An Approach to Estimating Building Layouts Using Radar and Jump-Diffusion Algorithm

768 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 57, NO. 3, MARCH 2009

An Approach to Estimating Building LayoutsUsing Radar and Jump-Diffusion AlgorithmMarija M. Nikolic, Mathias Ortner, Arye Nehorai, Fellow, IEEE, and Antonije R. Djordjevic

Abstract—Estimating building layouts using exterior radarmeasurements is a challenging task involving electromagneticmodeling, many unknown parameters, and a limited numberof sensors. We propose using the jump-diffusion algorithm as apowerful stochastic tool that can be used to estimate the numberof walls, their unknown positions and other parameters. Wedevelop an efficient iterative procedure that first uses low-fre-quency transmissions to obtain rough estimates of the buildinglayout. These estimates are then used to initialize the estimationat higher frequencies, thus obtaining more accurate results. Weshow that with proper frequency selection, the building layout canbe estimated using radar measurements at only a few frequencies.Numerical examples are used to illustrate the performance of theproposed algorithm in practical situations.

Index Terms—Building layout estimation, frequency hopping,inverse scattering, jump-diffusion, multiple frequencies, through-the-wall sensing.

I. INTRODUCTION

W E address the problem of estimating building layoutsusing exterior electromagnetic sensing, which is impor-

tant in urban warfare. Solving this problem is challenging dueto the complex and unknown environment. At the same time,inner walls have some a priori known information that wouldbe useful to utilize as we showed in [1].

Reconstruction of the building layout belongs to the broadarea of inverse scattering. Procedures for solving inversescattering problems can be classified into deterministic andstochastic. Among deterministic algorithms, wide applicationshave iterative techniques based on the linearization around acurrent estimate [2]–[4]. Because of the intrinsic ill-posednessof the inverse scattering problems [5]–[7], the reconstructionmethods may diverge or converge to a false solution.

In [8] and [9], the authors compute the images of the targetshidden behind walls using the Born approximation. A globally

Manuscript received September 12, 2007; revised October 09, 2008. Currentversion published March 27, 2009. This work was supported in part by the De-partment of Defense under the Air Force Office of Scientific Research (AFOSR)MURI Grant FA9550-05-1-0443, the AFOSR Grant FA9550-05-1-0018, Officeof Naval Research (ONR) Grant N000140810849, and by COST Action IC0603(Assist).

M. M. Nikolic is with the Department of Electrical and Systems Engineering,Washington University, St. Louis, MO 63130 USA, and also with the Schoolof Electrical Engineering, University of Belgrade, Belgrade, Serbia (e-mail:[email protected]).

M. Ortner and A. Nehorai are with the Department of Electrical and SystemsEngineering, Washington University, St. Louis, MO 63130 USA (e-mail: [email protected]).

A. R. Djordjevic is with the School of Electrical Engineering, University ofBelgrade, Belgrade, Serbia (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAP.2009.2013420

convergent stochastic algorithm for the inverse scattering basedon the simulated annealing is given in [10]. The algorithm of[10] is extremely time consuming, and cannot be applied forthe estimation of electrically large structures such as buildings.

In this paper, we exploit the knowledge on the typical buildinglayouts to improve the estimation. Prior information reduces theparameter space, and consequently the number of iterations. Weuse the stochastic algorithm that enables efficient inclusion ofthe prior knowledge—the jump-diffusion algorithm [11], [12].We showed the basic principles of this approach in [1]. Herein,we consider more elaborate applications.

We use the jump-diffusion algorithm to estimate the numberof the inner walls and their configuration. Instead of dividingthe unknown area into uniform cells with unknown properties[10], we consider more natural division to “walls” and “rooms.”The jump-diffusion algorithm has been widely used for solvingproblems where the dimensionality of the parameter space isnot known. However, most of its applications lie in the range ofimage processing [13]–[16]. In [15] the authors use the jump-diffusion to the extract rectangular buildings from aerial imagesof dense urban areas.

We use precise physical models that take into account thesignal propagation through the walls. Hence, the forward mod-eling has an important role in the process of estimation. Ourelectromagnetic modeling is based on the method of moments[17]. The forward modeling of electrically large structures,such as buildings, is computationally intensive, thus limitingthe practical aspects of the geometry estimation. Hence, theefficient usage of the bandwidth is critical for real-life im-plementations. Moreover, the selection of frequencies hasimportant role on the convergence of inverse scattering algo-rithms [7], [18]–[24].

We investigate the application of the jump-diffusion algo-rithm with multiple frequency data. There are several strategiesfor multifrequency-data processing such as frequency hopping,joint processing, and hybrid approach (e.g. [7], [18]). We ob-tain rough estimates at the low-frequency part of the bandwidththat are later used to initiate more accurate estimation at higherfrequencies. We show that accurate estimates may be obtainedusing only a few frequency samples.

