2810 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, …

2810 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 6, JUNE 2007

Acoustic Multitarget Tracking UsingDirection-of-Arrival Batches

Volkan Cevher, Member, IEEE, Rajbabu Velmurugan, Student Member, IEEE, and James H. McClellan, Fellow, IEEE

Abstract—In this paper, we propose a particle filter acousticdirection-of-arrival (DOA) tracker to track multiple maneuveringtargets using a state space approach. The particle filter determinesits state vector using a batch of DOA estimates. The filter likeli-hood treats the observations as an image, using template modelsderived from the state update equation, and also incorporates thepossibility of missing data as well as spurious DOA observations.Multiple targets are handled using a partitioned state-vectorapproach. The particle filter solution is compared with threeother methods: the extended Kalman filter, Laplacian filter, andanother particle filter that uses the acoustic microphone outputsdirectly. In addition, we demonstrate an autonomous system formultiple target DOA tracking with automatic target initializationand deletion. The initialization system uses a track-before-detectapproach and employs matching pursuit to initialize multipletargets. Computer simulations are presented to compare theperformance of the algorithms.

Index Terms—Batch measurement, bearings tracking, multipletarget tracking, particle filter, template matching.

I. INTRODUCTION

ACHALLENGING signal processing problem arises whenone demands automated tracking of the direction-of-ar-

rival (DOA) angles of multiple targets using an acoustic arrayin the presence of noise or interferers [1]–[5]. In the litera-ture, DOA tracking problems are usually formulated using state-space models [6]–[8], with an observation equation that relatesthe state vector (i.e., target DOAs and possibly motion states) tothe acoustic microphone outputs and a state equation that con-strains the dynamic nature of the state vector. The performanceof the tracking algorithms using state spaces relies heavily onhow accurate the models represent the observed natural phe-nomena. Hence, in many cases, it is important to use nonlinearand non-Gaussian state-space models despite their computa-tional complexity [9].

The presence of multiple targets increases the tracking com-plexity because data association is needed to sort the received

Manuscript received July 25, 2005; revised September 14, 2006. The asso-ciate editor coordinating the review of this manuscript and approving it for pub-lication was Dr. Peter Handel. This work was prepared through collaborativeparticipation in the Advanced Sensors Consortium supported by the U.S. ArmyResearch Laboratory under the Collaborative Technology Alliance Program Co-operative Agreement DAAD19-01-02-0008.

V. Cevher is with the Center for Automation Research, University of Mary-land, College Park, MD 20742 USA (e-mail: [email protected]).

R. Velmurugan and J. H. McClellan are with the Center for Signal and ImageProcessing, School of Electrical and Computer Engineering, Georgia Instituteof Technology, Atlanta, GA 30332 USA (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2007.893962

data for each target. In the literature, the association problemis handled in several different ways: 1) probabilistic data as-sociation methods estimate the states by summing over all theassociation hypothesis weighted by the probabilities obtainedby the likelihood [10]–[13]; 2) smoothness assumptions on thetarget (motion) states allow a natural ordering of the data [14];3) computationally costly maximum-likelihood (ML)/expecta-tion–maximization (EM) methods use the likelihood functionto search for a global maximum; or 4) nearest neighbor methodsprovide easy heuristics to perform measurement updates. Mostof these methods use the mean and covariance approximationof the sufficient statistics for the state, which may be estimatedwith a Kalman filter; however, for nonlinear state spaces withgeneral noise assumptions, Monte Carlo methods should beused to adequately capture the dynamic, possibly multimodal,statistics.

In a particle filter, where the observations arrive in sequence,the state probability density function is represented by discretestate samples (particles) distributed according to the under-lying distribution either directly or by proper weighting [8],[9]. Hence, the filter can approximate any statistics of thedistribution arbitrarily accurately by increasing the numberof particles with proven convergence results. In the particlefiltering framework, data association is undertaken implicitlyby the state-space model interaction. However, the particlefilter suffers from the curse of dimensionality, as the number oftargets increases [15]. To increase the algorithm’s efficiency,various methods are proposed, such as the partitioning approach[2], [16] or other Bayesian approaches [17].

In this paper, we present a particle filter algorithm to trackthe DOAs of multiple maneuvering targets, using an acousticnode that contains an array of microphones with known posi-tions. Each particle in the filter is created by concatenating par-titions, i.e., the state vector for each target. For example, thepartition of the th target has a state vector that consists of theDOA , the heading direction , and the logarithm ofvelocity over range of the th target.The total number of targets (or partitions) is determined by amode hungry Metropolis–Hastings block. Hence, given tar-gets, a particle has partitions where each target, and henceeach partition, is assumed to be independent. Target motions aremodeled as locally linear, i.e., each target has a constant velocitywithin an estimation period of duration .

The particle filter uses multiple DOAs to determine the statevector, based on an image template matching idea. We denotethe collection of DOAs a batch, where is the batch size. Inour problem, a DOA image is first formed when a batch of DOAobservations are received from a beamformer that processes thereceived acoustic data at -second intervals (see Fig. 1).

1053-587X/$25.00 © 2007 IEEE

CEVHER et al.: ACOUSTIC MULTITARGET TRACKING USING DOA BATCHES 2811

Fig. 1. Observation model uses a batch of DOA measurements. The DOA mea-surements are not necessarily ordered at the output of the beamformer. However,when the batch of DOAs is treated as a bearing versus time image, there is nat-ural ordering, which aids in the multiple target tracking by the particle filter.

Fig. 2. Template matching idea is illustrated. The solid line represents the trueDOA track. Black dots represent the noisy DOA estimates. The dashed anddotted lines represent the DOA tracks for two proposed particles, x and x .These tracks are calculated using the state update function h. Visually, the ithparticle is a better match than the jth particle; hence, its likelihood is higher.

Then, image templates for target tracks are created using thestate update function and the target partition state vectors (seeFig. 2). By determining the best matching template (e.g., mostprobable target track), the target state vectors are estimated. Be-cause the observations are treated as an image, the data associa-tion and DOA ordering problems are naturally alleviated. More-over, by assuming that the DOA observations are approximatelynormally distributed around the true target DOA tracks, withconstant DOA miss probability and clutter density, a robust par-ticle filter tracker is formulated.

The particle filter importance function proposes particlesfor each target independently to increase the efficiency of thealgorithm. To derive the proposal function for each target,we use Laplace’s method to approximate the posterior of thecorresponding partition by a Gaussian around its mode [8],[18], [19]. We calculate the partition modes using a robust

Newton–Raphson search method that imposes smoothnessconstraints on the target motion. We also present a slower butmore robust method for determining the partition posteriors,based on the mode hungry Metropolis–Hastings algorithm [20].Details of the partition posteriors are given in Section IV.

