# ML-PDA: Advances and a New Multitarget Approach.

**ABSTRACT** Developed over 15 years ago, the Maximum Likelihood-Probabilistic Data Association target tracking algorithm has been demonstrated to be effective in tracking Very Low Observable (VLO) targets where target signal-to-noise ratios (SNR) require very low detection processing thresholds to reliably give target detections. However this algorithm has had limitations, which in many cases would preclude use in real- time tracking applications. In this paper we describe three recent advances in the ML-PDA algorithm which make it suitable for real-time tracking. First we look at two recently reported techniques for finding the ML-PDA track estimate which improves computational efficiency by one order of magnitude. Next we review a method for validating ML-PDA track estimates based on the Neyman-Pearson Lemma which gives improved reliability in track validation over previous methods. As our main contribution, we extend ML-PDA from a single-target tracker to a multi-target tracker and compare its performance to that of the Probabilistic Multi-Hypothesis Tracker (PMHT).

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**ABSTRACT:**The Gaussian Mixture Cardinalized Probabil-ity Hypothesis Density (GM-CPHD) Tracker and the Maxi-mum Likelihood-Probabilistic Data Association (ML-PDA) Tracker were applied to the Metron simulated multi-static sonar dataset created for the MSTWG (Multistatic Tracking Working Group). The large number of measurements at each scan was a problem for the GM-CPHD. Winnowing by a de-tection test on SNR and Doppler followed by predetection fusion (contact sifting) had to be performed before tracking with the GM-CPHD to obtain good performance. Due to the nature of its likelihood formulation, ML-PDA was able to process the measurements directly and reasonable results for all five scenarios were achieved. Plots of the tracks ob-tained on the five scenarios the Metron dataset with both trackers are provided.01/2010; - SourceAvailable from: P. Willett[Show abstract] [Hide abstract]

**ABSTRACT:**The Gaussian Mixture Cardinalized PHD (GM-CPHD) Tracker was applied to the SEABAR07 and to the ldquoblindrdquo TNO dataset from the MSTWG (Multistatic Tracking Working Group). The Maximum-Likelihood Probabilistic Data Association (MLPDA) batch tracker was applied to the TNO dataset only. The tracking results (plots and MOPs) are given.Information Fusion, 2009. FUSION '09. 12th International Conference on; 08/2009 - SourceAvailable from: Mahendra Mallick[Show abstract] [Hide abstract]

**ABSTRACT:**Bearings-only tracking (BOT) using a single maneuvering platform has been studied extensively in the past. However, only a few studies exist in the open literature that deal with measurement origin uncertainty. Most publications are concerned with finding the best filtering approach, since BOT is inherently nonlinear, or finding the optimal maneuver strategy for the sensor platform to improve observability. We consider measurement origin uncertainty due to the existence of multiple targets in the surveillance region, non-unity detection probability, and false alarms. Our algorithm uses the multiframe assignment (MFA) to solve the data association problem, and filtering is performed using the unscented Kalman filter (UKF). We employ both the modified and log polar coordinate systems. Simulation results show that the proposed algorithm is very effective in terms of tracking accuracy and track maintenance capability, especially when formulated in the log polar coordinate system.Information Fusion (FUSION), 2010 13th Conference on; 08/2010

Page 1

Hindawi Publishing Corporation

EURASIP Journal on Advances in Signal Processing

Volume 2008, Article ID 260186, 13 pages

doi:10.1155/2008/260186

ResearchArticle

ML-PDA: Advances and a New Multitarget Approach

Wayne Blanding,1Peter Willett,2and Yaakov Bar-Shalom2

1Physical Sciences Department, York College of Pennsylvania, York, PA 17405, USA

2Department of Electrical and Computer Engineering, University of Connecticut, 371 Fairfield Road,

Storrs, CT 06269-2157, USA

Correspondence should be addressed to Wayne Blanding, wblandin@ycp.edu

Received 30 March 2007; Accepted 23 September 2007

Recommended by Roy L. Streit

Developed over 15 years ago, the maximum-likelihood-probabilistic data association target tracking algorithm has been demon-

strated to be effective in tracking very low observable (VLO) targets where target signal-to-noise ratios (SNRs) require very low

detection processing thresholds to reliably give target detections. However, this algorithm has had limitations, which in many cases

would preclude use in real-time tracking applications. In this paper, we describe three recent advances in the ML-PDA algorithm

which make it suitable for real-time tracking. First we look at two recently reported techniques for finding the ML-PDA track

estimate which improves computational efficiency by one order of magnitude. Next we review a method for validating ML-PDA

track estimates based on the Neyman-Pearson lemma which gives improved reliability in track validation over previous methods.

As our main contribution, we extend ML-PDA from a single-target tracker to a multitarget tracker and compare its performance

to that of the probabilistic multihypothesis tracker (PMHT).

Copyright © 2008 Wayne Blanding et al. This is an open access article distributed under the Creative Commons Attribution

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

cited.

1.INTRODUCTION

The problem of tracking very low observable (VLO) targets

in clutter has been an active area of research for a number

of years. The term VLO refers to targets with low signal-

to-noise ratio (SNR), either because the target is stealthy

or because of elevated background noise which masks the

target. A key difficulty lies in the relationship between tar-

get detection probability (Pd) and false alarm probability

(Pf a). In order to achieve a value of Pd sufficient to reli-

ably track, one must lower the detection threshold which

has the undesirable consequence of increasing Pf a. As the

false alarm rate increases (increasing clutter), conventional

Kalman filter-based tracking algorithms, such as multihy-

pothesis trackers (MHTs) which explicitly form track hy-

potheses based on hard measurement-to-target associations,

rapidly lose efficiency and effectiveness. The number of hy-

pothesesinMHTgrowsexponentiallyasthenumberofmea-

surements increases.

Therefore, new techniques have been developed to track

VLOtargets.Onemajorclassconsistsoftrack-before-declare

(TBD)—also called track-before-detect—techniques (see,

e.g., [1–3]). TBD refers to the fact that these techniques si-

multaneously performtrack estimation and track acceptance

(validation or detection) functions. These techniques share

common traits. They typically use either unthresholded sen-

sor data or thresholded data with significantly lower thresh-

oldsthanusedwithconventionaltrackers,therebyincreasing

the measurement data set by one or more orders of magni-

tude. They usually operate on measurement data over sev-

eral scans or frames in a batch algorithm to obtain a track

estimate. Note that single-frame Bayesian TBD techniques,

including those based on particle filters, also exist, for exam-

ple, [4, 5].

As a consequence of the very low or no detection level

thresholding, the computational complexity of TBD algo-

rithms is generally much higher than that of conventional

(i.e., Kalman-filter based) trackers. TBD algorithms are

therefore better suited to those VLO problems where con-

ventional trackers are unable to initiate or sustain a track.

Additionally, as the computational cost is already high, these

trackersarealsobetterfocusedonproblemsinwhichcontact

Page 2

2EURASIP Journal on Advances in Signal Processing

density is relatively low (i.e., the number of interacting con-

tacts is limited). An example of such an application is in very

long range sonar tracking.

