On the effects of the electromagnetic source modeling in the iterative multiscaling method

Authors


Abstract

[1] The validation against experimental data is a fundamental step in the assessment of the effectiveness of a microwave imaging algorithm. It is aimed at pointing out the limitations of the numerical procedure for a successive application in a real environment. Towards this end, this paper evaluates the reconstruction capabilities of the Iterative Multiscaling Approach (IMSA) when dealing with experimental data by considering different numerical models of the illuminating setup. In fact, since the incident electromagnetic field is usually collected in a limited set of measurement points and inversion methods based on the use of the “state” equation require the knowledge of the radiated field in a finer grid of positions, an effective numerical procedure for the synthesis of the electromagnetic source is generally needed. Consequently, the performances of the inversion process may be strongly affected by the numerical model and, in such a case, a great care should be devoted to this key issue to guarantee suitable and reliable reconstructions.

1. Introduction

[2] Within the framework of the medicine [Louis, 1992] and biomedical engineering [see, e.g., Liu et al., 2003, and references therein], without forgetting the industrial quality control in industrial processes [Hoole, 1991] and the subsurface sensing [Dubey, 1995; Daniels, 1996], many different applications require a noninvasive sensing of inaccessible areas. Towards this end, microwave imaging methodologies [Steinberg, 1991] have recently gained a growing attention since they allow to retrieve information on the environment probed with electromagnetic fields by fully exploiting the scattering phenomena [Colton and Krees, 1992].

[3] Unfortunately, the inverse problem to be faced is intrinsically nonlinear, ill-posed, and nonunique [Denisov, 1999]. In particular, the ill-posedness and the nonuniqueness arise from the limited amount of information collectable during the acquisition of the scattered field. The number of independent scattering data is limited [Bertero et al., 1995; Bucci and Franceschetti, 1989] and they can only be used to retrieve a finite number of parameters of the unknown contrast function. To fully exploit such an information and to achieve a suitable resolution accuracy, several multiresolution strategies have been proposed [Miller and Willsky, 1996a, 1996b; Bucci et al., 2000a, 2000b; Baussard et al., 2004a, 2004b].

[4] The Iterative Multiscaling Approach belongs to this class of algorithms [Caorsi et al., 2003]. The unknown scatterers are iteratively reconstructed by considering initially a rough estimate of the dielectric distribution and by enhancing successively the spatial resolution in a set of regions-of-interest (RoIs) where the objects have been localized. (The IMSA is initialized by considering the free space distribution, then no a priori information on the scenario under test is exploited. Moreover, the initialization of the intermediate steps is obtained from the reconstruction of the previous step with a simple mapping of the retrieved profile in the new discretization of the RoI.) Such a strategy is mathematically formulated by defining a suitable multiresolution cost function whose global minimum is assumed as the estimated solution. The functional is iteratively minimized by using a conjugate-gradient-based procedure [Kleinman and Van den Berg, 1992], but stochastic [Massa, 2002] or hybrid algorithms can be suitably applied.

[5] In order to validate such an approach, the multiresolution algorithm has been tested against experimental data [Caorsi et al., 2004a] collected in a controlled environment [Belkebir and Saillard, 2001], since synthetically generated data can give only limited indications and they model an ideal scenario.

[6] In dealing with real data, one of the key issue is the modeling of the electromagnetic source or of the related radiated field. In general, the electromagnetic field emitted by the probing system is measured only in the observation domain. However, iterative methods based on “Data” and “State” equations require the knowledge of the incident field (i.e., the field without the scatterers) generated from the source in the investigation domain. Towards this end, an accurate but simply model (i.e., requiring a reasonable computational burden) of the source should be developed. Complicated numerical models accurately reproduce real data, but they are difficult to be implemented starting from a limited number of samples of the radiated electromagnetic field collected in a portion of the observation domain. On the other hand, a rough model could introduce erroneous constraints to the reconstruction process. Nevertheless, whatever the source model, an effective inversion procedure should be able to reconstruct the scatterer under test with an acceptable accuracy according to its robustness to the noise.

[7] In this framework, to assess the effectiveness and the robustness of the IMSA, the results of a set of experiments, where different models for approximating the illuminating source are considered, will be shown.

