RADAR ESTIMATION OF BUILDING LAYOUTS USING JUMP-DIFFUSION

Marija M. Nikolic', Mathias Ortner', Arye Nehorai', Antonije R. Djordjevic2

Department of Electrical and Systems Engineering, Washington University, St. Louis, MO 63130,USA, nikolic2 e. wust , nehoraiaese. wusti. edu

2 School of Electrical Engineering, University of Belgrade, P.O. Box 35-54, 11120 Belgrade, Serbia,edjoda )ef~ C. yu

ABSTRACTIn Section 4, we develop the scalable approach were

Estimating buildings layouts using exterior radar low resolution estimates are used to initiate a more refinedmeasurements is a challenging task involving the estimation. Finally, in Section 5, we present some results ofelectromagnetic modeling, many unknown parameters, and the simulationslimited number of sensors. We propose using the jump-diffusion [1-4] algorithm as a powerful stochastic tool that 2. ELECTROMAGNETIC MODELINGcan be used to determine the number of walls, estimate their

unknown poiin an ote paaees.e mrv h In our modeling, we use a 2D method of moments (MoM)unknown p s acode since electromagnetic modeling of electrically largeconvergence rate of the jump-diffusion algorithm by 3D structures...3Dstructures (such as buildings) iS still inadmissibly timedeveloping an iterative procedure that first finds low- consuming. However, the approach we propose to estimateresolution estimates, which are then used to initiate our the interior of the building is general and can be easilymore accurate estimation. Our efficient usage of the applied to 3D problems.available frequency bandwidth, improves the computational We have developed the MoM code by generalizing thespeed that otherwise would be hampered by the forward program from [5] to include arbitrarily shaped lossy-electromagnetic modeling. dielectric objects. The program uses the equivalenceIndex Terms-Jump-diffusion, through the wall-sensing. principle to divide the system under consideration into a

number of subsystems (entities), each of them being filled1. INTRODUCTION with a homogeneous medium.

In our case, the system consists of probes (conductorsWe address the problem of estimating building layouts using whose cross section is electrically small), and lossy-exterior electromagnetic sensing, which is important in dielectric walls of the building. The program excites oneurban warfare. Solving this problem is challenging due to probe at a time by an impressed electric field, calculatesthe complex and unknown environment. The computational equivalent electric and magnetic currents on the surfaces ofspeed of the electromagnetic modeling of the electrically all entities, and finally evaluates the net electric currents inlarge structures also limits the practical aspects of the all probes. The result is a matrix relation of the formgeometry estimation. We show that the number and I = YE, where I is M x 1 vector of electric currents indisplacements of inner walls can be efficiently determined probes, Y is M xM admittance matrix, and E is M x 1by the jump-diffusion algorithm [1-4]. We develop an vector of induced electric field in probes (M is the numberiterative procedure considering trade-offs between accuracy of probes.).and computational speed. We improve the solution Increasing the number of probes improves the radarconvergence and reduce the processing time by efficient performance. However, the number of probes is technicallybandwidth usage. limited in real systems. One way to resolve this problem is

Our proposed solution is simulated using a 2D to use moving sensors [6]. We assume that one movingelectromagnetic forward modeler, which is described sensor consists of several probes. Accurate modelingin Section 2. In Section 3, we describe the application of the necessitates independent simulations for each position ofthe

sensor. However, this is computationally costly even in thejump-diffusion algorithm for building estimation. .M 't,nrnTc,-nlvr,-onibrvb~rjumpdiffsionalgoithmfor uildng etimaion.2D scenario. Instead, we consider the case where severalThis work was supported in part by the Department of Defense under the static sensors are simultaneously present, so that each sensorAir Force Office of Scientific Research MUMI Grant FA9550-05-1-0443, location corresponds to one position of the moving sensor.and the AFOSR Grant FA9550-05-1-0018 In general, such a model introduces an error due to coupling

among the static sensors. Nevertheless, if the probes are thin

978-1-4244-1714-8/07/$25.OO ©007 IEEE 177

and if the impedance (z) parameters are used instead of the The moves have fixed probabilities p1, p2,... I,p7admittance (y) parameters, the modeling error is negligibly 7small [7]. Therefore, we determine impedance coefficients where pPi = 1, except for the first iteration where P1 = 1,by inverting the admittance matrix to obtain E = ZI. If thesensor consists ofm probes, we set: P2 = ... = p7 = 0, since we assume there are no inner wallsZ. =0, li- .l>m, i, 1,, M. (1) inthebeginning. IneachiterationoneoftheabovemovesisWe calculate z matrix at N discrete frequencies, as we randomly selected according to adopted probabilities,explain letter. p1, i = 1,... ,7. The move is applied to a randomly selected

wall (walls have the same probability to be drawn). Wedenote by X the current estimate of the scene and by X',

