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Keywords:

  • microwave imaging;
  • inverse scattering;
  • Iterative Multiscaling Approach

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Mathematical Formulation
  5. 3. Numerical Analysis
  6. 4. Validation Against Experimental Data
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[1] In a reconstruction procedure based on the iterative solution of inverse scattering integral equations, the quality of the final image depends on both the numerical and experimental noise. Numerical noise is related to the accuracy of the numerical representation of the microwave imaging apparatus and system geometry. Experimental noise refers to the nonideal electromagnetic conditions in which the data acquisition is performed. This paper provides a systematic evaluation of the impact of the most significant sources of numerical and experimental noise on the reconstruction quality when the Iterative Multiscaling Approach (IMSA) is used. The assessment of the robustness and stability of the IMSA is carried out by considering synthetic as well as real data. The achieved results provide detailed indications on the range of applicability of the IMSA for qualitative and/or quantitative imaging purposes.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Mathematical Formulation
  5. 3. Numerical Analysis
  6. 4. Validation Against Experimental Data
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[2] Applications of imaging and detection by using scattered field data range from far-field imaging [Franceschetti and Lanari, 1999] to near-field imaging by considering microwave [Bolomey, 1991, 1995; Zoughi, 2000] as well as other electromagnetic frequencies [Kak and Slaney, 1988; Baltes, 1980]. Many of these imaging applications can be dealt within the same methodological framework. Then it turns out to be very attractive and justified to study numerical approaches able to solve fundamental problems of inverse scattering since any scientific advance in a particular application automatically (or in an indirect way) provides a useful contribution for the progress in related fields.

[3] A brief review of the more recent literature shows that a large number of very effective iterative nonlinear procedures has been proposed. Starting from an initial guess, the parameters of interest are iteratively updated by minimizing a suitably defined cost function involving the measured scattered field data. Generally speaking, two methodological approaches can be highlighted depending on whether the field inside the investigation domain is computed as a solution of the direct problem (in correspondence with the best estimate of the dielectric distribution) at each iteration [Joachimowicz et al., 1991; Chew and Wang, 1990; Franchois and Pichot, 1997] or as another unknown to be determined during the minimization procedure [Kleinman and van den Berg, 1992; Van den Berg and Abubakar, 2001]. The IMSA [Caorsi et al., 2003] belongs to the second class of iterative procedures. It is devoted to fully exploit all the available information content of scattered data. Owing to the limited amount of information content in the input data, it would be problematic to parameterize the investigation domain in terms of a large number of pixel values (in order to achieve a satisfying resolution level in the reconstructed image). In order to overcome this drawback, an iterative parameterization of the test domain, performing a synthetic zoom on the region to whom the scatterer belongs, allows to achieve the required reconstruction accuracy only in the “significant” region under test.

[4] Such a technique has yielded very promising results in processing preliminary synthetic test cases [Caorsi et al., 2003]. However, in order to develop a reliable reconstruction method robust to both modeling errors and uncertainties on data, it is mandatory to evaluate its limitations through accurate investigations. Consequently, a better understanding of the operational capabilities of the IMSA requires an extended assessment of the noise robustness as well as a systematic study of the impact of both experimental and model errors. Toward this end, this paper is aimed at providing an assessment of the effects of the major sources of experimental and model noise on the quality of the reconstruction. Accordingly, synthetic as well as laboratory-controlled experiments are taken into account in order to evaluate the effectiveness of the approach in dealing with customized scenarios as well as reference benchmarks.

[5] The paper is organized as follows. First, an outline of the iterative multiscaling approach will be concisely described (section 2). Sections 3 and 4 will present selected representative results for illustrating the effects of the most critical experimental and numerical parameters on the reconstruction accuracy. Final comments and conclusions will be drawn in section 5.

2. Mathematical Formulation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Mathematical Formulation
  5. 3. Numerical Analysis
  6. 4. Validation Against Experimental Data
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[6] Let us take as a starting point the nonlinear multiresolution inversion approach proposed in the work of Caorsi et al. [2003], briefly recalled in the following. At each fixed frequency f, the two-dimensional inverse scattering problem can be mathematically described through the data equation and the hstate equation [Jones, 1964]

  • equation image
  • equation image

v = 1,…, V being the index indicating different multiillumination/multiview positions and ρ = equation image. Moreover, Eequation image(.) and Eequation image(.) are the problem data namely the incident electric field measured inside the investigation domain DI where the unknown scattering object is supposed to be located, and the scattered electric field, collected in the observation domain (DM) located outside DI. As far as the problem unknowns are concerned, they are the contrast function τ(x, y), (x, y) ∈ DI describing the dielectric properties of the scenario under test and the electric field inside the investigation domain, Eequation image(x, y), v = 1,…, V, (x, y) ∈ DI.