One algorithm within this class is the maximum

likelihood-probabilistic data association (ML-PDA) tracker.

ML-PDA uses low-thresholded measurement data over a

batch of measurement frames and computes track estimates

using a sliding window. It is a parameter estimation tech-

nique which assumes deterministic target motion (no pro-

cess noise). Originally developed in 1990 [14], it was later

enhanced by incorporating measurement amplitude as a fea-

ture into the ML-PDA likelihood function [16]. It has been

used to track sonar targets using bearings-only and bear-

ing/frequency information [14, 16] as well as tracking an air-

craft in an optical data set (infrared) [9]. More recently it

has been used on active sonar data sets, including multistatic

tracking [5, 30].

Despite its ability to effectively track VLO targets, ML-

PDA has suffered from some limitations. As with most TBD

algorithms, it has high-computational complexity and as the

clutter level increases beyond a certain (problem-specific)

point it can no longer perform real-time tracking without

resortingtoparallelprocessing.Second,becauseML-PDAal-

waysprovidesatrackestimate,someformoftrackvalidation

must take place to determine if the estimate is the result of

an actual target or from noise-due measurements. The chal-

lenge for track validation lies in obtaining the appropriate

statistical distributions from which to perform the correct

hypothesis test. And finally ML-PDA, in its original formu-

lations, is restricted to single-target tracking. In this paper,

we review recent advances that alleviate the first two limita-

tions which brings context to the major contribution of this

paper—extending ML-PDA to a multitarget tracking algo-

rithm.

First, we briefly describe two recently reported tech-

niques for obtaining the ML-PDA track estimate which have

been shown to be significantly more efficient than the pre-

vious method used. These techniques are later used in the

multitarget version of ML-PDA.

Second, we describe work recently reported on an ML-

PDA track validation procedure based on application of the

Neyman-Pearson lemma. We show using extreme value the-

ory that the statistics of the LLR global maximum under the

“no-target” hypothesis is more closely approximated by the

GumbeldistributionasopposedtotheGaussiandistribution

used by earlier researchers.

As our main contribution, we extend the ML-PDA al-

gorithm to jointly estimate the parameters of multiple tar-

gets in a joint ML-PDA (JMLPDA) algorithm. By use of

measurement validation gating techniques, we incorporate

ML-PDAandJMLPDAintoamultitargetML-PDA(MLPDA

(MT)) tracking system. Comparisons are made between

MLPDA (MT) and the probabilistic multihypothesis tracker

(PMHT). PMHT is a multitarget tracking algorithm which

has good computational efficiency characteristics [26, 30].

The remainder of this paper is organized as follows.

Section 2 defines the terminology and gives a summary of

the ML-PDA algorithm. Section 3 describes the computa-

tional efficiency improvements in ML-PDA by use of the

genetic search and the directed subspace search techniques.

Section 4 summarizes the ML-PDA track validation proce-

dure. Section 5 derives the JMLPDA for multitarget track-

ing. Section 6 outlines the ML-PDA (MT) procedure to

track multiple targets and presents the comparison between

MLPDA (MT) and PMHT in a 2-target scenario. Section 7

summarizes.

2.ML-PDA PROBLEM FORMULATION

The ML-PDA algorithm was originally developed for use in

passive narrowband target motion analysis for LO targets

[14] and was later extended to incorporate amplitude infor-

mation to handle VLO targets [16]. In the window-based

ML-PDA algorithm, designed for use in real-time applica-

tions, a subset of the Nwmost recent data frames is used to

compute the track estimate. When a new frame of data is re-

ceived, the ML-PDA algorithm is repeated after adding the

new frame and deleting one or more of the oldest frames

from the data set, in effect creating a variable sliding window

for track detection and update.

2.1.ML-PDAderivation

A detailed derivation of the ML-PDA algorithm incorporat-

ing amplitude information in a 2D measurement space can

be found in [16]. A summary of the ML-PDA algorithm in-

corporating amplitude information is presented in this sec-

tion, generalized to arbitrary sized measurement and param-

eter spaces. The ML-PDA algorithm uses the following as-

sumptions.

(1) A single target is present in each data frame with a

given detection probability (Pd) and detections are in-

dependent across frames.

(2) At most one measurement per frame corresponds to

the target.

(3) The target operates according to deterministic kine-

matics (i.e., no process noise).

(4) Falsedetectionsaredistributeduniformlyinthesearch

volume (U).

(5) The number of false detections is Poisson distributed

according to probability mass function μf(m), with

parameter λ (spatial density), a function of the detec-

tor Pf ain a resolution cell, independent across frames.

(6) The amplitudes of target originated and false detec-

tions are distributed according to pdf p1(a) and p0(a),

respectively. The target SNR, which affects p1(a), is ei-

ther known or estimated in real time.

(7) Target originated measurements are corrupted with

additive zero-mean white Gaussian noise.

(8) Measurements obtained at different times are, condi-

tioned on the target state, independent.

The target parameter, xr, consists of the target kinematic

state at a given reference time and is related to the target state

at any time using the (possibly nonlinear) relation

x(i) = F?xr,i?.

(1)

Page 3

Wayne Blanding et al.3

The measurement set is given by

(Z,a) =??Zi,ai

i = 1,2,...,Nwframe number,

j = 1,2,...,mi measurement number,

??=??zij,aij

??.

(2)

where zijconsists of the kinematic measurement and aijthe

measurement amplitude.1Amplitude refers to the envelope

output of the detector in a single resolution cell [18, 28].

Measurements with a single subscript refer to all measure-

ments in a single data frame. Measurements with two sub-

scripts identify a specific measurement.

There are some cases where assumption (2) above breaks

down and the target may appear in more than one measure-

ment cell in a single frame of data. This may occur in an ac-

tive sonar or radar sensor when the target extent exceeds one

resolution cell, or for either passive or active sensors when

the target signal strength is high enough such that detectable

energy above the detector threshold is received in adjacent

cells or beams. In such cases, one can use redundancy elim-

ination logic [3] such as centroiding detections to eliminate

or consolidate the multiple target-originated measurements.

Such logic however may have the undesirable effect in a mul-

tiple interacting target scenario of masking the weaker target

when its detections are adjacent to a stronger target—the de-

tections would be combined into a single centroided detec-

tion.

A measurement, assuming it is target originated, is re-

lated to the parameter xrusing the (possibly nonlinear) rela-

tion

z = H?xr,xs(i),i?+wi, (3)

where wiis a zero-mean white Gaussian noise with known

covariance matrix R. The sensor kinematic state, xs(i), is in-

cluded to account for (known) sensor motion. From this we

obtain for a target originated measurement

p?zij| xr

?= N?zij;H?xr,xs(i),i?,R?.

(4)

The maximum likelihood approach finds the target pa-

rameter that maximizes the likelihood function, p(Z,a | xr).

