CHAPTER ELEVEN

Microwave Imaging Strategies, Emerging Techniques, and Future Trends

11.1 INTRODUCTION

Research activity on microwave imaging has evolved rapidly, and it is almost impossible to mention the plethora of new ideas, methods, and strategies that continuously appear in the scientific literature and are proposed by a number of very active research teams working around the world and discussed at many conferences devoted to applied electromagnetics. Nevertheless, some more recent ideas concerning possible new imaging strategies and emerging solutions can be summarized, in an attempt to delineate the trend in scientific research in this field.

One trend is certainly represented by the development of hybrid techniques, which allow utilization of the specific features of the different methods applied in combination. This topic has been discussed in Chapter 8. It should be mentioned once again that the ability to devise specific techniques for any particular application is of paramount importance from an engineering perspective, since it is absolutely evident that general-purpose approaches cannot be developed.

However, besides hybrid techniques, other strategies, approaches, and particular applications have been proposed by the research community. Some of them are discussed in the following sections, without any claim of completeness.

11.2 POTENTIALITIES AND LIMITATIONS OF THREE-DIMENSIONAL MICROWAVE IMAGING

Most of the approaches described in the previous sections concern two-dimensional imaging configurations, although it has been mentioned that, theoretically, most of them can be immediately extended to directly accommodate three-dimensional configurations.

Although three-dimensional inversion techniques were proposed in the 1980s or so, the limitations due to the computer powers made it possible to inspect only very simple scatterers with very rough discretizations and reduced meshes (Ghodgaonkar et al. 1983, Guo and Guo 1987). As an example, the pseudoinverse matrix [equation (5.5.11)] was used to invert a discretized form of the data equation [equation (3.2.1)] in order to reconstruct dielectric cubes (Ney et al. 1984, Caorsi et al. 1988). After these rather naive attempts, the microwave imaging community focused attention on two-dimensional approaches that allow spending all the available computer resources for obtaining acceptably fine discretizations. Concerning this point, also of interest is the discussion in the paper by Hagmann and Levin (1990), in which some of the previously reported results concerning three-dimensional configurations are judged to be “overly optimistic,” because “sizable errors in the forward solution were largely cancelled by errors in the inverse calculations”; that is, the so-called inverse crime perhaps could not be avoided.

Although research activity on microwave imaging was previously focused, as mentioned, on two-dimensional methods, some other techniques based on approximations have been extended to handle three-dimensional problems. Among them, we can mention the distorted Born iterative method (Section 6.6), which has been applied together with a solving procedure based on a biconjugate-gradient method and fast Fourier transform (FFT) by Gan and Chew (1995), whereas the extension of diffraction tomography (based on the Born and Rytov approximations) to three-dimensional problems has also been proposed (Vouldis et al. 2006). Moreover, three-dimensional reconstructions of buried objects have also been obtained (at low frequencies) by using the very-early-time electromagnetic (VETEM) system mentioned in Section 10.3 (Cui et al. 2003).

Obviously, when the objective of the investigation is simply the shape of the targets, qualitative methods can be applied without difficulties to three-dimensional problems, due to the reduced computational complexity (see Chapter 5). The linear sampling method, for example, has been used for three-dimensional scatterers several times (Giebermann et al. 1999; Brignone et al. 2009) and, recently, the level-set method has been extended to shaping three-dimensional metallic objects (Ferrayé et al. 2009).

Very recently, however, the direct application of nonlinear inspection techniques to reconstruct three-dimensional object has been reconsidered. The main reason is the terrific improvement in the computer power of modern PCs and also the wide availability of multiprocessor computers. In addition, it should be consider that an improved capability of handling the ill-posedness of the inverse scattering problem had a strong impact on this new research direction. In fact, the dramatic increase in the number of unknowns in three-dimensional problems, with respect to two-dimensional problems, is reflected in a significant increase in the ill-posedness of the inverse problem.

