TGRS-2013-00867

1

Abstract— Through-wall imaging (TWI) requires dealing with targets embedded in a complex obscuring environment as the walls of

a building. This obscuring layout is often composed by many simple elements (possibly interacting) such as slabs, corners, and T-like

structures. Most of the existing literature on TWI has focused on slab-like walls, which is reasonable when the targets are relatively far

from corners. This paper, instead, concerns the through-wall imaging in the more challenging situation where the targets are in close

proximity (inside and/or outside) of a building corner. The aim is to gain insight into how propagation through the corner impacts on

the imaging problem. To keep simple the study, a preliminary analysis is presented for a 2D geometry under the linearized Born

approximation. Firstly the Green’s function, as well as the kernel of the relevant scattering operator, is evaluated by using a high-

frequency analytical approach based on the geometrical optics and the uniform theory of diffraction. This allows one to take into

account the multipath propagation phenomena and provide thus an expression of the scattering operator more accurate than that

viable under the assumption of a simple slab wall. Then, the imaging is achieved by solving the relevant linear inverse scattering

problem with a regularizing truncated singular value decomposition algorithm. The filtering introduced by the inversion procedure,

which is dependent on the considered background scenario, is highlighted and linked to the achievable performance while imaging

targets both internal and external with respect to the corner. Finally, reconstruction results obtained from synthetic data are reported

to assess the approach.

Index Terms—radar imaging, through-wall, linear inverse scattering.

I. INTRODUCTION

hrough Wall Imaging (TWI) and urban sensing are very attractive research topics because of their civilian and military

applications ranging from homeland security (e.g. urban warfare and counter terrorism actions) to rescue missions in natural

disasters, etc. [1-2]. In these settings, the detection and localization of targets concealed behind visually opaque building

walls are demanded to get situational awareness and keep at the same time reduced risks for the on field operators.

Manuscript received August, 2013. G. Gennarelli and F. Soldovieri are with the Institute for Electromagnetic Sensing of the Environment, National Research Council of Italy, 80124, Napoli, Italy. G. Riccio is with the Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica Applicata, University of Salerno, Fisciano, Salerno, 84084, Italy. R. Solimene is with the Dipartimento di Ingegneria Industriale e dell’Informazione, Seconda Università di Napoli, Aversa, Caserta, 81031, Italy. Corresponding author: gennarelli.g@irea.cnr.it.

Radar Imaging Through a Building Corner

Gianluca Gennarelli, Giovanni Riccio, Raffaele Solimene, and Francesco Soldovieri

T

TGRS-2013-00867

2

TW radars exploit the capability of electromagnetic (EM) waves to penetrate through walls at frequencies in the band from some

hundreds to few thousands MHz. They radiate an ultra-wideband EM pulse and the signal scattered by building walls and targets

in the probed scene is collected and subsequently processed by proper algorithms to obtain images that should be easily

interpretable by end-users.

In order to succeed in such a task, a nonlinear inverse scattering problem should be solved [3]. Despite of the progress in

nonlinear inversion methods, they do not fit well yet some constraints typical in TWI. Apart from the limited reliability due the

occurrence of false solutions, the main drawback resides in the high computational burden they demand; this makes unpractical

their use in realistic scenarios involving very large (in terms of probing wavelength) spatial regions. However, as in TWI the

primary task is to detect and locate the obscured scatterers (hence quantitative reconstructions are not generally needed),

linearized imaging methods [4-11] can be successfully adopted to obtain qualitative reconstructions, i.e. object’s position and

approximate shape, with a much lower computational effort and without the occurrence of false solutions. A few examples are

the delay and sum beamforming proposed in [4, 5], which is based on a ray tracing model to compensate for the wave

propagation through a dielectric slab wall, the time-reversal MUSIC tailored for a multistatic radar system [6], the diffraction

tomographic algorithm in [10], and the Singular Value Decomposition (SVD)-based methods [8, 9, 11]. All the above methods

have been successfully applied in the case of a relatively simple background scenario consisting of a homogeneous wall and

some of them have been experimentally tested also in controlled conditions [7, 10, 11].

However, real scenarios can be more complex. For instance, the targets can be hidden behind layered [12] or inhomogeneous

cinderblock walls [13], or inside an enclosed room [14]. As a consequence, the radar echo consists of multipath contributions

whose phase delay must be necessarily compensated to enable accurate target reconstructions free from artifacts and distortions.

This turns into the need of a correct forward scattering model incorporating the multipath propagation phenomena arising in the

background scenario. Note that the imaging of targets in multipath environments by inverse scattering approaches has been

recently considered in urban canyon scenarios even when targets are not located in the line of sight (LOS) region of the radar

[15-17].

In this paper, we analyze a 2D TW scenario where the radar stands in proximity of a building corner and targets are

simultaneously located inside and outside the structure. The aim is to investigate how the propagation phenomenon arising for

the presence of the corner impacts on the imaging problem. A linearized inverse scattering model valid under Born

approximation is herein adopted. The kernel of the relevant integral equation is not available in a closed form for this non-

canonical scenario; therefore it is evaluated by employing the Uniform Theory of Diffraction (UTD) [18]. The relevant integral

equation is inverted by using the Truncated Singular Value Decomposition (TSVD) approach [19]. This allows one to obtain

regularized reconstructions as well as to analyze the spatial filtering introduced by the regularized imaging procedure and hence

TGRS-2013-00867

3

to appraise the imaging performance in terms of resolution limits. Finally, synthetic data provided by a Finite-Difference Time-

Domain (FDTD) forward solver are exploited to carry out the study.

It is worth noting that, differently from the technique for multipath exploitation developed in [14] that allows one to associate the

ghosts in the images to the true target positions, herein multipath is accounted for explicitly in the inversion model since the most

significant mechanisms (reflection, transmission, diffraction) are embodied in the relevant background Green’s function.

Furthermore, while the imaging of targets inside a “room” is considered in [14], we investigate the possibility to image targets

inside and outside a building corner.

The paper is structured as follows. Section II presents the mathematical formulation of the inverse scattering problem. Section III

concerns the ray-based solution for evaluating the kernel of the scattering operator. The imaging performance is investigated in

Sec. IV, whereas reconstructions are reported in Sec. V. Concluding remarks follow in Sec. VI.