The organization of the paper is as follows. In Section II, wedescribe the measurement model and the 2D electromagneticforward modeler. In Section III, we describe the applicationof the jump-diffusion algorithm for building estimation. InSection IV, we discuss the role of the frequency on the con-vergence and develop the frequency iterative approach. InSection V, we present results of the simulations, and in theAppendix, we detail the implementation of the algorithm.

0018-926X/$25.00 © 2009 IEEE

Authorized licensed use limited to: WASHINGTON UNIVERSITY LIBRARIES. Downloaded on June 11, 2009 at 14:46 from IEEE Xplore. Restrictions apply.

NIKOLIC et al.: AN APPROACH TO ESTIMATING BUILDING LAYOUTS 769

Fig. 1. Building layout and the measurement locations.

II. MOVING SENSORS

A. Measurement Model

We consider a 2D case since 3D modeling of buildings ina wide frequency range is inadmissibly time-consuming. Tomodel the measurements, we suppose a uniform linear arrayof sensors placed outside of the building. We assume thateach sensor in the array works both as transmitter and receiver.The array takes measurements at known positions, atfrequencies. An example of the building layout and a movingtwo-sensor array is shown in Fig. 1, where the discrete mea-surement locations are represented by dots.

The antenna array can be considered as a multiport network.Therefore, measurements may be represented by impedance

, admittance , or scattering parameters [25]. In thispaper, we use the impedance parameters. In 2D modeling, in-stead of induced voltages at sensor ports, we utilize the inducedelectric field in the sensors. Hence, the model reads

(1)

where is the operating frequency, is the index of the mea-surement position, is the measured induced electricfield in the -th sensor, is the mutual impedance coeffi-cient between the -th and -th sensors, is the feeding currentof the -th sensor, and is the measurement noise and mod-eling error. We assume that the noise is a complex zero-meanGaussian random variable with variance . The excitation isthe same for all sensors, i.e., we set , .

B. Electromagnetic Modeling

We determine the impedance parameters of the consideredsystem using the method of moments [17], [26]. We havegeneralized the program from [26] to include arbitrarily shapedlossy-dielectric objects. The program uses the equivalence prin-ciple to divide the system under consideration into a number ofsubsystems (entities), each of them being filled with a homoge-neous medium. In our case, the system consists of sensors (fil-ament conductors whose cross section is electrically small) andlossy-dielectric walls of the building. The program excites one

Fig. 2. Merge transformation.

sensor at a time by an impressed electric field, calculates equiva-lent electric and magnetic currents on the surfaces of all entities,and finally evaluates the net electric currents in all sensors. Thecalculations result in the admittance parameters , which afterinversion give the impedance parameters, .

III. BUILDING LAYOUT ESTIMATION USING THE JUMP

DIFFUSION ALGORITHM

A. Jump-Diffusion

The building layout estimation belongs to the group ofproblems with unknown dimensionality, since besides positionand characteristics of the walls, their number is also unknown.The application of the jump-diffusion algorithm in the caseswhere the dimensionality of the parameter space is not fixedwas studied in [11], [12]. The jump-diffusion allows traversingthrough parameter space by two types of transformations:jumps between subspaces of different dimensionality andstochastic diffusions within continuous subspaces.

In the problem we consider, the dimensionality of the sub-space is equal to the number of walls. Within each subspace,the unknown parameters for each wall are: center coordinates

, length , and orientation . We assume that the thick-ness and the dielectric permittivity of the walls are known. How-ever, we also consider cases in which these parameters are notexactly known. In the subspace with walls, the vector of theunknown parameters is . Following the notationfrom [1], we represent the pair as . Generally,varies within parameter space , where

and is the maximum number of walls.We seek for the solution that maximizes the likelihood

function. To explore the parameter space , we use two typesof transformations: transformations that change the number ofwalls (jumps), and transformations that only perturb the param-eters of the existing walls (diffusion). Some of the transforma-tions are inspired by [15], where the segmentation of radar im-ages of urban areas is considered. In the first group of transfor-mations, we have:

• Birth: A new wall is added to the current configuration. Theparameters of the new wall are . We assumewalls with orientation: .

• Death: A randomly selected inner wall is removed from thecurrent configuration.

• Merge: A randomly selected inner wall is merged with anearby wall (Fig. 2). The number of the walls is decreased,and the parameters of the merged wall are .