For the initialization of the particle filter, we describe a modehungry Metropolis–Hastings (MHMH) sampling algorithm. Forcomparison purposes, we also present an approximate Bayesianfilter based on the particle filter’s proposal function, and an ex-tended Kalman filter (EKF) [10]. The EKF is derived using asliding window implementation, because the Kalman filter ini-tial data association becomes prohibitive, while handling thecomplex observation model based on DOA batches in the caseof multiple targets, using the sufficient statistics. In addition,we qualitatively compare the computational complexity of thefilters: the particle filter solution outperforms the approximateBayesian solution with no significant increase in the computa-tional requirements.

This paper is organized as follows. Section II explains themechanics of the automated tracking system for multiple targetDOA tracking. Sections III–V elaborate on the individual ele-ments of the tracking system. The computational complexity ofthe algorithms is given in Section VI. Alternative tracking ap-proaches are discussed in Section VII. Computer simulations areshown in Section VIII to compare the performance of differentalgorithms.

II. TRACKER DESIGN

The tracker mechanics in this paper are constructed such thatthe tracker: 1) compensates DOA estimation biases due to rapidtarget motion [3]; 2) results in higher resolution DOA estimatesthan just beamforming [2]–[4]; 3) is robust against changes intarget signal characteristics; and 4) automatically determines thenumber of targets. The tracker system consists of three blocks asillustrated in Fig. 3. Details of the individual blocks are given inthe following sections. Below, we discuss the technicalities thatlead to this design. Note that the main objective of our trackeris to report multiple target DOAs at some period , after ob-serving the acoustic data at the node microphones. Quite often,target DOAs can change more than a few degrees during an es-timation period, e.g., due to rapid target motion. Hence, if wewere to just use conventional snapshot DOA estimation methods(e.g., MUSIC, MVDR, etc.) for tracking the targets, the bearingestimates become biased, because the received data is not sta-tionary [21]. This is intuitive, because these methods estimatean average of the target angular spread during their estimationperiods [3].

In general, locally linear motion models eliminate this biasby simultaneously estimating the bearing and the target motionparameters for the estimation period . Conceptually, this isequivalent to aligning the received acoustic data with the mo-tion parameters so that the data becomes stationary for bearingestimation purposes. But, this alignment process relies heavilyon the observation model and is computationally costly becauseof the high volume of the acoustic data used for estimation. Inthis paper, we propose to use a beamformer block to buffer thevariability in the observed acoustic signals and to create a com-pressed set of invariant statistics for our particle filter block.


Fig. 3. Tracker mechanics demonstrated. The beamformer block monitors the received acoustic signals to adapt to their frequency characteristics for better bearingestimation. It outputs a batch of DOAs that form a sufficient statistic for the particle filter block. The particle filter estimates the tracking posterior and can makevarious inferences (e.g., mean and mode estimates). It acquires the new partition information from the mode hungry MHMH block. It can also delete a partition,if there is no data in the respective partition gate (explained in the text). The MHMH block determines the new partitions and their distributions from the residualDOAs coming from the particle filter (i.e., the gating operation).

Therefore, the beamformer block (see Fig. 3) processes theacoustic data at shorter time intervals of (e.g.,

), where the targets are assumed to be relatively stationary.This reduces the number of acoustic data samples available forprocessing, resulting in a sequence of noisier DOA estimates.However, these noisier DOA estimates can be smoothed in theparticle filter block, because a motion structure is imposed onthe batch of DOAs. Moreover, for the state vector defined inthe introduction, a batch of more than three DOAs is sufficientfor observability of the state [3], [7]. Since the filter is built onthe compressed statistics which are also sufficient to observe thestate vector, it achieves a significant reduction in computation.

When the received signal characteristics change, the particlefilter tracker formulation is not affected because the beamformerblock can absorb variabilities in the acoustic signals. In the lit-erature, various trackers use similar state-space formulationsas in this paper [2]–[4]. The trackers presented in [2] and [3]directly employ the classical narrowband observation model,where targets exhibit constant narrowband frequency charac-teristics [21], [22]. The tracker in [4] tries to adapt to varyingtime-frequency characteristics of the target signals, assumingthat the varying frequencies are narrowband. The tracker in [5]also incorporates an amplitude model for the signals. Becausetheir probability density equations explicitly use an observa-tion equation, these trackers are hardwired to their observationmodel. Hence, a complete rework of the filter equations wouldbe required to track targets with wideband signal characteristics(e.g., [5]–[23]).

The third block in Fig. 3 addresses a fundamental issue fortrackers: initialization. It is an accelerated Metropolis–Hastingsalgorithm modified specifically for unimodal distributions. It

Fig. 4. Gating operation illustrated. Solid line is the true DOA track. The DOAobservations are shown with dots. Given a partition gate size � , the DOAs areignored (i.e., gated out) if they are more than � away from the solid line.

takes the ungated DOAs from the particle filter as inputs (seeFig. 4), or if there are no partitions in the particle filter, it takesdata from the beamformer block directly. It generates a particledistribution for each potential new target one at a time, i.e., itconverges on the strongest mode in the, possibly, multimodaltarget posterior and sifts out the corresponding DOA data. Itthen iterates to find other modes until stopping criteria are met.This block creates new partitions for the particle filter, whichcan also delete its own partitions when conditions described inSection V are met. Conceptually, this initialization idea is equiv-alent to the track-before-detect approach used in the radar com-munity [24], [25].

III. BEAMFORMER BLOCK

Beamformers use the collected acoustic data to determinetarget DOAs and are called narrowband beamformers if theyuse the classical narrowband array observation model [21], [22].


Fig. 5. Circles and squares denote the DOA estimates at two different frequencies f and f , calculated using the acoustic data received during a period of length� . In this example, the maximum number of beamformer peaks P is 2. Given the observations y , the objective of the particle filter is to determine the statex (t) which completely parameterizes the solid curve.

Beamformers are wideband if they are designed for target sig-nals with broadband frequency characteristics [26]; others aredesigned for signals with time-varying narrowband frequencycharacteristics [27], [28].