[7] To retrieve the unknown functions, the IMSA performs a synthetic zoom to fully exploit the limited information content of inverse scattering data [Bucci and Franceschetti, 1989] allowing an accurate resolution of the scatterer under test. More in detail, the method starts (s = 1, s being the index related to the scaling step) from a coarse (R = 1, R being the index of the resolution level) representation of the investigation domain DI and iteratively defines a subgridding of the area where the scatterer is located. By using the “knowledge” of the scenario under test achieved at the previous steps, the iterative approach defines an estimate of the scatterer location and occupation. This enables an efficient re-allocation of the unknowns to the “focused” area according to a multiresolution strategy. Consequently, an higher resolution level (R = s) is adopted only for the reduced investigation domain where the clustering procedure has estimated the presence of the scatterer. Mathematically, the multiscaling process is implemented by considering the following algorithmic procedure. At each step (s = 1,…, Sopt) of the iterative process, a multiscaling multiresolution function is defined

  • equation image

where w is the weighting function

  • equation image

and where number of discretization domains belonging to the “focused” area (n(R) = 1, …, N(R)) is chosen equal to the essential dimension of the scattered data [Bucci and Franceschetti, 1989] according to the criterion defined in the work of [Isernia et al., 2001].

[8] After the minimization of (3), where a set of conjugate-gradient iterations (k being the iteration index) is performed not modifying the discretization grid, a new focused investigation domain, D(s−1) is defined. Such a squared area is centered at

  • equation image
  • equation image
  • equation image

and L(s−1)-sided

  • equation image
  • equation image

where ℜ stands for the real or the imaginary part and ρequation image = equation image.

[9] Then a noise filtering is performed in order to eliminate some artifacts in the reconstructed image. The new dielectric distribution turns out to be:

  • equation image

where τth is a fixed threshold heuristically defined (τth = 0.2 maxequation image and τ0 is the value of the contrast function for the background medium. Moreover, J is the dimension of the complete neighborhood system of the subdomain located at (equation image) and (xj, yj) indicates a neighboring position.

[10] The multiresolution procedure is iterated until a “stationary condition” for the quantitative imaging of the scatterer under test is achieved (s = Sopt). This condition holds when

  • equation image

where ηu, u = xc, yc, L are fixed thresholds.

[11] Such a procedure can be easily extended to multiple-scatterers geometries by considering a suitable clustering procedure [see Jain, 1989, and the references therein] aimed at defining the number of scatterers belonging to the investigation domain and the regions where the synthetic zoom will be performed at each step of the iterative process.

3. Numerical Analysis

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Mathematical Formulation
  5. 3. Numerical Analysis
  6. 4. Validation Against Experimental Data
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[12] The complexity of the inverse scattering problem considerably increases if one considers a real environment in which the data are collected within a certain degree of approximation due to unavoidable measurement errors, the uncertainty in the location of the interrogating source, and the mechanical positioning of the field receivers. In this framework, different situations can be identified:

[13] 1. The measures of the incident and of the scattered fields (i.e., the inverse scattering data) are affected by the experimental noise.

[14] 2. The interrogating source is not completely known (or cannot be modeled accurately) or of stochastic type.

[15] 3. The experimental system (and in particular the electromagnetic sensors) is moved by means of a mechanical apparatus with some tolerances in the positioning.

[16] The analysis of these situations (equivalent to noisy scenarios, compared to the ideal situation, perturbed by various error sources) is carried out by considering a reference test case. The scattering scenario is constituted by an off-centered squared (Lref = 0.48 λ) scatterer characterized by an object function τref(x, y) = 0.5 and located at (xequation image = −0.24 λ, yequation image = 0.48 λ) (Figure 1a). Nevertheless, in order to allow for more general interpretations, some variations of the reference geometry (in terms of larger/smaller dimensions and inhomogeneous characteristics) will be taken into account as well. The actual scatterer is enclosed in a square investigation domain LI = 2.4 λ-sided and illuminated by a set of V = 4 TM-polarized plane waves impinging with different incident angles θv = equation image, v = 1, …, V. The scattered field data are collected in M(v) = 21, v = 1,…, V equally spaced measurement points lying on a circular observation domain ρDM = 1.7 λ in radius. Note that, according to [Isernia et al., 2001], properties of scattered fields make it possible to acquire essentially all the information available from scattering experiments by simply choosing for each view a number of measurements M(v) slightly larger than NM = 4πρDM and that the number of independent data, arising from equation imageM(v) measurements, will be given by the minimum between equation image and equation image. Consequently, the “focused area” has been discretized in N(R) = 6 domains.

image

Figure 1. Impact of the experimental noise on the reconstruction accuracy, estimated permittivity distributions of a square homogeneous dielectric cylinder (τ = 0.5). (a) Reference profile. Retrieved profile at S = Sopt for (b) noiseless conditions ((SNR)scatt = (SNR)inc = ∞), (c) (SNR)scatt = ∞ and (SNR)inc = 20 dB, (d) (SNR)scatt = ∞ and (SNR)inc = 5 dB, (e) (SNR)scatt = (SNR)inc = 20 dB, and (f) (SNR)scatt = (SNR)inc = 5 dB.

