A Multiarray Tomographic Approach for Through-Wall Imaging

A Multiarray Tomographic Approachfor Through-Wall ImagingFrancesco Soldovieri,Raffaele Solimene,and Giancarlo PriscoAbstract—A through-wall imaging problem for a2-D scalar geometry is addressed.It is cast as an inverse scattering prob-lem and tackled under the linear Born model by means of the truncated singular value decomposition inversion scheme.A mul-tiarray-based inversion strategy is considered.In particular,first the data collected by each single array are processed to obtain different tomographic images of the same scene under test.Then, the different images are suitably combined to obtain the overall image.The inversion scheme is tested for the challenging case of objects located within a complex environment resembling a room in a building.Index Terms—Microwave tomography,singular value decom-position,through-wall imaging(TWI).I.I NTRODUCTIONT HROUGH-W ALL imaging(TWI)consists of imaging ob-jects hidden behind an obstacle by using electromagnetic (EM)waves at the microwave frequencies[1],[2].This problem is of great interest,as the aim of detecting and localizing hidden objects is shared in many applicative contexts(i.e.,both military and civilian)such as in search and rescue,security, peacekeeping,and law enforcement operations.In this framework,the research and technological develop-ment activities concern innovative observation instrumentations and processing techniques complying the necessities related to surveillance and imaging in complex environments.The overall aim is to achieve reliable and high-resolution“images”of the investigated scene in almost real time.To this end,the modeling of the EM scattering in complex environments is needed.In particular,the crucial point to be addressed is the presence of the obstacle representing in some cases a wall structure.Indeed,the presence of a wall can hinder a successful imaging procedure due to the absorption and refraction losses that the illuminating radiation undergoes when passing through the wall.This restricts the exploitable frequency range from a few hundreds of megahertz to2–3GHz for which most of the building materials are relatively transparent[3],[4]. Furthermore,if the presence of the wall is not accounted for,blurred images,where the objects appear distorted and delocalized with respect to their actual positions,arise fromManuscript received May4,2007;revised September12,2007.F.Soldovieri is with the Institute for Electromagnetic Sensing of the Environ-ment(IREA),National Research Council(CNR),80124Naples,Italy(e-mail: soldovieri.f@r.it).R.Solimene and G.Prisco are with the Dipartimento di Ingegneria dell’Informazione,Seconda Universitàdi Napoli,81031Aversa,Italy.Color versions of one or more of thefigures in this paper are available online at .Digital Object Identifier10.1109/TGRS.2008.915754the refraction phenomenon.To avoid this circumstance,the propagation path through the wall has to be properly accounted for while processing the scatteredfield data to obtain the image. This has been done,for example,in[5]where a beamforming imaging procedure accommodating the refraction introduced by a single wall structure has been presented.The effect of the wall is implicitly accounted for,and hence, blurring is avoided,if the imaging is cast as an inverse scattering problem.Indeed,in this case,the problem amounts to solving a pair of coupled integral equations where the presence of the wall is embodied within the so-called background Green’s func-tion[6].However,it is known that inverse scattering problems are nonlinear.This leads to iterative inversion schemes[7], which are time consuming and generally require a great deal of computational resources.Moreover,these inversion strategies might prove unreliable due to the occurrence of the false solutions corresponding to the local minima of the nonconvex cost functional that they attempt to minimize[8].The problem is drastically simplified if the scattering equa-tions are linearized,for example,by means of the Born or the Kirchhoff approximations[6].It is known that for realistic cases,linear inversion schemes do not permit to obtain quanti-tative reconstructions of the scattering object features.Instead, they generally allow one to determine the scatterers’locations and their geometrical features[9].However,linear inversion schemes do not suffer from the local minima problem and are computationally effective.This allows one to deal with large (in terms of wavelength)investigation domain in a reasonable amount of time.Furthermore,well-assessed numerical and theoretical tools to achieve stability against noise can be ex-ploited[10].Therefore,if the main aim is to detect and localize scatterers by inspecting a large area with some constraints on the time required to achieve the image(as in TWI applications), it is worth investigating the performance achievable by a linear inversion scheme[11].The computational advantages of the linear model-based inversion schemes are also shared by the so-called qualitative method[12]–[14].Such methods directly aim at only retrieving the scatterer supports and do not require simplified models of the scattering phenomenon.In particular,they identify the spatial support of the scatterers by adopting certain indicator functions that assume very different values in norm depending whether they are evaluated within or outside