Automatic Estimation of Multiple Target Positions and Velocities Using Passive TDOA Measurements of Transients

8/2/2019 Automatic Estimation of Multiple Target Positions and Velocities Using Passive TDOA Measurements of Transients

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Automatic Estimation of Multiple Target Positions

and Velocities Using Passive TDOA Measurements

of TransientsDragana Carevic

Abstract This paper considers the problem of the estimationof the motion parameters of multiple targets moving linearly in athree-dimensional (3-D) observation area contaminated by clutter.The measurements are limited to time differences of arrival(TDOA) of short-duration acoustic emissions, or transients, gen-erated by the targets. This problem can arise in situations wherethe level of continuous broadband target-related noise is verylow. Owing to the fact that transient emissions are nonstationaryand can have low signal-to-noise ratio (SNR), the correspondingTDOA measurement errors are usually non-Gaussian. Therefore

Gaussian mixture distributions are used to appropriately modelthese errors.

An iterative maximum likelihood optimization technique basedon a modified deterministic annealing expectation-maximization(MDAEM) algorithm is applied to this problem. In each iterationthe algorithm uses a nonlinear least squares (LS) technique incomputing the motion parameters for each target. It generalizesthe variance deflation method previously used for the initializationof target tracking algorithms and increases the possibility ofattaining a globally optimal solution for random initial conditions.Simulation results are presented for several different numbers oftargets, clutter densities, and probabilities of gross error of thetarget related measurements and are found to be comparable tothe estimates obtained when the measurement-to-target assign-ments are exactly known.

Index Terms Source localization, EM algorithm, deterministicannealing, relative time delay estimation, underwater acoustictransients, Gaussian mixture model.

I. INTRODUCTION

THE problem of estimating motion of underwater targetsusing passive sensors is of considerable interest in anumber of surveillance applications. Sensors can be mounted

on board a ship, towed, or deployed in water, and their position

is assumed to be known. The measurements are based on the

differences in propagation travel time of an emitted signal from

the source to each of the sensors [1], [2] and are commonly

referred to as the time differences of arrival (TDOA) or relativetime delays.

Transients belong to a particular class of wide-band acoustic

emissions that are characterized by relatively short durations

(from few tens of milliseconds to several seconds) and arbi-

trary waveform shapes. Examples of these signals include a

deck hatch slamming, a hull reverberation, and a momentary

vibration caused by a pump. They are deemed to be important

for detection and passive localization of quiet targets that would

otherwise be hardly detectable.

This work was supported by the Defence Science and Technology Organi-sation, Australia.

The estimation of TDOA of a passive transient involves

the computation of a cross-correlation between outputs from

a given pair of receiving sensors [3], [4]. In the presence of

multipath propagation the estimate is taken to be the time lag

of an appropriately chosen peak in the crosscorrelogram [5],

[6], [7]. The accuracy of the TDOA measurement depends on

individual characteristics of the received signal spectrum [8]. It

is also sensitive to low signal-to-noise ratios (SNRs) [8], [9]

and to signal distortions resulting from acoustic propagationin spatially varying environments [10], [11]. Large errors in

the TDOA estimation are possible and, as a consequence,

the TDOA measurement error distribution is usually non-

Gaussian (has wide tails). A two-component Gaussian mixture

probability density function (pdf) is proposed as a general

model for this error distribution [12], [13]. Moreover, transients

radiated by targets are typically observed in the presence of

clutter that can originate from other sources, such as nearby

shipping traffic, biological noise, etc., or as a consequence of

very large (gross) measurement errors caused by environmental

effects such as multipath propagation. Besides, there may be a

number of false detections returned by the transient detection

algorithm.A number of techniques for tracking single and multiple

targets using measurements of uncertain origin have been

proposed. The probabilistic data association (PDA) [14] prob-

abilistically associates measurements with targets; it defines a

joint likelihood function and computes the target motion pa-

rameters by a direct maximization of this function. It is success-

fully applied to tracking targets in underwater acoustics [15],

[16], [17]. Alternative approaches [18], [19], [20], [21] formu-

late multitarget tracking as an incomplete data problem and

apply an iterative maximum likelihood (ML) estimator based

on the expectation-maximization (EM) algorithm [22]. This

algorithm maximizes the likelihood function of incomplete data

indirectly by iterating the expectation and maximization stepsuntil some appropriate convergence conditions are satisfied. In

[18], [23] the authors apply the EM-based methods to tracking

targets in cluttered underwater environments. The problem of

the localization of a single target in clutter based on passively

sensed transients in a 2-dimensional (2-D) observation area is

considered in [12], [13]. Other approaches to localizing and

tracking a single moving source in an underwater environment

using TDOA measurements are discussed in [24], [25], [26].

Recently, Vo et al. [27], [28] and Ma et al. [29] proposed

a method for tracking unknown number of speakers in 2-D

multipath environments using a sequential Monte Carlo (SMC)


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implementation based on TDOAs. Several other interesting

approaches to tracking single and multiple targets are presented

in [30], [31], [32], [33].

The PDA and the methods based on the EM algorithm are

sensitive to the choice of the initial values of the parameters.

The initialization of the algorithms using the values that are

different from the true parameters may result in convergence

to a suboptimal solution, i.e., the algorithm may convergeonly to a local ML solution or to a saddle point, and not

to a globally optimal solution [16], [18], [23]. A technique

commonly applied to improve the convergence of these al-

gorithms is to deflate the measurement variance: the initial

iteration is carried out with a large variance and the variance is

decreased (or deflated) in the subsequent iterations [18], [34].

An alternative approach is to use the deterministic annealing

EM (DAEM) algorithm recently proposed by Ueda and Nakano

[35], [36]. It formulates the ML estimation as the minimization

of an effective cost function based on the thermodynamics free

energy and incorporates a deterministic annealing process that

is characterized by simultaneous reduction in both the entropy

and the cost function of the system with gradual decrease ofa global control parameter called the computational tempera-

ture. At high temperatures the range of possible solutions of

the algorithm is considerably widened whereby its dependance

on the initial conditions is decreased.

Both the EM algorithm implemented with the variance

deflation and the DAEM algorithm increase the log-likelihood

in each iteration. Consequently they are not guaranteed to

obtain a global optimum in the cases where the initial guess

is poor (e.g., when the initial values are set at random).

Also, they are highly susceptible to the presence of false

measurements [13], [18]. To improve the quality of the result

Takada and Nakano [37], [38] propose the use of multiple-thread search with the DAEM algorithm. Each time a saddle

point of the likelihood function is reached, which is indicated

by detecting the eigenvalues of the Hessian matrix of the

free energy function that are smaller than a given threshold,

new search paths are instigated along the directions of the

corresponding eigenvectors. The number of search paths that

are simultaneously investigated in this way can be very large

which significantly increases the computational complexity of

the algorithm.

An alternative approach that introduces stochastic pertur-

bation in the DAEM algorithm, called the modified DAEM

(MDAEM) algorithm, is proposed in this paper. This algorithm

uses a stochastic imputation principle whereby, in each itera-tion, pseudo-complete data is simulated by sampling from a

posterior distribution conditioned on the measurements and the

current approximation of the parameters [39]. This data is used

to obtain the updated parameter values. Unlike the standard

DAEM algorithm that guarantees that the log-likelihood is

increased in each iteration, the MDAEM algorithm has a non-

zero probability of accepting a solution with a lower likelihood

value than that in the previous iteration. In this way it avoids

saddle points or insignificant local maxima of the likelihood

function. Contrary to the approach using the multiple-thread

search described in [37], [38], the computational complexity

of the MDAEM algorithm is close to that of the standard

DAEM algorithm. Moreover, while in a general case techniques

based on stochastic simulated annealing [40] in each iteration

accept parameter updates with a probability that depends on

the temperature, the MDAEM algorithm accepts such changes

with probability 1. In this way the total search is executed more

efficiently than in the simulated annealing.

This paper considers the problem of estimating position andvelocity of multiple targets observed in clutter based on TDOAs

of passive transients. To increase robustness with respect to

unknown initial conditions and number of false measurements

the proposed technique uses the MDAEM algorithm and jointly

solves two different problems: 1) measurement-to-target asso-

ciation, and 2) estimation of the target motion parameters. The

TDOA measurement errors are modelled as having Gaussian

mixture pdfs. The Gaussian mixture modelling is solved so as

to enable the use of a nonlinear least squares (LS) technique

in computing the motion parameters for each target.