[8] The paper is organized as follows. In section 2, the statement of the inverse problem and the mathematical formulation of the IMSA will be briefly resumed, while in section 3 the numerical models used to synthesize the probing electromagnetic source will be described. A numerical validation and an exhaustive analysis of the dependence of the reconstruction accuracy on the modeling of the radiated field will be carried out in section 4 by considering some experimental test cases. Finally, some conclusions will be drawn (section 5).

2. Mathematical Formulation

[9] The inversion procedure will be illustrated referring to a two-dimensional geometry (Figure 1). Let us consider an investigation domain DI, where an unknown scatterer is supposed to be located. The embedding medium is assumed lossless, nonmagnetic, and characterized by a dielectric permittivity equation image0. Such a scenario is illuminated by a set of V incident monochromatic electromagnetic fields Eincv(x, y), v = 1,…, V, and the corresponding scattered fields Escattv(equation image), v = 1,…, V, are available (computed as the difference between the field with Etotv and without the scatterer Eincv, Escattv = EtotvEincv) in m(v) = 1,…, M(v), v = 1,…, V, positions belonging to the observation domain DM. The object is described by a contrast function τ(x, y) = ɛr(x, y) − 1 − jequation image, (x, y) ∈ DI, ɛr(x, y) and σ(x, y) being the dielectric permittivity and the electric conductivity, respectively.

Figure 1.

Geometry of the problem.

[10] The arising scattering phenomena are mathematically described through the well-known Lippmann-Schwinger integral equations [Colton and Krees, 1992]:

equation image
equation image

where G2d denotes the Green function of the background medium [Jones, 1964].

[11] Since the problem associated with (1) is ill-posed [see Groetsch, 1993; Vogel, 2002] the system matrix after discretization of the Data Equation (according to the Richmond's procedure [Richmond, 1965]) is highly ill-conditioned, and, hence the problem is extremely sensitive to the the noise. To remedy this ill-conditioning, a regularization is needed. Thus, the problem is then reformulated in finding the unknown contrast function that minimizes a suitable cost function generally defined as follows

equation image

where G2dint and G2dext indicate the discretized forms of the internal and external Green's operators [Colton and Krees, 1992], equation image = equation image, ρun = equation image and An (Au) is the area of the n-th (u-th) square discretization domain. In particular, the first term of (3) enforces fidelity to the scattered data in the observation domain (Escattv(equation image), (equation image) ∈ DM) and it amounts to the residual error with respect to the scattered field computed from the Data Equation(1). The second term is a regularization term equal to the residual error with respect to the incident field in the investigation domain (Eincv(xn, yn), (xn, yn) ∈ DI) computed from the State Equation(2).

[12] However, due to the limited amount of information content in the input data [Bucci and Franceschetti, 1989], it would be problematic to parametrize the investigation domain in terms of a large number N of pixel values (in order to achieve a satisfying resolution level in the reconstructed image). To overcome this drawback, an initial uniform (coarse) discretization is used and successively an iterative parametrization of the test domain allows to adaptively increase the resolution level only in the region-of-interest of the investigation area thus achieving the required reconstruction accuracy [Caorsi et al., 2003].

[13] To retrieve the unknown scatterer (i.e., an object function that better fits the problem data, (Escattv(equation image), Eincv(x, y)), Equations (1) and (2) are discretized according to the Richmond's procedure [Richmond, 1965]. Moreover, to better exploit the limited information content of the scattering data, an adaptive multiresolution strategy is adopted [Caorsi et al., 2003].

[14] More in detail, such an adaptive multiresolution algorithm can be briefly described as follows. Firstly, the IMSA considers (i = 0, i being the step index) an homogeneous discretization of the investigation domain with a number of discretization domains N(0) equal to the essential dimension of the scattered data and computed according to americanthe criterion defined in Isernia et al. [2001]. Then, a “coarse” reconstruction of the investigation domain is yielded by minimizing (3) starting from the free-space configuration [τ(equation image) = 0.0 and Etotv(equation image) = Eincv(equation image)] in order to assess the robustness of the overall approach with respect to the “starting guess” in “worst-case”. After the minimization, where a set of conjugate-gradient iterations (k being the iteration index) is performed not modifying the discretization grid, a new focused investigation domain (RoI), DO(i), i = 0, is defined. Such a squared area is centered at

