THE LEVEL SET SHAPE RECONSTRUCTION ALGO- RITHM APPLIED TO 2D PEC TARGETS HIDDEN BE- HIND A WALL

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1 Pogess In Electomagnetics Reseach B, Vol. 25, , 2010 THE LEVEL SET SHAPE RECONSTRUCTION ALGO- RITHM APPLIED TO 2D PEC TARGETS HIDDEN BE- HIND A WALL M. R. Hajihashemi and M. El-Shenawee Univesity of Akansas Fayetteville, USA Abstact The level set algoithm is extended to handle the econstuction of the shape and location of objects hidden behind a dielectic wall. The Geen s function of statified media is used to modify the method of moments and the suface integal equation fowad solve. Due to the oscillatoy natue of the Sommefeld integals, the stationay phase appoximation is implemented hee to achieve fast and accuate econstuction esults, especially when the tagets ae located adequately fa fom the wall. Tansvese Magnetic (TM) plane waves ae employed fo excitation with limited view fo tansmitting and eceiving the waves in the fa field at one side of the wall. The esults show the capability of the level set method fo etieving the shape and location of multiple 2D PEC objects of abitay shapes even when thee ae located at a small distance fom the wall. To educe the computational expenses of the algoithm in the case of multiple hidden objects, the MPI paallelization technique is implemented leading to a eduction in the CPU time fom hous on a single pocesso to few minutes using 128 pocessos on the NCSA Supecompute Cente. 1. INTRODUCTION The electomagnetic waves have the ability to penetate though nonmetallic walls made of wood, glass, bick, and/o concete blocks. The eflected waves could be analyzed to econstuct the pofile and the location of hidden objects behind these walls. The though-wall imaging is challenging because of the complex multiple scatteing mechanisms that occu between the objects and the dielectic wall. Received 26 July 2010, Accepted 23 August 2010, Scheduled 25 August 2010 Coesponding autho: M. R. Hajihashemi (mhajihas@uak.edu).

2 132 Hajihashemi and El-Shenawee Examples of applications include eathquake and fie escue opeations, police seach opeations, homeland secuity, and militay applications as epoted in seveal wok in the liteatue [1 19]. The epoted esults in [1 17] wee based on pemittivity econstuction of hidden objects behind a wall, while woks in [18 20] wee focused on the shape econstuction of hidden objects. In [1], a subspace-based optimization method was applied fo though-wall imaging. The taget objects of diffeent pofiles wee well econstucted even in the pesence of high level of noise in the data. In [2], a though-the-wall imaging technique based on time evesal method was intoduced fo the detection of moving tagets in a clutteed envionment. In [3], a simple pocedue to detect changes in a though-the-wall imaging scenaio was epoted validated against synthetic and expeimental data. In [4], a micowave imaging technique that combined the FDTD method and the Polak-Ribièe algoithm fo econstucting undegound multiple scattees was pesented. In [5], the chaacteization of inclusions in concete stuctues using the matched-filte-based evese-time (MFBRT) migation algoithm and the paticle swam optimization (PSO) was investigated. Most of the accomplished woks in this field wee based on synthetic apetue ada (SAR) and ulta-wideband (UWB) ada techniques [8, 9]. In [10], a 2-D contast souce invese scatteing method was applied in a multilayeed medium and a high-quality image econstuction was achieved using multi-fequency data with a limited aay view. The Bon appoximation was employed fo qualitative econstuction of hidden 2-D objects behind the wall [7, 17]. The algoithm was tested using synthetic and expeimental data; howeve, the method was esticted to canonical scattees. A synthetic apetue aay technique was pesented fo imaging the tagets behind the wall using ulta wideband antennas and wide ange of incidence angles [8]. A 3-D ulta wideband SAR technique fo suface econstucting of hidden objects using eal data was implemented achieving a eduction in the calculation time [9]. Anothe 3-D though-the-wall beam fome based on applying a tansfomation to bing the taget and the tansmitte/eceive to the same height was designed [11]. A two-step imaging pocedue by means of a linea invese scatteing technique was pesented fo imaging the objects behind a wall whose paametes wee not completely known [12]. A wideband beam foming-based technique was poposed to pefom imaging with wall paamete ambiguities and diffeent standoff distances [13, 14]. The signatue of a metallic taget behind a thick bick wall was obtained using low fequencies and a monostatic

3 Pogess In Electomagnetics Reseach B, Vol. 25, appoach [15]. The application of spatial filtes to suppess, o dastically mitigate the wall eflections, was poposed in [16]. Some wok suggested estimating the wall paametes based on the model of a dielectic slab and the ealy aival time [12]. In [18], the Kichhoff appoximation was employed fo shape econstuction of pefectly conducting objects when the scatteing data wee collected in a finite egion aound the tagets. The linea sampling method was employed fo the invese poblem of non-accessible tagets concealed into a wall o unde a floo [19]. Most of the above woks povided only the contast of the hidden object with espect to the backgound medium but did not povide the shape of these tagets. Although the cuent wok assumes a pioi knowledge of the dielectic wall s constitutive paametes and thickness, which could be a ealistic assumption in seveal applications, the poposed level set algoithm povides the exact shape of the hidden tagets and also thei locations. The level set invesion method has shown a potential in shape econstuction as epoted in the liteatue [20 24]. In a elevant wok by Ramananjaona et al., the level set technique was used fo econstucting the shape of 2-D obstacles buied in a half-space of low dielectic mateial using both tansvese magnetic (TM) and tansvese electic (TE) polaizations [20]. Most of the epoted esults wee esticted to monochomatic data and ectangula shaped objects. The level set is an implicit mathematical famewok fo the shape econstuction poblems in electomagnetics, moe in depth details given in [20 29]. The numeical esults demonstate the capability of the method fo handling the topological changes (i.e., beaking and meging of the egion). A simple initial guess can evolve to seveal objects duing the econstuction scheme. The level set technique is implemented in this wok to econstuct the location and the shape of multiple PEC objects of abitay shapes hidden behind a known dielectic wall. The challenge in the cuent wok is the implementation of the Geen s function in statified media in the level set algoithm [26, 27]. An appoximation of the Geen s function using the stationay phase method is implemented to educe the CPU time without much sacificing of the accuacy. The stationay phase method (saddle point method) is found to be an effective appoach fo the fast calculation of the Sommefeld integals in statified media. The calculation time fo the MoM impedance matix in statified media is almost in the same ode of that in fee space. The accuacy of the method is validated using diffeent examples shown in Section 2.4. The obtained econstuction esults demonstate the efficiency of the algoithm especially when the