The proposed algorithm estimates motion parameters for Ptargets, where P is set by the operator. Often the true number

of targets in the measurements Pt is not known. In these casesP should be selected so as to be equal or greater than themaximum expected number of the true targets, so that P Pt.The algorithm performs robustly under such conditions and is

capable of estimating motion parameters for the Pt true targetsin addition to computing parameters for the P Pt dummytargets.

The paper is organized as follows. Background information

on the problem of localizing multiple targets observed in

clutter based on TDOAs of passive transients is presented in

Section II. The derivation of the DAEM algorithm for multiple

target localization using Gaussian mixture pdfs is described in

Section III. This section also presents the MDAEM algorithm

for target localization that uses stochastic imputations. Theresults of numerical simulations for several target geometries

are presented in Section IV. Finally, some concluding remarks

are given in Section V.

I I . BACKGROUND INFORMATION

A. Problem Formulation

We consider a scenario where P targets are assumed to bemoving in a three-dimensional (3-D) observation region. Let

Vp = [Xp(t)T,vTp ]

T (1)

denote a six-dimensional position-velocity parameter vec-tor that corresponds to the pth target where: Xp(t) =[Xp(t), Yp(t), Zp(t)]

T is the position of the target at time t,vp = [vpx, vpy, vpz]

T is the constant target velocity and t isthe reference time at which the target position is estimated.

Also, denote by

V = [V1,V2, . . . ,VP] (2)

the motion parameters for all P targets.At different discrete time instants the targets emit transient

signals asynchronously and independently of each other. The

signals may propagate through the environment along different


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paths and are passively sensed using a number of omnidirec-

tional sensors. For simplicity we assume an isovelocity acoustic

propagation model. In a general case the number of sensors

may vary over time and it is assumed to be known. To ensure

unique solution in 3-D sensors are not allowed to lie on a

straight line or on a plane [41].

B. Measurements of TDOAs of Transient SignalsLet there be N transient detections during an observation

time period which includes both target- and nontarget-related

signals. Targets are assumed to have emitted a limited number

of transients that may convey information sufficient for the

estimation of their positions and velocities. We also define

a set of generation time instants associated with the detected

transients1 {ti}Ni=1. Note that, in general, these time instantsare not evenly spaced.

Let the ith transient be emitted by the pth target at the timeti and received by Ri + 1 sensors. The exact TDOA betweenthe reference sensor l and the kth sensor, k = 1, . . . , l 1, l +1, . . . , Ri + 1, is given by

i,j,p =rl,p(ti) rk,p(ti)

c, j =

j = k, k < l

j = k 1, k > l (3)

where rk,p(ti) is the distance between the pth target and thekth sensor at the time ti

rk,p(ti) = |Xp(ti)Sk(ti)| =

(Xp(ti) Sk(ti))T(Xp(ti) Sk(ti)),(4)

Sk(ti) = [Xk(ti), Yk(ti), Zk(ti)]T is the position of the kth

sensor and c is the velocity of sound in water.The standard approach to estimating TDOA of a transient

signal is to select the time lag that maximizes the cross-

correlation between the outputs from the corresponding pair of

sensors [3], [4]. In a multipath environment there are typically

a number of paths reaching each sensor. In this situation the

cross-correlation function has many peaks and the maximal

peak may not correspond to the TDOA between arrivals

travelling along direct paths [5]. One approach is to select

a number of peaks (local maxima) of the cross-correlation

assuming that one of them corresponds to the direct path signal

arrivals. A similar technique is used in [28], [29] that considers

the problem of tracking multiple speakers in the presence of

multipath reflections. Here, TDOAs are measured based on

(continuous) speech signals and are sampled at regularly spaced

update times. At each update time the measurement set may

contain the direct path TDOA measurements related to one ormore targets in addition to false measurements.

By contrast, in our application target transient emissions

are asynchronous and the signals received at the time ti arerelated to only one target. The emission (update) time instants

ti are random (are not known in advance) and can be quiteseparated in time. A relatively small number of transients is

1The generation times can be approximated by observer times, as is acommon practice in underwater target tracking. In this case the observer timesare the times of arrival of the transients at the reference sensor. The maineffect for tracking of relatively slow targets (compared to the velocity of soundpropagation in water) is the introduction of a near constant time offset intotarget location. For very slow targets this effect is negligible.

expected to be emitted by each target. Besides, there may

be a number of detected transient emissions that correspond

to true locations within the observation area but are emitted

by sources other then the desired targets. For these reasons,

in this paper we propose to select TDOA measurements of

transients using the method described by Spiesberger [5], [6],

[7]. This method identifies robustly and efficiently the cross-

correlation peak that corresponds to the difference betweenfirst (direct path) transient arrivals in the presence of multipath

signal propagation. In addition to the cross-correlation between

the two receivers it uses the two auto-correlation functions and

assumes that the received signals contain only replicas from a

single transient of unknown waveform and, also, that the spatial

coordinates of the multipaths are unknown or are impractical

to estimate.

In this way, at the time ti only one measurement per a pair ofreceiving sensors is selected. Consequently, there are Ri TDOAmeasurements {Zi,j}Rij=1 at time ti obtained using Ri + 1sensors. Under the pth target assumption the measurement Zi,jis given by

Zi,j = i,j,p + ei,j (5)

where i,j,p is the exact TDOA for the pth target motion model(see Eq. (3)). In the cases where the measurement Zi,j iscorrectly taken using the transients that propagated along the

direct paths, the TDOA measurement error ei,j is modelledas an independent random variable (rv) that is distributed

according to a two-component Gaussian mixture pdf

p(ei,j) = (1 i,j)N(ei,j ; 0, (1)i,j ) + i,jN(ei,j; 0, (2)i,j ) (6)where N(x; , ) denotes a Gaussian (normal) pdf in variablex with mean and standard deviation (std) , and i,j 1defines the proportion of the two pdfs in the mixture. The

stds are related by (2)

i,j > (1)

i,j . In some situations Zi,j maycontain a gross measurement error. It may be far removed from

the exact TDOA i,j,p so that the corresponding error ei,j isnot distributed according to the pdf in Eq. (6). This may occur

in the cases where e.g., overlapping multipath reflections are

received by the sensors.

As a result, for the N transients, a set of cumulative (batch)measurements Z = {Zi,1, Zi,2, . . . , Z i,Ri}Ni=1 is obtained.It contains both the target related TDOA measurements and

clutter. Clutter is assumed to arise under one of the following

conditions: 1) if an existing transient is correctly detected by

the transient detection algorithm but it originates from a source

other than the desired targets (considered random) (clutter

type 1); 2) if the transient detection algorithm recorded thedetection when no transient was present (transient false alarm)

(clutter type 2); 3) if an existing target-related transient is

correctly detected, but the TDOA measurement between a pair

of sensors resulted in a gross error (clutter type 3).

Our goal is to estimate position and velocity of the P targetsusing the cumulative measurements Z. However, there is noprior information about the origin of the measurements nor

of the position and velocity of any of the targets. In addition

to the specific TDOA measurement characteristics described

above, the computational problems are related to the fact that

the TDOA measurement errors are non-Gaussian and that the


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relationship between the TDOA measurements and the target

motion parameters is highly nonlinear. Under these limitations

the following algorithm is deemed to be suitable.

III. THE ALGORITHM FOR MULTIPLE TARGET

LOCALIZATION USING TDOAS OF TRANSIENTS

In this section a likelihood function of the observed data

is first formulated. Using this formulation a DAEM algorithmfor multiple target localization is derived. Finally, a stochastic

modification of the DAEM algorithm with an aim to increase

the robustness with respect to unknown initial conditions and

number of false measurements in the data set is presented.

A. Likelihood Structure of the Observed Data

The cumulative measurements Zrepresent incomplete (ob-served) data [22], [18]. The complete data can be obtained

by associating with the batch Z a set of discrete valuedindices or labels that uniquely assign each measurement to

one of the targets or to clutter. However, these assignment

indices are unobservable and are called missing data. The

standard approach described in [18], [21] estimates the motion

parameters for the P targets and the missing assignment indicesby maximizing a likelihood function of the observed data Z.The derivation of the likelihood function used by the standard

method is presented in Appendix I. An alternative approach

that is tailored more specifically to our problem is discussed

next.