equation image

where equation image and equation image are defined as

equation image
equation image

and its side L(i) is defined as follows

equation image
equation image

where equation image stands for the real or the imaginary part and ρn(r)c(i) = equation image. Successively, the iterative process starts (ii + 1). According to the multiresolution strategy, an higher resolution level denoted by R (R = i) is adopted only for the RoI. DO(i−1) is discretized in N(i) square subdomain which number is always chosen equal to the essential dimension of the scattered data [Bucci and Franceschetti, 1989]. A finer object function profile is then retrieved, starting from the coarser reconstruction achieved at the (i-1)-th step, by minimizing the multiresolution cost function, Φ(i), defined as follows:

equation image

where

equation image

and R indicates the resolution level and DO(i) denotes the area of the RoI defined at the i-th step of the iterative procedure. It should be pointed out that the definition of (9) requires not only the knowledge of the available scattered field in the observation domain [Escattv(equation image) = Etotv(equation image) − Eincv(equation image), (equation image) ∈ DM], but also that of the incident field in DO(i) [Eincv(equation image), (equation image) ∈ DO(i−1)]. This latter information is generally not available from measurements [since, in general, only the samples of Eincv(equation image) other than Etotv(equation image) are experimentally measured], therefore it should be synthetically generated by means of a suitable model of the electromagnetic source.

[15] The multistep process continues by computing a new RoI according to (4)(7) and by estimating a new dielectric distribution through the minimization of the updated version of (9) until a “stationary reconstruction” is reached [Caorsi et al., 2003] (i = Iopt).

[16] Such a procedure can be extended to multiple-scatterers geometries by considering a suitable clustering procedure [Caorsi et al., 2004b] aimed at defining the number of scatterers Q belonging to the investigation domain and the regions DO(i)(q), q = 1,…, Q, where the synthetic zoom will be performed at each step of the iterative process.

3. Modeling the Incident Field

[17] The incident field data play a crucial role in the imaging process since the knowledge/availability of Eincv(x, y) in the investigation domain adds new information. In fact, as it can be noticed in the equation defining the multiresolution cost function (9), it allows to define another constraint (2) for the problem solution then reducing the ill-posedness of the inverse problem [Bertero and Boccacci, 1998] since such a term can be also considered as a sort of “regularization term”. Clearly, an erroneous or imprecise knowledge of the incident field could considerably affect the reliability of the functional and consequently of the overall imaging process since (9) controls the minimization procedure. As a matter of fact, in many practical situations, the incident field is only available at the measurement points belonging to the observation domain, Eincv(equation image), (equation image) ∈ DM. Such a situation is commonly encountered when dealing with real data because of the complexity and difficulties in collecting reliable and independent measures in a dense grid of points. Hence, to fully exploit the knowledge of the incident field and before facing with the data inversion, it is mandatory to develop a suitable model able to predict the incident field radiated by the actual electromagnetic source in the investigation domain, Eincv(x, y), (x, y) ∈ DI. Towards this aim, in the reference literature [see Belkebir and Saillard, 2001, and references therein], different solutions have been proposed. They are mainly based on plane or cylindrical waves expansions, since far-field conditions are usually satisfied. In this paper, such models will be analyzed and a new distributed model will be proposed. More in detail, let us consider (1) the Plane-Waves Model (PW-Model) where the incident field is modeled as the superposition of a set of W plane waves

equation image

θv being the incident angle, k0 the free-space propagation constant, and Aw the amplitude of w-th wave; (2) the Concentric-Cylindrical-Waves Model (CCW-Model) where the radiated field is represented through the superposition of cylindrical waves according to the following expansion

equation image

where Aw is an unknown coefficient, Hw(2) indicates the second kind w-th order Hankel function, ρ is the distance between the observation point located at (x, y) and the phase center of the radiating system where the w-th line source is placed and ϕv the corresponding angle; (3) the Distributed-Cylindrical-Waves Model (DCW-Model) where the actual source is replaced with a linear array of equally spaced line-sources, which radiates an electric field given by

equation image

where A(xw, yw) is the unknown coefficient related to the w-th element and ρw the distance between the observation point and the w-th line source.