II. THE INVERSE SCATTERING MODEL

Let us refer to the scenario depicted in Fig. 1, featuring a building corner consisting of a right-angled junction formed by two

non-magnetic ( 0µ µ= ) homogenous dielectric slabs (wall 1 and 2) having thickness d, relative permittivity εw and electrical

conductivity σ w . The junction divides the space into internal and external regions denoted by labels i and e, respectively. The

targets are supposed to reside in the domains ,min ,max ,min ,max[ , ] [ , ]i i i i ix x z zΩ = × and ,min ,max ,min ,max[ , ] [ , ]e e e e ex x z zΩ = × ,

which are probed by an array of antennas located at distance h from the wall 1 and operating in a

multimonostatic/multifrequency arrangement. Each radiating element, modeled as a line source polarized along the y-axis, is

located in the interval [ , ]a bx xΓ = with , 0a bx > , and works as transmitter and receiver in the frequency range

min max[ , ]=B f f . According to Fig. 1, the signal collected at each measurement point sx is given by the superposition of

multipath contributions depending on the considered investigation domain. A comprehensive analysis of the significant wave

propagation phenomena is presented in Sec. III.

A linearized scattering model valid under the Born approximation is adopted. Accordingly, the relation between the scattered

field ( )⋅sE and the targets in Ωi and Ωe is given by

( )

( ) ( ) ( ) ( ) ( ) ( )

20, ,

, , , , , , , , , , , , , , , , , ,inc inci e

i e

s s

s i i i i i s i i i i s e e e e e s e e e e

E x h f k

G x h x z f E x z x h f x z d G x h x z f E x z x h f x z dχ χΩ Ω

− = ⋅

− − Ω + − − Ω

∫∫ ∫∫

(1)

TGRS-2013-00867

4

In the above expression, 0k is the free-space wavenumber, ( ), ⋅inci eE denote the incident fields at the points , , ,( , )∈Ωi e i e i ex z ,

( ), ⋅i eG are the background Green’s functions accounting for the propagation from , ,( , )i e i ex z to ( , )sx h− , and

( ) ( )( )0 0, , , , ,, ,i e i e i e i e i ex z x zχ ε ε ε= − are the unknown contrast functions representing the targets in terms of permittivity

variations with respect to free-space. Accounting for the reciprocity [20], ( ), ⋅i eG are related to the incident fields and eq. (1) can

be rewritten as

( ) ( ) ( ) ( ) ( )2 20

0, , , , , , , , , , , ,inc inc

i e

s s i i i s i i i i e e e s e e e ejk

E x h f E x z x h f x z d E x z x h f x z dZ

χ χΩ Ω

− = − Ω + − Ω ∫∫ ∫∫ (2)

where 0Z is the free-space impedance.

More synthetically in operator notation, eq. (2) can be cast as

( ) ( ) ( )2 2, , [ ] [ ] [ ] :s s i i e eE x h f L L Bχ χ χ− = + = Ω → Γ×L L L L (3)

where χ is the unknown contrast accounting for targets in Ωi and Ωe , and L is a linear operator mapping the space of square

integrable contrast functions ( ) ( ) ( )2 2 2i eL L LΩ = Ω × Ω into the data space ( )2L BΓ × . In the following, ,Li e will be referred as

internal and external operators. The inverse problem defined by eq. (3) is ill-posed [19] and a suitable regularization strategy is

provided by the TSVD inversion scheme. Consequently, the contrast is recovered by the following formula:

( )0

,', '

TNs n

nnn

E vR x z uχ

σ=

=∑ (4)

where TN is a truncation index which determines the “degree of regularization” of the solution, 0n nσ ∞= is the set of the

singular values in a non-increasing order, 0n nv ∞= and 0n nu ∞

= represent orthonormal bases in ( )2 Γ ×L B and ( )2 ΩL ,

respectively [19].

The imaging performance achievable by inversion of the operators L , iL and eL will be analyzed in Sec. IV.

TGRS-2013-00867

5

III. RAY -BASED ANALYTICAL MODEL

In the following analysis, the geometrical and EM properties of the background scenario depicted in Fig. 1 are supposed known

a-priori. Of course, in realistic applications, the parameters of the building corner can be unknown and should be estimated

before applying imaging algorithms. As a matter of fact, inaccurate wall parameters (in terms of geometrical and EM properties)

are known to produce defocused and shifted target reconstructions. To overcome such drawbacks, several techniques are

available to recover the building layout [21]-[23] as well as the wall permittivity [24]-[26]. It is further assumed that the wall

surfaces are flat since surface roughness is negligible at the typical operating frequencies of TW radars.

According to eqns. (2) and (3), the evaluation of L involves the knowledge of ( ), ⋅inci eE . These quantities, in absence of a readily

available closed form solution, are here calculated by using the Geometrical Optics theory (GO) [27]. In particular, the incident

field is expressed as superposition of the contributions related to the multiple paths from the source to the observation point. As a

result, the total GO incident field is formulated as:

0 0

00

e e( , )

m d dm

mm

jk L jk LGO

dmm M

E x z EL L

− −

∈

=+∑ (5)

where M is the set of all possible ray paths. In eq. (5), 0mE is a complex amplitude term accounting for local reflection and

transmission mechanisms, 0mL is the total path in air, dmL is the total path inside the walls, and β α= −d d dk j is the complex

propagation constant of the walls with [27]

2

00

1 12 w

d w w

dk

β ε σα ω ε ε

= + ±

(6)

It must be stressed that eq. (5) implicitly assumes that the signal propagates as a uniform wave and this allows one to simplify the

GO field evaluation. The above assumption relies on the fact that the non-uniform behavior of EM waves originating from

propagation in a lossy medium [27] can be neglected for typical frequencies and material properties of interest in TWI

applications [28].

TGRS-2013-00867

6

A more accurate evaluation of ( ), ⋅inci eE would require to refine the GO modeling by also accounting for the contribution of the

field ( )dE ⋅ diffracted by the edges in the structure. In our model, only the most significant contribution due to the external edge

O is included. Eventually, the scattering model consists of the GO plus the first order diffraction contribution.

A. Evaluation of inciE

The calculation of the incident field in the internal region Ωi is performed by considering three different types of GO

contributions in the model (see Figs. 2a, b, and c). We denote by iθ and 1tθ the incidence angle and the transmission angle

inside wall 1, respectively.