In the second group are the transformations that change oneor more parameters of a randomly selected wall from the currentlayout:



• Translation: The position of the wall is perturbed .• Elongation: The length of the wall is perturbed .• Rotation: The orientation of the wall is perturbed .• Regeneration: All parameters of the wall are perturbed

.• Optimization: The parameters of the wall are

optimized using simplex algorithm [27].In each iteration, one of the transformations is selected ac-

cording to the adopted probability distribution. The probabili-ties for transformations are chosen empirically and depend onthe current scene estimate, . The details for the above trans-formations and their probabilities are given in Appendix. Theselected transformation is applied on a randomly chosen wall(unless birth is selected), where the probability of drawing isthe same for all walls. The result of the transformation is a pro-posed layout . This layout may be accepted or rejected, de-pending on the definition of transition kernel. The popular so-lution is the Metropolis-Hastings algorithm [28], [29] that en-ables escaping from local minima. However, we decided for thegreedy algorithm since it is faster and the speed of the algorithmis critical for practical applications of building layout estima-tion. Nevertheless, the probability for falling into local minimais reduced by proper frequency selection and wide explorationof the parameter space. The proposed layout will be accepteddepending on the likelihood function only: if the likelihood forthe proposed layout is larger than the likelihood for the cur-rent layout , the proposed layout will be accepted. Otherwisethe current layout remains.

B. Likelihood Function

We assume independent measurements and Gaussian noise.Hence, the likelihood function reads

(2)

where is a single dimensioned vector where we stackall the measurements . We seek for the solution

(3)

C. Algorithm Summary

For a given current estimate , the summary of the algorithmis as follows:

• Draw a transformation according to the probability distri-bution .

• Draw a wall from , where the probability is the same forall walls (unless the birth is selected).

• Calculate the proposed layout, .• Compare the likelihood functions for both layouts. If

, accept the proposition. Reject itotherwise. (The algorithm also rejects absurd layouts suchas building with two crossing walls.)

• Repeat the procedure until the maximum number of itera-tions has been exceeded or the error is acceptably small.

IV. ESTIMATION USING MULTIPLE FREQUENCIES

A. The Influence of Frequency

For sensing through the walls, a wideband approach is nec-essary to achieve the desired feature resolution. The positiveinfluence of the multiple frequencies on the inverse scatteringalgorithms has been widely studied [7], [18]–[24]. The inverseproblems are intrinsically ill-posed due to the finite dimension-ality of the scattered field [7]. Consequently, the number of pa-rameters that can be estimated from the scattered field is lim-ited. If the number of sought parameters exceeds the numberof degrees of freedom (dimensionality) of the scattered field,the solution is not unique and the procedure diverges. The di-mensionality increases with frequency; nevertheless the inversescattering at high frequencies is easily trapped at a false solution.The multifrequency data processing: (i) mitigates the ill-posed-ness, (ii) reduces ill-conditioning, and (iii) improves robustnessagainst false solutions. For processing of multifrequency dataseveral strategies have been employed [7], [18]–[24]. The mostcommon is frequency hopping where the low-frequency solu-tion is used to initialize the high-frequency solution [6], [18]. Atsufficiently low frequencies, the considered physical phenom-enon can be described by a linear model [2]. Therefore, no falsesolution occurs, but the information content is reduced. How-ever, the frequency hopping technique is limited to the prob-lems where the low-frequency spectrum of the unknown profilehas significant contribution to the overall spectrum content [7].Otherwise the procedure may yield erroneous result. The otheroption is to perform simultaneous analyses of multifrequencydata [7], [24]. Finally, a hybridization of previous methods ispossible.

Results from [7], [18]–[24] were derived using determin-istic approach. However, similar conclusions hold when theunknown profile is reconstructed using the jump-diffusionalgorithm. The low-frequency radar measurements are notvery sensitive to small perturbations of the building layout.Consequently, the probability of acceptance of the proposedtransformation is high. The jump-diffusion algorithm appliedat low frequencies traverses quickly through different buildinglayouts, converging to the solutions that are close to the exactlayout. The method of moments models are also very efficientat low frequencies. However, the low-frequency estimationwith jump-diffusion lacks desired resolution, might detect falsewalls, etc. At high frequencies, the probability of acceptance ofthe proposed layout is small unless it is very close to the exactsolution (probability of adding a new wall is particularly low).Therefore, the proposed algorithm stays too long in one stateunless the proposed state is very close to the exact solution.Consequently the convergence at high frequencies is veryslow. In addition, the forward calculations are exceptionallytime-consuming. Hence, we use low frequencies for the initialestimation.

We investigated the application of frequency hopping, jointprocessing, and hybrid approach for the building layout estima-tion. The best results were achieved using hybrid approach. The



estimation was performed in steps, where we use the estimatesobtained at lower frequencies to feed the estimation at higherfrequencies. We start at the lowest frequency in the availablefrequency band, initially assuming there are no inner walls. Thealgorithm converges to the first-step estimate . We use thisestimate to initiate the second-step jump-diffusion, which is per-formed simultaneously at frequencies , where .The selection of the next frequency has important role in the esti-mation process. If the next frequency is too close to the previousfrequency, the estimate improvement is negligible. On the otherhand, if it is much higher, the analysis diverges. In the latter case,the algorithm removes walls from the scene unless they are al-most identical to true walls. At the same time, the acceptancerate for transformations other than death is very low. Therefore,the next frequency should be chosen adaptively based on thetransformation acceptance rate.