The beamformer block chooses a beamformer for processingthe acoustic data depending on the local characteristics of theacoustic signals. That is, given the observed acoustic signal andits time-frequency distribution, we choose an optimal beam-former to calculate target DOAs. For example, we can choosemultiple beamformers if the received acoustic signal shows bothnarrowband and wideband characteristics. The output of thebeamformer is a DOA data cubecontaining the -highest DOA peaks of the beamformer pat-tern (see Figs. 1 and 5). In general, the number of DOA peaks

at batch index and each frequency index should begreater than or equal to the number of targets so that we canalso detect new targets while tracking the existing targets. InSection VIII, we fix but the derivations below ex-plicitly show the dependence on and . Note that the inputof the particle filter has the same structure regardless of thetarget signal characteristics. Finally, the choice of the param-eter for beamforming is determined by various physical con-straints, including: 1) target frequency spread, 2) target speed,and 3) a target’s affinity to maneuver. For reasonable beam-forming, at least two cycles of the narrowband target signalsmust be observed. This stipulates that , whereis the minimum beamforming frequency for the target. More-over, to keep the worst case beamforming bias1 bounded foreach DOA by an angle threshold denoted by , approximately,we have , where is the target velocityover range ratio. Last, the target motion should satisfy the con-stant velocity assumption during the output period . For slowmoving ground targets, 1 s is a reasonable choice. Notethat at least three DOA estimates are necessary to determine thestate vector. To improve the robustness of the tracker, we use

to decrease the probability that the state is not observ-

1The bias is calculated by taking the angular average of the target track.Hence, this bias also depends on the heading direction. The worst case biashappens when the target heading and DOA sum up to �. Moreover, it is alsopossible to analytically find an expected bias by assuming uniform heading di-rection, using a similar analysis done in [3].

able due to missing DOAs. Hence, the parameter is boundedby the following:

(1)

IV. PARTICLE FILTER

In this section, the details of the particle filter block in Fig. 3are discussed. In Section VII, we provide alternative algorithmsto replace this block based on the Kalman filter and Laplace’smethod and compare their performance in terms of computa-tional requirements.

A. State Equation

The particle filter state vector consists of the concatenationof partitions for each target, indexed by , and representedas , where is the number2

of targets at time . Each partition has the corresponding targetmotion parameters , as definedin the introduction. The angle parameters and aremeasured counterclockwise with respect to the -axis.

The state update equation can be derived from the geometryimposed by the locally constant velocity model. The resultingstate update equation is nonlinear

(2)

where withand is

(3)

The analytical derivations of (3) can be found in [3] and [4],where [4] also discusses state update equations based on a con-stant acceleration assumption.

2Explicit time dependence is not shown in the formulations. The parameterK is determined by the MHMH block (see Section V).


B. Observation Equation

The observations consist of all thebatch DOA estimates from the beamformer block indexed by

. Hence, the acoustic data of length is segmented intosegments of length . The batch of DOAs, , is assumed toform an approximately normal distributed cloud around the truetarget DOA tracks (see Fig. 1). In addition, only one DOA ispresent for each target at each or the target is missed. Mul-tiple DOA measurements imply the presence of clutter or othertargets. We also assume that there is a constant detection prob-ability for each target denoted by , where dependence onis allowed. If the targets are also simultaneously identified, anadditional partition dependency is added, i.e., .3

The particle filter observation model also includes a cluttermodel because beamformers can produce spurious DOA peaksas output (e.g., the sidelobes in the power versus angle patterns)[21]. To derive the clutter model, we assume that the spuriousDOA peaks are random with uniform spatial distribution on theangle space, and are temporally as well as spatially indepen-dent. In this case, the probability distribution for the numberof spurious peaks is best approximated by the Poisson distribu-tion with a spatial density [10], [29]. Moreover, the probabilitydensity function (pdf) of the spurious peaks is the uniform dis-tribution on . However, since the number of peaks in thebeamformer output can be user defined and since the beam-former power versus angle pattern has smoothness properties,we use the following pdf for the spurious peaks:

is spurious (4)

where is a constant that depends on the maximumnumber of beamformer peaks , the beamformer itself (i.e.,the smoothness of the beamformer’s steered response), and thenumber of targets . Equation (4) implies that the natural spaceof the clutter is reduced by a factor of because of the charac-teristics of our specific system.

We now derive the data likelihood function using the jointprobabilistic data association (JPDA) arguments found in [10].Similar arguments for active contour tracking relevant to thispaper are found in [30]. Consider the output of one batch period

, where for each and. The DOAs may belong to none, or some combination,

or all of the targets in the particle filter partitions. Hence, we firstdefine a notation to represent possible combinations betweenthe data and the particle filter partitions to effectively derive theobservation density.

Define a set that consists of -unordered combination ofall -partitions of the particle filter state vector: ,where is the set of all -unordered outcomes from pos-sibilities. Define as the number of elements of the set .Hence, each element of has numbers, and there are a totalof elements. For example, when and ,then , each element referring tosubset of the individual partitions of the particle state vector.

3Recognition/identification may have an impact on the choice of the acousticfrequencies as discussed earlier. Hence, some targets may have a lower detectionprobability at the recognized target frequencies. The partition dependence of thedetection probability takes care of this issue.

We refer to the individual elements of this set using the nota-tion , where . Hence, .Then, denote as asingle realization from the set . Using the same example,

. Hence, the setcontains the same elements of the set , reindexed

sequentially from .We denote as the probability

density function of the data, where only DOAs belong to thetargets defined by the partitions of . Hence, when ,all data is due to clutter

(5)

The probability density can be calculated by notingthat: 1) there are ordered ways of choosingDOA’s to associate with the -subset partitions and 2) the re-maining -DOAs are explained by the clutter. There-fore

(6)

where is in , and the function is derived from theassumption that the associated target DOAs form a Gaussiandistribution around the true target DOA tracks

(7)

where the superscript on the state update function refers onlyto the DOA component of the state update and can besupplied by the beamformer block.

Note that the DOA distribution (7) is not a proper circulardistribution for an angle space. For angle spaces, the von Misesdistribution is used as a natural distribution [31]. The von Misesdistribution has a concentration parameter with a correspondingcircular variance. It can be shown that for small (highconcentration), the von Mises distribution tends to the Gaussiandistribution in (7) [32]. Because the von Mises distribution hasnumerical issues for small DOA variances, the Gaussian approx-imation (7) is used in this paper. Hence, special care must betaken in the implementation to handle angle wrapping issues.

The Gaussians in (6) are directly multiplied, becausethe partitions are assumed to be independent. To elaborate, con-sider and for the set considered earlier

(8)


Hence, the density is a Gaussian mixture that peakswhen the updated DOA components of partitions 2 and 3

are simultaneously close to the observed data. Notethat (8) guarantees that no measurement is assigned to multipletargets simultaneously.

Given the densities , the observation density function canbe constructed as a combination of all the target association hy-potheses. Hence, by adding mixtures that consist of the data per-mutations and the partition combinations, we derive the obser-vation density

(9)

In (9), the parameters are the elements ofa detection (or confusion) matrix. For example, when ,

is the probability that no target DOA is in the beamformeroutput, whereas means that one (two) target DOA(s)are present in the beamformer output at each . These fixedvalues have to be provided by the user. However, they shouldbe changed adaptively to improve robustness of the particle filteroutput. For example, when two partitions and have closeDOA tracks and are about to cross, it is possible that the beam-former’s Rayleigh resolution is not sufficient to resolve separateDOAs for the two targets. Then, we change the confusion ma-trix to indicate the possibility that one of the targets will likelybe missed.