Path I refers to a ray undergoing multiple reflections inside wall 1 before reaching the observation point ( , )x z . Each ray-path is

characterized by the index p, i.e. 2p is the number of internal reflections, and can be identified by applying the Snell’s laws. As

for the multipath model in [14], this corresponds to determine iθ (for a given p) by solving the following equation:

2

(2 1) sintan ( ) tan 0, 1,

sin

ii i

si

w

p dh z d x x p P

θθ θε θ

++ + − = − = …−

(7)

The value of P is determined by imposing that, after 2P reflections, the amplitude of the transmitted field through wall 1 is lower

than an assigned threshold value. Once iθ has been estimated by eq. (7), the terms involved in eq. (5) can be evaluated by means

of the following expressions:

0cos

p ih z d

Lθ

+ −= (8)

( )2

2 1

sind p

wi

w

p dL

ε

ε θ

+=

− (9)

( ) ( ) ( )20 1 1- - -p

i i p t ta w w a w aE E T R Tθ θ θ= (10)

TGRS-2013-00867

7

where iE is the field amplitude at the source, -a wT , -w aR , and -w aT are the standard Fresnel’s reflection and transmission

coefficients for perpendicular polarization whose input argument is the local incidence angle [27]. The subscript a-w (w-a)

denotes the EM wave interaction at the air-wall (wall-air) interface.

Path II is shown in Fig. 2b and deals with a ray that, after 2p internal reflections in wall 1, is reflected from the internal surface of

the wall 2, and finally reaches the point ( , )x z . In this case, the equation for iθ reads as

2

(2 1) sintan ( ) tan 2 0, 1,

sin

ii i

si

w

p dh z d x x d p P

θθ θε θ

++ + − = + − = …−

(11)

Eqns. (8) and (9) can be again used for 0pL and d pL , and

( ) ( ) ( ) ( )20 1 1- - - - π 2p

i i p t t ia w w a w a a wE E T R T Rθ θ θ θ= − (12)

The Path III (see Fig. 2c) concerns a ray that undergoes multiple reflections in wall 1, travels in the internal region and then

undergoes reflections in wall 2 before reaching the observation point. Each path is identified by the indexes p and q, which are

relevant to the reflection mechanisms in wall 1 and 2, respectively. For a fixed q, the number of internal reflections in wall 2 is

equal to 2q+1. The equation to be solved for iθ is:

2 2

(2 1) sin 2( 1) sintan ( ) tan 2 0, 1, 0, 1,

sin cos

i ii i

si i

w w

p d q dh z d x x d p P q Q

θ θθ θε θ ε θ

+ ++ + − − = + − = … = …− −

(13)

being Q a truncation index analogous to P. Moreover,

02

2( 1)

cos cospq i i

w

h z d q dL

θ ε θ

+ − += −−

(14)

2 2

(2 1) 2( 1)

sin cosd pq

w wi i

w w

p d q dL

ε ε

ε θ ε θ

+ += +

− − (15)

TGRS-2013-00867

8

( ) ( ) ( ) ( ) ( ) ( )2 2 10 1 1 2 2- - - - - -π 2 q

pqi i p t t i t t

a w w a w a a w w a w aE E T R T T R Tθ θ θ θ θ θ+= − (16)

where ( )12 sin cost i

wθ θ ε−= is the transmission angle in wall 2.

To summarize, the incident field in the internal investigation domain Ωi is given by

Path I Path II Path III( , ) ( , ) ( , ) ( , )inciE x z E x z E x z E x z= + + (17)

B. Evaluation of inceE

This subsection is concerned with the evaluation of the incident field in the external region Ωe and the pertinent propagation

mechanisms, illustrated in Fig. 2d, account for GO and diffraction contributions. The GO contribution is associated to Path IV,

which differs from Path III by the last stretch occurring in the external region. For this GO contribution, the number of internal

reflections in the wall 2 is even and the angle iθ can be found by solving the following equation:

2 2

(2 1) sin (2 1) sintan ( ) tan | | 0,1, 0,1,

sin cos

i ii i

si i

w w

p d l dh z d x x d p P l L

θ θθ θε θ ε θ

+ ++ + − − = + − = … = …− −

(18)

being L a truncation index analogous to Q. Moreover,

02

(2 1)

cos cospl i i

w

h z d l dL

θ ε θ

+ − += −−

(19)

2 2

(2 1) (2 1)

sin cosd pl

w wi i

w w

p d l dL

ε ε

ε θ ε θ

+ += +

− − (20)

( ) ( ) ( ) ( ) ( ) ( )2 20 1 1 2 2- - - - - -π 2 l

pli i p t t i t t

a w w a w a a w w a w aE E T R T T R Tθ θ θ θ θ θ= − (21)

TGRS-2013-00867

9

If the observation point is in line of sight with the source, the incident ray contribution is included in the model

0e( , )

ijkLOS i

iE x z E

ρ

ρ

−= (22)

where iρ is the distance between the source and observation point. Since this contribution can be shadowed by the structure, it is

necessary to account for the diffraction contribution by the external edge O. According to [29], the diffracted field is given by

0( , ) ( , ') ( )

jk sd LOS e

E x z D E Os

φ φ−

= (23)

where ( , ')D φ φ is the diffraction coefficient for a right-angled finite conductivity wedge illuminated by a cylindrical wave, s is

the distance between the edge and the observation point, and ( , ')φ φ fix the incidence and diffraction directions, respectively.

The final expression for the incident field in the external investigation domain Ωe is given by

Path IV( , ) ( , ) ( , ) ( , )inc LOS deE x z E x z E x z E x z= + + (24)

IV. APPRAISAL OF THE IMAGING PERFORMANCE

This Section is devoted at analyzing the spatial filtering introduced by the regularized imaging procedure according to the TSVD

scheme of eq. (4). This point is crucial since the spatial filtering dictates the resolution achievable in the regularized

reconstructions. In particular, here, we compare the performance obtainable while inverting the operator L to the one

achievable by the inversion of the single iL and eL operators. The comparison is performed by considering two well-known

figures of merit. The first one is the spectral content defined as [30, 31]:

( ) ( )0

ˆ, ,TN

nn

sp uη ζ η ζ=

= ∑ (25)

( , )η ζ being the spectral variables, and ( )ˆnu ⋅ the 2D Fourier transform of the corresponding singular function ( )nu ⋅ . From the

TGRS-2013-00867

10

physical perspective, ( )sp ⋅ provides a representation of the spatial harmonics of the target’s contrast which are retrievable by the

regularized inversion. The second figure of merit is the Point Spread Function (PSF), which is defined as the regularized

reconstruction of a point-like target [19].