We repeat those steps, each time including a new higher fre-quency into the set of the operating frequencies. Keeping lowerfrequencies instead of just jumping to higher frequencies doesnot increase the calculation time significantly, but it is essentialfor the robustness of the algorithm. Similar behavior is recordedin [7] where deterministic approach is applied. We end the es-timation when the error between the measurements and the es-timate is sufficiently small or when the maximum number offrequency steps is reached.

As we increase the frequency, we accordingly reduce thespace for the parameter search. We draw the proposed valuesfor the wall parameters from Gaussian distribution, whichis centered at the current parameter values: ,

, . The variances , , aredecreased as the order of step increases. The probabilities ofthe transformations are also step dependent, e.g., as the stepnumber increases, the probability of fine transformations, suchas optimization, becomes higher.

B. Algorithm Summary

Given the estimate from the step , in the step :• Select the frequency , where .• Update the transformations and their probabilities.• Apply the jump-diffusion algorithm at frequencies

where the initial estimate is .• Repeat the procedure, unless the maximal frequency is

reached or the error is small enough.

V. RESULTS

A. Simulations

We examined the performance of the proposed methodthrough many simulations. We estimated the number, positions,orientations, and lengths of the inner walls in a 4 m 4 mbuilding. For simplicity, we assumed that the thickness and therelative permittivity of the walls were known. However, weaddressed the cases in which those parameters were not exactlyknown. Besides walls, there were unknown clutter objects inthe building simulating furniture. The adopted thicknessesof the exterior and interior walls were 20 cm and 15 cm,

respectively. The clutter objects had square cross section (side20–30 cm) and circular cross sections (diameter 20–30 cm).We tested the algorithm in a wide range of wall permittivities.We considered both simplified and realistic models for wallpermittivity. The permittivity of the clutter objects was thesame as the permittivity of the walls.

In our simulations, measurements were taken by a 4-sensorarray . There were 25 measurement locations at eachside of the building. We assumed white Gaussian noise, and set

. We define the signal-to-noise ratio using thepower of the signal reflected from the building. Due to the prop-agation attenuation, multiple reflections, etc., the electric fieldscattered from the building is weak compared with the electricfield induced in the sensors due to their direct coupling. The ap-proximate field scattered from the building is

(4)

where is the stacked vector of the induced electric fields inthe sensors when there is no building, and the

(5)

As the first example, we considered the layout with five un-known walls and three square clutter objects (Fig. 3). The as-sumed wall permittivity was . The first-step esti-mate was computed at , the second-step estimateat and , and the third-step esti-mate at , , and . Thenumber of iterations for each step was, respectively, 400, 200,and 100. We performed 10 independent estimations. For eachstep, we overlapped the results obtained in different simulationsand showed them jointly in Fig. 3. The estimated walls are asrepresented as gray rectangles, while the black lines denote thetrue layout. We did not estimate the clutter objects. Dark areascorrespond to more probable layouts. The parameters of the es-timated walls fluctuate around their true values. As frequencyincreases, the variances of the estimated parameters decrease.The third-step layout is very close to the exact solution, and itsuffices to perform deterministic optimization at higher frequen-cies if more accurate estimate is needed.

In the second example, we considered the layout with obliquewalls. Since the layout is more complex, there are more un-known parameters. We initiated the estimation at ,but the algorithm converged slowly. We added a higher fre-quency , which is sufficiently separated from .Therefore, we computed the first-step estimate atand , the second-step estimate at ,

, and , and the third-step esti-mate at , , , and

. The number of iterations for each step was400, 200, and 100. The results are shown in Fig. 4. The param-eters of the walls oscillate around their true values, due to thepresence of clutter.

The influence of the clutter on the measured electric field isillustrated in Fig. 5. Results show measured scattered field for



Fig. 3. (a) First step: � � �� , 400 iterations, (b) second step: � � �� , � � �� , 200 iterations, (c) third step: � � �� , � �

��, � � ��, 100 iterations. Results are shown jointly for 10 independent simulations. Wall permittivity is 3-j0.45.

Fig. 4. (a) First step: � � �� , � � �� , 400 iterations, (b) second step: � � �� , � � �� , � � ��, 200 iterations, (c) thirdstep: � � ��, � � ��, � � ��, � � ��, 100 iterations. Results are shown jointly for 10 independent simulations. Wall permittivityis 3-j0.45.