We first note that the Gaussian mixture model for the

measurement error ei,j at time ti implies that ei,j can be

generated either from the normal pdf N(ei,j; 0, (1)i,j ) or fromthe pdf N(ei,j ; 0, (2)i,j ), with probabilities (1 i,j) and i,j ,respectively (see Eq. (6)). Therefore, two models for the

measurement Zi,j conditioned on i,j,p are possible, and thesecorrespond to each of the Gaussian pdfs N(Zi,j; i,j,p, (k)i,j ),k = 1, 2.

Consequently, we introduce a set of discrete valued as-

signment indices L = {li,1, li,2, . . . , li,Ri}Ni=1 and define thecomplete data by {Z, L}. The set L is unobservable andrepresents missing data. Each index li,j is modelled as a rvthat takes a value from the discrete set {n : 1 n 2P + 1}. It expresses an assignment hypothesis at time ti asfollows. Setting li,j = 2p 1, p = 1, 2, . . . , P , assigns themeasurement Zi,j to the pth target motion model, and chooses

the measurement model N(Zi,j ; i,j,p, (1)i,j ). Setting li,j = 2p,however, selects the pth target model and the measurementmodel N(Zi,j; i,j,p, (2)i,j ). The relative prior probabilities ofthe two assignments are (1 i,j) and i,j, respectively. Whenthe assignment index is set to ki,j = 2P+ 1, the measurementZi,j is assigned to clutter.

Let i = {i,j,1, i,j,2, . . . , i,j,2P+1}Rij=1 denote the proba-bilities associated with the assignment indices {li,j}Rij=1 for theith transient, where i,j,n = Prob[li,j = n]. Also denote by

= {1, 2, . . . , N} (7)the corresponding assignment probabilities for the batch Z.The complete data probability conditioned on the target motion

parameters V and the batch assignment probabilities isdefined by

P(Z, L |V, ) =Ni=1

Rij=1

i,j,n qi,j,n|n=li,j(8)

where

qi,j,n =

N(Zi,j; i,j,n, (1)i,j ) n = 2p 1, p = 1, . . . , P N(Zi,j; i,j,n, (2)i,j ) n = 2p, p = 1, . . . , P

j n = 2P + 1(9)

and where j is a constant that depends on the sensor pair j.The observed data pdf conditioned on the parameters V

and is computed as the marginal distribution (over allpossible measurement to target/clutter assignments in L) ofP(Z, L |V, ) in Eq. (8), i.e., as

P(Z |V, ) =L

P(Z, L |V, ) =N

i=1Ri

j=12P+1

n=1i,j,n qi,j,n.

(10)It is required that the marginal pdf P(Z |V, ) and the pdf

P(Z |V, ) in Eq. (37) are equal as they are derived for thesame data. This condition is satisfied when the assignment

probabilities i and i are related as follows

i,j,2p1 = i,p (1 i,j) (11)i,j,2p = i,p i,j (12)

for p = 1, 2, . . . , P , and, also, when i,j,2P+1 = i,P+1.Eqs. (11) and (12) ensure that the measurement-to-target as-

signment probabilities for all measurements {Zi,j}Rij=1 at timeti as per Eq. (10) are effectively equal to i,p in Eq. (37).

In this way the probability that the jth measurement at timeti, Zi,j, originates from the target model p is equal for all

measurements {Zi,j}Rij=1 taken at that time, that is, theseprobabilities are independent of the index j. Besides, since theprobabilities of clutter measurement i,j,2P+1 do not dependon the measurement index j, it also follows that

i,j,2P+1 = i,2P+1 = i,P+1 for j = 1, . . . , Ri. (13)

B. The DAEM Algorithm for Multiple Target Localization

The proposed algorithm computes a ML estimate of the

motion parameters for the P target models V and the batch

assignment probabilities = {ti}N

i=1. It applies the DAEMalgorithm [35], [36] which is an iterative procedure that

attempts to minimize the effective cost function based on

thermodynamic free energy given by

F(Z |V, ; ) = 1

logL

P(Z, L |V, ) . (14)

The iterative minimization is performed similarly as the max-

imization of the incomplete log-likelihood function in the

conventional EM algorithm [22]. Additionally, it incorporates

an annealing loop due to the dependance of the cost function

in Eq. (14) on the global control parameter that is, by


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analogy to statistical mechanics, interpreted as the inverse of

the computational temperature [35].

Each iteration of the DAEM algorithm comprises an expecta-

tion step (E step) and a maximization step (M step). For brevity

we denote the parameters to be estimated by = {V, }. Alsodenote by m the parameters estimated during the mth iterationobtained via the DAEM algorithm. The iteration m m+1

is carried out as follows:E-step: Compute the expectation Q( | m ; ) as a function

of m defined by

Q( | m; ) =L

log[P(Z, L | )] P(L |Z, m ; ). (15)

M-step: Find m+1 that maximizes Q( | m ; ).The function P(L |Z, ; ) in Eq. (15) represents the poste-

rior distribution of the missing data L given the observed dataZand the parameters and . Using the analogy to statisticalmechanics and the principle of maximum entropy the posterior

P(L |Z, ; ) is defined by [35], [36]

P(L |Z, ; ) = P(Z, L | )

L P(Z, L | ). (16)

This distribution depends on the parameter as follows.For = 0 the temperature is high and the distributionP(L |Z, ; ) is uniform. At a low temperature, when = 1,P(L |Z, ; ) reduces to standard posterior and the DAEMalgorithm is equivalent to the EM algorithm. For 0 < < 1an increase of means a change in the form of P(L |Z, ; )from uniform to the standard posterior. Usually, is initiallyset to a small value min 0 (for m = 0) and is graduallyincreased in each iteration.

Using Eqs. (8) and (10), and substituting m = {Vm, m},it can be shown that Eq. (16) produces

P(L |Z, m; ) =Ni=1

Rij=1

wmi,j,n|n=li,j (17)

where

wmi,j,n =(mi,j,n q

mi,j,n)

k(

mi,j,k q

mi,j,k)

(18)

denotes the conditional posterior probability that the measure-

ment Zi,j originates from the target/measurement model n,

conditioned on the measured data, the parameters m

estimatedin the mth iteration, and the global control parameter . InEq. (18), qmi,j,n for n {2p 1, 2p} is a Gaussian pdf givenby

qmi,j,n =N(Zi,j; mi,j,p, (n)i,j ) (19)

where mi,j,p is a function of the pth target motion parametersV

mp estimated in the mth iteration (see also Eqs. (3), (8), (9)).

The computation of the expectation Q( | m; ) in the E-step is analogous to the procedure described in [18], [21].

Accordingly this function is broken into several independent

sub-functions by separating the variables in the parameter

vector = {V, } as follows

Q( | m; ) =Ni=1

Q(i | m; ) +P

p=1

QV(Vp | m; ) + O(20)

where

Q(i | m; ) =Rij=1

2P+1n=1

wmi,j,n log i,j,n (21)

for i = 1, 2, . . . , N ,

QV(Vp | m; ) =Ni=1

Rij=1

2pn=(2p1)

wmi,j,n log qi,j,n (22)

for p = 1, 2, . . . , P , and O is a remainder that dependson

(q)i,j,n and j. In this way the M-step of the algorithm

is decoupled into a maximization problem for each set of

assignment probabilities i = {i,j,1, i,j,2 . . . , i,j,2P+1}Rij=1at the time ti, i = 1, 2, . . . , N , and for each target motionmodel Vp, p = 1, 2, . . . , P .

The maximization ofQ(i | ; ) in Eq. (21) with respect toi is constrained by the requirement that

2P+1n=1 i,j,n = 1 for

j = 1, 2, . . . , Ri. Additional constraints are given by Eqs. (11)and (12) for p = 1, 2, . . . , P and j = 1, 2, . . . , Ri. The detailsof the maximization of Q(i | ; ) under these constraints arepresented in Appendix II. The resulting update equations for

the assignment probabilities in i are given by

m+1i,j,2p1 =1 i,j

Ri

Ris=1

2pn=2p1

wmi,s,n (23)

m+1i,j,2p = i,jRi

Ris=1

2pn=2p1

wmi,s,n (24)

for the pth target motion model, p = 1, 2, . . . , P , and for j =1, 2, . . . , Ri, and by

m+1i,2P+1 =1

Ri

Ris=1

wmi,s,2P+1 (25)

for clutter.