C. Particle Filter Proposal Function

In our problem of DOA-only multiple target tracking, the pro-posal function poses difficult challenges because 1) the statevector dimension is proportional to the number of targets ,hence, the number of particles to represent posterior can in-crease significantly as increases (the curse of dimension-ality); 2) in many cases, the targets maneuver, hence full poste-rior approximations are required for robust tracking; and 3) forfull posterior approximations, robustly determining the DOA-only data-likelihood is rather hard. We will address each of thesechallenges in this section. Note that once the proposal functionis formulated, the rest of the particle filter structure is well-de-fined: weighting and resampling.

1) Partitioned Sampling: A partitioned sampling approach isused to reduce the curse of dimensionality in the particle filter.The basic idea is as follows. Suppose that we (wrongfully) factorthe tracking posterior density as

(10)

In this case, the target posterior is conveniently a product ofthe partition posteriors (i.e., individual target posteriors) .

We can then generate samples for each partition according toits posterior (i.e., ) and mergethem to represent . It can be proven that the resulting particledistribution is the same as when we generate directly fromthe full posterior . However, in the partitionedsampling case, the computational complexity of the state vectorgeneration is linear with respect to the number of targets asopposed to exponential when the state vector is sampled fromthe full posterior.

In our problem, we can approximately factor out the trackingposterior to exploit the computational advantage of the par-titioned sampling. Note that in our case, the target dynamicscan already be factored out because we assume the targets aremoving independently.4 Unfortunately, the observation densitydoes not factor out, because the observed DOA data cannot beimmediately associated with any of the partitions. However, fora given partition, if we assume that the data is only due to thatpartition and clutter (hence, the DOA data corresponding toother partitions are treated as clutter), we can do the followingapproximate factorization on the observation likelihood (9):

(11)

Hence, for our problem, an approximate partition posterior isthe following:

(12)

where is given in (11) andwith . Note

that (11) will not be used as the data likelihood of the particlefilter. The previous approximate factorization of the data likeli-hood is to make use of the partitioned sampling strategy to pro-pose particles. To calculate the particle filter weights, the fullposterior uses the observation density (9).

2) Gaussian Approximation for Partition Posteriors: Now,we derive a Gaussian approximation to (12) to use as a proposalfunction for the particle filter. Hence, we use the current ob-served data to propose the filter’s particle support, also incor-porating target maneuvers if there are any. This approximationcan be done in two ways: 1) Laplace’s method or 2) the MHMHmethod. The first method is an order of magnitude faster thanthe second, but it is not as robust. The filter monitors its trackingand can decide to switch between methods to find the partitionposteriors.

4When targets are moving closely in tandem, there is a possibility that thebeamformer block may not resolve them. Hence, they can be treated as a singletarget. In other cases, the independence assumption still works. However, if highresolution observations (e.g., top down video images of the target plane) areavailable, it is better to also model the interactions of targets. A good exampleusing Monte Carlo Markov chain methods can be found in [33].


Fig. 6. Particle filter is susceptible to shadow tracking, when only theNewton–Raphson recursion is used to determine the data likelihood function.(Top) DOA tracking results with MHMH correction (solid line) and without(dotted line). At t = 6 s, the MHMH automatically corrects the heading bias.(Bottom Left) True track (inside), the shadow track (middle) and the divergedtrack (outside, dotted line) which lacks the MHMH correction. (Bottom Right)Filter heading estimates. Note the MHMH correction at time t = 6 s.

By default, the filter uses Laplace’s method to approximatein (12) and thereby derive the partition proposal

functions of the particle filter, denoted as. Laplace’s method is an analytical approximation of prob-

ability density functions based on a Gaussian approximationof the density around its mode, where the inverse Hessian ofthe logarithm of the density is used as a covariance approx-imation [19]. It can provide adequate approximations to pos-teriors that are sometimes more accurate than approximationsbased on third-order expansions of the density functions [18].The computational advantage of this approach is rather attrac-tive because it only requires first- and second-order derivatives.The condition for accurate approximation is that the posterior bea unimodal density or be dominated by a single mode. Hence,it is appropriate for approximating the partition posteriors ofthe particle filter. Laplace’s approximation requires the calcu-lation of the data statistics. The Laplacian approximation is de-scribed in detail in [34]. For this paper, it is implemented withthe Newton–Raphson recursion with backtracking for compu-tational efficiency. Because of the constrained nature of the al-gorithm’s modified cost function, this method is sometimes sus-ceptible to “shadow tracking” and can diverge under certain con-ditions (refer to Fig. 6). Shadow tracking refers to the scaledtarget tracks in – that can lead to the same DOA track as illus-trated in Fig. 6. Shadow tracks may cause the Laplacian approxi-mation deviate from the truth because of the motion smoothnessconstraints used to calculate it.

Any divergence in the Newton–Raphson mode calculationcan be detected by monitoring the number of DOA measure-ments that fall into the gate of the mode estimate (see Fig. 4).If there are less than a threshold number of DOAs (typicallyM/2) within a neighborhood of the Newton–Raphson mode’sDOA track, then the particle filter uses the MHMH method fordetermining the mode. After the MHMH iterations are over, ifthe MHMH corrected mode fails to have the required number of

DOAs within its gate, the partition is declared dead. Details ofthe MHMH method are omitted due to lack of space. The finalexpression for the partition proposal functions to be used in theparticle filter is given by

(13)

where the Gaussian density parameters are

(14)

where is the mode of , and is theHessian of at , calculated by either one ofthe methods in this section.

D. Algorithm Details

Pseudo-code of the particle filter algorithm is given inTable I. The filter implementation employs an efficient resam-pling strategy, named “deterministic resampling,” first outlinedby Kitagawa [35]. This resampling strategy is preferred be-cause of 1) the efficient sorting of the particles and 2) thenumber of random number generations. The deterministicresampling strategy also has known convergence properties[35]. Faster resampling schemes without convergence proofsare also available [36] and these could make a difference inthe filter computation, especially when . Effects of theresampling stage can be seen at the computational analysisdone in Section VI.

Finally, the partitions are managed by the specific interactionbetween the particle filter and the MHMH block. New parti-tions are introduced into the particle filter, using the distribu-tion supplied by the MHMH block. First, the particle filter blockdeletes the DOAs from the observed data that belong to the par-titions that it is currently tracking (gating) after estimation (seeFig. 4). Gating here is a simple distance thresholding operationand does not incur significant computation. Then, the remainingDOAs are used by the MHMH block to determine any new par-tition distributions. For the particle filter to stop tracking a giventarget, a deletion operation is necessary. The particle filter candelete partitions at either the proposal stage or after estimation.Last, note that our implementation of the particle filter does notmake partition associations such as partition split or merge. Weleave these decisions to a higher level fusion algorithm in thesensor network.