As an example of full-wave three-dimensional imaging, a quantitative iterative approach based on a Gauss-Newton method (see Chapter 6) has been applied to inspect a cube and a sphere by using both synthetic and real data (De Zaeytijd et al. 2007), in which the total electric field has been computed with a fast-forward solver at each iteration, whereas in the inverse solution, only the values of the dielectric permittivities have been considered as unknowns. Quite good results have been obtained with a three-dimensional mesh of 15 × 15 × 15 cells, for the inverse problem, and with a finer mesh for the (well-posed) direct computation (30 × 30 × 30 cells) (De Zaeytijd et al. 2007). A quasi-Newton minimization has also been proposed for reconstruction of the three-dimensional distribution of electrical conductivity for applications related to the detection of oil reservoirs (which is the same problem considered in Figs. 10.14–10.16). For these applications, imaging techniques can suitably exploit the high contrast in resistivity between saline-filled rocks and hydrocarbons (Abubakar et al. 2009). An iterative Newton method has also been used (Rubaek et al. 2009), in particular for processing the data collected by a cylindrical multistatic antenna setup constituted by 32 horizontally oriented antennas (see Chapter 9). In this approach, a multiplicative functional has been assumed.

In the medical field, an interesting application of the distorted Born iterative method has been proposed in the study by Winters et al. (2009), where, in order to solve a complex three-dimensional inverse problem, the use of patient-specific basis functions is proposed for discretization of the continuous model (Section 3.4). In this way, a reduced parameterization of the problem is obtained with a significant computational saving. Such an approach confirms once again the importance of developing specific application-oriented systems and reconstruction techniques.

It should be also mentioned that, although computationally expensive, stochastic optimization methods have been proposed for three-dimensional imaging as well. An example is represented by the application of the micro-genetic algorithm (Huang and Mohan 2005), which involves a limited population of individuals (Section 7.3).

The perspective of three-dimensional imaging can, of course, be of interest for research on hybrid methods (described in Chapter 8), since the reduction in numerical complexity is one of the main reasons for developing hybrid imaging approaches. An example of hybrid approaches for two-dimensional problems successively extended to three-dimensional configurations is represented by the combined strategy of linear sampling method and ant colony optimization (Section 8.3). It must be noted that the use of stochastic optimization methods (quite computationally heavy), although combined with fast qualitative procedures, still requires some additional assumptions in order to treat realistic configurations. For this reason, the bodies to be inspected are assumed, for example, to be homogeneous (Brignone et al. 2008). Under this hypothesis, the proposed hybrid method provided very accurate reconstructions, as can be seen in Figure 11.1.

FIGURE 11.1 Reconstruction of the shape (a) the relative dielectric permittivities (b) and electric conductivity (c) versus the iteration number for a homogeneous dielectric parallelepiped; (d) number of cost function evaluations. [Reproduced from M. Brignone, G. Bozza, A. Randazzo, R. Aramini, M. Piana, and M. Pastorino, “A hybrid approach to 3D microwave imaging by using linear sampling and ant colony optimization,” IEEE Trans. Anten. Propag. 56(10), 3224–3232 (Oct. 2008), © 2008 IEEE.]

It should be recalled that qualitative approaches, which are able to provide the support of the scatterer, can be also combined with deterministic quantitative techniques (Section 8.3). An example is the method proposed by Morabito et al. (2009), who combined the linear sampling method with a technique based on the extended Born approximation (Section 4.7) and applied it to a contrast source formulation (Section 3.2).

11.3 AMPLITUDE-ONLY METHODS

Although measurement of the phase of the electromagnetic field at microwave frequencies no longer represents a technical problem, there is still significant interest in developing imaging approaches based on the amplitude of the scattered field only. This assumption results in less expensive instrumentation, which is particularly suitable at the highest frequencies of the microwave range and at millimeter waves. There are essentially two possible conceptual approaches:

1. Application of phase retrieval techniques that are able to compute the phase of the field in the observation domain starting from measured values of the field amplitude (Hislop et al. 2007). Once amplitudes and phases of the scattered field at the measurement points are available, one of the methods described in Chapters 5–8 can be directly applied. This approach has been previously mentioned at the end of Section 9.2 and some references are provided there.
2. Exploring inversion procedures that reconstruct the scatterers directly, starting with amplitude-only data. In principle, a function similar to the one reported in Section 6.7 can be constructed on the bases of the amplitudes of the fields. In particular, the residual [equation (6.7.2)] between the measured data and the computed data (i.e., the data calculated at each iteration on the basis of the current trial solution) can be formed by using the difference of the amplitudes between the two fields, instead of their complex values (amplitudes and phases). An approach of this kind was followed by Caorsi et al. (2003), who optimized a phaseless functional by using the memetic algorithm (Section 8.2).