The numerical analysis is carried out for a building corner characterized by walls with 6wε = , 0.01wσ = , and 0.2d = m. The

work frequency range [0.5, 2.0]B = GHz, sampled with a step of 0.1 GHz, is considered for the inversion. The measurement

array is composed by 11 sources uniformly spaced with a step of 0.2 m in the interval [0.5, 2.5]Γ = m at a distance of 0.5 m

from wall 1. The investigation domains 2[0.2,3.0] [1.0,2.0]miΩ = × and 2[ 2.0,0.0] [1.0,2.0]meΩ = − × have been discretized into

square pixels with side 0.04 m. Note that eΩ mostly falls within the shadow region of the radar, so that there is a negligible LOS

contribution to the incident field in eΩ .

The behavior of the normalized singular values for the operator iL , eL , and L is presented in Fig. 3. In Fig. 3a, the three

singular values curves have been normalized with respect to the maximum singular value of L , so that it is possible to

appreciate that the singular values of eL decay faster than those of iL , and their amplitude is also significantly smaller

compared to the ones of iL .

The smaller amplitude of the singular values of eL can be clarified by considering that the singular values are quantities that

account for the amplitude of the scattered field at the observation point due to a contrast function with unitary energy in the

investigation domain eΩ . This concept is expressed in a quantitative way by the following eigenvalue integral equation:

( ) ( ) ( ) ( ) ( )2

0

0

Path IV, , , , , , , , , , , , , , ,

e

LOS dne ne s e e s e e s e e s ne e e e

jkv x h f E x z x h f E x z x h f E x z x h f u x z d

Zσ

Ω

− = − + − + − Ω ∫∫

(26)

According to eq. (26), the singular value neσ depends on the intensity of the incident field in eΩ which, unlike the inner

region, is mainly determined by the field ( )Path IVE ⋅ transmitted through the walls 1 and 2. Note that the LOS contribution

does not play a significant role in the considered example since, as already pointed out, eΩ mostly falls in the shadow region of

the radar, whereas the field diffracted from the corner is a “second-order” contribution.

The above considerations can be also quantified by recalling that [19]

TGRS-2013-00867

11

( )2 42 020

,

, , , , ,, , , ,inc

i e

ni e s i e i e i e s i eBn

kdf dx E x z x h f d

Zσ

ΓΩ

= − Ω∑ ∫ ∫ ∫∫ (27)

where niσ and neσ are the singular values associated to the operators iL and eL , respectively. In addition, by exploiting the

eq. (27) and considering that the investigation domains iΩ and eΩ are disjoint, we have that

2 2 2n ni ne

n n n

σ σ σ= +∑ ∑ ∑ (28)

being nσ the singular values of the operator L .

For the considered example, the calculation of each quantity involved in eq. (28), through the evaluation of the integrals in eq.

(27), provides: 2 1.227e 7nin

σ = +∑ , 2 6.5e 5nen

σ = +∑ and 2 1.292e 7nn

σ = +∑ . These results confirm that the sum of squares of the

singular values of eL is almost two order of magnitude smaller compared to its counterpart associated to iL , which is fully

consistent with the trends of singular values in Fig. 3a. Accordingly, it is possible now to have a justification supporting the fact

that the singular values of L , except for small changes, are very similar in magnitude to those of iL . This implies that, for a

given SVD threshold, the reconstruction capabilities for the imaging of iΩ are expected to be very similar when inverting L and

iL , being the number TN of singular functions retained in eq. (4) almost the same. Differently, as shown below, when imaging

targets in eΩ for the same TSVD threshold, the number TN of singular functions retained for inversion of L and eL can be

very different thus having an impact on the imaging performance.

It must be noted that the choice the maximum of the singular value of L as a normalization factor for the curves in Fig. 3a is

based on the intention to highlight the different weight played by iL and eL in building the operator L . However, in the

following analysis, the TSVD threshold is assumed as normalized to the maximum singular value of the corresponding operator

to be inverted (see Fig. 3b, where the curves are normalized with respect to the first singular value of the corresponding

operator). Two TSVD thresholds respectively set at -40 dB and -20 dB are considered and the corresponding number TN is

summarized in Table I for the three operators.

TGRS-2013-00867

12

TABLE I - NUMBER OF SINGULAR VALUES ABOVE THRESHOLD

Threshold (dB) iL eL L

-40 174 119 175

-20 122 49 124

Figures 4a and b depict the spectral content related to the investigation domain iΩ for the operators iL and L when the TSVD

threshold is fixed at -40 dB. The half-circles with radii 0min2k and 0 max2k ( 0mink and 0maxk are the wavenumbers in free-

space at the frequencies minf and maxf ), are also reported to show the spectral set retrievable in free-space with a

multimonostatic/multifrequency configuration in the ideal condition of infinitely long measurement line. The spectral contents

are nearly indistinguishable and, similarly to the cases presented in [30], both exhibit a low-pass filtering behavior along η and a

band-pass filtering one along ζ . However, the actual spectral coverage is only a subset of the spectral set enclosed by white

circles due to the limited angle under which the investigation domain is “viewed”. A slight asymmetry along the η -axis is also

observed; this behavior is imputable to the fact that even if the investigation domain iΩ is centered with respect to the

measurement domain, an asymmetry arises in the background scenario due to the presence of the side wall 2.

Figures 4c and d refer to the spectral content achievable in the external investigation domain eΩ by considering the operators

eL and L , respectively, when the threshold of the corresponding TSVD is again fixed at -40 dB. Both figures show that the

spectral contents are narrower and asymmetric compared to the ones of the domain iΩ . In particular, only some negative η -

components of the spectrum can be reconstructed owing to the fact that the investigation domain is viewed by rays incoming

with propagation direction opposite to the x-axis. Furthermore, as predictable from the SVD curves in Fig. 3a, the spectral

content obtained by the SVD of L (Fig. 4d) has a smaller support compared to the one achieved by the operatoreL (Fig. 4c).

Now, we report the results when the TSVD threshold is fixed at -20 dB. For this case, the spectral contents plotted in Figs. 5a

and b confirm how the reconstruction performance in iΩ is very similar when inverting iL (Fig. 5a) or L (Fig. 5b). Of course,

less singular functions are now retained in the TSVD scheme (see Tab. I) and, accordingly, we observe a narrowing of the

support where the spectral contents are significant. As for the spectral contents relevant to eΩ , the tighter TSVD threshold

allows to stress the difference in the spectral coverage feasible by inversion of eL (Fig. 5c) and L (Fig. 5d).

