing an image representing some physical properties of the considered scenario. To this end, a possible solving strategy is to adopt electromagnetic inverse-scattering techniques, which are potentially able to retrieve the full distributions of the dielectric properties (dielectric permittivity and electric conductivity) starting from scattered-field measurements [
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29]. Differently from methods belonging to the first class, it is however necessary to adopt an exact model of the electromagnetic propagation inside the scenario. This is complicated by the fact that, from a mathematical point of view, the electromagnetic scattering phenomena are described by non-linear equations (with respect to the dielectric properties of the region). Moreover, the inverse problem of retrieving the dielectric distribution from electric field measurements is also strongly ill-posed. In order to address these issues, several inverse-scattering approaches have been proposed in the scientific literature. In particular, both qualitative and quantitative imaging techniques have been devised. In the first case, the retrieved image often represents an indicator function, which may give information about the positions, extents, and number of targets in the scene of interest. Some common qualitative approaches of this kind are the linear sampling method [
30], the level set algorithm [
31], and MUSIC [
32]. Conversely, the images generated by quantitative methods also provide an estimation of the dielectric properties in any point of the inspected area, thus, allowing a characterization of the targets. Some examples of methods that belong to this class are the contrast source inversion technique [
33] and various Newton-type schemes [
34,
35,
36].
In this paper, which is an extended version of [
37], the use of a new inverse-scattering technique belonging to the class of quantitative imaging methods is considered. Such a technique, adopted for producing images of the scenario in which the sensors are located, is preliminarily investigated by using numerical simulations in a simplified two-dimensional environment. Moreover, by employing the reference data provided by the Institut Fresnel [
38], a simplified experimental scenario is also considered for preliminarily testing the developed procedure against real measurements. The inversion of the scattering model is performed by using a recently proposed technique developed in the framework of the
Banach spaces [
39,
40,
41], which is applied for the first time for addressing WSN imaging problems. In particular, a two-loop Newton-type scheme is adopted, in which the linearized problem obtained at each Newton step is solved in a regularized sense by using a truncated conjugate-gradient-like iterative algorithm in Banach spaces [
42,
43]. It has been found that such a class of approaches is able to address the non-linearity and the ill-posedness of the problem in an effective way. Furthermore, it is proven that in other imaging applications (e.g., medical and subsurface imaging) these methods are capable of providing better reconstruction performance than classical inversion techniques working in Hilbert spaces. In particular, the main advantages are related to the capabilities of
-based procedures (especially when the norm parameter
p is lower than 2) of reducing the ringing and oversmoothing effects usually associated to the low-pass filtering that is usually introduced by standard regularization techniques. Such a reduction may allow to better reconstruct the shape and position of the targets, as well as to improve the estimation of their dielectric properties. Moreover, thanks to these advantages, small and localized objects can be retrieved with good accuracy.
The paper is organized as follows. In
Section 2 the problem is formalized and a possible strategy for its solution is detailed. Some preliminary results, pertaining to an analysis carried out in a simplified 2D simulated environment, and an initial experimental validation in a simple test case, are presented in
Section 3. Finally, in
Section 4, conclusions are drawn.
2. Problem Description and Solution Strategy
The problem configuration is shown in
Figure 1. A predefined region of space on the
plane, denoted as the inspection domain
, contains some unknown targets. This region is surrounded by
N microwave transceiver sensors, located in points
. The background scenario is known, and a free-space configuration is considered. It is assumed that each sensor acts as a transceiver and can radiate or receive a time-harmonic electromagnetic field at a fixed frequency
f. Furthermore, each sensor is able to collect measurements of the
z-component of the electric field. The acquired field data are shared with an external control device, not discussed here, whose function is also to synchronize sensor measurements. During the acquisition phase, one device at a time operates in transmission mode, while all the other ones (i.e.,
elements) are employed for electric field measurements. Such a measurement process is repeated until all the
N sensing elements have been used in transmission and reception modes.