The building layout estimation belongs to the group of the proposed state due to the chosen move. The proposedproblems with unknown dimensionality, since besides state will be accepted if P(X') < P(X), where P(X') isdisposition and characteristics of the walls, their number is the error power of proposed state, and P(X) is the erroralso unknown. The application of jump-diffusion (JD) power of the current state. We define the power of the erroralgorithm in the cases where the dimensionality of the as,parameter space is not fixed was studied in [1, 2]. The jump- Ndiffusion has already been used for image segmentation P(X)= Zp(X, f)-Zn(fi) , (2)[3, 4]. Jump diffusion allows traversing through parameter i=I

space by two types of moves: reversible jumps between where N is the number of frequencies, Zp(x,fj) issubspaces of different dimensionality and stochastic simulated z matrix for the layout given by state X atdiffusions within each continuous subspace. In the problem frequencyf., Zm(/i) is measured z matrix at frequencyf.,we consider, dimensionality of the subspace is equal to the and 'fro' denotes Frobenious measure. We assume thatnumber of walls. Within each subspace, the unknown measurements are corrupted by white, zero-mean Gaussianparameters for every wall are: center coordinates (x,y), noise, which is uncorrelated with received signals:length (l) and orientation (cc). We assume that the thickness z =Z (3)and the dielectric permittivity of the walls are known. For m mOlayout estimation we adopted the following moves: where ZmO is z matrix for the exact building layout and W is

1. Birth: the number of the walls is increased. The additive noise. We summarize the noise assumptions in thecenter of the new wall is randomly selected within usual way:the interior of the building, i.e., E(wj) = 0, i,j = 1 ... Mx Unif[xmin,xXmax],y Unif[yminIymax]* The E(w, wk)=0, i.j,k. 1, (4)length of the new wall is a uniform random * 2variable within adopted limits / Unif[lmin,Imax] E(wj/ wj/)-O,E(wj/ w,) =

2. Death: the number of the walls is decreased. A We are interested in the signals that are reflected backselected inner wall is removed from the scene. from the building simce they contain information about the

3.Translation: the center of a selected wall is building interior, and not in the total field induced in thetranslate within.thebuildi. The newcoosensors. Due to the propagation attenuation, multiple

transltednithlxin t builIng.The n coin ate reflections etc., the electric field scattered from the buildingare x Unif[xminn, Xaxoy UniflymineYmax]l is weak compared to the electric field induced in the sensors

4. Rotation: the orientation of a selected wall is due to their direct coupling. Therefore, we define the powerflipped .The walls are assumed to be parallel to the of "useful" signal (P0), i.e., the signal that is reflected backexterior walls, i.e. their orientation is assumed to be from the buildingeither 00 or 900 degrees. N

5. Elongation: the length of a selected wall is Io = E Zm(fJ -Zs (f )fro (5)changed. The new wall length is/ Unif[ln I/x] where Zs is the z matrix calculated for the case where only6 Regeneration: mall parameter aselectedwallar sensors are in the scene and not the building. We define the

6. Regeneration: all parameter a selected wall arechanged. The new parameters are signal-to-noise ratio (SNR) as

x Unif[xmin IXmax] y UnifYminI Ymax], and SNR -PO (6)/I Unif[l/ ,l /0]2

7. Optimization. The length of a selected wall and itscenter coordinates are optimized using simplexalgorithm. The simplex is initiated by the currentvalues of those parameters.

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4. FREQUENCY SELECTION where X is the final state calculated using the frequency

For sensing through the walls, a wideband approach is pair (j0 I). The procedure is continued until the error is

necessary to achieve the desired feature resolution. Forward smaller than desired value.modeling of electrically large structures is computationallyintensive, hence efficient usage of the bandwidth is critical 5. RESULTSfor practical applications ofjump-diffusion algorithm.

At low frequencies, the probability of accepting the We consider a 4m x 4m building with five unknownproposed move is very high due to the insensitivity of the inner walls that are shown in Figure 1 (line with squaremeasurements to small perturbations in wall positions, markers). The goal is to determine the number ofthe walls,length, etc. Hence, the algorithm traverses quickly through their positions, and lengths. For simplicity, we assumed that

different states (building layouts), converging to the the thickness and the relative permittivity of the walls are

solutions that are close to the exact layout. The forward known. We assume that the measurements are taken by theelectromagnetic modeling is also very efficient at low radar system that moves around the building. The measuring

1nHeiks system consists of 2 probes. The measurement places arefrqunie. oevr,lw-rqunc stmain c denoted by dots around the building in Figure 1. We havedesired resolution, might detect false walls, etc. In contrast, shown in [7] that the permibuivity and thickness of a similarhigh-frequency estimation is extremely sensitive, and problem can be efficiently estimated bythe means of MLEconsequently the probability of rejecting the proposed move and range gating technique. We adopted w = 15 cm for theis high unless we are close to the exact solution. Hence, thickne oftinterior Wa wotw20 cm for the

convergence at the high frequencies is very slow. Also, theforward calculations are exceptionally time-consuming. walls, and er =3 for their relative permittivity.Therefore, we perform the estimation in two stages, namely .........................