The update for the pth target motion parameter vector Vpis defined by

V

m+1

p = argmaxVp QV(Vp | m

; ). (26)

Rearranging Eq. (22) yields

QV(Vp | m; ) Ni=1

Rij=1

logN(Zi,j; i,j,p, mi,j,p) (27)

whereN(Zi,j ; i,j,p, mi,j,p) is a Gaussian with the effective std

mi,j,p =

wmi,j,2p1

((1)i,j )

2 +wmi,j,2p

((2)i,j )

2

1/2

. (28)


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The update for the pth target motion parameter vector Vm+1pcan be obtained by using a standard weighted nonlinear LS

procedure, such as the Levenberg-Marquardt algorithm [42].

This algorithm iterates between the computation of the

conditional posterior probabilities wmi,j,n in Eq. (18) usingthe current parameter estimate m = {Vm, m} and thecomputation of the updated parameter estimates m+1 =

{Vm+1

, m+1

} using the probabilities wm

i,j,n as per Eqs. (23)-(28). In each iteration the computational temperature 1/ isdecreased by a small value.

Remark 1: To understand the relationship between the

DAEM algorithm and the variance deflation approach [18],

[34] consider the posterior probability of missing data

P(L |Z, m; ) in Eqs. (17)-(18). The numerator of the ex-pression in Eq. (18), for a case where n = 2p 1, can berewritten as follows (similar expression can be obtained for

n = 2p)

(mi,j,2p1 qmi,j,2p1)

=

mi,j,2p1

2(1)i,j

exp

(Zi,j mi,j,p)

2

1

(1)i,j

2

.

(29)

The denominator of the expression in Eq. (18) also consists

of the subparts of the form as in Eq. (29). It can be seen

that the variance in the exponent in Eq. (29) is multiplied

by 1/. Since 1/ is initially a large number and graduallydecreases in each iteration, this, similar to the variance deflation

method, has an effect of initially increasing the measurement

variance. Consequently, the distribution P(L |Z, m; ) isuniform (flat) for high temperatures, so all data configurations

defined by L are equally probable. This makes the algorithmsless dependant on the initial conditions. However, the approach

based on variance deflation increases variances used in the

maximization of Eq. (27), whereas the DAEM algorithm does

not. Besides, contrary to the DAEM algorithm that has a

theoretical justification as its cost function is derived based on

statistical mechanics and the principle of maximum entropy

[35], [36], variance deflation is an ad hoc method.

Remark 2: The conventional EM algorithm in [22], obtained

by setting = 1 in Eq. (18) in all iterations, can not be appliedto our problem. The relationship between the TDOA measure-

ments and the target motion parameters is highly nonlinear

and the values of the TDOA measurement error variances in

Eq. (6) are very small. As a consequence, using, in Eq. (18),

the parameter values that have even a small offset from the

true target motion parameters for = 1 results in setting the

exponential functions in Eq. (29) to zero, deeming Eq. (18)undetermined. For similar reasons the DAEM algorithm is

not applicable to the maximization of the standard likelihood

function P(Z |V, ) in Eq. (37) that is based on Gaussianmixture pdfs.

C. Modified DAEM Algorithm for Multiple Target Localization

The algorithm described in the previous section usually

attains the desired ML solution when the initial values of the

parameters are chosen to be close to this solution. As the

temperature 1/ gets lower, the solution surface becomes morecomplicated, and the number of saddle points and suboptimal

maxima increases. In the cases where the initial guess is poor

(e.g., when the initial values are set at random) this algorithm

may not obtain the desired global optimum.

In order to make this method more robust for different initial

conditions we propose the MDAEM algorithm that randomizes

the DAEM algorithm by using stochastic imputations [39]. In

each iteration the algorithm generates pseudo-complete data

by sampling from a posterior distribution conditioned on themeasurements and the current approximation of the parameters.

This data is used to obtain the updated parameter values in the

subsequent iteration.

The motivation for the MDAEM algorithm comes from the

papers that describe stochastic imputation methods based on

the EM algorithm [43], [44]. Celeux and Diebolt [43] apply

a single stochastic imputation, and are concerned with the

problem of estimating parameters from finite mixture densities.

The approach described in [44] utilizes multiple imputations

to compute the expectation in the E-step of the EM algorithm,

followed by the maximization step. This approach is useful in

situations where the computation of the expectation function

is analytically intractable.The pseudo-complete data in the MDAEM algorithm is

generated based on the assumption that the signals received by

the sensors at the time ti contain replicas of transient emissionfrom a single target (this assumption is also used by the TDOA

measurement process described in Section II-B). In this way all

TDOA measurements {Zi,j}Rij=1 at the time ti are associatedwith the same target. The probability that the measurement

subset {Zi,j}Rij=1 is generated by the pth target is computed as

umi,p =1

Ri

Rij=1

wmi,j,2p1 + w

mi,j,2p

for p = 1, 2, . . . , P ,

(30)and as umi,P+1 =1Ri

Rij=1 w

mi,j,2P+1 for clutter. An auxiliary

set of target assignment indices G = {gi}Ni=1 is next introduced.In the mth iteration of the algorithm an index gmi = p,gmi Gm assigns the measurements {Zi,j}Rij=1 to the pthtarget model. The value of the assignment index gmi Gm israndomly simulated from the distribution umi,p, p = 1, . . . , P +1in Eq. (30) and this is done for all i, i = 1, 2, . . . , N .

The resulting assignments are used to obtain the functions

QV(Vp | m; ) in the E-step of the algorithm as follows

QV(Vp | m; ) iSmpRi

j=1logN(Zi,j; i,j,p, mi,j,p) (31)

where Smp = {r : gmr = p}. That is, only the measurementsassigned to the pth target motion model via Gm are used tocompute the new estimate Vm+1p . These measurements are now

assigned to the pth target motion model with probability 1:they can either belong to one of the target/measurement models

associated with the pth target or they are clutter obtained as theresult of a gross measurement error (clutter type 3, as described

in Section II-B). This implies that the probabilities wmi,j,2p1and wmi,j,2p used to compute the effective stds of the Gaussiansmi,j,p in Eq. (31) need to be appropriately renormalized.Equivalently, using the existing probabilities wmi,j,n computed


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7

1. Initialization: Set m = 0, = minSet initial value 0

2. Iteration m m+1E-step: For all i, j, and n compute wmi,j,n using Eq. (18)M-step: Compute update m+1

Update for {i}Ni=1 for all i, compute m+1i using Eqs. (23)-(25)

Update for{Vp}

P

p=1 generate Gm = {gmi }Ni=1 by randomly simulatingsamples from umi,p

for p = 1, . . . , P if Smp = {r : gmr = p} =

compute Vm+1p using Eqs. (26), (31)-(32)

else set Vm+1p = Vmp

3. Set m = m + 1; increase 4. If > max end procedure

Else go to 2.

Fig. 1. Pseudo-code of the MDAEM algorithm for motion parameterestimation of P targets in clutter.

as per Eq. (18) the effective stds can be obtained as

mi,j,p =

2p

s=2p1

wmi,j,s + wmi,j,2P+1

1wmi,j,2p1(

(1)i,j )

2 +wmi,j,2p

((2)i,j )

2

1/2

(32)

for all {(i, j) : i Smp , j = 1, 2, . . . , Ri} and p = 1, 2, . . . , P .Remark 3: The values (1/mi,j,p)

2 act as weights in the LS

procedure used to maximize the function QV(Vp | m; ) inEq. (31). It can be seen from Eq. (32) that, for the correctly

estimated motion parameters of the pth target, the weight that

corresponds to a clutter measurement assigned to the pth targetwill be close to zero, since in this case wmi,j,2p1 0 andwmi,j,2p 0.

Remark 4: In a situation where the true number of targets

Pt is not known the number of targets used by the algorithmP should be selected so as to be greater than the maximumexpected number of targets in the data, P Pt. The algorithmestimates target positions and velocities for the Pt true targetsand for the PPt dummy targets. If the pth target is a dummytarget it may happen that in the iteration m m+1 nomeasurement is associated with its estimated parameters using

stochastic imputation, so that Smp = . Consequently, for thistarget the parameters Vp are not updated but the estimates

obtained in the previous iteration are retained, that is Vm+1p =V

mp .

The processing steps of the proposed algorithm are summa-

rized in Fig. 1. The assignment probabilities i are computedusing the update formulas in Eqs. (23)-(25). The details of the

procedure used to randomly simulate the auxiliary assignment

index set Gm in the mth iteration of the algorithm are shownin Fig. 2.