Finally, it should be mentioned that amplitude-only data have been inverted by extending other classical methods. For example, diffraction tomography (Section 5.11) has been used (Devaney 1992), an approach based on the Rytov approximation (Section 4.8) has been proposed (Zhang et al. 2009), and a so-called multiscale approach has been followed (Franceschini et al. 2006).

11.4 SUPPORT VECTOR MACHINES

In the previous sections, it was mentioned several times that the image of the target is not always the final objective of the inspection process. In some cases, sufficient information is represented by the knowledge of some useful parameters of the target, such as, position or dimensions. In certain applications, the object under test can be a canonical object or can be approximated by a canonical object. As a priori information, one can also assume, in specific cases, that the target is homogeneous or that its dielectric parameters exhibit a specific dependence on the spatial coordinates. Fast and simplified techniques can be applied in these cases, as described in Chapter 5. Moreover, when few parameters have to be detected, the global optimization methods discussed in Chapter 7 can be suitably applied. Beside these approaches, the determination of few parameters of a scattering configuration can be efficiently addressed by using neural network approaches as well.

Conceptually, the network must first be trained, in particular by using, as input data, two sets of data. The first one includes several arrays of parameters, each of them able to describe one of the known scatterers belonging to a given set. The second set of data contains the values of the scattered field (at the measurement points) produced by the previously considered known scatterers. Both sets constitute the training data of the network. The previously considered known scatterers might be canonical objects (e.g., the ones mentioned in Section 3.5), for which the produced scattered field can be calculated in a simple and fast way. Once the offline training phase is completed, the network should be able to almost instantaneously retrieve the key parameters of an unknown scatterer just starting from the values of the field that it scatters. This constitutes the test phase.

The use of neural networks has been proposed for inverse scattering purposes in several papers (e.g., Rekanos 2002, Bermani et al. 2002, and references cited therein). However, more recently, support vector machines (SVMs) (Vapnik 1998) have attracted notable attention due to their excellent generalization capabilities (i.e., the ability of dealing, in the test phase, with data quite different from those considered in the training phase) in several applications, such as pattern recognition, time prediction, and regression. Support vector machines have been also successfully applied in other areas of electromagnetics (e.g., for the determination of the arrival angles of incident waves (Hines et al. 2008)).

In order to retrieve an unknown target from scattering data, the functional to be minimized, F [equation (6.7.1)] can be successfully approximated by using a support vector regression (SVR) approach. To this end, let l = 1,..., L be the array of data characterizing the lth known scatterer of the training set (which includes L known targets). In addition, l = 1,..., L, is the array containing the samples of the scattered electric field at the measurement points. These field samples are produced by solving a direct scattering problem involving the lth known scatterer. Thus, the training set Ω_train is given by

The values contained in the training set Ω_train are used to evaluate the functional approximates the unknown actual functional F. In particular, the approximating relationship is given by

where φ is a nonlinear function used to transform the input data z from their original space, namely, Σ, to a higher-dimensional space H (usually called feature space) and denotes the inner product of H; w and b are parameters whose optimal values are obtained by minimizing the so-called regression risk function associated with SVR, which is given by

where C is a constant and is the ε-insensitive loss function, that is

Where ε is the allowed error during the training phase. According to Vapnik (1998), the regression risk is minimized by using a dual formulation. In particular, a standard dualization method utilizing Lagrange multipliers is employed. The following quadratic optimization problem, to be solved with respect to the Lagrange multipliers α_i and α′_i, is obtained:

subject to

In both equations (11.4.5) and (11.4.6), α′_l, α_l ∈ [0, C]. The parameter C [equation (11.4.3)] has been found to control the generalization properties of the support vector machine (Smola and Schölkopf 2004). In particular, high values of C produce small estimation errors for configurations equal to those contained in the training set, but they result in a reduced generalization capability. Finally, in equation (11.4.5), κ is the kernel function, working on the original space Σ and defined as