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[17] As far as the inversion method is concerned, the parametric configuration, defined in the work of Caorsi et al. [2003] and heuristically selected, has been adopted (equation image = 1% and ηL = 5%).

[18] In order to quantify the impact of the equivalent noise sources on the reconstruction accuracy of the IMSA, some error figures are defined. Namely, the center location error (12) and the occupation area error (13) aimed at quantitatively evaluating the effectiveness in the qualitative imaging

  • equation image
  • equation image

and the reconstruction errors (14) which give a measure of the quantitative imaging

  • equation image

where Nequation image ranges over the whole investigation domain (i [RIGHTWARDS DOUBLE ARROW] tot), or over the area where the actual object is located (i [RIGHTWARDS DOUBLE ARROW] int), or over the background belonging to the investigation domain (i [RIGHTWARDS DOUBLE ARROW] ext).

3.1. Effects of the Experimental Noise

[19] The impact of the experimental noise on the effectiveness of the IMSA has been evaluated by considering synthetically generated data to avoid the effects of other error sources. The experimental noise has been simulated by adding to the problem data additive Gaussian noises, ηscatt and ηinc

  • equation image
  • equation image

with zero mean value and standard deviations given by

  • equation image

and

  • equation image

SNR being the signal-to-noise ratio.

[20] For determining the effects of the noise on the reconstruction, several simulations have been performed with different values of the signal-to-noise ratio ranging between 5 and 100 dB. Since the nondeterministic nature of the Gaussian noise, ten independent realizations of the noise process with the same SNR have been performed. For a set of selected SNR values, the tomographic images have been reconstructed and samples of the reconstructions compared with the image of the noiseless case ((SNR)inc = (SNR)scatt = ∞; Figure 1b), where the dashed lines indicate the region occupied by the actual structure). As expected, the quality of the reconstructed image is a decreasing function of the SNR. Such a behavior is also confirmed by the values of the error figures (Table 1) and from the color-level representation of the error figures given in Figure 2 where each sample point is the average of the results of repeated realizations. However, the IMSA demonstrates its effectiveness showing a good accuracy in the estimate of the location (γ ≤ 5.50 × 10−2 and Av{γ} = 1.13 × 10−2 with a wide region ((SNR)scatt ≥ 20 dB, (SNR)inc ≥ 20 dB) in which γ ≅ 5.0 × 10−3 (Figure 2a) as well as of the shape of the unknown scatterer (Av{Φ} = 4.40%). Acceptable errors in the quantitative imaging of the investigation domain (Avtot} = 0.72% and Avext} = 0.48%) are obtained as well. However, it should be pointed out that there are not significant differences on the reconstruction if the additive Gaussian noise is added to the scattered data or to the incident field.

image

Figure 2. Impact of the experimental noise on the reconstruction accuracy, color-level representation of the (a and b) qualitative and (c–e) quantitative error figures for different values of the signal-to-noise ratio.

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Table 1. Impact of the Experimental Noise on the Reconstruction Accuracy: Error Figures
(SNR)scatt(SNR)incγΦξtotξintξext
6.30 × 10−40.200.116.643.0 × 10−8
20 dB6.48 × 10−410.030.8810.810.67
5 dB2.12 × 10−26.071.2414.830.75
20 dB2.23 × 10−311.890.868.420.64
5 dB1.48 × 10−24.021.1010.860.73
20 dB20 dB1.07 × 10−212.920.9711.130.66
5 dB5 dB5.50 × 10−216.115.4117.692.99

[21] In order to further assess the effectiveness of the IMSA in dealing with the experimental noise of Gaussian type, going from the reference geometry to more general scenarios, different positions (Figure 3), various dimensions of the square object (Figure 4), and some variations in the conductivity (Figure 5) of the scatterer under test when (SNR)scatt = (SNR)inc = 20 dB have been analyzed.

image

Figure 3. Impact of the experimental noise on the reconstruction accuracy, values of the error figures for different positions in the investigation domain of the reference scatterer ((SNR)scatt = (SNR)inc = 20 dB).

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image

Figure 4. Impact of the experimental noise on the reconstruction accuracy, behavior of the error figures versus the dimensions of the scatterer under test ((SNR)scatt = (SNR)inc = 20 dB).