Fig. 5. Scattered field at one sensor position (denoted by arrow).

the case with and without clutter in a sensor at one position. Theaverage error for 10 independent simulations is shown in Fig. 6.

We also considered concrete walls with realistic models forpermittivity. The permittivity of concrete changes significantlyat low frequencies [30], [31]. We used the Cole-Cole model [31]

(6)

to fit the experimental data for concrete provided in [30]. Weobtained values: , , , and .The algorithm performed similarly as in the case with the con-

Fig. 6. Convergence of the estimation steps from Fig. 4.

stant permittivity. Moreover, the algorithm detected clutter ob-jects, although they were not subject of the estimation. Theestimation results are shown in Fig. 7. The first step estimatewas computed at , the second-step estimate at

, and , and the third-step estimateat , , and . Thenumber of iterations for each step was 400, 300, and 200.

B. The influence of Errors in Wall Thickness

We examined the robustness of the algorithm to the errors inthe wall thickness. Various experiments showed that the algo-



Fig. 7. (a) First step: � � �� , 400 iterations, (b) second step: � � �� , � � �� , 300 iterations, (c) third step: � � �� , � �� , � � �� , 200 iterations. Wall permittivity is � � � ��

Fig. 8. (a) Estimation with erroneous wall thickness: true and assumed valuesare 0.2 m and 0.15 m, respectively. Third-step estimate, � � �� , � �� , � � �� , and � � �� . (b) Estimation with erro-neous wall permittivity: true and assumed values are 3-j0.45 and 3.45-j0.55,respectively. Second-step estimate, � � �� , � � �� , and � �� .

rithm is capable to estimate properly the layout in the case whenthe wall thickness is known only approximately. We consideredthe case in which the true thickness of the walls is 20 cm, whilein simulations we assumed that it is 15 cm. We used the samefrequencies and the same number of iterations as in the case ofexact wall parameters. The results for the third-step estimate,obtained for 5 independent simulations, are shown in Fig. 8(a).The estimated layouts provide a good initial representation ofthe exact solution. Due to the thickness error, the variance ofestimated parameters is larger. However, the algorithm is ableto provide useful insight into the building layout.

C. The Influence of Errors in Wall Permittivity

We investigated the case when the dielectric permittivity isknown only approximately. In the considered example, the exactpermittivity was and the assumed permittivitywas . (The relative error was 15 %.) In the firststep, computed at and , the algo-rithm performed similarly to the case without the error. How-ever, in the second step, calculated at ,

, and , the number of accepted moveswas extremely low. Therefore we decreased the third frequency

to . In the third step we choose new frequency tobe . However, the improvement in the third stepwas insignificant. In Fig. 8(b) we show results for the second-step estimate, obtained for 5 independent simulations. Com-pared to the case with the exact wall parameters, lower frequen-cies are used and less accuracy is achieved.

D. Computational Data

We calculate the forward electromagnetic model using themethod of moments [26]. The contours of the walls are uni-formly divided into segments. The length of each segment isapproximately , where is the wavelength in the wall. Theequivalent electric and magnetic currents have constant valuesalong each segment. The sensors are assumed to be perfectlyconducting, and they are uniformly divided into segments withconstant electric currents. The sufficient number of segmentsper sensor that yielded numerically stable result is found to be4. The total number of unknowns for one sensor position is

, where is the number of sensors. Inde-pendent simulation should be started for each sensor position(Fig. 9(a)). However, this is extremely computationally costlyeven in the 2D case. Instead of running the forward model foreach sensor position, we considered the case in which static sen-sors are placed all around the building (Fig. 9(b)). Each posi-tion of a static sensor corresponded to one position of a sensorfrom the moving array. In general, such a model introduces anerror. Nevertheless, if the sensors are modeled as conductorswith electrically small cross-section, and if the impedance pa-rameters are used, the modeling error is negligibly small. Athin conductor (filament) with zero net current has negligibleinfluence on the field distribution in other conductors. Hencethe relevant impedance coefficients remain practically the samewhether they are calculated for a group of sensors treated oneat a time, or for all sensors analyzed simultaneously. This is notthe case for parameters because currents are induced in allconductors when one conductor is excited. Therefore, the totalnumber of unknowns is , where is thenumber of static sensors.

We show in Table I the average number of unknowns at eachfrequency for the example from Fig. 7. The computation time



Fig. 9. (a) Impedance matrix �� is computed for each position of themoving array with � sensors. (b) Impedance matrix �� is computedfor all static sensors. Impedance matrix �� of a moving array at a givenposition is a submatrix of �� .