When incorporating amplitude into the likelihood function,

it is convenient to define an amplitude likelihood ratio as

ρij=p1

?aij| τ?

p0

?aij| τ?, (5)

whereτ isthedetectorthreshold(ineachresolutioncell)and

theconditioningisontheamplitudeexceedingthethreshold,

aij > τ. For many applications (including those used in this

paper), the Rayleigh distribution is used which corresponds

to a Swerling-I target fluctuation model.

1Any other feature with a probabilistic model can also be used.

From these assumptions and definitions, the likelihood

function becomes

p?Z,a | xr

=

i=1

Nw

?

+Pd

mi

j=1

Nw

?

Pdμf

Umi−1mi

j=1

?

Nw

?

p?Zi,ai| xr

??1 − Pd

?

=

i=1

?mi

mi

?

?

p?zij,aij| xr

?mi

?mi−1?

j=1

p?zij,aij| “clutter”?

??

?mi

mi

?

l / =j

p?zil,ail| “clutter”??

?aij| τ?

?aij| τ?mi

=

i=1

?1 −Pd

Umiμf

?

j=1

p0

+

p0

?

j=1

p?zij| xr

?ρij

?

.

(6)

The above equation represents the weighted sum of all the

likelihoods of associating a specific measurement (or no

measurement)tothetargetwithallothermeasurementsfalse

detections. This is obtained using the total probability theo-

rem and is the essence of the PDA approach [1].

Dividing (6) by the likelihood function given that all

measurements are false detections, namely,

?

Nw

?

i=1

1

Umiμf

?mi

?mi

?

j=1

p0

?aij| τ??

, (7)

and taking the logarithm of the resulting function, a more

compact form (the log-likelihood ratio (LLR)) is obtained

and is given by

Λ?Z,a | xr

The ML-PDA track estimate, ? x(k), is given by the LLR global

? x(t) = argmax

where t is the reference time corresponding to the target pa-

rameter xr.

The reference time for parameter xrcan be selected arbi-

trarily. Referencing the parameter to the middle of the ML-

PDA batch yields a track estimate which minimizes track er-

rors, but which also induces a time latency in the track esti-

mate. Referencing the parameter to the end of the ML-PDA

batch (i.e., the most recent time) yields larger estimation er-

rors, but without the time latency.

As with any tracking algorithm, in order for a track esti-

mate to have a finite covariance matrix the system must have

the property of observability; see, for example, [19].

?=

Nw

?

i=1

ln

??1 −Pd

?+Pd

λ

mi

?

j=1

ρijp?zij| xr

??

. (8)

maximum in the parameter space

xr

Λ?Z,a | xr

?, (9)

3. ML-PDA EFFICIENCY IMPROVEMENTS

Because the ML-PDA algorithm is essentially a maximum-

likelihood method, the track estimate is the parameter value

Page 4

4EURASIP Journal on Advances in Signal Processing

20

21

22

23

24

25

×102

ηf(m)

−8

−6

−4

−2

0

2

4

Log likelihood ratio

0

1

2

3

4

5

×102

ξf(m)

Figure 1: Representative LLR surface at a velocity maximizing the

center peak. This figure is repeated from [6].

whichmaximizestheML-PDALLR.TheLLRisahighlynon-

convexfunctionwhichcontainsmany(fromseveralhundred

to over a thousand) local maxima. Additionally the LLR sur-

face contains large regions where the LLR is near its mini-

mum value and a near-zero gradient. Figure 1 illustrates the

complexity of the LLR surface by showing a small represen-

tative region of the parameter space over position with ve-

locity fixed, for a 4-dimensional parameter (two position di-

mensions,twovelocitydimensions).Themeasurementspace

consistsofrange,bearing,andrangeratesimulatinganactive

sonar problem.

Three methods to compute the LLR global maximum

have been reported in the literature. Prior researchers used

a multipass grid (MPG) search to find the LLR global max-

imum [14, 16]. The genetic search (GS) and directed sub-

space search (DSS) have been shown to perform better than

the MPG in terms of reduced computational complexity (av-

eraging an order of magnitude) and increased ability to dis-

tinguish the LLR global maximum (3dB improved perfor-

mance) [6]. As these two methods are used in later simula-

tions, they are summarized below.

3.1.Geneticsearch

The GS is a stochastic search technique which is motivated

from evolutionary biology and “survival of the fittest.” While

this technique has seen little use in the tracking community,

it has been used effectively to solve many complex optimiza-

tion problems.

More fully described in [12], the genetic search mim-

ics biological evolution in that the ML-PDA parameter (de-

scribed by a binary bit string) is analogous to biological

DNA. Starting with a (randomized) population of param-

eter values, each population member is assigned a fitness

value based on the LLR evaluated at each population mem-

ber’s parameter. Population members, the parents of the cur-

rent generation, are selected for reproduction based on their

fitness value. The “fitter” population members (those with

higher LLR values) are more likely to reproduce with the

bestmembersreproducingmultipletimes(i.e.,bearingmore

children).

Population members selected for reproduction are ran-

domly paired with other population members selected for

reproduction. With a given probability, the two parents will

producetwochildrenwhicheachsharecharacteristics(pieces

of the parameter bit string) of each parent; otherwise the

children will be clones of the parents. Parents selected to re-

produce multiple times will be paired with different mates.

The children then become the parents of the next gen-

eration. As this process is propagated over generations, the

population becomes more fit and will eventually converge to

asingleparameterwhichisthentakenastheLLRglobalmax-

imum.

The GS implemented in this paper2is an “off the shelf”

implementation using the techniques from [12]. While there

are few theoretical results which guarantee the convergence

oftheGStotheglobalmaximumofanarbitraryfunction,by

tuning our algorithm for both speed of execution as well as

effectiveness at finding the global maximum of the ML-PDA

LLR, results, described more fully in [6], show improvement

in both speed of execution as well as effectiveness compared

to the MPG search.

3.2.Directedsubspacesearch

The second method for maximizing the ML-PDA LLR is a

recently developed technique called the directed subspace

search. The DSS is motivated from the desire to use infor-

mation from measurement data to help guide the search for

the LLR global maximum. Grid searches and GSs are general

optimization tools that do not take advantage of the struc-

ture of the objective function to guide the search process. By

using the structure of the objective function to identify areas

in the parameter space that are more likely to contain local

or global maxima, a more efficient search is possible. Con-

sidering that in the active sonar application presented here,

about 70% of the LLR surface is at the floor value (mean-

ing?Nw

desirable.

First we observe that in many tracking applications the

measurement space is a subspace of the parameter space. In

the 3D measurement space described in this section (bear-

ing, range, range rate), one can map (bearing, range) to the

Cartesianparameterpositions.Rangerateisequivalenttora-

dial velocity (referenced to the sensor), leaving tangential ve-

locity as a “free” parameter.

Next we observe that LLR maxima can only result from

the aggregation of one or more measurements in the Nw-

frame data set that closely fit a parameter xr. In regions of

the parameter space where no measurements influence the

LLR (i.e., where p(zij| xr) ≈ 0∀i, j from (8)), the LLR will

be at the floor value. In regions of the parameter space where

only a single measurement influences the LLR, the LLR will

i=1log(1 − Pd) in (8)), bypassing these areas becomes

2A more detailed description of the specific GS algorithm used can be

found in [6].