V. MHMH BLOCK

The objective of the MHMH block5 is to find any new tar-gets in the observed data and determine their probability densitydistributions. The MHMH block uses the matching pursuit ideaby Mallat [37] by consecutively applying the MHMH algorithmon the ungated DOAs until stopping conditions described in the

5The MHMH block is not to be confused with the MHMH correction of theproposal stage.


TABLE IPARTICLE FILTER TRACKER PSEUDO-CODE

TABLE IIMHMH SAMPLING ALGORITHM

following are satisfied. Each iteration of the MHMH block re-sults in a new target state vector distribution, whereas the totalnumber of MHMH block iterations gives the number of newtargets.

The MHMH block employs the mode hungry Metropolis–Hastings sampling algorithm to determine the distributionfor the target that has the highest mode in the multi-targetobservation density. The MHMH sampling algorithm [20] isan accelerated version of the Metropolis–Hastings algorithm[38]–[41] that generates samples around the modes of a targetdensity. The pseudo-code for the MHMH sampling algorithmis given in Table II (reproduced from [20]). For our system,we use a random walk for the candidate generating function

in Table II with the walk noise variances set to ,s , and . Cross sampling is applied every

iterations on subpartitions of size , which istypically half the number of MHMH particles . Forour problem, the sampling algorithm is iterated a total of 150

iterations. Finally, the target density is obtained by settingon the observation density (9)

(15)

The pseudo-code of the MHMH block is given in Table III.The algorithm creates a set of particles to input the MHMH sam-pling algorithm. This initial set consists of some of the residualDOAs and a grid of values. After the MHMH samplingis over, the mode of these particles is monitored by two condi-tions. The first condition checks if there are sufficient DOAs inthe mode’s DOA gate. This ensures that the mode is relativelyaccurate when used by the particle filter. The second condition


TABLE IIIMHMH BLOCK PSEUDO-CODE

TABLE IVNOTATION FOR COMPUTATIONAL COMPLEXITY ANALYSIS

makes sure that mode belongs to a physical target as opposedto some alignment of the clutter. If these conditions are not sat-isfied, the MHMH block exits and no new partitions are intro-duced. Otherwise, we resample particles with replacementfrom the MHMH output and declare a new partition in the par-ticle filter. The DOAs belonging to the new mode are gated out,and the procedure is repeated.

VI. COMPUTATIONAL COMPLEXITY

This section describes the computational requirements of theparticle filter and MHMH blocks described in Sections IV andV, respectively. The preprocessing beamforming block has sev-eral established efficient implementation methods based on Q-Rmatrix decomposition [21] and is not considered here. Table IVsummarizes the notation for important parameters needed to de-rive the implementation complexities.

A. Computational Analysis of the Particle Filter Block

The choice of our proposal function drives the complexityand execution time of the tracking algorithm. As expected com-plexity increases with the number of targets and the number ofparticles used. The complexity analysis and simulation results inTable V suggest that the overall complexity of the algorithm isof . This is also apparent from the linear increase inthe total computational time in Fig. 7. The computational timewith respect to the number of targets is approximately quadratic.Moreover, the complexity of the particle proposal stage (Step 1in Table I) and the weight evaluation stage (Steps 2 and 3 inTable I) are comparable. For a small number of particles, up to

, the execution time of the Newton iteration stage isconsiderably more compared to other stages. Note that this stagedoes not depend on the number of particles used, but depends onthe number of targets. The number of iterations in the Newton

search is typically between 10 and 20 because of the adaptivestep size and the stopping conditions.

B. Computational Analysis of the MHMH Block

The computational requirements of the MHMH block areshown in Table VI. The main complexity of the MHMH al-gorithm consists of generating the particles and identifying thedata mode. The MHMH complexity analysis is similar to thatof the particle filter block and is based on the algorithms inTables II and III. The algorithm complexity is directly propor-tional to the number of burn-in iterations, denoted by . Typ-ically, is in the range 150 to 400, and is proportional to thenumber of particles in the initial set. During each it-eration, significant time is spent in evaluating the target density

, shown in (15). As can be seen from Table VI, this dependson both and .

VII. OTHER TRACKER SOLUTIONS

This section discusses alternative DOA tracking algorithmsto replace the particle filter in Fig. 3, namely the EKF and theLaplacian filter. These algorithms incur less computational costthan the particle filter. The Laplacian filter solution can even beimplemented in a parallel fashion. However, the EKF solution isdifficult to extend to multiple targets. In addition, the Laplacianfilter performance is sensitive to its initialization.

A. Extended Kalman Filter

The EKF solution is an -scan multiple hypothesis tracking(MHT) filter. Unlike the standard Kalman filter-based ap-proaches, the observation equation (9) requires snapshots ofdata to perform state estimates. Hence, the usage of MHT herediffers from traditional approaches.6 Compared to the particlefilter approach that updates the state every seconds, the MHTapproach estimates the states every seconds, based on thesliding -DOA snapshots. This is similar to a sliding windowapproach [10], but assumes that the number of targets is fixedand known since it is estimated by the MHMH block.

In the EKF approach explicit measurement-to-target associa-tions have to be made. We use the JPDA approach to generate aset of hypotheses, relating measurements to targets, during eachsnapshot [10]–[13], [42]. Similar hypotheses are generated for

6In traditional MHT, theM -scan associations from one snapshot t+(m�1)�are carried over to the next snapshot t+m� . Here, the associations during eachsnapshot t+m� depend only on the predicted measurement, based on state x ,and do not depend on associations at t + (m � 1)�


TABLE VCOMPUTATIONAL COMPLEXITY ANALYSIS—PARTICLE FILTER BLOCK

Fig. 7. Execution times for various particle filter stages. (a) One target.(b) Three targets.

each of the DOA’s, starting from time . The best hypoth-esis at each snapshot is chosen, based on the probabilities ob-tained from the JPDA. These -best hypotheses are then usedto associate the measurements to targets at each snapshot. If ameasurement-to-target association cannot be found for a target,it is assigned a probability of zero. For each target, a modifiedinnovation term is obtained as

(16)

where is the measurement associated with the targetat time instant and is the probability corre-

sponding to the hypothesis at the th snapshot obtained usingthe JPDA. The EKF covariance update is

(17)

where , is the Kalman gain,is the probability that the associations are correct,

is the state prediction covariance, and is the measure-ment covariance obtained as in the standard EKF [10].

B. Laplacian Filter

The Laplacian filter is a Bayesian filter that uses the productof the partition proposal functions (13) as the tracking posterior.Hence, the posterior of the Laplacian filter is a combination ofmultitarget Gaussian approximations

(18)

This approximation asymptotically has an estimationerror [18] for a single target. However, it is difficult to analyzethe asymptotic effects on the multitarget posterior.