[18] Such models are completely defined when the set of unknown coefficients, Aw or A(xw, yw), have been determined. Therefore, the solution of an inverse source problem, where the known terms are the values of the incident field measured in the observation domain Eincv(equation image), is required. More in detail, the following system has to be solved:

equation image

or in a more concise form

equation image

where (a) for the PW-Model Gms = equation image, dm = xmcosequation imagev + ymsinequation imagev, and Is = As, s = 1,…, S, S = W; (b) for the CCW-model Gms = Hs(2)(k0ρm)ejsequation imagev, ρm = equation image, (xsource, ysource) being the location of the source, and Is = As−1−W, s = 1,…, S, S = 2W + 1; (c) for the DCW-model Gms = −equation imageH0(2)(k0ρms), ρms = equation image, and Is = A(xs, ys), s = 1,…, S, S = W.

[19] Unfortunately, (13) involves the limitations typical of an inverse-source problem [see, e.g., Devaney and Sherman, 1982]. In particular, [equation image] is ill-conditioned and the solution is usually nonstable and nonunique. Now, the problem of determining [I] from the knowledge of the incident field can be recast as the inversion of the linear operator [equation image] through the SVD-decomposition [Natterer, 1986]

equation image

where

equation image

and

equation image

[20] Owing of the properties of [equation image], the sequence of singular values {γs}s=1S will be decreasing and convergent to zero. Consequently, the solution of equation (14) does not continuously depend on problem data and the unavoidable presence of the noise, due to measurement errors as well as to an inaccurate model of the experimental setup, could produce an unreliable source synthesis.

[21] In the next section, an exhaustive numerical analysis will be carried out to assess the robustness of the IMSA against the error in the incident field data and to better understand “how” and “how much” the model of the actual electromagnetic source affects the IMSA performances.

4. Numerical Analysis

[22] In this section, such an assessment will be performed by considering different targets and starting from experimental data. The scattered data refers to the data set available at the “Institute Fresnel” - Marseille, France”. As described in Belkebir and Saillard [2001], Testorf and Fiddy [2001], and Marklein et al. [2001] and sketched in Figure 2, the bistatic radar measurement system consists of an emitting antenna placed at rs = 720 ± 3 mm from the center of the experimental setup and a receiver which collects equally spaced (5°) measurements of Etotv(equation image) and Eincv(equation image) on a circular investigation domain of radius rm = 760 ± 3 mm. Note the presence of a blind-sector of θl = 120 around the emitting antenna (Figure 2). The scatterers considered in the following experiments are shown in Figures 2a–2c for reference.

Figure 2.

Numerical Experiments: (a) off-centered homogeneous circular cylinder (Real data set “Marseille” [Belkebir and Saillard, 2001] - “dielTM_dec8f.exp”), (b) two homogeneous circular cylinders (Real data set “Marseille” [Belkebir and Saillard, 2001] - “twodielTM_8f.exp”), and (c) U-shaped metallic cylinder (Real data set “Marseille” [Belkebir and Saillard, 2001] - “uTM_shaped.exp”).

[23] In the first example (Figure 2a), we will consider the circular dielectric profile (Lref = 30 mm in diameter) positioned about 30 mm from the center of the experimental setup (xcref = 0.0, ycref = −30 mm) and characterized by a homogeneous permittivity ɛr(x, y) = 3.0 ± 0.3 [τ(x, y) = 2.0 ± 0.3]. The square investigation domain, LDI = 30 cm sided, is partitioned in N = 100 homogeneous discretization domains and the reconstruction is performed by exploiting all the available measures (M(v) = 49, v = 1,…, V) and views (V = 36), but using mono-frequency data (f = 4 GHz).

[24] The performances of the IMSA in terms of quantitative as well as qualitative imaging have been assessed considering necessarily the State Term during the minimization of the cost function (9) and thus introducing the information-content of the incident electric field. (Some examples of algorithms employing only the Data Term can be found in the special section [Belkebir and Saillard, 2001]).

[25] To do this, two simple models for the field emitted by the probing antenna have been preliminary taken into account. The first one represents the radiated field with a plane wave (W = S = 1), the other with a cylindrical wave (W = 0, S = 1). The amplitudes of the modeled incident waves are estimated according to the SVD-based procedure detailed in section 3 starting from the knowledge of the values of the incident field measured in the forward direction and available directly from the experimental data set. They turn out to be ∣Aw=1(PW-Model)∣ = 1.23 and ∣Aw=0(CCW-Model)∣ = 17.27, respectively.