the scatterer supports.To this class of inversion algorithms belongs the linear sampling[12]and the factorization[13]methods.They require a multistatic/multiview/single-frequency configuration, i.e.,a configuration different with respect to the one of concern herein(see subsequent sections for details about the adopted0196-2892/$25.00©2008IEEEFig.1.Geometry of the problem.configuration),and their performances can be reduced when an aspect-limited configuration(such as the one considered here)is adopted[15].The time-reversal multiple signal clas-sification(MUSIC)imaging algorithm[14]also works in the multifrequency case but still requires a multistatic/multiview configuration.In this paper,we are concerned with a TWI problem where the unknown scatterers are located within an environment resembling a building’s room(see Fig.1)for a2-D and scalar case.In particular,we consider a multiarray configu-ration where data processing is performed by a microwave tomographic approach divided in two steps.First,the scattered field collected by each single array is processed to obtain a microwave imaging of the scene under test by exploiting a Born linear inverse scattering algorithm as described in[11]. Then,the various reconstructions are“combined”to achieve the overall image.As to thefirst step of the procedure,i.e.,the single array TWI reconstruction,we outline that the problem is cast as an inverse scattering one where the“background”scenario is identified by a three-layered1-D medium(the second layer being representative of the wall)and the object to be detected resides within the third layer.This allows one to simplify the formulation curtailing the computational burden and the time required to achieve the images.In particular,the illuminating source is located in thefirst layer and the scattered field is collected only at the same position of the source while the latter moves.In addition,the inversion scheme exploits multifrequency data.Hence,a multimonostatic/multifrequency configuration is addressed.This paper is organized as follows.In Section II,the inverse problem is formulated under the Born approximation and the inversion algorithm based on the singular value decomposition is briefly recalled.Furthermore,the procedure to combine single array images is described.In Section III,we present the numerical results that show the validation of the multiarray configuration in the challenging cases of objects located in a closed environment resembling a parison between the reconstructions obtained by the two-step-based combining procedure and the ones obtained by simultaneously inverting the data collected over the two arrays is provided as well. Conclusions follow.II.F ORMULATION OF THE P ROBLEM AND THEM ULTIARRAY I NVERSION P ROCEDUREThe actual geometry of the problem is depicted in Fig.1.Tar-gets are hidden within a known structure resembling the walls of a room.The walls have thickness d,dielectric permittivity b,and electric conductivityσb.The medium inside and outside the wall structure is assumed as free space,and 0is the cor-responding dielectric permittivity.The magnetic permeability µ0is everywhere equal to that of the free space.We consider a2-D geometry where both the obstacle and the targets are assumed to be infinitely long and invariant along the y-axis. Furthermore,we assume that the targets are located within an investigation domain D=[−x M,x M]×[z min,z max].The source of the incidentfield E inc(·)is afilamentary current located at the air/obstacle interface(see Fig.1)and is radiating within a frequency bandwidth f∈[f min,f max].The scattered field is collected at the same position as the source while the latter moves along a rectilinear domain.In particular,two differ-ent measurement arrays are synthesized on two different sides of the room,i.e.,Σ1=[−x O,x O]along the x-axis andΣ2= [z1,z2]along an axis parallel to the z-axis.Thus,a multiarray multimonostatic/multifrequency configuration is exploited.In the following formulation,time dependence exp(j2πft) is assumed and understood.The problem of detecting and localizing scatterers located within the room is cast as a linear inverse scattering one,and the Born approximation is used to achieve the linearization of the scattering equations.Under this hypothesis,the problem amounts to inverting the following mathematical relationship linking the scatteredfield E S(·)(the data of the problem)and the unknown object[6]:E S(r O,ω)=k20DG(r O,x,z,ω)E inc(r O,x,z,ω)×χ(x,z)dx dz,r O∈Σ1Σ2(1)whereω=2πf,r O is the source/observation position,G(·) is the background Green’s function,andχ(·)is the so-called contrast function,which is the actual unknown of the problem. In particular,it is defined asχ(x,z)=eq(x,z)− 0(2) witheq= D(x,z) 0−jσD(x,z)ω(3)and accounts for the relative difference between the background EM parameters and D andσD of the scatterers.To solve the inverse problem(1),a two-step-based procedure is adopted.First,we separately process the data collected by each single array to obtain two different images of the sameFig.2.Geometry considered while achieving the reconstruction correspond-ing to arrayΣ1.scene from two different points of view.Then,the two images are combined to achieve the overall image of the investigation domain.Step one means that the inverse problem outlined in(1) is split in two similar problems,one for each array,i.e.,E S(r Om ,ω)=k20DG(r Om,x,z,ω)E