Page 5

Wayne Blanding et al.5

Table 1: Directed subspace search algorithm.

Step

1

2

Action

Set grid density for free parameter(s)

Map one measurement to parameter space

Using the measurement, compute LLR values

over grid of free parameter(s)

Repeat steps 2, 3 for all measurements in data set

Pass best result to local optimization routine

3

4

5

be at a local maximum. That is for the bearing, range, range

rate measurement, the LLR will be at a constant (local max-

imum) value for all values of tangential velocity. In areas of

the parameter space where two or more measurements influ-

ence the LLR, each measurement (bearing, range, range rate)

will lie in the vicinity of the local maximum produced by the

measurements.

Therefore, by mapping measurements to the parameter

space and searching only those regions of parameter space

where measurements exist and can contribute to an increase

in the LLR, one narrows down the parameter space of in-

terest and bypasses those regions where it is known (from

the lack of supporting measurements) that the LLR is near

the floor value. This effect produces the first advantage of the

DSSsearchovertheMPGsearchortheGS—areducedsearch

volume containing a subset of the full parameter space and

consequently improved computational efficiency.

The DSS search algorithm, outlined in Table 1, is de-

signed to search this reduced parameter space. In the DSS

search, we take each measurement in the Nw-frame data set,

map it to parameter space, and compute the LLR. Since this

mapping leaves one or more free parameters which can take

on any value for a given measurement, the LLR is computed

over a set of values defined by the measurement and a grid

of values of the free parameter(s). For example, using a mea-

surement space of bearing, range, range rate, one computes

the LLR at the bearing, range, range rate given by the mea-

surement over a grid of tangential velocities. This process

is repeated for every measurement in the measurement set.

Figure 2 illustrates the DSS search. Three measurements are

shown plotted in position subspace along with their corre-

sponding radial velocity vectors. The grid of tangential ve-

locity points is overlaid on each measurement. The LLR is

evaluated at each tangential velocity and for each measure-

ment.

Once the LLR is computed over the set of grid points of

the free parameter(s) for each measurement in the Nw-frame

data set, the parameter that gives the maximum LLR value is

taken and used to initialize a local optimization algorithm

(we use the Davidon-Fletcher-Powell algorithm [21]). The

reason a local optimization algorithm is needed is that while

the DSS grid search will return the local maximum from any

single-measurementmaximum,itwillonlyreturnparameter

values in the vicinity of local maxima caused by two or more

measurements, not the maximum itself. The final, converged

parameter from the local optimization algorithm is the DSS

estimate of the LLR global maximum.

0200 4006008001000

ξ (east position) (meters)

0

100

200

300

400

500

600

700

800

η (north position) (meters)

Measurement position

Measured radial velocity

Tangential velocity grid points

Figure 2: Three measurements (position, radial velocity) overlaid

with their respective DSS search grid points of tangential velocity.

Vectors represent velocities. Sensor is at origin.

4.ML-PDA TRACK VALIDATION

Since the ML-PDA track estimate is the location of the ML-

PDA LLR global maximum in the parameter space, ML-

PDA will always return a track estimate even when a target

is not present. Therefore, a reliable means of validating the

track estimate as target-originated is required. This becomes

a hypothesis testing problem—given the value of the LLR

global maximum, is this value more consistent with a “tar-

get present” (H1) or a “target absent” (H0) hypothesis?

According to the Neyman-Pearson lemma [17], the most

powerful test of H0 versus H1 is given by comparing the

likelihood ratio (or log-likelihood ratio) to a threshold. If a

valid track estimate exists, then by using ML principles, it is

givenbythelocationoftheLLRglobalmaximum.Therefore,

the test becomes determining if the LLR global maximum is

more likely to have been formed from only noise-originated

measurements(H0)ortarget-plusnoise-originatedmeasure-

ments (H1). The threshold for this test is selected to maxi-

mize the power of the test, PDT(true track detection proba-

bility), at a given size or level of significance, PFT(false track

acceptance probability). Thus the threshold value, γ, is cho-

sen based on the statistics of the global LLR maximum under

H0,

γ = F−1

w

?1 −PFT

?,(10)

where Fw(w) is the cumulative distribution function of the

LLR global maximum. If Fw(w) is known exactly, this hy-

pothesis test becomes the optimal test as it obeys all con-

ditions required by the Neyman-Pearson lemma. However,

since we do not a priori know this distribution, optimality of

this test is not guaranteed.

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6EURASIP Journal on Advances in Signal Processing

4.1.TheLLRglobalmaximumunderH0

Previous researchers assumed that the distribution of the

LLR global maximum under H0was Gaussian based on the

centrallimittheorem(CLT)[14,16].Morerecentlyusingex-

treme value theory, the Gumbel distribution has been shown

to be both a better theoretical model and a closer match to

the empirical distribution obtained from Monte Carlo simu-

lations [7]. The theoretical analysis is summarized next.

The LLR global maximum can be viewed as the maxi-

mum from the set of all LLR local maxima. Define the ran-

dom variable y with cumulative distribution function (cdf)

Fy(y) to be the value of an LLR local maximum and the ran-

dom variable w with cdf Fw(w) to be the value of the LLR

global maximum. Then using the formula for the distribu-

tionofthemaximumorderstatisticfromasetofM LLRlocal

maxima (samples) [20],

Fw(w) =?Fy(w)?M.

(11)

Implicit is the assumption that the LLR local maxima

are i.i.d. The independence assumption is not strictly valid

in that LLR local maxima which share measurements will

be correlated to some extent. However, it can be consid-

ered a good approximation because the maxima are gener-

ally well separated in the parameter space (see Figure 1) and

noise-related maxima will principally result from groupings

of a small number (one or two) of measurements. The small

number of measurements contributing to an LLR maximum

limits the correlation between maxima for a given measure-

ment data set. These assumptions remain valid over a wide

range of problem formulations and typical values of Pf a.

Further one can consider the pdf of the LLR local maxi-

mum to be a mixture distribution. Let the random variable y

representthevalueofanLLRlocalmaximumwithpdf fy(y).

Each component of the mixture is distributed according to

fi

surements that associate to form the LLR local maximum.

The probability of an LLR local maximum consisting of i as-

sociatedmeasurementsisdenoted pi.Theoreticallyicantake

on values from 1 to the total number of measurements in the

data set.Thedistribution fy(y)canthereforebeexpressed by

?

y(y) with the superscript indicating the number of mea-

fy(y) =

i

pifi

y(y).

(12)

Absent conditioning on the number of measurements asso-

ciated with an LLR local maxima, the LLR local maxima can

be considered to be identically distributed according to the

mixture distribution described in (12).

EVT describes the asymptotic (large sample size) behav-

ior of the largest value from an i.i.d. sample of size M from

a distribution with a cdf Fy(y), and is well developed in the

statistical literature [10, 13]. Let

w = max?y1, y2,..., yM

EVT states that if a limiting cdf of w exists as M→∞, then

that distribution must belong to one of three forms (Gum-

bel, Weibull, or Frechet). The distribution appropriate to a

?.