It is also noteworthy to compare the PSFs for the two TSVD thresholds at -40 dB (Fig. 6) and -20 dB (Fig. 7). In particular, we

consider the point target in iΩ located at (1.5,1.5)m, while the one in eΩ is placed at ( 1.0,1.5)− m. The results provided by this

TGRS-2013-00867

13

analysis are consistent with the above considerations on the spectral contents. First, for a fixed TSVD threshold, resolution limits

in iΩ are almost the same regardless of the operator inverted (see Figs. 6a, b and Figs. 7a, b). On the other hand, the regularized

reconstruction of the point target in eΩ when inverting L is worse compared to the one provided by the inversion of eL (see

Figs. 6c, d and Figs. 7c, d). As a matter of fact, the main lobe which is now slanted appears slightly wider and, most notably,

small side lobes appear in the image. These differences are even more evident when the threshold is at -20 dB, as confirmed by

Figs. 7c and d. Note that the asymmetric nature of the side lobes in crossrange direction depends exclusively on the asymmetry

of inceE , which becomes more significant when approaching the LOS region of the radar.

V. NUMERICAL TESTS

This Section is devoted at assessing the effectiveness of the inversion strategy. The building corner has the geometry and

electromagnetic properties considered in previous section, and the TW scenario is depicted Fig. 8a for reader’s convenience. The

array exploits 11 antennas located over the line Γ = [0.5, 2.5] m at a stand-off distance of 0.5 m from wall 1, and operates in the

frequency range B = [0.5, 2.0] GHz. Each source of the array radiates a Ricker wavelet centered at the frequency of 1 GHz.

Synthetic scattered field data are generated by means of the FDTD forward solver GPRMAX2D [32]; therefore, this code

provides data that allow avoiding the “inverse crime”. Such data are obtained by a background subtraction operation (total field

in presence of the targets minus field when no target is present) in order to mitigate the effects of wall reflections and corner

diffraction.

As shown in Fig. 8a, targets are present in iΩ and eΩ . In particular, the two red squares with side 0.2 m and centered at the

points (1.0, 1.5) m and (2.0,1.5) in iΩ are lossless dielectric objects with relative permittivity 1.5. As regards the blue cylinder

in eΩ centered at ( 1.0,1.5)− m, it is a perfectly conducting target having radius 0.1 m. Note that inversion schemes based on the

Born approximation have been proved to be effective also beyond the limit of validity on which they are based, e.g. in the case of

perfectly conducting objects [33].

The B-scan representing the scattered field data is displayed in Fig. 8b. As can be seen, it contains information about the inner

and outer targets; however it is impossible to readily achieve information about their position and geometrical shape. The time

domain scattered field data are transformed into frequency domain to apply the inversion procedure. The same parameters

considered in the previous Section are adopted and the TSVD threshold is set to -40 dB. The reconstructions in both domains iΩ

and eΩ are given in terms of the modulus of the corresponding contrast function.

Results illustrated in Figs. 9a and b refer to the tomographic reconstruction in iΩ as obtained when inverting the operators iL

and L , respectively. As expected, in accordance with Born model and due to the peculiar measurement arrangement, only the

TGRS-2013-00867

14

upper and lower edge of the objects can be identified in both cases. Moreover, in agreement with the former analysis on spectral

contents and PSFs, reconstructions are very similar save for the fact that the one provided by the inversion of L appears better

focused and cleaner. This fact is reasonable since, when inverting iL , the contribution of the field scattered by the cylinder in

eΩ is not accounted for in the scattering model and this gives rise to several weak ghosts targets (see Fig. 9a). Therefore, the

presence of ghost targets in the image is not due to an incorrect modeling of multipath propagation; rather it is exclusively

caused by the temporal superposition of radar signals originated by targets in different domains. This statement is confirmed by

the fact that, when no target is present in the external domain eΩ , the tomographic reconstructions in iΩ obtained inverting the

operator iL and L are almost coincident and do not exhibit ghosts targets (see Figs. 10a and b). Note that in both Figs. 9 and

10, the intensity of the reconstructions at z=1.7 m is larger than at z=1.3 m. Actually, this phenomenon may be expected since for

a lossless dielectric target, the reflection echo from the back interface may be stronger compared to the one originated by the

upper side.

Reconstructions shown in Figs. 11a and b concern the perfectly conducting cylinder in eΩ . Specifically, Fig. 11a is obtained by

inverting the operator eL . Although a low intensity spot appears in correspondence of actual target position, the image is not

clean and a stronger ghost target appears at the right edge of the investigation domain due to the signals scattered from targets in

iΩ . On the contrary, the reconstruction achieved by the inversion of L mainly produces a strong spot in correspondence of the

illuminated portion of the target, making it clearly detectable (see Fig. 11b).

We report in Fig. 12 a tomographic reconstruction in iΩ based on an incorrect model to evaluate kernel of the scattering

operator L . This last assumes a single homogeneous wall as background scenario. As can be noticed, compared to Fig. 9b,

clearly visible ghosts, with magnitude even comparable to the true targets, appear at the edges of the investigation domain, thus

complicating the detection process.

The robustness of the inversion approach is now tested for targets located at a larger downrange from the radar. In particular, we

consider a corner with the same properties of the previous examples and investigation domains 2[0.2,3.0] [1.0,4.2]miΩ = × and

2[ 2.0,0.0] [1.0, 4.2]meΩ = − × , which are discretized with square pixels having side 0.04 m. The TW radar is the same as in

previous examples, save for the smaller frequency step of 30 MHz, which has been chosen to avoid aliasing errors along range.

The TSVD threshold is set to -30 dB.

The first test bed concerns targets each located at a different downrange from the radar: the square dielectric objects (inside the

corner) are centered at (1.0, 3.0) m and (2.0, 1.5) m, while the perfectly conducting cylinder (outside the corner) is at (-1.0, 2.0)

m. The reconstructions based on the inversion of the operator L are reported in Figs. 13a and b. As can be seen, it is effectively

TGRS-2013-00867

15

possible to image targets at different ranges in both inner and outer regions. Note that the lateral spots around the cylinder in Fig.

13b are an effect of TSVD regularization, as also observed in the PSF reported in Figs. 6d and 7d.

The next two test cases refer to targets that are simultaneously located at z=2.5 m and z=3.5 m, respectively. The corresponding

reconstructions provided by the inversion of the operator L are depicted in Figs. 14 and 15. As can be seen, save for small

artifacts caused by the multiple interactions between the targets inside the corner (a reconstruction procedure based on a linear

model of the EM scattering is not able to account for the multiple interactions between the scatterers), the true objects can be

quite well detected and localized in both cases.