4 134 Hajihashemi and El-Shenawee taget objects ae not placed too close to the wall. Plane wave illumination of the wall with TM polaization indicates that the electic field is paallel to the cylindes axes. Seveal multiple scatteing mechanisms contibute to the eceived waves such as the scatteing fom the objects, the scatteing fom the dielectic wall, and the multiple scatteing between the objects and the wall as discussed in Section 2. The constitutive paametes and the thickness of the wall ae assumed to be a pioi known which allows the offline calculation of thei effect on the scatteed waves. Howeve, the multiple scatteing between the unknown objects and the wall cannot a pioi be pedicted. In the case of inhomogeneous wall, closed-fom expessions to take into account the wall effect ae not geneally available. In this case, numeical techniques such as finite diffeence methods could be applicable. As epoted in [12], thee ae some statistical methods to deal with the though-the-wall imaging unde ambiguous wall paametes. 2. METHODOLOGY 2.1. Level Set Repesentation We assume that the moving contou Γ(t) is epesented implicitly as the zeo level of a two-dimensional function Φ( ), as shown in the two dimensional configuation (2-D) in Fig. 1. At each time t, the inteface is epesented as [28]: Γ(t) = {(x, y) Φ(x, y, t) = 0} (1) Upon obtaining the deivative of (1) with espect to the evolving time t, we have the following expession fo tacking the motion of the inteface known as the Hamilton-Jacobi equation [28, 29]: Φ(x, y, t) + F ( ) Φ(x, y, t) = 0 t (2a) Φ 0 = Φ(x, y, t = 0) (2b) whee F ( ) is the nomal component of the defomation velocity on the contou (see Fig. 1). The objective hee is to minimize the cost function which is the mismatch eo between the simulated scatteed fa-field of the evolving objects duing the invesion pocess and the scatteed fa-field of the tue object (data) [21]. The appopiate fom of the defomation velocity, making a deceasing cost function, was given in [22]. It was based on the fowad and adjoint cuents induced on the suface of the evolving objects [30]. The PDE in (2) is solved numeically using the highe ode finite diffeence schemes elaboated in [28]. If the value of the level set

5 Pogess In Electomagnetics Reseach B, Vol. 25, Figue 1. Implicit epesentation of the evolving contou using level set. function Φ at any gid point (x i, y j ), at any time n t, is denoted by the symbol Φ n ij = Φ(x i, y j, n t), the updated value of the level set function at time (n + 1) t is calculated as follows [28]: Φ n+1 ij = Φ n ij t [ max(f ij, 0) + + min(f ij, 0) ] (3a) whee F ij = F (x i, y j ) epesents the velocity function at (x i, y j ). The symbols + and ae given as follows [28]: ( ) 1/2(3b) + = max(dij x, 0)+min(Dx+ ij, 0)+max(Dy ij, 0)+min(Dy+ ij, 0) ( ) 1/2 = min(dij x, 0)+max(Dx+ ij, 0)+min(Dy ij, 0)+max(Dy+ ij, 0) (3c) The diectional deivatives of the level set functions in (3) ae calculated, fo example, as follows: D y+ ij = Φ(x i, y j + y) Φ(x i, y j ) (3d) y Dij x = Φ(x i, y j ) Φ(x i x, y j ) (3e) x The function Φ 0 ( ) is the signed distance function, shotest distance between a gid point and the contou, coesponding to the initial guess. The choice of the signed distance function avoids steep gadients and apidly changing featues duing the invesion algoithm [29]. The time step equied fo solving the PDE given in (1) is chosen accoding to the Couant-Fiedichs-Lewy (CFL) stability condition which assets that the numeical wave should popagate at least as fast as the physical waves [28, 29]. The level set shape

6 136 Hajihashemi and El-Shenawee econstuction algoithm in the statified media is simila to that of the fee space epoted in [21] except that the fowad scatteing poblem is moe computationally demanding in the statified media due to the need to numeically evaluate the Sommefeld integation. The modified fowad scatteing poblem and the choice of the defomation velocity ae discussed as follows Fowad Scatteing Poblem in a Statified Media This wok is based on the configuation of Fig. 2 whee a known homogenous dielectic wall with the elative pemittivity ε and conductivity σ is located at h < x < 0 in the x-y plane. A time-vaying line souce caying a cuent I = 1 A with angula fequency ω = 2πf is located at (x, y ) in the half space x > 0 paallel to the z-axis. The electic field, which is the Geen s function of the statified media, at any point (x, y) in x > 0 egion is poduced as follows, assuming time convention of e jωt [26, 27]: G(x,y; x, y )= ωµ + 0 e jξ(y y ) ( ) e u(ξ) x x +R (k 0,ξ)e u(ξ)(x+x ) dξ 4πj u(ξ) = ωµ ) 0 (k 0 (x x ) 2 + (y y ) 2 4 H(2) 0 + e jξ(y y ) + ωµ 0 4πj u(ξ) R (k 0, ξ)e u(ξ)(x+x ) dξ (4a) T + Y ( x, y) ξ R + ε σ PEC θ Incident Plane Wave inc Multiple Scatteing X E T PEC sc θ h R Figue 2. Configuation of hidden objects behind a dielectic wall.