TABLE IAVERAGE NUMBER OF UNKNOWNS �� FOR THE LAYOUT FROM FIG. 7

TABLE IINUMBER OF ITERATIONS AND CPU TIME PER FREQUENCY STEP

FOR THE LAYOUT FROM FIG. 7

TABLE IIIPROBABILITY DISTRIBUTIONS FOR TRANSFORMATIONS

(on a standard single-processor PC) for each step is shown inTable II.

VI. CONCLUSION

We developed a scheme for estimating unknown buildinglayouts. The proposed approach uses the jump-diffusion algo-rithm with a moving sensor system to determine the numberand parameters of the walls (position, length, orientation). Theproblem is challenging due to many unknown parameters anddifficulties in electromagnetic modeling of electrically largestructures such as buildings. We showed that the jump-diffusionalgorithm is a powerful tool for solving inverse electromagneticscattering problems for which we have some prior knowledge.

We designed an adaptive procedure that uses low-frequency(low-resolution) estimates to initiate high-frequency (high-res-olution) estimation. We used accurate electromagnetic forward

models based on the method of moments since the buildinglayout estimation needs wide frequency bandwidth which en-compasses relatively low frequencies. The choice of the oper-ating frequency has significant influence on the convergenceof the jump-diffusion algorithm. We demonstrated through nu-merous simulations that accurate estimates may be obtainedusing only a few frequencies, thus making possible to use thejump-diffusion in real scenarios. We examined the robustnessof our approach in the presence of unknown clutter objects. Thealgorithm proved capable of discriminating between walls andsmaller-size clutter objects. We also showed that the algorithmis not sensitive to small errors in the wall thickness or materialrelative permittivity. We have demonstrated the accuracy and ro-bustness of our approach through numerical examples. In futurework we will investigate these aspects further through analysissuch as computing Cramer-Rao bounds.

APPENDIX

We describe here in detail the transformations fromSection III.

• Birth: A new wall is added to the current configu-ration. The parameters of the new wall are drawnas , ,

, where , , arethe boundaries of the building, and are adoptedlimits for the wall length. For the 4 m 4 building weset: , , ,

.• Death: A randomly selected inner wall is removed from the

current configuration.• Translation: New coordinates of the center of the wall are

drawn as , , where ,are the current center coordinates, is the frequency step,and are the adopted constants. We set

.• Elongation: The new length is drawn as ,

where is the length of the wall. We set .• Rotation: The orientation of a randomly selected wall is

flipped . The assumed wall orientations are 0 ,or 90 .

• Regeneration: All parameters of the randomly selectedwall are perturbed. The new wall parameters are drawnas , ,

, .• Merge: A randomly selected inner wall is merged with

a nearby wall of the same orientation. The center of themerged wall is the centroid of the tightest rectangle encom-passing the original two walls, and its length is drawn as

, where are defined inFig. 2.

• Optimization. The parameters of a randomly selected wallare optimized using simplex algorithm.

We adopted the probabilities for the above transformations asin Table III, where is the number of the walls in the layout

. In this table, is a constant that assures .



REFERENCES

[1] M. M. Nikolic, A. Nehorai, and A. R. Djordjevic, “Radar estimation ofbuilding layout estimation using jump-diffusion,” presented at the 2ndIEEE Int. Workshop on Computat. Adv. Multi-Sensor Adapt. Process.,St. Thomas, U.S. Virgin Islands, Dec. 2007.

[2] W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional per-mittivity distribution using the distorted Born iteration method,” IEEETrans. Med. Imag., vol. 9, no. 2, pp. 218–225, Jun. 1990.

[3] Y. M. Wang and W. C. Chew, “An iterative solution of the two-dimen-sional electromagnetic inverse scattering problem,” Int. J. Imag. Syst.Technol., vol. 1, pp. 100–108, 1989.

[4] N. Joachimowicz, C. Pichot, and J. P. Hugonin, “Inverse scattering: Aniterative numerical method for electromagnetic imaging,” IEEE Trans.Antennas Propagat., vol. 39, pp. 1742–1752, 1991.

[5] A. G. Tijhuis, K. Belkebir, A. Litman, and B. P. de Hon, “Theoret-ical and computational aspects of 2-D inverse profiling,” IEEE Trans.Geosci. Remote Sens., vol. 39, no. 6, pp. 1316–1330, Jun. 2001.

[6] O. M. Bucci and T. Isernia, “Electromagnetic inverse scattering: Re-trievable information and measurement strategies,” Radio Sci., vol. 32,no. 6, pp. 2123–2137, Nov. 1997.

[7] O. M. Bucci, L. Crocco, T. Insernia, and V. Pascazio, “Inverse scat-tering problems with multifrequency data: Reconstruction capabilitiesand solution strategies,” IEEE Trans. Geosci. Remote Sens., vol. 38, no.4, Jul. 2000.