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image

Figure 5. Impact of the experimental noise on the reconstruction accuracy, dependence of the error figures on the conductivity σ of the scatterer under test ((SNR)scatt = (SNR)inc = 20 dB).

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[22] Concerning the dependence of the reconstruction on the scatterer position (equation image being the distance of the scatterer center from the reference system origin), the achieved results confirm the effectiveness of the IMSA in terms of localization (Figure 3a) as well as accuracy in the reconstruction of the dielectric profile (Figure 3b). As expected, the error parameters slightly increase for the off-centered targets, but their values (γ ≤ 1.60 × 10−2, Φ ≤ 16.00%, ξ(tot) ≤ 1.2%) seem to be acceptable in the range 0 < d < 1.107 λ and allow accurate reconstructions.

[23] Similar conclusions can be carried out from the analysis of the sensibility of the method to the object dimensions. More in detail, it should be observed that the accuracy improves in correspondence with an increment of the area of the object (Figure 4). However, not-so-significant differences turns out to be in the qualitative (11.0 ≤ Φ ≤ 13.40%; Figure 4a) as well as in the quantitative imaging of the scenario under test (ξ(tot) ≅ 1.0%; Figure 4b). On the contrary, the performances of the approach reduces when the conductivity of the scatterer increases (Figure 5). In particular, the object area is overestimated (Φ∣σ=1.0 ≅ 13.40%) and the value of the reconstruction error is greater than ξ(tot) ≥ 5% when σ ≥ 0.5 S/m.

[24] Since the experimental error represents one of the main problem in dealing with real inverse scattering problem, for completeness the effects of a systematic (but non-Gaussian) additive noise are analyzed as well. Toward this aim, a random white noise has been added to the data samples according to the following procedure

  • equation image
  • equation image

where αscatt and αinc are two random numbers varying from −1 to 1.

[25] The inversion results when the synthetic data are corrupted by increasing the amount of random additive white noise are presented. As representative parameters, the localization error and the total reconstruction error are shown in Figures 6a and 6b, respectively. The IMSA is able to correctly localize the reference scatterer with an error γ ranging from 1.27 × 10−2 to 7.50 × 10−1. Moreover, the reconstruction error ξ(tot) turns out to be lower than 10% in a large set of values (0.0 ≤ αscatt ≤ 0.6, 0.0 ≤ αinc ≤ 0.7). It seems to assess the good ability of the method in suppressing the data noise.

image

Figure 6. Impact of the experimental noise (non-Gaussian model) on the reconstruction accuracy, color-level representation of the (a) center-location error γ and of the (b) total reconstruction error ξtot.

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3.2. Effects of the Numerical Noise

3.2.1. Receiver Positioning

[26] During the inversion procedure, the knowledge of the inverse scattering data is required as well as the coordinates where these measurements are collected. However, because of the tolerance of the mechanical positioning apparatus, some differences occur between the actual positions of the measurement points and those simulated in the numerical procedure. Numerically, such a situation is equivalent to a numerical noise corrupting scattering data. In order to avoid incorrect reconstructions, it should be taken into account. From a numerical point-of-view, it is convenient to simulate this scenario by considering, instead of the original measurement points (xp, yp) ∈ DP, p = 1,…, P, (where p can range over the investigation domain (p [RIGHTWARDS DOUBLE ARROW] n, DP [RIGHTWARDS DOUBLE ARROW] DI) or over the observation domain (p [RIGHTWARDS DOUBLE ARROW] m(v), DP [RIGHTWARDS DOUBLE ARROW] DM)), alternative (or noisy) observation points (equation imagep, equation imagep) whose positions are randomly chosen so that the following condition be fulfilled

  • equation image

where CΔ(xp, yp) is a circle, Δ = t · equation image in radius (t ∈ (0, 1) being a random number and ρi,j = equation image), centered at the pth measurement point. More in detail, Δ approximates the maximum error of the measurement system. It is related to the accuracy of the experimental positioning apparatus (which strongly depends on the application, but generally it is lower than few millimeters [Franchois et al., 1998]). Then measured data collected at (equation imagep, equation imagep) are assumed as input data at (xp, yp).