7 Pogess In Electomagnetics Reseach B, Vol. 25, whee u(ξ) = ξ 2 k0 2 is the phase facto and H(2) 0 ( ) is the zeo ode Hankel s function of the second kind. The symbol k 0 epesents the fee space wave numbe. The symbol of ε 0 epesents the pemittivity of the fee space and the symbol of µ 0 epesents the pemeability of the fee space. The tem R (k 0, ξ), the wall eflection coefficient at x > 0, is given by [26]: ( R u1 u 3 u 2 ) 2 sinh(u2 h) (k 0, ξ) = u 2 (u 1 + u 3 ) cosh(u 2 h) + ( u 1 u 3 + u2) 2 (4b) sinh(u2 h) with u 1 = u 3 = ξ 2 k0 2 and u 2 = ξ 2 k0 2 ε. The symbol ε is the effective dielectic pemittivity of the dielectic wall with thickness h given by [26] ε = ε + σ (5) jωε 0 In this wok, we assume that multiple pefectly electic conducting (PEC) infinite cylindes with abitay coss-sections and total contou C (that epesents the contous of all objects) ae located in the half space x > 0. The induced cuent on the suface of the conducting cylindes has only the z-component Jz ind (x, y ). Upon enfocing the bounday condition fo the total electic field to vanish on the suface of the PEC objects, the following integal equation is obtained to be solved fo the induced cuent using the Method of Moments (MoM) [27]. C J ind z ( x, y ) G ( x, y; x, y ) dl = E t z(x, y) (6) whee G(x, y; x, y ) is the Geen s function in the statified media given by (4a) and E t z(x, y) is the tansmitted incident electic field at any point (x, y) C given by [27]: E t z(x, y) = T + (k 0, k 0 sin θ i )e jk 0(x cos θ i +y sin θ i ) whee T + (k 0, k 0 sin θ i ) is the tansmission coefficient though the dielectic wall given by [26]: T + (k 0, ξ) = 2u 1 u 2 e u 1h u 2 (u 1 + u 3 ) cosh(u 2 h) + ( u 1 u 3 + u 2 2) sinh(u2 h) Upon calculating the induced cuent on the suface of conducting cylindes, the scatteed electic field due to the PEC objects, Ez,O scat(x, y ), at any eceive point (x, y ) at x < h egion is given (7) (8)

8 138 Hajihashemi and El-Shenawee by [27]: E sc z,o(x, y ) = ωµ 0 4πj + C J ind z ( x, y ) T (k 0, ξ)e jξ(y y ) e u(ξ)(h+x x ) dξdl (9) u(ξ) whee T (k 0, ξ) is the tansmission coefficient though the dielectic wall given by [26]: T (k 0, ξ) = 2u 1 u 2 u 2 (u 1 + u 3 ) cosh(u 2 h) + ( u 1 u 3 + u 2 2) sinh(u2 h) (10) The total scatteed field, which is due to the wall and the objects, in (x, y ) is given by [27]: E sc z,total (x, y ) = E sc z,wall (x, y ) + E sc z,o(x, y ) (11) whee E sc wall (x, y ) is the scatteed field fom the dielectic wall in the absence of the PEC objects and it is given by [26] Ewall sc (x, y ) = R + (k 0, k 0 sin θ i )e jk 0(x cos θ i y sin θ i ) (12) whee R + (k 0, k 0 sin θ i ) is the eflection coefficient fom the dielectic wall given by [26]: R + (k 0, ξ) = R (k 0, ξ)e 2u 1h (13) The fa field patten of the scatteed field fom the objects epesented by EO scat ( ) will be used in the level set algoithm. Since the constitutive paametes of the dielectic wall and its thickness ae a pioi known, the contibution of the wall to the total scatteed field is Ewall scat(x, y ) fo cetain incident diection θ inc can be calculated offline using (12). If the point eceive (x, y ) is adequately fa fom the dielectic wall (300λ is assumed in this wok), the fa-field patten fom objects, Pz,O sc (θinc, θ sc ) in the diection of θ sc is calculated as follows [21]: Pz,O(θ sc inc, θ sc ) = ρ e jk0ρ Ez,O(x sc, y ) (14) whee ρ = x 2 + y 2 and θ sc = tan 1 ( y x ) (see Fig. 2). The defomation velocity F ( ) in (2) is the same as in [22] whee the conducting cylindes wee immesed in fee space with no walls. In geneal, the fomulation of the defomation velocity was obtained upon minimizing the mismatch between the scatteed fa fields of the