[8] F. Soldovieri and R. Solimene, “Through-wall imaging via a linear in-verse scattering algorithm,” IEEE Geosci. Remote Sens. Lett., vol. 4,no. 4, pp. 513–517, Oct. 2007.

[9] R. Solimene, F. Soldovieri, and G. Prisco, “A multiarray tomographicapproach for through-wall imaging,” IEEE Trans. Geosci. RemoteSens., vol. 46, no. 4, pp. 1192–1199, Apr. 2008.

[10] L. Garnero, A. Franchois, J.-P. Hugonin, C. Pichot, and N. Joachi-mowicz, “Microwave imaging-complex permittivity reconstruction-bysimulated annealing,” IEEE Trans. Microw. Theory Tech., vol. 39, no.11, pp. 1801–1807, Nov. 1991.

[11] P. J. Green, “Reversible jump Markov chain Monte Carlo computa-tion and Bayesian model determination,” Biometrika, vol. 82, no. 4,pp. 711–732, 1995.

[12] U. Grenander and M. Miller, “Representation of knowledge in complexsystems,” J. Roy. Statist. Soc., vol. B56, pp. 97–109, 1994.

[13] S. C. Zhu, “Stochastic jump-diffusion process for computing medialaxes in Markov random field,” IEEE Trans. Pattern Analysis MachineIntell., vol. 21, no. 11, pp. 1158–1169, Nov. 1999.

[14] Z. Tu and S. C. Zhu, “Image segmentation by data-driven Markov chainMonte Carlo,” IEEE Trans. Pattern Analysis Machine Intell., vol. 24,no. 5, pp. 657–673, May 2002.

[15] M. Ortner, X. Descombes, and J. Zerubia, “Building outline extractionfrom digital elevation models using marked point processes,” Int. J.Comput. Vis., vol. 72, no. 2, pp. 107–132, Apr. 2007.

[16] M. I. Miller, U. Grenander, J. A. O’Sullivan, and D. L. Snyder, “Au-tomatic target recognition organized via jump-diffusion algorithms,”IEEE Trans. Image Process., vol. 6, pp. 157–174, 1997.

[17] R. F. Harrington, Field Computation by Moment Methods. Malabar,FL: Krieger, 1982.

[18] W. C. Chew and J. H. Lin, “A frequency-hopping approach formicrowave imaging of large inhomogeneous bodies,” IEEE Microw.Guided Wave Lett., vol. 5, no. 10, pp. 439–442, Dec. 1995.

[19] R. Pierri, R. Persico, and R. Bernini, “Information content of Bornfield scattered by an embedded slab: Multifrequency, multiview andmultifrequency-multiview cases,” J. Opt. Soc. Amer. A, vol. 16, pp.2392–2399, 1999.

[20] M. Lambert, D. Lesselier, and B. J. Kooij, “The retrieval of a buriedcylindrical obstacle by a constrained modified gradient method in theH-polarization case and for Maxwellian materials,” Inv. Probabil., vol.14, pp. 1265–1283, 1998.

[21] A. Litman, D. Lesselier, and F. Santosa, “Reconstruction of a two-di-mensional binary obstacle by controlled evolution of a level set,” Inv.Probabil., vol. 14, pp. 685–706, 1998.

[22] A. G. Tijhuis, K. Belkebir, A. C. S. Litman, and B. P. de Hon, “Mul-tiple-frequency distorted-wave Born approach to 2D inverse profiling,”Inv. Probabil., vol. 17, pp. 1635–1644, 2001.

[23] L. Song, C. Yu, and Q. H. Liu, “Through-Wall Imaging (TWI) by radar:2-D tomographic results and analyses,” IEEE Trans. Geosci. RemoteSens., vol. 43, no. 12, pp. 2793–2798, Dec. 2005.

[24] T. Isernia, V. Pascazio, and R. Pierri, “A nonlinear estimation methodin tomographic imaging,” IEEE Trans. Geosci. Remote Sens., vol. 35,pp. 910–923, 1997.

[25] D. M. Pozar, Microwave Engineering, 3rd ed. New York: Wiley,2005.

[26] A. R. Djordjevic, T. K. Sarkar, and S. M. Rao, “Analysis of finiteconductivity cylindrical conductors excited by axially-independentTM electromagnetic field,” IEEE Trans. Microw. Theory Tech., vol.MTT-33, no. 10, pp. 960–966, Oct. 1985.

[27] J. Nelder and R. Mead, “A simplex method for function minimization,”Comput. J., vol. 7, pp. 308–313, 1965.

[28] W. K. Hastings, “Monte Carlo sampling methods using Markov chainsand their applications,” Biometrika, vol. 57, no. 1, pp. 97–109, 1970.

[29] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, andE. Teller, “Equations of state calculations by fast computing machines,”J. Chem. Phys., vol. 21, no. 6, pp. 1087–1092, 1953.