[27] Figure 7 shows the values of the averaged error figures for different values of the positioning parameters (t)inc, (t)scatt ∈ [0, 1]. The pictures show that the reconstruction capabilities strongly depends on the positioning of the scattered-field sensors. As far as the localization error is concerned, it turns out to be that 8.03 × 10−3 ≤ γ ≤ 6.78 × 10−1 and on average its value is equal to Av{γ} = 1.88 × 10−1 (greater of about one order in magnitude as compared with the average value related to the experimental noise, 1.88 × 10−1 versus 1.13 × 10−2) with a large variance (Var{γ} = 2.36 × 10−2). Larger errors occur when (t)scatt ≥ 0.65 and a wrong location of the scatterer in the region defined by ((t)scatt ≥ 0.85, (t)inc ≥ 0.30). However, Φ (Figure 7b), ξtot (Figure 7c), and ξext (Figure 7e) present a step-like behavior (with the amplitude of the errors almost independent of (t)inc) confirming the strong impact of (t)scatt on the retrieval process. More in detail, it happens that 0.17% ≤ Φ ≤ 14%, 0.27% ≤ ξtot ≤ 3%, and 1.34 × 10−3% ≤ ξext ≤ 0.9% for (t)scatt ≤ 0.65 with a nonnegligible increase otherwise.

image

Figure 7. Impact of the numerical noise on the reconstruction accuracy (receiver positioning), color-level representation of the (a and b) qualitative and (c–e) quantitative error figures for different values of the positioning tolerance (t).

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[28] For completeness, Figure 8 shows some examples of the reconstructions for different tolerances of the mechanical positioning system. As previously indicated, the reconstruction accuracy decreases in a more significant way with the increase of (t)scatt (Figures 8a–8c) if compared to the corresponding increase in (t)inc (Figures 8d–8f) as confirmed from the values of the error figures reported in Table 2. As an example, let us consider the images related to ((t)scatt = 1.0, (t)inc = 0.0) (Figure 8c) and to ((t)scatt = 0.0, (t)inc = 1.0) (Figure 8f). Figure 8f shows that even though the shape and the geometric informations are not accurately estimated (Φ = 25.24% and ξint = 12.97%), a structure can be identified in roughly the correct location (γ = 1.58 × 10−2, ξtot = 2.09% and ξext = 1.42%). On the contrary, a collection of small squares exist in a region close the actual object and multiple artifacts are present in the background. This produces a poor reconstruction (equation image ≅ 17 × equation image ≅ 4.5 × equation image and equation image ≅ 6 × equation image) (Figure 8c).

image

Figure 8. Impact of the numerical noise on the reconstruction accuracy (receiver positioning), estimated permittivity distributions of a square homogeneous dielectric cylinder (τ = 0.5). Retrieved profile at S = Sopt when (a) ((t)scatt = 0.2, (t)inc = 0.0), (b) ((t)scatt = 0.5, (t)inc = 0.0), (c) ((t)scatt = 1.0, (t)inc = 0.0), (d) ((t)scatt = 0.0, (t)inc = 0.2), (e) ((t)scatt = 0.0, (t)inc = 0.5), (f) ((t)scatt = 0.0, (t)inc = 1.0), (g) ((t)scatt = (t)inc = 0.2), and (h) ((t)scatt = (t)inc = 0.5).

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Table 2. Impact of the Numerical Noise on the Reconstruction Accuracy (Receiver Positioning): Error Figures
(t)scatt(t)incγΦξtotξintξext
0.20.05.92 × 10−28.021.359.940.25
0.50.01.82 × 10−211.151.4215.650.11
1.00.02.79 × 10−131.349.5023.468.92
0.00.21.99 × 10−25.420.436.210.032
0.00.54.99 × 10−28.390.8911.440.55
0.01.01.58 × 10−225.242.0912.971.42
0.20.26.37 × 10−28.451.3913.710.17
0.50.53.77 × 10−213.842.1517.430.66
1.01.06.78 × 10−133.3312.3033.5612.97

[29] To give an idea of the connection between tolerances and positions of the receivers as well as amplitudes of inverse scattering data, Figures 9 and 10 show the plots of the locations of the field sensors (Figures 9a and 10a) and related measured data for different values of (t)scatt and (t)inc, respectively.

image

Figure 9. Impact of the numerical noise on the reconstruction accuracy (receiver positioning), receivers positions (a) and scattered electric field data (b and c) Re{Eequation image}, Im{Eequation image}, v = 1) for various values of (t)scatt.

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image

Figure 10. Impact of the numerical noise on the reconstruction accuracy (receiver positioning), receivers positions inside the investigation domain (a) and incident electric field data, (b) Re{Eequation image}, (c) Im{Eequation image}, v = 1) for various values of (t)inc.