(13)

specific application is based on the support of the underlying

distribution of Fy(y). The Gumbel cdf is the appropriate dis-

tributioninourapplicationbecausethesupportofthedistri-

bution of the LLR local maximum is restricted to 0 < y < ∞,

and is of the form

Fw(w) = exp?−exp?−an

where anand unare the scale and location parameters for the

distribution and which depend on the number of samples

used in (13).

The level of accuracy to which the Gumbel distribution

approximates the distribution of the LLR global maximum is

affected by two important issues.

?w −un

???,(14)

(1) There is no guarantee that an asymptotic distribution

exists for the given fy(y). Specifically the structure of

fy(y) as a mixture distribution described by (12) may

preclude the existence of an asymptotic distribution.

(2) While the number of LLR local maxima is large, Fw(w)

may not have reached its asymptotic distribution. It

has been noted for example that while the maximum

fromsamplesofanexponentialdistributionattainsthe

asymptotic distribution with a relatively small number

of samples (fast convergence), for a Gaussian distribu-

tion a much larger sample size is required to attain the

asymptotic distribution (slow convergence) [13].

4.2.MethodstoestimateFw(w)

Two methods were used in [7] to estimate the Gumbel dis-

tribution parameters, and to thereby estimate the distribu-

tion of the LLR global maximum—an offline method and a

real-time method. In the offline method used in the simula-

tions described later, the tracking problem is repeatedly sim-

ulated under H0to obtain a set of LLR global maxima. Then

maximumlikelihoodtechniques[13]areusedtoestimatethe

Gumbel parameters.

This method has the advantage of yielding an optimal

(in the ML sense) estimate of the Gumbel distribution pa-

rameters, although as has been previously stated the Gum-

bel distribution is only an approximation to the true dis-

tribution of Fw(w). This approach may be impractical due

to the extensive offline simulations required. For a general-

purpose tracking system using this methodology, separate

sets of Gumbel distribution parameters must be estimated

for the full range of possible Pf a, target SNR, and Nwas well

as for variations in the boundaries and volumes of the mea-

surement and parameter spaces since each of these factors

affect either the number of local LLR maxima or the distri-

bution fy(y) or both. If the system were designed for a single

special purpose use, this method may be advantageous.

5. JOINT ML-PDA

In this section, we derive the multitarget version of ML-

PDA, called joint ML-PDA (JMLPDA). The derivation of the

JMLPDA algorithm is similar to that of ML-PDA. In this sec-

tion, the JMLPDA formulation for obtaining the joint track

estimate of K = 2 targets is presented. JMLPDA can be

Page 7

Wayne Blanding et al.7

further extended to jointly estimate any number of targets by

extending the JLLR framework in this section to K targets.

5.1.JMLPDAderivation

The assumptions from Section 2 used in ML-PDA are also

used in JMLPDA and are supplemented by the following ad-

ditional assumptions.

(1) K previously confirmed targets exist.

(2) At most one measurement per frame corresponds to

each target.

(3) Ameasurementcannotbeassociatedtomorethanone

target.

(4) Measurements originating from different targets are

independent.

(5) Target originated measurement errors have the same

distribution for each target (i.e., are a function of the

sensor, not the target).

The parameter to be estimated is the kinematic state of

all targets at a given reference time

xr=?x1T

r

··· xKT

r

?T,(15)

where xk

motion model is described by

ris the kinematic state of the kth target whose target

xk(i) = Fk?xk

r,i?.

(16)

The measurement set is given by

(Z,a) =??Zi,ai

i = 1,2,...,Nw frame number,

j = 1,2,...,mi measurement number,

??=??zij,aij

??,

(17)

where zijconsists of the kinematic measurement and aijthe

measurement amplitude. Amplitude refers to the envelope

output of the detector in a single resolution cell [18, 28].

Measurements with a single subscript refer to all measure-

ments in a single data frame. Measurements with two sub-

scripts identify a specific measurement.

Because the SNR of each target can be different, the mea-

surement amplitude likelihood ratio must be defined for

each target. The amplitude likelihood ratio for the kth tar-

get is now given by

?aij| τ,Hk

p0

ρk

ij=p1

?

?aij| τ? ,(18)

whereτ isthedetectorthreshold(ineachresolutioncell)and

the pdf is conditioned on the amplitude exceeding τ and that

the measurement originates from the kth target (hypothesis

Hk).

A measurement, assuming it is target originated, is re-

lated to the kth target xk

tion

z = Hk?xk

rusing the (possibly nonlinear) rela-

r,xs(i),i?+wi,(19)

where wiis a zero-mean white Gaussian noise with covari-

ance matrix R and xs(i) is the sensor kinematic state such

that for a target originated measurement

p?zij| xk

r

?= N?zij;Hk?xk

r,xs(i),i?,R?.

(20)

InthecasewhereK = 2targets,thejointlikelihoodfunc-

tion, p(Zi,ai| xr), for a single frame of data is formed as the

weighted sum of four terms corresponding to the four possi-

ble target detection events. Let

L0

i= p?Zi,ai| xr,no target detections?,

L1

i= p?Zi,ai| xr,only target2detected?,

L12

i

i= p?Zi,ai| xr,only target1detected?,

L2

= p?Zi,ai| xr,both targets detected?.

Then p(Zi,ai| xr) is given by

p?Zi,ai| xr

+?1 −P1

where Pk

target.

Theindividualterms,Lk

cific measurement to each detected target with all other mea-

surements considered as false detections and are given by

(21)

?=?1 −P1

d

??1 − P2

d

d

?L0

i+P1

d

?1 −P2

dL12

d

?L1

i

?P2

dL2

i+P1

dP2

i,

(22)

dis the single frame detection probability of the kth

i,areformedbyassociatingaspe-

L0

i=

μf

?mi

Umi

?mi−1?

Umi−1mi

?mi−1?

Umi−1mi

?mi−2?

Umi−2mi

?

mi

?

j=1

p0

?aij| τ?,

mi

?

mi

?

L1

i=

μf

j=1

p0

?aij| τ?mi

?aij| τ?mi

mi

?

?p?zil| x2

?

?

j=1

p?zij| x1

p?zij| x2

r

?ρ1

?ρ2

ij,

L2

i=

μf

j=1

p0

j=1

r

ij,

L12

i

=

μf

?m1−1?

p?zij| x1

j=1

p0

?aij| τ?

×

mi

?

j=1

mi

?

l / =j

l=1

rr

?ρ1

ijρ2

il.

(23)

The joint likelihood function considering all Nwframes

of data is the product of the single frame joint likelihood

functions

p?Z,a | xr

?=

Nw

?

i=1

p?Zi,ai| xr

?.

(24)

The joint log-likelihood ratio (JLLR) is obtained by divid-

ing (24) by the likelihood that all measurements are noise

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8EURASIP Journal on Advances in Signal Processing

originated,?Nw

of the result yielding

i=1(1−P1

d)(1−P2

d)L0

i,and taking the logarithm

Λ?Z,a | xr

?=

Nw

?

i=1

ln

?