9 Pogess In Electomagnetics Reseach B, Vol. 25, evolving objects and the tue tagets (defined as the cost function). The expession of the defomation velocity F is given in [22]: [ N I N M (i) F ( ) = α Re e i ( π 4 E sc, θ meas i=1 Ez,meas(θ sc i inc j=1 z,sim(θi inc, θ meas ) ) Jz ( ).J z( ) ij ] ij ) (15) whee N I is the numbe of incident waves, N M (i) is the numbe of measuements unde the ith incidence, θi inc is the ith incident angle, θij meas is the mth measuement angle fo θi inc, J z ( ) is the induced cuent solution of the fowad poblem and J z( ) is the induced cuent solution of the adjoint poblem [30]. The symbols Ez,sim sc and Ez,meas sc epesent the calculated and the measued scatteed fields, espectively, and α is a positive nomalization coefficient. The main constaint fo the shape econstuction of the objects behind the wall is that the incident and scatteed waves ae limited to cetain views as shown in Fig. 2, unlike the fee space configuation. These constaints ae given by: π 2 < θinc < π 2, and π 2 < θsc < 3π 2 (16) The angles ae measued with espect to the positive x-diection as shown in Fig. 2. Theefoe, it is challenging to etieve the details in pats of the tagets whee the illumination is not available simila to the concept of shadowing Stationay Phase method The calculation of the Sommefeld integal in statified media is computationally challenging, but the stationay phase method is employed to appoximate the integation in (4a) [31]. The integand in (4a) has a apidly oscillating behavio due to the exponential tem e jξ(y y ) u(ξ)(x+x ) while the eflection coefficient R (k 0, ξ) has a slowly vaying behavio compaed to the exponential pat. Theefoe, the latte can be eplaced by its value at the stationay point ξ = ξ 1 whee, [ jξ ( y y ) ( ξ ξ 2 k0 2 x + x )] = 0 (17a) ξ=ξ1 k 0 (y y ) ξ 1 = (17b) (x + x ) 2 + (y y ) 2

10 140 Hajihashemi and El-Shenawee The second tem in (4a) can be appoximated as follows: ωµ 0 4πj + e jξ(y y ) R (k 0, ξ 1 ) ωµ 0 4πj = R (k 0, ξ 1 ) ωµ 0 4πj u(ξ) + + R (k 0, ξ)e u(ξ)(x+x ) dξ e jξ(y y ) ξ 2 k 2 0 e ξ 2 k 2 0 (x+x ) dξ e jξ(y y ) ξ 2 k 2 0 e ξ 2 k 2 0 (x+x ) dξ (k 0 (x + x ) 2 + (y y ) 2 ) = ωµ 0 R (k 0, ξ 1 )H (2) 0 (18) 4 A Simila appoach is employed to appoximate the integal of (9). The fist step is to find the stationay point of the apidly vaying exponential pat as: ξ ξ 2 = [ jξ ( y y ) k 0 (y y ) (h + x x ) 2 + (y y ) 2 ( ξ 2 k0 2 h + x x )] =0 (19a) ξ=ξ2 (19b) The slowly vaying pat in (9) can be eplaced by its value at the stationay point as follows: Ez,O(x sc, y ) ωµ 0 4πj = ωµ 0 4 C C Jz ind (x, y )T (k 0, ξ 2 ) J ind z (x, y )T (k 0, ξ 2 )H (2) 0 + e jξ(y y ) u(ξ) e u(ξ)(h+x x ) dξdl ) (k 0 (h+x x ) 2 +(y y ) 2 dl (20) The scatteed field is calculated in the fa field zone which justifies the appoximation in (19). Compaed with othe numeical methods, the level-set algoithm based on the MoM povides moe efficient econstuction esults, since calculating the scatteed fields and the defomation velocity equies the calculation of the induced cuents only on the contou of the evolving objects. Futhemoe the defomation velocity is diectly calculated on the moving contous and then extended to the whole computational domain [21 25]. In most of the invese scatteing techniques, the scatteed field in a single fequency does not povide enough infomation fo etieving the details of the taget objects. In this wok afte a pe-assigned numbe of iteations (e.g., 1000 iteations), the woking fequency hops to a highe one to etieve fine details of the unknown objects [21].

11 Pogess In Electomagnetics Reseach B, Vol. 25, Validation of the Fowad Solve The accuacy of the fowad solve in the statified medium is investigated using (i) the full Geen s function in (4a), (ii) the same expession but with ignoing the second tem in (4a), and (iii) using the stationay phase method (19), (20). The validation is conducted at diffeent fequencies and diffeent distances of the object fom the wall. The numeical esults ae compaed with the commecial EMsimulato FEKO [32]. In the fist example, the scatteed electic field due to a cicula PEC cylinde with a adius of c = 20 cm centeed at (x c, y c ) = (50 cm, 0) is shown in Fig. 3. Nomal incidence is consideed θ inc = 0 at fequency of f = 1 GHz. The thickness of the wall is assumed 20 cm made of a mateial with ε = 2.2 and loss tangent tan δ = The point eceives ae placed at x = 2m and 1 m < y < 1 m. Fig. 4 shows the magnitude of the scatteed electic fields due to the objects Ez,O sc (x, y ) (9) using the full Geen s function (4a), the Geen s function upon ignoing the second tem in (4a), the stationay phase method (20), and FEKO. The esults show that the stationay phase method demonstates vey good ageement with using both FEKO and the full Geen s function (4a). Howeve, ignoing the inteaction between the dielectic wall and the object (i.e., ignoing the second tem in (4a)) causes an eo of 3% compaed to the full Geen s function in this case. As expected, the multiple scatteing between the objects and the wall, which is epesented by the second tem in (4a), inceases as the distance between the wall and the objects deceases. The phase esults show simila validation but ae not pesented hee. Anothe validation was conducted using the same cicula PEC cylinde but when located close to the wall at (x c, y c ) = (25 cm, 0) with the pemittivity of the wall ε = 4.5. In this case and compaed with the esults of Fig. 4, the 20 (cm) Y ε = 2.2 tan( δ) = cm Ignoing R - in (4a) X (cm) Figue 3. Cicula cylinde with the adius of 20 cm. Figue 4. Scatteed electic field fom the cicula cylinde.