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[30] In order to generalize previous indications and according to the assessment strategy used in section 3.1, several numerical experiments have been carried out by assuming a noisy scenario characterized by (t)scatt = 0.1 and (t)inc = 0.2. Figure 11 summarizes the obtained results in terms of reconstruction errors. As can be observed, the total reconstruction error ξtot is almost constant both for different positions (0.5% ≤ ξtot ≤ 0.6%) and for different dimensions (4.1% ≤ ξtot ≤ 5%) of the reference scatterer. On the contrary, larger differences turn out to be by increasing the scatterer conductivity with an increment of the error values of about one order in magnitude (1.0% ≤ ξtot ≤ 12.5% and 0.9% ≤ ξext ≤ 10.1%).

image

Figure 11. Impact of the numerical noise on the reconstruction accuracy (receiver positioning), values of the reconstruction errors ξ(i), j = tot, int, ext (a) for different positions, (b) for different dimensions, and (c) for different values of the conductivity of the reference scatterer ((t)scatt = 0.1, (t)inc = 0.2).

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3.2.2. Interrogating Source Location

[31] To evaluate the effects of a not completely known or stochastic source on the incident electric field and consequently on the reconstruction algorithm (which assumes the incident field radiated by a deterministic source), let assume that the plane wave radiated by the electromagnetic source instead of propagating in the direction

  • equation image

propagates in another direction

  • equation image

where k0 = 2πfequation image and equation imageν is a random variable uniformly distributed in the range ζ(−equation image, equation image) (ζ being a random number in the range (0, 1)).

[32] The results of a large set of simulations (repeated ten times for each value of ζ in order to give accurate statistical informations) are presented in Figure 12 in terms of qualitative (Figure 12a) and quantitative error figures (Figure 12b). In general, the errors increase proportionally to ζ, except for the total reconstruction error. In such a case, a knee is present for a threshold value equal to ζth ≅ 0.4, which separates two saturation regions (ξext ≅ 0.7% for ζth < 0.4 and ξext ≅ 1.5% otherwise). Nevertheless, the same level of performances as was seen in the examples related to the experimental noise (section 3.1) can be observed here (Table 3). An accurate localization is achieved (γ = 5.81 × 10−3 for ζ = 0.25 and γ = 8.76 × 10−2 for ζ = 1.0) as pictorially resumed in Figure 13. However, with the increment of ζ, noticeable errors in the object shaping as well as in the dielectric reconstruction occur and the scatterer cannot be exactly shaped. However, the IMSA is always (also for ζ = 1.0) able to retrieve a structure that occupies a large subset of the true scatterer.

image

Figure 12. Impact of the numerical noise on the reconstruction accuracy (interrogating source location). (a) Qualitative and (b) quantitative error figures for different values of ζ.

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image

Figure 13. Impact of the numerical noise on the reconstruction accuracy (interrogating source location), estimated permittivity distributions of a square homogeneous dielectric cylinder (τ = 0.5). Retrieved profile at S = Sopt when (a) ζ = 0.25, (b) ζ = 0.50, (c) ζ = 0.75, and (d) ζ = 1.0.

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Table 3. Impact of the Numerical Noise on the Reconstruction Accuracy (Interrogating Source Location): Error Figures
StatisticsMinimumMaximumAverageVariance
γ1.62 × 10−48.76 × 10−23.25 × 10−26.20 × 10−4
Φ1.9512.136.969.07
ξtot0.781.521.220.09
ξint8.0719.8413.4510.37
ξext0.341.020.680.05

[33] Finally, for completeness, selected results of a “generalization” study are presented in Figure 14. The values of the total error are presented versus ζ and for different scattering scenarios. As expected (according to the conclusions drawn in previous subsections) largest variations occur for dissipative objects (e.g., 6.0% ≤ ξtot ≤ 12.7% when ζ = 1.0; Figure 14c). As far as other test cases are concerned, ξtot ranges between 0.5% and 4.0% (Figure 14a) and from 0.5% up to 1.5% (Figure 14b) whatever the scatterer position and dimension, respectively.

image

Figure 14. Impact of the numerical noise on the reconstruction accuracy (interrogating source location), behavior of the total reconstruction error ξ(tot) versus ζ (a) for different positions, (b) for different dimensions, and (c) for different values of the conductivity of the reference scatterer.

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4. Validation Against Experimental Data

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Mathematical Formulation
  5. 3. Numerical Analysis
  6. 4. Validation Against Experimental Data
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[34] Through the numerical analysis carried out in section 3, it has been shown that the IMSA is effective in reconstructing the contrast of penetrable objects and an accurate qualitative imaging of the scenario under test can be obtained. Moreover, the stability of the method with respect to realistic sources of electromagnetic noise has been verified. However, previous tests are concerned with synthetic (i.e., numerically simulated) scattering experiments. To fully assess the imaging capabilities of the IMSA when a realistic scenario is dealt with, some experiments with real data have been also considered. More in detail, the validation against experimental data is motivated by:

[35] 1. The need of validating the proposed approach on a benchmark and to judge its contribution in the framework of inverse scattering algorithms completing the validation performed with synthetic data (undoubtedly real data does not allow one to evaluate the performances of a method in an exhaustive set of possible scenario as in the case of numerically generated data but state a common benchmark).