[30] J. Baker-Jarvis, M. D. Janezic, B. F. Riddle, R. T. Johnk, P. Kabos, C.L. Holloway, R. G. Geyer, and C. A. Grosvenor, “Measuring the per-mittivity and permeability of lossy materials: Solids, liquids, metals,building materials, and negative-index materials,” NIST, TechnicalNote 1536.

[31] A. Robert, “Dielectric permittivity of concrete between 50 MHz and1 GHz and GPR measurements for building materials evaluation,” J.Appl. Geophys., vol. 40, pp. 89–94, 1998.

Marija M. Nikolic received the B.Sc., M.Sc.,and Ph.D. degrees from the University of Bel-grade, Belgrade, Serbia, in 2000, 2003, and 2007,respectively.

In 2001, she joined the School of ElectricalEngineering, University of Belgrade, as a TeachingAssistant. In 2008, she was promoted to AssistantProfessor. Currently, she is with the University ofWashington, Saint Louis, MO, pursuing the Ph.D.degree in signal processing. She is interested innumerical electromagnetics applied to signal pro-

cessing, antennas analyses and design, and MIMO systems.

Mathias Ortner received the B.Sc. degree in aero-nautics and space sciences from the French Schoolof Aeronautics and Space, Toulouse, France, in 2001,and the Ph.D. degree in computer science and imageprocessing from the University of Nice, Sophia-An-tipolis, France, in 2004.

He is with the Department of Electrical and Sys-tems Engineering, Washington University, St. Louis,MO, where, from 2004 to 2006, he was a Postdoc-toral Research Visitor on Prof. Nehorai’s team.

Arye Nehorai (S’80–M’83–SM’90–F’94) receivedthe B.Sc. and M.Sc. degrees in electrical engineeringfrom the Technion, Haifa, Israel, and the Ph.D. de-gree in electrical engineering from Stanford Univer-sity, Stanford, CA.

From 1985 to 1995, he was a faculty member withthe Department of Electrical Engineering at YaleUniversity. In 1995, he joined as Full Professor theDepartment of Electrical Engineering and ComputerScience, University of Illinois at Chicago (UIC).From 2000 to 2001, he was Chair of the department’s

Electrical and Computer Engineering (ECE) Division, which then became anew department. In 2001, he was named University Scholar of the Universityof Illinois. In 2006, he became Chairman of the Department of Electrical andSystems Engineering, Washington University, St. Louis (WUSTL), MO. He isthe inaugural holder of the Eugene and Martha Lohman Professorship and theDirector of the Center for Sensor Signal and Information Processing (CSSIP)at WUSTL since 2006.



Dr. Nehorai was Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL

PROCESSING during the years 2000 to 2002. In the years 2003 to 2005, he wasVice President (Publications) of the IEEE Signal Processing Society, Chair ofthe Publications Board, member of the Board of Governors, and member of theExecutive Committee of this Society. From 2003 to 2006, he was the foundingeditor of the special columns on Leadership Reflections in the IEEE SignalProcessing Magazine. He was co-recipient of the IEEE SPS 1989 Senior Awardfor Best Paper, co-author of the 2003 Young Author Best Paper Award, andco-recipient of the 2004 Magazine Paper Award. He was elected DistinguishedLecturer of the IEEE SPS for the term 2004 to 2005 and the 2006 IEEESPS Technical Achievement Award. He is the Principal Investigator of thenew multidisciplinary university research initiative (MURI) project “AdaptiveWaveform Diversity for Full Spectral Dominance.” He has been a Fellow ofthe Royal Statistical Society since 1996.

Antonije R. Djordjevic was born in Belgrade,Serbia, on April 28, 1952. He received the B.Sc.,M.Sc., and D.Sc. degrees from the School of Elec-trical Engineering, University of Belgrade, in 1975,1977, and 1979, respectively.

In 1975, he joined the School of Electrical Engi-neering, University of Belgrade, as a Teaching Assis-tant. He was promoted to an Assistant Professor, As-sociate Professor, and Professor, in 1982, 1988, and1993, respectively. In 1983, he was a Visiting Asso-ciate Professor at Rochester Institute of Technology,

Rochester, NY. Since 1992, he has been an Adjunct Scholar with Syracuse Uni-versity, Syracuse, NY. In 1997, he became a Corresponding Member of theSerbian Academy of Sciences and Arts, and he became a full member in 2006.His main area of interest is numerical electromagnetics, in particular applied tofast digital signal interconnects, wire and surface antennas, microwave passivecircuits, and electromagnetic-compatibility problems.


An Approach to Estimating Building Layouts Using Radar and Jump-Diffusion Algorithm

Documents

Transcript of An Approach to Estimating Building Layouts Using Radar and Jump-Diffusion Algorithm