12 142 Hajihashemi and El-Shenawee eo between using the full Geen s function and the stationay phase method has inceased to 4%; the eo between using the full Geen s function and FEKO has inceased to 11%; and the eo between using the full Geen s function and the same expession but with ignoing the R tem in (4a) has inceased to 25% (plots ae not shown). As obseved in Fig. 10(d), if the taget is placed close to the wall, the econstuction esults ae not vey accuate; this is a common poblem in see though wall imaging. In this case, the assumption that the souce (wall) and the obsevation point (taget) to be adequately fa fom each othe does not hold. Altenatively, othe moe time consuming techniques, such as complex image method, could be used [34]. The CPU time equied when using the full Geen s function is 38 min total, 16 min when ignoing the second tem in (4a), while it is only 2 sec when using the stationay phase method. Fo FEKO, the simulation of the stuctue of Fig. 3 equied seveal hous since it was modeled as a 3-D electomagnetic scatteing poblem. Based on the above, the stationay phase method will be used in all esults of Section 4 fo geneating the synthetic data and fo calculating the scatteed fields of the evolving objects duing the invesion pocess. 3. NUMERICAL RESULTS 3.1. Reconstuction of Two Elliptical Cylindes In the fist case, the econstuction of two elliptical PEC cylindes located behind a lossless dielectic wall is examined. The thickness of the wall is assumed h = 50 cm with the pemittivity of ε = 2.2. The semi-majo and semi-mino axes lengths of the ellipises ae a = 6 cm and b = 2 cm, espectively. The sepaation distance between the two ellipses is 40 cm measued between thei centes. The two ellipses ae located at (40 cm, 20 cm) and (40 cm, 20 cm), espectively. In this example, nineteen diections of the incident plane waves ae used with nineteen diections of the eceived waves pe each incidence (16). A step of 10 deg. is used. This configuation esults in 361 synthetic data at each fequency. Six fequencies ae used in the fequency hopping scheme as 10 MHz, 200 MHz, 500 MHz, 1 GHz, 3 GHz and 5 GHz. The initial guess of the two unknowns is assumed a cicula cylinde of the adius of c = 10 cm centeed at (x c, y c ) = (40 cm, 0) (see Fig. 6(a)). The fequency hopping technique is combined with the level set method to avoid dopping the algoithm in local minima as explained in [21, 33]. Lowe fequencies of 10 MHz and 200 MHz wee employed at the beginning of the fequency hopping scheme to find the tagets locations, followed by elatively highe fequencies

13 Pogess In Electomagnetics Reseach B, Vol. 25, to etieve the details of the objects. Notice the pogess of the econstuction vesus the fequency shown in Fig. 6. Notice also that the computational domain should include the initial guess, the two tagets with adequate space fo the evolving objects. In this wok, the computational domain is a squae with the dimensions of 80 cm centeed at (40 cm, 0). Figue 5 shows the nomalized cost function vesus the invesion iteations. The cost function is nomalized with espect to the synthetic data at each fequency. The expession of the cost function is given as: Cost function = N inc N meas i Ez,sim sc (θinc i i=1 j=1, θij meas ) E sc z,meas(θi inc, θ meas ij ) 2 N inc N meas i Ez,meas(θ sc i inc, θ meas ij ) 2 i=1 j=1 (21) Due to the fact that the cost function at each fequency is not nomalized to the pevious value, the plots in Fig. 5 show jumps once the algoithm hops to a new fequency as discussed in [21 25]. The esults in Fig. 5 show that the algoithm conveged afte about 7000 iteations with esidual eo of The obseved fluctuations in the cost function plot ae not undestood yet as they ae noticed at lowe and highe fequencies; howeve, they do not affect the final econstuctions. Fig. 6(a) shows the initial guess, Fig. 6(b) shows the econstuction afte 3700 iteations at 500 MHz, and Fig. 6(c) shows the econstuction afte 7000 iteations at 5 GHz. These esults showed the capability of the level set algoithm when the two ellipses wee located fa fom the wall. Figue 5. Nomalized cost function fo econstuction of two elliptical cylindes (d = 40 cm).

14 144 Hajihashemi and El-Shenawee Based on ou esults in 2D cases [24], using the low fequency of 10 MHz educes the eo between the simulated and measuement data. The defomation velocity is well-behaved and points towads the location of the taget objects. A coase discetization of the evolving contous is used in the lowest fequency compaed with the highe ones. In fee space case, the lowest fequency successfully helps etieving the unknown location of the taget [24]. The main poblem aises due to the fa-field citeia fo collecting the measuement data in eal applications. The second case uses the same data of Figs. 5, 6 but with deceasing the distance to wall to d = 20 cm instead of d = 40 cm. In this case, the two ellipses ae located at (20 cm, 20 cm) and (20 cm, 20 cm), espectively. The same initial guess of Fig. 6 is assumed hee. Fig. 7 shows the final econstuction esults. The cost function which demonstates the convegence of the algoithm in this case (not shown hee) is simila to that shown in Fig. 5. To investigate the effect of the distance between the wall and the two ellipses on the econstuction esults, the same data of Fig. 6 is epeated except with moe decease of the distance to only d = 8 cm. The same initial guess of Fig. 6 is used hee. The final econstuction ε = cm Initial guess ε = cm Evolving (a) (b) ε = cm Final Figue 6. Reconstuction of two elliptical cylindes behind the dielectic wall when (d = 40 cm). (a) Initial guess, (b) afte 3700 iteations, (c) afte 7000 iteations. (c)

15 Pogess In Electomagnetics Reseach B, Vol. 25, Initial guess ε = cm ε = cm Evolving (a) (b) ε = cm Final Figue 7. Reconstuction of two elliptical cylindes behind the dielectic wall when (d = 20 cm). (a) Initial guess, (b) afte 3700 iteations, (c) afte 7000 iteations. (c) esults ae shown in Fig. 8. In this case, the cost function (not shown hee) is simila to that shown in Fig. 5. The esults of Figs. 5 8 show that the level set econstuction algoithm successfully etieved the two ellipses even when the distance to the wall was 40 cm, 20 cm, and 8 cm. The CPU time equied fo the above examples was 40 minutes total. The SUN platfom with AMD Opteon (tm) Pocesso 850 of 2393 MHz and 8 GB of RAM was used Reconstuction of a Defected Pipe The econstuction of a defected pipe hidden behind a wall is investigated as shown in Figs. 9 and 10. The length of the cack is 2.8 cm with width of 3 cm as depicted in Fig. 10. The defect is located on the suface of a cicula pipe that has a adius of 10 cm and it is centeed 60 cm away fom the wall. The thickness of the wall is assumed 20 cm made of dy concete of pemittivity ε = 4.5 and loss tangent tan δ = The total numbe of synthetic data used in this example is the same with 19 incident and 19 scatteing diections pe each incidence (16). The same initial guess of Fig. 6 is used hee with fou diffeent location of the defected pipe as shown in Figs. 10(a) (d).