The sequence {m}m0 produced by the algorithm is anirreducible, inhomogeneous Markov chain [39]. If{m}m0 isalso ergodic then its distribution converges to a unique station-

ary distribution of this Markov chain. Owing to the non-linear

For i = 1 to N do {For p = 1, 2, . . . , P compute probabilities umi,p

using Eq. (30)

Generate a uniform random variable u on {0, 1},u U(0, 1)

Set Q = 0, p = 0Repeat {

set p = p + 1compute Q = Q + umi,p

} until Q uSet gmi = p

}

Fig. 2. Simulation of the assignment indices Gm in the mth iteration of thealgorithm.

TABLE I

TARGET POSITIONS AT THE TIME tsq , q = 1, 2, 3, AND VELOCITIES FOR THE

TRUE TARGETS MOTION MODELS USED IN THE SIMULATIONS.

Target tsq X(tsq) Y(t

sq) Z(t

sq) vx vy vz

q (s) (m) (m) (m) (m/s) (m/s) (m/s)1 0 4000 -3000 -1900 -3.0 5.5 0.1

2 270 -3000 -400 -1200 4.2 3.8 -0.2

3 420 -5000 1200 -50 5.0 -4.0 0.0

maximization of the expression in Eq. (31) and the presence

of clutter in the measurements, ergodicity of this chain is very

hard to study. The results obtained by numerical simulations

using test geometries with Pt {1, 2, 3} targets indicate thatwith the appropriately chosen cooling schedule and for random

initial conditions this algorithm usually converges to the desired

optimal solution. Some of these results are presented in the

following section.

IV. NUMERICAL SIMULATIONS AND DISCUSSION

Three target motion models are used in the numerical

simulations. For each model the transient emission starts at a

different time instant tsq , q = 1, 2, 3. The positions of the targetsat the times tsq and their velocities are listed in Table I. Thecoordinate system is oriented such that xy plane overlaps withthe water surface and negative z axis corresponds to depth.The simulations are performed for the test scenarios where

Pt {1, 2, 3} targets are observed in clutter. The test scenariowith Pt = 1 target uses the motion model q = 1 (Target 1) in

Table I; for the scenario with Pt = 2 targets the motion modelsare q = 1 and q = 2 (Targets 1 and 2), and in the case wherePt = 3 the motion models are q = 1, q = 2 and q = 3.

An array of five receiving sensors is used in the time delay

analysis. The sensors are stationary and are positioned at

U1 = [0, 0, D2]T, U2 = [D1, D1, D1 + D2]T, U3 =

[D1, D1, D1 + D2]T, U4 = [D1, D1, D1 + D2]T andU5 = [D1, D1, D2 D1]T where D1 = 7 0/

3) and

D2 = 300 m.For each test scenario the transient emissions that correspond

to the qth target, q {1, . . . , P t}, start at tsq and the timeincrements between two consecutive emissions are randomly


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8

drawn from the interval [tmin, tmax], tmin = 10 s,tmax = 180 s. All TDOAs are measured with respect tothe reference sensor U1 and the number of measurements at

the time ti is Ri = 4. The velocity of sound in water is takento be c = 1500 m/s. The observation area is limited to have amaximum range of 6000 m relative to the reference sensor andconstant depth z = 2300 m. The parameters of the TDOAerror distribution (see Eq. (6)) are:

(1)

i,j = (1)

= 40 s and(2)i,j = 200 s, and i,j = 0.20. The measurements that fall

outside the observation area are discarded so the real number

of transients is NTq 12 for q = 1, . . . , P t.A target-originated data set obtained in this way is combined

with a set of clutter TDOA measurements based on real tran-

sients emitted by sources other than the desired targets (clutter

type 1 described in Section II-B). The total number of clutter-

related emissions per km3 per hour is taken to have a Poisson

distribution with mean . These emissions occur randomly inthe time interval that starts 8 min. before and finishes 8 min.

after the observation time for the target-originated measurement

set, and the corresponding source locations are taken to be

uniformly distributed within the observation area. Each emis-

sion of this type is assigned TDOA measurements computed as

per Eqs. (3) and (4). Clutter type 3 is simulated by assuming

that every measurement in a target-related measurement set

can result in a gross error with probability . Clutter type 2caused by false transient detections is not considered. Such

measurements can be easily identified and removed as they

usually do not correspond to meaningful physical locations.

The MDAEM algorithm estimates P target motion models.No previous knowledge of the motion parameters is assumed.

Therefore, on initialization the target position at the reference

time t, Xp(t), and target velocity vp for the pth motion

model, p {1, . . . , P }, are randomly selected from appropriateuniform distributions. In particular, x and y components of theposition are taken to be uniformly distributed within 3000 mto 3000 m, the z component within 0 m to 2300 m, and eachcomponent of the velocity within 5 m/s to 5 m/s. Similarly,initially, assignments to any of the target models or to clutter

are assumed to be equally probable.

Lowering the computational temperature (which is inverse

to the control parameter in Eq. (18)), or cooling, in theMDAEM algorithm should be performed relatively slowly to

enable search through the entire solution space and to allow for

the convergence to globally optimal solution. In this paper, in

the mth iteration of the algorithm, the parameter is computed

as = min(m + 1)

2 m = 0, 1, . . . (33)

is limited to min < max 1, where min =C1(

(1))2

, C1 = const. The algorithm usually attains thesteady state condition for < 1 so the computation can bestopped before = 1 is reached. The stopping criterion isdefined when > max = C2 (

(1))2 where C2 = const,C2 C1. In this paper, C1 = 3 and C2 = 22 106. Theparameter j in Eq. (9) is set to 10

7 and the reference time

at which the target positions Xp(t) are estimated is t = 900s.

The nonlinear LS procedure based on the Levenberg-

Marquardt algorithm [42] is used in each iteration of the

MDAEM algorithm to compute the target motion parameters

by maximizing the expression in Eq. (31). This LS algorithm

is in itself iterative and requires convergence to obtain the

solution. In our application, however, it is not necessary for

this procedure to converge, that is, only few passes through

this algorithm in each iteration of the MDAEM algorithm are

sufficient for the overall convergence.A simulation that corresponds to the scenario with Pt =

3 targets is shown in Fig. 3 (a). The figure shows thetarget-related transient-emission positions and clutter-related

emissions generated as described above. The noisy TDOA

measurements Zi,2 and Zi,4, i = 1 . . . , N are shown inFig. 4. The probability of gross measurement error is =0.1. The MDAEM algorithm is run by setting the numberof the estimated target motion models to P = Pt = 3. Thetarget trajectories obtained using the true target positions and

velocities and the trajectories estimated by the algorithm are

shown in Fig. 3 (b). True and estimated positions of the targets

at the reference time t are denoted by the star symbols. Also,

the symbol denotes the starting position of the target (trueand estimated) at the time tsq, for q = 1, 2, 3.

Often the number of targets Pt in the measurement set is notknown. In this situation the number of target models estimated

by the algorithm, P, should be chosen so as to be equal orgreater then the maximum expected number of targets, so that

P Pt. The proposed algorithm performs robustly under suchmodel mismatch conditions. It estimates the motion parameters

for the Pt true targets in addition to computing parameters forthe P Pt dummy targets.

An example where P = 4 target motion models are esti-mated by the MDAEM algorithm using a data set that contains

measurements from Pt = 1 target is shown in Table II. Themeasurements are derived using N = 53 transients of which 12are related to the target and the rest are clutter type 1 simulated

as described above. The probability of gross measurement error

is = 0.1. The last row in Table II shows the true targetparameters. It can be seen that the estimated parameters for

the target model p = 2 are close to the true target parameterswhile the parameters for other models are far removed.