[26] In spite of the inaccuracy in reproducing the values of the incident field collected at the measurement points (Figures 3a–3d), starting from such rough models the IMSA is able to localize the unknown target with a satisfactory degree of accuracy as shown in Figure 4 and confirmed by the geometric parameters reported in Table 1.

Figure 3.

Comparisons between the incident field measured in DM and the values synthesized by means of the PW-Model ((a) amplitude and (b) phase), and CCW-Model ((c) amplitude and (d) phase).

Figure 4.

Reconstructions of an off-centered homogeneous circular cylinder (Real data set “Marseille” [Belkebir and Saillard, 2001] - “dielTM_dec8f.exp”) achieved at the convergence step of the inversion procedure by modeling the radiated field through (a) the single PW-Model and (b) the single CCW-Model.

Table 1. Reconstruction of an Off-Centered Homogeneous Circular Cylinder (Real Data Set “Marseille” [Belkebir and Saillard, 2001] - “dielTM_dec8f.exp”) - Estimated Geometrical Parameters
 Ioptequation image, mmequation image, mmequation image, mm
Data Equation Only43.00−16.0058.00
PW-Model (W = S = 1)4−2.00−26.1034.00
PW-Model (W = S = 20)4−2.41−22.7345.44
CCW-Model (W = 0, S = 1)2−1.81−26.1035.20
CCW-Model (W = 5, S = 11)21.57−10.2360.08
DCW-Model (W = S = 15)3−1.90−26.1027.40

[27] As far as the single-plane-wave model is concerned, it should be pointed out that the reconstructed contrast is characterized by an average value of the object function equal to equation image = 2.1, then very close to the actual value of the real target. (If not specified, the IMSA is used to reconstruct the real part of the object function.) However, several pixels belonging to the area of the reference profile present a larger object function values [τ(xn, yn) = 2.5] and the retrieved object contour does not accurately reproduce a circular shape.

[28] With respect to the PW model, a better reconstruction is obtained when a little more complex source model (i.e., the single CW-Model) is used as it can be observed in Figure 4b and inferred from the values of the error figures (which quantify the qualitative imaging of the scatterer under test) given in Table 2 and defined as follows

equation image
equation image

where the subscript “ref” refers to the actual profile.

Table 2. Reconstruction of an Off-Centered Homogeneous Circular Cylinder (Real Data Set “Marseille” [Belkebir and Saillard, 2001] - “dielTM_dec8f.exp”) - Error Figures
 ρΔequation image
PW-Model (W = S = 1)0.04617.32.1
CCW-Model (W = 0, S = 1)0.04513.31.7
DCW-Model (W = S = 15)0.0458.71.8

[29] According to the indications drawn from these experiments, which point out that even a rough representation of the incident field significantly benefits the inversion of the scattered field data, the successive procedural step will be aimed at refining the numerical model of the electromagnetic source to further improve the effectiveness of the retrieval process. However, it should be noticed out that using a wrong, even though complex, model might actually degrade the reconstruction, thus great care is needed in defining the most suitable complex model. In order to point out such a concept, the problem has been studied considering the previous scattering geometry, but using numerical “measured” data with a controllable degree of noise. More in detail, the following analysis has been carried out. Different electromagnetic sources have been considered to illuminate the scenario under test (i.e., “PW-Source”, “CCW-Source”, and “DCW-Source”) and starting from the values of the incident field synthetically computed in the observation domain Eincv(equation image), (equation image) ∈ DM, various source models (i.e., “PW-Model”, “CCW-Model”, and “DCW-Model”) have been synthesized. Then, a noise characterized by a SNR = 20 dB has been superimposed to the data and the reconstruction process has been carried out starting from the different source models previously determined. The obtained results in terms of qualitative (18)–(19) and quantitative error figures ξ(j) defined as

equation image

where N(r)(j) can range over the whole investigation domain (jtot), or over the area where the actual scatterer is located (jint), or over the background belonging to the investigation domain (jext), are reported in Table 3. As expected, the use of a model corresponding to the actual source turns out to be the most suitable choice and more complex modeling cause larger errors. As an example, let us consider the PW-source. When the profile retrieval is performed using the PW-model then the reconstruction error is equal to ξtot = 0.30. Otherwise, ξtot(DCW-Model) = 13.30 and ξtot(CCW-Model) = 20.53. Similar conclusions hold true also for other illuminations and source models in terms of quantitative error figures, as well.