16 146 Hajihashemi and El-Shenawee d=8 cm ε = cm Initial guess ε = cm Evolving (a) (b) ε = cm Final Figue 8. Reconstuction of two elliptical cylindes behind the dielectic wall when (d = 8 cm). (a) Initial guess, (b) afte 3700 iteations, (c) afte 7000 iteations. (c) Figue 9. Nomalized cost function fo econstuction of the defected pipe (d = 60 cm). In this case, six fequencies ae used in the fequency hopping scheme as 10 MHz, 1 GHz, 3 GHz, 5 GHz, 7 GHz and 9 GHz. The nomalized cost function is shown in Fig. 9 which clealy shows the convegence of the algoithm afte 9990 Iteations. The esults in Figs. 10(a) (c) show satisfactoy esults of the defected pipe; howeve Fig. 10(d) did not show a good econstuction of the defect due to the

17 Pogess In Electomagnetics Reseach B, Vol. 25, inceased multiple scatteing with the wall at the educed distance in this case. The highe fequencies 5 GHz, 7 GHz and 9 GHz ae needed to etieve the defect in the pipe as shown in Figs. 10(a) (c), although the tansmission coefficient in the wall was educed to 0.71, 0.63, 0.53, espectively, at these fequencies when consideing the wall alone. The esults show that as the distance to the wall deceases, the econstucted pofile stats to deteioate. In the last case, when the cack is only 1 cm away fom the dielectic wall, it was not etieved successfully as shown in Fig. 10(d). This can be explained by the fact that the multiple scatteing between the wall and the object inceases leading to inaccuate calculations using the stationay phase method. On the othe hand, when the same defected pipe is placed in fee space, the defect was pefectly econstucted as epoted in [24] whee fequencies up to 15 GHz wee used. The econstuction time fo the cases of Fig. 10 was 4 hous total on the same SUN platfom. It is known that inveting limited view data imposes a challenge on the econstuction accuacy. Howeve, the level set has shown a success in econstucting the shape and location of tagets using limited view data compaed with othe methods such as the linea sampling method [19, 23]. Fo example, in the econstuction of a defected pipe Initial guess Tue object ε = cm tan δ = Tue object ε = cm tan δ = Initial guess (a) (b) Tue object Tue object ε = cm tan δ = Initial guess ε = cm tan δ = Initial guess (c) (d) Figue 10. Reconstuction of the defected pipe at diffeent distances fom the wall. (a) d = 40 cm, (b) d = 30 cm, (c) d = 15 cm, (d) d = 11 cm.

18 148 Hajihashemi and El-Shenawee in fee space when illumination was available at one diection, the level set was successful in etieving a patial pofile of the defect [24] Reconstuction of the Thee Objects Using Noisy Data In this example, the econstuction of thee objects of abitay cosssections is shown when noisy data was added. This example deals with an unsymmetical stuctue of the objects behind the wall. The objects ae ectangula cylinde, tiangula cylinde, and elliptical cylinde. The thickness of the wall is assumed 20 cm with pemittivity of ε = 9.0 and loss tangent tan δ = The same numbe of data and the same initial guess ae used hee. Two diffeent noise levels ae examined in this example with the signal to noise atio (SNR) SNR = 20 log( Ems signal Enoise ms ), whee Esignal ms and Enoise ms epesent the oot-mean-squae of the signal and the Gaussian noise, espectively. A fixed SNR is used at all fequencies which mean that the level of the noise depends on the amplitude of the synthetic data which is changing with fequency. In this case, nine fequencies ae used in the fequency hopping scheme as 10 MHz, 100 MHz, 200 MHz, 500 MHz, 750 MHz, 1 GHz, 3 GHz, 5 GHz and 7 GHz. As mentioned ealie, the lowe fequencies helped to etieve the geneal pofile of the tagets while the highe fequencies helped to econstuct the details of the thee objects. The nomalized cost function is shown in Fig. 11. The econstuction esults of the thee tagets using noisy data coesponding to SNR = 10 db ae shown in Fig. 12(a), in Fig. 12(b) afte 6500 iteations at 200 MHz, and in Fig. 12(c) afte iteations at 7 GHz showing the final econstuction. Figue 11. Nomalized cost function fo econstuction of the thee objects (SNR = 10 db).