1+

P1

d

λ?1 − P1

P2

d

λ?1 −P2

λ2?1 −P1

?

l / =j

d

?

mi

?

mi

?

j=1

p?zij| x1

p?zij| x2

r

?ρ1

?ρ2

ij

+

d

P1

?

dP2

??1 −P2

p?zij| x1

j=1

r

ij

+

d

dd

?

×

mi

j=1

mi

?

l=1

r

?p?zil| x2

r

?ρ1

ijρ2

il

?

.

(25)

The global maximum of the JLLR defines the parameter

estimate, ? xr, and gives the track estimate of the two targets.

estimatesaretheresultofnoiseortargetoriginated measure-

ments (a test for target existence).

The extension of JMLPDA to an arbitrary number of tar-

gets is straightforward in that one must extend (22) to be-

come the weighted sum of all possible target detection events

for the given number of targets. The number of terms in this

function, however, increases exponentially with the number

of targets, K, according to 2K. An additional consideration

is that one is maximizing the JLLR over 4K dimensions (as-

suming4-dimensionalstatevectorforeachtarget).Thecom-

putational cost of maximizing this function therefore grows

with the number of targets. Some level of separation of the

4K-dimensional problem could be exploited in that for a

given frame of data in the batch, not all targets may be inter-

acting with each other. This would lead to a reduced compu-

tationalcomplexityrangingfromthatofK four-dimensional

problems to one 4K-dimensional problem depending upon

the level of separation achieved. Based on these considera-

tions, application of JMLPDA to more than 3 targets may

not be practical. In this paper, we consider only the 2-target

JMLPDA case.

The JMLPDA algorithm also assumes that the number of

targets is known. In the context of a multitarget application,

this knowledge comes from the prior target state estimates.

The presence of a new (previously undetected) or spawned

target in the measurement set will cause JMLPDA to behave

in an unpredictable way but will generally not give accurate

track estimates. If the number of targets is fewer than that as-

sumed in JMLPDA (i.e., a target death event occurs), the tar-

get validation procedure will correctly not validate the track

estimate for the nonexistent target(s).

A separate test must be performed to determine if the track

5.2.JMLPDAtrackvalidation

A procedure for JMLPDA track validation along the same

lines as ML-PDA track validation appears feasible. How-

ever, the computational complexity associated with JMLPDA

track estimates could make implementation difficult. Fur-

ther, one must account for all possible combinations of track

validation results for each target (e.g., target 1 valid/target

Table 2: JMLPDA track validation procedure.

Step

1

2

3

Action

Find global JLLR maximum

Identify measurement-to-target associations for all targets

Select a target

Edit out measurements (using the complete measurement set)

associated with all other targets

Compute single-target LLR at selected target’s parameter

estimate

Validate estimate using the off-line track validation threshold

Repeat steps 3–6 for all targets

4

5

6

7

2 invalid). Therefore, for simplicity, we apply directly the

ML-PDA track validation technique to the JMLPDA track

estimates using an adjusted measurement data set described

next.

The procedure for obtaining JMLPDA track estimates is

summarizedinTable 2.Toperformtrackvalidation,onefirst

obtains the joint track estimates for each target using the

JMLPDA algorithm. Then based on the track estimates at

each frame in the batch, one obtains the posterior likelihood

that each measurement in the data set is associated with each

target, similar to the PDA approach [1]. From these associ-

ation probabilities, the measurement with the highest asso-

ciation probability is associated with each target. If multi-

ple targets share the same “most likely” measurement, the

measurement is associated to the target with the largest as-

sociation probability between the two targets (a greedy ap-

proach) and the remaining targets associate with their next

most likely measurement. As the posterior association prob-

abilities account for the possibility of associating none of the

measurements to a target, a measurement is associated to a

target only if the posterior association probability for that

measurementexceedstheposteriorprobabilityofassociating

none of the measurements to the target.

Once these hard measurement-to-target associations are

made, to validate the track estimate for a single target the

measurement data set is modified by editing out those mea-

surements that are associated to all other targets. Then the

LLR is computed at the track estimate of the target under

test using the ML-PDA LLR of (8) and compared to the ML-

PDA track validation threshold. The ML-PDA track valida-

tion threshold is obtained using the procedure outlined in

Section 4. Thus the JMLPDA track validation problem is re-

formulated into an ML-PDA track validation problem.

6.MULTITARGET ML-PDA

Multitarget ML-PDA is a tracking system which incorpo-

ratesallphasesofthetrackingproblem:trackinitiation,track

maintenance/update, and track termination functions and

uses the ML-PDA and JMLPDA algorithms for track update.

Figure 3 shows a flowchart of the actions taken by the track-

ingsystemuponreceiptofanewframeofdata.Thefollowing

subsections describe in more detail how the measurement

gating is carried out, the track validation for ML-PDA and

Page 9

Wayne Blanding et al.9

Delete tracks that meet

track termination criteria

No

Valid

estimate

?

Yes

Initiate

new track

Track estimate (MLPDA)

Form residual

measurement set

End for loop

Validate track estimate

MLPDA

track estimate

JMLPDA

track estimate

No Yes

Jointly

associable

measure-

ments

?

For each existing target

Apply gating, obtain

(Zk,ak) for each target

Form (Z,a) from most

recent Nwframes

Receive new data frame

Figure 3: Flowchart for one iteration of the MLPDA(MT) tracking

system.

JMLPDA track estimates, the formation of the residual mea-

surement set, and the track termination criteria.

6.1. Measurementgating

Measurement gating, or using a subset of the full mea-

surement set to obtain a track estimate for a single tar-

get, is a well-known technique in multitarget tracking (see,

e.g., [1, 4]). Since the JMLPDA algorithm is computation-

ally more complex than the ML-PDA algorithm (particularly

when the joint estimates of more than two targets is being

performed), use of measurement gating becomes vital. The

advantage lies in using JMLPDA only for those cases where

targets share gated measurements. Further, by reducing the

size of the measurement set computation time for the ML-

PDA algorithm is reduced as well.

In this application, a measurement gate is set up based

on the prior track estimate and its associated covariance us-

ing the Mahalanobis distance whereby measurements are in-

cluded which satisfy the relation

?zij− ? zi

?TS−1

i

?zij− ? zi

?≤ γ,(26)

where ? ziis the predicted measurement at frame i of the cur-

ML-PDA or JMLPDA algorithm. Track estimates are propa-

gated forward in time as necessary according to the constant

velocity model. The Siterm is the innovation covariance at

frame i and is given by

rent batch based on the last validated track estimate from the

Si= HPiHT+R,(27)

where Piis the covariance of the track estimate on the ith

frame of the current batch based on the last validated track

estimate and H is the measurement matrix where z = Hx,

assuming a linear measurement model. The limiting thresh-

old γ is set based on a desired probability of containing the

target-originated measurement within the gate volume (PG).