inc(r Om,x,z,ω)×χ(x,z)dx dz,r Om∈Σm(4)with m∈{1,2}.Note that the aforementioned choice entails that not all the data has to be processed simultaneously.This allows one to deal with lower computational resources and to also speed up the reconstruction procedure.We further simplify the problem by considering for each single array the background geometry in Fig.2.For such a geometry,a three-layered background medium is considered. In particular,as far as the inversion of(4)with m=1is concerned,we consider the background medium to be three-layered along the z-axis(and invariant along x and y)with the second layer being representative of the wall and the scatterers embedded in the third layer.Thefirst and third layers are free space(see Fig.2).Analogously,the inversion of(4)with m=2is achieved by considering a three-layered background medium along the x-axis(and invariant along y and z).In other words,in the inversion procedure,only the walls over which the antennas move are considered and assumed infi-nitely long along x(for m=1)and along z(for m=2). Under this assumption,the Green’s function relevant to the layered background medium and the corresponding incident field have the plane-wave expansion as reported,for example, in[11].According to the previous arguments,the imaging problem amounts to solving(4)(separately for m=1and m=2)for χ(·)when approximated expressions of the Green’s function and of the incidentfield are considered.We emphasize that the same three-layered background Green’s functions are exploited to compute the so-called unperturbedfield,i.e.,thefield scat-tered by the scenario under test when the scattering objects areTABLE IP ARAMETERS OF THE CONFIGURATIONabsent.As the scatteredfield is the difference between thefield scattered by the obstacle plus the objects and the unperturbed one,this also means that the scatteredfield is approximated and different from that of the actual.The linear inverse problem is solved by resorting to the trun-cated singular value decomposition(TSVD)inversion scheme after the integral operators appearing in(4)have been converted in matrix form by means of the method of moments as done in[11].Thus,regularized solutions of integral(4)can be writ-ten asR1χ=Nn=0E S1,v n1σn1u n1,for m=1(5) R2χ=Nn=0E S2,v n2σn2u n2,for m=2(6)where E Smis the scatteredfield collected over arrayΣm, and R mχ’s are the corresponding regularized reconstructions. The sets{σnm}∞n=0denote the singular values ordered in a nonincreasing sequence,whereas{u nm}∞n=0and{v nm}∞n=0 form orthonormal bases in the unknown and data spaces,and the truncation index N is chosen according to the available signal-to-noise ratio(SNR)[10].At this point,we have two different reconstructions of the same scene.According to the previous discussion,this is the first step of the multiarray inversion procedure.The second step,which leads to the overall image,con-sists of combining these reconstructions.This is achieved by comparing the two reconstructions pixel by pixel within the investigation domain and creating a new image by retaining for“each pixel”the larger value attained by the modulus of the retrieved contrast function between the two reconstructions achieved under the single array configurations.III.R ECONSTRUCTION R ESULTSIn this section,some numerical examples are given to assess the reconstruction capabilities of the inversion algorithm.In particular,the following examples refer to the configuration depicted in Fig.1whose parameters,which are maintained throughout this section,are listed in Table I.Moreover,for all the following reconstruction examples,the TSVD procedure is implemented by retaining the singular functions corresponding to the singular values larger than0.1times that of maximum. The exact“total”scatteredfield,i.e.,thefield scattered by the room’s wall plus the scatters,is computedfirst in theFig. 3.Tomographic reconstruction of a square object using the single arrayΣ1.time domain by means of afinite-difference time-domain-based code[16],and then,it is Fourier transformed to obtain data in the frequency domain within which the inversion scheme works.Afilamentary source,which radiates a Ricker wavelet with central frequency equal to800MHz,is used.Then,the scatteredfield data are obtained as the difference between the total scatteredfield for the actual situation under investigation and the unperturbedfield,i.e.,thefield collected when only the obstacle is present in the scene.As aforementioned,the latter is the one predicted by the approximate scenarios considered in (4),i.e.,by considering as background medium a three-layered medium(see Fig.2for the case of m=1).It is important to outline that,as the synthetic data have been generated independently of the model adopted to develop the inversion algorithm,the so-called inverse crime is avoided.As aforementioned,by exploiting a linear inversion scheme, only a qualitative reconstruction of the geometrical features of the targets can be obtained.Thus,for all the following cases,we consider as relevant quantity the modulus of the reconstructed contrast function normalized to its maximum.To better“interpret”the outcome of the multiarray recon-struction procedure,wefirst look into how the reconstructions obtained by exploiting each single array,i.e.,R1χor R2χ, appear.As an example,in Fig.3,the reconstruction of a lossless homogeneous square dielectric object of