The measurement-to-target assignment probabilities that cor-

respond to the estimated target models p = 1, . . . , 4 inTable II and the measurement-to-clutter assignment proba-

bilities computed by the MDAEM algorithm are shown in

Figure 5 (a) and (b), respectively. In both figures the abscissa

denotes the measurement index i (in time), i = 1, 2, . . . , N ,and the vertical lines mark the position of the target-related

measurements. For the pth estimated target motion model theassignment probabilities shown in Figure 5 (a) are computed

as i,j,2p1/(1 i,j) for all i (see Eq. (11)). As discussedin Section III-A these probabilities do not depend on index j.It can be seen from Figure 5 (a) that the target assignment

probabilities that correspond to the model p = 2 are large forthe target-related measurements. They have value 1 for those

time indices where the TDOA measurements from all sensor

pairs associated with one target-related transient emission are

correct and value 0.75 in the cases where one of the TDOA


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9

60004000

20000

20004000

6000

60004000

20000

20004000

6000

2200

2000

1800

1600

1400

1200

1000

800

600

400

200

0

X (m)Y (m)

Z(m)

Target 1Target 2Target 3ClutterSensors

(a)

60004000

20000

20004000

6000

6000

400020000

20004000

6000

2200

2000

1800

1600

1400

1200

1000

800

600

400

200

0

X (m)Y (m)

Z(m)

True Target 1Estimated Target 1True Target 2Estimated Target 2True Target 3Estimated Target 3

(b)

Fig. 3. Simulation that corresponds to the tracking scenario with Pt = 3targets: (a) true target trajectories; (b) true and estimated target trajectories.True and estimated positions of the targets at the reference time t are denotedby the star symbol. The symbol o denotes the target position at the time tspfor p = 1 , 2, 3.

measurements has a gross measurement error. The models

p = 1, p = 3 and p = 4 correspond to dummy targets.They are associated only with a few (clutter) measurements

in the data set and the association probabilities are in most

cases small. Figure 5 (b) shows the measurement-to-clutter

assignment probabilities i,2P+1 estimated by the algorithmfor this data set.

A. Comparison to Other Methods

The standard PDA method and the methods based on the EM

algorithm that can be applied to this problem are very sensitive

to initial conditions and would require a similar cooling scheme

in order to assure convergence to a global optimum. Other

approaches that treat related, but not identical, problems such

as those described in [30], [31], [32], [33] are also inapplicable.

Recently, Vo et al. [27], [28] and Ma et al. [29] describe

a method for the localization of unknown number of speakers

500 0 500 1000 1500 2000 25000.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04

0.05

Time (s)

Zi,2

(s)

Target 1Target 2Target 3Clutter

(a)

500 0 500 1000 1500 2000 25000.05

0.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04

Time (s)

Zi,4

(s)

Target 1Target 2Target 3Clutter

(b)

Fig. 4. The TDOA measurements used to obtain the results in Fig. 3 (b):(a) {Zi,2}Ni=1 (b) {Zi,4}

Ni=1

. Shown are both clutter and the target relatedmeasurements where = 0.1.

TABLE II

TARGET POSITIONS AT THE REFERENCE TIME t

AND VELOCITIES FOR

P = 4 TARGETS MOTION MODELS ESTIMATED BY THE MDAEM

ALGORITHM WHEN THE TRUE NUMBER OF TARGETS IS Pt = 1 . THE TRUE

TARGET PARAMETERS ARE SHOWED IN THE LAST ROW.

Target Xp(t) Yp(t) Zp(t) vx vy vz

p (m) (m) (m) (m/s) (m/s) (m/s)

1 4495.2 1165.7 -3960.8 7.207 9.212 -5.014

2 1317.4 1976.1 -1830.1 -3.035 5.570 0.097

3 -792.1 807.1 -743.2 0.581 1.877 -2.907

4 2213.8 -519.6 1216.2 -1.677 -7.232 7.654

true 1300 1950 -1810 -3.0 5.5 0.1

in multipath environment using TDOA measurements. They

employ the random finite set (RFS) theory and a SMC imple-

mentation to develop a Bayesian RFS filter that simultaneously

tracks the time-varying speaker location in 2-D and the number

of speakers. This approach treats a problem similar to the one

considered in this paper and offers a possibility of an extension

to tracking targets with non-constant velocities. Given that our

measurements are sparse in time, computationally extensive

tracking algorithms based on particle filters [45] could be

applied. However the approach in [28], [29] assumes a known

environment that is confined to a room enclosure. It uses a


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10

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Measurement Index

Target

AssignmetProbabilities

Model 1

Model 2

Model 3

Model 4

(a)

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Measurement Index

ClutterAssign

metProbabilities

Clutter

(b)

Fig. 5. Assignment probabilities estimated by the MDAEM algorithm forP = 4 target motion models using a data set that contains measurementsfrom Pt = 1 target in clutter: (a) target assignment probabilities for the targetmodels p = 1, . . . , 4 (b) clutter assignment probabilities. Vertical lines markthe position of the target-related measurements.

Gaussian pdf for the TDOA measurement error and does not

apply Gaussian mixture model. Other important differences

between our problem and the problem described in [28], [29]

are related to the TDOA measurement process as discussed in

Section II-B. Most notably, the approach in [28], [29] assumes

regularly spaced measurement update times, whereas the up-

date times in our application are random and the measurements

from different targets are asynchronous. For these reasons it is

conjectured that the approach in [28], [29] can not be applied

to our data without additional adjustments.

B. Average Performance of the MDAEM Algorithm

We test the ability of the MDAEM algorithm to correctly

estimate motion parameters for Pt targets in a situation wherethe true number of targets in the data set is not exactly known.

The number of the estimated target models P is taken tobe greater than the maximum expected number of targets, so

that P > Pt. For all tracking scenarios with Pt {1, 2, 3}targets, the MDAEM algorithm is run by setting the number

of estimated target motion models to P = 4. For each scenarioaverage performance of the algorithm is obtained by using

100 sets of target-related measurements generated as described

above. These sets are combined with clutter measurements that

are simulated for several different values of the parameters (for clutter type 1) and (for clutter type 3). For each runthe best fitted results are selected by comparing the parameters

estimated by the algorithm to the known parameters that corre-

spond to the true targets. In particular, mean (per sample) abso-

lute distances (MADs) are computed between the P trajectoriesthat correspond to the estimated target motion models and

the trajectories obtained using the Pt true target parameters.The MADs are computed using the distances evaluated for

every target-related measurement time (for more information

on how MAD is computed see [13]). The known true target

motion models are then associated with those estimated target

parameters for which the corresponding MADs are found to

be minimized.

The means of the MADs related to the estimated target

models associated with the true targets, over the 100 setsof simulated measurements and for the scenarios with Pt {1, 2, 3} targets, are shown in Tables III-V. The results arepresented as a function of the parameters and . For each

value of the parameter Tables III-V also show the meannumber of clutter points (MNC) (for clutter type 1) in the

simulated batch measurements over the 100 data sets.

A ML estimate of the motion parameters is computed for

each of the Pt {1, 2, 3} targets in a data set where themeasurement to target assignments are assumed to be exactly

known, i.e., where only the (correct) measurements generated

by a specific target are used to estimate the corresponding

motion parameters. This technique is denoted as the indepen-

dent ML estimation (IMLE). The results obtained using the

IMLE are compared to the results obtained using the proposed

MDAEM algorithm.

The optimization in the IMLE is done using constrainednonlinear optimization [42]. This technique is also very sen-

sitive to the initial parameter values that are, similarly as for

the MDAEM algorithm, determined randomly. To increase the

accuracy of the estimation this technique is applied iteratively

in conjunction with variance deflation.

The last row in Tables III-V shows the mean MAD computed

for different tracking scenarios using the IMLE. It can be

seen from Tables III-V that the results obtained using the

MDAEM algorithm are close to those obtained using the IMLE.

Moreover, the estimation accuracy of the MDAEM algorithm

does not change much with the parameters and .

The results of the experiments show that the MDAEM

algorithm for multiple target localization performs robustly

under the above model mismatch conditions (see also Table II).

It provides correct estimates of the Pt target motion modelsthat correspond to the true targets, in addition to estimating

parameters of P Pt dummy targets. In a more generalsituation these estimates can be passed on to an appropriately

designed detection algorithm that can be used to determine

the number of true targets in the data set and to select the

corresponding motion parameter estimates. A technique that

can verify the presence of a single target in the data set has

recently been proposed in [13]. This technique can be extended

and used for the detection in scenarios with multiple targets.


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11

TABLE III

MEA N MAD FOR INDIVIDUAL TARGETS COMPUTED USING 100

SIMULATIONS THAT CORRESPOND TO THE TEST SCENARIO WITH Pt = 1

TRUE TARGET. THE ESTIMATION METHODS ARE THE MDAEM

ALGORITHM AND THE IMLE (LAST ROW).

