Multi-Frequency GPR Microwave Imaging of Sparse Targets through a Multi-Task Bayesian Compressive Sensing Approach

An innovative inverse scattering (IS) method is proposed for the quantitative imaging of pixel-sparse scatterers buried within a lossy half-space. On the one hand, such an approach leverages on the wide-band nature of ground penetrating radar (GPR) data by jointly processing the multi-frequency (MF) spectral components of the collected radargrams. On the other hand, it enforces sparsity priors on the problem unknowns to yield regularized solutions of the fully non-linear scattering equations. Towards this end, a multi-task Bayesian compressive sensing (MT-BCS) methodology is adopted and suitably customized to take full advantage of the available frequency diversity and of the a-priori information on the class of imaged targets. Representative results are reported to assess the proposed MF-MT-BCS strategy also in comparison with competitive state-of-the-art alternatives.

Introduction: During the last decades, many efforts have been devoted to the development of microwave imaging (MI) techniques for retrieving reliable and easy-to-interpret images of subsurface regions starting from the radargrams collected above the interface with a ground penetrating radar (GPR) [1]- [9].The solution of the arising subsurface inverse scattering (IS) problem poses several challenges mainly related to the intrinsic non-linearity (NL) and the illposedness (IP) [10].On the one hand, the NL can be avoided by introducing Born-like approximations of the scattering equations [5], provided that weak scatterers are at hand and assuming that qualitative guesses (i.e., location and shape) are sufficient for the targeted application.Otherwise, multi-resolution strategies, integrated with both deterministic [7] [9] and stochastic [8] optimization techniques, proved to be effective in mitigating the NL by reducing the ratio between unknowns and non-redundant informative data.On the other hand, the IP issue can be tackled by collecting the maximum amount of information from the scattering experiments.For instance, the wide-band nature of GPR measurements above the interface [1] provides an intrinsic frequency diversity in the collectable data.Such an information on the scenario under test can be profitably exploited with both frequency-hopping (FH) [6][7] and multifrequency (MF) [8][9] MI techniques by processing each spectral component in a cascaded fashion or jointly, respectively.Another effective recipe against the IP is the use of the a-priori information on the class of imaged targets.As a matter of fact, Compressive Sensing (CS)-based techniques [6][11]- [14] faithfully retrieved sparse objects (i.e., objects representable with few non-null expansion coefficients with respect to a suitably-chosen representation basis).In such a framework, Bayesian CS (BCS) solvers have emerged as effective, computationally-fast, and also feasible tools since they do not require the compliancy of the scattering operator with the restricted isometry property (RIP), whose check is often computationally unaffordable [11].Following this line of reasoning, this letter presents a novel MF approach for reliably, robustly, and efficiently solving the GPR-MI of pixel-sparse subsurface objects.The proposed approach is based on a fully non-linear contrast source (CSI) formulation of the scattering equations, then solved by means of a customized multi-task BCS (MT-BCS) solver [13] [14] based on a joint marginal likelihood maximization strategy that enforces the correlation between multi-static/multi-view wide-band GPR data.
Mathematical Formulation: Let us consider a 2D half-space scenario where the investigation domain D is a subsurface region within a lossy soil with relative permittivity rs  and conductivity s  (Fig. 1).By considering a multi-static/multi-view measurement system, D is illuminated by V z-oriented line sources placed in an observation domain  at distance H above the interface (Fig. 1).The v-th ( where v i e and v s e are the incident and scattered fields, respectively, while T is the duration of the ] the 3 [dB] bandwidth of the transmitted waveform, the scattered field at the p-th ( P p ,..., where is the Fourier transform.Moreover, it is related to the contrast function,   modeling the unknown dielectric distribution in the investigation domain where ) equivalent current induced within the investigation domain.