6.2.Trackvalidation

Because ML-PDA and JMLPDA will always return a track es-

timate (the global maximum of the LLR or JLLR), one must

test the track estimate for validity. The ML-PDA track vali-

dation procedures are described in Section 4. JMLPDA track

validation procedures are described in Section 5.2.

6.3.Residualmeasurementset

Once track estimates are obtained for all targets currently

in track, a search for new targets must take place. JMLPDA

is unsuitable for this task since it requires knowledge of the

number of new targets. Therefore, ML-PDA is used.

In order to eliminate the effects of known targets on the

ML-PDA LLR, one must account for those measurements

that can be associated with known targets. To do this, one as-

sociates at most one measurement in each data frame to each

target using the same PDA approach described in Section 5.2

for track validation. The residual data set is then the origi-

nal data set with those measurements associated to known

targets edited out.

Using the residual data set the ML-PDA algorithm is ap-

plied and the resulting track estimate validated. If a new tar-

get is validated, its associated measurements are also edited

out to form a new residual data set and the process repeated

until the ML-PDA algorithm returns a track estimate that

fails the validation test. This technique assumes that new tar-

gets are well separated in which target-originated measure-

ments from one new target do not affect the LLR at the track

estimate for any other new target.

6.4.Tracktermination

If taken in isolation, the failure of ML-PDA to validate a

trackestimateissufficienttodeclarethereisnotrackpresent.

However, this does not account for any prior knowledge that

a track had previously existed based on measurement data

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10EURASIP Journal on Advances in Signal Processing

outside of the window of the current ML-PDA batch. When

tracking VLO contacts, in order to limit the false track ac-

ceptance rate, the true target acceptance rate (based on the

track validation threshold value) can be relatively low (in the

50–70% region) even when target detections are present in

the batch. Therefore, in the MLPDA (MT), we have elected

to incorporate an additional higher level track termination

test beyond the ML-PDA track validation.

Existing targets are tested for termination using an M/N

rule. A track is terminated if fewer than M validated track

estimates were obtained from the most recent N applica-

tions of MLPDA (MT) on that target. Based on the operating

characteristic of the ML-PDA algorithm, one can obtain the

probability of detecting a true track (PDT) using ML-PDA.

Then for given values of M and N, one can obtain the track

termination statistics (probability of correctly terminating a

track that is lost and the probability of incorrectly terminat-

ing a track that is still held). In making this calculation, one

mustalsoaccountforthefactthatPDTrelatestoindependent

track estimates. When using a sliding window ML-PDA im-

plementation in which a new track estimate is obtained from

the most recent Nwframes of data, track estimates separated

by fewer than Nwframes are correlated in that they use com-

mon data frame(s).

6.5.ML-PDA(MT)performance

A 2-target crossing scenario was developed to test the per-

formance of the MLPDA (MT) tracking system. The surveil-

lance region consists of a 12km-by-12km square region (the

origin is located at the southwest corner) in which two tar-

gets are placed. Target 1 is initially located near the southwest

corner of the region moving northeast and target 2 is ini-

tially located near the northwest corner of the region moving

southeast. Target motion is constant velocity in an x-y plane.

Measurements consist of x-y position. The parameters for

this scenario are listed in Table 3. Figure 4 shows the target

trajectories and a representative single frame of clutter over

the surveillance region.

The simulations are intended to test the ability of the

MLPDA (MT) algorithm to maintain track when multiple

targets are present in the surveillance region. The ability of

theMLPDA(MT)toinitiatenewtracksanddeletelosttracks

will be explored in future work. Monte Carlo simulations

were performed over a range of Pdand Pf avalues, with 100

simulations conducted at each operating point. To establish

a performance comparison, each simulation was run using

the MLPDA (MT) algorithm and the probabilistic multiple

hypothesis tracker (PMHT) algorithm [24–26].

PMHT is a capable multiple target tracking algorithm

thatusesdatainabatchof Nwframesandcomputesthejoint

track estimates at each frame of data using the expectation-

maximization (EM) algorithm [11] which returns the max-

imum a posteriori (MAP) estimate. The EM algorithm is a

general estimation technique for incomplete or missing data

problems and is guaranteed to converge to at least a local

maximum of the objective function. For PMHT the missing

dataarethespecificmeasurement-to-targetassociations.The

Table 3: Scenario parameters.

Parameter

Initial x1

Initial x2

Target Pd

Target SNR

False Alarm Density (λ)

False Alarm Rate (Pfa)

Avg. Number of False Alarms

(Surveillance Region)

Avg. Number of False Alarms

(Validation Gate)

Batch Size (Nw)

R

Sample Period (T)

Scenario Duration

Value

r(in m and m/s)

r(in m and m/s)

[2000 2000 5 5]T

[2000 10000 −5 5]T

0.7/0.9

5–15dB

4–20 × 10−7

0.05–0.25

57–285

0.33–1.64

7 frames

diag[10121012] m2

20 sec

80 frames

02468 1012

×103

x-position

0

2

4

6

8

10

12

×103

y-position

Target 1

Target 2

Clutter

Figure4:Simulationscenarioshowingtargettrajectoriesandarep-

resentative frame of noise measurements, Pfa= .15.

specific version of PMHT used in the simulations is the ho-

mothetic PMHT [29].

In our implementation, the PMHT uses a continuous

white noise acceleration (CWNA) target motion model [2]

with the process noise spectral density set at values of 0.0125

and 0.05m2/s3. The process noise spectral density was set

so that the tracker could accommodate small target velocity

changes on the order of 0.5 and 1.0m/s over a sample inter-

val.

As we are comparing two distinctly different trackers, it

is worthwhile to highlight some of the key distinctions or ad-

vantages one tracker inherently has over the other and which

make direct comparisons more difficult.

Page 11

Wayne Blanding et al.11

4681012 14 16

SNR (dB)

0

10

20

30

40

50

60

70

80

90

100

Percent in track at simulationend

JMLPDA, Pd= 0.9

JMLPDA, Pd= 0.7

PMHT, Pd= 0.9

PMHT, Pd= 0.7

Figure 5: Percent of simulations, each tracker remained in track to

the end of the run.

(1) The simulations do not test the track initiation abil-

ity of each tracker. This is a key advantage to ML-

PDA in that it simultaneously performs track initia-

tionandtrackupdatefunctions.PMHTrequiresasep-

arate track initiation module. In a heavy clutter envi-

ronmentsuchastheonesexaminedinthispaper,track

initiation becomes problematic for MHT-based track

initiation modules (including PMHT) due to the ex-

plosion of hypotheses resulting in the testing of tenta-

tive (unconfirmed) tracks which are mostly clutter.

(2) Since PMHT computes the MAP estimate (a Bayesian

tracker), it incorporates prior information on the tar-

get state into the current track estimate. Being a

maximum-likelihood tracker, ML-PDA and JMLPDA

do not take advantage of any prior information in

computing the current track estimate. Consequently,

for the same batch length (Nw) for each tracker, one

would expect PMHT to provide more accurate track

estimates than ML-PDA/JMLPDA.