19 Pogess In Electomagnetics Reseach B, Vol. 25, SNR= 10 db SNR= 10 db ε = 9 20cm tan δ = Initial guess ε = 9 20cm tan δ = Evolving (a) (b) SNR= 10 db ε = 9 20cm tan δ = final Figue 12. Noisy data of SNR = 10 db. (a) Initial guess, afte (b) 6500 iteations and (c) iteations. (c) Despite the noisy data and the limited view configuation of the incident and scatteing diections, the level set algoithm successfully etieves the thee objects behind the wall with a single initial guess. Howeve since thee is no illumination fom the ight side of the wall, the econstucted pofile is a little bit distoted on that side. Due to the moe complex configuation in this example, lage numbe of fequencies and iteations wee needed compaed with the pevious examples. The same case is epeated but when the data is moe noisy with SNR = 5 db. The esults show less satisfactoy econstucted shapes due to the highe level of noise (not pesented hee). The CPU time was 3 hous fo iteations on the same SUN platfom. The numbe of employed fequencies is inceased accoding to the complexity of the tagets objects. In the example of Fig. 11, the numbe of fequencies is inceased (9 fequencies) compaed with the ealie examples (6 Fequencies). Howeve, using moe fequencies does not degade the econstucted pofile but leads to inceasing the equied CPU time. In geneal, the woking fequency should jump to a highe one, when the cost function dops in local minima and no futhe details of the taget is etieved using the data at the cuent fequency. In this wok, a pe-assigned numbe of iteations

20 150 Hajihashemi and El-Shenawee (mostly 1000 iteations) ae used, while in [23], the stagnancy of the cost function is used as a signal fo the fequency hopping. The late scheme avoids the inceased CPU time when the cost function dops in local minima. In [23], the scheme was based on collecting the most ecent 20 samples of the cost function and implementing the aveage window technique of each five samples. If the diffeence between the aveages is less than a theshold (e.g., 1%), the algoithm hops to the highe fequency Paallelization of the Reconstuction Algoithm In geneal, the poblem of the shape econstuction in the statified media is moe computationally intensive compaed with the fee space case. Theefoe, it is necessay to implement algoithm paallelization to speed up the computations. Using the MPI paallelization, the computational load is distibuted between seveal pocessos. The level set algoithm was paallelized fo the econstuction of multiple 2D PEC objects immesed in fee space [22] with achieved maximum speedup ange of 53X to 84X using 256 pocessos on the San Diego Supe Compute Cente (SDSC) facilities. In this wok, the paallelized code is tested on the National Cente fo Supecomputing Applications (NCSA) at the Univesity of Illinois. NCSA TeaGid IA- 64 Linux Cluste is employed to un the code. The machine consists of 887 IBM cluste nodes: 256 nodes with dual 1.3 GHz Intel R Itanium R 2 pocessos. The paallelized algoithm in [22] is modified hee to accommodate the Geen s function in statified media (4). The paallelization is based on thee main bottlenecks; (i) the domain decomposition appoach fo updating the level set function in the whole computational domain, (ii) the distibution of calculating the defomation velocity, and (iii) the invesion of the MoM impedance matix using Scalapack libay available on NCSA supecomputes. The details of these bottlenecks ae epoted and discussed in [22]. The paallelized level set algoithm using diffeent numbe of pocessos is tested to obtain the same econstuction esults of Fig. 12. The econstuction CPU time consumed afte invesion iteations using a single pocesso is 7.5 hous while it is 15minutes when using 128 pocessos leading to achieve a maximum speedup of 29X as shown in Fig. 13. The coesponding paallelization efficiency is 22% in this case as shown in Fig. 14. Note that speedup is defined as the CPU time when using a single pocesso divided by the CPU time when using multiple pocessos on the same platfom. The paallelization efficiency quantitatively descibes the effectiveness of the paallelized code and is defined as the speedup divided by the

21 Pogess In Electomagnetics Reseach B, Vol. 25, Figue 13. Speedup vs. the numbe of pocessos. Figue 14. Efficiency vs. numbe of pocessos. numbe of pocessos [22]. The esult of Fig. 13 shows a maximum speedup of 29 when using 128 pocessos; howeve, moe inceasing of the numbe of pocessos was not helpful due to the ovehead in the communications and othe factos as discussed in [22]. The maximum numbe of 256 pocessos is used to show the decease in the speed up cuve in Fig. 13. It is impotant to emphasize that the pesented algoithm is not limited to symmetic simple shapes of the tagets o to the 2-D configuations as epoted in [23]. Also the algoithm was tested successfully on both TM and TE polaizations [21, 25]. 4. CONCLUSIONS The Level set algoithm is implemented to econstuct the shape and location of multiple 2-D PEC objects hidden behind a dielectic wall with a pioi known paametes. The stationay phase method is implemented to appoximate the Sommefeld integals to speed up the calculations and to avoid the inaccuacy geneated due to the abitay tuncation in the integation limits when numeically evaluating the full Geen s function. The esults demonstate that the level set method is capable of econstucting the shapes and locations of multiple objects hidden behind a wall even with (i) limited view data, (ii) coupted data up to SNR = 10 db, and (iii) nea poximity to the wall. Moe investigations ae necessay to incease the accuacy of the algoithm when econstucting fine featues in tagets located vey close to the wall. Also moe wok is needed when the wall s paametes ae not a pioi known. REFERENCES 1. Lu, T., K. Agawal, Y. Zhong, and X. Chen, Though-wall imaging: Application of subspace-based optimization method,