The Laplacian filter handles multiple targets using the par-tition approach. Each target posterior is assumed independent.Hence, the filter can be parallelized for faster implementation. Inaddition, because the Laplacian filter uses the Newton–Raphsonrecursion to calculate the data likelihood, its complexity is sim-ilar to the particle filter for . However, the particle filtercannot be implemented in a parallel structure because there arecross terms in the data likelihood. For the EKF, we need esti-mates from the EKF compared to one estimate from the particlefilter or Laplacian filter every seconds. Furthermore, usingJPDA steps to make explicit target-data associations during eachestimate of the EKF also increases its complexity. Consideringthis, the Laplacian filter complexity is less than that of the EKFimplementation.

VIII. SIMULATIONS

We first give a multitarget, multifrequency tracking example;then the automated system performance is demonstrated. Fi-nally, we compare the tracking performance of the algorithmspresented in the paper. Table VII summarizes the simulation pa-rameters used.

A. Multifrequency Tracking

Fig. 8 shows a challenging three-target scenario, where thetargets cross in the bearing space as they are maneuvering. Forthis simulation, two independent layers of DOA estimates are


TABLE VICOMPUTATIONAL COMPLEXITY ANALYSIS—MHMH BLOCK

Modulo operations. Random number generation. Compare operations. Sort operations.

TABLE VIISIMULATION PARAMETERS

Fig. 8. (Top) DOAs, represented by diamonds and dots, are generated by inde-pendent noise and are input to the filter unsorted. Particle filter DOA estimates(solid line) are close to the ground truth (dashed line) and cannot be discerned,except when the targets cross. (Bottom Left) Ground truth (thin solid line) andestimated (thick dotted line) tracks. (Bottom Right) True target heading (thinline) compared to estimated heading (thick line).

used each with the correct mean and a variance of . Theparticle filter does a good job of keeping the target association aswell as determining the maneuvers, which are most prominentin the heading plot. In Fig. 8, the dashed lines are ground truth,whereas the solid lines are the particle filter estimates whichare almost identical to the ground truth. There is a small bias inthe filter DOA estimates between 4 s and 6 s wherethe targets are crossing. This bias is, in part, also due to thetarget maneuvers, which start at 4 s. The filter maintains thetrack coherence in this difficult case by using the independent

Fig. 9. Two targets are successfully initialized and tracked by the automatedsystem. The system can successfully detect the short target tracks at 40 < t <

45 s as well. (a) System input. (b) System result.

Fig. 10. (a) Probability of missed detections in the MHMH block with varyingnoise variance in the measurement data. A partition gate size of � = 3 isused. (b) A sample realization with � = 2 from the Monte Carlo run thatcalculates the probability of miss in part (a).

frequency observations (with only one frequency, we observedthat the filter can confuse the targets). Although the estimatesdeteriorate in the region where targets are crossing as well asmaneuvering, the particle filter locks back on the targets afterthe transient region.

B. System Simulation

This subsection simulates the automatic system, described bythe block diagram in Fig. 3. Two targets are being tracked by thesystem, where one target appears in the beamformer output be-tween 5 s and 32 s, and the other one between 11 sand 37 s (see Fig. 9). The targets are successfully initializedby the MHMH block and then deleted by the particle filter. Thetrack coherence is also maintained around 20 s, because theparticle filter also keeps the target motion parameters. In thiscase, there were no false target initializations by the MHMHblock. The short bearing tracks between times 40 s and45 s were also embedded into the data set to demonstrate theability to initiate and kill tracks. In Fig. 10, we provide a simu-lation to demonstrate the performance of our detection scheme.


Fig. 11. Monte Carlo run results for each filter using 100 independent noise realizations using Matlab’s boxplot command. The ground truth is shown with thedashed lines. The last row shows the track estimates and their mean. In this example, the Laplacian performs quite well and hence there were no MHMH iterationsdone in the proposal stage of the particle filter. This is due to the smooth movement of the targets. (a) Particle filter. (b) EKF. (c) Laplacian filter.

In this simulation, we randomly picked a target track with a du-ration of 1 s. We generated the corresponding bearing track for

with varying bearing errors. We then added a spurioustrack with by generating random bearing estimatesthat are uniformly distributed in . We repeated this for aMonte Carlo run of size 100. During each simulation, we alsorandomly replaced any bearings in the input data with randomclutter with probability to simulate missing bearings. To de-termine the probability of missed detections in our initializationalgorithm, we counted the number of times our initialization al-gorithm failed to detect the target and then divided that by thesize of the Monte Carlo run. For DOA error variances less than

, the detection scheme is quite successful, i.e., the probabilityof miss is less than 0.1. As the DOA noise increases above thegate size, the detection performance loses its monotonicity.

C. Comparisons of DOA Trackers

In Figs. 11–13, Monte Carlo simulations compare the esti-mates obtained using the particle filter, the EKF, and the Lapla-

cian filter. The state estimation performances of these filters arecomparable. The Laplacian filter tends to have less variance thanthe particle filter in general, because the actual posterior is widerthan the Gaussian approximation. Hence, we can consider theLaplacian filter as a mode tracker. The estimates of the particlefilter can be further improved by also adding cross partition sam-pling as suggested in [2] or auxiliary sampling [43].

The filters’ tracking performances in most cases are similar,when initialized correctly around the true target stateas in Fig. 11.However, the EKF is more sensitive to errors in initializationcompared to the other two filters. The EKF may have difficultytracking the DOAs correctly, when initialized with the mean ofthe particle set generated by the MHMH block. The EKF cancompletely lose the DOA tracks, if the initial estimates are notaccurate, whereas the other two filters can still absorb the discrep-ancies. There is also a notable decrease in the EKF performance,when targets maneuver as in Figs. 12 and 13. The particle filter isthe most robust to target maneuvers compared to the other twomethods due to the diversity provided by the particles.


Fig. 12. Monte Carlo run results for each filter using 100 independent noise realizations for a target that maneuvers rapidly. The last row shows track estimatesand their mean. In this example, the MHMH iterations in the Laplacian calculation were necessary to pull the Laplacian towards the true track because of the abruptmaneuvers at t = 11 s and t = 12 s. Note that the filter track estimates shadow the true track due to the constant velocity assumption. (a) Particle filter. (b) EKF.(c) Laplacian filter.

In Fig. 14, we compare the particle filter, the EKF, and theLaplacian filter presented here with the particle filter that usesthe received acoustic signals directly (denoted as DPF) [4]. Forthe comparison, we use the classical narrowband observationmodel to generate the acoustic data as described in [4]; theacoustic SNR at the array is approximately 7 dB in the simu-lation. For the simulation, we use a constant frequency targetat 150 Hz, an eight- microphone circular array with 0.45 wave-length separation, and a sampling frequency of 1000 Hz. Forthe synthetic target track, a process noise is added with vari-ances , , and . The process noiseexplains the abrupt changes in the heading estimates in Fig. 14at each time period.