As for the computational cost of the imaging approach, Table II summarizes the computation times to perform the simulations

for both (small and large) domains on a standard laptop equipped with an Intel Core I7 processor and 8 GB RAM. For the

problem at hand, the major cost is related to the computation of the operator L , i.e. the incident field, which is dependent on the

size of the investigation region and number of data (number of radiating elements times number of frequencies).

Generally speaking, the inversion approach can be computationally demanding since the complexity grows exponentially with

problem size, and the SVD of a large size matrix is also a time consuming task if compared to efficient implementations based

on the Fast Fourier Transform (FFT) algorithm. However, we have adopted the SVD based approach for different reasons. First

of all, the application of FFT-based algorithms such as those falling in the framework of diffraction tomography (e.g. see [10])

can be readily performed only in canonical scenarios, where the exact Green’s function of the problem is available in closed

spectral form (single wall, multilayered single wall, etc.) under the assumption of targets located in the far-zone. On the other

hand, the use of FFT inversion approaches does not allow the flexibility of the inversion procedure ensured by the TSVD

scheme, where one can choose the regularization parameter so to control the propagation of the noise and model error on data to

the error on the reconstructed unknown. In addition, for the problem at hand, the Green’s function is not readily available in

closed form, and thus an analytical solution based on the GO and the UTD to incorporate all significant multipath contribution in

the forward model has been ad-hoc developed. As for the inversion stage, the SVD approach is a valuable tool to predict the

achievable reconstruction performance of the proposed approach in this non-canonical scenario. Last but not least, if the scenario

under investigation does not change, the operator and its SVD can be computed only one time (off-line) and stored in memory; in

this way, one could think to build a database of different operators (matrices) for different scenarios and use each of them for the

specific problem at hand. As a result, the inversion procedure amounts only to scalar products of data vectors and projection

along the basis vectors in the unknowns space, so that the reconstructions can be attained only in few seconds.

TGRS-2013-00867

16

TABLE II – COMPUTATION TIMES OF THE IMAGING APPROACH

SCENARIO DATA UNKNOWNS L SVD

small domains 176 3000 11232 s 0.3 s

large domains 561 9600 109800 s 7.4 s

VI. CONCLUSIONS

The radar of imaging of targets located in close proximity of a building corner has been dealt with. The aim was to investigate

the positive role of accounting for the propagation through the corner on the reconstruction performance of the imaging problem.

This analysis has been carried out in the framework of the linearized inversion based on the Born approximation. For such an

inverse modeling, we have exploited a closed form ray-based solution to model the multipath phenomena and correctly calculate

the kernel of the integral equation. The inversion of the scattering operator has been accomplished via TSVD regularization, and

the imaging performances have been appraised in the internal and external regions with respect to the building corner.

As demonstrated by numerical tests based on synthetic FDTD data, the implemented inversion approach makes it possible to

identify targets simultaneously located inside and outside the structure at different downrange distances from the radar. In

particular, the inversion of the operator linking the contrast in the inner and outer domain to measurement data permits to reduce

the effect of clutter and the arising of artifacts due to the simultaneous presence of the targets in the internal and external

domains, when compared to the strategy to image each domain separately. Furthermore, for the considered measurement

configuration, targets in the internal region are generally better resolved than the external ones.

The preliminary results of this research are very encouraging and may represent a first step towards the development of more

sophisticated imaging algorithms in complex scenarios of interest for TWI and urban sensing. The experimental validation of the

imaging strategy, as well as the problems of retrieving the corner properties and clutter filtering in the radar signal will be subject

of future investigations.

REFERENCES

[1] E. J. Baranoski, “Though-wall imaging: historical perspectives and future directions,” J. Franklin Inst., vol. 345, no. 6, pp. 556-569, 2008.

[2] M. G. Amin, Through-the-wall radar imaging, New York, CRC Press, 2010.

[3] L. P. Song, C. Yu, and Q. H. Liu, “Through-wall imaging (TWI) by radar: 2-D tomographic results and analyses,” IEEE Trans. Geosci. Remote Sens., vol.

43, no. 12, pp. 2793-2798, 2005.

[4] F. Ahmad, M. G. Amin, and S. A. Kassam, “Synthetic aperture beamformer for imaging through a dielectric wall,” IEEE Trans. Aerospace and Electronic

Systems, vol. 41, no. 1, pp. 271-283, 2005.

TGRS-2013-00867

17

[5] F. Ahmad, Y. Zhang, and M. G. Amin, “Three-dimensional wideband beamforming for imaging through a single wall,” IEEE Geosci. Remote Sens. Let,

vol. 5, no. 2, pp.176-179, 2008.

[6] W. Zhang, A. Hoorfar, and L. Li, “Through-the-wall target localization with time reversal music method,” Progress In Electromagnetics Research, vol.

106, pp. 75-89, 2010.

[7] M. Dehmollaian, and K. Sarabandi, “Refocusing through building walls using synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 6,

pp. 1589-1599, 2008.

[8] F. Soldovieri and R. Solimene, “Through-wall imaging via a linear inverse scattering algorithm,” IEEE Geosci. Remote Sens. Let., vol. 4, no. 4, 2007.

[9] R. Solimene, F. Soldovieri, G. Prisco and R. Pierri, “Three-dimensional through-wall imaging under ambiguous wall parameters,” IEEE Trans. Geosci.

Remote Sens., vol. 47, no. 5, pp. 1310-1317, 2009.

[10] W. Zhang, and A. Hoorfar, “Three-dimensional real-time through-the-wall radar imaging with diffraction tomographic algorithm,” IEEE Trans. Geosci.

Remote Sens., vol. 51, no. 7, pp. 4155-4163, 2013.

[11] F. Soldovieri, F. Ahmad, and R. Solimene, “Validation of microwave tomographic inverse scattering approach via through-the-wall experiments in semi

controlled conditions,” IEEE Geosci. Remote Sens. Let., vol. 8, no.1, pp. 123-127, 2011.

[12] R. Solimene, R. Di Napoli, F. Soldovieri, R. Pierri, “TWI for an unknown symmetric lossless wall,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 8, pp.

2876-2886, 2011.

[13] P. C. Chang, R. J. Burkholder, and J. L. Volakis, “Model-corrected microwave imaging through periodic wall structures,” International Journal of

Antennas and Propagation, vol. 2012, article ID 948365, 7 pages, 2012.

[14] P. Setlur, M. Amin, and F. Ahmad, “Multipath model and exploitation in through-the-wall and urban radar sensing,” IEEE Trans. Geosci. Remote Sens.,

vol. 49, no. 10, pp. 4021-4034, 2011.