22 152 Hajihashemi and El-Shenawee Pogess In Electomagnetics Reseach, Vol. 102, , Maaef, N., P. Millot, X. Feièes, C. Pichot, and O. Picon, Electomagnetic imaging method based on time evesal pocessing applied to though-the-wall taget localization, Pogess In Electomagnetics Reseach M, Vol. 1, 59 67, Soldoviei, F., R. Solimene, and R. Piei, A simple stategy to detect changes in though the wall imaging, Pogess In Electomagnetics Reseach M, Vol. 7, 1 13, Rekanos, I. T. and A. Räisänen, Micowave imaging in the time domain of buied multiple scattees by using an FDTDbased optimization technique, IEEE Tansactions on Magnetics, Vol. 39, No. 3, , May Tavassos, X. L., D. A. G. Vieia, N. Ida, C. Vollaie, and A. Nicolas, Invese algoithms fo the GPR assessment of concete stuctues, IEEE Tansactions on Magnetics, Vol. 44, No. 6, , June Boek, S. E., An oveview of though the wall suveillance fo homeland secuity, Poceedings of the 34th Applied Imagey and Patten Recognition Wokshop (AIPR05), Soldoviei, F. and R. Solimene, Though-wall imaging via a linea invese scatteing algoithm, IEEE Geo. and Rem. Sens. Lettes, Vol. 4, No. 4, , Octobe Dehmollaian, M. and K. Saabandi, Refocusing though building walls using synthetic apetue ada, IEEE Tansactions on Geoscience and Remote Sensing, Vol. 46, No. 6, , June Hantsche, S., A. Reisenzahn, and C. G. Diskus, Thoughwall imaging with a 3-D UWB SAR algoithm, IEEE Signal Pocessing Lettes, Vol. 15, , Song, L. P., C. Yu, and Q. H. Liu, Though-wall imaging (TWI) by ada: 2-D tomogaphic esults and analyses, IEEE Tansactions on Geoscience and Remote Sensing, Vol. 43, No. 12, Decembe Ahmad, F., Y. Zhang, and M. G. Amin, Thee-dimensional wideband beamfoming fo imaging though a single wall, IEEE Geo. and Rem. Sens. Lettes, Vol. 5, No. 2, , Apil Piei, R., G. Pisco, F. Soldoviei, and R. Solimene, Thee-dimensional though-wall imaging unde ambiguous wall paametes, IEEE Tansactions on Geoscience and Remote Sensing, Vol. 47, No. 5, , May Wang, G. and M. G. Amin, Imaging though unknown walls

23 Pogess In Electomagnetics Reseach B, Vol. 25, using diffeent standoff distances, IEEE Tans. Signal Pocess., Vol. 54, No. 10, , Octobe Debes, C., M. G. Amin, and A. M. Zoubi, Taget detection in single- and multiple-view though-the-wall ada imaging, IEEE Tansactions on Geoscience and Remote Sensing, Vol. 47, No. 5, , May Yacoub, H. and T. K. Saka, A homomophic appoach fo though-wall sens., IEEE Tansactions on Geoscience and Remote Sensing, Vol. 47, No. 5, , May Yoon, Y. S. and M. G. Amin, Spatial filteing fo wallclutte mitigation in though-the-wall ada imaging, IEEE Tansactions on Geoscience and Remote Sensing, Vol. 47, No. 9, , Septembe Soldoviei, F., R. Solimene, and G. Pisco, A multiaay tomogaphic appoach fo though-the wall imaging, IEEE Tansactions on Geoscience and Remote Sensing, Vol. 46, No. 4, , Apil Soldoviei, F., A. Bancaccio, G. Leone, and R. Piei, Shape econstuction of pefectly conducting objects by multiview expeimental data, IEEE Tansactions on Geoscience and Remote Sensing, Vol. 43, No. 1, 65 71, Januay Catapano, L. and L. Cocco, An imaging method fo concealed tagets, IEEE Tansactions on Geoscience and Remote Sensing, Vol. 47, No. 5, , May Ramananjaona, C., M. Lambet, D. Lesselie, and J.-P. Zol esio, Shape econstuction of buied obstacles by contolled evolution of a level set: fom a min max fomulation to numeical expeimentation, Invese Poblems, Vol. 17, , Hajihashemi, M. R. and M. El-Shenawee, TE vesus TM fo the shape econstuction of 2-D PEC tagets using the levelset algoithm, IEEE Tansactions on Geoscience and Remote Sensing, Vol. 48, No. 3, , Mach Hajihashemi, M. R. and M. El-Shenawee, High pefomance computing fo the level-set econstuction algoithm, Jounal of Paallel and Distibuted Computing, Vol. 70, No. 6, , June Hajihashemi, M. R., Invese scatteing level set algoithm fo etieving the shape and location of multiple tagets, Ph.D. Dissetation, Univesity of Akansas, Hajihashemi, M. R. and M. El-Shenawee, Shape econstuction using the level set method fo micowave applications, IEEE

24 154 Hajihashemi and El-Shenawee Antennas and Wieless Popagation Lettes, Vol. 7, 92 96, Hassan, A. M., M. R. Hajihashemi, D. A. Woten, and M. El- Shenawee, Dift de-noising of expeimental TE measuements fo imaging 2D PEC cylindes, IEEE Ant. and Wieless Popagation Lettes, Vol. 8, , Wait, J. R., Electomagnetic Waves in Statified Media, IEEE Pess, Cui, T. J. and W. Wiesbeck, TM wave scatteing by multiple two-dimensional scattees buied unde one-dimensional multilayeed media, Poc. IEEE Intenational Geo. and Rem. Sens. Symposium, 27 31, Lincoln, May Sethian, J. A., Level Set Methods and Fast Maching Methods, Cambidge Univesity Pess, Oshe, S. J. and R. P. Fedkiw, Level Set Methods and Dynamic Implicit Sufaces, Spinge-Velag, Roge, A., Recipocity theoem applied to the computation of functional deivatives of the scatteing matix, Electomagnetics, Vol. 2, No. 1, 69 83, Chew, W. C., A quick way to appoximate a sommefeldweyl-type integal, IEEE Tans. on Antennas and Popagation, Vol. 36, No. 11, , Novembe FEKO Use s manual, Suite 5.3, July Chew, W. C. and J. H. Lin, A fequency-hopping appoach fo micowave imaging of lage inhomogeneous bodies, IEEE Micowave and Guided Wave Lettes, Vol. 5, No. 12, , Decembe Aksun, M. I., A obust appoach fo the deivation of closedfom Geen s functions, IEEE Tans. Micowave Theoy Tech., Vol. 44, , May 1996.