A commonly used kernel is the Gaussian kernel (Smola and Schölkopf 2004, Pastorino and Randazzo 2005), which is defined as

where γ is a constant to be selected by the user. By using the dual-optimization problem, one can express w in terms of the input data z as follows:

By substituting (11.4.9) into (11.4.2), one can rewrite as

The resulting quadratic optimization problem can be efficiently solved by applying, for example, the sequential minimal optimization (SMO) algorithm (Keerthi et al. 2001). It is worth noting that only a subset of the Lagrange multipliers have nonzero values (i.e., those satisfying the condition thus a sparse solution is obtained. The samples associated with the nonzero Lagrange multipliers are called support vectors (Vapnik 1998).

A very preliminary example of the use of support vector machines for detecting a discontinuity in a biological body is presented in Figure 11.2. In particular, a quite rough two-dimensional model of a human abdomen (Caorsi et al. 2004) includes a circular discontinuity in one of the tissues. The support vector machine has been trained by using various elliptic scatterers (Section 3.5), and the forward problem has been solved by using the method of moments. The final image is provided in Figure 11.2b. As can be seen, for this simple configuration, the mass in the abdomen is correctly located and shaped.

The application of support vector machines has been also proposed by Zhang et al. (2007), with some differences in implementation for the localization of passive scatterers. In particular, the application considered concerns detection of the dielectric parameters (problem A) and, separately, of the position (problem B) of an infinite circular cylinder in free space starting from measurements of the scattered electric field. The reconstruction errors for the two problems are briefly summarized in Tables 11.1 and 11.2.

11.5 METAMATERIALS FOR IMAGING APPLICATIONS

Another challenge is use of metamaterials for microwave imaging applications, since important steps have been taken toward the development of devices involving such materials. In particular, left-handed metamaterials, which were theoretically studied for the first time by the Russian scientist V. G. Veselago (Veselago 1968), have more recently been considered in some advanced technical areas of electromagnetics.

In certain frequency bands, these materials have a negative dielectric permittivity and a negative magnetic permeability. This property results in a number of significant effects and consequences. However, in terms of imaging applications, the most relevant aspect is the possibility of realizing focusing lens made by flat slabs. Because of the negative values of ε_r, and μ_r, a microwave field, generated by a localized source (ideally, a point source), can be focused on a specific point by a flat slab. In this way it is possible to scan the focal point inside a target to be inspected by simply moving the source (in front of the slab) in the lateral and longitudinal directions (Wang et al. 2007).

FIGURE 11.2 Localization of a dielectric discontinuity by using a support vector machine: (a) original configuration (rough model of a human abdomen); (b) final image. (Simulation performed by A. Randazzo, University of Genoa, Italy.)

TABLE 11.1 Error Analysis for Problem A

TABLE 11.2 Error Analysis for Problem B

11.6 THROUGH-WALL IMAGING

Recently there has also been increasing interest in through-wall imaging (Chang et al. 2009), in which a target is to be localized and reconstructed by using an imaging system positioned behind a wall or another blinding structure. The main purpose is for security, but other surveillance applications are possible. Radar techniques have been proposed mainly for this purpose, but the field can benefit from the development of inverse scattering-based imaging methods, as well. From a theoretical perspective, the presence of the hidden obstacle can be detected by using the proper Green function for the inhomogeneous structure, which can be assumed to be approximately known. Other approaches are also possible. For example, a combination of 2D images obtained by using a linear approach based on the Kirchhoff approximation (Section 4.9) and solved by a truncated singular value decomposition (SVD) (Chapter 5) has been proposed (Solimene et al. 2007, Kidera et al. 2008).

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