Table 3. Reconstruction of an Off-Centered Homogeneous Circular Cylinder (SNR = 20 dB) for Different Illuminations and Considering Various Electromagnetic Sources - Quantitative Error Figures ξtot, ξint, and ξext
ξtotPW-ModelCCW-ModelDCW-Model
PW-Source0.3020.5313.30
CCW-Source16.610.370.45
DCW-Source16.440.360.34
ξintPW-ModelCCW-ModelDCW-Model
PW-Source13.7958.6444.66
CCW-Source20.3116.3817.00
DCW-Source19.9825.2215.29
ξextPW-ModelCCW-ModelDCW-Model
PW-Source0.2019.7113.06
CCW-Source16.580.250.32
DCW-Source16.420.170.22

[30] Consequently, the more complex source configurations described in section 3, which consider the superposition of plane waves or of cylindrical waves, have been taken into account in order to define the most suitable source model. In such a framework since the numerical description of the actual source in the real measurement setup is only partially or not generally available, the optimal model has to be defined by looking for the most suitable number of the unknown source coefficients, S, and corresponding values, As, s = 1,…, S. For each of the source models, S has been chosen by looking for the configuration that provides a satisfactory matching between measured and numerically computed values of the incident field in the observation domain. Such a matching has been evaluated by computing the following parameter

equation image

where Re{·} and Im{·} stand for the real and imaginary part, respectively, and the superscript equation image indicates a numerically estimated quantity.

[31] In Figure 5, the behavior of the “matching parameter” is displayed for different source models. As can be observed, μ reduces when S increases. Thus, the optimal number of source coefficients, Sopt, has been heuristically defined as the value belonging to a stability region. Consequently, the optimal values have been set to: Sopt(PW-Model) = 20 (where μ ≃ 4 × 10−4) and Sopt(CCWModel) = 11 (where μ ≃ 10−4). The amplitudes of the weighting source coefficients are shown in Figure 6. The magnitudes of the CCW-Model coefficients (Figure 6b) are very large when compared to those of the single PW-Model or single CCW-Model. As expected, the corresponding radiated-field distributions inside the investigation domain DI (Figures 7c and 7d) turn out to be unacceptable (for comparison purposes, the plot of the incident electric field computed by means of the single CCW-Model is given in Figures 7e and 7f. Moreover, Figures 7a and 7b show how even the incident field synthesized by means of the PW-Model presents rather high values with respect to the distribution of Figures 7e and 7f. Since the incident field is the guess value for the optimization of the internal field, a completely wrong starting distribution may considerably affect the whole retrieval procedure. Accordingly, the adopted inversion strategy is not able to correctly estimate neither the shape nor the dielectric distribution of the unknown scatterer (Figure 8). As far as the case related to the PW-Model is concerned, it should be noted that the iterative process is stopped at the fourth step (Table 1) and the quality of the reconstructed profile (Figure 8a) turns out to be strongly reduced (if compared to that of Figure 4a) in terms of qualitative as well as quantitative imaging. Similar indications can be drawn from the analysis of the retrieved distribution obtained with the CCW-Model. However, reducing the number of terms in the expansion could lead to better results like, for example, those presented in the special section [Belkebir and Saillard, 2001] and those obtained in this work by using S = 1. Notwithstanding this, the value suggested by the indicator has been used in the proposed experiments.

Figure 5.

Fitting between computed and measured values of the radiated field in the observation domain versus various numbers of source coefficients, S, and for different source models.

Figure 6.

Behavior of weighting source coefficients as a function of the index w for (a) the PW-Model (S = 20) and for (b) the CCW-Model (S = 11).

Figure 7.

Plots of the radiated fields (V = 1) computed by means of the PW-Model (S = 20) (amplitude (a) and phase (b) distributions), the CCW-Model (S = 11) (amplitude (c) and phase (d) distributions), and the single CCW-Model (S = 1) (amplitude (e) and phase (f) distributions).

Figure 8.

