The problem of determining and understanding thenature of buried objects by means of nondestructive and non-invasive techniques represents an interesting issue for a greatvariety of applications. In this framework, the theory of electro-magnetic inverse scattering problems can help in such an issueby starting from the measures of the scattered field collectedon a surface. What will be presented in this communication isa two-dimensional (2-D) technique based on the so-called Bornapproximation (BA) combined with a compressive sensing (CS)approach, in order to improve reconstruction capabilities for aproper class of targets. The use of a multiview-multistatic configu-ration will be employed together with a multifrequency approachto overcome the limited amount of data due to the single-frequencytechnique. Therefore, after a first numerical analysis of the per-formance of the considered algorithm, some numerical examplesfor 2-D aspect-limited configurations will be presented. The sce-nario is composed of a simplified scene, which consists of twohalf-spaces, and with the probes located close to the interfacebetween the two media. As proposed in the following, it is easyto observe that the use of CS for this kind of problems mayimprove reconstruction capabilities, confirming the validity of thepresented approach.
The capability of electromagnetic fields to pene-trate different materials makes them very attractive toreconstruct, both in a qualitative and quantitative way, the mor-phological and electrical features of the unknown objects bymeans of a ondestructive technique, which starts from the mea-sures of the scattered field. Such a technique may be appliedin several field, including geophysics [1], characterization ofmaterials [2], monitoring in biomedical engineering [3], anddemining applications [4].In the framework of the so-called aspect-limited problems,an interesting application is related to ground penetrating radar (GPR), which employs signals whose frequencies vary froma few hundreds of megahertz till to some gigahertz accordingto the considered scenario. As most of the techniques basedon radar approaches, target information is retrieved from thetwo-travel time of a pulse radiated by a source and gatheredby a fixed-offset system between transmitter and receiver [5].The image obtained by joining the radar echoes collected whilemoving the antennas is referred to as a raw data “radargram.”Unfortunately, what radar techniques usually allow is onlythe extraction of qualitative information on the investigatedregion that is based on a subjective interpretation of the rawdata and on user experience. Therefore, since in many cases, itis not possible to provide detailed information on the targets,several data processing techniques have been proposed, amongwhich one can find focalization procedures [6] and tomographictechniques [7].Among all these techniques, tomographic imaging seems tobe a promising approach to overcome the limitations related tostandard procedures, since it makes possible to achieve not onlyinformation about the shape and localization of buried objects,but it also allows a quantitative electromagnetic characteriza-tion of these targets in the imaging domain under test [8]. https://vssewingmachine.in/ Nevertheless, the detection performance of GPR largelydepends on a lot of factors, which can partially or totally hideor distort the response of the buried targets. Among all thesefactors, it is remarkable to cite the coupling between antennasand soil, the electromagnetic features of the background, thespeed and scattering of wave propagation, and the electromag-netic contrast of the buried objects on which the intensity of thescattered fields depends [9].Therefore, there is a need to develop appropriate techniquesfor clutter reduction and subsurface imaging.
In this category,detection techniques are employed on a subsurface image builtfrom a full GPR scan after clutter reduction. They includeadvanced algorithms for hyperbola detection [10]–[13], andmigration approaches [14], [15]; their performance mainlydepends on the data set quality and on the preprocessing methodused to subtract the contribution of the background.In this framework, a great variety of tomographic techniquesmay be found in the literature to find a solution to the electro-magnetic inverse scattering problem previously described [16]–[20]. More in detail, it is possible to divide these approachesin two main classes: the first one faces the inverse scatteringproblem without any approximation and, in principle, this classcan provide an accurate reconstruction of the region under test,but it drives into a nonlinear ill-posed inverse problem, and a second class of approximated approaches which simplifiesthe model. Even though the first class of algorithms mayrealize better quantitative reconstructions, the nonlinearity ofthe model makes the solution of such problems be usually verysensitive to the availability of adequate information about thereference scenario, so that any sensible information about thescene has to be dealt with: such a feature may limit their appli-cability since an accurate knowledge of the electromagneticfeatures of the scenario is required to model correctly the prob-lem. Despite the increasing interest in this class of approaches,the inaccuracy in the knowledge of the reference scenario (soilpermittivity and conductivity) and the not-perfect knowledgeof the antennas radiation characteristics in the presence of thesoil affect the quality of the reconstruction problems. Moreover,the nonlinearity of the relationship between data and unknownsmay drive into false solutions that still exist, thus affectingthe reliability of the overall solution strategy [21], [22], andincreasing the computational time.
The second class of solutionapproaches exploits simplified models of the electromagneticscattering to develop linear inversion approaches [23]–[25].What has been proposed in this paper is an inversion strategywhich belongs to the second class of the approaches presentedbefore, since it is based on the so-called Born approxima-tion (BA). Due to the linearity of the problem, the solution issearched as the global minimum of a quadratic cost function,for which no false-solutions exist. Moreover, for linear inverseproblems, the adoption of well-assessed regularization schemes[26] is possible and reconstruction capabilities can be foreseen.Despite of the advantages said above by the adoption of suchmodels, the class of targets that may be recovered is limitedto those objects for which the BA is still valid, as for smallobjects whose electromagnetic features are very close to thoseof the background. In addition, since the aspect-limited natureof data implies that single-frequency data are not sufficient torealize effective inversions, a multifrequency approach will beconsidered throughout this communication. In order to assessthe actual performance of the approach in a relatively simplesituation, the canonical and significant two-dimensional (2-D)geometry is considered, together with the use of a regularizationtechnique based on compressive sensing (CS), which makes itpossible to reduce the number of data considerably.