With this set of field measurements available, the objective is to retrieve a map of the dielectric properties of the targets inside the inspection domain . For the sake of simplicity, only a 2-D model, characterized by (with respect to the z axis) electromagnetic fields and object properties invariant along the same axis, is considered. In addition, non-magnetic materials (i.e., characterized by the same magnetic permeability as the vacuum, ) are assumed in this paper. However, extending the present formulation to a three-dimensional problem is, in principle, quite straightforward, even though the computer implementation can be challenging.
Basically, when an unknown target is present inside the investigation area, the electric field is perturbed with respect to the absence of objects. Consequently, this field perturbation (usually referred as the scattered electric field
), if properly measured, can be used to reconstruct the target properties [
44]. In this work, we focus on retrieving a point-by-point map of the complex dielectric permittivity inside
, that is:
In this equation, the term represents the real part of the relative dielectric permittivity, is the electric conductivity, is the angular frequency, and finally stands for the vacuum dielectric permittivity.
In general, the scattered electric field
in the
n-th sensor location is related to the dielectric properties of the investigation area
by means of an integral equation, i.e.,
where
is the vacuum wave number,
is the so-called
contrast function (a free-space background, modeled as vacuum, is assumed),
is the total electric field (in the presence of the unknown targets), and
g is the Green’s function of the considered configuration [
44].
In order to implement such a model within a computer code, the inspection domain is subdivided into I cells of square area , whose center is positioned at , and where both the contrast function c and fields are supposed to be constant (in other words, the continuous model has been discretized by using piecewise-constant basis functions).
Assuming that the
m-th sensor (positioned at
) is the only one in transmitting mode for the current view, and that, for each location of the transmitting device, the
z-component of the scattered electric field
is known at the positions of all the other
sensors, the discrete version of (
2) can be written as [
39]
where the integral has been replaced by a summation, and the term
is defined as
Equation (
3) can be reformulated in a matrix form by introducing the following vectors:
and it results that
The term
is a rectangular matrix of size
, and its elements are defined as
. Actually, the total electric field in
in the presence of the targets is another unknown quantity, represented by the vector
, and it can be found with the aid of another equation (whose derivation is similar to the previous one), that is
in which the vector
contains the values of the electric field inside
without the unknown targets, and
is a square matrix of size
I whose elements are given by
, with
.
By combining (
6) and (
7) together, we have:
This equation represents a nonlinear relationship which links the contrast function values of the vector to the measured scattered electric field values contained in . Retrieving starting from the knowledge of is a well-known inverse scattering problem. Unfortunately, due to the particular mathematical properties of this kind of problem, the previous equation turns out to be ill-posed. As a consequence, its solution is not trivial and should be accomplished with suitable inversion procedures, which should be capable of addressing the problem of ill-posedness as well as its non-linearity.
The solution approach adopted in this work is an iterative deterministic scheme, based on a Newton-conjugate-gradient (NCG) method. In particular, this method operates in the mathematical framework of
Banach spaces, and is composed by two nested iterative loops. In the external loop (whose iterations are indicated by the index
l), an inexact-Newton approach is applied to linearize (
8) around the value of the contrast function which is currently reconstructed, denoted as
. The main steps of the external loop are summarized in
Figure 2.
Once the problem is linearized, the resulting linear equation is solved in an inner loop by means of a regularizing non-conventional conjugate-gradient-like algorithm operating in
Banach spaces, outlined in
Figure 3 [
43]. The parameter
p can be tuned inside the inversion process, and is chosen as a fixed value before starting the iterations. Both the inner and the outer loops are terminated when the relative variation of the minimized residual functional falls below a predefined threshold.
In particular, the aim of the inversion method is to minimize the residual functional
where
is the norm of the considered Banach space. Conversely, standard Hilbert-space regularization algorithms usually minimize the residual functional
and the application of the conjugate gradient method is straightforward. In
Banach spaces, instead, this is no longer true, since the usual iteration scheme of the CG is not well defined. To overcome this problem, the method used in this paper is based on the concept of duality maps in Banach spaces, which, for the considered
spaces, are defined as
where
is a vector of
N components, and
Note that, in usual Hilbert spaces with norm, the duality maps reduce to identity operators.