Mean MAD (m) MNC Pt = 1

q = 1

0 0 0 -

0 1 105 18.44 83.58

0 2 105 31.94 86.45

0.1 0 0 89.38

0.1 1 105 18.78 93.07

0.1 2 105 33.51 99.32

- - - 74.23

TABLE IV



TRUE TARGETS. THE ESTIMATION METHODS ARE THE MDAEM


Mean MAD (m)

MNC Pt = 2

q = 1 q = 2

0 0 0 80.71 65.35

0 1 105 20.04 87.31 72.07

0 2 105 37.46 89.32 68.43

0.1 0 0 97.83 80.80

0.1 1 105 20.43 91.21 73.74

0.1 2 105 37.15 100.25 74.17

- - - 79.11 57.03

TABLE V



TRUE TARGETS. THE ESTIMATION METHODS ARE THE MDAEM


Mean MAD (m)

MNC Pt = 3

q = 1 q = 2 q = 3

0 0 0 83.18 74.78 93.01

0 1 105 19.95 90.22 77.59 91.67

0 2 105 39.93 89.20 76.20 94.16

0.1 0 0 93.59 77.86 99.13

0.1 1 105 19.93 96.29 83.13 95.15

0.1 2 105 40.43 90.10 83.44 100.73

- - - 76.79 62.00 86.06

V. CONCLUSION

Under consideration has been the problem of estimating

motion parameters of multiple targets using TDOA measure-

ments based on target-emitted transient signals. The targets

are moving linearly in a three-dimensional (3-D) observation

area contaminated by clutter. The TDOA measurement errors

are modelled as having (possibly different) Gaussian mixture

probability density functions. An iterative ML optimization

technique based on a deterministic annealing version of the

EM algorithm is applied to this problem. For each target a ML

estimate of the target motion parameters is obtained using a

nonlinear LS method. The proposed algorithm generalizes the

variance deflation method previously used for the initialization

of several target tracking algorithms.

The performance of the proposed algorithm is tested using

simulated measurements related to tracking scenarios with one,

two and three targets. The number of targets in the measure-

ment data is supposed not to be known and the algorithm is

run under the conditions that the number of estimated targets

is greater than the number of the true targets. The simulationresults are presented for several different clutter densities and

probabilities of gross error of the target related measurements.

The proposed algorithm performs robustly under the model

mismatch conditions. The results are found to be comparable

to the estimation results obtained when the measurement-to-

target assignments are exactly known.

APPENDIX I

Denote by {Z, K} the complete data where K ={ki,1, ki,2, . . . , ki,Ri}Ni=1 are (unobservable) discrete valuedassignment indices associated with the batch measurements Z.The indices {ki,j}Rij=1 K are modelled as independent andidentically distributed rvs that take a value from the discrete set

{m : 1 m P + 1}. Following the standard method [18],[21], each index ki,j assigns the corresponding measurementZi,j either to the target motion model m, m = 1, 2, . . . , P (when ki,j = m) or to clutter (for ki,j = P+1). Next define byi = {i,1, . . . , i,P+1} the probability vector associated withthe assignment indices {ki,j}Rij=1, where i,m = Prob[ki,j =m], and where [18]

{i,m 0}P+1m=1 andP+1

m=1

i,m = 1. (34)

Also, denote the assignment probabilities for the batch Zby = {i}Ni=1. The probability i,m that the jth measurementat time ti, Zi,j, originates from the model m is equal for all

measurements {Zi,j}Rij=1 taken at that time.The complete data probability of{Z, K}, under the assump-

tion that the motion parameters V and the batch assignment

probabilities are known, is given by

P(Z, K |V, ) =Ni=1

Rj=1

i,m pi,j,m|m=ki,j(35)


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12

where pi,j,m is the measurement model pdf corresponding tothe target assignment m, m = 1, 2, . . . , P ,

pi,j,m = p(Zi,j; i,j,m) = (1i,j)N(Zi,j; i,j,m, i,j(1))+i,jN(Zi,j ; i,j,m, (2)i,j ).(36)

For clutter assignment ki,j = P + 1 and the correspondingprobability is pi,j,P+1 = j , where j is a constant.

The marginal distribution of P(

Z,

K |V, ) over

Kis

defined as

P(Z |V, ) =K

P(Z, K |V, ) =Ni=1

Rij=1

P+1m=1

i,m pi,j,m.

(37)

APPENDIX II

The maximization of the function Q(i | ; ) in Eq. (21)with respect to i = {i,j,1, i,j,2, . . . , i,j,2P+1}Rij=1 is con-strained by the requirement that

2P+1

n=1

i,j,n = 1, for j = 1, 2, . . . , Ri (38)

i.e., these probabilities must sum to unity, as each represents the

fraction of the measurement produced by a particular model.

Additional constraints are obtained by using Eqs. (11) and (12),

asi,j,2p11 i,j =

i,j,2pi,j

(39)

for p = 1, 2, . . . , P and j = 1, 2, . . . , Ri. From Eqs. (11) and(12) it also follows that

i,1,2p11 i,1 =

i,2,2p11 i,2 = . . . =

i,Ri,2p11 i,Ri

(40)

and

i,1,2pi,1

= i,2,2pi,2

= . . . = i,Ri,2pi,Ri

. (41)

Using the (P + 1)Ri constraints in Eqs. (38) and (39) theLagrangian for this maximization problem is formulated as

Li = Q(i | ; )+Rij=1

Pp=1

j,p [(1 i,j) i,j,2p i,ji,j,2p1]+Rij=1

j,P+1

1

2P+1n=1

i,j,n

(42)

where j,p are Lagrange multipliers. Differentiating the La-grangian with respect to i,j,2p1 and i,j,2p and setting theresults to zero yields

i,j,2p1 =wi,j,2p1

i,jj,p + j,P+1(43)

i,j,2p =wi,j,2p

j,P+1 (1 i,j) j,p (44)

for the P target motion models p = 1, 2, . . . , P . For the cluttermodel we have

wi,j,2P+1 = i,j,2P+1j,P+1. (45)

Summing both sides of Eq. (45) over j and using the assump-tion in Eq. (13) it follows that

i,2P+1 =

Rij=1 wi,j,2P+1

Rij=1 j,P+1

. (46)

We next eliminate i,j,2p1 and i,j,2p from Eqs. (43) and(44) using Eq. (39) and apply the result to express j,p as afunction of j,P+1 as follows

j,p = j,P+1i,jwi,j,2p1 (1 i,j)wi,j,2p

i,j(1 i,j)(wi,j,2p1 + wi,j,2p) . (47)

The substitution of Eqs. (43), (44), (46) and (47) in Eq. (38),

after some manipulation, results inP

p=1

(wi,j,2p1 + wi,j,2p) + j,P+1

Rij=1 wi,j,2P+1Rij=1 j,P+1

= j,P+1.

(48)

Summing both sides of this expression over j we obtain that

Rij=1

j,P+1 =

Rij=1

2P+1n=1

wi,j,n = Ri (49)

where we noted that2P+1

n=1 wi,j,n = 1 (according to Eq. (18)).We next substitute Eq. (47) in Eqs. (43) and (44) and obtain

j,P+1 i,j,2p11 i,j = j,P+1 i,j,2pi,j = wi,j,2p1 + wi,j,2p (50)Again, after summing Eq. (50) over j and by using Eqs. (40),(41) and (49), the expressions for the target assignment prob-

abilities i,j,2p1 and i,j,2p are obtained as

i,j,2p1 =1 i,j

Ri

Ris=1

2pn=2p1

wi,s,n (51)

i,j,2p =i,jRi

Ris=1

2pn=2p1

wi,s,n (52)

The expression for the probability of clutter assignment i,2P+1is obtained by substituting Eq. (49) in Eq. (46) as

i,2P+1 =1

Ri

Ris=1

wi,s,2P+1. (53)

REFERENCES

[1] G. C. Carter, Time delay estimation for passive sonar signal processing,IEEE Trans. Acoust., Speech, Signal Processing, vol. 29, no. 3, pp. 463470, 1981.

[2] C. H. Knapp and G. C. Carter, The generalized correlation methodfor estimation of time delay, IEEE Trans. Acoust., Speech, SignalProcessing, vol. 24, no. 4, pp. 320327, 1976.

[3] B. G. Ferguson and J. L. Cleary, In situ source level and source positionestimates of biological transient signals produced by snapping shrimp in

an underwater environment, J. Acoust. Soc. Am., vol. 109, no. 6, pp.30313037, 2001.

[4] J. C. Hassab and R. E. Boucher, Optimum estimation of time delay by ageneralized correlator, IEEE Trans. Acoust., Speech, Signal Processing,vol. 27, no. 4, pp. 373380, 1979.