Reconstructions of an off-centered homogeneous circular cylinder (Real data set “Marseille” [Belkebir and Saillard, 2001] - “dielTM_dec8f.exp”) achieved at the convergence step of the inversion procedure by modeling the radiated field through (a) the PW-Model (S = 20) and (b) the CCW-Model (S = 11).

[32] The obtained discouraging results can be properly motivated by observing the singular-values spectrum (Figure 9) and by computing the condition number η of the linear matrix operator [equation image] (defined as follows η = equation image), which clearly point out an intrinsic instability of the system and the ill-conditioning of the problem. In more detail, the ill-conditioning index turns out to be equal to η(PW-Model) = 41.07 and to η(CCW-Model) = 5.62 × 107, respectively.

Figure 9.

Normalized behavior of the singular values of [equation image] for (a) the PW-Model (S = 20) and for (b) the CCW-Model (S = 11).

[33] A possible solution for suitably defining the source model and, consequently, for improving the resolution accuracy of the retrieval process (alternative to employ a truncated-SVD regularization algorithm as suggested by the step-like behavior of the singular-values spectrum), is to define a spatially distributed line-source model as described in section 3.

[34] According to the procedure for choosing the number as well as the magnitude of the source weights previously described, a reasonable configuration is Sopt(DCWModel) = 15 (Figure 5) with the coefficients distributed as shown in Figure 10a. For completeness, in order to give an idea of the fitting between measured and computed data, Figures 10b and 10c display the values of the amplitude and phase of the radiated-field computed in the observation domain. Moreover, Figure 11 gives a gray-level representation of the incident electric field synthesized in the investigation domain.

Figure 10.

Radiated-field modeling: DCW-Model (S = 15). (a) Behavior of weighting source coefficients as a function of the index w. Comparison between the incident field measured in DM and the numerically computed values ((b) amplitude and (c) phase).

Figure 11.

Plots of the radiated field (V = 1) computed by means of the DCW-Model (S = 15) (amplitude (e) and phase (f) distributions).

[35] The use of such a model for the incident field allows a significant improvement in the reconstruction. Such a result can be appreciated in Figure 12 where the gray-level representation of the object function is given. In particular, for this representative configuration, also the intermediate reconstructions (Figures 12a–12c) of the multiscaling process are reported in order to show how the profile improves during the iterative procedure. As it can be noticed, even though the computational domain is not finely discretized at the first step (Figure 12a), the IMSA iteratively increases the resolution in the RoI in order to obtain an accurate discretization at the convergence step (Figure 12c) where a meaningful profile is obtained. As a matter of fact, the localization as well as the dimensioning error of the convergence step (Figure 12c) reduces with respect to the other source models (ρ(DCW-Model) = 0.045λ0, Δ(DCW-Model) ≈ 9 - Table 2) and the homogeneity of the actual scatterer is better reproduced. As far as the explanation of the better performance of such an approach with respect to the other source-synthesis modalities is concerned, it is mainly motivated by the faithful and stable reproduction (Figures 10b and 10c) of the actual values of the field measured in the observation domain.

Figure 12.

Reconstruction of an off-centered homogeneous circular cylinder (Real data set “Marseille” [Belkebir and Saillard, 2001] - “dielTM_dec8f.exp”) achieved at (a) i = 1, (b) i = 2 and (c) at the convergence step (i = 3) of the inversion procedure by modeling the radiated field through the DCW-Model (S = 15).

[36] To further assess the robustness and the effectiveness of the IMSA, by validating the radiated-field synthesis as well, the second example considers a multiple-scatterers scenario (“twodielTM_8f.exp” - [Belkebir and Saillard, 2001]). Under the same assumptions of the previous example in terms of measures, radiation frequency, and views as well as extension and partitioning of the investigation domain, two dielectric (τ(q) = 2.0 ± 0.3, q = 1,…, Q, Q = 2) circular (Lref(q) = 30 mm in diameter) cylinders are placed 90 mm from each other (Figure 2b).