[15] M. Gustafsson, “Positioning of objects behind corners using X-band radar,” Proc. of XXXth URSI General Assembly and Scientific Symposium, pp. 1-4,

2011.

[16] G. Gennarelli and F. Soldovieri, “A linear inverse scattering algorithm for radar imaging in multipath environments,” IEEE Geosci. Remote Sens. Let., vol.

10, no.5, pp. 1085-1089, 2013.

[17] G. Gennarelli, I. Catapano, and F. Soldovieri, “RF/microwave imaging of sparse targets in urban areas,” IEEE Antennas Wireless Propag. Let., vol. 12, pp.

643-646, 2013.

[18] R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE, vol. 62, pp.

1448–1461, 1974.

[19] M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging, Bristol, U. K.: Inst. Phys., 1998.

[20] W. C. Chew, Waves and fields in inhomogeneous media, IEEE Press, 1995.

[21] C. Le, et al. “Ultrawideband (UWB) radar imaging of building interior: measurements and predictions,” IEEE Trans. Geoscience and Remote Sensing, vol.

47, no. 5, pp. 1409-1420, 2009.

[22] P. C. Chang, R. J. Burkholder, J. L. Volakis, and R. J. Marhefka, “High-frequency EM characterization of through-wall building imaging,” IEEE Trans.

Geoscience and Remote Sensing, vol. 47, no. 5, pp. 1375-1387, 2009.

[23] W. Zhang; A. Hoorfar, and C. Thajudeen, "Building layout and interior target imaging with SAR using an efficient beamformer,” IEEE International

Symposium on Antennas and Propagation (APSURSI), 2011, vol., no., pp. 2087-2090, 3-8 July 2011.

[24] L. Li, W. Zhang, and F. Li, “A novel autofocusing approach for real-time through-wall imaging under unknown wall characteristics,” IEEE Trans.

Geoscience and Remote Sensing, vol. 48, no. 1, pp. 423-431, 2010.

TGRS-2013-00867

18

[25] T. Jin; B. Chen, and Z. Zhou, “Image-domain estimation of wall parameters for autofocusing of through-the-wall SAR imagery,” IEEE Trans. Geoscience

and Remote Sensing, vol. 51, no. 3, pp. 1836-1843, 2013.

[26] R. Solimene, F. Soldovieri, G. Prisco, and R. Pierri, “Three-dimensional through-wall imaging under ambiguous wall parameters,” IEEE Trans. Geoscience

and Remote Sensing, vol. 47, no. 5, pp. 1310-1317, 2009.

[27] C. A. Balanis, Advanced engineering electromagnetics. New York: Wiley, 1989.

[28] D. Pena, R. Feick, H. D. Hristov, and W. Grote, “Measurement and modeling of propagation losses in brick and concrete walls for the 900-MHz band,”

IEEE Trans. Antennas Propagat., vol.51, no.1, pp. 31-39, 2003.

[29] R. J. Luebbers, “Finite conductivity uniform GTD versus knife edge diffraction in prediction of propagation path loss,” IEEE Trans. Antennas Propagat.,

vol. 32, no. 1, pp. 70–76, 1984.

[30] R. Persico, R. Bernini, and F. Soldovieri, “The role of measurement configuration in inverse scattering from buried objects under the Born approximation,”

IEEE Trans. Antennas Propagat., vol. 53, no. 6, pp. 1875-1887, 2005.

[31] G. Leone and F. Soldovieri, “Analysis of the distorted Born approximation for subsurface reconstruction: truncation and uncertainties effect”, IEEE Trans.

Geosci. and Remote Sensing,” vol. 41, no. 1, pp. 66-74, 2003.

[32] A. Giannopoulos, GprMax2D/3D, User’s Guide, 2002. [Online]. Available: www.gprmax.org.

[33] T. B. Hansen, and P. M. Johansen, “Inversion scheme for ground penetrating radar that takes into account the planar air-soil interface,” IEEE Trans.

Geosci. Remote Sens., vol. 38, no.1, pp. 496-506, 2000.

TGRS-2013-00867

19

List of Figures

x

Γ

O

z

xa xb h

d

ε0 µ0

εw µ0 σw

Ω i

(xi,zi)xi,min

zi,min

zi,max

xi,max

ε0 µ0

Ω e

(xe,ze)xe,min

ze,min

ze,max

xe,max

d

i

e

1

2

Fig. 1 The TWI problem: an antenna array stands in proximity of a building corner and the signal collected at each antenna is given by the superposition of

multipath contributions depending on the location of the targets in the investigation domains.

TGRS-2013-00867

20

(x ,-h)

iθ

θ

p

1

2

iθ

(x, z)

s

Path I

x

z

O

iθ

(x ,-h)

iθ

θ

p

1

2

(x, z)

s

iθPath II

x

z

O

a) b)

iθ

(x ,-h)

iθ

θ

p

1

2 (x, z)

s

Path III

x

z

O

q

θ

iθ

(x ,-h)

iθ

θ

p

1

2

(x, z)

s

θ

shadowboundary φ'

φ

l

Path IV

diffractedray

x

z

O

s

c) d)

Fig. 2. Sketch of the ray paths considered for the evaluation of the incident field in the inner region (panels a, b, c) and outer region (panel d). a) Path I: a ray

emitted by the source undergoes multiple reflections in wall 1. b) Path II: a ray emitted by the source undergoes multiple reflections in wall 1 and then is

reflected by the internal surface of wall 2. c) Path III: a ray emitted by the source undergoes multiple reflections in wall 1 and then multiple reflections in wall 2.

d) Path IV: the ray propagation mechanisms are the same as in c) but the observation point is in the outer domain. The edge diffracted ray is also shown.

TGRS-2013-00867

21

0 20 40 60 80 100 120 140 160 180-60

-50

-40

-30

-20

-10

0

i

σ i/ σm

ax (d

B)

Li

Le

L

a)

0 20 40 60 80 100 120 140 160 180-60

-50

-40

-30

-20

-10

0

i

σ i/ σm

ax (d

B)

L

i

Le

L

b)

Fig. 3 Normalized singular values relevant to ,iL ,eL and L . a) The maximum singular value of L is adopted as normalization factor for each curve. b) Each

curve is normalized to its maximum singular value. Two truncation thresholds are set at -20 dB and -40 dB.