(3) PMHT gets the best state estimate, ? x, at frame Nwsub-

sumptions, it fits the best constant velocity solution

to the entire batch. Consequently, ML-PDA/JMLPDA

position errors for the target are smallest in the middle

of the batch and largest at frame Nw.

(4) Because it incorporates process noise into its target

motion model, PMHT should perform better in sit-

uations where the target is maneuvering. ML-PDA has

a reduced capability to track maneuvering targets be-

cause its target motion model is deterministic. Since

the true target motion is constant-velocity, one would

expect (examining this factor in isolation) ML-PDA to

have reduced rms tracking errors and possibly a longer

mean track life.

ject to its assumptions. Due to ML-PDA/JMLPDA as-

Examining these factors in advance of the simulations,

it is unclear which tracker is expected to perform better—

ML-PDA or PMHT. Factors (2) and (3) above give PMHT

an advantage. Factor (4) gives ML-PDA an advantage. Factor

(1) was not exercised in the simulations.

ThesesimulationsanalyzetheabilityoftheMLPDA(MT)

and PMHT tracking systems to maintain track on the targets

as they cross. In addition to monitoring tracking errors over

time, statistics on the track life (time until the track diverges

from the target) are evaluated for each tracker. Each tracker

was provided with randomized initial target states with ac-

curacy similar to that of the steady-state tracker errors. The

initial covarianceprovidedtothePMHTwasartificiallylarge

(making the initial track estimate uninformative). However,

once PMHT processed its first set of data, it used its own

track estimate covariance for subsequent batches thereby in-

cluding the effects of an informative prior for PMHT.

MLPDA (MT) does not use prior information except to

establish a measurement validation gate. However, MLPDA

(MT) does have the ability to accept or reject track esti-

mates based on track validation criteria. In the case where

the MLPDA (MT) track estimate is not validated, the current

track estimate and covariance reported by the algorithm is

the previous track estimate and covariance propagated for-

ward in time by applying the motion model equations.

The simulations yield the performance of the PMHT and

MLPDA (MT) algorithms in terms of their ability to keep

track of the targets without track divergence. A track esti-

mate was considered to have diverged from the target if the

combined rms position errors exceeded 800m, equating to

a position error in each dimension exceeding 4 standard de-

viations. Additional statistics are provided to illustrate rms

errors over the course of the scenario.

Figure 5 shows the percent of simulations where a target

remained in track for the full 80 frames of the scenario for a

variety of Pdand SNR values for each tracker. Since the in-

track percentage for each target and a given tracker was ap-

proximately the same, the results for the two targets are com-

bined such that the in-track percentage is determined over

200 opportunities (100 simulations and 2 targets).

Figures 6–8 present results for the Pd = 0.9 and Pf a =

0.1 set of simulations plotted as a function of time so that

performance of each tracker can be observed. Figure 6 shows

the percent of simulations each tracker is in track of a target

overtime.Figures7and8showtheevolutionofrmsposition

and velocity errors of each tracker over time. As before, the

results for the two targets are combined to yield 200 samples

(100 simulations and 2 targets) for each tracker.

In these simulations, MLPDA (MT) consistently outper-

forms PMHT in terms of in-track percentage. This can be at-

tributed to the ability of MLPDA (MT) to reject “bad” track

estimated through its validation criteria. An increased track

loss rate is observed for MLPDA (MT) when the targets are

interacting and JMLPDA is used to jointly estimate the target

tracks.

It can also be observed that the performance of MLPDA

(MT) is relatively independent of Pd in that performance

decreases approximately linearly with SNR along the same

line for the two Pd values shown in Figure 5. In contrast,

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12EURASIP Journal on Advances in Signal Processing

010 20 30405060 7080

Frame number

0

10

20

30

40

50

60

70

80

90

100

Percent of simulationin track

MLPDA (MT)

PMHT

Figure 6: Percent of simulations, each tracker is in track of each

target.

010 20 30 405060 70 80

Frame number

80

90

100

110

120

130

140

150

160

RMS position error

MLPDA (MT)

PMHT

Figure 7: Evolution of rms position errors over time for each

tracker and target.

PMHT performance improves significantly when the detec-

tor threshold is raised, simultaneously reducing the number

of clutter points and reducing target Pd. Further analysis is

required to fully quantify this effect.

When considering tracking over the full scenario,

changes in the PMHT process noise power spectral density

haveasignificanteffectonPMHT’sabilitytoremainintrack.

Simulations were conducted where the PMHT modeled pro-

cess noise was reduced from 0.05 to 0.0125m2/s3. With the

reduced process noise, PMHT performance improved to the

point that for each Pd considered and at low clutter levels

PMHTin-trackperformancewassuperiortothatofMLPDA

010 20 304050 607080

Frame number

0.6

0.8

1

1.2

1.4

1.6

1.8

2

RMS velocity error

MLPDA (MT)

PMHT

Figure8:Evolutionofrmsvelocityerrorsovertimeforeachtracker

and target.

(MT). As clutter increased, PMHT performance degraded

fasterthan thatof MLPDA(MT)suchthattherewasaclutter

lever beyond which MLPDA (MT) outperformed PMHT.

As a final comment, it is reemphasized that the strength

of ML-PDA and MLPDA (MT) lies in its ability to success-

fully track VLO targets. A consequence of this performance

is a higher computational complexity than most other track-

ing algorithms. When extended to the multitarget arena,

MLPDA(MT)willbelimitedinitsabilitytotracklargenum-

bers of interacting targets. However, its computational com-

plexity in tracking noninterfering targets does remain linear

in target number.

Another situation where MLPDA(MT) may fail is in a

relatively common passive sonar problem wherein a high

SNR target lies in the same (or adjacent) resolution cell as

the VLO target of interest. In this case, since ML-PDA as-

sumes at most one measurement corresponds to the target,

ML-PDA will preferentially assign the measurement to the

stronger target, which has a higher Pd. In order to success-

fully track the weaker target, one would need to increase the

size of the ML-PDA batch to include frames where the in-

teraction is less. Analysis of this type of situation, including

guidelines on the adaptive sizing of the ML-PDA batch is a

subject of future research.

7. CONCLUSIONS

While ML-PDA has been demonstrated to be an effective

tracking algorithm when tracking VLO targets, no multitar-

get version ofML-PDAhasbeen reported in theliterature.In

this paper, we first described several recent advances in the

ML-PDA target tracking algorithm which make this tracker

feasible for implementation in real-time tracking systems.

Next, we extended the ML-PDA framework to jointly

track multiple targets using JMLPDA. Incorporating both

Page 13

Wayne Blanding et al.13

ML-PDA and JMLPDA into a multitarget tracking system

yields the ML-PDA (MT) target tracking system. Compar-

isons were made with the PMHT using a 2-target cross-

ing scenario which showed that as clutter density increased,

ML-PDA performed better at maintaining track through the

problem than did PMHT.

More work remains in extending the comparison of

MLPDA (MT) to PMHT (or other multitarget algorithms)

to situations commonly found in passive sonar tracking as

well as to maneuvering targets.

ACKNOWLEDGMENT

This research was supported by the Office of Naval Research.

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