[36] 2. The need of avoiding an inverse crime [Colton and Kress, 1992] consisting in testing the inversion algorithm on a data set generated by a forward solver closely related to that used during the inversion process.

[37] 3. The need of evaluating the accuracy of the approach in dealing with independent laboratory-controlled experiments. It is evident that the transfer of an inversion procedure to industrial applications greatly depends on demonstrating them successfully with experimental data. This not only guarantees that the scattering data are fully independent to the inverse scattering algorithm, but also check the sensitivity of the algorithm to realistic noise and modeling errors.

[38] 4. The need of estimating the feasibility of synthetic scenarios with respect to experimental ones in order to state the significance of numerical testing.

[39] In order to perform the experimental validation, the multiple-frequency angular-diversity bistatic data provided by the Institut Fresnel, Marseille, France [Belkebir et al., 2000] represent a significant benchmark. The data are obtained in a controlled environment where the target and the electromagnetic source are known and instrumental and experimental errors jointly occur. In particular, the availability of data measured at different frequencies provides a good opportunity to evaluate various noisy conditions on the same scattering geometry. Toward this end, all the reconstructions have been computed starting from data of one single frequency, avoiding multifrequency [Belkebir et al., 1997] or frequency-hopping approaches [Ferraye et al., 2003] that certainly could greatly improve the retrieval accuracy.

[40] More in detail, the experimental setup consists of a 2D bistatic measurement system with an emitter placed at a fixed position, while a receiver is rotating with an arm along the vertical cylindrical scatterer under test. The antennas are double ridged horn antennas linearly polarized with a frequency range from 1 GHz to 18 GHz. The distances between emitter center and receiver center of the experimental setup are set to 720 mm ± 3 mm and 720 mm ± 3 mm, respectively. The target rotates from 0 to 350 in steps of 10. The rotation of the receiver is from 60 to 300 in steps of 5. Obviously, the aspect-limited nature of the measurement setup leads to a reduction of the available information. Consequently, all V = 36 available views and, for each of them, M(v) = 49, v = 1, …, V, are used for the reconstruction.

[41] The first test case is related to a single circular dielectric cylinder and the corresponding data set “dielTM_dec8f.exp” is considered. The cylinder (characterized by an estimated object function equal to τref(x, y) = 2.0 ± 0.3) of circular cross section (Lref = 30 mm in diameter), is located at (xequation image = 0.0, yequation image = −30 mm) by assuming the coordinate reference system centered at the center of the experimental setup (a detailed description of the underlying experimental setup as well as of the data sets is given in [Belkebir and Saillard, 2001] and [Belkebir et al., 2000]). Moreover, it is assumed to belong to a square investigation domain of 30 × 30 cm2. The results of the reconstructions, starting from the free-space configuration, are shown in Figure 15. As can be observed, the reconstruction accuracy is very low in correspondence with the smallest frequency (f = 1 GHz) and for the higher frequencies (f = 7–8 GHz) where the reconstructed objects are diffused over a large part of the investigation domain (causing an overestimate of the occupation area, Lequation imagef=7 GHz = 97.06 mm and Lequation imagef=8 GHz = 76.46 mm). On the contrary, the location and the size of the object are satisfactorily retrieved at the intermediate frequencies (implicitly, the similarity on the geometric and dielectric estimates (Table 4) seems to indicate stable noisy conditions of the experimental setup in such a frequency range) with a notable improvement in the resolution accuracy as confirmed in Table 4. The maximum value of the estimated object function slightly exceeds the expected one to compensate the unavoidable smoothing in the reconstruction and it turns out to be within the given tolerance of the reference profile. These results further confirms the effectiveness of the IMSA also in considering aspect-limited data (as for the case of “Marseille” experiments).

image

Figure 15. Experimental validation, reconstruction of an off-centered homogeneous circular cylinder (real data set “Marseille” [Belkebir and Saillard, 2001], “dielTM_dec8f.exp”): (a) f = 1 GHz, (b) f = 2 GHz, (c) f = 3 GHz, (d) f = 4 GHz, (e) f = 5 GHz, (f) f = 6 GHz, (g) f = 7 GHz, and (h) f = 8 GHz.

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Table 4. Experimental Validation: Estimated Geometric and Dielectric Parametersa
fequation imageequation imageequation imageequation image (x, y)
MinimumMaximumAverage
  • a

    Here f [GHz]; equation image, equation image, equation image [mm].