To use the filters presented in this paper, we applied a simpleMVDR beamformer on the acoustic data at the center frequencyand calculated DOA estimates. The filters being compared usethe same noise variances for the state update equation. The DPFruns three orders of magnitude slower than the particle filter pre-

sented here and performs better because the observation modelis perfectly matched by the data. The particle filter has a largervariance partly because of the clutter handling mechanism thatcreates a noise floor for the pdf (for a single target, the pdf isa raised Gaussian). Not surprisingly, the Laplacian filter showsa similar estimation variance as the DPF in bearing estimationbecause it tracks the mode of the target track with a narrowerposterior approximation.

IX. CONCLUSION

In this paper, we present an automated framework formultiple target DOA tracking based on a batch measurementmodel. The DOA batches are treated as images to naturallyhandle the data association and ordering issues. The presenceof multiple targets is handled using a partition approach. Theobservation likelihoods are calculated jointly and are assignedby using the templates created by the state vectors and the state


Fig. 13. Same example as in Fig. 12 without the MHMH correction. In this case, the Laplacian filter cannot handle the maneuver. The particle filter can still trackthe target due to its particle diversity. However, the estimates of the particle filter are now worse than the estimates with the MHMH correction at the proposalstage. (a) Particle filter. (b) EKF. (c) Laplacian filter.

Fig. 14. Monte Carlo comparison of the DOA filters with the DPF filter presented in [4]. (a) DPF. (b) Particle filter. (c) EKF. (d) Laplacian filter.

update equation. We present three filters for the DOA trackingproblem: the particle filter, the Laplacian filter, and the EKF.These filters offer computationally attractive alternative to

filters based on acoustic sensor outputs directly. The filters arecompared in terms of computational cost, sensitivity to initial-ization, and robustness. For the initialization of the automated


DOA tracking system, we also discuss a sampling algorithmbased on the matching pursuit idea, using the MHMH method.

REFERENCES

[1] C. Sword, M. Simaan, and E. Kamen, “Multiple target angle trackingusing sensor array outputs,” IEEE Trans. Aerosp. Electron. Syst., vol.26, no. 2, pp. 367–372, Mar. 1990.

[2] M. Orton and W. Fitzgerald, “A Bayesian approach to tracking mul-tiple targets using sensor arrays and particle filters,” IEEE Trans. SignalProcess., vol. 50, no. 2, pp. 216–223, Feb. 2002.

[3] Y. Zhou, P. Yip, and H. Leung, “Tracking the direction-of-arrival ofmultiple moving targets by passive arrays: Algorithm,” IEEE Trans.Signal Process., vol. 47, no. 10, pp. 2655–2666, Oct. 1999.

[4] V. Cevher and J. H. McClellan, “General direction-of-arrival trackingwith acoustic nodes,” IEEE Trans. Signal Process., vol. 53, no. 1, pp.1–12, Jan. 2005.

[5] J. R. Larocque, J. P. Reilly, and W. Ng, “Particle filters for trackingan unknown number of sources,” IEEE Trans. Signal Process., vol. 50,no. 12, pp. 2926–2937, Dec. 2002.

[6] T. Kailath, A. Sayed, and B. Hassibi, Linear Estimation. EnglewoodCliffs: Prentice-Hall, 2000.

[7] W. Brogan, Modern Control Theory. Englewood Cliffs, NJ: Prentice-Hall, 1991.

[8] A. Doucet, N. Freitas, and N. Gordon, Eds., Sequential Monte CarloMethods in Practice. New York: Springer-Verlag, 2001.

[9] J. Liu and R. Chen, “Sequential Monte Carlo methods for dynamicsystems,” J. Amer. Statistical Association, vol. 93, pp. 1032–1044, Sep.1998.

[10] Y. Bar-Shalom and T. Fortmann, Tracking and Data Association.New York: Academic, 1988.

[11] Y. Bar-Shalom, “Tracking methods in a multitarget environment,” IEEETrans. Autom. Control, vol. AC-23, no. 4, pp. 618–626, Aug. 1978.

[12] K. Chang and Y. Bar-Shalom, “Joint probabilistic data association formultitarget tracking with possibly unresolved measurements,” IEEETrans. Autom. Control, vol. AC-29, no. 7, pp. 585–594, Jul. 1978.

[13] L. Ng and Y. Bar-Shalom, “Multisensor multitarget time delayvector estimation,” IEEE Trans. Acoust., Speech, Signal Process., vol.ASSP-34, no. 4, pp. 669–677, Aug. 1986.

[14] R. Karlsson and F. Gustafsson, “Monte Carlo data association for mul-tiple target tracking,” in Proc. IEE Target Tracking: Algorithms andApplications, 2001, pp. 13/1–13/5.

[15] D. Crisan and A. Doucet, “A survey of convergence results on particlefiltering methods for practitioners,” IEEE Trans. Signal Process., vol.50, no. 3, pp. 736–746, Mar. 2002.

[16] J. MacCormick and M. Isard, “Partitioned sampling, articulated ob-jects, and interface-quality hand tracking,” in Proc. Eur. Conf. Comput.Vision, 2000, pp. 3–19.

[17] M. Isard and J. MacCormick, “BraMBLe: A Bayesian multiple-blobtracker,” in Proc. 8th Int. Conf. Comput. Vision, 2001, pp. 34–41.

[18] L. Tierney and J. B. Kadane, “Accurate approximations for posteriormoments and marginal densities,” J. Amer. Statist. Assoc., no. 81, pp.82–86, 1986.

[19] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian DataAnalysis. London, U.K.: Chapman Hall/CRC, 2004.

[20] V. Cevher and J. H. McClellan, “Fast initialization of particle filtersusing a modified Metropolis-Hastings algorithm: Mode-hungry ap-proach,” presented at the ICASSP, Montreal, QC, Canada, 2004.

[21] D. Johnson and D. Dudgeon, Array Signal Processing: Concepts andTechniques. Englewood Cliffs, NJ: Prentice-Hall, 1993.

[22] P. Stoica and A. Nehorai, “Music, maximum likelihood, andCramér-Rao bound,” IEEE Trans. Acoust., Speech, Signal Process.,vol. 37, no. 5, pp. 720–741, May 1989.

[23] W. Ng, J. P. Reilly, T. Kirubarajan, and R.-R. Larocque, “Widebandarray signal processing using MCMC methods,” IEEE Trans. SignalProcess., vol. 53, no. 2, pp. 411–426, Feb. 2005.

[24] Y. Boers and J. N. Driessen, “Multitarget particle filter track beforedetect application,” IEE Proc. Radar, Sonar, Navigation, vol. 151, no.6, pp. 351–357, Dec. 2004.