[5] J. L. Spiesberger, Identifying cross-correlation peaks due to multipathswith the application to optimal passive localization of transient signalsand tomographic mapping of the environment, J. Acoust. Soc. Am., vol.100, pp. 910917, 1996.

[6] , Linking auto- and cross-correlation functions with correlationequations: Application to estimating the relative travel times and am-plitude in multipath, J. Acoust. Soc. Am., vol. 104, no. 1, pp. 300312,1998.

[7] , Finding the right cross-correlation peak for locating sounds inmultipath environments with a fourth-moment function, J. Acoust. Soc.

Am., vol. 108, no. 3, pp. 13491352, 2000.


13/13

13

[8] B. G. Ferguson, Improved time-delay estimates of underwater acousticsignals using beamforming and prefiltering techniques, IEEE J. Oceanic

Eng., vol. 14, no. 5, pp. 238244, 1989.

[9] J. P. Ianniello, Time delay estimation via cross-correlation in thepresence of large estimation errors, IEEE Trans. Acoust., Speech, SignalProcessing, vol. 30, no. 6, pp. 9981003, 1982.

[10] L. A. Pflug, G. E. Ioup, J. W. Ioup, R. L. Field, and J. H. Leclere, Time-delay estimation for deterministic transients using second- and higher-order correlations, J. Acoust. Soc. Am., vol. 94, no. 3, pp. 13851399,1993.

[11] B. G. Ferguson, Variability in passive ranging of acoustic sources inair using a wavefron curvature technique, J. Acoust. Soc. Am., vol. 108,no. 4, pp. 15351543, 2000.

[12] D. Carevic, Robust estimation techniques for target-motion analysisusing passively sensed transient signals, IEEE J. Oceanic Eng., vol. 28,no. 2, pp. 250261, 2003.

[13] , Tracking target in cluttered environment using multilateral time-delay measurements, J. Acoust. Soc. Am., vol. 115, no. 3, pp. 11981206,2004.

[14] Y. Bar-Shalom and T. E. Fortman, Tracking and data association. NewYork: Academic Press, 1988.

[15] H. M. Shertukde and Y. Bar-Shalom, Detection and estimation formultiple targets with two omnidirectional sensors in the presence of falsemeasurements, IEEE Trans. Acoust., Speech, Signal Processing, vol. 38,no. 5, pp. 749763, 1990.

[16] C. Jauffret and Y. Bar-Shalom, Track formation with bearing andfrequency measurements in clutter, IEEE Trans. Aerosp. Electr. Syst.,vol. 26, no. 6, pp. 9991010, 1990.

[17] T. Kirubarajan and Y. Bar-Shalom, Low observable target motionanalysis using amplitude information, IEEE Trans. Aerosp. Electon.Syst., vol. 32, no. 4, pp. 13671384, 1995.

[18] H. Gauvrit, J. P. Le Cadre, and C. Jauffret, A formulation of multitargettracking as an incomplete data problem, IEEE Trans. Aerosp. Electron.Syst., vol. 33, no. 4, pp. 12421257, 1997.

[19] D. Avitzour, A maximum likelihood approach to data association, IEEETrans. Aerosp. Electron. Syst., vol. 28, no. 2, pp. 560566, 1992.

[20] R. L. Streit and T. E. Luginbuhl, Maximum likelihood method forprobabilistic multi-hypothesis tracking, in Proc. SPIE, vol. 2235, 1994,pp. 394405.

[21] , Probabilistic multi-hypothesis tracking, Naval Undersea WarfareCenter, Tech. Rep. 10428, February 1995.

[22] A. P. Dempster, N. M. Liard, and D. B. Rubin, Maximum likelihoodfrom incomplete data via EM algorithm, J. Royal Stat. Soc. B, vol. 39,

pp. 138, 1977.[23] R. G. Hutchins and D. T. Dunham, Evaluation of a probabilisticmultihypothesis tracking algorithm in cluttered environment, in Proc.Thirtieth Asilomar Conf. on Signals, Systems and Computers, vol. 2, LosAlamitos, CA, 3-6 Nov. 1996, pp. 12601264.

[24] J. C. Hassab, B. W. Guimond, and S. C. Nardone, Estimation of locationand motion parameters of a moving source observed from a linear array,

J. Acoust. Soc. Am., vol. 70, no. 4, pp. 10541061, 1981.

[25] S. K. Katsikas, A. K. Leros, and D. G. Lainiotis, Underwater tracking ofmaneouvring target using time delay mesurements, Signal Processing,vol. 41, no. 1, pp. 1729, 1995.

[26] B. Xerri, J.-F. Cavassilas, and B. Borloz, Passive tracking in underwateracoustics, Signal Processing, vol. 82, no. 8, pp. 10671085, 2002.

[27] B. Vo, S. Singh, and W. K. Ma, Tracking multiple speakers with randomsets, in Proc. ICASSP 2004, Montreal, Canada, 2004.

[28] B. Vo, W. K. Ma, and S. Singh, Localizing an unknown time-varyingnumber of speakers: a Bayesian random finite set approach, in Proc.

ICASSP 2005, Philadelphia, USA, 2005.[29] W. K. Ma, B. Vo, S. Singh, and A. Baddley, Tracking an unknown time-varying number of speakers using TDOA measurements: a random finiteset approach, IEEE Trans. Signal Processing, accepted for publication.

[30] L. Frenkel and M. Feder, Recursive expectation-maximization (EM)algorithms for time-varying parameters with applications to multipletarget tracking, IEEE Trans. Signal Processing, vol. 47, no. 2, pp. 306320, 1999.

[31] G. Pulford and B. La Scala, MAP estimation of target manoeuvresequence with the expectation-maximization algorithm, IEEE Trans.

Aerosp. Electron. Syst., vol. 38, no. 2, pp. 367377, 2002.

[32] D. B. Reid, An algorithm for tracking multiple targets, IEEE Trans.Automatic Control, vol. 24, no. 6, pp. 843854, 1979.

[33] Y. Barniv, Dynamic programming solution for detecting dim movingtargets, IEEE Trans. Aerosp. Electron. Syst., vol. 21, no. 1, pp. 144156, 1985.

[34] P. Willett, Y. Ruan, and R. Streit, PMHT: Problems and some solutions,IEEE Trans. Aerosp. Electron. Syst., vol. 38, no. 3, pp. 738754, 2002.

[35] N. Ueda and R. Nakano, Deterministic annealing EM algorithm, NeuralNetworks, vol. 11, no. 2, 1998.

[36] , Mixture density estimation via EM algorithm with deterministicannealing, in Proc. IEEE Workshop Neural Networks for Signal Pro-cessing, Ermioni, Greece, 6-8 Sept. 1994, pp. 6977.

[37] M. Takada and R. Nakano, Multi-thread search with deterministicannealing EM algorithm, in Proc. Int. Joint Conf. Neural Networks,

IJCNN 2002, vol. 1, Honolulu, HI ,USA, 12-17 May 2002, pp. 1034

1038.[38] , Threshold-based dynamic annealing for multi-thread DEAM and

its extreme, in Proc. Int. Joint Conf. Neural Networks, IJCNN 2003,vol. 1, Oregon, PO, 20-24 July 2003, pp. 501506.

[39] M. A. Tanner, Tools for statistical inference. Springer-Verlag New YorkInc., 1996.

[40] S. Kirkpatrick, J. C. D. Gelatt, and M. P. Vecchi, Optimization bysimulated annealing, Science, vol. 220, no. 4598, pp. 671679, 1983.

[41] K. C. Ho and W. Xu, An acurate algebraic solution for moving sourcelocation using TDOA and FDOA measurements, IEEE Trans. SignalProcessing, vol. 52, no. 9, pp. 24532463, 2004.

[42] P. R. Gill, W. Murray, and M. H. Wright, Practical optimization.London: Academic Press, 1981.

[43] G. Celeux and J. Diebolt, The SEM algorithm: a probabilistic teacheralgorithm derived from the EM algorithm for mixture problem, Com-

putational Statistics Quaterly, vol. 2, pp. 7382, 1985.[44] G. C. G. Wei and M. A.Tanner, A Monte Carlo implementation of the

EM algorithm and poor mans data augmentation algorithm, J. Am. Stat.Assoc., vol. 85, no. 411, pp. 699704, 1990.

[45] M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, A tutorialon particle filters for online nonlinear-non-Gaussian Bayesian tracking,

IEEE Trans. Signal Processing, vol. 50, no. 2, pp. 174188, 2002.

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