TGRS-2013-00867

22

η [m-1]

ζ [m

-1]

-50 0 500

20

40

60

80

η [m-1]

ζ [m

-1]

-50 0 500

20

40

60

80

a) b)

η [m-1]

ζ [m

-1]

-50 0 500

20

40

60

80

η [m-1]

ζ [m

-1]

-50 0 500

20

40

60

80

c) d)

Fig. 4 Normalized spectral contents obtained with a TSVD threshold at -40 dB. a) Investigation domain iΩ and SVD of iL . b) Investigation domain iΩ and

SVD of L . c) Investigation domain eΩ and SVD of eL . d) Investigation domain eΩ and SVD of L . The half-circles with radius min2k and max2k

defining the retrievable spectral set in free-space with a multifrequency/multimonostatic measurement configuration are also shown. The color scale corresponds

to the interval [-20, 0] dB.

η [m-1]

ζ [m

-1]

-50 0 500

20

40

60

80

η [m-1]

ζ [m

-1]

-50 0 500

20

40

60

80

a) b)

η [m-1]

ζ [m

-1]

-50 0 500

20

40

60

80

η [m-1]

ζ [m

-1]

-50 0 500

20

40

60

80

c) c)

Fig. 5 Normalized spectral contents obtained with a TSVD threshold at -20 dB. a) Investigation domain iΩ and SVD of iL . b) Investigation domain iΩ and

SVD of L . c) Investigation domain eΩ and SVD of eL . d) Investigation domain eΩ and SVD of L . The color scale corresponds to the interval [-20, 0] dB.

TGRS-2013-00867

23

x [m]

z [m

]

0.5 1 1.5 2 2.5 3

1

1.5

2

x [m]

z [m

]

0.5 1 1.5 2 2.5 3

1

1.5

2

a) b)

x [m]

z [m

]

-2 -1.5 -1 -0.5 0

1

1.2

1.4

1.6

1.8

2

x [m]

z [m

]-2 -1.5 -1 -0.5 0

1

1.2

1.4

1.6

1.8

2

c) d)

Fig. 6 Magnitude of the regularized PSFs for a TSVD threshold at -40 dB. a) Investigation domain iΩ and SVD of iL . b) Investigation domain iΩ and SVD

of L . c) Investigation domain eΩ and SVD of eL . d) Investigation domain eΩ and SVD of L . The color scale corresponds to the interval [-10, 0] dB.

x [m]

z [m

]

0.5 1 1.5 2 2.5 3

1

1.5

2

x [m]

z [m

]

0.5 1 1.5 2 2.5 3

1

1.5

2

a) b)

x [m]

z [m

]

-2 -1.5 -1 -0.5 0

1

1.2

1.4

1.6

1.8

2

x [m]

z [m

]

-2 -1.5 -1 -0.5 0

1

1.2

1.4

1.6

1.8

2

c) d)

Fig. 7 Magnitude of the regularized PSFs for a TSVD threshold at -20 dB. a) Investigation domain iΩ and SVD of iL . b) Investigation domain iΩ and SVD

of L . c) Investigation domain eΩ and SVD of eL . d) Investigation domain eΩ and SVD of L . The color scale corresponds to the interval [-10, 0] dB.

TGRS-2013-00867

24

x [m]

z [m

]

-2 -1 0 1 2 3

-2

-1

0

1

2

3

Ωe Ωi

a)

x [m]

time

[ns]

0.5 1 1.5 2 2.5

0

5

10

15

20

25

30

35

40-6

-4

-2

0

2

4

6

b)

Fig. 8 a) Geometry of the TWI scenario. Red squares denote dielectric targets with relative permittivity 1.5. The blue cylinder represents the perfectly conducting

target. b) Scattered field versus time and measurement position.

TGRS-2013-00867

25

x [m]z

[m]

0.5 1 1.5 2 2.5 3

1

1.5

2 -15

-10

-5

0

a)

x [m]

z [m

]

0.5 1 1.5 2 2.5 3

1

1.5

2 -15

-10

-5

0

b)

Fig. 9 Tomographic reconstructions obtained in iΩ with a TSVD threshold equal to -40 dB. a) SVD of iL . b) SVD of L .

x [m]

z [m

]

0.5 1 1.5 2 2.5 3

1

1.5

2 -15

-10

-5

0

a)

x [m]

z [m

]

0.5 1 1.5 2 2.5 3

1

1.5

2 -15

-10

-5

0

b)

Fig. 10 Tomographic reconstructions obtained in iΩ with a TSVD threshold equal to -40 dB. a) SVD of iL . b) SVD of L . No target is present in eΩ .

TGRS-2013-00867

26

x [m]

z [m

]

-2 -1.5 -1 -0.5 0

1

1.5

2 -15

-10

-5

0

a)

x [m]

z [m

]

-2 -1.5 -1 -0.5 0

1

1.5

2 -15

-10

-5

0

b)

Fig. 11 Tomographic reconstructions obtained in eΩ with a TSVD threshold equal to -40 dB. a) SVD of eL . b) SVD of L .

x [m]

z [m

]

0.5 1 1.5 2 2.5

1

1.2

1.4

1.6

1.8-15

-10

-5

0

Fig. 12 Tomographic reconstructions obtained in iΩ with a TSVD threshold equal to -40 dB. The operator L is calculated by considering a single homogenous

wall background scenario.

x [m]

z [m

]

0.5 1 1.5 2 2.5

1

1.5

2

2.5

3

3.5

4-15

-10

-5

0

x [m]

z [m

]

-2 -1.5 -1 -0.5 0

1

1.5

2

2.5

3

3.5

4-15

-10

-5

0

a) b)

Fig. 13 Tomographic reconstructions of targets located at different downranges for a TSVD threshold equal to -30 dB. a) Inner domain iΩ . b) Outer domain

eΩ .

TGRS-2013-00867

27

x [m]

z [m

]

0.5 1 1.5 2 2.5

1

1.5

2

2.5

3

3.5

4-15

-10

-5

0

x [m]

z [m

]

-2 -1.5 -1 -0.5 0

1

1.5

2

2.5

3

3.5

4-15

-10

-5

0

a) b)

Fig. 14 Tomographic reconstructions of targets located at z=2.5 m for a TSVD threshold equal to -30 dB. a) Inner domain iΩ . b) Outer domain eΩ .

x [m]

z [m

]

0.5 1 1.5 2 2.5 3

1

1.5

2

2.5

3

3.5

4-15

-10

-5

0

x [m]

z [m

]

-2 -1.5 -1 -0.5 0

1

1.5

2

2.5

3

3.5

4-15

-10

-5

0

a) b)

Fig. 15 Tomographic reconstructions of targets located at z=3.5 m for a TSVD threshold equal to -30 dB. a) Inner domain iΩ . b) Outer domain eΩ .