[25] M. G. Rutten, B. Ristic, and N. J. Gordon, “A comparison of particle fil-ters for recursive track-before-detect,” in Proc. 8th Int. Conf. Inf. Fus.,2005, pp. 169–175.

[26] H. Wang and M. Kaveh, “On the performance of signal-subspace pro-cessing-part II: Coherent wide-band systems,” IEEE Trans. Acoust.,Speech, Signal Process., vol. ASSP-35, no. 11, pp. 1583–1591, Nov.1987.

[27] A. Gershman and M. Amin, “Wideband direction-of-arrival estimationof multiple chirp signals using spatial time-frequency distributions,” inProc. 10th IEEE Workshop Statistical Signal Array Process., 2000, pp.467–471.

[28] A. Gershman, M. Pesavento, and M. Amin, “Estimating the parame-ters of multiple weideband chirp signals in sensor arrays,” IEEE SignalProcess. Lett., vol. 7, no. 6, pp. 152–155, Jun. 2000.

[29] A. Papoulis and S. Pillai, Probability, Random Variables and StochasticProcesses. New York: McGraw-Hill, 2002.

[30] J. MacCormick and A. Blake, “A probabilistic exclusion principle fortracking multiple objects,” in Proc. 7th Int. Conf. Comput. Vision, 1999,pp. 572–578.

[31] M. Evans, N. Hastings, and B. Peacock, Statistical Distributions, 3rded. New York: Wiley, 2000.

[32] T. Edgoose, L. Allison, and D. L. Dowe, “An MML classification ofprotein sequences that knows about angles and sequences,” in Proc.Pacific Symp. Biocomput., 1998, pp. 585–596.

[33] Z. Khan, T. Balch, and F. Dellaert, An MCMC-based particle filterfor tracking multiple interacting targets College Comput., Georgia Inst.Technol., Atlanta, Tech. Rep. GIT-GVU-03-35, 2003.

[34] V. Cevher and J. H. McClellan, “An acoustic multiple target tracker,”presented at the IEEE SSP, Bordeaux, France, 2005.

[35] G. Kitagawa, “Monte Carlo filter and smoother for non-Gaussian non-linear state space models,” J. Comp. Graph. Stat., vol. 5, no. 1, pp.1–25, Mar. 1996.

[36] M. Bolic, P. M. Djuric, and S. Hong, “New resampling algorithms forparticle filters,” in Proc. ICASSP, 2003, pp. 589–592.

[37] S. Mallat and S. Zhang, “Matching pursuits with time-frequency dic-tionaries,” IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3397–3415,Dec. 1993.

[38] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller,“Equations of state calculations by fast computing machines,” J. Chem.Phys., vol. 21, pp. 1087–1092, 1953.

[39] W. Hastings, “Monte Carlo sampling methods using Markov chainsand their applications,” Biometrika, vol. 57, pp. 97–109, 1970.

[40] S. Chib and E. Greenberg, “Understanding the Metropolis–Hastingsalgorithm,” Amer. Statistician, vol. 49, no. 4, pp. 327–335, 1995.

[41] L. Tierney, “Markov chains for exploring posterior distributions,” Ann.Statist., vol. 22, no. 4, pp. 1701–1728, 1994.

[42] S. S. Blackman, Multiple-Target Tracking With Radar Application.Norwood, MA: Artech House, 1986.

[43] M. Pitt and N. Shephard, “Filtering via simulation: Auxiliary particlefilters,” J. Amer. Statist. Assoc., vol. 94, pp. 590–599, 1999.

Volkan Cevher (M’99) was born in Ankara, Turkey,in 1978. He received the B.S. degree in electrical en-gineering from Bilkent University, Ankara, Turkey,in 1999, as a valedictorian, and the Ph.D. degree inelectrical engineering from Georgia Institute of Tech-nology, Atlanta, in 2005.

He is currently working with Dr. R. Chellappa asa Research Associate in the Center for AutomationResearch, University of Maryland, College Park,where he works on computer vision problems.During the summer of 2003, he was employed by

Schlumberger Doll Research, Ridgefield, CT. He worked as a Postdoctoral Re-searcher at the Georgia Institute of Technology under the supervision of Dr. J.H. McClellan till the end of 2005. His research interests include structure frommotion, sensor network management problems (sensor build and placementstrategies), Monte Carlo Markov chain methods (specifically particle filters),target tracking models, adaptive filters, time frequency distributions, fractionalFourier transform, and array signal processing.

Dr. Cevher was a corecipient of the Center for Signal and Image ProcessingOutstanding Research Award in the fall of 2004.

Rajbabu Velmurugan (S’03) received the B.E. de-gree from the Government College of Technology,Coimbatore, India, in 1995, and the M.S. degree fromClarkson University, Potsdam, NY, in 1998, both inelectrical engineering. He is currently pursuing thePh.D. degree in electrical engineering at the GeorgiaInstitute of Technology, Atlanta, under the advise-ment of Prof. J. H. McClellan.

He worked in Larsen and Toubro Ltd., Madras,India, from 1995 to 1996, and at The MathWorks,Natick, MA, from 1998 to 2001, before joining

Clarkson University. His research interests include particle filtering techniques,design and implementation of real-time signal processing systems, and arraysignal processing.


James H. McClellan (F’85) received the B.S.degree in electrical engineering from LouisianaState University (LSU), Baton Rouge, in 1969, andthe M.S. and Ph.D. degrees from Rice University,Houston, TX, in 1972 and 1973, respectively.

Since 1987, he has been a Professor in the Schoolof Electrical and Computer Engineering, GeorgiaInstitute of Technology, Atlanta, where he presentlyholds the John and Marilu McCarty Chair. From1973 to 1982, he was a member of the research staffat Massachusettes Institute of Technology, Lincoln

Laboratory, Lexington, MA, and then a Professor at Massachusettes Institute

of Technology, Cambridge. From 1982 to 1987, he was employed by Schlum-berger Well Services, Austin, TX. He is a coauthor of the textbooks NumberTheory in Digital Signal Processing, (Prentice-Hall, 1979), Computer Exercisesfor Signal Processing (Prentice-Hall, 1997), DSP First: A Multimedia Approach(Prentice-Hall, 1997), and Signal Processing First (Prentice-Hall, 2003).

Dr. McClellan was a recipient of the McGraw-Hill Jacob Millman Award foran outstanding innovative textbook in 2003, the W. Howard Ector OutstandingTeacher Award at Georgia Institute of Technology in 1998, the Education Awardfrom the IEEE Signal Processing Society in 2001, the Technical AchievementAward for work on FIR filter design in 1987, the Society Award in 1996, bothfrom the IEEE Signal Processing Society, and the IEEE Jack S. Kilby SignalProcessing medal in 2004. He is a member of Tau Beta Pi and Eta Kappa Nu.

2810 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, …

Documents

Transcript of 2810 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, …