we use the rough estimates obtained at the lower frequenciesto initiate the more accurate estimation at higherfrequencies. We select the initial frequency according to thesampling criterion in the frequency domain:

4D, ~~~~~~~~~~(7)where Dmax is the largest dimension in the considereddomain, and c is the speed of the light. The power of theerror decreases as number of iterations increases until itreaches the order of noise, as we show later. In the second

step~~~~~~~~~ ......

iastmt Xl bane tsarig ............step, we use the final estimate (XI) obtained at starting Figure 1. Joint results for the positions and lengths of the inner walls (solidfrequency to initiate the estimation at a higher frequency, line) by the first-step JD algorithm at fi= 10 MHz. Results are obtained

f2. In order to preserve robustness, we perform after 5 simulations of 500 trials at high signal-to-noise ratio. The truepositions and lengths are denoted by lines with square markers.simultaneous calculations at initial frequency, fl. The error ... ... ...... ..........

power is therefore

P(X;f2,fl)= Zp(X;f2) Zm(X;f2) +

Zp(X;f)-Zm(X;f ) fro (8)

The frequency 12 is selected adaptively based on the error

f2= argmin(P(X ;fi,f)>Pmin), (9)f

where P is the adopted value for the error power that isabove the noise level. We repeat the procedure, and at step#i we have:

(fro (1 0) Figure 2. Joint results for the positions and lengths of the inner walls (solidZ (X; a )A_ Z (X; a) line) by the second-step JD algorithm at f = 10 MHz andf2 = 100 MHz.

+ Ip V J J m V lfro'^ Results are obtained after 5 simulations of250 trials at high signal-to-noise{ {A\ \ ~~~~~~~~~~ratio.The true positions and lengths are denoted by lines with square

fi = argmin tPtX11l; fi, f)> Pmin)}X (1 1) markers.f

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Firstly, we consider the case with high signal-to-noise .....................

ratio (SNR =30 dB). The initial estimation is performed at10 Mfiz. The joint results after 500 iterations for five

independent simulations are shown in Figure 1 (solid lines).The estimated layout represents rough image of the originallayout and it is used for the initialization of the second-stepestimation performed at (f = 10 MHz, f2 100Hz). Theestimated layout in the second step for 5 independentsimulations and after 250 trials is shown in Figure 2. Theestimated walls are very close to the original walls. If finetuning is required, only one more step would have to beperformed. We illustrate the decrease of the error poweraveraged over the number of simulations in Figure 3 for .........................

both estimations steps. Figure 5. Estimated positions and lengths of the inner walls obtained by theFirst step third-step JD estimation at fi 10 MHz and f2 = 200 MHz (line with30-Second step . .....circles). Results are obtained at SNR= 10 dB, after 50 trials. The true

25-~~ ' = r rpositions and lengths of the walls are denoted by lines with squares.

5.......... 1. .6. CONCLUSION10 .k

We developed an efficient estimation scheme to estimate0 -------3------------------------------------------------------- unknown building layouts. The proposed approach uses the%E 0 2030 400 500 jump-diffusion algorithm with moving the radar system to

Figure 3. Normalized error power as a function of trials for the first-step JD determine the number and parameters of the walls (e.g.estimation (shown in Figure 1), and the second-step JD estimation (shown position, length, orientation). We designed an iterativein Figure 2). procedure that uses low-frequency (low resolution)

estimates to initiate high-frequency (high resolution)estimation. We increased the computational speed by a

proper frequency management. The accuracy and efficiencyof the proposed method were confirmed by varioussimulations.

r_4e r7.REFERENCES

[1] P.J. Green, "Reversible Jump Markov Chain Monte CarloComputation and Bayesian Model Determination," Biometrika,vol. 82, no. 4, pp.711- 732, 1995.[2] U. Grenander and M. Miller, "Representation of Knowledge in

......................... Complex Systems," J. Royal Statistics Soc., vol. B56, pp. 97-109,Figure 4. Estimated positions and lengths of the inner walls obtained by the 1994.first-step JD estimation atf1 = 10 MHz (line with asterisks) and the second- [3] S.C. Zhu, "Stochastic Jump-Diffusion Process for Computingstep JD estimation at fi = 10 MHz and f2= 50 MHz (line with circles). Medial Axes in Markov Random Field," IEEE Trans. PatternResults are obtained at SNR= 10 dB, after 500 trials and 100 trials, Analysis and Machine Intelligence vol. 21 no. 11 1158-1169respectively. The true positions and lengths of the walls are denoted by nov. 19egelines with squares. Nov. 1999.

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