side0.4m,dielectric permittivity9 0,and located at the center of the investigation domain is shown when arrayΣ1is exploited(the configuration reported in Fig.2has been considered).As can be seen,the reconstruction allows one to identify the upper and lower sides of the object,the latter being delocalized in depth(along z)due to the difference between the actual electromagnetic velocity in the object(0.1m/ns)and the one assumed by the model (0.3m/ns).In addition,the second side is at a lower level in the reconstruction due to the refraction occurring within the object.This result is consistent with the theoretical expecta-tion concerning the spatialfiltering introduced by the TSVD inversion scheme for a configuration such as the one considered in Fig.3.Indeed,the TSVD inversion scheme acts as a low-pass spatialfilter along the direction where the measurement array is synthesized(hence,in the specific case along x)and as a bandpassfilter along the direction orthogonal to the array line(hence,along z)[11].Analogously,the reconstruction R2χthat can be obtained by exploiting arrayΣ2would allow one to identify the object’s edge,which is orthogonal to the x-direction.Hence,by adopting the multiarray strategy,we mainly expect to be able to reconstruct the two sides of the scatterers,which directly face the two arrays.Now,we turn to consider the case depicted in Fig.1,which is our main concern in this contribution.In particular,in all the following TSVD reconstruction examples,the data are corrupted in the frequency domain by adding to the real and imaginary parts of the scatteredfield two independent zero-mean additive white Gaussian noise processes with a standard deviationη0equal toαtimes each of the root mean square (rms)of the scatteredfield,so that an SNR(evaluated as 10log10[rms2/(2η20)])of20dB is obtained.As previously explained,in this case,the model that the single array reconstructions rely on does not account for all the room’s walls as it considers only the wall over which the array is synthesized.Moreover,thefiniteness of this is also neglected.To mitigate the effect due to thisfiniteness,we consider arrays whose extents are smaller than the side of the square investigation domain and,hence,than the room wall length.Thefirst multiarray TWI reconstruction example is shown in Fig.4and refers to the same square object as considered in Fig.3,which is now enclosed in a room(see Fig.1) whose external perimeter is represented by the square hav-ing sides of3.8m;the parameters of the configuration are listed in Table I.For comparison,besides the reconstruction obtained by the combining procedure[Fig.4(c)]described in the previous section,the two individual reconstructions(i.e., the ones obtained by exploiting separately the arraysΣ1and Σ2)and the reconstruction obtained by inverting(1),i.e.,by simultaneously exploiting the data collected overΣ1andΣ2, are also reported.First,it can be noted that,although the data are noisy and despite of having adopted an approximate Green’s function,the single array reconstructions[Fig.4(a)and(b)] allow one to localize the edges of the scatterer with a quality comparable to the one obtained in Fig.3.Furthermore,as expected,in the combined reconstruction[Fig.4(c)]the directly illuminated sides of the objects are correctly localized with a quality superior to that obtainable by simultaneously inverting the data collected over the two arrays.This can be roughly jus-tified by the following reasons.While the combining procedure picks up,for each pixel,the“best”of the amplitude of the two single array reconstructions,simultaneously inverting the two sets of data approximately corresponds tofirst adding the two reconstructions and then taking the overall amplitude. Therefore,the reconstruction will be emphasized where the reconstructions add“constructively”(at the left top corner of the object for the example at hand)and will be lowered where the summation is not in“phase.”Finally,also in this case,the spots at lower intensity account for the sides of the object not directly“visible”by the two arrays.As a further example,in Fig.5,we report the multiarray to-mographic reconstruction for the case of two dielectric objects with different shapes:an object of square cross section havingobtained by exploitingΣ2.(c)Reconstruction obtained by the combining procedure.(d)Reconstruction obtained by simultaneously exploitingΣ1andΣ2.Fig.5.TSVD tomographic reconstruction of square and circular dielectric objects located within a room.(a)Reconstruction obtained by exploitingΣ1.(b)Reconstruction obtained by exploitingΣ2.(c)Reconstruction obtained by the combining procedure.