1−4.39−31.0184.720.320.690.58
2−1.65−26.0033.381.062.171.98
3−2.07−26.7932.981.072.201.97
4−1.78−26.1530.721.472.151.99
5−1.87−25.9028.581.332.311.98
6−1.20−33.2032.761.502.102.02
7−3.59−14.2397.060.562.281.68
8−3.68−38.4676.460.322.301.77

[42] In order to validate the multiscaling inversion procedure on metallic-type targets, two sets at a fixed frequency (f = 4 GHz) from the database are then considered. These examples are concerned with a centered (Figure 16a; “rectTM_cent.exp”) and an off-centered (Figure 16b; “rectTM_dece.exp”) rectangular cylinders whose dimensions are 25.4 × 12.7 mm2. Under the assumption that the scatterers are highly conducting, the real part of the contrast has been ignored. Moreover, following the indications reported in the work of Van den Berg et al. [1995], if at some point in the iterative minimization the reconstructed Im{τ} is larger than Γmax the contrast is replaced by Γmax. In these examples, Γmax = 10.5. The reconstructed profiles are shown in Figure 16 where the exact locations of the boundaries of the actual objects are indicated by the dashed lines. These results seem to indicate that the shape of metal objects can be recovered with high quality by using the iterative multiscaling approach. However, future works will be needed to fully evaluate (by exploiting all the features of the IMSA as well) the effectiveness of such a technique in dealing with metallic objects in a large frequency range.

image

Figure 16. Experimental validation (real data set “Marseille” [Belkebir and Saillard, 2001]), reconstruction of metallic cylinders at the frequency f = 4 GHz: (a) data set “rectTM_cent.exp” and (b) data set “rectTM_dece.exp”.

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[43] For completeness, the last experiment deals with the reconstruction of a multiple-dielectric-scatterers configuration. The data related to the frequency f = 4 GHz of the “twodielTM_8f.exp” collection are used. In such a case, the actual configuration is composed of two dielectric cylinders, with the same dielectric properties described in the first experimental example, located 90 mm one far from the other and placed 30 mm from the center of the experimental setup. The reconstructed image (Figure 17) gives a clear location of the two cylinders without any artifacts or ghost targets. Such a result further assess the effectiveness of the IMSA in dealing with dielectric target. Moreover, it confirms the possibility to use the approach in complex scenarios and for single as well as multiple scatterers configurations.

image

Figure 17. Experimental validation, reconstruction of multiple dielectric cylinders at the frequency f = 4 GHz (real data set “Marseille” [Belkebir and Saillard, 2001], “twodielTM_8f.exp”).

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5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Mathematical Formulation
  5. 3. Numerical Analysis
  6. 4. Validation Against Experimental Data
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[44] In this paper, a systematic evaluation of the impact of the most significant sources of numerical and experimental noise on the reconstruction capabilities of the iterative multiscaling approach has been carried out. The performances of the approach, in terms of image quality and quantitative error parameters, have been shown for different conditions, both simulated and real. Starting from these experiments it may be stated that globally the IMSA demonstrated an acceptable stability and robustness to noisy conditions (in particular, the method turns out to be more sensitive to the accurate positioning of the receivers than to the other noise sources and to the conductivity of the scatterer under test) allowing, also in presence of large errors in the scattering data, an accurate localization of the scatterers under test. As far as the quantitative imaging is concerned, acceptable results have been achieved thanks to the synthetic zoom allowed by the iterative multiscaling procedure. Certainly, further refinements could be achieved by adding some penalty functions to the multiscaling multiresolution cost function in order to fully exploit all the available a priori knowledge of the solution or by considering state-of-the-art edge-preserving regularizations leading to significant enhancements in the image reconstruction.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Mathematical Formulation
  5. 3. Numerical Analysis
  6. 4. Validation Against Experimental Data
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[45] The experimental data used herein are by courtesy of K. Belkebir and M. Saillard (Institut Fresnel, Marseille, France). The authors wish to thank E. Vico for helpful discussions on IMSA and Ing. D. Franceschini for providing some numerical results of computer simulations.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Mathematical Formulation
  5. 3. Numerical Analysis
  6. 4. Validation Against Experimental Data
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
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  • Belkebir, K., S. Bonnard, F. Sabouroux, and M. Saillard (2000), Validation of 2D inverse scattering algorithms from multi-frequency experimental data, J. Electromagn. Waves Appl., 14, 16371668.
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Mathematical Formulation
  5. 3. Numerical Analysis
  6. 4. Validation Against Experimental Data
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
FilenameFormatSizeDescription
rds5049-sup-0001tab01.txtplain text document0KTab-delimited Table 1.
rds5049-sup-0002tab02.txtplain text document1KTab-delimited Table 2.
rds5049-sup-0003tab03.txtplain text document0KTab-delimited Table 3.
rds5049-sup-0004tab04.txtplain text document1KTab-delimited Table 4.

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