Inverse Problem Solution Approach: To numerically solve the inverse problem at hand, the equation ( 4) is first recast into the following matrix expression by partitioning D into N square sub-domains centered at   , and . stands for the transpose operator and     ./ .  denotes the real/imaginary part.Successively the solution of ( 5) is found with a customized multi-frequency multi-task BCS (MF-MT-BCS) technique [14] by jointly enforcing the spatial sparsity of the unknown components of the equivalent currents, and their correlation among the different illuminations and spectral components, the number of "tasks" solved in parallel being equal to ) frequency is computed as by applying a fast relevant vector machine (RVM) method [14] to solve the following optimization problem for retrieving the set of N 2 hyper-parameters shared among the V views and P frequencies.In (8), , I being the identity matrix, while 1  and 2  are BCS control parameters.Moreover, H . and .indicates the conjugate transpose and the determinant, respectively.Finally, the contrast distribution ( N n ,..., 1  ) at the central frequency, [m] above the interface (Fig. 1), has been chosen for the sensing setup to collect the time-domain GPR radargrams.These latter have been simulated with the GPRMax2D SW [15], while the scattered spectrum has been sampled at 9  P uniformlyspaced frequencies within the 3 [MHz]) [6].As for the setting of the MF-MT-BCS control parameters (8), the optimal trade-off values     ).The MF-MT-BCS data inversion gives a very accurate image of D independently on the data signal-to-noise ratio (SNR) and it faithfully recovers the support as well as the contrast value of the two buried scatterers [Figs.2(b)-2(c) vs. Fig.2(a)].To better point out the advantage of jointly processing all spectral components of the scattered field, as done by the proposed MF inversion scheme, the results of two FH-based state-of-art BCS solution strategies are reported in Fig. 2 for comparison purposes.It is worth reminding that these methods process each p-th ( P p ,..., 1  ) frequency in a cascaded fashion, from the lowest to the highest one, by either enforcing the correlation between multiple views ( V L  -FH-MT-BCS method [6]) or considering each view as a single task ( 1  L -FH-ST-BCS method [6]).As it can be observed, the MF-MT-BCS outperforms both FH strategies, the worst inversion being performed by the FH-ST-BCS [Figs.2(f)-2(g)].Such outcomes are quantitatively confirmed by the values of the total error, tot  , computed as in [9] and reported in Fig. 3 versus the SNR.The MF-MT-BCS does not only provide the lowest errors, but it is also significantly more robust against the data noise since, for instance, 2(e) vs. Fig.2(c)] and Similar conclusions can be drawn also when dealing with a more complex-shaped scatterer.As a matter of fact, the "S-shaped" object [ 0 . 1   , Fig. 4(a)] has been imaged by the MF-MT-BCS [Fig.4(b) vs. Fig.4(a)] remarkably better than the FH-MT-BCS [Fig.4(c)] and the FH-ST-BCS [Fig.4(d)], both FH methods failing in retrieving the actual support of the scatterer.The MF-MT-BCS is more effective to recover objects with a higher conductivity than the hosting medium, as well.Indeed, despite the increased complexity due to the presence of a non-null imaginary part of the contrast and the non-negligible amount of noise, it is the only method able to provide an accurate guess of both the real part [Fig.5(c) vs. Fig.5(a)] and the imaginary one [Fig.5, the performance of each inversion method have been quantified in terms of the total, the internal (i.e., within the target support), and the external (i.e., in the background) errors [9], the corresponding values being reported in Tab.I. Finally, it is worth pointing out the higher efficiency exhibited by the MF-MT-BCS thanks to the "one-shot" inversion of all P frequency components of the GPR spectrum.As a representative example, let us consider that the reduction of the inversion time on a standard laptop with Intel(R) Core(TM) i5-8250U CPU @ 1.60GHz and 16 [GB] of RAM amounts to , respectively (Tab.I).

Conclusion:
A novel sparsity-promoting strategy has been proposed to effectively solve the 2D GPR-MI problem.Thanks to the adopted MF strategy, the MF-MT-BCS method allows a computationally-efficient exploitation of the frequency-diversity of the GPR data by correlating all the multi-chromatic components extracted from the measured radargrams.As a result, it outperforms available FH-based solution strategies formulated within the BCS framework by exhibiting remarkably higher accuracy, robustness, and computational efficiency.[dB]) -Total, internal, and external reconstruction errors [9] and inversion time for the MF-MT-BCS, the FH-MT-BCS, and the FH-ST-BCS methods.
from a preliminary calibration performed by blurring the time-domain total field data samples with different levels of white Gaussian noise.The first test case is concerned with the "Two-Bars" scattering profile of Fig.2

Figure 3 .
Numerical Assessment ("Two-Bars" Scatterer, Behavior of the total integral error as a function of the SNR on time-domain total field for the MF-MT-BCS, FH-MT-BCS, and FH-ST-BCS methods.

Table Captions Table I .
Numerical