(d)Reconstruction obtained by simultaneously exploitingΣ1andΣ2.Fig.6.Normalized radargram obtained by collecting the data on the horizon-tal arrayΣ1and corresponding to the case shown in Fig.5.sides equal to0.4m and another one of circular cross sectionof radius0.3m.Both the objects have a dielectric permittivityequal to30 0.As for the previous example,the reconstructionallows one to determine the positions of the two objects andto deduce the information about their dimensions.In addition,in this case,the combined reconstruction[Fig.5(c)]appearsbetter than the one reported in Fig.5(d).In particular,previousdiscussion applies well to the case of the square object forwhich the two single reconstructions overlap once again at theleft top corner.For the circular object,the single reconstructionsdo not overlap and the outcome of the combining procedure isalmost equal to the reconstruction in Fig.5(d).We furthermore note that the square object is reconstructedat a lower level than the circular one.This is because it is fartherfrom the arrays than the circular object and outside the apertureof the vertical arrayΣ2.According to the analysis reportedin[17],this result is to be expected.Indeed,in the paper bySolimene and Pierri,it is shown that,for a configuration such asthe one at hand,the TSVD inversion scheme introduces a spa-tially varyingfiltering,which leads to reconstructing at lowerlevel objects that are more deeply located with respect to themeasurement array.For the sake of comparison,we also reportthe radargram corresponding to the previous case(see Fig.6),i.e.,the noise-free scatteredfield in the time domain,which isobtained by exploiting the horizontal measurement arrayΣ1.Itis evident that the radargram is very difficult to interpret.The next example reported in Fig.7concerns the case ofan object elongated along x resembling an internal wall.Inparticular,it consists of a rectangular dielectric object of2.7×0.1m2in size with dielectric permittivity equal to3 0.Thereconstruction obtained by the combining procedure(the oneobtainable by simultaneously inverting all the sets of data isalmost equal to it)permits to obtain information about thelocation of the longer side of the object due to the contributionof the horizontal arrayΣ1.The spot reconstructed at about −1.35m along the x-axis,instead,results from the contribution due to the vertical arrayΣ2and permits to have informationabout the location of the object’s side parallel and closer toarrayΣ2.As a concluding example,we present the reconstruction(see Fig.8)corresponding to the case in which theobjects Fig.7.Multiarray tomographic reconstruction by exploiting the combiningprocedure of an object resembling an internalwall.Fig.8.Multiarray tomographic reconstruction by exploiting the combining procedure of two dielectric objects located in two different internalrooms. Fig.9.Normalized radargram obtained by collecting the data on the horizon-tal arrayΣ1and corresponding to the case shown in Fig.8.considered in Figs.5and7are both present.This situation is meant as a representative of the objects located in two different internal rooms.The multiarray tomographic reconstruction allows one to retrieve both the objects,and the internal wall can be discerned as well.Furthermore,the aforementioned spatially varying behavior of the reconstruction can be still observed. Finally,it is instructive to present the time-domain scattered field(radargrams)concerning this configuration(no noise are considered in the radargrams).The radargrams are shown in Figs.9and10.From thesefigures,the presence and theFig.10.Normalized radargram obtained by collecting the data on the vertical arrayΣ2and corresponding to the case shown in Fig.8.locations of the two scatterers can be roughly deduced,mainly from the radargram obtained by using the vertical arrayΣ2for which the effect of the internal wall is negligible.However, the reconstruction reported in Fig.8proves to be more detailed and objective.IV.C ONCLUSIONWe have tackled the TWI problem of imaging scattering objects that are hidden by a known obstacle resembling a building’s room.In particular,we have presented and checked against synthetic data a tomographic approach based on the Born approximation for the scattering phenomenon and based on a multiarray configuration.The latter consists of suitably combining the reconstruction obtained by each single array. The reconstruction results we have shown allow one to state that the proposed approach also works well in detecting and localizing the objects’“edges”in the complex scenario we have considered and even under the simplifications adopted to de-velop the inversion procedure.Indeed,the inversion procedure has proven to be able to achieve more objective and detailed reconstructions of the investigated zone as compared to the raw data in the time domain(radargram).Many future developments can be identified.In the inversion scheme to obtain each single tomographic image,we have adopted a Green’s function of an ideal situation represented by a three-layered background medium(nonethe-less,the inversion scheme has worked well in the same sense stated previously).This,on the one hand,is of great importance in view of practical situations where the background is un-known or known with some degree of uncertainty[18].Indeed, the previous results suggest that it is sufficient to estimate only the parameters of the wall[19]over which the antennas move. On the other hand,it is worth investigating if better results can be achieved when a more accurate expression of the Green’s function,accounting for all the walls,is exploited.While showing the reconstructions,we have remarked that objects located in different positions are reconstructed differ-ently(in terms of resolution and level).As aforementioned, this is a consequence of the spatially varyingfiltering that the TSVD inversion procedure introduces for configurations such as the one considered herein.This leads the more“weakly”reconstructed objects to be more sensitive to the effect of noise in the reconstruction and,in general,to be less detectable. 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