[37] Figure 13 shows the results of the reconstruction process in correspondence with different source models. As can be seen, whatever the stable source synthesis the two targets are correctly located and dimensioned with a satisfactory accuracy. Certainly, the more sophisticated synthesis approach (DCW-Model - S = 15) allows to obtain a better reconstruction as confirmed by the geometric parameters of the retrieved profiles resumed in Table 4. In order to show the capabilities of the IMSA in estimating the lossless nature of the dielectric scatterers, the reconstruction corresponding to the DCW-Model has been run using a blind inversion scheme, that is without a priori information of its characteristics. Such assumption does not exploit the alternative definition of the solution space, which allows to reconstruct only the real part of the object function. Accordingly, Figure 13d points out that the minimum of the imaginary part of the object function is 0.08 (corresponding to σ = 1.78 × 10−3equation image).

Figure 13.

Reconstructions of two homogeneous circular cylinders (Real data set “Marseille” [Belkebir and Saillard, 2001] - “twodielTM_8f.exp”) achieved at the convergence step of the inversion procedure by modeling the radiated field through (a) the single PW-Model, (b) the single CCW-Model and the DCW-Model (S = 15) [(c) real part and (d) imaginary part].

Table 4. Reconstruction of Two Homogeneous Circular Cylinders (Real Data Set “Marseille” [Belkebir and Saillard, 2001] - “twodielTM_8f.exp”) - Estimated Geometrical Parameters (d(Iopt) = equation image
 PW-Model (W = S = 1)CCW-Model (W = 0, S = 1)DCW-Model (W = S = 15)
equation image,mm12.4212.8913.17
equation image,mm40.7742.9645.87
equation image,mm46.9440.5032.70
equation image,mm2.252.231.88
equation image,mm−45.48−44.91−45.27
equation image,mm43.7040.8632.76
equation image,mm86.8488.5091.84
Iopt333

[38] Finally, in order to complete the validation of the approach, the last example deals with a metallic structure. The scatterer is an U-shaped metallic cylinder (Figure 2c) and the reconstruction is performed starting from the complete data collection of the data set “uTM_shaped.exp” [Belkebir and Saillard, 2001] at the working frequency of f = 4 GHz. According to the trategy proposed in Van den Berg et al. (1995), only the imaginary part of the object function has been retrieved considering a lower bound in the reconstructed contrast and if at some iteration the estimated Im{τ(x, y)} is lower than τImmax = −15.0, then the contrast is replaced by τImmax. As a result, the imaginary part of the retrieved profile in the configuration with the DCW-Model for the synthesis of the radiated field, is depicted in Figure 14. At the convergence step (Iopt = 4), the reconstruction clearly reveals that we are dealing with a U-shaped target. The outer and the inner contour of the “U” are well reproduced (even though little artifacts appear) confirming the effectiveness of approach in shaping and locating dielectric as well metallic scatterers.

Figure 14.

Reconstruction of an U-shaped metallic cylinder (Real data set “Marseille” [Belkebir and Saillard, 2001] - “uTM_shaped.exp”) achieved at the convergence step of the inversion procedure by modeling the radiated field through the DCW-Model (S = 15).

5. Conclusions

[39] The Iterative Multiscaling Approach has been tested against experimentally acquired data by focusing the attention on its robustness as regards different mathematical models used to synthesize the incident electric field. The effectiveness of the iterative minimization of the cost functional in reconstructing unknowns scatterers presents a certain degree of sensitivity to the model of the incident field used to formalize the constraint stated by the State Equation. By considering a more complex approximation model (DCM-Model), satisfactory localizations and reconstructions have been carried out by indicating the positive effect of a suitable synthesis methodology on the inversion process.

[40] However, even though an accurate approximation model generally might result in a more accurate reconstruction, which complex model is more appropriate for the incident field may depend on the measurement setup, especially the microwave source configuration. For example, for simple plane-wave incident field, using the PW-model might reduce artifacts which result from measurement noise. So future investigations are needed by considering other experimental data sets (currently not-available, but under development) to generalize the conclusions of such an analysis.

[41] Moreover, the results of the numerical analysis carried out in the paper and the comparison with the reconstructions obtained in the related literature suggest that improved imaging techniques (e.g., multifrequency techniques) or additional regularization terms may probably diminish the impact of the incident field model. Since this point has not directly investigated other researches will be aimed at further improving the effectiveness of the IMSA by considering multifrequency strategies, further regularization terms and more effective optimization algorithms for the minimization of the multiresolution cost function in order to verify the above hypothesis.

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